One-loop effective action of the holographic antisymmetric Wilson loop

One-loop effective action of the holographic antisymmetric Wilson loop

Alberto Faraggi,1,* Wolfgang Muck,2,3,† and Leopoldo A. Pando Zayas1,‡

1Michigan Center for Theoretical Physics Randall Laboratory of Physics, The University of Michigan Ann Arbor,Michigan 48109-1040, USA

2Dipartimento di Scienze Fisiche, Universita degli Studi di Napoli ‘‘Federico II’’ Via Cintia, 80126 Napoli, Italy3INFN, Sezione di Napoli, Via Cintia, 80126 Napoli, Italy(Received 7 February 2012; published 29 May 2012)

We systematically study the spectrum of excitations and the one-loop determinant of holographic

Wilson loop operators in antisymmetric representations of N ¼ 4 supersymmetric Yang-Mills theory.

Holographically, these operators are described by D5-branes carrying electric flux and wrapping an S4 �S5 in the AdS5 � S5 bulk background. We derive the dynamics of both bosonic and fermionic excitations

for such D5-branes. A particularly important configuration in this class is the D5-brane with AdS2 � S4

world volume and k units of electric flux, which is dual to the circular Wilson loop in the totally

antisymmetric representation of rank k. For this Wilson loop, we obtain the spectrum, show explicitly that

it is supersymmetric and calculate the one-loop effective action using heat kernel techniques.

DOI: 10.1103/PhysRevD.85.106015 PACS numbers: 11.25.Tq, 11.25.Uv

I. INTRODUCTION

Wilson loop operators play a central role in gaugetheories, both as formal variables and as important orderparameters. In the context of the AdS/CFT correspondenceexpectation values of Wilson loops were first formulatedby Maldacena [1] and Rey-Yee [2].

One of the most exciting developments early on was therealization that the expectation value of the BPS circularWilson loop can be computed using a Gaussian matrixmodel [3,4]. This conjecture was later rigorously provedin [5]. In a beautiful, now classic work by Gross andDrukker, the matrix model was evaluated and its leadingN, large ’t Hooft coupling limit was successfully comparedwith the string theory answer. One of the most intriguingwindows opened by this problem is the question of quan-tum corrections it their entire variety. For example, havingan exact field theory answer (Gaussian matrix model)prompted Gross and Drukker to speculate that the exactmatrix model result was the key to understanding highergenera on the string theory side. The quantum correctionson the string theory side have been the subject of muchinvestigation starting with earlier efforts in [6,7] and con-tinuing in more recent works such as [8,9]. Despite theseconcerted efforts, it is fair to say that a crisp picture ofmatching the BPS Wilson loop at the quantum level onboth sides of the correspondence has not yet been achieved.

More recently the question of tackling BPS Wilsonloops in more general representations has been success-fully tackled at leading order. The introduction of generalrepresentations gives a new probing parameter, thus ex-panding the possibilities initiated in the context of thefundamental representation. In the holographic framework,

a half BPS Wilson loop in N ¼ 4 supersymmetric Yang-Mills (SYM) theory in the fundamental, symmetric orantisymmetric representation of SUðNÞ is best describedby a fundamental string, a D3-brane or a D5-brane withfluxes in their world volumes, respectively. Drukker andFiol computed in [10], using a holographic D3 branedescription, the expectation value of a k-winding circularstring which, to leading order, coincides with thek-symmetric representation. A more rigorous analysis ofthe role of the representation was elucidated in [11,12].Some progress on the questions of quantum corrections tothese configurations immediately followed with a strongemphasis on the field theory side [13–15]. Developing thegravity side of this correspondence is one of the mainmotivations for this work. In particular, we derive thespectrum of quantum fluctuations in the bosonic and fer-mionic sectors for a D5-branewith k units of electric flux initsAdS2 � S4 world volume embedded inAdS5 � S5. Thisgravity configuration is the dual of the half BPS Wilsonloop in the totally antisymmetric representation of rank k inN ¼ 4 SYM.Although our main motivation comes from the study of

Wilson loops, there is another strong motivation for ourstudy of quantum fluctuations. String theory has heavilyrelied on the understanding of extended objects in thecontext of the gauge/gravity correspondence. They haveplayed a key role in interpreting and identifying varioushadronic configurations (quarks, baryons, mesons,k-strings). A more general approach on the quantizationof these objects is a natural necessity. The long history offailed attempts at quantizing extended objects around flatspace might have found its right context. Although largelymotivated by holography, it is important by itself that thequantum theory of extended objects in asymptotically AdSworld volumes seems to be much better behaved thannaively expected. In our simplified setup we are faced

*[email protected]†[email protected]‡[email protected]

PHYSICAL REVIEW D 85, 106015 (2012)

1550-7998=2012=85(10)=106015(20) 106015-1 � 2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevD.85.106015

with various divergences, but many of them allow for somequite natural interpretations. Although we do not attack thegeneral problem of divergences in a general context, wehope that our analysis could serve as a first step in this morefundamental direction of quantization of extended objects.

In this paper, we systematically study small fluctuationsof D5-branes embedded in asymptoticallyAdS5 � S5, withflux in its world volume and wrapping an S4 � S5 [16–19].The formalism we develop readily applies to more generalbackgrounds than just the holographicWilson loop, includ-ing holographic Wilson loop correlators [20,21] and re-lated finite-temperature configurations [22,23]. Using thisgeneral formalism, we obtain the spectrum of both bosonicand fermionic excitations of D5-branes dual to the halfBPS circular Wilson loop. Our analysis is explicit bynature and falls nicely in the group theoretic frameworkput forward in [24]. We also compute the one-loop effec-tive action using heat kernel techniques.

The paper is organized as follows. In Sec. II, we introducethe class of D5-brane configurations for which our analysisapplies. For completeness, the bulk background geometriesand the main features of the D5-brane background configu-rations are reviewed in sections II A and II Brespectively.Section III contains the general analysis of the bosonic andfermionic excitations of these D5-branes. The second-orderactions for the bosonic and fermionic degrees of freedomare constructed in Secs. III A and III Brespectively, and theirclassical field equations are analyzed in secs. III C and III D.Sections IV and V deal with the holographic Wilson loop.The spectrum of fluctuations is obtained in Sec. IV.Section V presents the calculation of the one-loop effectiveaction using the heat kernel method. We conclude inSec. VI. Technical material pertaining to our notation, thegeometry of embeddings and to aspects of the heat kernelmethod are relegated to a series of appendices.

Note: While our work was in progress, the paper [25]appeared, in which the spectrum of the bosonic fluctuationswas derived.

II. BACKGROUND GEOMETRYAND CLASSICALD5-BRANE SOLUTIONS

We begin by briefly reviewing the bulk geometry andclassical D5-brane configurations we are interested in.Although we will eventually focus on AdS5 � S5, weemphasize that the methods developed in this paper aremore general and apply to other solutions of type-IIBsupergravity, including the near horizon limit of blackD3-branes. Throughout the paper we will work inLorentzian signature and switch to Euclidean signatureonly to discuss functional determinants in Sec. V. We referthe reader to Appendix A for notation and conventions.

A. Bulk background

We are interested in probe D5-branes embedded in thefollowing solution of type-IIB supergravity,

ds2 ¼ �fðrÞdt2 þ dr2

fðrÞ þr2

L2

X3i¼1

ðdxiÞ2 þ L2ðd#2

þ sin2#d�24Þ;

Cð4Þ ¼ r4

L4dt ^ d3xþ L4Cð#Þd4�; (2.1)

where

fðrÞ ¼ r2

L2

�1� r4þ

r4

�;

Cð#Þ ¼ 3

2# � 3

2sin# cos# � sin3# cos#: (2.2)

All the other background fields vanish. The function Cð#Þsatisfies dC=d# ¼ 4sin4#, so that the 5-form is

Fð5Þ ¼ dCð4Þ ¼ 4

Lð1þ �Þd5�L; (2.3)

where d5�L ¼ L5d5� is the volume measure of a 5-sphereof radius L, as it appears in the metric (2.1).With

L4 � 4�gsN�02 ¼ ��02; (2.4)

where � is the ’t Hooft coupling, (2.1) describes N D3-branes, generically at finite temperature. The blackhole horizon ‘‘radius’’, rþ, is related to the inverse tem-perature by

rþ ¼ �L2

�: (2.5)

The zero temperature AdS5 � S5 solution is recoveredby setting rþ ¼ 0. In this case, we can make the replace-ment r ! L2=z to obtain the AdS5 metric in the standardPoincare coordinates with boundary at z ¼ 0. Anticipatingthe embedding of the D5-branes, the metric of S5 has beenwritten in terms of an S4 at some azimuth angle, # 2½0; ��.

B. Background D5-branes

In the background (2.1), the bosonic part of the D5-braneaction is

SðBÞD5 ¼ �T5

Zd6�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detðgþF Þab

qþ T5

ZCð4Þ ^F ;

(2.6)

where T�15 ¼ ð2�Þ5�03gs is the D-brane tension, gab is the

induced metric on the world volume, and F ¼ dA is thefield strength living on the brane.We consider D5-brane configurations such that four

coordinates �� wrap the S4 � S5 at a constant angle #and the remaining two coordinates �� ¼ ð�; �Þ span aneffective string world sheet, with induced metric g��, in

the aAdS5 part of the bulk. By symmetry, the only non-vanishing components of F are1

1With a slight abuse of notation we use F to denote also theantisymmetric component.

FARAGGI, MUCK, AND PANDO ZAYAS PHYSICAL REVIEW D 85, 106015 (2012)

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F �� ¼ F ��; (2.7)

and we can fix a gauge such that only A� is nontrivial. Itfollows that

� detðgþF Þab ¼ � detðgþF Þ�� detg�

¼ L8sin8#ð1�F 2Þð� detg��Þ: (2.8)

Hence, the action (2.6) can be written as

SðBÞD5 ¼ � N

3�2�0Z

d�d�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg��

q½sin4#

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�F 2

q� Cð#ÞF �: (2.9)

The prefactor arises from T5V4L4 ¼ N

3�2�0 , where V4 ¼8�2=3 is the volume of the unit S4.

Quantization of 2-form flux (which is an integral of theequation of motion forA�), and the equation of motion for# are solved by [16,17]

� n

N¼ 1

�ð# � sin# cos#Þ; ðn ¼ 0; . . . ; NÞ;

(2.10)

and

F ¼ cos#: (2.11)

Although n, the fundamental string charge dissolved on theD5-brane, is quantized, we can consider 2 ð0; 1Þ as acontinuous variable in the large-N limit.

One must add to (2.9) appropriate boundary terms[10,26]

IðBÞbound ¼ �Z

d� sgnr0ðr�r þA��AÞ; (2.12)

where

�r ¼ @LD5

@r0; �A ¼ @LD5

@A0�

¼ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg��p @LD5

@F;

(2.13)

and the prime denotes a derivative with respect to �.Putting everything together, one finds that the action ofthe background D5-brane can be reduced to that of aneffective string living in the aAdS5 portion of the 10-dimensional geometry [18]

SðBÞD5 þ IðBÞbound ¼ � N

3�2�0 sin3#

�Zd�d�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg��

q�

Zd� sgnr0r

@ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg��

p@r0

�: (2.14)

III. FLUCTUATIONS

In this section, we consider the fluctuations of the bo-sonic and fermionic degrees of freedom of the D5-branesolutions described above. We construct the quadraticactions and derive the classical field equations. As a firstresult, the spectrum of fluctuations of the circular Wilsonloop of operators in the antisymmetric representations,

predicted in [24], is fully derived. Our formalismreadily applies to more general backgrounds, includingholographic Wilson loop correlators [20,21] and relatedfinite-temperature configurations [22,23].

A. Bosonic fluctuations

Let us start by defining the dynamical variables thatparameterize the physical fluctuations. We will make useof well-known geometric relations for embedded mani-folds [27], which are reviewed in appendix B. The fieldspresent in (2.6) are the target-space coordinates of the D5-brane xm and the gauge field componentsAa living on thebrane. Both are functions of the D5-brane world volumecoordinates �a.We now recall a few facts from differential geometry

that, although known to the reader, we bring to bearexplicitly in our calculations. We shall parameterize thefluctuations of xm around the background coordinates bythe generating vector y of an exponential map [27]

xm ! ðexpxyÞm ¼ xm þ ym � 1

2�m

npynyp þOðy3Þ;

(3.1)

thereby obtaining a formulation that is manifestly invariantunder bulk diffeomorphisms. Recall that, as familiar fromGeneral Relativity, the differences of coordinates are notcovariant objects, but vector components are. Here andhenceforth, all quantities except the fluctuation variablesare evaluated on the background. Locally, the vector com-ponents ym coincide with the Riemann normal coordinatescentered at the origin of the exponential map. Riemannnormal coordinates are also helpful for performing thecalculations, because of a number of simplifying relationsthat hold at the origin. For example, one can make use of

�mnp ¼ 0; �m

np;q ¼ � 2

3Rm

ðnpÞq; (3.2)

while the expression for a covariant tensor of rank k is, upto second order in y,

Am1...mk! Am1...mk

þ Am1...mk;nyn þ 1

2

�Am1...mk;np

þ 1

3

Xkl¼1

Rqnpml

Am1...q...mk

�ynyp: (3.3)

In the equations that follow, we will implicitly assume theuse of a Riemann normal coordinate system. Moreover, weshall drop terms of higher than second order in y. Thetangent vectors along the world volume (see Appendix B),which serve to calculate the pullback of bulk tensor fields,are given by

xma ! xma þraym � 1

3Rm

pnqxnay

pyq: (3.4)

Reparametrization invariance allows us to gauge away thefluctuations that are tangent to the world volume. Thisleaves us with

ONE-LOOP EFFECTIVE ACTION OF THE HOLOGRAPHIC . . . PHYSICAL REVIEW D 85, 106015 (2012)

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ym ¼ Nmi �

i; (3.5)

where the �i parameterize the fluctuations orthogonal tothe world volume, or normal fluctuations, and the index iruns over all normal directions. The expression above is thenatural geometric object related to fluctuations; it hasappeared in previous works, for example, [28] and, moreexplicitly, in [24]. We found it appropriate to provide anexplicit account of the origin of this parametrization ofthe fluctuations. Using the relations summarized inAppendix B, this gives rise to

raym ¼ �Hia

bxmb �i þ Nm

i ra�i; (3.6)

where Hiab is the second fundamental form of the back-

ground world volume, and ra denotes the covariant de-rivative including the connections in the normal bundle.

The fluctuations of the gauge field are introduced by

F ab ! F ab þ fab; (3.7)

and we recall that the background gauge field only lives onthe 2-d part of the world volume as shown in (2.7).

Following these preliminaries, we now consider fluctua-tions of the degrees of freedom of the D5-brane. The goal isto expand the action (2.6) to second order in the fields �i

and aa. For the Born-Infeld term, we make use of theformulaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detM

p ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detMp �

1þ 1

2trX þ 1

8½trX�2

� 1

4trðXXÞ þOðX3Þ

�; (3.8)

where X denotes the matrix X ¼ M�1�M, and we haveintroduced

Mab ¼ gab þF ab: (3.9)

Combining (3.3), (3.4), (3.5), and (3.6) to obtain the in-duced metric, we have

�Mab ¼ �2Hiab�i þ fab þra�

irb�j�ij þ ðHia

cHjbc

� Rmpnqxma x

nbN

pi N

qj Þ�i�j: (3.10)

Substituting (3.10) into (3.8) and using the backgroundrelations, one obtains after some calculation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detMab

p ! L4sin3#ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg��

q �sin2# þ 4

Lcos# sin#�5 þ 1

2cos#��f�� þ 2cos2#

L sin#�5��f��

þ 1

2gabð�ijra�

irb�j þra�

5rb�5Þ þ 1

4sin2#gabgcdfacfbd þ 2

L2ð3cos2# � sin2#Þð�5Þ2

� 1

2ðHi��Hj

�� þ Rmpnqg��xm�x

n�N

pi N

qj Þ�i�j

�; (3.11)

where gab is the inverse of the 6-d metric

d s2 ¼ g��d��d�� þ L2d�2

4 (3.12)

which is independent of#. Henceforth, the normal index i in(3.11) refers only to the three normal directions within theAdS5 part of the bulk, as we have indicated explicitly �

5 forthe normal direction within S5. Note that (3.12) is not theinducedmetric on theworld volume. Rather, the backgroundflux deforms themetric and the fluctuations see the deformedgeometry as appropriate for open string fluctuations.

In order to expand the Chern-Simons term in (2.6), wemake use of (3.3), (3.4), and (3.7), and the backgroundrelations. One soon finds that to quadratic order in thefluctuations, only the components of the four-form Cð4Þthat live on S4 contribute. After some calculation one obtains

C ^F ! d4�d�d�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detg��

qL4

�Cð#Þ cos#

� 1

2Cð#Þ��f�� þ 4

Lsin4# cos#�5

� 2

Lsin4#�5��f�� þ 8

L2sin3#cos2#ð�5Þ2

�:

(3.13)

Replaceing (3.11) and (3.13) in (2.6), the linear terms in�5 are found to cancel as expected for an expansion around

a classical solution. The linear term in f�� is a total

derivative and is cancelled by a boundary term similar to(2.12). Thus, one ends up with the following quadraticterms in the action,

SðB;2ÞD5 ¼ T5sin3#

Zd6�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgabp �

� 1

2gabð�ijra�

irb�j

þra�5rb�

5Þ þ 2

L2ð�5Þ2 � 1

4sin2#gabgcdfacfbd

� 2

L sin#�5��f�� þ 1

2ðHi��Hj

��

þ Rmpnqg��xm�x

n�N

pi N

qj Þ�i�j

�: (3.14)

The dynamical fields present in (3.14) are the scalar �5,the scalars �i transforming as a triplet under the SOð3Þsymmetry of the normal bundle, and the gauge fields aa,fab ¼ @aab � @baa.

B. Fermionic fluctuations

We now consider fluctuations of the fermionic degreesof freedom of the D5-brane. This is somewhat easier thanthe bosonic part, because one just needs the fermionic partof the action, in which all bosonic fields assume theirbackground values.


106015-4

Our starting point is the fermionic part of the D5-braneaction with -symmetry, which was derived in [28]2

SðFÞD5 ¼ T5

2

Zd6�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detMab

p��ð1� �D5Þ½ð ~M�1Þab�bDa��;

(3.15)

where Mab ¼ ðgþF Þab as before, ~Mab ¼ gab þ�3�ð10ÞF ab, � is a doublet of 32-component left-handed

Majorana-Weyl spinors (�ð10Þ� ¼ �, the �3 acts on the

doublet notation) and �� ¼ i�y�0. Moreover, in the back-ground (2.1), the derivative operator Da is given by

Da ¼ ð@axmÞ�rm þ i�2

16 � 5!Fnpqrs�npqrs�m

�; (3.16)

where rm ¼ @m þ 14!m

np�np is the 10-d spinor covariant

derivative. We give a derivation of its pullback onto theworld volume in Appendix B. Using (B14) and the back-ground relations (2.1), one finds

D� ¼ r� þ 1

4Aij��

ij � 1

2Hi��

��i � 1

4L��

5��9ði�2Þ� ð1þ �ð10ÞÞ (3.17)

and

D� ¼ r� � 1

2Hi��

�5 þ 1

4L��

5��9ði�2Þð1þ �ð10ÞÞ:(3.18)

The matrix �D5 in (3.15) is3

�D5 ¼ 1

sin#~��6��9�1ð1þ �3

~� cos#Þ; (3.19)

where we have defined

~� ¼ 1

2��

��: (3.20)

It is useful to rewrite (3.19) as

�D5 ¼ eR�3~�~��6��9�1e

�R�3~�; sinhð2RÞ ¼ � cot#:

(3.21)

Finally, the inverse of the matrix ~Mab is found to be

ð ~M�1Þ� ¼ g�;

ð ~M�1Þ�� ¼ 1

sin2#ðg�� cos#��3�ð10ÞÞ:

(3.22)

Now, owing to the fact that � is left-handed, we canreplace �ð10Þ byþ1 in (3.17) and (3.18) and by�1 in (3.22)

, when they are substituted into (3.15). Because �� ¼�~��, we also find

ð ~M�1Þ�� ¼ 1

sin#eR�3

~��e�R�3~� (3.23)

when acting on a left-handed spinor. Putting everythingtogether and using also that the extrinsic curvatureterms in (3.17) and (3.18) are H5�

¼ �ðcot#=LÞ�� and

Hi�� ¼ 0, because the 2-d part of the background is a

minimal surface, we find after some calculation that theaction (3.15) becomes

SðFÞD5 ¼ T5

2sin4#

Zd6�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� detgabp

��eR�3~�ð1� ~��6��9�1Þ

��r� þ �� ~r� þ 1

L�5��9ði�2Þ

�e�R�3

~��;

(3.24)

where we have abbreviated

~r � ¼ r� þ 1

4Aij��

ij (3.25)

to denote the covariant spinor derivative including the

connections in the normal bundle, �� ¼ sin#�� are thecovariant gamma matrices normalized for a 4-sphere ofradius L, and gab is the 6-d metric (3.12), which we usedalso for the bosons.To simplify (3.24) slightly, we introduce the rotated

double spinor �0 ¼ e�R�3~��. Its conjugate is easily found

as ��0 ¼ i�ye�R�3~��0 ¼ ��eR�3

~�, which is just the combi-nation that appears in (3.24). Henceforth, we shall workwith the rotated spinor and drop the prime for brevity.Now we fix the -symmetry. The covariant gauge-fixing

condition �ð10Þ�3� ¼ � [28] reduces to �3� ¼ �, because� is left-handed. The terms in the action that survive thisprojection are

SðFÞD5 ¼ T5

2sin4#

Zd6�


��

��r� þ �� ~r�

þ 1

L~��5

��; (3.26)

where � is now a single, 32-component spinor.Next, let us write (3.26) in terms of 6-d spinors. For this

purpose, we choose the following, chiral representation ofthe 10-d gamma matrices,

�� ¼ I4 � �� I2 � �2 ð� ¼ 0; 1Þ�i ¼ I4 � �01 � �i � �2 ði ¼ 2; 3; 4Þ�5 ¼ �5 � I2 � I2 � �1

�� ¼ i�5�� I2 � I2 � �1 ð� ¼ 6; 7; 8; 9Þ; (3.27)

where ��, �� and �i are Euclidean 4-d gamma matrices,Lorentzian 2-d gamma matrices and a set of Pauli matrices,respectively, satisfying

2We thank L. Martucci for pointing out to us that, in order tocorrectly interpret the symbol Wm in the gauge-fixed action (30)of [28], one should start from their equation (17). We haveomitted the term �, which vanishes in the background (2.1).

3The matrix �ð0ÞD5 of [28] is given by �ð0Þ

D5 ¼ �~��6��9. Noticethe minus sign due to their definition of the epsilon symbol,which differs from ours.


106015-5

f��; �g ¼ 2��; f��; ��g ¼ 2��;

f�i; �jg ¼ 2�ij; (3.28)

and �5 ¼ �6789. Notice the peculiar representation of ��,which will turn out to be handy for reconstructing 6-dgamma matrices. It follows from (3.27) that the 10-dchirality matrix is simply

�ð10Þ ¼ �0��9 ¼ I4 � I2 � I2 � �3: (3.29)

Then, after reconstructing the 6-d gamma matrices forthe D5-brane world volume by

� � ¼ �5 � ��; �� ¼ �� I2; (3.30)

and using the left-handedness of �, the action (3.26) be-comes

SðFÞD5 ¼ T5

2sin4#

Zd6�


��

��ara

þ 1

4��Aij��

ij � i

L�01

��; (3.31)

where � now represents a doublet of 8-component, 6-dDirac spinors that stems from the spinor components that

survive the chiral projection. The 6-d gamma matrices �a

act on the spinors, and the matrices �ij ¼ iijk�k act on thedoublet.

Performing a chiral rotation, � ¼ ei��ð6Þ�0, one can ob-tain other, equivalent ways of writing the action (3.31), inwhich the ‘‘mass’’ term changes its appearance. In particu-lar, for � ¼ �=4, one obtains

SðFÞD5 ¼ T5

2sin4#

Zd6�


��0��ara

þ 1

4��Aij��

ij þ 1

L�6789

��0: (3.32)

In contrast to (3.31), in which the mass term commuteswith the 4-d part of the kinetic term and anticommutes withthe 2-d part, in (3.32) it commutes with the 2-d part of thekinetic term and anticommutes with the 4-d part. In Sec. V,(3.31) and (3.32) will give rise to two different ways ofcalculating the heat kernel, with slightly different results.

To conclude the 6-d reduction of the fermionic action,we consider the 10-d Majorana condition. In the decom-position (3.27), the intertwiner is given by4

Bþð9;1Þ ¼ Bþð5;0Þ � B�ð1;1Þ � B�ð3;0Þ � I2: (3.33)

Notice that the choice of intertwiners is unique for the odd-dimensional parts, but we have indicated the signs forclarity. Moreover, from (3.27) it is evident that Bþð5;0Þ ¼B�ð4;0Þ when acting on the ��. Therefore, the first two

factors on the right hand side of (3.33) just form the 6-dintertwiner

B�ð5;1Þ ¼ B�ð4;0Þ � B�ð1;1Þ: (3.34)

Thus, if we write the 6-d spinor doublet as

� ¼ Xr¼1

�r � �r; (3.35)

where �r is a doublet of constant (and normalized) 3-dsymplectic Majorana spinors satisfying

��r ¼ rB�ð3;0Þ��r; (3.36)

then the 10-d Majorana condition �� ¼ Bþð9;1Þ� gives rise

to the symplectic Majorana condition on the two 6-dspinors,

�r� ¼ rB�ð5;1Þ��r: (3.37)

A similar analysis can be done for the chirally rotatedspinor �0. This completes the 6-d formulation of the fer-mionic action.

C. Classical field equations-bosons

We shall work in the Lorentz gauge

r aaa ¼ 0; (3.38)

where ra denotes the covariant derivative with respect tothe metric (3.12) and, if acting on fields with indices i,contains also the appropriate connections for the normalbundle. Condition (3.38) leaves the residual gauge symme-

try aa ! aa þ @a� with rara� ¼ 0. Taking this intoaccount, the field equations that follow from (3.14) are

½�ijrara þHi

��Hj�� þ Rmpnqg

��xm�xn�N

ipNqj ��j ¼ 0;

(3.39)�rara þ 4

L2

��5 � 4

L sin#��r�a� ¼ 0; (3.40)

�rara � 1

2Rð2Þ

�a� � 4 sin#

L��r��

5 ¼ 0; (3.41)

�rara � 3

L2

�a� ¼ 0: (3.42)

Note that rara ¼ r�r� þr�r�, where r�r� is the

Laplacian on an S4 of radius L, while the covariant deriva-tive r� on the 2-d part of the world volume includes theconnections for the normal bundle in the case of (3.39). In(3.41), Rð2Þ denotes the curvature scalar of the 2-d part of

the world volume. So far, the components a� and a� of the

gauge fields are not entirely decoupled from each other,because of the gauge condition (3.38). However, we canuse the residual gauge freedom to set r�a

� ¼ 0 on-shell.To see this, contract (3.41) with r�, which yields

r arar�a� ¼ 0: (3.43)

4It is irrelevant whether we use Bþð9;1Þ or B�ð9;1Þ, because theydiffer by a factor of �ð10Þ.


106015-6

Thus, for any a� satisfying (3.43), one can find a residualgauge transformation � satisfying r�r��þr�a

� ¼ 0making the fields a� and a� transverse,

r�a� ¼ r�a

� ¼ 0: (3.44)

This still leaves us with the residual gauge transformationssatisfying

r �r�� ¼ r�r�� ¼ 0: (3.45)

To continue, we decompose the fields into5

�j ¼ X1l¼0

�j

l ð�; �ÞYlð�Þ; �5 ¼ X1l¼0

�5l ð�; �ÞYlð�Þ;

a� ¼ X1l¼0

a�l ð�; �ÞYlð�Þ; a� ¼ X1l¼0

alð�; �ÞY�lþ1ð�Þ;

(3.46)

where Ylð�Þ and Y�lþ1ð�Þ are scalar and transverse vector

eigenfunctions of the Laplacian on S4, respectively. Thecorresponding eigenvalues and their degeneracies aregiven by [29]

r�r�Ylð�Þ ¼ � lðlþ 3ÞL2

Ylð�Þ;

Dlð4; 0Þ ¼ 1

6ðlþ 1Þðlþ 2Þð2lþ 3Þ; (3.47)

r�r�Ylþ1ð�Þ ¼ � l2 þ 5lþ 3

L2Ylþ1ð�Þ;

Dlþ1ð4; 1Þ ¼ 1

2ðlþ 1Þðlþ 4Þð2lþ 5Þ: (3.48)

Substituting (3.46), (3.47), and (3.48), into the field Eqs.(3.39), (3.40), (3.41), and (3.42) yields��

r�r� � lðlþ 3ÞL2

��ij þHi

��Hj��

þ Rmpnqg��xm�x

n�N

ipNqj

��j

l ¼ 0; (3.49)

�r�r� � lðlþ 3Þ � 4

L2

��5l �

4

L sin#��r�a�l ¼ 0;

(3.50)

�r�r� � lðlþ 3Þ

L2� 1

2Rð2Þ

�a�l � 4 sin#

L��r��

5l ¼ 0;

(3.51)

�r�r� � ðlþ 2Þðlþ 3Þ

L2

�al ¼ 0: (3.52)

The dynamics of the two components a� is contained inthe field strength f ¼ ��r�a�. After decomposing f into

spherical harmonics on S4, one can proceed to diagonalize(3.50) and (3.51), which gives rise to the 2-d Klein-Gordonequations�

r�r� � 1

L2ðlþ 3Þðlþ 4Þ

��l ¼ 0;

�l ¼�fl þ sin#

Lðl� 1Þ�5

l

�;

(3.53)

�r�r� � 1

L2lðl� 1Þ

��l ¼ 0;

�l ¼�fl � sin#

Lðlþ 4Þ�5

l

�:

(3.54)

We should exclude the l ¼ 0 case of (3.54), because in thiscase one can rewrite (3.51) identically as

��r��0 ¼ 0; (3.55)

which implies that this particular mode is not dynamical. Asimilar result was found in [17]. This matches with the factthat the residual gauge transformation (3.45) is given by a2-d massless field with SOð5Þ angular momentum l ¼ 0.To summarize, the classical field equations for the

bosonic fluctuations have been reduced to the 2-d fieldEqs. (3.49), (3.52), (3.53), and (3.54).

D. Classical field equations-fermions

Let us consider the classical field equations for thefermions. We shall be agnostic about the symplecticMajorana condition (3.37), which can be imposed after-wards. This has the advantage that the following argumentshold also if we switch to Euclidean signature. The Diracequation following from the action (3.31) is�

�a ~ra � i

L�01

�� ¼ 0; (3.56)

where now

~r � ¼ r� þ 1

4Aij��

ij: (3.57)

Instead of using (3.30), it is easier to use the alternative 4þ2 decomposition

� � ¼ I4 � ��; �� ¼ �� 01: (3.58)

Thus, (3.56) becomes�� ~r� þ �01

��r� � i

L

�� ¼ 0: (3.59)

Let c �s be a doublet of 2-d spinors and �ls a 4-d spinor

satisfying the following 2-d and 4-d Dirac equations,respectively,

5Notice the index shift for the vector harmonics, which is usedto have all sums start from l ¼ 0. The sums over other quantumnumbers are implicit.


106015-7

� � ~r�c �s ¼ s�c �s; ðs ¼ 1; � 0Þ; (3.60)

� �r��ls ¼ islþ 2

L�ls; ðs ¼ 1; l ¼ 0; 1; 2; . . .Þ:

(3.61)

The �ls are just the eigenfunctions of the Dirac operator onthe 4-sphere [30]. Then, expanding � as

� ¼ X�;l;s;s0

a�lss0�ls0 � c �s; (3.62)

and using the property �01c �s ¼ c ��s, (3.59) leads to the

following relation for the coefficients,

s�La�lss0 þ i½s0ðlþ 2Þ � 1�a�l�ss0 ¼ 0: (3.63)

For (3.63) to have a nontrivial solution, it is necessary that

�L ¼�lþ 1 for s0 ¼ 1

lþ 3 for s0 ¼ �1: (3.64)

Summarizing, the classical field equation for the fermionicfluctuations have been reduced to the 2-d Dirac Eq. (3.60)with the eigenvalues (3.64). Notice, however, that c �s is a

doublet of 2-d Dirac spinors, and ~r� contains the normalbundle connection term.

IV. SPECTRUM OF OPERATORS ON CIRCULARWILSON LOOPS

We can verify now the predictions for the spectrum ofoperators on the circular Wilson loop made in [24] bymatching the diagonalized fluctuations with the multipletslisted in Table 3 of that paper.

Consider the D5-brane dual to the circular half-BPSWilson loop in the totally antisymmetric representation.The 10-d space-time geometry is AdS5 � S5. To make useof the rotational symmetry of the circular Wilson loop, wewrite the AdS5 metric as

d s2 ¼ L2

z2ð�dt2 þ dz2 þ dx2 þ d�2 þ �2d�2Þ; (4.1)

where z 2 ð0;1Þ is the radial coordinate. The 2-d part ofthe background D5-brane world volume is given by theembedding

� ¼ i�; � ¼ �; z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � �2

q;

ð0 � � � RÞ;(4.2)

where R is the radius of the Wilson loop. This embeddingdescribes AdS2 � AdS5. Notice that we have included afactor of i in the equation for� in order to formally achieveLorentzian signature on the world-sheet.

When evaluating the solution (4.2) in the classical action(2.14), the result is

SðBÞD5 þ IðBÞbound ¼ � N

3�2�0 2�L2sin3# ¼ � 2N

ffiffiffiffi�

p3�

sin3#:

(4.3)

In the second line we have written the gravity answer usingfield theory variables to illustrate that it matches perfectlywith the leading result for the expectation value of thek-antisymmetric Wilson loop computed using the matrixmodel [13,14].Let us determine the geometric quantities needed for the

field equations. First, because the bulk is AdS5 � S5, thecurvature term in (3.49) simply contributes a mass term of�2=L2. Second, as AdS2 is maximally symmetric, thesecond fundamental forms Hi

�� must be proportional to

the 2-d induced metric, g��. But because they are also

traceless (the effective string world-sheet is minimal), weconclude that Hi

�� ¼ 0. Third, an explicit calculation

using the formulas in appendix B shows that the SOð3Þgauge fields Aij� vanish identically.6 Therefore, the modes

of the independent bosonic fields, �j

l , al, �l and �l, satisfymassive Klein-Gordon equations on AdS2,

ðr�r� �m2Þ’ ¼ 0: (4.4)

The masses can be read off from (3.49), (3.52), (3.53), and(3.54), and are related to the conformal dimensions of thedual operators by the standard formula

h ¼ 1

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4þm2L2

s: (4.5)

For the fermions, the field Eq. (3.60) is a massive Diracequation on AdS2,

7

ð��r� �mÞc ¼ 0; (4.6)

and the (dimensionless) masses mL are given by (3.64).They are related to the conformal dimenions of the dualoperators by

h ¼ jmjLþ 1

2: (4.7)

We present our results in Table I in a form similar toTable 3 of [24]. The predictions of the multiplet structuremade in that paper are fully confirmed.

V. ONE-LOOP EFFECTIVE ACTION

Having found the full spectrum of excitations of thehalf-BPS D5-brane in AdS5 � S5 dual to the circularWilson loop, we now proceed to compute the correspond-

6This last statement depends, obviously, on the choice of thenormal vectors. In general, one gets a pure gauge Aij�.

7Actually, (3.60) is a doublet of Dirac equations, but the(symplectic) Majorana condition that still must be imposedmakes it equivalent to a single Dirac equation with an uncon-strained spinor.


106015-8

ing one-loop effective action using � function techniques[31,32]. Eigenfunctions of the Laplace and Dirac operatorsin maximally symmetric spaces and their associated heatkernels have been extensively studied [30,33–37]. We shallfollow in spirit the recent calculations of logarithmic cor-rections to the entropy of black holes in [38,39], especiallywith regard to the treatment of zero modes.

We start by providing a general review of the � functionmethod in section VA, focussing for simplicity on a singlemassive scalar field and highlighting the scaling propertiesof the functional determinant. In Sec. VB, the expansion ofthe bosonic fields into eigenfunctions on AdS2 and S4 isdone explicitly, so that we can proceed with the calculationof the bosonic heat kernels in section VC. The fermionicheat kernels are calculated in Sec. VD, and the results areput together in Sec. VE.

A. Computing functional determinants

Let �S denote the 1-loop correction to the effectiveaction for a single, real massive scalar field. It is given by

e��S ¼Z

D�e�ð1=2ÞR

ddxffiffiffiffiffiffiffidetg

p�ð�þm2Þ�: (5.1)

where the functional integration measure is defined by

1 ¼Z

D�e�ð1=2Þ�2R

ddxffiffiffiffiffiffiffidetg

p�2

: (5.2)

The constant � of dimension inverse length is needed for

dimensional reasons, because ½�� ¼ L1�d=2, so that ½S� ¼1. Formally, the functional integral (5.1) is written as afunctional determinant

e��S ¼ ½Detð�hþm2Þ��1=2: (5.3)

To give an operational definition to these formal expres-sions, introduce an orthonormal set of eigenstates of hsatisfying

�hfn ¼ �nfn;Z

ddxffiffiffiffiffiffiffiffiffidetg

pfnfm ¼ �nm: (5.4)

If the spectrum of h is continuous, the sum is to beunderstood as an integral with the appropriate spectralmeasure. In this basis, the field � can be expanded as

� ¼ Xn

�nfn: (5.5)

Notice the units ½fn� ¼ L�d=2 and ½�n� ¼ L. The integra-tion measure satisfying (5.2) is

D� ¼ Yn

��ffiffiffiffiffiffiffi2�

p d�n

�; (5.6)

and a short calculation shows that (5.1) gives rise to

�S ¼ 1

2

Xn

ln�n þm2

�2: (5.7)

In our case, the masses are proportional to 1=L, where Lis the radius of the AdS2 and S4 factors. Hence, we canwrite

�n þm2 ¼ 1

L2ð~�n þ ~m2Þ; (5.8)

where the ~�n are the eigenvalues of the Laplacian � ~hcorresponding to AdS2 � S4 with unit radius, and ~m rep-resent dimensionless numbers. Defining the � function

�ðsÞ ¼ Xn

ð~�n þ ~m2Þ�s; (5.9)

(5.7) can be expressed as

�S ¼ � 1

2� 0ð0Þ � lnð�LÞ�ð0Þ ¼ � lnðL=L0Þ�ð0Þ:

(5.10)

In the last equation, we have traded the inverse length� fora renormalization length scale L0 absorbing also the firstterm.In order to study the � function, it is convenient to

introduce the heat kernel

Kðx; y; tÞ ¼ Xn

e�ð�nþm2ÞtfnðxÞfnðyÞ: (5.11)

Here and henceforth, we have dropped the tilde and im-plicitly assume unit length L ¼ 1. By construction, (5.11)satisfies the heat equation�

@

@t�hþm2

�Kðx; y; tÞ ¼ 0; (5.12)

with the initial condition Kðx; y; 0Þ ¼ �ðx; yÞ. Setting x ¼y and integrating over the manifold gives the trace

YðtÞ �Z

ddxffiffiffiffiffiffiffiffiffidetg

pKðx; x; tÞ ¼ X

n

e�ð�nþm2Þt: (5.13)

Then, the � function is related to the integrated heat kernelby the Mellin transform,

TABLE I. Matching of the bulk fields with multiplets ofOSpð4�j4Þ, cf. Table 3 of [24]. The quantum numbers have thefollowing meaning: h is the conformal dimension, n ¼ 0, 1

2 , 1

stand for SOð3Þ singlets, doublets and triplets, respectively, m ¼0, 1, 2 for scalar, spinor and vector fields on S5, respectively, andl is the S5 angular momentum. In general, l 0, except for thefield �l, see the discussion at the end of section III A.

bosons

field m2L2 ðh; nÞ � ðm; lÞ�lðl 1Þ lðl� 1Þ ðl; 0Þ � ð0; lÞ�l ðlþ 3Þðlþ 4Þ ðlþ 4; 0Þ � ð0; lÞal ðlþ 2Þðlþ 3Þ ðlþ 3; 0Þ � ð2; lÞ�il ðlþ 1Þðlþ 2Þ ðlþ 2; 1Þ � ð0; lÞ

fermions

field mL ðh; nÞ � ðm; lÞc lþ (lþ 1) ðlþ 3

2 ;12Þ � ð1; lÞ

c l� (lþ 3) ðlþ 72 ;

12Þ � ð1; lÞ


106015-9

�ðsÞ ¼ 1

�ðsÞZ 1

0dtts�1YðtÞ: (5.14)

Notice that since AdS2 � S4 is noncompact, the � functionwill diverge; Kðx; x; tÞ is independent of x for a homoge-neous space. Thus, YðtÞ and �ðsÞ are proportional to thevolume of unit AdS2 � S4, which must be regularized.

We can separate the integral in (5.14) into align

�ðsÞ ¼ 1

�ðsÞ�Z 1

0dtts�1YðtÞ þ

Z 1

1dtts�1YðtÞ

�: (5.15)

The second term converges for any s since YðtÞ �e�ð�0þm2Þt for large t. On the other hand, it can be shownthat YðtÞ has the asymptotic expansion

YðtÞ ffi X1n¼0

antðn�dÞ=2: (5.16)

as t ! 0þ. Substituting this in the first term of (5.15) gives

1

�ðsÞXn

ansþ ðn� dÞ=2 : (5.17)

This shows that �ðsÞ will have poles at s ¼ d; d� 1; . . . ; 1.The pole at s ¼ 0, however, is removed by the gammafunction. Inverting (5.14) gives

YðtÞ ¼ 1

2�i

Idst�s�ðsÞ�ðsÞ; (5.18)

where the integration contour encircles all the poles of theintegrand. In particular, (5.18) implies that

�ð0Þ ¼ ad: (5.19)

Thus, the problem of computing functional determinants ismapped to the problem of computing the t independentcoefficient in the asymptotic expansion of the integratedheat kernel.

The above derivation can be extended to higher spinfields with analogous results. Each field has its own heatkernel and thus its own � function. The total effectiveaction is obtained by simply adding the contribution ofthe integrated heat kernels from all the fields present in thetheory. In the case of massless fields, special attention mustbe paid to possible zero modes of the corresponding kineticoperators as they must be excluded from the definition ofthe heat kernel. This can be done in an elegant fashion bysubtracting from the final heat kernel its value for large t[38]. It turns out, however, that the pieces of the heat kernelthat would have to be subtracted have cancelled betweenthe contributions from various fields. Moreover, it can beargued that, as far as the logarithmic corrections are con-cerned, the full heat kernel yields the correct result [38].For the fluctuations of the D5-brane, a further complicationstems from the fact that some modes are coupled and mustbe diagonalized. We shall deal with these issues at duemoment.

B. Mode decomposition for the bosons

We want to calculate the one-loop effective action forthe bosons in the background of the holographic Wilsonloop. Let us start with the action (3.14). There are twopoints we have to address before doing the path integral.First, our fields have physical dimensions ½�� ¼ ½a� ¼ L.Thus, to obtain the canonical dimensions used in the lastsubsection, we must absorb a square root of T5 into eachfield.Second, (3.14) involves the metric gab defined in (3.12),

which is AdS2 � S4, with both factors of radius L. Thefluctuation fields, however, were defined on the back-ground world volume, which has a metric Mab ¼ gab þF ab, as defined by the Born-Infeld part of the action. Thischange has an influence on the functional integration mea-sures, which is easily accounted for by a suitable rescalingof the fields. Consider the norms for scalar and vector fieldson the background world volume, which are used to definethe integration measures,

jj�jj2 ¼Z

d6�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidetMab

p�2 ¼ sin5#

Zd6�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffidetgab

p�2;

jjajj2 ¼Z

d6�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidetMab

pMabaaab

¼ sin3#Z

d6�ffiffiffiffiffiffiffiffiffiffiffiffiffiffidetgab

pgabaaab: (5.20)

The integrals on the right hand sides of (5.20) are the normsthat are used to define the integral measures for the pathintegral on a manifold with metric gab. Therefore, in orderto write the action (3.14) in terms of integration variableswith standard measure and canonical units, we must re-scale the fields by

� ! �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT5sin

5#q ; aa ! aaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T5sin3#

p : (5.21)

Thus, (3.14) gives rise to the Euclidean action

SðB;2ÞD5;E ¼ � 1

sin2#

Zd6�


p �1

2�ij�

i

�rara � 2

L2

��j

þ 1

2�5

�rara þ 4

L2

��5 � 2i

L�5��f��

� 1

4f�f� � 1

2f��f�� 1

4f��f��

�: (5.22)

Notice the i in the first term on the second line, whichstems from switching the explicit Lorentzian �� to

Euclidean signature. The additional factor 1=sin2# in frontcan be absorbed by rescaling gab ! sin2#gab giving riseto AdS2 and S4 of radius L sin#.There are two difficulties we have to address in the

calculation of the heat kernels. First, there is the gaugeinvariance, aa ! aa þ @a�. Second, the sector consistingof the gauge field a� and the scalar �5 must be diagonal-ized. This problem does not allow us to factorize the heat


106015-10

kernel in a straightforward fashion. Therefore, we chooseto do a complete mode expansion of the action into eigen-states on S4 and AdS2, which will also allow us to performthe gauge fixing on a state-by-state basis.

Let us start with the mode expansion of the fields ap-pearing in (5.22). Fields that are scalars on S4 can bedecomposed into spherical harmonics, such as

�5 ¼ X1l¼0

Ylð�Þ�5l ð�; �Þ: (5.23)

We do not explicitly write the sum over the minor angularmomentum quantum numbers, which are easily accountedfor by remembering the degeneracies Dlð4; 0Þ given in(3.47).

The gauge field a�, which is a vector on S4, decomposesinto

a� ¼X1l¼1

�Y�l ð�Þalð�;�Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2

lðlþ3Þ

sðr�Ylð�ÞÞblð�;�Þ

�:

(5.24)

In contrast to the expansion of the gauge-fixed classicalfield, we have to include the longitudinal modes. Thesquare root factor in the second term is necessary in orderfor the eigenfunctions multiplying the coefficients bl to beproperly normalized.

For the AdS2 part, we work in the Poincare metric (C1).The normalized scalar eigenfunctions of the Laplacian arethen given by (C12), with eigenvalues �r�r� ! � ¼ð2 þ 1=4Þ. Hence, the AdS2 scalars decompose like

�5 ¼Z 1

�1dk

Z 1

0dfðk;Þðx; yÞ�5

ðk;Þð�Þ: (5.25)

For the AdS2 vector a� we have to be more careful [38].

Locally, an eigenfunction of the vector Laplacian can be

written as a� ¼ ��1=2ðr�f1 þ ��r�f2Þ, where f1 and

f2 are eigenfunctions of the scalar Laplacian with the sameeigenvalue �. In doing so, we must take care to include azero mode, which is not a normalizable scalar mode, butwhich gives rise to a normalizable vector mode. This modecomes from the ¼ i=2 case of the scalar eigenfunctionsand reads (for unit L)

~f kðx; yÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffi2�jkjp eikx�jkjy: (5.26)

The full expansion of the vector a� reads, therefore,

a� ¼Z 1

�1dk

Z 1

0d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2

2 þ 1=4

s½ðr�fðk;ÞÞcðk;Þð�Þ

þ ð��r�fðk;ÞÞdðk;Þð�Þ� þZ 1

�1dkðr� ~fkÞ~ckð�Þ:

(5.27)

One can check that the eigenfunctions in front of the modecoefficients cðk;Þ, dðk;Þ and ~ck are orthonormal with re-

spect to the normRd2x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidetg��

pa�a�. Remember that now

�� ¼ þ2, because we are in Euclidean signature.

After doing the mode expansion in (5.22), one obtains

SðB;2ÞD5;E ¼ 1

2L2sin2#

Z 1

�1dk

Z 1

0d

X1l¼0ð1Þ

��2 þ 1

4þ lðlþ 3Þ þ 2

��ij�

i

lðk;Þ�j

lðk;Þ þ�2 þ 1

4þ lðlþ 3Þ � 4

�ð�5

lðk;ÞÞ2

þ 8i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 1

4

s�5

lðk;Þdlðk;Þ þ�2 þ 1

4þ lðlþ 3Þ

�d2lðk;Þ þ

�2 þ

�lþ 3

2

�2�a2lðk;Þ

þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lðlþ 3Þp

clðk;Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 1

4

sblðk;Þ

�2�þ 1

2L2sin2#

Z 1

�1dk

X1l¼0

lðlþ 3Þ~c2l;k: (5.28)

The summation over l starts with 0 for the first two lines,but with 1 for the third line. The last line is the contributionfrom the special AdS2 vector modes.

C. Bosonic heat kernels

Triplet The calculation is simplest for the tripletfields �i. The contribution of each triplet field to the heatkernel is

Y�iðtÞ ¼ e�2�tYscAdS2ð�tÞYsS4ð�tÞ; (5.29)

where �t ¼ t=ðL sin#Þ2, and the heat kernels on dAdS2 and

S4 (the hats indicate that these are AdS2 and S4 of unitradii) are given, respectively, by [38]

YscAdS2ðtÞ ¼VcAdS22�

e�t=4Z 1

0d tanhð�Þe�2t (5.30)

and

YsS4ðtÞ ¼ X1

l¼0

Dlð4; 0Þe�lðlþ3Þt

¼ X1l¼0

1

6ðlþ 1Þðlþ 2Þð2lþ 3Þe�lðlþ3Þt: (5.31)

VcAdS2 ¼ VAdS2=L2 denotes the regulated volume of unit

AdS2. The superscript s on the heat kernels indicates thatthey are for scalar fields.


106015-11

Let us rewrite (5.31) by completing the square in theexponent, including the value l ¼ �1 in the sum (this doesnot alter the sum) and shifting the summation index by one.This yields

YsS4ðtÞ ¼ � 1

12e9t=4

�1þ 4

@

@t

��sðtÞ; (5.32)

with

�sðtÞ ¼ X1l¼0

�lþ 1

2

�e�ðlþ1=2Þ2t: (5.33)

The evaluations of the integral in (5.30) and the infinitesum in (5.33) are carried out in Appendix D. Substitutingthe results into (5.29) we obtain

Y�iðtÞ ¼VcAdS22�

�� 1

12ð1þ 4@�tÞ�sð�tÞ

�½��sð��tÞ�

¼VcAdS22�

�1

12�t3� 1

36�t2� 1

756þ � � �

�: (5.34)

Transverse gauge modes Let us integrate over the trans-verse modes alðk; Þ, where l 1. From (5.28) we can readoff the contribution to the heat kernel

YaðtÞ ¼ YscAdS2ð�tÞYvS4ð�tÞ; (5.35)

where

YvS4ðtÞ ¼ X1

l¼1

Dlð4; 1Þe�ðlþ1Þðlþ2Þt

¼ X1l¼1

1

2lðlþ 3Þð2lþ 3Þe�ðlþ1Þðlþ2Þt; (5.36)

while YscAdS2ð�tÞ is the scalar heat kernel (5.30). The infinitesum in (5.36) can be rewritten as

YvS4ðtÞ ¼ �et=4

�@

@tþ 9

4

��sðtÞ þ 1; (5.37)

where the 1 can be traced back to a missing l ¼ 0 sum-mand after shifting the summation index.

The action (5.22) is invariant under the gauge symmetrya� ! a� þ 1

sin# @��, where the factor 1sin# in front of the

gauge parameter stems from the different rescalings ofscalar and vector fields needed to get standard integrationmeasures, as discussed at the beginning of Sec. VB.Expanding also � into modes, this translates into

blðk;Þ ! blðk;Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ 3ÞpL sin#

�lðk;Þ ðl 1Þ;

clðk;Þ ! clðk;Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 1=4

pL sin#

�lðk;Þ ðl 0Þ:(5.38)

Invariance of (5.28) under (5.38) is immediate upon in-spection of the third line in (5.28).

We can now impose a gauge on a mode-by-mode basis.An obvious choice is to fix the coefficients clðk;Þ, whichcan be done using Faddeev-Popov. Hence, we must intro-duce

�ðclðk;ÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 1=4

pL sin#

(5.39)

into the functional integral, where the second factor is theFaddeev-Popov determinant. However, performing the in-tegral over blðk;Þ, we obtain L sin#ffiffiffiffiffiffiffiffiffiffiffiffi

2þ1=4p , but only for l 1.

Hence, the net result of gauge fixing, the trivial integrationover clðk;Þ and the integration over blðk;Þ is minus the

contribution of an AdS2 scalar,

Ygf;b;cðtÞ ¼ �YscAdS2ð�tÞ: (5.40)

This compensates the 1 in (5.37). Hence, after gauge fixing,the heat kernel for the vector fields a� is

Ya�ðtÞ ¼ YaðtÞ þ Ygf;b;cðtÞ

¼VcAdS22�

��9

4þ @�t

��sð�tÞ

�½��sð��tÞ�

¼VcAdS22�

�1

4�t3� 7

12�t2� 19

1260þ � � �

�: (5.41)

Mixed sector To integrate over �5

lðk;Þ and dlðk;Þ, we haveto deal with the matrix

M ¼2 þ 1

4 þ lðlþ 3Þ � 4 4iffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 1

4

q4i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 1

4

q2 þ 1

4 þ lðlþ 3Þ

0B@1CA:(5.42)

Its eigenvalues are

ð 2iÞ2 þ 1

4þ lðlþ 3Þ þ 2; (5.43)

but its determinant can also be written in terms of realfactors,

detM ¼�2 þ 1

4þ lðl� 1Þ

��2 þ 1

4þ ðlþ 3Þðlþ 4Þ

�:

(5.44)

The two factors on the right hand side of (5.44) are pre-cisely what one would expect from the classical spectrum.It is possible to calculate the heat kernel either from the

eigenvalues (5.43) or the factors in (5.44). The results of thecalculations differ in the scheme dependent divergentterms 1=t2 and 1=t, but we shall perform both calculations,because a similar ambiguity will be encountered for thefermions.The heat kernel calculation using the eigenvalues (5.43)

is similar to the situation encountered in [38], and we shall


106015-12

follow the treatment of that paper. The effect of the mixingbetween the scalar and the gauge field is a complex shift ofthe AdS2 eigenvalue compared to (5.30). Hence, the inte-grated heat kernel for the �5 and d integration is

Y�5;d1 ðtÞ ¼ e�2�tYs

S4ð�tÞ½2YscAdS2ð�tÞ þ �YscAdS2ð�tÞ�; (5.45)

where

�YscAdS2ðtÞ ¼VcAdS22�

e�t=4Z 1

0d tanhð�Þ½e�ð�2iÞ2t

þ e�ðþ2iÞ2t � 2e�2t�: (5.46)

For the first two terms in the integrand of (5.46), we shiftthe integration variables to � 2i and þ 2i, respectively,such as to obtain the same exponent as in the third term.Then, we deform the integral contours such that we haveintegrals from �2i (þ 2i) to 0 (staying to the right of theimaginary axis) and from 0 to 1. The latter cancel againstthe third term in (5.46). Finally, switching the sign of theintegration variable in one of the two remaining integrals,they can be combined into

�YscAdS2ðtÞ ¼VcAdS22�

e�t=4I

dð� 2iÞ tanhð�Þe�2t;

(5.47)

where the integration contour circles clockwise around thepoles at ¼ i=2 and ¼ 3i=2. The residue theorem thenyields

�YscAdS2ðtÞ ¼ �VcAdS22�

ðe2t þ 3Þ: (5.48)

Putting everything together, we get

Y�5;d1 ðtÞ ¼ Ys

S4ð�tÞ�2e�2�tYscAdS2ð�tÞ �

VcAdS22�

ð3e�2�t þ 1Þ�

¼VcAdS22�

�� 1

12ð1þ 4@�tÞ�sð�tÞ

�� ½�2�sð��tÞ � 3e�t=4 � e9�t=4�

¼VcAdS22�

�1

6�t3� 13

18�t2� 1

3�t� 551

1890þ � � �

�: (5.49)

Strictly speaking, we should have removed a zero mode bysubtracting the value of the integrated heat kernel at t ¼ 1.One easily finds from (5.31) that Ys

S4ð1Þ ¼ 1, so the last

term in the brackets on the first line of (5.49) contains azero mode. We shall ignore this for the moment, becausethe subtraction can be done at the very end [38].

Let us now consider the alternative choice, namely, weperform the calculation using the factors of the determinant(5.44). In this case we get

Y�5;d2 ðtÞ ¼ YscAdS2ð�tÞ

X1l¼0

Dlð4; 0Þ½e�lðl�1Þ�t þ e�ðlþ3Þðlþ4Þ�t�:

(5.50)

The infinite sum can be rewritten as

X1l¼0

Dlð4; 0Þ½e�lðl�1Þt þ e�ðlþ3Þðlþ4Þt�

¼ 2

3et=4

��@t þ 47

4

��sðtÞ þ 2; (5.51)

where the 2 stems from extra terms due to shifts of thesummation index. Thus, the final result is

Y�5;d2 ðtÞ ¼

VcAdS22�

�2

3

�47

4� @�t

��sð�tÞ þ 2e��t=4

�½��sð��tÞ�

¼VcAdS22�

�1

6�t3þ 35

18�t2þ 1

�t� 551

1890þ � � �

�: (5.52)

As anticipated, the results (5.49) and (5.52) differ in thescheme dependent 1=t2 and 1=t terms. It is worth notingthat the relevant terms for the final result, that is, theleading 1=t3 terms and the constant terms, are identicalin both choices.Special modes Finally, let us integrate over the special

modes ~c. The AdS2 part of their heat kernel is obtainedfrom the wave functions (5.26) as

K~ccAdS2ðtÞ ¼Z 1

�1dkðr� ~fkÞðr�

~fkÞ

¼Z 1

�1dk

2k2y2

2�jkj e�2jkjy ¼ 1

2�: (5.53)

This is independent of t, because the special modes arezero modes on AdS2.Thus, the integrated heat kernel for the special AdS

vector modes is

Y~cðtÞ ¼VcAdS22�

YsS4ð�tÞ ¼

VcAdS22�

�1

6�t2þ 1

3tþ 29

90þ � � �

�:

(5.54)

Note that we have not subtracted the zero mode l ¼ 0.However, one can recognize that Y~cðtÞ cancels the thirdterm in the brackets on the first line of (5.49), whichcontains the zero mode from the ð�5; dÞ sector, as discussedabove. Thus, all bosonic zero modes cancel precisely, andno further subtraction is necessary.All bosonic modes Let us put together the results for all

bosonic fields, (5.34), (5.41), and (5.49), [or (5.52) and(5.54)],

YbosðtÞ ¼ 3Y�iðtÞ þ Ya�ðtÞ þ Y�5;dðtÞ þ Y~cðtÞ: (5.55)

Using (5.49) for the mixed sector, we obtain

Ybos1 ðtÞ ¼

VcAdS22�

�2

3�t3� 11

9�t2þ 11

945þ � � �

�; (5.56)

while using (5.52) gives rise to


106015-13

Ybos2 ðtÞ ¼

VcAdS22�

�2

3�t3þ 13

9�t2þ 4

3�tþ 11

945þ � � �

�: (5.57)

D. Fermionic heat kernels

We have seen in Sec. III B that there are equivalent waysof writing the 6-d fermionic action that are related to eachother by chiral rotations. As is well-known [40], the fer-mion integration measure is, in general, not invariant undera chiral rotation in the presence of curvature or gaugefields. To detect whether this is an issue here, let uscalculate the fermionic heat kernels corresponding to theactions (3.31) and (3.32), in both cases using a the standardmeasure for the fermions. We will find that the resultingheat kernels differ in the scheme-dependent 1=t2 and 1=tterms, but the leading 1=t3 term and the constant term areidentical, just as we found in the mixed sector of thebosons. This implies that we can safely ignore genericproblems with the measure under chiral rotations.Remember that in (3.31) and (3.32) the mass term com-mutes with either the 2-d or the 4-d part of the kinetic termand anticommutes with the other. In both cases, the two 6-dspinors in the doublet are not coupled, because Aij� ¼ 0,

and the symplectic Majorana condition (3.37) reduces thedoublet to a single independent Dirac spinor, giving rise toa factor of 2 in the action.

Let us start with (3.32). Arguing as for the bosons, wefind that � must be rescaled like a scalar, so that (3.32)gives rise to

SðFÞD5;E ¼ 1

sin#

Zd6�


p��

��ara þ 1

L�6789

��:

(5.58)

Again, the factor 1= sin# in front can be absorbed byrescaling the metric. Analogous arguments hold for theaction (3.31).

Writing the Dirac operator in the brackets of (5.58) as

D ¼ ��r� þ��r� þ 1

L�6789

�; (5.59)

one can verify that the two terms on the right hand side

anticommute. The 4-d Dirac operator on S4, ��r�, has

eigenvalues iðlþ 2Þ=Lðl ¼ 0; 1; 2; . . .Þ with degeneracyDlð4; 12Þ ¼ 2

3 ðlþ 1Þðlþ 2Þðlþ 3Þ [30]. The 2-d Dirac op-

erator on AdS2, ��r� has a continuous spectrum i�=L

(� 0; the spectral measure can be found in [30,38]).Taking the square of D, we get

D2 ¼ ð��r�Þ2 þ��r� þ 1

L�6789

�2: (5.60)

Because �6789 commutes with ��r� and has eigenvalues1, we obtain the integrated heat kernel as

Yf1 ðtÞ ¼ �Yf

S4ð�tÞ½2YfcAdS2ð�tÞ þ �YfcAdS2ð�tÞ�; (5.61)

where

Yf

S4ðtÞ ¼ �X1

l¼0

Dl

�4;1

2

�e�ðlþ2Þ2t; (5.62)

YfcAdS2ðtÞ ¼ �VcAdS22�

2Z 1

0d�� cothð��Þe��2t (5.63)

and

�YfcAdS2ðtÞ ¼ �VcAdS22�

2Z 1

0d�� cothð��Þ

� ½e�ð�þiÞ2t þ e�ð��iÞ2t � 2e��2t�: (5.64)

The S4 part (5.62) is rewritten as

Yf

S4ðtÞ ¼ 2

3ð@t þ 1Þ�fðtÞ; (5.65)

where we have introduced

�fðtÞ ¼ X1l¼0

le�l2t: (5.66)

The explicit evaluation of �f and the integral in (5.63) arerelegated to Appendix D. The expression (5.64) is obtainedalong the lines of the first mixed sector calculation inSec. VC. One obtains the contour integral

�YfcAdS2ðtÞ¼VcAdS22�

2I 0�

0þd�ð�� iÞcothð��Þe��2t; (5.67)

where the contour runs from 0þ to i along the right of theimaginary axis and back to 0� along the left. Note thatthere are no poles inside the contour, and the integrand isregular at � ¼ i. However, we cannot close the contour dueto the pole at � ¼ 0, so that the value of the integral mustbe defined as the principal value (half of the residue value),

�YfcAdS2ðtÞ ¼VcAdS22�

2�iRes�¼0½ð�� iÞ cothð��Þe��2t�

¼VcAdS22�

2: (5.68)

Collecting everything together, we obtain

Yf1 ðtÞ ¼ �

VcAdS22�

�2

3ð@�t þ 1Þ�fð�tÞ

�½4�fð��tÞ þ 2�

¼ �VcAdS22�

�2

3t3� 11

9t2þ 2

3t� 271

3780þ � � �

�: (5.69)

Starting, instead, with the action (3.31), we have theDirac operator8

8In Euclidean signature, the 2-d chirality matrix is �01 ¼i�0�1, so that the property ð�01Þ2 ¼ 1 is maintained.


106015-14

D ¼��r� � i

L�01

�þ ��r�: (5.70)

Analogous arguments as above lead to the integrated heatkernel

Yf2 ðtÞ ¼ YfcAdS2ð�tÞ

X1l¼0

Dl

�4;1

2

�½e�ðlþ1Þ2 �t þ e�ðlþ3Þ2 �t�:

(5.71)

After a short calculation, the infinite sum can be rewrittenas X1

l¼0

Dl

�4;1

2

�½e�ðlþ1Þ2t þ e�ðlþ3Þ2t� ¼ 4

3ð2� @tÞ�fðtÞ:

(5.72)

Hence, after substituting the results into (5.71), we obtain

Yf2 ðtÞ ¼ �

VcAdS22�

�8

3ð@�t � 2Þ�fð�tÞ

��fð��tÞ

¼ �VcAdS22�

�2

3t3þ 13

9t2� 271

3780þ � � �

�: (5.73)

As already anticipated from the results of the mixed sectorbosons, the two ways of calculating the heat kernel lead toresults that differ in the scheme-dependent 1=t2 and 1=tterms, but yields identical results for the leading 1=t3 andthe constant terms.

E. Combining bosons and fermions

We are now in a position to give the full answer for theheat kernel. As we have two slightly different expressionsfor the bosons and two for the fermions, there would befour different combinations. One can readily see that theleading 1=t3 term cancels in all of them, and the constantterm, which is responsible for the scaling, is always thesame. We can, however, make the following nice observa-tion, which indicates that supersymmetry does more thanjust cancelling the leading term. It appears natural tocombine (5.56) with (5.69), because the heat kernels ofthe mixed sector bosons and the fermions were calculatedwith a shift of the eigenvalues on the AdS2 part. Similarly,we should add (5.57) and (5.73), for which the eigenvalueshifts happend on the S4 part. In these combinations, alsothe 1=t2 terms cancel, and we obtain

Y1ðtÞ ¼ Ys1ðtÞ þ Yf

1 ðtÞ ¼VcAdS22�

�� 2

3tþ 1

12þ � � �

�;

(5.74)

Y2ðtÞ ¼ Ys2ðtÞ þ Yf

2 ðtÞ ¼VcAdS22�

�4

3tþ 1

12þ � � �

�: (5.75)

It remains to regularize the infinite volume VcAdS2 , forwhich we follow the treatment of [39] complemented witha field theory prescription due to Polyakov [41]. For the

circular Wilson loop, it is appropriate to describe unitAdS2by the metric

d s2 ¼ d�2 þ sinh2� d�2: (5.76)

To regularize the volume we introduce a cut-off �0, so thatthe regularized volume of AdS2 is 2�ðcosh�0 � 1Þ. In thecontext of corrections to the entropy of black holes [38] theinterpretation of the regularization is as follows. Whensubstituted in the effective action, the term proportionalto cosh�0 gives rise, up to a term that vanishes when�0 ! 1, to a divergent contribution ��E, where ��2� sinh�0 is the inverse temperature and �E is the shiftin the ground state energy due to the introduction of thecut-off. This regularization has a simple interpretation onthe field theory side as well. In [41], Polyakov studied theevaluation of vacuum expectation values of general Wilsonloops and determined a divergent term that is proportionalto the length of the contour and can be interpreted as themass renormalization of the test particle traveling aroundthe contour. Either interpretation leads, for the one-loopcorrection, to

VcAdS2 ¼ �2�: (5.77)

It is worth pointing out that the above prescription can alsobe justified directly as a regularization of the string, as donein the classical work [26].Let us now collect the various pieces and give the final

result. Using (5.10), (5.19), (5.74), and (5.77), taking intoaccount also that the appropriate radius of the manifold forcanonically normalized fields is L sin#, as discussed after(5.22), we find for the one-loop effective action

�S ¼ 1

12lnL sin#

L0

: (5.78)

The simplicity of this result certainly elicits thinkingabout some powerful symmetry principle behind it. Wehave explicitly witnessed how supersymmetry was crucialin various cancellations of divergences; it could be that italso constrains the form of the final answer. Anothertantalizing possibility is the interpretation of the aboveresult in terms of a one-loop beta function computationthat controls the effective D5-brane tension.

VI. CONCLUSIONS

In this paper, we have explicitly treated the D5-braneconfiguration dual to the half-BPS circular Wilson loop inthe totally antisymmetric representation. We have derivedthe fluctuations in both, the bosonic and the fermionicsectors. We have also verified that the excitations fallprecisely in the expected supermultiplets of OSpð4�j4Þ.Lastly, we computed the one-loop determinants and pro-vided an answer for the effective action at the one-looplevel.Our work is largely motivated by the applications to the

Wilson loops and the potential to take the correspondence


106015-15

beyond the classical ground state by incorporating quan-tum corrections. This provides a step towards being able todirectly compare one-loop corrections from the field theory(Matrix model) and gravity (D-brane) sides. More gener-ally, our work represents a systematic exploration of thevarious issues that can arise during the quantization ofextended objects in the context of the AdS/CFT correspon-dence. We have encountered and resolved various ambi-guities and in the process shed some light on the type ofissues that need to be resolved if a coherent quantization ofextended objects in curved backgrounds is to be achieved.For example, we hope to have fully clarified the, at timesad hoc, process of computing the action for the quadraticfluctuations by explicitly highlighting the differential geo-metric nature of the fluctuations. We also resolved varioustechnical issues in the computation of the heat kernel forfermions and showed a natural way to determine a scheme.More importantly, at least in our example, we witness thatthe role of supersymmetry seems to go beyond the ex-pected cancellation of the leading divergence.

There are a few very interesting problems that follownaturally from our work, and we finish by highlightingsome of them.

(i) A natural direction is the calculation and comparisonwith the matrix model. We hope to report on thisinteresting issue in an upcoming publication. Thetask at hand, although conceptually clear, is plaguedwith many technical issues. Some of these issues aregeneric to the whole program of comparing expec-tation values of operators in the field theory and inthe gravity dual. We mentioned in the introductionthat, even in the apparently simple case of theWilsonloop in the fundamental representation, an agree-ment has not been found [6–9]. Hopefully, the extraknob that constitutes the representation might lead tosome simplifications.

(ii) In this paper we did not discuss the field theory dualbeyond the mere mentioning of the role as half BPSWilson loops. An important interpretation is pro-vided by the D5-branes as a dual to a one-dimensional defect CFT and has been quoted inrecent works as a model for interesting condensedmatter phenomena related to quantum impuritymodels [25,42,43]. A similar interpretation of D6-branes as dual descriptions of fermionic impuritiesin N ¼ 6 supersymmetric Chern-Simons-mattertheories in 2þ 1 dimensions has been advanced in[44]. In such contexts, uncovering the precise role ofthe spectrum of excitations should lead to a deeperunderstanding of the interactions of the system.

(iii) More generally, our paper provides a first solid stepin the direction of analyzing extended objects at thequantum level. It seems that the analysis of confor-mal branes, that is branes whose world volumecontains AdS factors, avoids dealing with the

daunting issues encountered in the quantization ofextended objects in asymptotically flat spacetimes.We plan to pursue this analysis in the future.

(iv) Recently, Sen and collaborators have studiedcorrections to the entropy of various black holeconfigurations using techniques similar to thoseutilized here. The key technical fact that the nearhorizon geometry of various black holes containsAdS factors seems to provide a tantalizing play-ground for our methods. We hope that understand-ing the quantization of such structures at a deeperlevel might help clarify difficult issues in black holephysics.

ACKNOWLEDGMENTS

We are grateful to A. Tirziu for collaboration in relevantmatters and comments. We are also thankful to M.Kruczenski, L. Martucci and A. Ramallo for variouscomments. A. F. is thankful to Fulbright-CONICYT. Thework of A. F. and L.A. P. Z. is partially supported by theDepartment of Energy under Grant No. DE-FG02-95ER40899 to the University of Michigan.W.M. acknowl-edges partial support by the INFN research initiative TV12.

APPENDIX A: CONVENTIONS

We summarize the conventions used throughout thepaper. The 10-dimensional curved coordinates are denotedby Latin indices from the middle of the alphabet, m; n ¼0; . . . ; 9. Latin indices from the beginning of the alphabet,a; b ¼ 0; . . . ; 5, denote generic coordinates of the D5-branes. Greek indices from the beginning of the alphabet,�, � ¼ �, � are used for the coordinates of the effectivestring embedded in the aAdS5 part of the backgroundgeometry. Greek indices from the middle of the alphabet,�; ¼ 6; . . . ; 9, denote the coordinates of the S4 part of theD5-brane world volume. The corresponding flat indices areunderlined. In contrast to [28], the Levi-Civita symbolsa1...an are tensors, i.e., they include the appropriate factors

offfiffiffiffiffiffiffiffiffiffiffiffiffij detgjp

. With the exception of section V, we assumeLorentzian signature for the 2-d part of the world sheet,which implies ��

�� ¼ �2.

APPENDIX B: GEOMETRY OFEMDEDDED MANIFOLDS

To describe the embedding of the D5-brane world vol-ume in the bulk, we shall use the structure equations ofembedded manifolds [27]. Deviating from the generalnotation of the main text and of Appendix A, we shalldenote with Latin indices m; n; . . . the curved bulk coor-dinates and with Greek indices �;�; . . . the world-volumecoordinates. Latin indices i, j are used for the directionsnormal to the world-volume. The corresponding flat indi-ces are underlined.


106015-16

A d-dimensional Riemannian manifold M embedded in

a ~d-dimensional Riemannian manifold ~M (d < ~d) is de-

scribed by ~d differentiable functions xm (m ¼ 1 . . . ~d) of dvariables �� (� ¼ 1 . . .d). The �� are coordinates on M(the world volume), whereas xmð�Þ specify the location in~M (the bulk). The tangent vectors to the world volume aregiven by xm� ð�Þ � @�x

mð�Þ. They provide the pullback ofany bulk quantity onto the world volume. For example, theinduced metric is

g�� ¼ xm�xn�gmn: (B1)

In addition, there are d? ¼ ~d� d normal vectors Nmi , i ¼

1; . . . ; d?. Together with the xm� , they satisfy the orthogo-nality and completeness relations

Nmi x

n�gmn ¼ 0; Nm

i Nnj gmn ¼ �ij;

g��xm�xn� þ �ijNm

i Nnj ¼ gmn:

(B2)

We shall adopt a covariant notation raising and loweringindices with the appropriate metric tensors. The freedom ofchoice of the normal vectors gives rise to a group OðnÞ oflocal rotations of the normal frame.

The geometric structure of the embedding is determined,in addition to the intrinsic geometric quantities, by thesecond fundamental form Hi

��, which describes the ex-

trinsic curvature, and the gauge connection in the normal

bundle, Aij� ¼ �Aji

�. They are determined by the equa-tions of Gauss and Weingarten, respectively,

r�xm� � @�x

m� þ �m

npxn�x

p� � ��

��xm� ¼ Hi

��Nmi ;

(B3)

r�Nmi � @�N

mi þ �m

npxn�N

pi � Aj

i�Nmj ¼ �Hi�

�xm�:

(B4)

As is evident here, by using the appropriate connections,r� denotes the covariant derivative with respect to allindices. The integrability conditions of the differentialEqs. (B3) and (B4) are the equations of Gauss, Codazziand Ricci, which are, respectively,

Rmnpqxm�x

n�x

p�x

q� ¼ R�� þHi

��Hi�� Hi��Hi��;

(B5)

Rmnpqxm�x

n�N

pi x

q� ¼ r�Hi�� r�Hi��; (B6)

Rmnpqxm�x

n�N

pi N

qj ¼ Fij�� Hi�

�Hj�� þHi��Hj��;

(B7)

where Fij�� is the field strength in the normal bundle,

Fij�� ¼ @�Aij� � @�Aij� þ Aik�Akj� � Aik�A

kj�: (B8)

As mentioned before, the covariant derivative in (B6)

contains also the connections Aji�.

Let us derive the expression for the pullback of thespinor bulk covariant derivative on the world volume ofthe brane, which is needed in Sec. III B,

xm�rm ¼ xm�

�@m þ 1

4!m

np�np

�: (B9)

The bulk spin connections can be obtained by

!mnp ¼ �e

pqð@meqn þ �q

mpepnÞ; (B10)

and similarly for the world volume spin connections.Let us pick a local frame adapted to the embedding,

emn ¼8<: xm�e

�� for n ¼ �;

Nmi for n ¼ i:

(B11)

Then, using (B3) and (B4), it is straightforward to showthat

xm� ð@meqn þ �qmpe

pnÞ

¼8<:H

i��N

qi e

�� þ!��e

��xq� for n ¼ �;

�Hi��xq� þ Aj

i�Nqj for n ¼ i:

(B12)

Hence, one finds for the pullback of the bulk spin con-nections

xm�!m�� ¼ !��; xm�!m�i ¼ �Hi��e��;

xm�!mij ¼ Aij�: (B13)

Consequently, (B9) becomes

xm�rm ¼ r� � 1

2Hi��

��i þ 1

4Aij��

ij: (B14)

APPENDIX C: SCALAR HEAT KERNEL ON AdS2

Here we show an explicit derivation of the scalar heatkernel on AdS2 using Poincare coordinates and verify thatit coincides with the calculation done in global coordinates[38].We begin by finding the eigenfunctions and eigenvalues.

Consider the AdS2 metric in Poincare coordinates

ds2 ¼ dx2 þ dy2

y2: (C1)

The Laplacian reads

h ¼ y2ð@2x þ @2yÞ: (C2)

Assuming a dependence of the form eikx, the spectralproblem becomes

� y2ð@2y � k2Þ�ðk;ÞðyÞ ¼�2 þ 1

4

��ðk;ÞðyÞ; (C3)

where we have written the eigenvalues as 2 þ 1=4.


106015-17

The two independent solutions to this equation are

�ð1Þðk;ÞðyÞ ¼

ffiffiffiy

pLiðjkjyÞ and �ð2Þ

ðk;ÞðyÞ ¼ffiffiffiy

pKiðjkjyÞ

(C4)

where

L�ðzÞ ¼ i�

2

I��ðzÞ þ I�ðzÞsinð��Þ (C5)

K�ðzÞ ¼ �

2

I��ðzÞ � I�ðzÞsinð��Þ (C6)

and I� is the modified Bessel function of the first kind. Ofcourse, K� is the usual modified Bessel function of thesecond kind. It is better to consider L� and K� (as opposedto I� and K�) as independent solutions since they are bothreal when the order is imaginary and the argument real.

If is purely imaginary, both solutions fail to be squareintegrable. For real , the asymptotic behavior as y ! 0þis

LiðyÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�

sinhð�Þr

½ cosð lnðy=2Þ � cÞ þOðy2Þ�;(C7)

KiðyÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�

sinhð�Þr

½ sinð lnðy=2Þ � cÞ þOðy2Þ�;(C8)

where c is a constant, and

LiðyÞ ¼ 1

sinhð�Þffiffiffiffiffi�

2y

sey�1þO

�1

y

��; (C9)

KiðyÞ ¼ffiffiffiffiffi�

2y

se�y

�1þO

�1

y

��; (C10)

when y ! 1. From this we see that both solutions vanish

as we approach the boundary y ¼ 0, but only �ð2Þðk;Þ van-

ishes as y ! 1. In other words, only �ð2Þðk;Þ is square

integrable.The relation (see Kontorovich-Lebedev transform)Z 1

0dy

Ki�ðyÞKiðyÞy

¼ �2

2� sinhð��Þ�ð�� Þ; (C11)

sets the normalization of the eigenfunctions as

fðk;Þðx; yÞ ¼ 1ffiffiffiffiffiffi�3

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinhð�Þp

eikxffiffiffiy

pKiðjkjyÞ; (C12)

where k 2 R and 0.Now, the diagonal heat kernel is

Kððx; yÞ; ðx; yÞ; tÞÞ ¼Z

dkdeð2þð1=4ÞÞtf�ðk;Þðx; yÞfðk;Þðx; yÞ(C13)

Using the above eigenfunctions this is

Kððx; yÞ; ðx; yÞ; tÞÞ ¼ 1

�3

Z 1

0de�ð2þð1=4ÞÞt sinhð�Þ

�Z 1

�1dkyKiðjkjyÞ2

(C14)

¼ 2

�3

Z 1

0de�ð2þð1=4ÞÞt sinhð�Þ

Z 1

0dkKiðkÞ2

(C15)

This does not depend on y, as expected. The norm of themodified Bessel function isZ 1

0dxKiðxÞ2 ¼ �

4�

�1

2þ i

��

�1

2� i

�(C16)

¼ �2

4 coshð�Þ (C17)

Therefore

Kððx; yÞ; ðx; yÞ; tÞÞ ¼ 1

2�R2

Z 1

0de�ð2þð1=4ÞÞt tanhð�Þ

(C18)

This is the same expression one gets when working withglobal coordinates on the disk.

APPENDIX D: INTEGRALS AND INFINITE SUMS

We will perform here the evaluation of the integrals andinfinite sums needed for the heat kernel calculations ofbosons and fermions in section V.For the bosons, let us start with the infinite sum (5.33),

�sðtÞ ¼ X1l¼0

�lþ 1

2

�e�ðlþ1=2Þ2t: (D1)

Converting the sum into a contour integral that picks upsuitable poles, as outlined in [2], one obtains

�sðtÞ ¼ ImZ ei 1

0d tanð�Þe�2t: (D2)

Here, 0< � 1, so that Im > 0 in the integrand. Now,we write tanð�Þ ¼ i tanhð�i�Þ and expand the tanh as

tanhð�Þ ¼ 1� 2X1k¼1

ð�1Þkþ1e�2�k (D3)

to obtain

tanð�Þ ¼ i

�1� 2

X1k¼1

ð�1Þkþ1e2�ik�: (D4)

The integral in (D2) can be done exactly for the first term ofthe expansion, while in the remaining terms we expand

e�2t as a power series in t, integrate and perform thesummation over k. The result is [cf. (2.18) of [38]]


106015-18

�sðtÞ ¼ 1

2tþ 2

X1n¼0

ð2nþ 1Þ!n!ð2�Þ2nþ2

tnð1� 2�2n�1Þ�ð2nþ 2Þ

¼ 1

2tþ 1

2

X1n¼1

tn�1

n!ð1� 21�2nÞjB2nj

¼ 1

2tþ 1

24þ 7

960tþ 31

16128t2 þOðt3Þ: (D5)

On the first line, �ðsÞ denotes the Riemann zeta function,which we expressed in terms of the Bernoulli numbers B2n

on the second line.Consider now the integral in (5.30). In analogy with the

calculation above, we expand the tanh using (D3) such thatthe leading term is captured by the integral over the firstterm of the expansion. For the remaining terms, expand

e�2t as a power series in t, integrate and perform thesummation over k. The result is [cf. (2.15) of [38]]Z 1

0dtanhð�Þe�2t¼ 1

2t�1

2

X1n¼1

ð�tÞn�1

n!ð1�21�2nÞjB2nj

¼��sð�tÞ: (D6)

Again, we have expressed the Riemann zeta functions interms of Bernouuli numbers, and the last equality resultsfrom a direct comparison with the second line of (D5).

Similar calculations must be done for the fermion con-tributions. Consider the infinite sum (5.66)

�fðtÞ ¼ X1l¼0

le�l2t: (D7)

Converting the sum into a contour integral, one obtains

�fðtÞ ¼ �ImZ ei 1

0d cotð�Þe�2t: (D8)

Write cotð�Þ ¼ �i cothð�i�Þ and expand the coth as

cothð�Þ ¼ 1þ 2X1k¼1

e�2�k (D9)

to obtain

cotð�Þ ¼ �i

�1þ 2

X1k¼1

e2�ik�: (D10)

Continuing as for �sðtÞ, we obtain [cf. (3.3.16) of [38]]

�fðtÞ ¼ 1

2t� 1

2

X1n¼1

tn�1

n!jB2nj

¼ 1

2t� 1

12� 1

120t� 1

504t2 þOðt3Þ: (D11)

Finally, an analogous calculation for the integral in(5.63) yields

Z 1

0d cothð�Þe�2t ¼ 1

2tþ 1

2

X1n¼1

ð�tÞn�1

n!jB2nj

¼ ��fð�tÞ: (D12)

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