arX

iv0

711

0866

v2 [

hep-

th]

17

Dec

200

7

MPP-2007-163

Gauge Thresholds and Kahler Metrics for

Rigid Intersecting D-brane Models

Ralph BlumenhagenMaximilian Schmidt-Sommerfeld

Max-Planck-Institut fur Physik

Fohringer Ring 6

80805 Munchen

Germany

blumenhamppmumpgde

pumucklmppmumpgde

Abstract

The gauge threshold corrections for globally consistent Z2timesZprime2 orientifolds

with rigid intersecting D6-branes are computed The one-loop correctionsto the holomorphic gauge kinetic function are extracted and the Kahlermetrics for the charged chiral multiplets are determined up to two con-stants

Contents

1 Introduction and Motivation 2

2 Setup and partition functions 4

3 Gauge threshold corrections 7

4 Holomorphic gauge kinetic function 10

5 Universal gauge coupling corrections 13

6 Summary of results 16

A Anomaly analysis 17

1 Introduction and Motivation

Gauge threshold corrections in D-brane models [1 2 3 4 5] have lately attractedrenewed interest This is mainly due to the fact that they were shown to beequal to one-loop amplitudes appearing in superpotential couplings generated byeuclidean D-brane instantons [6 7] As the gauge threshold corrections genericallyare not holomorphic functions of the moduli fields it is not a priori clear howthese couplings can be incorporated in a superpotential It can however be shownthat a cancellation between the non-holomorphic terms in the thresholds andterms arising from non-trivial moduli-dependent Kahler metrics takes place thusrendering both the instanton-generated superpotential and the one-loop correctedgauge kinetic function holomorphic [8 9 10]

Gauge threshold corrections in D6-brane models on toroidal backgrounds haveso far only been computed for so-called bulk branes which are uncharged undertwisted sector RR-fields [11 12] Here the prototype example is the Z2 times Z2

orbifold with h21 = 3 which only has eight three-cycles all of which are bulkcycles Twisted three-cycles occur for the Z2 times Z

prime2 orbifold with h21 = 51 This

orbifold is a particularly interesting background for intersecting D6-brane modelsbecause the CFT is free and thus explicit calculations can be performed and moststrikingly there exist rigid three-cycles These latter ones allow one to constructmodels without phenomenologically undesirable adjoint fields1 and therefore inparticular asymptotically free gauge theories [16] Moreover these rigid cyclesare important when studying non-perturbative effects because euclidean D2-brane(short E2-brane) instantons wrapping these cycles have the zero mode structureneeded for a contribution to the superpotential [17 18] This means that due to

1Similar constructions without these fields can be made in the Type I string [13] or usingshift orientifolds [14 15]

2

the aforementioned relation between one-loop threshold corrections and the one-loop instanton amplitudes the results of this paper are relevant when determiningE2-instanton effects in toroidal intersecting D6-brane models

In this paper gauge threshold corrections for D6-brane models on the Z2timesZprime2

orbifold with h21 = 51 [16] in which branes can be charged under the twisted RRfields are computed Furthermore it is shown that also on this background acancellation between the non-holomorphic parts of the gauge thresholds and theterms involving the Kahler metrics occurs in an equation relating the holomorphicgauge kinetic function to the physical gauge coupling which is the one calculatedin string theory Since the main body of this paper is quite technical let usmention two of our main findings A summary of all the results including formulasis given at the end of the paper We will determine the one-loop corrections tothe holomorphic gauge kinetic function fa for the gauge theory on a brane stacklabelled a It is interesting to see that on this background in contradistinctionto the case of the Z2 times Z2 orbifold with h21 = 3 [8] there are corrections to thegauge kinetic function from sectors preserving N = 1 supersymmetry

The gauge threshold corrections computed in this paper allow for a determi-nation of Kahler metrics of charged matter on the background considered Themetric for the vector-like bifundamental matter arising from strings stretched be-tween two stacks of branes that are coincident but differ in their twisted chargescan be determined using holomorphy arguments and yields the expected resultsEquivalently one can determine up to two constants the Kahler metric for thechiral bifundamental matter arising at the intersection of two stacks of branesconfirming previous findings

The results of this paper extend those computed in the T-dual picture [9 10]insofar as to be valid in a global rather than local model and to include moregeneral twisted sector charges In addition a new contribution to the so-calleduniversal gauge coupling corrections [19 20 21 8] is found Their appearancetogether with the holomorphy of the gauge kinetic function implies a redefinitionof the twisted complex structure moduli at one loop

Before we dwell upon the technical details of our computation let us spellout the motivation for this project which is twofold Firstly the gauge thresholdcorrections are computed They are important in D-brane models as the gaugecouplings on different branes depend on the volumes of the cycles the branes wrapand therefore are generically not equal at the compactification or string scaleThis poses a potential problem with the apparent gauge coupling unification seenin the MSSM which could be solved upon taking the gauge threshold correctionsinto account

Secondly as already mentioned the present background allows for E2-ins-tanton contributions to the superpotential and is thus a good arena to explicitlystudy non-perturbative effects in intersecting D6-brane models Examples of sucheffects that are important are Majorana mass terms for the right-handed neutrinos[18 22] and the issue of moduli stabilisation [23] Note in this context that

3

the one-loop corrections determined here lead to a dependence of the instanton-induced terms on the Kahler moduli whereas at tree level they only depend onthe complex structure moduli

2 Setup and partition functions

The setup considered in this paper [16] is an orientifold of an orbifolded torusThe torus is a factorisable six-torus T 6 = T 2 times T 2 times T 2 and the orbifold groupis Z2 timesZ

prime2 where each Z2-factor inverts two two-tori There are three Z2-twisted

sectors with sixteen fixed points each The orientifold group is ΩR(minus1)FL whereΩ is world-sheet parity R an antiholomorphic involution and FL the left-movingspacetime fermion number The three two-tori have radii R

(i)12 along the xiyi-

axes i isin 1 2 3 The tori may (βi = 12) or may not (βi = 0) be tilted There

are four orbifold fixed points on each torus at (0 0) (0 R(i)2 2) (R

(i)1 2 βiR

(i)2 2)

and (R(i)1 2 (1 + βi)R

(i)2 2) They will be labelled fixed points 123 and 4 All

this geometrical data is shown in figure 1 for an untilted and a tilted torus

β =0i

x

y

R2

x

y

R2

RR1 1[a ]=[arsquo ]

[b ]

[a ]

[b ] [arsquo ]

1

2

3

4

1

2

4

3

i

i

i

i

(i)

i

i i (i)

(i)

i i

(i)i

iβ =12

Figure 1 Geometry of the two-tori orbifold fixed points and one-cycles

(Stacks of) D-branes on this background are described by the wrapping num-bers (ni mi) the charges under the twisted RR-fields ǫi isin minus1 1 the positionδi isin 0 1 and the discrete Wilson lines λi isin 0 1 The brane wraps the one-cycle ni[aprimei]+mi[bi] on the irsquoth torus the fundamental one-cycles [aprimei] = [ai]+βi[bi]and [bi] are shown in figure 1 The ǫi satisfy ǫ1 = ǫ2ǫ3 The position is describedby the three parameters δi where δi = 0 if the brane goes through fixed point1 on the irsquoth torus and δi = 1 otherwise An alternative way to characterise abrane is to use ǫikl isin minus1 0 1 i isin 1 2 3 k l isin 1 2 3 4 [16] instead of ǫi δi

and λi ǫikl is the charge of the brane under the fixed point labelled kl in the irsquothtwisted sector The ǫikl can be determined from ǫi δi and λi Note that for eachi only four out of the sixteen ǫikl are non-zero In both these descriptions there is

4

some redundancy [16] Rather than fixing some of the ǫikl charges to be 1 [16] itwill here be more convenient to choose n12 gt 0 (or mi positive if ni vanishes)

It is useful to define mi = mi + βini such that a brane wraps the one-cycleni[ai] + mi[bi] on the irsquoth torus The volume of this one-cycle is given by

V i =

radic(ni)2(R

(i)1 )2 + (mi)2(R

(i)2 )2 (21)

and the tree-level gauge coupling reads

1

g2tree= eminusφ10

prod

i

V i =eminusφ4

(T1T2T3)12

prod

i

V i =(SU1U2U3)

14

(T1T2T3)12

prod

i

V i (22)

Here φ10(φ4) is the 10(4)-dimensional dilaton in string frame S is the dilaton inEinstein frame Ui are the (real parts of the) complex structure moduli in Einstein

frame and Ti = R(i)1 R

(i)2 are the (real parts of the) Kahler moduli From (22)

one can using the supersymmetry condition (see below) derive the dependenceof the tree-level gauge kinetic function on the untwisted moduli [24]

ftree = Scn1n2n3 minus3sum

i 6=j 6=k=1

U ci n

imjmk (23)

Sc and U ci are the complexified dilaton and complex structure moduli the ax-

ions being RR-fields Similarly T ci are complexified Kahler moduli the axions

stemming from the NSNS 2-form-fieldThe D-brane is rotated by the angles θi defined via tan θi = miR

(i)2 niR

(i)1

with respect to the x-axes of the three tori Only supersymmetric configurationswill be considered in this paper ie

sum3i=1 θ

i = 02

The charges of the four orientifold planes are denoted ηΩR and ηΩRi i isin1 2 3 and have to satisfy

ηΩR

3prod

i=1

ηΩRi = minus1 (24)

in the present case of the Z2 times Zprime2 orbifold with h21 = 51 [16] The tadpole

cancellation conditions are given bysum

a

Nan1an

2an

3a = 16ηΩR (25)

sum

a

Naniam

jam

ka = minus24minus2βjminus2βk

ηΩRi i 6= j 6= k isin 1 2 3 (26)

sum

a

Nania(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (27)

sum

a

Namia(ǫ

iakl + ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (28)

2Branes at angles θi satisfying

sum3

i=1θi = plusmn2π are also supersymmetric but are for sim-

plicity not considered here

5

where R(k) = k in case of an untilted torus and R(1 2 3 4) = 1 2 4 3 inthe other case [16] and the sum is a sum over all stacks of branes Na denotesthe number of branes on stack a The wrapping numbers and twisted chargescarry an index a denoting the brane stack which they describe The orientifoldprojection acts on the wrapping numbers and twisted charges as follows

mI rarr minusmI (29)

ǫikl rarr minusηΩRηΩRiǫiR(k)R(l) (210)

The massless open string spectrum can be read of from the open string par-tition function given by annulus (and Mobius) amplitudes without any vertexoperators inserted These can be calculated from boundary (and crosscap) statesdescribing the D-branes (and O-planes) [25 26 27 28 29 30 31] Only the an-nulus amplitudes will be given here and four cases will be distinguished (Theseare not all possibilities there are but all needed here) that will also be importantin the rest of this paper

Case 1 The annulus has both boundaries on the same stack of branes a Theamplitude is

A(1)aa = minusN2

a

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)aa

+16

3sum

i=1

σiaa

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)aa

] (211)

where σiab =

14

sum4kl=1 ǫ

iaklǫ

ibkl ϑ = ϑ(0 2il) η = η(2il) and V i

a defined in (21)now carries an index a to denote the brane stack considered The followingKaluza-Klein and winding sum has been defined [32]

L(i)ab =

sum

mw

exp

[minusπl(V i

a )2

(m2

(R(i)1 R

(i)2 )2

+ w2

)+ iπm(δia minus δib) + iπw(λi

a minus λib)

]

Case 2 The annulus stretches between two stacks of D-branes a and b wrappingthe same submanifold of the covering six-torus of the internal space and havingthe same discrete Wilson lines turned on(λi

a = λib) This means in particular

θia = θibVia = V i

b δia = δib however not all twisted charges ǫi are equal

One can show that in this case σiab = plusmn1 The amplitude is

A(2)ab = minusNaNb

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)ab

+16

3sum

i=1

σiab

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)ab

] (212)

6

Case 3 The two stacks of branes wrap submanifolds that are homologicallyequal on the covering torus (This implies θia = θibV

ia = V i

b ) but do not satisfyλia = λi

b δia = δib for all i The amplitude looks as the one of case 2

Case 4 The annulus stretches between two branes that intersect at non-trivialangles on all three tori The amplitude reads

A(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

)sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

3prod

i=1

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

(213)

+32sum

i 6=j 6=k

I iab σiab

sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

ϑ[|αminus12|

β

](θjab)

ϑ[

012

](θjab)

ϑ[|αminus12|

β

](θkab)

ϑ[

012

](θkab)

]

where θiab = θiaminusθib and the intersection number I iab = (mian

ibminusmi

bnia) were defined

Upon transforming into the open string channel one finds the following mass-less spectrum Case 1 yields a massless vector multiplet in the adjoint represen-tation of U(Na) Case 2 gives a massless hypermultiplet in the bifundamentalrepresentation of U(Na) times U(Nb) In case 3 there are no massless fields Incase 4 one finds |Υab| Υab = 1

4

prod3i=1 I

iab +

sum3i=1 I

iabσ

iab chiral multiplets in the

bifundamental representation of U(Na)times U(Nb)

3 Gauge threshold corrections

The gauge threshold corrections will be computed using the background fieldmethod employed previously [33 34 11] This means that the one-loop correctionto the gauge coupling of brane a induced by brane b is determined as followsFirst one replaces the 4d spacetime part of the partition function in the aboveexpression as follows [11]

ϑ[αβ

](0 2il)

η3(2il)rarr 2iπBqa

ϑ[αβ

](minusǫa 2il)

ϑ[1212

](minusǫa 2il)

(31)

where πǫa = arctan(πqaB) qa is the charge of the open string ending on brane aand B is the background magnetic field in the 4d spacetime One then expandsthe resulting expressions in a series in B and the coefficient of B2 gives the desiredexpression for the correction to the gauge coupling which needs to be evaluatedfurther

For the four cases distinguished above one finds denoting the amplitudes after

7

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

the aforementioned relation between one-loop threshold corrections and the one-loop instanton amplitudes the results of this paper are relevant when determiningE2-instanton effects in toroidal intersecting D6-brane models

In this paper gauge threshold corrections for D6-brane models on the Z2timesZprime2

orbifold with h21 = 51 [16] in which branes can be charged under the twisted RRfields are computed Furthermore it is shown that also on this background acancellation between the non-holomorphic parts of the gauge thresholds and theterms involving the Kahler metrics occurs in an equation relating the holomorphicgauge kinetic function to the physical gauge coupling which is the one calculatedin string theory Since the main body of this paper is quite technical let usmention two of our main findings A summary of all the results including formulasis given at the end of the paper We will determine the one-loop corrections tothe holomorphic gauge kinetic function fa for the gauge theory on a brane stacklabelled a It is interesting to see that on this background in contradistinctionto the case of the Z2 times Z2 orbifold with h21 = 3 [8] there are corrections to thegauge kinetic function from sectors preserving N = 1 supersymmetry

The gauge threshold corrections computed in this paper allow for a determi-nation of Kahler metrics of charged matter on the background considered Themetric for the vector-like bifundamental matter arising from strings stretched be-tween two stacks of branes that are coincident but differ in their twisted chargescan be determined using holomorphy arguments and yields the expected resultsEquivalently one can determine up to two constants the Kahler metric for thechiral bifundamental matter arising at the intersection of two stacks of branesconfirming previous findings

The results of this paper extend those computed in the T-dual picture [9 10]insofar as to be valid in a global rather than local model and to include moregeneral twisted sector charges In addition a new contribution to the so-calleduniversal gauge coupling corrections [19 20 21 8] is found Their appearancetogether with the holomorphy of the gauge kinetic function implies a redefinitionof the twisted complex structure moduli at one loop

Before we dwell upon the technical details of our computation let us spellout the motivation for this project which is twofold Firstly the gauge thresholdcorrections are computed They are important in D-brane models as the gaugecouplings on different branes depend on the volumes of the cycles the branes wrapand therefore are generically not equal at the compactification or string scaleThis poses a potential problem with the apparent gauge coupling unification seenin the MSSM which could be solved upon taking the gauge threshold correctionsinto account

Secondly as already mentioned the present background allows for E2-ins-tanton contributions to the superpotential and is thus a good arena to explicitlystudy non-perturbative effects in intersecting D6-brane models Examples of sucheffects that are important are Majorana mass terms for the right-handed neutrinos[18 22] and the issue of moduli stabilisation [23] Note in this context that

3

the one-loop corrections determined here lead to a dependence of the instanton-induced terms on the Kahler moduli whereas at tree level they only depend onthe complex structure moduli

2 Setup and partition functions

The setup considered in this paper [16] is an orientifold of an orbifolded torusThe torus is a factorisable six-torus T 6 = T 2 times T 2 times T 2 and the orbifold groupis Z2 timesZ

prime2 where each Z2-factor inverts two two-tori There are three Z2-twisted

sectors with sixteen fixed points each The orientifold group is ΩR(minus1)FL whereΩ is world-sheet parity R an antiholomorphic involution and FL the left-movingspacetime fermion number The three two-tori have radii R

(i)12 along the xiyi-

axes i isin 1 2 3 The tori may (βi = 12) or may not (βi = 0) be tilted There

are four orbifold fixed points on each torus at (0 0) (0 R(i)2 2) (R

(i)1 2 βiR

(i)2 2)

and (R(i)1 2 (1 + βi)R

(i)2 2) They will be labelled fixed points 123 and 4 All

this geometrical data is shown in figure 1 for an untilted and a tilted torus

β =0i

x

y

R2

x

y

R2

RR1 1[a ]=[arsquo ]

[b ]

[a ]

[b ] [arsquo ]

1

2

3

4

1

2

4

3

i

i

i

i

(i)

i

i i (i)

(i)

i i

(i)i

iβ =12

Figure 1 Geometry of the two-tori orbifold fixed points and one-cycles

(Stacks of) D-branes on this background are described by the wrapping num-bers (ni mi) the charges under the twisted RR-fields ǫi isin minus1 1 the positionδi isin 0 1 and the discrete Wilson lines λi isin 0 1 The brane wraps the one-cycle ni[aprimei]+mi[bi] on the irsquoth torus the fundamental one-cycles [aprimei] = [ai]+βi[bi]and [bi] are shown in figure 1 The ǫi satisfy ǫ1 = ǫ2ǫ3 The position is describedby the three parameters δi where δi = 0 if the brane goes through fixed point1 on the irsquoth torus and δi = 1 otherwise An alternative way to characterise abrane is to use ǫikl isin minus1 0 1 i isin 1 2 3 k l isin 1 2 3 4 [16] instead of ǫi δi

and λi ǫikl is the charge of the brane under the fixed point labelled kl in the irsquothtwisted sector The ǫikl can be determined from ǫi δi and λi Note that for eachi only four out of the sixteen ǫikl are non-zero In both these descriptions there is

4

some redundancy [16] Rather than fixing some of the ǫikl charges to be 1 [16] itwill here be more convenient to choose n12 gt 0 (or mi positive if ni vanishes)

It is useful to define mi = mi + βini such that a brane wraps the one-cycleni[ai] + mi[bi] on the irsquoth torus The volume of this one-cycle is given by

V i =

radic(ni)2(R

(i)1 )2 + (mi)2(R

(i)2 )2 (21)

and the tree-level gauge coupling reads

1

g2tree= eminusφ10

prod

i

V i =eminusφ4

(T1T2T3)12

prod

i

V i =(SU1U2U3)

14

(T1T2T3)12

prod

i

V i (22)

Here φ10(φ4) is the 10(4)-dimensional dilaton in string frame S is the dilaton inEinstein frame Ui are the (real parts of the) complex structure moduli in Einstein

frame and Ti = R(i)1 R

(i)2 are the (real parts of the) Kahler moduli From (22)

one can using the supersymmetry condition (see below) derive the dependenceof the tree-level gauge kinetic function on the untwisted moduli [24]

ftree = Scn1n2n3 minus3sum

i 6=j 6=k=1

U ci n

imjmk (23)

Sc and U ci are the complexified dilaton and complex structure moduli the ax-

ions being RR-fields Similarly T ci are complexified Kahler moduli the axions

stemming from the NSNS 2-form-fieldThe D-brane is rotated by the angles θi defined via tan θi = miR

(i)2 niR

(i)1

with respect to the x-axes of the three tori Only supersymmetric configurationswill be considered in this paper ie

sum3i=1 θ

i = 02

The charges of the four orientifold planes are denoted ηΩR and ηΩRi i isin1 2 3 and have to satisfy

ηΩR

3prod

i=1

ηΩRi = minus1 (24)

in the present case of the Z2 times Zprime2 orbifold with h21 = 51 [16] The tadpole

cancellation conditions are given bysum

a

Nan1an

2an

3a = 16ηΩR (25)

sum

a

Naniam

jam

ka = minus24minus2βjminus2βk

ηΩRi i 6= j 6= k isin 1 2 3 (26)

sum

a

Nania(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (27)

sum

a

Namia(ǫ

iakl + ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (28)

2Branes at angles θi satisfying

sum3

i=1θi = plusmn2π are also supersymmetric but are for sim-

plicity not considered here

5

where R(k) = k in case of an untilted torus and R(1 2 3 4) = 1 2 4 3 inthe other case [16] and the sum is a sum over all stacks of branes Na denotesthe number of branes on stack a The wrapping numbers and twisted chargescarry an index a denoting the brane stack which they describe The orientifoldprojection acts on the wrapping numbers and twisted charges as follows

mI rarr minusmI (29)

ǫikl rarr minusηΩRηΩRiǫiR(k)R(l) (210)

The massless open string spectrum can be read of from the open string par-tition function given by annulus (and Mobius) amplitudes without any vertexoperators inserted These can be calculated from boundary (and crosscap) statesdescribing the D-branes (and O-planes) [25 26 27 28 29 30 31] Only the an-nulus amplitudes will be given here and four cases will be distinguished (Theseare not all possibilities there are but all needed here) that will also be importantin the rest of this paper

Case 1 The annulus has both boundaries on the same stack of branes a Theamplitude is

A(1)aa = minusN2

a

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)aa

+16

3sum

i=1

σiaa

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)aa

] (211)

where σiab =

14

sum4kl=1 ǫ

iaklǫ

ibkl ϑ = ϑ(0 2il) η = η(2il) and V i

a defined in (21)now carries an index a to denote the brane stack considered The followingKaluza-Klein and winding sum has been defined [32]

L(i)ab =

sum

mw

exp

[minusπl(V i

a )2

(m2

(R(i)1 R

(i)2 )2

+ w2

)+ iπm(δia minus δib) + iπw(λi

a minus λib)

]

Case 2 The annulus stretches between two stacks of D-branes a and b wrappingthe same submanifold of the covering six-torus of the internal space and havingthe same discrete Wilson lines turned on(λi

a = λib) This means in particular

θia = θibVia = V i

b δia = δib however not all twisted charges ǫi are equal

One can show that in this case σiab = plusmn1 The amplitude is

A(2)ab = minusNaNb

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)ab

+16

3sum

i=1

σiab

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)ab

] (212)

6

Case 3 The two stacks of branes wrap submanifolds that are homologicallyequal on the covering torus (This implies θia = θibV

ia = V i

b ) but do not satisfyλia = λi

b δia = δib for all i The amplitude looks as the one of case 2

Case 4 The annulus stretches between two branes that intersect at non-trivialangles on all three tori The amplitude reads

A(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

)sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

3prod

i=1

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

(213)

+32sum

i 6=j 6=k

I iab σiab

sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

ϑ[|αminus12|

β

](θjab)

ϑ[

012

](θjab)

ϑ[|αminus12|

β

](θkab)

ϑ[

012

](θkab)

]

where θiab = θiaminusθib and the intersection number I iab = (mian

ibminusmi

bnia) were defined

Upon transforming into the open string channel one finds the following mass-less spectrum Case 1 yields a massless vector multiplet in the adjoint represen-tation of U(Na) Case 2 gives a massless hypermultiplet in the bifundamentalrepresentation of U(Na) times U(Nb) In case 3 there are no massless fields Incase 4 one finds |Υab| Υab = 1

4

prod3i=1 I

iab +

sum3i=1 I

iabσ

iab chiral multiplets in the

bifundamental representation of U(Na)times U(Nb)

3 Gauge threshold corrections

The gauge threshold corrections will be computed using the background fieldmethod employed previously [33 34 11] This means that the one-loop correctionto the gauge coupling of brane a induced by brane b is determined as followsFirst one replaces the 4d spacetime part of the partition function in the aboveexpression as follows [11]

ϑ[αβ

](0 2il)

η3(2il)rarr 2iπBqa

ϑ[αβ

](minusǫa 2il)

ϑ[1212

](minusǫa 2il)

(31)

where πǫa = arctan(πqaB) qa is the charge of the open string ending on brane aand B is the background magnetic field in the 4d spacetime One then expandsthe resulting expressions in a series in B and the coefficient of B2 gives the desiredexpression for the correction to the gauge coupling which needs to be evaluatedfurther

For the four cases distinguished above one finds denoting the amplitudes after

7

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

the one-loop corrections determined here lead to a dependence of the instanton-induced terms on the Kahler moduli whereas at tree level they only depend onthe complex structure moduli

2 Setup and partition functions

The setup considered in this paper [16] is an orientifold of an orbifolded torusThe torus is a factorisable six-torus T 6 = T 2 times T 2 times T 2 and the orbifold groupis Z2 timesZ

prime2 where each Z2-factor inverts two two-tori There are three Z2-twisted

sectors with sixteen fixed points each The orientifold group is ΩR(minus1)FL whereΩ is world-sheet parity R an antiholomorphic involution and FL the left-movingspacetime fermion number The three two-tori have radii R

(i)12 along the xiyi-

axes i isin 1 2 3 The tori may (βi = 12) or may not (βi = 0) be tilted There

are four orbifold fixed points on each torus at (0 0) (0 R(i)2 2) (R

(i)1 2 βiR

(i)2 2)

and (R(i)1 2 (1 + βi)R

(i)2 2) They will be labelled fixed points 123 and 4 All

this geometrical data is shown in figure 1 for an untilted and a tilted torus

β =0i

x

y

R2

x

y

R2

RR1 1[a ]=[arsquo ]

[b ]

[a ]

[b ] [arsquo ]

1

2

3

4

1

2

4

3

i

i

i

i

(i)

i

i i (i)

(i)

i i

(i)i

iβ =12

Figure 1 Geometry of the two-tori orbifold fixed points and one-cycles

(Stacks of) D-branes on this background are described by the wrapping num-bers (ni mi) the charges under the twisted RR-fields ǫi isin minus1 1 the positionδi isin 0 1 and the discrete Wilson lines λi isin 0 1 The brane wraps the one-cycle ni[aprimei]+mi[bi] on the irsquoth torus the fundamental one-cycles [aprimei] = [ai]+βi[bi]and [bi] are shown in figure 1 The ǫi satisfy ǫ1 = ǫ2ǫ3 The position is describedby the three parameters δi where δi = 0 if the brane goes through fixed point1 on the irsquoth torus and δi = 1 otherwise An alternative way to characterise abrane is to use ǫikl isin minus1 0 1 i isin 1 2 3 k l isin 1 2 3 4 [16] instead of ǫi δi

and λi ǫikl is the charge of the brane under the fixed point labelled kl in the irsquothtwisted sector The ǫikl can be determined from ǫi δi and λi Note that for eachi only four out of the sixteen ǫikl are non-zero In both these descriptions there is

4

some redundancy [16] Rather than fixing some of the ǫikl charges to be 1 [16] itwill here be more convenient to choose n12 gt 0 (or mi positive if ni vanishes)

It is useful to define mi = mi + βini such that a brane wraps the one-cycleni[ai] + mi[bi] on the irsquoth torus The volume of this one-cycle is given by

V i =

radic(ni)2(R

(i)1 )2 + (mi)2(R

(i)2 )2 (21)

and the tree-level gauge coupling reads

1

g2tree= eminusφ10

prod

i

V i =eminusφ4

(T1T2T3)12

prod

i

V i =(SU1U2U3)

14

(T1T2T3)12

prod

i

V i (22)

Here φ10(φ4) is the 10(4)-dimensional dilaton in string frame S is the dilaton inEinstein frame Ui are the (real parts of the) complex structure moduli in Einstein

frame and Ti = R(i)1 R

(i)2 are the (real parts of the) Kahler moduli From (22)

one can using the supersymmetry condition (see below) derive the dependenceof the tree-level gauge kinetic function on the untwisted moduli [24]

ftree = Scn1n2n3 minus3sum

i 6=j 6=k=1

U ci n

imjmk (23)

Sc and U ci are the complexified dilaton and complex structure moduli the ax-

ions being RR-fields Similarly T ci are complexified Kahler moduli the axions

stemming from the NSNS 2-form-fieldThe D-brane is rotated by the angles θi defined via tan θi = miR

(i)2 niR

(i)1

with respect to the x-axes of the three tori Only supersymmetric configurationswill be considered in this paper ie

sum3i=1 θ

i = 02

The charges of the four orientifold planes are denoted ηΩR and ηΩRi i isin1 2 3 and have to satisfy

ηΩR

3prod

i=1

ηΩRi = minus1 (24)

in the present case of the Z2 times Zprime2 orbifold with h21 = 51 [16] The tadpole

cancellation conditions are given bysum

a

Nan1an

2an

3a = 16ηΩR (25)

sum

a

Naniam

jam

ka = minus24minus2βjminus2βk

ηΩRi i 6= j 6= k isin 1 2 3 (26)

sum

a

Nania(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (27)

sum

a

Namia(ǫ

iakl + ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (28)

2Branes at angles θi satisfying

sum3

i=1θi = plusmn2π are also supersymmetric but are for sim-

plicity not considered here

5

where R(k) = k in case of an untilted torus and R(1 2 3 4) = 1 2 4 3 inthe other case [16] and the sum is a sum over all stacks of branes Na denotesthe number of branes on stack a The wrapping numbers and twisted chargescarry an index a denoting the brane stack which they describe The orientifoldprojection acts on the wrapping numbers and twisted charges as follows

mI rarr minusmI (29)

ǫikl rarr minusηΩRηΩRiǫiR(k)R(l) (210)

The massless open string spectrum can be read of from the open string par-tition function given by annulus (and Mobius) amplitudes without any vertexoperators inserted These can be calculated from boundary (and crosscap) statesdescribing the D-branes (and O-planes) [25 26 27 28 29 30 31] Only the an-nulus amplitudes will be given here and four cases will be distinguished (Theseare not all possibilities there are but all needed here) that will also be importantin the rest of this paper

Case 1 The annulus has both boundaries on the same stack of branes a Theamplitude is

A(1)aa = minusN2

a

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)aa

+16

3sum

i=1

σiaa

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)aa

] (211)

where σiab =

14

sum4kl=1 ǫ

iaklǫ

ibkl ϑ = ϑ(0 2il) η = η(2il) and V i

a defined in (21)now carries an index a to denote the brane stack considered The followingKaluza-Klein and winding sum has been defined [32]

L(i)ab =

sum

mw

exp

[minusπl(V i

a )2

(m2

(R(i)1 R

(i)2 )2

+ w2

)+ iπm(δia minus δib) + iπw(λi

a minus λib)

]

Case 2 The annulus stretches between two stacks of D-branes a and b wrappingthe same submanifold of the covering six-torus of the internal space and havingthe same discrete Wilson lines turned on(λi

a = λib) This means in particular

θia = θibVia = V i

b δia = δib however not all twisted charges ǫi are equal

One can show that in this case σiab = plusmn1 The amplitude is

A(2)ab = minusNaNb

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)ab

+16

3sum

i=1

σiab

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)ab

] (212)

6

Case 3 The two stacks of branes wrap submanifolds that are homologicallyequal on the covering torus (This implies θia = θibV

ia = V i

b ) but do not satisfyλia = λi

b δia = δib for all i The amplitude looks as the one of case 2

Case 4 The annulus stretches between two branes that intersect at non-trivialangles on all three tori The amplitude reads

A(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

)sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

3prod

i=1

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

(213)

+32sum

i 6=j 6=k

I iab σiab

sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

ϑ[|αminus12|

β

](θjab)

ϑ[

012

](θjab)

ϑ[|αminus12|

β

](θkab)

ϑ[

012

](θkab)

]

where θiab = θiaminusθib and the intersection number I iab = (mian

ibminusmi

bnia) were defined

Upon transforming into the open string channel one finds the following mass-less spectrum Case 1 yields a massless vector multiplet in the adjoint represen-tation of U(Na) Case 2 gives a massless hypermultiplet in the bifundamentalrepresentation of U(Na) times U(Nb) In case 3 there are no massless fields Incase 4 one finds |Υab| Υab = 1

4

prod3i=1 I

iab +

sum3i=1 I

iabσ

iab chiral multiplets in the

bifundamental representation of U(Na)times U(Nb)

3 Gauge threshold corrections

The gauge threshold corrections will be computed using the background fieldmethod employed previously [33 34 11] This means that the one-loop correctionto the gauge coupling of brane a induced by brane b is determined as followsFirst one replaces the 4d spacetime part of the partition function in the aboveexpression as follows [11]

ϑ[αβ

](0 2il)

η3(2il)rarr 2iπBqa

ϑ[αβ

](minusǫa 2il)

ϑ[1212

](minusǫa 2il)

(31)

where πǫa = arctan(πqaB) qa is the charge of the open string ending on brane aand B is the background magnetic field in the 4d spacetime One then expandsthe resulting expressions in a series in B and the coefficient of B2 gives the desiredexpression for the correction to the gauge coupling which needs to be evaluatedfurther

For the four cases distinguished above one finds denoting the amplitudes after

7

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

some redundancy [16] Rather than fixing some of the ǫikl charges to be 1 [16] itwill here be more convenient to choose n12 gt 0 (or mi positive if ni vanishes)

It is useful to define mi = mi + βini such that a brane wraps the one-cycleni[ai] + mi[bi] on the irsquoth torus The volume of this one-cycle is given by

V i =

radic(ni)2(R

(i)1 )2 + (mi)2(R

(i)2 )2 (21)

and the tree-level gauge coupling reads

1

g2tree= eminusφ10

prod

i

V i =eminusφ4

(T1T2T3)12

prod

i

V i =(SU1U2U3)

14

(T1T2T3)12

prod

i

V i (22)

Here φ10(φ4) is the 10(4)-dimensional dilaton in string frame S is the dilaton inEinstein frame Ui are the (real parts of the) complex structure moduli in Einstein

frame and Ti = R(i)1 R

(i)2 are the (real parts of the) Kahler moduli From (22)

one can using the supersymmetry condition (see below) derive the dependenceof the tree-level gauge kinetic function on the untwisted moduli [24]

ftree = Scn1n2n3 minus3sum

i 6=j 6=k=1

U ci n

imjmk (23)

Sc and U ci are the complexified dilaton and complex structure moduli the ax-

ions being RR-fields Similarly T ci are complexified Kahler moduli the axions

stemming from the NSNS 2-form-fieldThe D-brane is rotated by the angles θi defined via tan θi = miR

(i)2 niR

(i)1

with respect to the x-axes of the three tori Only supersymmetric configurationswill be considered in this paper ie

sum3i=1 θ

i = 02

The charges of the four orientifold planes are denoted ηΩR and ηΩRi i isin1 2 3 and have to satisfy

ηΩR

3prod

i=1

ηΩRi = minus1 (24)

in the present case of the Z2 times Zprime2 orbifold with h21 = 51 [16] The tadpole

cancellation conditions are given bysum

a

Nan1an

2an

3a = 16ηΩR (25)

sum

a

Naniam

jam

ka = minus24minus2βjminus2βk

ηΩRi i 6= j 6= k isin 1 2 3 (26)

sum

a

Nania(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (27)

sum

a

Namia(ǫ

iakl + ηΩRηΩRiǫ

iaR(k)R(l)) = 0 (28)

2Branes at angles θi satisfying

sum3

i=1θi = plusmn2π are also supersymmetric but are for sim-

plicity not considered here

5

where R(k) = k in case of an untilted torus and R(1 2 3 4) = 1 2 4 3 inthe other case [16] and the sum is a sum over all stacks of branes Na denotesthe number of branes on stack a The wrapping numbers and twisted chargescarry an index a denoting the brane stack which they describe The orientifoldprojection acts on the wrapping numbers and twisted charges as follows

mI rarr minusmI (29)

ǫikl rarr minusηΩRηΩRiǫiR(k)R(l) (210)

The massless open string spectrum can be read of from the open string par-tition function given by annulus (and Mobius) amplitudes without any vertexoperators inserted These can be calculated from boundary (and crosscap) statesdescribing the D-branes (and O-planes) [25 26 27 28 29 30 31] Only the an-nulus amplitudes will be given here and four cases will be distinguished (Theseare not all possibilities there are but all needed here) that will also be importantin the rest of this paper

Case 1 The annulus has both boundaries on the same stack of branes a Theamplitude is

A(1)aa = minusN2

a

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)aa

+16

3sum

i=1

σiaa

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)aa

] (211)

where σiab =

14

sum4kl=1 ǫ

iaklǫ

ibkl ϑ = ϑ(0 2il) η = η(2il) and V i

a defined in (21)now carries an index a to denote the brane stack considered The followingKaluza-Klein and winding sum has been defined [32]

L(i)ab =

sum

mw

exp

[minusπl(V i

a )2

(m2

(R(i)1 R

(i)2 )2

+ w2

)+ iπm(δia minus δib) + iπw(λi

a minus λib)

]

Case 2 The annulus stretches between two stacks of D-branes a and b wrappingthe same submanifold of the covering six-torus of the internal space and havingthe same discrete Wilson lines turned on(λi

a = λib) This means in particular

θia = θibVia = V i

b δia = δib however not all twisted charges ǫi are equal

One can show that in this case σiab = plusmn1 The amplitude is

A(2)ab = minusNaNb

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)ab

+16

3sum

i=1

σiab

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)ab

] (212)

6

Case 3 The two stacks of branes wrap submanifolds that are homologicallyequal on the covering torus (This implies θia = θibV

ia = V i

b ) but do not satisfyλia = λi

b δia = δib for all i The amplitude looks as the one of case 2

Case 4 The annulus stretches between two branes that intersect at non-trivialangles on all three tori The amplitude reads

A(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

)sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

3prod

i=1

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

(213)

+32sum

i 6=j 6=k

I iab σiab

sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

ϑ[|αminus12|

β

](θjab)

ϑ[

012

](θjab)

ϑ[|αminus12|

β

](θkab)

ϑ[

012

](θkab)

]

where θiab = θiaminusθib and the intersection number I iab = (mian

ibminusmi

bnia) were defined

Upon transforming into the open string channel one finds the following mass-less spectrum Case 1 yields a massless vector multiplet in the adjoint represen-tation of U(Na) Case 2 gives a massless hypermultiplet in the bifundamentalrepresentation of U(Na) times U(Nb) In case 3 there are no massless fields Incase 4 one finds |Υab| Υab = 1

4

prod3i=1 I

iab +

sum3i=1 I

iabσ

iab chiral multiplets in the

bifundamental representation of U(Na)times U(Nb)

3 Gauge threshold corrections

The gauge threshold corrections will be computed using the background fieldmethod employed previously [33 34 11] This means that the one-loop correctionto the gauge coupling of brane a induced by brane b is determined as followsFirst one replaces the 4d spacetime part of the partition function in the aboveexpression as follows [11]

ϑ[αβ

](0 2il)

η3(2il)rarr 2iπBqa

ϑ[αβ

](minusǫa 2il)

ϑ[1212

](minusǫa 2il)

(31)

where πǫa = arctan(πqaB) qa is the charge of the open string ending on brane aand B is the background magnetic field in the 4d spacetime One then expandsthe resulting expressions in a series in B and the coefficient of B2 gives the desiredexpression for the correction to the gauge coupling which needs to be evaluatedfurther

For the four cases distinguished above one finds denoting the amplitudes after

7

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

where R(k) = k in case of an untilted torus and R(1 2 3 4) = 1 2 4 3 inthe other case [16] and the sum is a sum over all stacks of branes Na denotesthe number of branes on stack a The wrapping numbers and twisted chargescarry an index a denoting the brane stack which they describe The orientifoldprojection acts on the wrapping numbers and twisted charges as follows

mI rarr minusmI (29)

ǫikl rarr minusηΩRηΩRiǫiR(k)R(l) (210)

The massless open string spectrum can be read of from the open string par-tition function given by annulus (and Mobius) amplitudes without any vertexoperators inserted These can be calculated from boundary (and crosscap) statesdescribing the D-branes (and O-planes) [25 26 27 28 29 30 31] Only the an-nulus amplitudes will be given here and four cases will be distinguished (Theseare not all possibilities there are but all needed here) that will also be importantin the rest of this paper

Case 1 The annulus has both boundaries on the same stack of branes a Theamplitude is

A(1)aa = minusN2

a

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)aa

+16

3sum

i=1

σiaa

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)aa

] (211)

where σiab =

14

sum4kl=1 ǫ

iaklǫ

ibkl ϑ = ϑ(0 2il) η = η(2il) and V i

a defined in (21)now carries an index a to denote the brane stack considered The followingKaluza-Klein and winding sum has been defined [32]

L(i)ab =

sum

mw

exp

[minusπl(V i

a )2

(m2

(R(i)1 R

(i)2 )2

+ w2

)+ iπm(δia minus δib) + iπw(λi

a minus λib)

]

Case 2 The annulus stretches between two stacks of D-branes a and b wrappingthe same submanifold of the covering six-torus of the internal space and havingthe same discrete Wilson lines turned on(λi

a = λib) This means in particular

θia = θibVia = V i

b δia = δib however not all twisted charges ǫi are equal

One can show that in this case σiab = plusmn1 The amplitude is

A(2)ab = minusNaNb

int infin

0

dl

[ϑ43 minus ϑ4

4 minus ϑ42 + ϑ4

1

η12

3prod

i=1

(V ia )

2

R(i)1 R

(i)2

L(i)ab

+16

3sum

i=1

σiab

ϑ23ϑ

22 minus ϑ2

4ϑ21 minus ϑ2

2ϑ23 + ϑ2

1ϑ24

η6ϑ24

(V ia )

2

R(i)1 R

(i)2

L(i)ab

] (212)

6

Case 3 The two stacks of branes wrap submanifolds that are homologicallyequal on the covering torus (This implies θia = θibV

ia = V i

b ) but do not satisfyλia = λi

b δia = δib for all i The amplitude looks as the one of case 2

Case 4 The annulus stretches between two branes that intersect at non-trivialangles on all three tori The amplitude reads

A(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

)sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

3prod

i=1

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

(213)

+32sum

i 6=j 6=k

I iab σiab

sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

ϑ[|αminus12|

β

](θjab)

ϑ[

012

](θjab)

ϑ[|αminus12|

β

](θkab)

ϑ[

012

](θkab)

]

where θiab = θiaminusθib and the intersection number I iab = (mian

ibminusmi

bnia) were defined

Upon transforming into the open string channel one finds the following mass-less spectrum Case 1 yields a massless vector multiplet in the adjoint represen-tation of U(Na) Case 2 gives a massless hypermultiplet in the bifundamentalrepresentation of U(Na) times U(Nb) In case 3 there are no massless fields Incase 4 one finds |Υab| Υab = 1

4

prod3i=1 I

iab +

sum3i=1 I

iabσ

iab chiral multiplets in the

bifundamental representation of U(Na)times U(Nb)

3 Gauge threshold corrections

The gauge threshold corrections will be computed using the background fieldmethod employed previously [33 34 11] This means that the one-loop correctionto the gauge coupling of brane a induced by brane b is determined as followsFirst one replaces the 4d spacetime part of the partition function in the aboveexpression as follows [11]

ϑ[αβ

](0 2il)

η3(2il)rarr 2iπBqa

ϑ[αβ

](minusǫa 2il)

ϑ[1212

](minusǫa 2il)

(31)

where πǫa = arctan(πqaB) qa is the charge of the open string ending on brane aand B is the background magnetic field in the 4d spacetime One then expandsthe resulting expressions in a series in B and the coefficient of B2 gives the desiredexpression for the correction to the gauge coupling which needs to be evaluatedfurther

For the four cases distinguished above one finds denoting the amplitudes after

7

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

Case 3 The two stacks of branes wrap submanifolds that are homologicallyequal on the covering torus (This implies θia = θibV

ia = V i

b ) but do not satisfyλia = λi

b δia = δib for all i The amplitude looks as the one of case 2

Case 4 The annulus stretches between two branes that intersect at non-trivialangles on all three tori The amplitude reads

A(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

)sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

3prod

i=1

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

(213)

+32sum

i 6=j 6=k

I iab σiab

sum

αβ

(minus1)2(α+β)ϑ[αβ

](0)

η3

ϑ[αβ

](θiab)

ϑ[1212

](θiab)

ϑ[|αminus12|

β

](θjab)

ϑ[

012

](θjab)

ϑ[|αminus12|

β

](θkab)

ϑ[

012

](θkab)

]

where θiab = θiaminusθib and the intersection number I iab = (mian

ibminusmi

bnia) were defined

Upon transforming into the open string channel one finds the following mass-less spectrum Case 1 yields a massless vector multiplet in the adjoint represen-tation of U(Na) Case 2 gives a massless hypermultiplet in the bifundamentalrepresentation of U(Na) times U(Nb) In case 3 there are no massless fields Incase 4 one finds |Υab| Υab = 1

4

prod3i=1 I

iab +

sum3i=1 I

iabσ

iab chiral multiplets in the

bifundamental representation of U(Na)times U(Nb)

3 Gauge threshold corrections

The gauge threshold corrections will be computed using the background fieldmethod employed previously [33 34 11] This means that the one-loop correctionto the gauge coupling of brane a induced by brane b is determined as followsFirst one replaces the 4d spacetime part of the partition function in the aboveexpression as follows [11]

ϑ[αβ

](0 2il)

η3(2il)rarr 2iπBqa

ϑ[αβ

](minusǫa 2il)

ϑ[1212

](minusǫa 2il)

(31)

where πǫa = arctan(πqaB) qa is the charge of the open string ending on brane aand B is the background magnetic field in the 4d spacetime One then expandsthe resulting expressions in a series in B and the coefficient of B2 gives the desiredexpression for the correction to the gauge coupling which needs to be evaluatedfurther

For the four cases distinguished above one finds denoting the amplitudes after

7

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

the manipulations described with an additional superscript g

Ag(1)aa = 32π N2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

L(i)aa (32)

Ag(2)ab = 32π NaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (33)

Ag(3)ab = 32π NaNb

int infin

0

dl3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

L(i)ab (34)

Ag(4)ab = NaNb

int infin

0

dl

[8(prod3

i=1 Iiab

) 3sum

i=1

ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+sum

i 6=j 6=k

32I iab σiab

(ϑprime1(θ

iab 2il)

ϑ1(θiab 2il)+

ϑprime4(θ

jab 2il)

ϑ4(θjab 2il)

+ϑprime4(θ

kab 2il)

ϑ4(θkab 2il)

)] (35)

The overall normalisation will later be fixed by demanding the running of thegauge coupling with the correct beta function coefficient but the relative normal-isation is taken into account correctly The full correction to the gauge couplingon brane a is given by summing over all annuli (and Mobii) with at least oneboundary on brane a

The above expressions can be evaluated analogously to the corresponding onesin the Z2 times Z2 orbifold with h21 = 3 [11 12] The results are

Ag(1)aa = 32πN2

a

int infin

0

dl

3sum

i=1

σiaa

(V ia )

2

R(i)1 R

(i)2

(36)

+32πN2a

(σaa ln

[M2

s

micro2

]minus

3sum

i=1

σiaa ln

[(V i

a )2])

(37)

minus32πN2a

3sum

i=1

4 σiaa ln

[η(iR

(i)1 R

(i)2 )]

(38)

minus32πN2aσaa ln[4π] (39)

Here σab =sum3

i=1 σiab was defined and the divergence

intinfin dtt

arising from themassless open string modes was replaced by ln [M2

s micro2] [12] For the cases 2 and

3 we obtain

Ag(2)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(310)

+32πNaNb

3sum

i=1

σiab

(ln

[M2

s

micro2

]minus ln

[(V i

a )2]minus 4 ln

[η(iR

(i)1 R

(i)2 )])

8

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

(311)

minus32πNaNbσab ln[4π] (312)

Ag(3)ab = 32πNaNb

int infin

0

dl

3sum

i=1

σiab

(V ia )

2

R(i)1 R

(i)2

(313)

minus64πNaNb

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iR

(i)1 R

(i)2 )

η(iR(i)1 R

(i)2 )

(314)

Note that in case 2 there is again a divergence proportional tointinfin dt

t which was

replaced by ln[M2s micro

2] whereas there is none in case 3 due to the absence ofmassless modes in this sector Finally for case 4 the thresholds are

Ag(4)ab = NaNb

int infin

0

dl 8(prod3

i=1 Iiab

) 3sum

i=1

π cot[πθiab

](315)

+NaNb

int infin

0

dl

3sum

i=1

32 I iab σiab π cot

[πθiab

](316)

+16πNaNbΥab

3sum

i=1

(siab ln

[M2

s

micro2

]+ ln

[Γ(1minus |θiab|)Γ(|θiab|)

]siab

)(317)

+64πNaNb ln[2]sum

i

I iab (θia minus θib) σ

iab (318)

+16πNaNb

((ln[2]minus γE)Υab

sum

i

siab + ln[4]sum

i 6=j 6=k

I iabσiab(s

jab + skab)

)

(319)

where the abbreviation siab = sign(θiab) was usedThe contributions (39) (312) and (319) are just moduli independent finite

constants and as such of no further interest The terms in (36) (310) (313)(315) and (316) are divergent integrals and the sum over these contributionsfrom all annuli has to cancel The expression (315) also appears in the modelon the orbifold with h21 = 3 and the sum can be shown to vanish upon using theuntwisted tadpole cancellation conditions [11] 3

Using (V ia )

2 = (nia)

2(R(i)1 )2 + (mi

a)2(R

(i)2 )2 tan θia = mi

aR(i)2 ni

aR(i)1 as well as

the fact that in cases 1 and 2 (nia m

ia) = (ni

b mib) the three terms (36) (310)

3For this to happen the Mobius amplitudes have to be taken into account as well but theyare unchanged as the orientifold planes do not carry twisted charges

9

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

(313) and (316) can all be brought into the form4

ATTab = 8πNaNb

int infin

0

dl3sum

i=1

4sum

kl=1

ǫiaklǫibkl

nian

ib(R

(i)1 )2 + mi

amib(R

(i)2 )2

R(i)1 R

(i)2

(320)

where the superscript TT denotes that these are the contributions from (32)(33) (34) and (35) that vanish after using the twisted tadpole cancellationconditions Summing over this contribution from all annuli yields (denoting theorientifold image of brane b by bprime)

sum

b6=a

(ATTab +ATT

abprime ) +ATTaaprime (321)

= Na

sum

i

nia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)1 )2

sum

b

Nbnib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))

]

+Na

sum

i

mia

R(i)1 R

(i)2

sum

kl

ǫiakl

[(R

(i)2 )2

sum

b

Nbmib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))

]

which vanishes upon using the twisted tadpole conditions (27) and (28) Theother contributions to the gauge coupling corrections will be discussed in the nextsections

4 Holomorphic gauge kinetic function

In a supersymmetric gauge theory one can compute the running gauge couplingsga(micro

2) in terms of the gauge kinetic functions fa the Kahler potential K and theKahler metrics of the charged matter fields Kab(micro2) [35 36 21]

8π2

g2a(micro2)

= 8π2real(fa) + ∆0 +sum

r

∆r (41)

with

∆0 = T (Ga)

(minus3

2ln

[Λ2

micro2

]minus 1

2K + ln

[1

g2a(micro2)

])(42)

∆r = Ta(r)

(nr

2ln

[Λ2

micro2

]+

nr

2K minus ln

[detKr(micro

2)])

(43)

and Ta(r) = Tr(T 2(a)) (T(a) being the generators of the gauge group Ga) In addi-

tion T (Ga) = Ta(adjGa) and nr is the number of multiplets in the representation

4Note that due to the above choice of signs for n12 m12 and the supersymmetry conditionnot only the absolute values but also the signs of the wrapping numbers are equal

10

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

r of the gauge group and the sum in (41) runs over these representations Forthis paper only the gauge groups SU(Na) and its fundamental adjoint symmet-ric and antisymmetric representations (r = f adj s a) are relevant For theseT (Ga) = Ta(adj) = Na Ta(f) = 12 Ta(s) = (Na+2)2 and Ta(a) = (Naminus2)2In this context the natural cutoff scale for a field theory is the Planck scale ieΛ2 = M2

PlThe stringy one-loop correction to the LHS of (41) was calculated in the

previous section As in a supersymmetric theory fa is a holomorphic functionof the chiral superfields the non-holomorphic terms on the LHS of (41) betterbe equal to the non-holomorphic terms in ∆0 and ∆r It will be shown that thisis actually the case for the model under consideration apart from some universalthreshold corrections [19 20 21 8] to be discussed in the next section

∆0 must match the contribution of the annulus Ag(1)aa to the LHS of (41)

On the orbifold with h21 = 3 the latter vanishes [11] and the terms on theRHS cancel amongst each other [8] In the present case things are a little bitdifferent There are no chiral multiplets in the adjoint of the gauge group suchthat using K = minus ln(SU1U2U3T1T2T3) M

2Pl prop M2

s

radicSU1U2U3 and (22) the one

loop contribution in ∆0 becomes

∆0

T (Ga)= minus3

2ln

[M2

Pl

micro2

]minus 1

2K + ln

[1

g2atree

]= minus3

2ln

[M2

s

micro2

]+ ln

[prod

i

V ia

] (44)

The RHS of (44) matches (up to the overall normalisation the relative nor-malisation of the two terms is however correct) precisely the terms (37) Theterm (38) has no corresponding one on the RHS of (41) but upon complexify-ing the Kahler moduli Ti rarr T c

i (the axions stem from the NSNS 2-form-field) itcan be analytically continued to a holomorphic function of the complex Kahlermoduli One is thus lead to conclude that there is a one-loop correction to thegauge kinetic function of the form

δaf1minusloopa =

Na

4π2

3sum

i=1

ln [η(iT ci )] (45)

especially as there is a very similar correction in the case of the Z2 times Z2 orbifoldwith h21 = 3 arising there from a different open string sector [8] The normalisa-tion of the term on the RHS of (45) is determined from the relative normalisationof the different terms in (41) and (36) (37) (38) (39)

Next the contributions in case 2 will be discussed Note that in this case|σi

ab| = |σab| = 1 The terms in (311) give a contribution to the LHS of (41)

11

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

proportional to

Aprimeg(2)ab = 32πNaNbσab

(ln

[M2

s

micro2

]

minus2σab ln

[3prod

i=1

(V ia )

σiab

]minus 4σab

3sum

i=1

σiab ln [η(iTi)]

) (46)

where the prime on Aprimeg(2)ab denotes that the tadpole and constant contributions

have been subtractedThis is to be compared (up to the overall normalisation) with the contribution

of nf = 2Nb multiplets in the fundamental representation (Ta(f) = 12) of thegauge group to the RHS of (41)

∆f =2Nb

4

(ln

[M2

s

micro2

]minus 2 ln

[Kf(SU1U2U3)

14 (T1T2T3)

12

]) (47)

One concludes that the Kahler metric for the two chiral multiplets in this sectoris

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(48)

and that there is the following one-loop correction to the gauge kinetic func-tion

δb(2)f1minusloopa = minus 1

4π2

sum

b

Nb σab

3sum

i=1

σiab ln [η(iT

ci )] (49)

The normalisation of the terms on the RHS of (49) is determined from the relativenormalisation of the different terms in (41) and (46) There is an overall minussign in (49) as compared to (45) which essentially comes from the fact that thegauge multiplet itself contributes with a different sign to the beta function thanchiral multiplets Upon changing variables the Kahler metric (48) is identicalto the one for adjoint fields in the model with h21 = 3 [24 5] as one would expectfrom the fact that these fields are described by the same vertex operators in theworldsheet CFT

The contribution from case 3 to the LHS of (41) is finite after using thetadpole cancellation condition as one would expect from the fact that there areno massless open string states in this sector One concludes that the term (314)leads to the following correction to the gauge kinetic function

δb(3)f1minusloopa = minus 1

8π2

sum

b

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(410)

12

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

Finally there is the sector yielding the chiral bifundamentals (case 4) The termsin (317) contribute

Aprimeg(4)ab = 16πNaNbΥab

sum

i

sign(θiab)

(ln

[M2

s

micro2

]

+1sum

j sign(θjab)

ln

[3prod

k=1

(Γ(1minus |θkab|)Γ(|θkab|)

)sign(θkab)])

(411)

where the prime in Aprimeg(4)ab denotes omission of the tadpole and constant contribu-

tions and those from (318) to the LHS of (41) An equal contribution (up tothe overall normalisation) to the RHS results if the Kahler metric for the chiralbifundamentals is

KC(1)fab = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(412)

Note that this agrees with the result obtained in the case with h21 = 3 [8] it ishowever more general in that it allows for arbitrary signs of θjab

5 Universal gauge coupling corrections

At first sight it might seem that holomorphy implies that the exact Kahler metricfor the chiral bifundamentals is given by (412) However this is not true As inthe case of the Z2 times Z2 orbifold with h21 = 3 [8] an additional factor is allowedif at the same time the dilaton and complex structure moduli are redefined atone-loop This redefinition is related to sigma-model anomalies in the low energysupergravity theory [19 20]

The factor takes the form [5 24 8]

KC(2)fab =

3prod

i=1

Uminusξ sign(Υab)θ

iab

i Tminusζ sign(Υab)θ

iab

i (51)

where ξ and ζ are undetermined constants Upon summing over all chiral mattercharged under U(Na) and using both the twisted and untwisted tadpole cancella-tion conditions this term leads to the following contribution to the RHS of (41)

13

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

[8]

minussum

b

primeTa(f)

(ln detK

C(2)fab + lndetK

C(2)fabprime

)

minusTa(a) ln detKC(2)faaprime minus Ta(s) ln detK

C(2)faaprime

= minus1

4n1an

2an

3a

[sum

b

Nbm1bm

2bm

3b

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

4

3sum

i 6=j 6=k=1

niam

jam

ka

[sum

b

Nbmibn

jbn

kb

3sum

l=1

θlb (ξ lnUl + ζ lnTl)

]

minus1

8

sum

ikl

niaǫ

iakl

sum

b

Nbmib

(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

+1

8

sum

ikl

miaǫ

iakl

sum

b

Nbnib

(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l)

)sum

j

θjb(ξ lnUj + ζ lnTj)

(52)

The prime on the first sum indicates that it only runs over branes b that intersectbrane a at generic angles on all three tori

The first two terms on the RHS of (52) are cancelled by the correction arisingfrom the tree-level gauge kinetic function on the RHS of (41) upon redefiningthe dilaton and complex structure moduli as follows

S rarr S minus 1

32π2

sum

b

Nbm1bm

2bm

3b

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(53)

Ui rarr Ui +1

32π2

sum

b

Nbmibn

jbn

kb

(sum

l

θlb(ξ lnUl + ζ lnTl)

)(54)

In order to interpret the last two terms in (52) one notices that the tree-levelgauge kinetic function in addition to the dependence on untwisted moduli givenin (23) depends also on the twisted moduli An anomaly analysis sketched inthe appendix suggests that the full tree-level gauge kinetic function is

fatree = Sc

3prod

i=1

nia minus

3sum

i 6=j 6=k=1

U ci n

iam

jam

ka +

3sum

i=1

4sum

kl=1

nia

(ǫiakl

minusηΩRηΩRiǫiaR(k)R(l)

)W c

ikl + mia

(ǫiakl + ηΩRηΩRiǫ

iaR(k)R(l)

)W c

ikl(55)

whereW cikl and W c

ikl i isin 1 2 3 k l isin 1 2 3 4 are twisted sector fields Thereare h21 hypermultiplets coming from the closed string sector in the spectrumof type IIA string theory on a Calabi-Yau space In the present case h21 =

14

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

huntwisted21 + htwisted

21 acquires contributions from untwisted (huntwisted21 = 3) and

twisted (htwisted21 = 48) sectors W c

ikl and W cikl are the 2htwisted

21 = 96 complexscalars arising from the sixteen fixed points labelled by kl in each of the threetwisted sectors labelled by i The real parts of W c

ikl and W cikl are NSNS-sector

fields and the axions are RR-fieldsThe twisted moduli can also lead to sigma model anomalies in the low energy

supergravity theory and can therefore mix with the dilaton and the other complexstructure moduli One is thus lead to conclude that the twisted moduli are shiftedby

δ(1)Wikl = minus 1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (56)

and

δ(1)Wikl =1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb(ξ lnUj + ζ lnTj) (57)

respectively such that the last two terms in (52) cancelOne comment on (51) is in order From the worldsheet conformal field theory

point of view it is clear that the Kahler metrics for the chiral matter arising at abrane intersection should be the same on the orbifolds with h21 = 51 and h21 = 3At first sight it might seem that (51) differs from the corresponding expressionfor the other orbifold [8] in that the latter has sign(

prodi I

iab) in the exponent rather

than sign(Υab) as in (51) There is however a physical argument that shows thatthese signs must be equal The orbifold projection removes some of the stringstates but it cannot change their spacetime chirality As the aforementionedsigns determine the spacetime chirality they must be equal

There is one term (318) in the gauge threshold corrections that has so farbeen neglected Upon summing over all annuli with one boundary on brane stacka and using the tadpole cancellation condition it can be cast into

sum

i

sum

kl

(sum

b6=a

NbIiabǫ

iaklǫ

ibkl(θ

ia minus θib) +

sum

b

NbIiabprimeǫ

iaklǫ

ibprimekl(θ

ia minus θibprime)

)

=sum

i

sum

kl

naǫiakl

sum

b

Nbmibθ

ib(minusǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))

+sum

i

sum

kl

miaǫ

iakl

sum

b

Nbnibθ

ib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l)) (58)

where it was used that I iabprime = minus(niam

ib + mi

anib) ǫ

ibprimekl = minusηΩRηΩRiǫ

ibR(k)R(l) and

θibprime = minusθib This term resembles the last two terms in (52) One would thereforelike to conjecture that this term is also cancelled by a redefinition of the twisted

15

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

moduli In particular the shifts would have to be

δ(2)Wikl = minus 1

32π2

sum

b

Nbmibǫ

ibklθ

ib ln[2]

δ(2)Wikl =1

32π2

sum

b

Nbnibǫ

ibklθ

jb ln[2] (59)

Taking into account both the contributions (56)(57) and (59) the real parts ofthe twisted moduli acquire the redefinition (61) Note here that

sum

kl

ǫiakl(ǫibkl plusmn ηΩRηΩRiǫ

ibR(k)R(l)) =

sum

kl

ǫibkl(ǫiakl plusmn ηΩRηΩRiǫ

iaR(k)R(l)) (510)

6 Summary of results

Eventually even for the danger of repeating ourselves let us summarise themain results of this paper The gauge threshold corrections for intersecting D6-brane models on the Z2 times Z

prime2 orbifold with h21 = 51 which allows for rigid

three-cycles were computed It was shown that the results fulfil the non-trivialcondition that in a supersymmetric theory the gauge kinetic function must bea holomorphic function of the chiral superfields Let us emphasise that this isimportant because it shows that the D-instanton generated couplings can beincorporated in a holomorphic superpotential[8] For it to be true a mixingbetween the complex structure moduli of the twisted and untwisted sectors musttake place at one loop In particular the twisted complex structure moduli areredefined as

Wikl rarr Wikl minus1

64π2

sum

b

Nbmibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij)

Wikl rarr Wikl +1

64π2

sum

b

Nbnibǫ

ibkl

sum

j

θjb (ξ lnUj + ζ lnTj + ln[4] δij) (61)

In contrast to the Z2 times Z2 orbifold with h21 = 3 the gauge kinetic function doesin the present case receive one-loop corrections from sectors preserving N = 1supersymmetry Upon summing over all contributions one finds that the one-loop correction to the gauge kinetic function for the gauge theory on brane stacka is

f 1minusloopa =

1

4π

3sum

i=1

ln [η(iT ci )]minus

1

4π2

sum

b isin case 2

Nb

σab

3sum

i=1

σiab ln [η(iT

ci )]

minus 1

8π2

sum

b isin case 3

Nb

σab

3sum

i=1

σiab ln

ϑ[12(1minus|δiaminusδi

b|)

12(1minus|λiaminusλi

b|)

](0 iT c

i )

η(iT ci )

(62)

16

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

The Kahler metric for the vector-like bifundamental matter arising from stringsstretched between two stacks of branes that are coincident but differ in theirtwisted charges was using holomorphy arguments determined to be

KVf = (SU1U2U3)

minus 14 (T1T2T3)

minus 12

(3prod

i=1

(V ia )

σiab

) 1σab

(63)

Equivalently one can determine the Kahler metric for the chiral bifundamentalmatter arising at the intersection of two stacks of branes to be

KCfab = K

C(1)fab K

C(2)fab

= Sminus 14

3prod

i=1

Uminus14minusξ sign(Υab)θ

iab

i Tminus12minusζ sign(Υab)θ

iab

i times

[3prod

i=1

(Γ(1minus |θiab|)Γ(|θiab|)

)sign(θiab)]minus1[2

P

j sign(θj

ab)]

(64)

where ξ and ζ are undetermined constants It was argued [8] that they should beξ = 0 and ζ = plusmn12 These values do however not follow from the calculationsperformed in this paper

As already discussed in the introduction the results of this paper are impor-tant for the study of E2-instantons on the background considered as the one-loopamplitudes computed here are equal to one-loop amplitudes in the instanton back-ground They are therefore important ingredients when eg neutrino Majoranamasses [18 22] or moduli stabilisation is studied [23]

Acknowledgements

The authors would like to thank Dieter Lust Sebastian Moster StephanStieberger and especially Nikolas Akerblom and Erik Plauschinn for valuable dis-cussions This work is supported in part by the European Communityrsquos HumanPotential Programme under contract MRTN-CT-2004-005104 lsquoConstituents fun-damental forces and symmetries of the universersquo

A Anomaly analysis

Models on the background considered in this paper and described in the maintext do generically have an anomalous spectrum The anomalies are howevercancelled by a generalised Green-Schwarz mechanism [37]

In the following the U(1)aminusSU(Nb)2 anomalies will be considered and it will

be assumed that brane stacks a and b are an example of what was called case 4in the main text

17

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

Oriented case

It is convenient to start with the case in which there is no orientifold planeThe anomaly coefficient arising from the chiral multiplets can be computed to be[4]

minusNaΥab =Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

minus3sum

i 6=j 6=k=1

mibn

jbn

kbn

iam

jam

ka minus m1

bm2bm

3bn

1an

2an

3a

minussum

i

sum

kl

mibǫ

ibkln

iaǫ

iakl +

sum

i

sum

kl

nibǫ

ibklm

iaǫ

iakl

] (A1)

The following terms arise from the Chern-Simons actions for the brane stacks aand b

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

+

3sum

i 6=j 6=k=1

mibn

jbn

kb A

(0)i + m1

bm2bm

3b A

(0)0

](A2)

SCSa sup Na

intFa and

[minus n1

an2an

3a A

(2)0 minus

3sum

i 6=j 6=k=1

niam

jam

ka A

(2)i

+

3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A3)

and lead to a cancellation of the anomalies described by the first eight summandsin (A1) Fa is the U(1)a field strength and Fb the SU(Nb) field strength A

(0)0i

A(0)0i are axions arising from untwisted RR fields and A

(2)0i A

(2)0i their 4d dual

two-formsThe remaining anomalies are cancelled if the following couplings of the twisted

RR fields to the gauge fields arise in the low energy effective action [38 39]

SCSb =

inttr(Fb and Fb)

[nibǫ

ibkl A

(0)ikl + mi

bǫibkl A

(0)ikl

](A4)

SCSa = Na

intFa and

[minus ni

aǫiakl A

(2)ikl + mi

aǫiakl A

(2)ikl

](A5)

Here A(0)ikl and A

(0)ikl are axions arising from the twisted RR sectors and A

(2)ikl A

(2)ikl

their 4d dual two-forms

18

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

Unoriented case

Things are quite similar to the oriented case but the orientifold images have tobe taken into account and some of the axions are projected out of the spectrumThe anomaly coefficient becomes

Na

2(minusΥab +Υaprimeb) =

Na

4

[n1bn

2bn

3bm

1am

2am

3a +

3sum

i 6=j 6=k=1

nibm

jbm

kb m

ian

jan

ka

+1

2

sum

i

sum

kl

mibǫ

ibkln

ia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))

+1

2

sum

i

sum

kl

nibǫ

ibklm

ia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))

](A6)

and the Chern-Simons actions yield

SCSb sup

inttr(Fb and Fb)

[n1bn

2bn

3b A

(0)0 +

3sum

i 6=j 6=k=1

nibm

jbm

kb A

(0)i

](A7)

SCSa sup Na

intFa and

[3sum

i 6=j 6=k=1

mian

jan

ka A

(2)i + m1

am2am

3a A

(2)0

](A8)

to cancel the anomalies related to the first four summands in (A6) Note that A0i

is projected out whereas A0i remains in the spectrum Full anomaly cancellationoccurs if the couplings

SCSb =

inttr(Fb and Fb)

[nib(ǫ

ibkl minus ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

+mib(ǫ

ibkl + ηΩRηΩRiǫ

ibR(k)R(l))A

(0)ikl

](A9)

=

inttr(Fb and Fb)

[nib

2(ǫibkl minus ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl minus ηΩRηΩRiA

(0)iR(k)R(l))

+mi

b

2(ǫibkl + ηΩRηΩRiǫ

ibR(k)R(l))(A

(0)ikl + ηΩRηΩRiA

(0)iR(k)R(l))

](A10)

and

SCSa = Na

intFa and

[nia(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

+mia(ǫ

iakl minus ηΩRηΩRiǫ

iaR(k)R(l))A

(2)ikl

](A11)

19

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

= Na

intFa and

[nia

2(minusǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

+mi

a

2(ǫiakl minus ηΩRηΩRiǫ

iaR(k)R(l))(A

(2)ikl minus ηΩRηΩRiA

(2)iR(k)R(l))

](A12)

are present in the effective action Note that from (A10) and (A12) one can

infer which linear combinations of Aikl and Aikl are projected out To be precisethose ones that do not appear in (A10) and (A12) are projected out

(A9) leads one to conclude that the combinations W cikl = Wikl + iA

(0)ikl and

W cikl = Wikl + iA

(0)ikl (or rather some linear combinations thereof) are the appro-

priate complex scalars of the chiral multiplets in the low energy effective actionand that the holomorphic gauge kinetic function is indeed given by (55)

20

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

References

[1] A M Uranga ldquoChiral four-dimensional string compactifications withintersecting D-branesrdquo Class Quant Grav 20 (2003) S373ndashS394hep-th0301032

[2] F G Marchesano Buznego ldquoIntersecting D-brane modelsrdquohep-th0307252

[3] D Lust ldquoIntersecting brane worlds A path to the standard modelrdquoClass Quant Grav 21 (2004) S1399ndash1424 hep-th0401156

[4] R Blumenhagen M Cvetic P Langacker and G Shiu ldquoToward realisticintersecting D-brane modelsrdquo Ann Rev Nucl Part Sci 55 (2005)71ndash139 hep-th0502005

[5] R Blumenhagen B Kors D Lust and S Stieberger ldquoFour-dimensionalString Compactifications with D-Branes Orientifolds and Fluxesrdquo Phys

Rept 445 (2007) 1ndash193 hep-th0610327

[6] S A Abel and M D Goodsell ldquoRealistic Yukawa couplings throughinstantons in intersecting brane worldsrdquo hep-th0612110

[7] N Akerblom R Blumenhagen D Lust E Plauschinn andM Schmidt-Sommerfeld ldquoNon-perturbative SQCD Superpotentials fromString Instantonsrdquo JHEP 04 (2007) 076 hep-th0612132

[8] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoInstantons and Holomorphic Couplings in Intersecting D- brane ModelsrdquoJHEP 08 (2007) 044 arXiv07052366 [hep-th]

[9] M Billo et al ldquoInstantons in N=2 magnetized D-brane worldsrdquoarXiv07083806 [hep-th]

[10] M Billo et al ldquoInstanton effects in N=1 brane models and the Kahlermetric of twisted matterrdquo arXiv07090245 [hep-th]

[11] D Lust and S Stieberger ldquoGauge threshold corrections in intersectingbrane world modelsrdquo Fortsch Phys 55 (2007) 427ndash465 hep-th0302221

[12] N Akerblom R Blumenhagen D Lust and M Schmidt-SommerfeldldquoThresholds for intersecting D-branes revisitedrdquo Phys Lett B652 (2007)53ndash59 arXiv07052150 [hep-th]

[13] E Dudas and C Timirgaziu ldquoInternal magnetic fields and supersymmetryin orientifoldsrdquo Nucl Phys B716 (2005) 65ndash87 hep-th0502085

21

[14] I Antoniadis G DrsquoAppollonio E Dudas and A Sagnotti ldquoOpendescendants of Z(2) x Z(2) freely-acting orbifoldsrdquo Nucl Phys B565

(2000) 123ndash156 hep-th9907184

[15] R Blumenhagen and E Plauschinn ldquoIntersecting D-branes on shift Z(2) xZ(2) orientifoldsrdquo JHEP 08 (2006) 031 hep-th0604033

[16] R Blumenhagen M Cvetic F Marchesano and G Shiu ldquoChiral D-branemodels with frozen open string modulirdquo JHEP 03 (2005) 050hep-th0502095

[17] R Blumenhagen M Cvetic and T Weigand ldquoSpacetime instantoncorrections in 4D string vacua - the seesaw mechanism for D-branemodelsrdquo Nucl Phys B771 (2007) 113ndash142 hep-th0609191

[18] M Cvetic R Richter and T Weigand ldquoComputation of D-braneinstanton induced superpotential couplings - Majorana masses from stringtheoryrdquo Phys Rev D76 (2007) 086002 hep-th0703028

[19] J-P Derendinger S Ferrara C Kounnas and F Zwirner ldquoAll loop gaugecouplings from anomaly cancellation in string effective theoriesrdquo Phys

Lett B271 (1991) 307ndash313

[20] J P Derendinger S Ferrara C Kounnas and F Zwirner ldquoOn loopcorrections to string effective field theories Field dependent gaugecouplings and sigma model anomaliesrdquo Nucl Phys B372 (1992) 145ndash188

[21] V Kaplunovsky and J Louis ldquoOn Gauge couplings in string theoryrdquoNucl Phys B444 (1995) 191ndash244 hep-th9502077

[22] M Cvetic and T Weigand ldquoHierarchies from D-brane instantons inglobally defined Calabi-Yau Orientifoldsrdquo arXiv07110209 [hep-th]

[23] P G Camara E Dudas T Maillard and G Pradisi ldquoString instantonsfluxes and moduli stabilizationrdquo arXiv07103080 [hep-th]

[24] D Lust P Mayr R Richter and S Stieberger ldquoScattering of gaugematter and moduli fields from intersecting branesrdquo Nucl Phys B696

(2004) 205ndash250 hep-th0404134

[25] M R Gaberdiel and J Stefanski Bogdan ldquoDirichlet branes on orbifoldsrdquoNucl Phys B578 (2000) 58ndash84 hep-th9910109

[26] M R Gaberdiel ldquoLectures on non-BPS Dirichlet branesrdquo Class Quant

Grav 17 (2000) 3483ndash3520 hep-th0005029

22

[27] J Stefanski Bogdan ldquoDirichlet branes on a Calabi-Yau three-foldorbifoldrdquo Nucl Phys B589 (2000) 292ndash314 hep-th0005153

[28] M R Gaberdiel ldquoDiscrete torsion orbifolds and D-branesrdquo JHEP 11

(2000) 026 hep-th0008230

[29] N Quiroz and J Stefanski B ldquoDirichlet branes on orientifoldsrdquo Phys

Rev D66 (2002) 026002 hep-th0110041

[30] B Craps and M R Gaberdiel ldquoDiscrete torsion orbifolds and D-branesIIrdquo JHEP 04 (2001) 013 hep-th0101143

[31] J Maiden G Shiu and J Stefanski Bogdan ldquoD-brane spectrum andK-theory constraints of D = 4 N = 1 orientifoldsrdquo JHEP 04 (2006) 052hep-th0602038

[32] R Blumenhagen L Gorlich and B Kors ldquoSupersymmetric orientifolds in6D with D-branes at anglesrdquo Nucl Phys B569 (2000) 209ndash228hep-th9908130

[33] C Bachas and C Fabre ldquoThreshold Effects in Open-String Theoryrdquo Nucl

Phys B476 (1996) 418ndash436 hep-th9605028

[34] I Antoniadis C Bachas and E Dudas ldquoGauge couplings infour-dimensional type I string orbifoldsrdquo Nucl Phys B560 (1999) 93ndash134hep-th9906039

[35] M A Shifman and A I Vainshtein ldquoSolution of the anomaly puzzle insusy gauge theories and the Wilson operator expansionrdquo Nucl Phys B277

(1986) 456

[36] V Kaplunovsky and J Louis ldquoField dependent gauge couplings in locallysupersymmetric effective quantum field theoriesrdquo Nucl Phys B422 (1994)57ndash124 hep-th9402005

[37] G Aldazabal A Font L E Ibanez and G Violero ldquoD = 4 N = 1 typeIIB orientifoldsrdquo Nucl Phys B536 (1998) 29ndash68 hep-th9804026

[38] L E Ibanez R Rabadan and A M Uranga ldquoAnomalous U(1)rsquos in type Iand type IIB D = 4 N = 1 string vacuardquo Nucl Phys B542 (1999)112ndash138 hep-th9808139

[39] L E Ibanez R Rabadan and A M Uranga ldquoSigma-model anomalies incompact D = 4 N = 1 type IIB orientifolds and Fayet-Iliopoulos termsrdquoNucl Phys B576 (2000) 285ndash312 hep-th9905098

23