Covariant quantization of D-branes

Renata Kallosh*Physics Department, Stanford University, Stanford, California 94305-4060

~Received 15 May 1997!

We have found thatk symmetry allows covariant quantization provided the ground state of the theory isstrictly massive. ForD-p-branes a Hamiltonian analysis is performed to explain the existence of a manifestlysupersymmetric and Lorentz covariant description of the BPS states of the theory. The covariant quantizationof the D-0-brane is presented as an example.@S0556-2821~97!03518-2#

PACS number~s!: 04.60.Ds, 04.60.Gw, 11.25.Mj, 11.30.Pb

I. INTRODUCTION

Extended objects with global supersymmetry have localksymmetry. This symmetry is difficult to quantize in Lorentzcovariant gauges keeping a finite number of fields in thetheory. A revival of interest tok-symmetric objects is due tothe recent discovery ofD-p-branes@1# and D-p-brane ac-tions @2–5#.

Before the choice of the gauge is made the problem withcovariant quantization ofk symmetry can be seen as theimpossibility to disentangle covariantly the combination ofthe first and second class fermionic constraints. In the pastthe quantization of the superparticle, of the supersymmetricstring, and of the supermembrane was performed only in thelight-cone gauge for spinors:

G1u50. ~1!

In this gaugek symmetry is fixed, leaving only the secondclass fermionic constraints, whose Poisson brackets are in-vertible even for the massless ground state. In covariantgauges the major problem is that the constraints are not in-vertible.

Recently a covariant gauge fixingk symmetry ofD-branes has been discovered@3#. The fermionic gauge is ofthe form

u250, ~2!

where u2 is one of the chiral spinors of the 10d theory.Moreover, since there is a duality between theD-1-brane andthe fundamental type IIB string, a gauge fixing of the funda-mental string in covariant gauge has been achieved in@3# bypassing. Does this mean that the previous attempt to covari-antly quantize the Green-Schwarz~GS! string missed thepoint, or something else happened? Is the existence of cova-riant gauges a special property ofD-p-branes only, or dothey exist for allp-branes? To address these issues one hasto take into account that the covariant quantization per-formed in @3# also used the so-called static gaugesXm5sm

for fixing the bosonic reparametrization symmetry. The total

picture of covariant quantization ofk symmetry with the useof static gauges is difficult to analyze.

Here we will first switch to a Hamiltonian form of thetheory that will allow us to study the issues in quantization ofk symmetry before a choice of the reparametrization fixinggauge is made. We will analyze the quantization ofk-symmetric theories and we will establish connection of thequantized theory withd510, N52 supersymmetry algebrawith central extensions:

$Qa ,Qb%52~CGm!abS Pm1Qm

N

2pa8D , ~3!

$Qa ,Qb%52~CGm!abS Pm2Qm

N

2pa8D , ~4!

$Qa ,Qb%52(A

~CGA!abT~p!

QAR

p!. ~5!

We present here the supersymmetry algebra in the formgiven in @1#. HereQN and QR are Neveu-Schwarz–Neveu-Schwarz~NS-NS! and Ramond-Ramond~R-R! charges andA runs over antisymmetrized products of gamma matrices.We will find the covariant mass formula for the quantizedD-p-branes.

The quantization ofD-p-branes in static gauges@3# leadsto complicated nonlinear actions. In this paper we will per-form a covariant quantization of theD-0-brane that will givea simple quadratic action.

II. MAIN RESULTS

We have found that the covariant quantization ofD-branes is consistent and that the ground stateuC& of thesystem has a nonvanishing mass:

M2uC&52~GmPm!2uC&

5T2@det~G1F!ab1PaGabPb#uC&, ~6!

M2.0. ~7!

Here Pm is the momentum conjugate to the coordinateXm

and the eigenvalue of the operator2(GmPm)2 is given by the*Electronic address: kallosh@physics.stanford.edu

PHYSICAL REVIEW D 15 SEPTEMBER 1997VOLUME 56, NUMBER 6

560556-2821/97/56~6!/3515~8!/$10.00 3515 © 1997 The American Physical Society

positive definite expression@T2det(G1F)ab1PaGabPb#. T

is theD-brane tension, the indexa runs over the space com-ponents of the brane,Gab ,Fabare the space part of the metricinduced on the brane and the 2-form, respectively, andPa isthe momentum conjugate to the vector field on the brane.Both the metricGaband the 2-formFab are manifestly su-persymmetric under theN51 part of the fullN52 globalsupersymmetry.

For example, for the covariantly quantizedD-2-brane wefind from Eq.~6! that the mass of the ground state is

M ~D2!2 5T2$Pm,Pn%21~F12!

21PrGrsPs,

m,n50,1, . . . ,9; r ,s51,2. ~8!

Here the square of the Poisson brackets is defined as

$Pm,Pn%2[@~Xm,r2 l Gml ,r !~Xn

,s2 l Gnl ,s!2~m→n!#2,~9!

andl is one of the chirald510 spinors that remains in thetheory afterk symmetry is gauge fixed covariantly. This is aLorentz covariant generalization of the mass operator

M ~M2!2 5$Xi ,Xj%22 u G2G i$u,Xi%, i , j 51, . . . ,9,

~10!

which appears in the quantization of theM -2 supermem-brane@6# in the light-cone gauge@7,8#. In the form in whichr ,s are matrix indices this expression has been used in theM~matrix! model @9#.

The static gauge for theD-2 braneX15s,X25r corre-sponds to

$P1,P2%5~X1! ,s~X2! ,r1•••511•••. ~11!

This makes the constant part of the square of the momentumon the ground state nonvanishing as long as the tensionT isnonvanishing. Thus the covariant quantization of theD-2-brane performed in@3# indeed confirms our general con-clusion that the ground state of aD-brane has a nonvanishingmass as long as the tension of theD-brane is nonvanishing.

This observation also solves the apparent paradox withthe covariant quantization ofk symmetry of the type IIBfundamental string in@3#. For theD-1-brane we get

M ~D1!2 5T2~P1!21P1G11P

1. ~12!

The ground state of theD-1-brane is massive. The technicalreason for this~in Lorentz covariant gauge for spinors andthe static gaugeX15s for reparametrization symmetry@3#!is that (P1)25@(X1),s#21•••511•••. This corresponds tothe D-string wrapped around the circle. In case of a funda-mental string, the covariant gauge fixing proposed in@3# alsouses a static gauge. Thereforethe massless state of the fun-damental GS string is projected out, and in this way the twoquantized string theories,D-1-brane and type IIB fundamen-tal strings, are dual to each other. For theD-0-brane the massformula ~6! cannot be applied since there are no space direc-tions. However, the mass formula in this case is simplyM25Z2, whereZ equals the tension of theD-0-brane.

We will proceed with the derivation of the results statedabove.

III. IRREDUCIBLE k SYMMETRY ON D-BRANES

The k-symmetricD-brane action in the flat backgroundgeometry1 consists of the Born-Infeld-Nambu-Goto termS1and Wess-Zumino termS2:

SDBI1SWZ5TS 2E dp11sA2det~Gmn1Fmn!1E Vp11D .

~13!

HereT is the tension of theD-brane,Gmn is the manifestlysupersymmetric induced world-volume metric

Gmn5hmnPmmPn

n , Pmm5]mXm2 u Gm]mu, ~14!

and Fmn is a manifestly supersymmetric Born-Infeld~BI!field strength~for p even!2

Fmn[Fmn2bmn5@]mAn2 u G11Gm]mu~]nXm2 12 u Gm]nu!#

2~m↔n!. ~15!

When p is odd, G11 is replaced byt3^ I . The action hasglobal supersymmetry,

deu5e, deXm5 e Gmu, ~16!

local k supersymmetry,

dXm5 u Gmdu52d u Gmu, d u 5 k ~11G!, ~17!

and

G5ea/2G~0!8 e2a/2, ~18!

where

a5H 1 12 Yjkg jkG11 ~ type IIA!,

2 12 Yjkg jks3^ 1 ~ type IIB!.

~19!

HereG (0)8 is the product structure, independent of the BIfield, (G (0)8 )251,trG (0)8 50. All dependence on BI fieldF5 ‘ ‘ tan’’ Y is in the exponent@10#. The matrixG11 in typeIIA and s3^ 1 in type IIB theory anticommute withG (0)8 andwith G. Therefore in the basis whereG11 and s3^ 1 arediagonal,G (0)8 andG are off diagonal:

G~0!8 5S 0 g

g21 0D . ~20!

We introduced a 16316-dimensional matrixg that does notdepend on BI field. Now thek-symmetry generator can bepresented in a useful off-diagonal form

G5S 0 gea

~ gea!21 0D , ~21!

1We use the notation of@3#.2We define spinors for evenp as u5u11u2 where

u1[ 12 (11G11)u andu2[ 1

2 (12G11)u.

3516 56RENATA KALLOSH

where

a5H 1 12 Yjkg jk ~ type IIA!,

2 12 Yjkg jk ~ type IIB!.

~22!

The fact thatG is off diagonal and that the matrixgea isinvertible is quite important and the significance of this wasalready discussed in@3,10#. In particular this allows us toconsider only irreduciblek-symmetry transformations byimposing a Lorentz covariant condition onk of the form

k 150, k 25” 0 ~ type IIA!, ~23!

k 250, k 15” 0 ~ type IIB!, ~24!

In this way we have an irreducible 16-dimensionalk sym-metry since the matrixgea is invertible, acting as

d u 15 k 2gea, d u 25 k 2 ,

dXm52 k 2Gmu2 ~ type IIA!, ~25!

d u 15 k 1 , d u 25 k 1~ gea!21,

dXm52 k 1Gmu1 ~ type IIB!. ~26!

IV. FERMIONIC CONSTRAINTSPRIOR TO GAUGE FIXING

The Hamiltonian analysis of supersymmetric extendedobjects withk symmetry requires the knowledge of the fer-mionic constraints. We split the world-volume coordinatessm into a time s05t and a space partsa, wherea51, . . . ,p. In the first approximation we will neglect thespace dependence onsa of spinorsu.

To find fermionic constraints we observe that ourk-symmetric D-p-brane actions depend on the followingcombinations of the time derivatives of the fields:

LDBI~Xm2 u 1Gmu12 u 2Gmu2 ,Aa2@ u 1Gmu1

2 u 2Gmu2#Pma!2LWZ~ u 1TWZu21c.c.!, ~27!

andTWZ[GAZA in the WZ term andGA are given by an oddnumber of antisymmetrizedG matrices in type IIA theoryand an even number in the type IIB case.

We introduce canonical momentaPm ,P0,Pa,Pu1,Pu2

to

Xm,A0 ,Aa ,u1 ,u2. The fermionic constraints follow:

F15 Pu11 u 1~Pm1PaPa

m!Gm1 u 2GMZM , ~28!

F25 Pu21 u 2~Pm2PaPa

m!Gm1 u 1GMZM . ~29!

These 32 constraints can be shown to represent 16 first classconstraints and 16 second class ones. The Poisson bracketsof these constraints are given by@terms with possible deriva-tives of the delta functionsdp(sa2sa) are omitted#

$F1~t,sa!,F1~t,sa!%52~Pm1PaPam!CGmdp~sa2sa!

1•••, ~30!

$F2~t,sa!,F2~t,sa!%52~Pm2PaPam!CGmdp~sa2sa!

1•••, ~31!

$F1~t,sa!,F2~t,sa!%52GAZAdp~sa2sa!1•••.~32!

These brackets realized510, N52 supersymmetry algebrawith central extensions. The R-R charges in this algebraZAare due to the structure of the WZ part of the action. Forexample, inp50 case we haveGAZA5Z, whereZ is themass of theD-0-particle. Inp51 we getGAZA5GmPs

m , forp52 this term is GAZA5Gm1Gm2eab@Pam1

Pbm21Fab#,

a51,2, etc. The terms with thePaPam could be considered

as defining the NS-NS charge.The problem of covariant quantization of the fundamental

string was the impossibility to disentangle these constraintsinto the first class and the second class covariantly. For theD-branes the situation is different due to the presence of theR-R charges, which appear in the bracket ofF1 andF2 inEq. ~32!. We will see later that the reparametrization con-straints in presence of R-R central charges are such that thebracket for, e.g.,$F1 ,F1% is invertible.

V. GAUGE-FIXED k SYMMETRY

From now on we will study the possibilities to quantizethese actions. One of the possibilities that we will pursuehere is to immediately gauge fixk symmetry by choosing thegauge for theta. Here again we will follow@3# and simplytake

u250, u1[l ~ type IIA!, ~33!

u150, u2[l ~ type IIB!. ~34!

Note that our choice of irreduciblek symmetry~which is notunique! was made here with the purpose of explicitly elimi-nating u2 (u1) in the type IIA ~IIB ! case usingd u 25 k 2

(d u 15 k 1). The gauge-fixed action has one particularly use-ful property: the Wess-Zumino term vanishes in this gauge@3#. We are left with the reparametrization-invariant action

Sk fixed52E dp11sA2det~Gmn1Fmn!, ~35!

Gmn5hmnPmmPn

n , Pmm5]mXm2 l Gm]ml, ~36!

Fmn5@]mAn2 l Gm]ml~]nXm2 12 l Gm]nl!#2~m↔n!.

~37!

The justification for this procedure can be done either byshowing that the remaining action does not have fermionicgauge symmetries anymore~which was done in@10#! or bythe study of the Hamiltonian of the gauge fixed theory andthe constraints. In@3# the remaining reparametrization sym-metry of the theory was gauge fixed by choosing a static

56 3517COVARIANT QUANTIZATION OF D-BRANES

gauge. The resulting gauge-fixed action does not have fermi-onic degeneracy and this also shows that the choice of agauge-fixing made in@3# is acceptable.

We will take from now on a different approach and studythe Hamiltonian of the theory afterk symmetry is gaugefixed.

VI. HAMILTONIAN STRUCTURE OF THE THEORY

The set of canonical momenta and coordinates of thetheory in Eq. ~35! includes (Pm ,Xm),(P0,A0),(Pa,Aa),(Pl ,l). All expressions are relatively simple if as before weneglect in the first approximation terms with world-volumespace derivatives on spinors]l/]sa. The phase space action~35! can be brought to the canonical form

L5pi]0qi1jata~p,q!, ~38!

where ta(p,q) are some constraints of the first and secondkind:

Lcan5Pm]0Xm1PaF0a1 Pl]0l2j„PmPm1PaGabPb

1T2det@~G1F!ab#…2jBIP02ja~PmPa

m1PbFab!

1@ Pl1 l ~Pm1PaPam!Gm#c1•••. ~39!

Recently a Hamiltonian analysis of the bosonic part of theD-brane action was performed in@11#. Our k-symmetrygauge-fixed action is a supersymmetric generalization of thetheory, studied in@11#. Indeed we have one manifest 16-dimensional global supersymmetry:

del5e, deXm5 e Gml. ~40!

The second 16-dimensional global supersymmetry existssince the gauge u250 required k21e250 andd u 152 e 2gea in type IIA theory and analogous in the typeIIB case. At this stage it is important that we have a mani-festly realized 16-dimensional supersymmetry. This allowsus to use the structure of the bosonic Hamiltonian theory@11# and perform anN51 supersymmetrization of it. Thedots contain terms with space derivatives on spinors]l/]sa.There is also a secondary ‘‘Gauss law’’ constraint@11#. Thebosonic constraints related to the reparametrization gaugesymmetry, the time reparametrization, and space reparam-etrization constraints~with Lagrange multipliersj,ja) are

t5PmPm1PaGabPb1T2det@~G1F!ab#, ~41!

ta5PmPam1PbFab . ~42!

The constraint related to the Abelian gauge symmetry~withthe Lagrange multiplierjBI) is

tBI5P0. ~43!

These constraints are not much different from the constraintsin the bosonic theory@11#. The new feature here is the pres-ence of 16 fermionic constraints:

Fl5 Pl1 l ~Pm1PaPam!Gm. ~44!

If the gauge fixing of the previous section is correct, weshould find out that the fermionic constraints are secondclass, and there is no fermionic gauge symmetry left. For thisto happen, the Poisson bracket of fermionic constraints hasto be invertible when other constraints are imposed. Thus wehave to calculate

$Fl~t,sa!,Fl~t,sa!%5~Pm1PaPam!CGmdp~sa2sa!

1•••. ~45!

Here the ellipsis stands for terms with derivatives of thedp(sa2sa) function and terms with space derivatives onspinors. The square of the matrix in the right-hand side of theconstraint is

@~Pm1PaPam!Gm#25PmPm1PaGabP

b12PmPaPam .

~46!

When the reparametrization constraints are imposed, we get

„@~Pm1PaPam!CGm#2

…T5Ta505PmPm1PaGabPb

52T2det@~G1F!ab#. ~47!

This is quite remarkable:the invertibility of the second classconstraintsfor D-branes quantized in theLorentz covariantgaugerelies on two basic facts: the tension has to be non-vanishing and the space part of the determinant of the BIaction has to be nonvanishing:

T5” 0, ~48!

det@~G1F!ab#5” 0. ~49!

To understand the meaning of this restriction on the theory,we may consider the physical states satisfying all first classconstraints:

tuC&5tauC&5tBIuC&50. ~50!

The mass of any such state we define by the eigenvalue ofthe square of the momentum:

M2uC&52PmPmuC&5„PaGabPb1T2det@~G1F!ab#…uC&.

~51!

The mass of any physical state of the theory therefore con-sists of two positive contributions: the first and the secondterm in Eq.~51!. For the fermionic constraints to be secondclass this requirement means that all physical states of thetheory have to have strictly nonvanishing mass.

M2uC&>T2det@~G1F!ab#uC&5” 0uC&. ~52!

This applies equally well to the ground state of the theory.Thus the Hamiltonian analysis of the reason whyD-branesadmit Lorentz covariant gauge-fixing ofk symmetry leads tothe following conclusion: as long as there are no masslessstates in the theory,k symmetry admits covariant gauges.

The existence of R-R charges or equivalently the exis-tence of central extensions in supersymmetry algebra~32! isrelated toM2Þ0 condition „T2det@(G1F)ab#Þ0… as fol-lows. WhenTÞ0 and whenPamÞ0 and/orFÞ0, we have

3518 56RENATA KALLOSH

simultaneously the R-R charges in the supersymmetry alge-bra and strictly nonvanishing mass of all physical states inthe theory. IfT50 or if Pam50 andF50, there are no R-Rcharges and the massless state is not excluded. However, inthis case the covariant gauge is not acceptable, since thefermionic constraints are not invertible.

We may compare this Hamiltonian result with the gaugefixing of D-branes in static gaugesXm5sm, which was usedin @3#. The tension was equal to one there,T51. In staticgauges,

Gmn5hmn1•••, det~G1F!ab5~dethab1••• !5” 0,~53!

and therefore both conditions for invertibility of the bracketof covariant fermionic constraints~48! and~49! are satisfied.

VII. COVARIANT QUANTIZATION OF A D-0-BRANE

Consider the k-symmetric action of a D-0-brane.D-0-brane action does not have a Born-Infeld field sincethere is no place for an antisymmetric tensor of rank 2 inone-dimensional theory. The action~13! for the p50 casereduces to

S52TS E dtA2Gtt1E u G11u D . ~54!

This action can be derived from the action of the massless11-dimensional superparticle:

S5E dtAgttgtt~Xm2 u Gmu !2, m50,1, . . . ,8,9,10.

~55!

We may solve the equation of motion forX10 as P105Z,whereZ is a constant, and useG115G10. From this one candeduce a first order action

S5E dt@Pm~Xm2 u Gmu !1 12 V~P21Z2!

2Z u G11u1 x 1d2#. ~56!

We will show now that theD-0-brane action can be obtainedfrom this one upon solving equations of motion forPm , V,x1, and d2. Here V(t) is a Lagrange multiplier,Z5T issome constant parameter in front of the WZ term, andP2[PmhmnP

n. The chiral spinorsx1 and d2 are auxiliaryfields. They are introduced to close the gauge symmetry al-gebra off shell. To verify that this first order action is one ofthe D-p-brane family actions given in Eq.~13! we can useequations of motion forPm :

Pm521

V~Xm2 u Gmu !, ~57!

and for the auxiliary fieldsx150 andd250. The action~56!becomes

S5E dtS 21

2V~Xm2 u Gmu !21

1

2VZ22Z u G11u D .

~58!

The equation of motion forV is

V2521

Z2 ~Xm2 u Gmu !2, ~59!

and we can insertV52(1/Z)A2(Xm2 u Gmu)2 back intothe action~58! and get

S52ZS E dtA2~Xm2 u Gmu !21Z u G11u D . ~60!

This is the action~13! for D-0-brane atT5Z as given in Eq.~54!.

The action~56! is invariant under the 16-dimensional ir-reduciblek symmetry and under the reparametrization sym-metry. The gauge symmetries are~we denoteGmPm5P” )

d u 5 k 2~G11Z1P” !, ~61!

dXm52hPm2d u Gmu2 k 2Gmd, ~62!

dV5h14k 2u12x 1k2 , ~63!

d x 5 k 2P” , ~64!

dd5@P21Z2#k2 . ~65!

Hereh(t) is the time reparametrization gauge parameter and

k2(t)5 12 (12G11)k(t) is the 16-dimensional parameter of

k symmetry. The gauge symmetries form a closed algebra:

@d~k2!,d~k28!#5d~h52k 2P”k28!. ~66!

To bring the theory to the canonical form we introducecanonical momenta tou and toV and find, excluding auxil-iary fields,

L5PmXm1PVV1 Puu1 12 V~P21Z2!1PVw

1@ Pu1 u ~P”1ZG11!#c. ~67!

We have primary constraintsF[ Pu1 u (P”1ZG11)'0and PV50. The Poisson brackets for 32 fermionic con-straints are

$F,F%52C~P”1G11Z!. ~68!

We also have to require that the constraints are consistentwith the time evolution$PV ,H%50. This generates a sec-ondary constraint

t5P21Z2. ~69!

Thus the Hamiltonian is weakly zero and any physical stateof the system satisfying the reparametrization constraint is aBogomoln’yi-Prasad-Summerfield~BPS! stateM5uZu since

P21Z2uC&50⇒Z2uC&52P2uC&5M2uC&. ~70!

The (32332)-dimensional matrixC(P”1G11Z) is not invert-ible since it squares to zero when the reparametrization con-straint is imposed. This is a reminder of the fact that theD-0-brane is a d511 massless superparticle. The 32-

56 3519COVARIANT QUANTIZATION OF D-BRANES

dimensional fermionic constraint has a 16-dimensional partthat forms a first class constraint and another 16-dimensionalpart that forms a second class constraint. We notice that thePoisson brackets reproduce thed510,N52 algebra with thecentral charge, which can also be understood asd511,N51supersymmetry algebra with the constant value ofP115Z.

We proceed with the quantization and gauge fixk sym-metry covariantly by takingu250,u1[l and find

Lg fk 5Pm~Xm2 l Gml !1 1

2 V~P21Z2!. ~71!

The 16-dimensional fermionic constraint

Fl[~ Pl1 lP” !'0 ~72!

forms the Poisson brackets

$Fla ,Fl

b%52~P”C!ab. ~73!

The matrixP”C is perfectly invertible as long as the centralchargeZ is not vanishing. The inverse to Eq.~73! is

$Fa,Fb%21u t505@2~P”C!ab#215~CP” !ab

2P2. ~74!

This proves that the fermionic constraints are second classand that the fermionic part of the Lagrangian,

2 lP” l[2 ilaFablb, Fab52 i ~CP” !ab , ~75!

is not degenerate in a Lorentz covariant gauge. None of thiswould be true for a vanishing central charge. Note that in therest frameP05M ,P¢50; hence,

Fab5Mdab . ~76!

For aD-0-brane one can covariantly gauge fix the reparam-etrization symmetry by choosing theV51 gauge and includ-ing the anticommuting reparametrization ghostsb,c. Thisbrings us to the following form of the action:

Lg fk,h5PmXm2 lP” l1 1

2 ~P21Z2!1bc. ~77!

Now we can define Dirac brackets

$l, l %* 5$l,F%$F,F%21$F, l %5P”

2P252

P”

2Z2. ~78!

The generator of the 32-dimensional supersymmetry is

e Q5 e ~P”1G11Z!l. ~79!

It forms the Dirac brackets

@ e Q,Qe8#* 5 e ~P”1G11Z!P”

2P2~P”1G11Z!e8

5 e GmPme85 e ~P”1G11Z!e8. ~80!

We can also rewrite it ind511 Lorentz covariant form

@ e Q,Qe8#* 5 e GmPme85 e P”e8, m50,1, . . . ,8,9,10,

Z5P10, G115G10. ~81!

This Dirac bracket realizes thed511, N51 supersymmetryalgebra or, equivalently,d510, N52 supersymmetry alge-bra with the central chargeZ.

One can also take into account that the path integral in thepresence of second class constraints has an additional termwith ABer$Fl,Fl%;ABerFab @12#; see Appendix. It can beused to make a change of variables

Sa5Fab1/2lb. ~82!

The action becomes

L5PmXm2 iSaSa1bc2H ~83!

H52 12 ~P21Z2!. ~84!

The generators of global supersymmetry commuting with theHamiltonian take the form

e Q5 e ~P”1G11Z!F21/2S. ~85!

Taking into account that$Sa ,Sb%* 52( i /2)dab we haveagain realizedd510, N52 supersymmetry algebra in theform ~80! or ~81!.

The nilpotent Becchi-Rouet-Stora-Tyutin~BRST! opera-tor in this gauge where only reparametrization ghosts arepropagating is given by

QBRST5cH H5$b,QBRST% ~QBRST!250 ~86!

and here we used the fact that$b,c%51. In turn, the Hamil-tonian~and therefore the BRST operator! can be constructedfrom supersymmetry generators as

H5 12 $Qa ,Qb%* $Qa,Qb%* 5 1

2 ~CP” !ab~P”C!ab

52 12 P” 252 1

2 ~P” 21Z2!. ~87!

The terms with anticommuting fieldsSa can be rewrittenin a form where it is clear that they can be interpreted asworld-line spinors:

L5Pm]0Xm1 Sar0]0Sa1bc2H. ~88!

Here Sa5 iSar0 and (r0)2521, r05 i being a one-dimensional matrix.

Thus, we have the original 10 coordinatesXm and theirconjugate momentaPm , and a pair of reparametrizationghosts. There are also 16 anticommuting world-line spinorsS, describing 8 fermionic degrees of freedom. The Hamil-tonian is quadratic. The ground state withM25Z2 is thestate with the minimal value of the Hamiltonian. Thus for theD-superparticle one can see that the condition for the cova-riant quantization is satisfied in the presence of a centralcharge, which makes the mass of a physical state nonvanish-ing. The global supersymmetry algebra is realized in a cova-riant way, as different from the light-cone gauge.

3520 56RENATA KALLOSH

VIII. CONCLUSION

Thus, we have confirmed here the conclusion of the work@3# that one can covariantly quantizeD-p-branes. However,as different from@3#, we did not use the static gauge forfixing the reparametrization symmetry, and analyzed theHamiltonian structure of the theory. This allowed us toclarify the reason and the generic condition under which acovariant quantization ofD-p-branes is possible: the groundstate has to be strictly massive,Mground state

2 5Z2.0. Techni-cally, this reduces to two conditions:~i! The tension of theD-p-braneT has to be nonvanishing;~ii ! det(G1F)ab has tobe nonvanishing (a,b are space components of the brane!.

Those two conditions for theD-p-brane are basicallyequivalent to the definition of this object. Both these condi-tions have to be satisfied if there are nonvanishing centralcharges in the supersymmetry algebra that are due to theexistence of R-R charges of theD-p-brane. The cross term inthe left-right part of N52 supersymmetry algebra

$Qa ,Qb%52(A

~CGA!abTQA

R

p!~89!

does not vanish when conditions~i! and ~ii ! for covariantquantization are satisfied. These conditions provide thein-vertibility of the second class fermionic constraints in Lor-entz covariant gauges.

As the special case we have performed a covariant quan-tization of D-0-brane. The resulting supersymmetry genera-tor is d510 Lorentz covariant and the Dirac bracket of thequantized theory formd510, N52 supersymmetry algebrawith a central charge.

By announcing that theD1-brane can be covariantlyquantized we have to be able to deal covariantly with thetype IIB Green-Schwarz action since it is SL(2,R) dual tothe D1-brane. Our conclusion is that this is indeed possibleunder the condition that the massless state is projected outfrom the fundamental type IIB string. In@3# this was effec-tively demonstrated since the static gauge used there corre-sponds to the type IIB superstring wrapped around the circleand such an object does not have massless excitations. Thus,we conclude that the old problem of impossibility to covari-antly quantize the fundamental string can be avoided for typeIIB fundamental string for all states but massless. The dualpartner of it, theD-1 string, does not have massless states,and duality works only in the sector of massive excitations ofthese two theories, quantized covariantly. To confirm thispicture and better understand these issues one has to quantizethese two dual theories in the conformal gauge for the rep-arametrization symmetry and in covariant gauge fork symmetry.

One has to note here that the choice of the Lorentz cova-riant fermionic gauge is not trivial for objects dual toD-branes. It has been explained in@10# that the projectorGof k symmetry ofD-p-branes anticommutes with Lorentz-covariant chiral projectors in d510. This technically ex-plains why Lorentz covariant gauges are capable of remov-ing the degeneracy of the theory due tok symmetry onD-p-branes. This is not the case for the fundamental strings,and the choice of the acceptable Lorentz covariant fermionic

gauge has to be done properly. It is important that in type IIBtheory with two chiral spinors any gauge of the type

u15cu2 , ~90!

wherec is and arbitrary constant, is consistent with Lorentzsymmetry.

Similar observations apply to theD2-brane versus theeleven-dimensional supermembrane@6#, which are also re-lated to each other, this time via duality on the world vol-ume. Is it possible that we can covariantly quantize theD2-brane but not the dual d511 supermembrane? First no-tice that the chiral projectors12 (16G11) that have been usedfor the covariant quantization of aD-2-brane are covariant ind510 but not Lorentz covariant ind511. Secondly, ourrequirement of the absence of massless states ind510 forthe D-2-brane does not exclude the masslessd511 statesince we have a BPS ground state ind510. As we have seenin quantization of theD-0-brane in Sec. VII, the BPS condi-tion 2P10

2 5M102 5Z2 is in fact equivalent to the statement

that the 11-dimensional superparticle is massless,2P10

2 2Z252P112 50.

At this stage we have learned that the 10d Lorentz cova-riant quantization of theD-2-brane may be a step towardsunderstanding the spectrum of states of the fundamentaltheory. In particular, we have found the mass formula for theD-2-brane~8!, which is ad510 Lorentz covariant generali-zation of the mass formula of the supermembrane quantizedin the light-cone gauge@7,8#.

ACKNOWLEDGMENTS

I had stimulating discussions with Eric Bergshoeff, DavidGross, Gary Horowitz, Joe Polchinsky, Joachim Rahmfeld,and Wing Kai Wong. This work was supported by the NSFgrant PHY-9219345.

APPENDIX: QUANTIZATION IN CANONICAL GAUGES

Quantization of an arbitrary Bose-Fermi system with thefirst and second class constraints in canonical gauges wasperformed by Fradkin and collaborators@12#. One starts withthe classical Lagrangian of the form

L5piqi2H0~p,q!2jkQk~p,q!2jmTm~p,q!. ~A1!

The first class constraints obey the relations

$Tm ,Tn%uT50,Q5050, $H0 ,Tm%uT50,Q5050, ~A2!

and for the second class constraints we have

Ber$Qk ,Q l%uT50,Q505” 0. ~A3!

The symbol$ % stands for the Fermi-Bose extension of thePoisson bracket and Ber~Berezinian! is a superdeterminantof the matrix$Qk ,Q l%[Qkl . The Dirac bracket is defined as

$A,B%* 5$A,B%2$A,Qk%~Qkl!21$Q l ,B%. ~A4!

56 3521COVARIANT QUANTIZATION OF D-BRANES

The path integral in canonical gaugesFm(p,q)50 ~whereall ghosts are nonpropagating fields! is given by

Z5E exp$ iS@q,p,j,p#%) „Ber$F,T%* d~Q!

3~Ber$Q,Q%!1/2dqdpdjdp…, ~A5!

where the action is

S5E @piqi2H0~p,q!2jmTm~p,q!2pmFm#dt.

~A6!

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3522 56RENATA KALLOSH