Statistical mechanics of multiply woundD-branes

Gavin Polhemus*Physics Department and Enrico Fermi Institute, University of Chicago, 5640 Ellis Avenue, Chicago, Illinois 60637

~Received 3 February 1997!

The D-brane counting of black hole entropy is commonly understood in terms of excitations carryingfractional charges living on long, multiply wound branes~e.g., open strings with fractional Kaluza-Kleinmomentum!. This paper addresses why the branes become multiply wound. Since multiply wound branes areT dual to branes evenly spaced around the compact dimension, this tendency for branes to become multiplywound can be seen as an effective repulsion between branes in theT-dual picture. We also discuss how thefractional charges on multiply wound branes conspire to always form configurations with integer charge.@S0556-2821~97!03916-7#

PACS number~s!: 04.70.Dy

I. INTRODUCTION

The past year has seen considerable progress in under-standing black hole entropy through counting states ofD-brane configurations. This approach has been particularlyfruitful in the study of five-dimensional black holes@1#. Oneof the most carefully studied models is that of black holes inthe toroidal compactification of type IIB supergravity downto five dimensions@2#. When discussing the applications ofthe ideas below, this model will be used as an example.These five-dimensional black holes are characterized bymass, three charges, and two asymptotic fields. Since it is notknown how to calculate the entropy of these configurationsdirectly, the black hole is modeled by a configuration offive-branes, one-branes, their antibranes, and strings carryingthe same charges. In certain limits~all near theBogomol’nyi-Prasad-Sommerfield limit! the entropy of theD-brane configuration can be calculated directly by countingmicrostates. In these cases the entropy of theD-brane con-figuration has been found to agree with the Beckenstien-Hawking entropy for a black hole with the same charges,fields, and mass. Further calculations have shown agreementin the absorption cross sections, and low energy emissionspectra@3–6#.

In order for theD-brane to give the right entropy in thelimit of large charges and mass the configurations must haveexcitations that carry fractional charges. These fractionalcharges also lead to the large density of string and smallmass gap that are expected for a black hole. This has beencommonly understood as an indication that the branes whichwrap compact dimensions, rather than forming a large col-lection of singly wound branes, join together to form asingle, long, multiply wound brane@7–9#.

The notion ofD-brane winding is not as obvious as fun-damental string winding. WhenQ D-branes coincide, thegauge theory on the brane is enlarged to a U(Q) symmetry.If the brane wraps a compact dimension (x9 in this example!,then this gauge symmetry can be broken by adding a Wilsonline A95diag$u1 ,u2 , . . . ,uQ%/2pR9 @10#. This leads to aholonomy given by the matrix

U95S eiu1 0 ••• 0

0 eiu2 0

A � A

0 0 ••• eiuQD . ~1!

In the T-dual picture~which we will refer to as the ‘‘un-wound picture’’ to distinguish it from the ‘‘wound picture’’above!, this is a set of parallel branes at positionsR9(u1 ,u2 , . . . ,uQ) around the compact dimension.

Of course, we could also add Wilson lines that have off-diagonal elements. For example, one case that represents asingle, multiply wound brane is

U95S 0 1 0 ••• 0

0 0 1 0

A � � A

0 0 0 1

1 0 ••• 0 0

D . ~2!

However, one can always put such a matrix in diagonal formby a change of basis. So the Wilson lines with off diagonalelements break the gauge symmetryexactlylike the diagonalWilson line with the same eigenvalues. For example, theeigenvalues of Eq.~2! are theQth roots of 1. So a multiplywound brane isT dual to parallel branes that are evenlyspaced around the compact dimension. As described in@10,11#, this leads to fractionally charged excitations. How-ever, the total charge of a configuration is always an integer.This restriction is discussed in Sec. III below.

The issue of how branes become multiply wound so thatthey can carry fractional charges can be addressed by con-sidering how the ‘‘unwound’’D-branes spread out aroundthe compact dimension. If the branes were noninteracting,they could be expected to spread out to fill the compactdimension much the way particles of a noninteracting gasspread out to fill a container. However, we show in this paperthat the branes do not behave this way and, in fact, spreadout to fill the compact dimension with remarkable unifor-mity.*Electronic address: g-polhemus@uchicago.edu

PHYSICAL REVIEW D 15 AUGUST 1997VOLUME 56, NUMBER 4

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II. HOW BRANES REPEL

The most obvious reason for the branes to spread outevenly around the compact dimension would be to minimizesome sort of repulsive potential. However, there is absolutelyno way for a potential to be generated—Wilson lines do notbreak the supersymmetries so, in the absence of excitations,all Wilson lines must have zero energy. Likewise, in theunwound picture there are no forces pushing the branes apartbecause any distribution of parallel branes preserves some ofthe supersymmetries and, therefore, has zero energy.

Since all Wilson lines are zero energy, we assume that allWilson lines are equally probable when doingD-brane ther-modynamics. It may be that the process of creating theD-brane configurations leads to certain Wilson lines morefrequently than others, but that would be a rather exceptionalsituation. Certainly, one can carefully prepare a specific con-figuration, but in a realistic scenario we anticipate that inter-actions during preparation lead to every zero energy configu-ration with equal probability. Note that, since all of the statesare zero energy, the temperature of the system does not mat-ter.

While all Wilson lines are equally probable, not all posi-tions of the branes are equally probable. There is only oneelement of U(Q) with all its eigenvalues equal to 1, but thereare many whose eigenvalues are, for example, theQth rootsof 1. The probability that the branes will have positions(u1 ,u2 , . . . ,uQ) around the compact dimension is propor-tional to the volume of U(Q) with eigenvalues(eiu1,eiu2, . . . ,eiuQ).

A. An example with two branes „not two-branes…

As an example, consider the case of two coincidentbranes. In this case the gauge group is U~2!5U~1!3SU~2!.The U~1! subgroup is the center of mass position in the un-wound picture, so the brane’s relative position is given bythe eigenvalues of the SU~2!. The elements of SU~2! can beparametrized byj j using the three Pauli matricess j as gen-erators:

U5ei j js j . ~3!

The eigenvalues are thene6 i uju, resulting in a separation be-tween the branesu12u252uju. Notice that SU~2! is topo-logically S3 and the surfaces of constantuju are like thespheres of constant latitude running fromuju50 at the northpole to uju5p at the south pole onS3. The radius of eachsphere isr 5sinuju so the probability that the two branes areat positionsu1 andu2 is

P~u1 ,u2!5sin2S u12u2

2 D . ~4!

The probability density actually vanishes for the branes be-ing coincident. The peak of the probability distribution is atmaximum separation in the unwound picture, which corre-sponds to a doubly wound brane in the wound picture.

B. Many branes

The probability of a randomly chosen element of SU~Q!having eigenvalues (u1 ,u2 , . . . ,uQ) has been exactly solvedfor all Q @12#:

P~u1 ,u2 , . . . ,uQ!5C)k, l

Q

ueiuk2eiu lu2. ~5!

The maximum is clearly at equal spacing. Fixing the originat ^uQ&50 and putting the remainingu j in increasing ordergives ^u j&52p j /Q.

To gain an intuitive understanding of the branes’ behav-ior, it is useful to consider another physical system that hasthe same probability distribution: a collection ofQ particles,each with chargeq, living in two dimensions on a circle ofradiusR @12,13#. The two-dimensional world can be thoughtof as the complex plane. The two-dimensional Coulomb po-tential for a pair of charges atzk and zl isWkl52q2lnuzk2zlu. Restricting the charges to the unit circlezj5Reu j . The energy of the system is then

W52q2(k, l

Q

lnueiuk2eiu lu. ~6!

In a thermal ensemble of such systems at inverse tempera-ture b the probability distribution will be

P~u1 ,u2 , . . . ,uQ!5Ce2bW ~7!

5C)k, l

Q

ueiuk2eiu lubq2. ~8!

For b52/q2, this is exactly the probability distribution inEq. ~5!. Note that the inverse temperatureb has nothing todo with the temperature of the branes or the black hole. Thisis the temperature of the model system of electric chargesthat has the same probability distribution as theD-brane con-figuration at any temperature.

Clearly, the charges are going to have a very strong ten-dency to spread out evenly around the circle, forming a sortof crystal. In the case of only two particles it is easy to seethat the fluctuations in the spacing are roughly the same asthe average spacingD(u12u2)'^u22u1&5p. It turns outthat as the number of particles gets large and the averagespacing gets smaller, the fluctuations in nearest neighborspacing get proportionally smaller so thatD(u j2u j 11)'^u j2u j 11&52p/Q @12#.

Although the spacing between adjacent particles variessignificantly, long wavelength fluctuations in the averagespacing are minuscule. These long wavelength fluctuationscan be studied in the continuum limit. In this case the dis-crete index j is replaced by a continuous parameterj52p j /Q. The positions can be expanded about their meansin Fourier modes:

uj5^uj&1 (nÞ0

Q21

aneinj, ~9!

wherea2n5an* and^uj&5j. The energy of the system is theground state energy plus the energies of these modes:

56 2203STATISTICAL MECHANICS OF MULTIPLY WOUND D-BRANES

W5W01 (n52~Q21!

nÞ0

Q21

Wn . ~10!

Calculating these energies of these fluctuations is a simpleexercise in two-dimensional electrostatics:

Wn52p2r02nuanu2, ~11!

wherer0 is the average charge densityr05Qq/2p.As the number of charges gets large, all of the long wave-

length fluctuations actually freeze out. This is best seen byfixing the average charge density of the systemr0. Since weare interested in the temperature at which the charged par-ticles are distributed like the ‘‘unwound’’D-branes,b52/q25Q2/2p2r0

2, the temperature decreases like 1/Q2.To calculate the average energy in these modes at

b52/q2, we use the partition function

Zn5E0

`

dane2bWn ~12!

51

2r0A2pbn. ~13!

The mean energy in these modes is then

^Wn&52]

]blnZn5

1

2b. ~14!

From this we can easily find the average~root-mean-squared!amplitude of these macroscopic oscillations:

^uanu2&51

~2pr0!2nb~15!

51

2Q2n. ~16!

This can be used to find the fluctuations in the spacingbetween remote particles:

^~uj2uz!2&5K Fj2z1 (

nÞ0an~einj2einz!G2L ~17!

5~j2z!21 (n.0

^uanu2&~einj2einz!

3~e2 inj2e2 inz! ~18!

5^uj2uz&21

1

Q2(n.0

1

nsin2

n~j2z!

2. ~19!

The infinite sum is divergent and needs to be cut off atn5Q/2. The sum can be approximated by an integral; thesine cuts off the integral in the infrared atn5p/(j2z):

(n.0

1

nsin2

n~j2z!

2'

1

2Ep/~j2z!

Q/2 dn

n5

1

2ln~k2 l !, ~20!

where we have usedj52pk/Q andz52p l /Q. So the sizeof the fluctuations in spacing fork2 l @1 is

D~uk2u l !'1

QA1

2ln~k2 l !. ~21!

As we would expect, the fluctuations in spacing get bigger aswe consider particles that are farther apart. What is surpris-ing is how slowly the fluctuations increase.

Since the probability distribution for the particles is thesame as that for the unwound branes, all of the above state-ments about deviations from equal spacing apply to the branecase as well. However, the mechanism is different. The par-ticles spread out to minimize their energy while the branesspread out to find larger volumes of phase space. Because ofthis difference, the charged particles cannot be used to modelall aspects of the branes’ behavior—for example, the mo-mentum distribution of the branes in different.

All interestingD-brane models of black holes have morethan one compact dimension, each with its own~possiblytrivial! Wilson line. These Wilson lines all transform to-gether under the U(Q) symmetry, allowing a trace term inthe Lagrangian that gives nonzero energy to configurationswith Wilson lines that cannot be simultaneously diagonalized@14#:

V5T2

2 (m,n5D

9

Tr@Xm,Xn#2, ~22!

whereT is the fundamental string tension andD is the di-mension of the noncompact spacetime. Both the additionalWilson lines and the potential that come from the additionalcompact dimensions may be important in understandingD-brane models. However,D-brane calculations of blackhole entropy have only required that the branes become mul-tiply wound around a singleS1 @8#. Based on this we expectthat the remaining four compact dimensions do not affectthis winding.

III. INTEGER WINDING AND MOMENTUMOF D-BRANE CONFIGURATIONS

Nonextremal configurations have nonzero energy, allow-ing excitations away from zero potential in Eq.~22!. Theseexcitations correspond to loops of unexcited fundamentalstring connecting the various branes@14#. Since winding iseasier to visualize than momentum, we start by looking at thefractional winding states that connect the branes in the un-wound picture.T duality allows us to apply this understand-ing to the fractional momentum states in the woundD-brane picture.

In the unwound picture the excitations are strings con-necting theD-branes. Since there are a lot of branes around,the strings do not have to go all the way around the compactdirection—they are allowed to have fractional winding.However, the entire configuration must have integer wind-ing. This is a result of charge conservation on the branes.The charge on the brane that comes from the end of a stringmust be canceled by an opposite charge from another string.In terms of the orientation of the strings, each brane must

2204 56GAVIN POLHEMUS

have as many strings beginning on it as it has ending on it.This guarantees that the net winding of any configurationwill be zero.

T duality turns the fractional winding that appeared in theunwound picture into fractional momentum in the woundpicture. The excitations of wound branes are free to moveindependently. However, their ends are still charged, so theexcitations trail field lines in theD-brane as they move.Moving a single excitation around the compact directionwraps the field lines, changing the energy of the system.Since moving a single excitation around the compact dimen-sion does not restore the system to its original configuration,single excitations do not see the periodicity of the compactdimension and there is no reason for their momentum to bequantized in units of 1/R.

To avoid trailing any field lines, excitations must bandtogether into parties that are neutral in theD-brane theory.These parties can then circumnavigate the compact dimen-sion and discover its periodicity. It is the momentum of theseparties which is quantized in units of 1/R. It is easy to verifythat the sum of the fractional momenta that make up a neu-tral party is always an integer.

IV. DISCUSSION

In the models of five-dimensional black holes the excita-tions are strings connectingD-one-branes toD-five-branes.We expect that the ‘‘repulsion’’ described above encouragesboth the one-branes and the five-branes to link up into long,multiply wound branes, allowing fractionally charged exci-tations.~The total charge will still be an integer for the rea-sons given above.!

It may be possible to understand models of near extremalblack holes entirely in the languageD-branes~without anyfundamental strings attached! by replacing the fundamentalstrings with excitations away fromV50 in Eq. ~22!. How-ever, since the fundamental strings in black hole models con-nect branes of different dimensionality, it is less clear how toproceed.

It would also be interesting to see if these ideas could beextended to the branes ofM theory.

ACKNOWLEDGMENTS

I would like to thank Jeff Harvey for many enlighteningdiscussions on many matters and Juan Maldacena for a help-ful conversation about charge conservation on branes. Thiswork was supported by the Office of Naval Research.

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