Black hole partition function using the hybrid formalism of superstrings

Black hole partition function using the hybrid formalism of superstrings

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona, 145, 01405-900, Sa˜o Paulo, Brasil (Received 24 January 2011; published 20 July 2011)

The IIA superstring partition functionZIIA, on a EuclideanAdS2S2CY3computes the modified

elliptic genusZBH of the associated black hole. The hybrid formalism of superstrings onAdS2S2,

defined as a sigma model on the coset supermanifoldPSU_ð1;1_j2_Þ=U_ð1_Þ U_ð1_Þ with a Wess-Zumino term, together with Calabi-Yau and chiral boson conformal field theories, is used to calculate the partition function of IIA superstrings on the Euclidean attractor geometryAdS2S2CY3. Instead of the kappa

symmetry analysis used by Beaselyet al.in Ref. [33], we use world-sheet superconformal invariance to construct a nilpotent Becchi, Rouet, Stora, Tyutin (BRST) operator. The sigma model action is explicitly shown to be closed under this BRST operator. Localization arguments are then used to deform the world-sheet path integral with the addition of a BRST exact term, where contributions arise only from the center ofAdS2and the north and south poles ofS2. This leads to the Ooguri, Strominger, and Vafa result

ZBH¼ZIIA¼ jZtopj2, wherejZtopj2is the square of the topological string partition function.

DOI:10.1103/PhysRevD.84.026005 PACS numbers: 11.25.Mj, 04.70.Dy, 11.25.Hf

I. INTRODUCTION

In [1], Ooguri, Strominger, and Vafa (OSV) conjectured a relationship of the form

Z_{BH¼ j}Z_topj2_{; (1)}

where ZBH is the (indexed) entropy of four-dimensional Bogomol’nyi-Prasad-Sommerfeld (BPS) black holes in type II Calabi-Yau compactifications andZ_topis the square of the topological string partition function evaluated at the attractor point on the associated Calabi-Yau. One way to view the relationship is to think of it as an asymptotic expansion in the limit of large black hole charges. In this limit, ZBH receives all order perturbative contributions from the F-term corrections in the low energy effective action of theN ¼2supergravity. For the type IIA super-strings on Calabi-Yau 3-folds, these are of the general form

Z

d4_xd4_ðW

WÞgFgðXÞ; (2)

where X _{are the vector multiplet fields and the Weyl} superfield W _{involves the graviphoton field strength} and the Weyl tensor. TheseF-term corrections are in turn captured by the topological string amplitudes F_g [2,3]. Taking into account the fact that the BPS black hole entropy gets corrected in the presence of these terms via attractor mechanism [4–9], and that the supergravity par-tition function defines a mixed thermodynamic ensemble, provides the link (1) [1]. Several refinements of the rela-tionship (1) have been suggested in the literature [10–27]. In particular, in [28], an M-theory lift was used to calculate the black hole partition function at low temperatures as a dilute-gas sum over BPS wrapped membranes in AdS3S2CY3, via the Maldacena, Strominger, Witten (MSW) conformal field theory (CFT) [29] and agrees with the Gopakumar-Vafa partition

function [30,31] (see related discussion in [32]). The rela-tion (1) is then obtained after doing a modular transforma-tion to go to high temperatures. Further, in [33] a com-putation of the black hole partition function without the need for a modular transformation was presented. In this method, first anM-theory lift of the IIA attractor geometry leads to a quotient ofAdS₃S2_CY

3, whose asymptotic boundary is a torus. Then, usingAdS₃=CFT₂ duality, it is shown that the partition function of IIA theory on the attractor geometry AdS₂S2_CY

3, denoted as ZIIA, is equivalent to the partition function of the black hole. Starting from the Euclidean Calabi-Yau attractor geometry for the black hole, carrying D0-D2-D4 charges q₀, q_A (electric),pA_{(magnetic), respectively, it was argued that [33]}

Z_IIAð0_; A_{; p}A_{Þ ¼Z} BH

¼Tr_{Rð Þ}F_eð ð42₌0_ÞÞL

0 qAðA=0Þ; ₍₃₎

where0_,A_{are potentials conjugate to the D0-D2 charges.} The trace on the right-hand side runs over the black hole microstates computed in the CFT on the boundary torus dual to theM-theory lift of the IIA attractor geometry. In essence, the black hole partition function can be evaluated in terms of the type IIA partition function onAdS₂S2_CY

3 geome-try [34].

cycles in the Calabi-Yau geometry and sit at the center of AdS₂ and (north) south poles ofS2_{. The instantons} typi-cally break some supersymmetries and one has to integrate over the resulting zero modes in the path integral. The analysis of [33] involved regularizing the divergence com-ing from such integrals uscom-ing-symmetry (see [36–38] for a different approach and [39–41] for recent work). Furthermore, the Calabi-Yau part of the instanton partition function was assumed to produce the topological string partition function and using some input from supergravity, the relation (1) was obtained.

The purpose of this paper is to use the hybrid formalism [42] to calculate the partition function of IIA superstrings onAdS₂S2 _{with Ramond-Ramond (RR) flux. There are} several advantages of this approach over the Green-Schwarz formalism. First, the calculation of scattering amplitudes in the hybrid formalism can be performed in a super-Poincare´ covariant manner and is relevant for describing d_¼4, N_¼2 theories. The calculation of superspace low energy effective actions in Calabi-Yau compactifications and their connection to topological string amplitudes can be derived nicely [43,44]. Another important advantage is that studying string propagation in Ramond-Ramond backgrounds is in general difficult in the traditional Ramond-Neveu-Schwarz (RNS) formalism due to the need to introduce spin fields. In the Green-Schwarz formalism, covariant quantization is problematic, except in light-cone gauge. To address these perennial issues, several covariant quantization techniques have been developed over the last few years by Berkovits and collaborators. In ten dimensions, the pure spinor formalism [45] has led to a remarkable simplification of the calculation of superstring loop amplitudes in flat space and their equivalence to the RNS formalism has been fully demonstrated (see e.g. [46–50] and the nonminimal approach [51–54]). The for-mulation of the pure spinor superstring in AdS₅S5 background [45,55,56], following earlier studies on hybrid formalism in AdS₃S3 _{[57,58] and in AdS}

2S2 [42] backgrounds, is being actively used in a world-sheet ap-proach to the Maldacena conjecture [59–61].

There have been innumerable applications of the hybrid formalism of superstrings in the presence of RR back-grounds; to list a few, scattering amplitudes in lower di-mensions [44,62], in obtaining C-deformation [63] from superstrings in graviphoton backgrounds [64], emergence of non(anti)commutative superspace [65,66], for Calabi-Yau compactifications to two dimensions [67], and more recently in the study of flux vacua from the world-sheet point of view [68,69].

We use the formulation of hybrid superstrings in the near horizon geometry of extremal black holes in four dimen-sions, presented in [42]. It was shown that two-dimensional sigma models based on the coset supermanifold PSU_Uð1Þð1;1_Uj_ð2₁Þ_Þ with the Wess-Zumino term provide an elegant description of superstrings propagating on AdS₂S2 _background

with Ramond-Ramond flux. It is also desirable to have a world-sheet perspective on our understanding of the derivation of OSV relation (1); as mentioned earlier, the world-sheet approaches to largeNtopological string dual-ities are playing a special role in our understanding of the Maldacena conjecture [61]. For the present case, the con-nection of low energy F-terms and the topological string amplitudes can be understood nicely in the hybrid ap-proach [44] (see [70,71] for a review). More importantly, in the hybrid approach, the dilaton in type II theories couples to the N_{¼ ð}2;2_Þ world-sheet supercurvature via the Fradkin-Tseytlin term in the action [43]. This has the advantage that the scattering amplitudes have a well-defined dependence on the string coupling constant. This will be important while trying to get the right factors of topological string coupling in the IIA partition function.

The rest of the paper is organized as follows. In Sec.II, we review the hybrid formalism of superstrings in flat space. In Sec. III, we discuss the _UPSU_ð1Þð1;1_Uðj2₁Þ_Þ sigma model and write down the lowest order terms in the AdS2S2 background in theU_ð1_Þ U_ð1_Þnotation, using the expres-sions for left-invariant currents. In Sec. IV, we present various partition functions in AdS2S2CY3 related to the black hole partition function. In Sec.V, we present the computation of type IIA partition function over AdS₂S2_CY

3 by embedding the theory in N¼4 topological strings. We provide localization arguments using the Becchi, Rouet, Stora, Tyutin (BRST) method and obtain the OSV relation.

II. HYBRID FORMALISM OF SUPERSTRINGS

The fundamental definition of N_¼2 superconformal theory describing hybrid superstrings in four dimensions is by a field redefinition ofN _¼1RNS matter and ghost variables [70,72–76]. The N_¼2,c_¼6 system obtained after field redefinition is twisted and splits in to a four-dimensionalc_¼ 3,N_¼2superconformal field theory, coupled to an internal six-dimensionalc_¼9,N_¼2 super-conformal theory. Now, one can proceed in two ways to define physical states and calculate scattering amplitudes. Thec¼6,N ¼2generators can be untwisted and coupled to a c¼ 6,N¼2ghost system to write down a BRST operator. Another way is to embed this N¼2 system in small N¼4 algebra and use the N¼4 topological method, where there is no need to introduce ghost variables. The general four-dimensional and six-dimensional actions in arbitrary curved backgrounds with Ramond-Ramond flux have been discussed in [43,57], respectively. Below, we review the flat four-dimensional case and consider the AdS₂S2_{case in the next section.}

Flat space-time

[42]. The four-dimensional part consists of four bosonic and 16 fermionic space-time variables given as Xm _for m_¼0to 3 and_ð

L; pLÞ,ðL_;pLÞ ð_ R; pRÞ,ðR_;pRÞ_ , where , _ _¼1;2. The six-dimensional part has six bosonic variablesYj_,Y

jforj¼1to 3 and their associated superpartners, left-moving fermions_ðcj_L;c_LjÞ and right-moving fermions _ðcj_R;c_RjÞcorresponding to the Calabi-Yau CFT. Finally, one has a chiral boson _L and an antichiral boson_R. The world-sheet action corresponding to these fields in the superconformal gauge can be divided into three parts as

Sd¼4þSCYþSL;R: (4)

The four-dimensional part of the action in flat superspace is

S_{d¼4 ¼} 1 0

Z

dzdz

₁

2@X m_@X

mþpL@ LþpL_@L_

þp_R@

RþpR_@R_

: (5)

The Calabi-Yau part is

S_{CY ¼} 1 0

Z

dzdz_ð@Y j_@Y

jþcjL@cLjþcjR@cRjÞ; (6)

andS

L;R stands for the action of the chiral and antichiral bosons. For the present case, all the world-sheet fields satisfy periodic boundary conditions. The free field operator-product expansions (OPEs) corresponding to the above action are

Xm_ðyÞXn_ðzÞ! 0mn_{lnjy zj}2_;

p_Lðy_ÞLðz_Þ!0

y z; pLð_ yÞ _

Lðz_Þ!0 _

_ y z;

p_Rðy_ÞRðz_Þ!0

y z; pRð_ yÞ _

Rðz_Þ!0 _

_ y z;

_Lðy_ÞLðz_Þ! ln_ðy z_Þ; _Rðy_ÞRðz_Þ! ln_ðy z_Þ: (7)

The world-sheet action (4) is manifestly conformally in-variant from the conformal dimensions of the fields. The action (4) further has anN_¼2superconformal invariance realized nonlinearly, the left-moving generators of which are1

TL_¼1 2@X

m_@X

mþpL@LþpL_@L_ þ 0

2@L@LþT L CY

G L_¼ 1_{ffiffiffiffiffi} 0

p eLðdLÞ2þG L

CY GþL¼ 1

ffiffiffiffiffi

0

p e LðdLÞ2þGþL CY

JL_¼0_@

LþJLCY (8)

where

d_L¼p_Lþim

_L_@Xm 1 2ðLÞ

2_@ Lþ

1

4L@ðLÞ 2_;

dL_ ¼pL_ þim_L@Xm 1 2ðLÞ

2_{@L_ þ}1

4L_@ðLÞ 2_;

(9)

and _ðd_LÞ2 _means 1

2dLdL. The right-moving N¼2 generators are given as

TR_¼1 2@X

m_@X

mþpR@ RþpR_@R_ þ 0

2@ R@ RþT R CY

G R_¼ 1_{ffiffiffiffiffi} 0

p eR_ðdRÞ2_þGþR CY;

GþR_¼ 1_{ffiffiffiffiffi} 0

p e R_ðdRÞ2_þG R

CY; JR¼0@ R JRCY (10)

with

d_R¼p_Rþim

_R_@X m 1 2ðRÞ

2_@ Rþ

1

4R@ðRÞ 2_;

dR_ ¼pR_ þim_R@X m 1 2ðRÞ

2_{@R_ þ}1

4R_@ðRÞ 2_:

(11)

The signs on the fermionic generators GL;R _represent their U_ð1_Þ charges with respect to JL;R_{. Also,} ½T_CYL;R; GL;R_CY; GL;R_CY; JL;R_CYŠ are the left- and right-moving N _¼2 c_¼9 generators describing the Calabi-Yau 3-fold. We have written the left- and right-moving algebras explicitly for the type IIA superstring. The type IIB generators can be obtained by noting that under mirror symmetry,JR

CY! JCYR , resulting in interchangeGþCYR! G R

CY. The space-time part remains unaffected by this change.

The set of operatorsd_L,d_{L_}, as shown in [77], satisfy the algebra

dLðyÞdLð_ zÞ !2i0 _L_

y z; dLðyÞdLðzÞ !reg;

d_{Lð_} y_Þd_Lð_ zÞ !reg; (12)

with similar expressions for the right-moving part. Here

m L ¼@Xm

i 2

m

ð_ L@L_ þL_@LÞ;

m R ¼@X m

i 2

m

ð_ R@R_ þR_@ RÞ: (13)

The operatorsdL,dL_ also commute with the space-time supersymmetry generators of the action

1_{The superscriptsL,Ron various generators stand for left- and}

right-moving parts of the algebra. When these superscripts are not indicated explicitly, we always mean the left-moving part. We also adopt the notationjAj2_¼A

q_L¼Z dzQ_L

whereQ_L¼p_L im

_L_@Xm 1 4ðLÞ

2_@ L;

q_{L_ ¼}Z dzQ_{L_}

whereQ_{_ ¼}p_{L_} im

_L@Xm 1 4ðLÞ

2_{@L_: (14)}

Further,q_R andq_{R_} are obtained by replacingLwithR and@with@. Thus, Eqs. (9) and (14) contain the world-sheet versions of the space-time superspace covariant derivatives and supersymmetry generators, respectively.

III. HYBRID FORMALISM INAdS₂S2

The four-dimensional part of the actionSd¼4 in Eq. (4) can be replaced with the world-sheet action for the sigma model on AdS₂S2_{. Below, we discuss the AdS}

2S2 model in detail. In [42] it was shown that the two-dimensional sigma models based on the coset supermani-fold PSU_ð1;1_j2_Þ=U_ð1_Þ U_ð1_Þ can be used to quantize superstrings on AdS₂S2 _{in RR backgrounds. One of} the main advantages of the hybrid approach is that four-dimensional super-Poincare´ invariance and theN_¼2 tar-get space-time supersymmetry can be made manifest even in the presence of RR fields. For instance, like in Green-Schwarz formalism, the model has fermions which are spinors in target space and scalars on the world sheet allowing a simple treatment of RR fields. Second, like in RNS formalism, the action is quadratic in flat space allow-ing for quantization. As mentioned earlier, a first principle definition of the world-sheet variables in hybrid formalism is through a redefinition of RNS world-sheet variables. This follows from the fact that any criticalN _¼1 string can be embedded in anN _¼2string [73].

For type II superstrings, the four -symmetries of the Green-Schwarz action are replaced by a critical world-sheet N_{¼ ð}2;2_Þ superconformal invariance. These local N_¼2 superconformal generators are nothing but the twisted RNS world-sheet generators. The important differ-ence with the RNS approach is that the vertex operators for RR fields do not have square-root cuts with supersymmetry generators and there is no need to sum over spin structures on the world sheet [70].

A.AdS₂S2

sigma model

TheAdS₂S2 _{sigma model action is constructed as a} gauged principle chiral field [42,78]. Letg_ðx_{Þ 2}Gdenote a map from the world sheet into the supergroupG, with the currentJ_¼g 1_{dgvalued in the Lie algebraG. The sigma} model action on the coset space is constructed by gauging a subgroup H, whose Lie algebra is H0. To construct a string theory onAdS₂S2_{, one has to quotient the group} GPSU_ð1;1_j2_Þ by (the right action of ) bosonic

subgroupsHU_ð1_Þ U_ð1_Þ. This subgroup is in fact the invariant locus of a Z_{4 automorphism, the existence of} which plays a crucial role in the construction. In effect, a Z_{4decomposition of thePSUð1;1j2ÞLie algebraGcan be} written as [42]

G ¼H0H1H2H3;

where the subspaceHk is the eigenspace of theZ4 gen-eratorwith eigenvalue_ði_Þk_{. The subspacesH}

1andH3 contain all the fermionic generators of the algebraG and

H0,H2contain the bosonic generators ofG.

The AdS2S2 sigma model action is a sum of the action on the coset supermanifold PSU_ð1;1_j2_Þ=U_ð1_Þ Uð1Þand a Wess-Zumino term and is given as2[42]

S_AdS

2S2¼ 1 2

Z

d2_xStr1 2J2J2þ

3 4J1J3þ

1 4J1J3

; (15)

where the left-invariant currents are given as J_{2 ¼ ð}g 1_@gÞm_T

m;

J1 ¼ ðg 1@gÞRTRþ ðg 1@gÞR_TR_;

J3 ¼ ðg 1@gÞLTLþ ðg 1@gÞL_TL_;

(16)

where T_m,T_;R;L_{_} denote thePSU_ð1;1_j2_Þgenerators corre-sponding to translations and supersymmetry. J’s can be obtained by replacing@with@and Str stands for supertrace over the PSU_ð1;1_j2_Þ matrices, the algebra of which is given in Eq. (28). The action in Eq. (15) has invariance under globalPSU_ð1;1_j2_Þtransformations, which are real-ized by left-multiplication asg_{¼ ð}A_{TAÞg. There is also} an invariance under localU_ð1_Þ U_ð1_Þgauge transforma-tions, which is realized ong_ðx; _Þby a right-multiplication asg_¼g_ðx_Þ. The supersymmetry transformations (up to a local Lorentz rotation) of the superspace coordinates are achieved by a global left action of the superalgebra on the coset superspace, whereas the -supersymmetry corre-sponds to a local right action on the same coset superspace. The -supersymmetry is explicitly broken in the present case and is replaced by world-sheet superconformal invari-ance [79].

It is useful to write the action by introducing auxiliary fields as

S_AdS

2S2¼ 1 0

Z

d2_z1 2cdJ

c_Jd

1 4Ngs

LRðJ

L_JR _JL_JR_Þ

1

4Ng_s_L_RðJ _

L_JR_

JL_ _JR_ _Þþd LJ

L

þd_LJ _ L_þd

RJ R_{þd_}

RJ _ R_þNg

sdLdR LR

þNgsd_Ld_R _ LR_

: (17)

2_{We use the notationJJ}

This form of the action can be obtained by writing the flat space-time action in (5) in terms of manifestly supersym-metric variables of Eqs. (13) and (21), and then covarian-tizing in a curved background [42]. Using the following constraint equations

d_{L ¼} 1

Ng_sLRJ

R; d R ¼

1

Ng_sLRJ

L; ₍₁₈₎

and performing the scaling

Ec

M! ðNgsÞ 1EcM; E L;R M ;E

_ L;R

M ! ðNgsÞ ð1=2ÞE L;R M ;E

_ L;R M ;

(19)

one arrives at the action

S_AdS2S2 ¼ 1 0_N2_g2

s

Z

d2_z

₁

2cdJ c_Jd 1

4LRðJ L_JR

þ3 JLJRÞ 1

4_L_RðJ _

LJ_Rþ3 J_LJ_RÞ

:

(20)

The abovePSU_ð1;1_j2_Þinvariant action can be written in a more familiar form for type IIA superstrings on AdS₂S2 _{with Ramond-Ramond fluxes in a manifestly} space-time supersymmetric manner using the following identifications [42]:

Jc c_{; J}L;R L;R_{; J}L;R_ L;R_ _{; (21)}

and similarly forJ’s as well. Here,A

j ¼EAM@jZM,ZM ¼ ðXm_;

L; _ L;

^ R;

^_

RÞ, supervierbein EMA and A takes the tangent-superspace values _½c; _L;__L; _R;__RŠ. The full action is now

S_¼S_AdS

2S2þSCYþSL;R: (22) We now collect theN¼2generators in this notation

T_¼T_d¼4þT_CY; G _¼G_d¼4þG_CY;

Gþ¼Gþ_d¼4þGþ_CY; J¼Jd¼4þJCY; (23)

where

T_d¼4¼c

zczþdzþd_z_ þ 1 2@ @; G_d¼4¼e_ðdÞ2_{; G}þ

d¼4¼e ðdÞ2; Jd¼4¼@ (24) represent the generators of c_¼ 3, N_¼2 algebra describingAdS₂S2_{, and}

T_CY¼@Y j_@Yjþ1 2ðc

j_@cjþcj@c j_Þ;

G_CY¼cj@Y_j; Gþ

CY¼ þcj@Y j; JCY¼cjcj; (25) represent the c_¼9, N_¼2 generators describing the Calabi-Yau 3-fold. The generators ofAdS₂S2 _algebra, given in Eq. (24), are similar to the ones of flat space-time given in Eqs. (8) and (10), except for the constraints

satisfied by din Eq. (18). One can use the equations of motion in (18) to write the four-dimensional generators explicitly in terms of the left-invariant currents. We also note that the fermionic generators in (24) remain holomor-phic even at the quantum level [42]. The generators in Eq. (23) correspond to a c_¼6, N_¼2 algebra. In Sec.IV, we topologically twist this algebra and embed it in anN_¼4algebra.

B.AdS₂S2

action inU_ð1_ÞU_ð1_Þnotation

For the localization arguments to be presented in Sec.V C, it will further be useful to write the action (20) in a manner in which theU_ð1_Þ U_ð1_Þcharges are manifest. This notation is quite useful, as most of the algebraic relations can be easily obtained just from constructing U_ð1_Þ U_ð1_Þ invariant expressions. The sigma model on the coset space G=Hwith the Wess-Zumino term is one-loop conformally invariant provided that the group G is Ricci flat andHis the fixed locus of aZ4automorphism of G. For instance, J1 decomposes under H¼Uð1Þ Uð1Þ as3

J₁ Jþþ

R ; JR ; JRþ; JþR : (26) Now, in terms of the decomposition given in (26), the fermionic generators of theN¼2superconformal algebra inAdS₂S2_{can be constructed as [42]}

G R d¼4¼

Z

dz_ðeRJþþ

R JR Þ; Gþd¼R4¼

Z

dz_ðe RJþ R JRþÞ

(27)

and also the left-moving part involvingJ3_{. The} transforma-tions of various fields under these generators are obtained by a global right-multiplication ofg. Note that unlike in flat space, the currents have dependence on both holomorphic and antiholomorphic coordinates.

Let us start by writing down the PSU_ð1;1_j2_Þ algebra relations. The generators of the bosonic subgroup SO_ð2;1_Þ SO_ð3_ÞareM,Tþ

þ0,T 0,N,T0þþ,T0 , where the

subscripts denotes charges underU_ð1_Þ U_ð1_Þ. The fermi-onic generators are Ti

þþ, Ti and Tþi , Tiþ, where i,

j_¼L,R. Thus, the relevant relations of thePSU_ð1;1_j2_Þ algebra are

½Tþ

þ0; T 0Š ¼M; ½T0þþ; T0 Š ¼N;

fTi

þþ; Tþj g ¼ijTþ þ0; fT

i

þþ; Tjþg ¼ijT0þ þ;

fTi _{; T}j

þ g ¼ijT0 ; fTi ; Tj_þg ¼ijT 0;

fTi

þþ; Tj g ¼ijðN MÞ; fTiþ; Tþj g ¼ijðNþMÞ;

(28)

3_{The fermionic generators have appropriate flat-space limits;}

for instance, J_þþ; !d,Jþ ; þ!d_. We take the þ þ,

where LR_¼1¼ RL_{. Using the above generators, the} action in (20) can be written as

S_AdS

2S2

¼_0N12_g2

Z

d2_z1 2ðJ

þ

þ0J 0 J0þþJ0 þJ 0Jþþ0 J0 J0þþÞ

1

4ðJL JþþR þ3 JL JRþþþJþþL JR þ3 JþþL JR Þ 1

4ðJþL JRþþ3 JþL JRþþJLþJþR þ3 JLþJþR Þ

:

(29)

This action and its equations of motion are important to the localization arguments to be presented in Sec. V C. Using the variation of a general current J as

J_¼dX_{þ ½}J; X_Š, we obtain the equations of motion of fermionic currents, following from (29) as

rJ_{L ¼}0; _rJþþ

L ¼0; rJLþ¼0;

rJþ

L ¼0; rJR ¼0; rJþþR ¼0;

rJ þ

R ¼0; rJRþ ¼0;

(30)

where we have only written the equations for covariantly holomorphic and antiholomorphic currents. In Eq. (30), rJ_¼@J_{þ ½}J0; JŠwith J0 denoting the generators in the subspace H0, stands for the covariant derivative, and similarly for _r. Using the parameterization of the group element as g_¼exp_ðXT_þLT_LþRT_RÞ, the left-invariant currents are

Jþþ0¼@Xþþ0þ1

2ð@þþL þL þ@þL þþL þL$RÞ þ ;

J 0_¼@X 0_þ1

2ð@L Lþþ@LþL þL$RÞ þ ;

J0þ

þ¼@X0þþþ1

2ð@þþL Lþþ@LþþþL þL$RÞ þ ;

J0 _¼@X0 _þ1

2ð@L þL þ@Lþ L þL$RÞ þ ; (31)

and the barred ones can be obtained similarly. The left-handed currents for instance are

Jþþ

L ¼@þþL þ 1 2ð @X

þ

þ0_Rþþ@_RþXþþ0þ@X0þþþ_R @þ_R X0þþÞ þ ;

J_{L ¼}@_{L þ}1 2ð @X

0þ

R þ@þR X 0þ@X0 Rþ @RþX0 Þ þ ;

Jþ

L ¼@þL þ 1 2ð@X

þ

þ0_R @_R Xþþ0 @X0 þþ_R þ@þþ_R X0 Þ þ ;

J þ

L ¼@Lþþ 1 2ð@X

0þþ

R @þþR X 0 @X0

þ

þ_R þ@_R X0þþÞ þ ; (32)

and the right-handed currents and barred ones can be obtained similarly. Thus, the first few terms relevant for us in the action (29) are

S_¼Z d2_zð@Xþ

þ0@X 0 @X0þþ@X 0 @þþ_L @ _R @_L @ þþ_R þ@þ_L @ _Rþþ@_Lþ@ þ_R þ Þ: (33)

The OPEs corresponding to the action (33) are

Xþþ0ðyÞX 0ðzÞ lnjy zj; X0þþðyÞX0 ðzÞ þlnjy zj; þþ_L ðyÞ_R ðzÞ lnjy zj;

_{L ð}y_Þþþ

R ðzÞ lnjy zj; þL ðyÞRþðzÞ lnjy zj; LþðyÞþR ðzÞ lnjy zj: (34)

Notice that the auxiliary fields have been integrated out in the action in (33). There are new OPEs among the fermi-onic coordinates in the AdS₂S2 _{background, as in the} AdS3S3case [58].

IV. BLACK HOLE AND TOPOLOGICAL STRING PARTITION FUNCTIONS

In this section, we give details of the black hole and topological string partition functions to be used in the

A.N_¼2_{topological string partition function}

To define the N_¼2 topological strings, we use the generators of two copies of thec_¼9,N ¼2algebra of Calabi-Yau, given in Eq. (25):TL

CY, GCYL, GþCYL, JLCY and TR

CY, GCYR, GþCYR,JCYR , where all the fermionic generators GL;R_CY have spin3₂. Now, we perform a twist which shifts the U_ð1_Þ charges of the fields, leading to a topological theory. An A-twist corresponds to the choice of shifts, TL

new;CY¼TCYL 12@J CYL and Tnew;CYR ¼TCYR þ12@JRCY. This results in the shift of spinshL_,hR _{of the generators} of left- and right-moving algebras ashL

new¼hL 12qLand hR

new¼hRþ12qR, where qL, qR are the Uð1Þ charges, respectively. Note that T_new;CYðy_ÞT_new;CYðz_Þ OPEs have no central charge in the twisted N¼2 system, and all bosonic and fermionic world-sheet fields have integer spin. However,Gþ_{CYðyÞGCYðzÞOPEs have a central charge}

given asc^_¼3[2]. As a result of this twisting,G L CY,GþCYR acquire spin 2 andGþL

CY,GCYR acquire spin 1. TheN¼2 topological string amplitude is then given as [2]

F_g¼Z Mg

Y

3g 3

i¼1

G_CYðiÞ

2

; (35)

where_hjG_CYðÞj2_{icorresponds tohG} L

CYðÞGþCYRðÞi. The spin 2 generatorsG L

CY andGþCYR appear in the amplitude and behave similar to theb-ghost in the bosonic string. The spin 1 generators can be used to form a fermionic nilpotent operator, to study the cohomology of the theory. The formula (35) should also be understood as coming from coupling theN_¼2theory to topological gravity. One can define the full topological string free energy to be

F _top¼ X1

g¼0

g2g_top 2F_g; (36)

wheregtopis the coupling constant weighing the contribu-tions at different genera. Finally, the topological string partition function is defined as

Z_top¼expFtop: (37)

One can repeat the above discussion for anA-twist, corre-sponding to the choice of shifts, TL

new ¼TLþ12@J L and TR

new¼TR 12@JR, leading up to the antitopological string partition functionZtop.jZtopj2in Eq. (1) then stands for the product of the N_¼2 topological and its conjugate, the antitopological string partition functions, Z_top and Z_top, respectively.

B.N_¼4_{topological strings}

TheN_¼2topological strings discussed in the previous subsection were modeled after theN _¼0bosonic strings. In a similar vein,N_¼4topological strings consisting of a twisted small N_¼4 algebra are modeled after N_¼2 strings. The twistedN ¼4generators are

fT; Gþ_;G~þ_{; G ;G ; J}~ þþ_{; J; J g (38)}

and correspond to an algebra with central charge c^_¼2 [44]. The OPEs of the algebra are given in [44] and one sees that there are two doublets ðGþ_;G~ _{Þ andðG}~þ_{; G Þ.}

There is also anSUð2_Þfflavor which rotates these doublets and leaves the N_¼4 unchanged. Explicitly, the flavor rotation is

c

~

Gþ_ðu_{Þ ¼}u1G~þþu2Gþ; GcðuÞ ¼u1G u2G~

c

~

G _ðu_{Þ ¼}u

2G~ u1G ; GcþðuÞ ¼u2Gþþu1G~þ; (39) where ju1j2_{þ ju2j}2_{¼1and the complex conjugate of u}

a is ab_u

b [80]. The generators G~, appearing in Eq. (39), are defined as

~

G I J ; Gþ_{; G}~þI _Jþþ_{; G ; (40)}

where

Jþþ _expZz_{J; J exp} Zz_{J: (41)}

The flavor rotatedN_¼4topological string amplitude is given by [44]

Z_¼ X 2g 2

n¼ 2gþ2

ð4g 4_Þ!

ð2g 2_þn_Þ!_ð2g 2 n_Þ!F n

gu2g1 2þnu 2g 2 n 2

¼Z Mg

jGc_ð1Þ...Gc_ð3h 3Þj2

Z

jGc~þ_j2g 1Z _JJ;

(42)

where ’s denote the Beltrami differentials and the RJ type of insertions are needed to ensure that the path integral

does not vanish trivially. Also, the R cG~þ _{insertions in the}

path integral allow the use ofN_¼2topological strings in the calculation of amplitudes.

S_¼Z d4_xd4_ðW2_Þg_exp AðCÞ 20þi

Z

C B

;

where the sum over holomorphic curvesC, accompanied by the exponential factor denoting the area of the world sheet, defines theA-model amplitudeF_gat genusg, given in Eq. (35). This is equivalent to the computation in [44], since in flat space-time, one needs graviton-graviphoton insertions in the path integral to generate low energy F-terms.

C.Z_BH

ZBH stands for the partition function of extremal black holes in four dimensions, which are solutions of N¼2 supergravity coupled tonVvector multiplets with magnetic and electric chargesp_,q

. The asymptotic values of the moduli fieldsX_{, ¼0;1; ; n}

V of the vector multip-lets are arbitrary in the black hole solution. Near the horizon of the black hole, the values of the moduli fields are constant and fixed by the attractor equations [4]

p _{¼Re½CXŠ; q}

¼Re½CFŠ; (43) where F_¼ @F

@X and C is a complex constant. In the presence of higher derivative F-terms in the action, en-coded in the prepotential

F_ðX_{; W}2_{Þ ¼} X1 g¼0

F_gðX_ÞW2g_{; (44)}

where Fg are computed by Eq. (35) and W2 contains the square of the anti-self-dual graviphoton field strength, the attractor equations are modified due to [7–9]

C2_W2_{¼256; (45)}

in the gaugeK ¼0,C¼2Q. Here,Kdenotes the Ka¨hler potential andQis a complex combination of electric and magnetic graviphoton charges. Thus, the black hole parti-tion funcparti-tion takes the form [1]

lnZ_{BH ¼} 4Q2X g

F_g

_pþi 2Q

₈

Q

_2g

; (46)

where one usesX_{¼ ðpþi}

Þ=2Q. Here, are con-tinuous electric potentials, which are conjugate to integer electric charges q_{. As argued in [33,34], the partition} function in Eq. (46) is related toZ_IIA as in Eq. (3). The computation of Z_IIA involves taking in to account the contributions of world-sheet instantons which wrap non-trivial curves in the Calabi-Yau 3-fold [33–35].

V. TYPE IIA PARTITION FUNCTION ON

AdS₂S2

CY₃

In this section, we calculate the partition function of type IIA superstrings onAdS₂S2_CY

3 and show its connection to the N _¼2 topological string partition

function. There are at least two methods to compute the partition function of type IIA superstrings onAdS₂S2 CY₃. In the first method, using the fact that c_¼6is the critical central charge for anN_¼2matter system given in Eq. (23), one introduces a set of c_¼ 6, N_¼2ghosts, constructs an N_¼2 BRST operator, and calculates the partition function using standard N _¼2 techniques [84]. The second method is to twist the original c_¼6, N_¼2 algebra of Eq. (23), embed it in a smallN_¼4algebra and compute the partition function using theN _¼4topological techniques. This is the N_¼4 topological method we follow and is based on theN¼4topological strings dis-cussed in Sec.IV B. TheN ¼4topological method can be used to calculate the partition function of any N¼2 system which has a critical central charge ofc_¼6[44].

A. EmbeddingAdS₂S2

CY₃

inN_¼4_{topological strings}

As mentioned above, to use the N¼4 topological method in the present case, one has to embed theAdS₂ S2_CY

3 generators in the twistedN ¼4algebra. We start by explicitly denoting thec_¼6,N_¼2algebra generators given in Eq. (23) asTL_,G L_,GþL_,JL_andTR_, G R_{, G}þR_{, J}R_{, where all the fermionic generatorsG}L;R

have spin3₂. Now, there are two possible embeddings of the AdS2S2CY3 algebra in theN¼4topological alge-bra. In one case, we end up with anA-model in the Calabi-Yau geometry discussed in Sec.IVA, and in the other case, the conjugateA-model. We now discuss theA-model em-bedding. This can be achieved by the shifts TL

new¼TL 1

2@JL andTRnew¼TR 12@JR. This results in the shift of spins hL_{, h}R _{of the generators of left- and right-moving} algebras ashL

new¼hL 12qLandhRnew ¼hR 12qR, where qL_,qR _{are the Uð1Þcharges, respectively. As a result, the} generators which acquire spin 1 and spin 2 in the full AdS₂S2_CY

3 geometry are GþL, GþR and G L, G R_{, respectively. Using Eqs. (8), (10), and (23), this} translates to GþL

CY, GCYR acquiring spin 1 and GCYL, GþCYR acquiring spin 2 in the Calabi-Yau theory. This matches exactly with the A-twist discussed in Sec.IVA. Thus, we have an A-model topological string in the Calabi-Yau geometry. On the other hand, in theAdS₂S2 _geometry, it isGþL

d¼4,Gþd¼R4that acquire spin 1 andGd¼L4,Gd¼R4acquire spin 2. We will have more to say on this when we discuss the construction of BRST operator in Sec. V C. To sum-marize, the twisting of the algebra in the AdS₂S2 _is opposite to that in the Calabi-Yau geometry.

Now again the T_newðy_ÞT_newðz_Þ OPEs have no central charge in the twisted N _¼2system, and all bosonic and fermionic world-sheet fields have integer spin. However, Gþ_{ðyÞG ðzÞ OPEs have a central charge given asc^¼2.}

satisfy anSU_ð2_Þcalgebra, where the subscriptcstands for color. Under this SU_ð2_Þc, G and Gþ _{generate two new}

supercurrentsG~þandG~ . Using the definitions in Eq. (23) and the formulas in Eq. (40), we can explicitly write

~

G _¼e 2 JCYðdÞ2þe G~ CY; ~

Gþ _¼e2þJCYðdÞ2þeG~þþ

CY; (47)

whereG~_CY andG~þþ

CY are residues of the pole in the OPE of eJCY _with G

CY and e JCY with GþCY, respectively. Their explicit form will not be needed here, but can be con-structed using the methods in [68]. The full set of gener-ators for thisN¼4system is already given in Eq. (38).

B.Z_IIA

In the last subsection, we showed how the hybrid formal-ism inAdS2S2CY3 is a criticalN¼2string theory and can be embedded in the N_¼4 topological algebra. This embedding, in particular, implies that the partition function of IIA superstrings onAdS2S2CY3 can be calculated using Eq. (42).

One of the consequences of topological twisting is the existence of a nilpotent generatorQ. In terms of the twist-ing procedure discussed in last subsection, this generator can be constructed fromGþL_andGþR_{. The explicit form} of this generator will not be needed here. However, the AdS₂S2_{part of this generator is important for} localiza-tion arguments and will be discussed in Sec.V C. Because of the twisting, the type IIA path integralZ_IIAreduces to a sum of local contributions from the fixed points of the nilpotent generator Q [85]. In the Calabi-Yau geometry, these contributions come from configurations which are world-sheet instantons in theA-model and anti-instantons in the conjugateA-model. It is well known that the world-sheet instantons wrap nontrivial curves in the Calabi-Yau geometry and lead to holomorphic maps [86], given as

@Yi_{¼0: (48)}

The anti-instantons come from antiholomorphic maps in the Calabi-Yau geometry. In AdS₂S2 _{geometry, such} world-sheet instanton configurations break translational symmetries and associated supersymmetries. After a Euclidean rotation in the world-sheet action in Eq. (17), the instantonic configuration can be obtained by setting the world-sheet variablesXm_,,__{, and their right-moving} counterparts to zero.4The only nontrivial terms remaining in theAdS2S2 action (17) are

Z

ðd LdRW

LRþd _ L

d_R_ W_L_RÞ: ₍₄₉₎

To summarize,Z_IIAreduces to a sum of two contributions, coming from world-sheet instantons and anti-intantons,5 which wrap holomorphic and antiholomorphic curves in the Calabi-Yau geometry, respectively, and have legs in AdS₂S2_{. The action in Eq. (49) will be useful in the} evaluation of these contributions to the partition function. We now discuss the instanton contribution to ZIIA and denote it as ZI

IIA. Toward the end, we briefly discuss the anti-instanton contribution to the type IIA partition func-tion, denoted asZAI

IIA. This corresponds to an opposite twist to the one in Sec. VA. The final answer for Z_IIA will be obtained by joiningZI

IIA andZAIIIA.

1. World-sheet instantons

With the choice of twisting in Sec. VA, Eq. (50) below corresponds to the instanton contribution to the type IIA partition function, ZI

IIA. One way to see this is that in the Calabi-Yau geometry, we have anA-model topological string. The spin 2 generators G , G~ are like the b-ghost of the bosonic strings and appear in the partition function in Eq. (50), integrated against the3g 3Beltrami differentials. Thus using Eq. (42), the instanton contribution to the partition function of type IIA superstrings onAdS₂S2_CY

3 is

ZI IIA¼

Z

du X 2g 2

nI¼2 2g

ðu_2Þ2g 2þnI_ðu1Þ2g 2 nI

2

Y

3g 3

j¼1

Z

d2_m j

Yg

i¼1

Z

d2_v i

Y

g 1

i¼1

c

~

Gþ_{ðviÞJðvgÞ}

Z jGc

2

; (50)

where theGc~þ andGc are defined in Eq. (39). Just as in flat space [44], the integration overucan be done using

Z

du_ðu2g₁ 2þmu2g₂ 2 n_Þðu_2Þ2g 2þn_ðu1Þ2g 2 n

¼_mnð2g 2_þm_Þ!_ð2g 2 n_Þ!: (51)

To calculateZI

IIAin hybrid formalism, the world-sheet fields which we need to integrate over in the path integral are (a) AdS₂S2 _{fields: X}m_{, ð; dÞ, ð}__;d

_

Þ and their right-moving counterparts, for m_¼0; ;3 and , _ _¼ 1;2; (b) Calabi-Yau variables; and (c) the chiral boson and its right-moving counterpart.

Let us start by choosingn_{I ¼}g 1in Eq. (50), which was argued in [44], to be the piece relevant for computing

4_{For the four-dimensional part, we sometimes use the}

flat-space notation of Sec. , as the left-invariant currents of

AdS2S2 can always be written in terms of them, using the

expansions in Eqs. (31) and (32).

5_{Let us note that, to reproduce theF-terms in Eq. (2), one has}

to insert 2g 2 chiral graviphoton and two graviton vertex operators in the amplitude in flat space. In AdS2S2, the

the low energy F-terms in Eq. (2). Next, it is needed to write the partition function (50), transforming variables

from the hatted generators,Gc~þ andGc, using the defini-tions in Eq. (39). This can be done as follows. There are 3g 3Gc’s andg 1Gc~þ’s in the partition function (50). Thus, using (39),Gc’s can contribute3g 3 l G ’s and þlG~ ’s, whereasGc~þ’s can contributeg 1 l Gþ_{’s and}

þlG~þ_{’s, in the partition function. These contributions are}

further subject to the constraints coming from the back-ground charges of the U_ð1_Þ current J. For the partition function to not vanish trivially, one has to ensure thatG

andG~_{contributions contain1 gunits ofcharge and}

3g 3 units of J_CY charge [44]. This is only possible if each G contributes G_CY, each G~ contributes e 2 RJCY_ðdÞ2_{, each G}þ _{contributes ðdÞ}2_{e , and each}

~

GþcontributesG~þþ_CYe_{. Making these choices, we have}6

ZI IIA¼

Y

3g 3

j¼1

Z

d2_m

jdetðImÞjdet!kðviÞj2

Y g 1

i¼lþ1

e _ðdÞ2_{ðviÞJðvgÞ}

Y

l

j¼1

e _ðdÞ2Z jGCY

Y3g 3 k¼lþ1

Z

kGCY

ð:1Þ

2 : (52)

Here,is the period matrix and!k_{aregholomorphic} one-forms on _g. We have also inserted 1_¼R_ðGþ_;2_{eÞ in} Eq. (52). This makes the functional integral well defined and can be used to remove the factor ofJðvgÞ, as follows. Using the definitions in Eq. (23), notice thatGþ_{has OPEs}

only withJ_ðv_gÞ in Eq. (52). Thus, one can pull theRGþ

counter off the term2_{e, till it hitsJðvgÞ, giving}

ZI IIA¼

Y

3g 3

j¼1

Z

d2_m

jdetðImÞjdet!kðviÞj2

Yg

i¼lþ1

e _ðdÞ2_ðviÞY l

j¼1

e _ðdÞ2Z jGCY

Y

3g 3

k¼lþ1

Z

_kG_CY

ð2_eÞ

2 : (53)

In Eq. (53), apart from functional integral over various world-sheet coordinates, one also has to perform

integra-tions over the zero modes of these variables. The counting of zero modes is as follows. There are four bosonic zero modes ofXm _{and eight fermionic zero modes} correspond-ing to,’s. Further, on a genusgRiemann surface, each variable d contributes g-zero modes, giving in total 8g-zero modes. These zero modes are saturated as follows. In Eq. (53), four zero modes of’s are explicitly present together with4g-zero ofd’s. Another4g-zero modes ofd’s can also be pulled out from2gpowers of the first term in the instanton action in Eq. (49) as

Z

ðd LdR

LRNgsÞ2g; ₍₅₄₎

where in an extremal black hole background, the expecta-tion value of the graviphoton field strength is WLR ¼ LRNg

s. These exactly saturate the required zero modes ofd’s, in the partition function in Eq. (53). Thus we have

ZI IIA¼

Y

3g 3

j¼1

Z

d2_m

jdetðImÞjdet!kðviÞj2

Yg

i¼lþ1

e _ðdÞ2_ðviÞY l

j¼1

e _ðdÞ2Z jGCY

Y

3g 3

k¼lþ1

Z

_kG_CY

ð2_eÞ

2

ðg_topÞ2gY 2g

r¼1

Z

d2_z rd2

;

(55)

¼ Y

3g 3

j¼1

Z

d2_m

jdetðImÞjdet!kðviÞj2

Yg

i¼lþ1

e _ðdÞ2_ðviÞY l

j¼1

ðe _ðdÞ2_Þð2_eÞ

2 ðg_topÞ2g

Y

2g

r¼1

Z

d2_z rd2

Y3g 3 j¼1

Z

_jG_CY

2 ; (56)

where g_top¼Ng_s and in the last line of Eq. (56) the Calabi-Yau part is separated.

In Eq. (56), the integrations over various world-sheet coordinates can now be done exactly as in flat space-time [44]. The functional integrals over theandðd₁;1_Þfields contribute jZ_1j 2_{½detðImފ} 1_{, ðp}

2;2Þ fields contributes jZ₁det!_jðv_kÞj2_{, ðd}

; Þ contributejZ1j4ðdet½ImŠÞ2, and xm_{’s give a factor ofjZ1j} 4_{ðdet½ImŠÞ} 2_{, whereðZ1Þ} ð1=2Þ

is the partition function for a chiral boson [87]. Thus, we have

ZI IIA ¼g

2g top

Z

d4_Xd2 Ld2R

Y

3g 3

j¼1

Z d2_m j

Y3g 3 j¼1

Z

jGCY

2 ; (57)

6_{We omit the details of the regularization over the negative}

energy of the chiral boson, which is same as in flat space and discussed in detail in [44]. The regularization fixes the poles of e_G~þþ

CY coming from G~þ to be sewn in with those of

e 2 RJCY_ðdÞ2 _{coming from G}~ _{, giving the term} Ql

j¼1ðe ðdÞ2

R

jGCYÞin Eq. (52). It also contributes a factor

ofdet_ðIm_Þjdet!k_ðv

iÞj2, from the Jacobian of change of

where, explicitly, theA-model topological string amplitude as given in Eq. (35) is

Fg ¼

Y

3g 3

j¼1

Z

d2_m j

Y3g 3 j¼1

Z

L jGCYL

Z

R jGþCYR

: (58)

Let us emphasize that, out of the eight fermionic zero modes required for a nonzero result, the _jj2 _{term in} Eq. (56) has been used to absorb the _{ðL_};_{RÞ_} integrals. One is left with the integrations over four bosonic and fermionic variables in (57) as there is no action for these variables. In the following subsection, localization argu-ments are used to do the integrations over these variables.

2. Anti-instantons

A similar analysis as above can be repeated for the contribution of anti-instantons, which results in an

A-topological model in the Calabi-Yau geometry. In this case, the anti-instanton contribution to the IIA partition function withA-twist is

ZAI IIA¼

Z

du X 2g 2

nI¼2 2g

ðu_2Þ2g 2þnIðu1Þ2g 2 nI

2

Y

3g 3

j¼1

Z

d2_m j

Yg

i¼1

Z

d2_v i

Y

g 1

i¼1

c

~

G _ðv_iÞJ_ðv_gÞZ _jGcþ

2

; (59)

where theGc~ andGcþ_{are defined in Eq. (39) In this case,}

to do the integration over the_ðL; _RÞzero modes,1_¼

R

ðG ; 2_{e Þ is inserted in the path integral and the} counter-pulling argument overJ_ðv_gÞis repeated. The ab-sorption of 4g-zero modes of d’s requires pulling 2g powers of the second term in the instanton action in Eq. (49). Finally, one is left with the integration over fourX_m and_{ðL_};_{RÞ_} zero modes as

ZAI

IIA¼g 2g top

Z

d4_Xd2_Ld2_R3gY3 j¼1

Z

d2_m j

Y3g 3 j¼1

Z

_jGþ

CY

2

; (60)

where theA-model topological string amplitude is

F_g¼ Y 3g 3

j¼1

Z

d2_m j

Y3g 3 j¼1

Z

L jGþCYL

Z

R jGCYR

: (61)

For this case, the opposite twisting leads toGþL

CY andGCYR acquiring spin 2. A nilpotent operator in this case in the Calabi-Yau geometry is

Q_CY¼G L

CY þGþCYR: (62)

The full partition function of type IIA superstrings on AdS₂S2_CY

3 is obtained from Eqs. (57) and (60).

C. BRST localization

In this section, our aim is perform the integration over X_m,_ðL; _RÞand_{ðL_};_{RÞ_} , appearing in Eqs. (57) and (60). In the present situation, the world-sheet instanton in AdS₂S2 _{contributes four bosonic zero modes coming} from the breaking of translation isometries and their super-symmetric partners. From the PSU_ð1;1_j2_Þ sigma model point of view, the left-multiplication symmetry correspond-ing to the generatorsT_mandT_{;_} is broken by the instanton expectation value. Since there is no action for X_m in Eqs. (57) and (60), the integration gives rise to an infinite space-time volume ofAdS₂. Also, since there is no action for_ðL; _RÞand_{ðL_};_{RÞ_} , the result is zero. Below, we use a localization procedure similar to [33] to solve this problem.

Let us note that, in the Green-Schwarz treatment [33], due to the presence of-symmetry, a world-sheet instanton preserves four out of eight space-time supersymmetries. In particular, instantons and anti-instantons at the opposite poles ofS2_{preserve same amount of supersymmetry, but a} different set, namely, chiral and antichiral, respectively. This reflects in our analysis in Eqs. (57) and (60). In Eq. (57), one is left with integration over the chiral set of variablesðL; RÞof the instantons and in Eq. (60), over the antichiral variables corresponding to anti-instantons. In the hybrid description of the superstring such (anti-) instantons break all space-time supersymmetries, since one cannot perform a compensating -transformation to be in the gauge. Instead, one has superconformal invari-ance, which can be used to write down a operator which renders the partition function finite. The advantage of the hybrid description is that a unified BRST treatment can be given for AdS₂S2 _{and the Calabi-Yau parts, using the} nilpotent operatorQdiscussed at the beginning of Sec.V. Since the AdS2S2 and the Calabi-Yau CFTs decouple, Q contains four- and six-dimensional parts. Below, we explicitly give theAdS2S2part of this operator denoted asQd¼4for localizing world-sheet instantons andQd¼4for anti-instantons. These operators turn out to be BRST op-erators of theAdS₂S2 _{sigma model.}

In the present case, the essence of the localization proce-dure is to use these BRST operators to make a semiclassical expansion of the path integral around the saddle points. This is possible because thePSU_ð1;1_j2_Þsigma model action can be shown to be exact under these BRST operators, con-structed from the left-invariant currents. Thus, if the action S_AdS

2S2 is

Q_d¼4-closed and Q2

d¼4¼0, then one can de-form the action by adding another BRST exact term as

S_AdS

and the parametertcan be varied freely [88,89]. In particu-lar, ast_{! 1}, the theory localizes to the critical points of

Q_d¼4V. Thus, one can integrate over these critical points to render the path integral in Eq. (60) finite, by localizing the genusganti-instantons at the north pole ofS2_{. The analysis} can be repeated usingQ_d¼4for integration over variables in Eq. (60).

We now writeQ_d¼4and then show that the sigma model action isQd¼4-closed. For this purpose, it is useful to use theUð1Þ Uð1Þform of theAdS2S2 action derived in Sec.III Band write7

Q_d¼4 ¼Gd¼L4þGd¼R4: (64)

In terms of left-invariant currents, we have

G L d¼4¼

Z

eL_Jþþ

L JL ; Gd¼R4¼

Z

eR_Jþþ

R JR : (65)

The operatorQ_d¼4 in Eq. (64) is nilpotent, i.e.,Q2 d¼4¼0 as it is constructed from a sum ofG L

d¼4 andGd¼R4, which have regular OPEs of theN_¼2algebra. Also, using the equations of motion in (30), one can check that G L

d¼4 is

antiholomorphic and G R

d¼4 is holomorphic. The BRST operator is constructed from covariantly holomorphic and antiholomorphicH-invariant combination of currents [42]. The action ofQd¼4on the group elementgis to transform it by a right-multiplication as

Q_d¼4ðg_{Þ ¼}g_ððe_{J3Þ þðeJ1ÞÞ; (66)}

whereis a constant anticommuting parameter. The left-invariant currents transform under (66) as

Q_d¼4ðJ_{jÞ ¼jþ1;0}@_ðe_{J3Þ þ ½J}

jþ1;eJ3Š

þjþ3;0@ðeJ1Þ þ ½Jjþ3;eJ1Š;

Q_d¼4ðJ_{jÞ ¼jþ1;0}@_ðe_{J3Þ þ ½J}

jþ1;eJ3Š

þjþ3;0@ðeJ1Þ þ ½Jjþ3;eJ1Š; (67)

wherejis defined modulo 4, i.e.J_jJ_jþ4. Now, one can show that the PSU_ð1;1_j2_Þ sigma model action in (29) is closed under the BRST operator (64). To check this, we follow the procedure used in [56], and start by rewriting the sigma model action in Eq. (29) as

S_AdS

2S2 ¼

Z

d2_z1 2ðJ

þ

þ0J 0 J0þþJ0 þJ 0Jþþ0 J0 J0þþÞ þ1

2ð JL JþþR JL JRþþ JþþL JR JþþL JR Þ

þ1₂ðJþ

L JRþþJþL JRþþJLþJþR þJLþJþR Þ þ 1

4ðJL JþþR JL JRþþþJþþL JR JþþL JR Þ

þ1₄ð J_Lþ J_RþþJþ_L J_Rþ J_LþJþ_R þJ_LþJþ_R Þ

: (68)

Now, under the BRST operator (64), the variations of the terms in first and the third brackets in (68) can be shown to cancel each other. The variation of the terms in second bracket is

₁

2½rðe L_J

L ÞJþþR þJL rð eRJRþþÞ þrð eLJL ÞJRþþþJL rðeRJRþþÞ

þ rðeLJþþ

L ÞJR þJLþþrð eRJR Þ þrð eLJþþL ÞJR þJþþL rðeRJR ފ

; (69)

and the variations of the terms in the fourth and fifth brackets are respectively

₁

4½rðe LJ

L ÞJþþR þJL rð eRJþþR Þ rð eLJL ÞJþþR JL rðeRJRþþÞ þ rðeLJþþL ÞJR

þJþþ_L rð eR_J

R Þ rð eLJþþL ÞJR JþþL rðeRJR ފ

; (70)

and

₁

4½ ðJ

þ

þ0eRJ_R J0 eRJ_RþþÞJ_Rþ Jþ_L ðJ0þþeLJ_L J 0eLJþþ_L Þ ð J0þþeRJ_R þJ 0eRJþþ_R ÞJþ_R

J þ

L ðJ0 eLJþþL J

þ

þ0eLJ_L Þ þ ðJþþ0eRJ_R J0 eRJ_RþþÞJ_RþþJþ_L ðJ0þþeLJ_L J 0eLJþþ_L Þ

þ ð J0þ

þeRJ_R þJ 0eRJ_RþþÞJ_Rþ þJ_LþðJ0 eLJþþ_L Jþþ0eLJ_L ފ

: (71)

7_{The choice of this operator coincides with the fact that the instanton and the anti-instanton require integration over fermionic}

variables with a different set of indices, namely, undotted and dotted. This is also related to the fact that in EuclideanAdS2S2, the

symplectic Majorana condition on fermions, does not take dotted into undotted indices. For instance,_ð

iÞy¼ijj andði_Þy¼

__

Now one makes use of the Maurer-Cartan equations com-ing from@J @J _{þ ½}J;J_{Š ¼}0. For instance, we have

rJ_{L r}J_{L þ ½}J 0_;Jþ

R Š þ ½JRþ ;J 0Š

þ ½J0 _;J þ

R Š þ ½JRþ;J0 Š ¼0: (72) Using these relations in (71), one can show that the Eqs. (70) and (71) are exactly equal. Thus, adding Eqs. (69)–(71) and doing some partial integrations, we get the full variation of (68) to be

h ðeL_J

L ÞrJþþR þ rJL ðeRJþþR Þ

þ ðeLJþþ

L ÞrJR þ rJLþþðeRJR Þi: (73) This is identically zero, using the equations of motion in (30). Thus we have shown that theAdS₂S2sigma model action in (29) is closed under the BRST operator (64).

Now, to do theX_m andð_{L_};_{RÞ_} integrations in (60), a BRST exact operator is constructed as follows. We require an operatorI, such thatQ_d¼4I_¼0andI_¼Q_d¼4V, which facilitates a simple Gaussian integration over the variables. Below, it is shown that the general form of such an operator is given by

V¼ ðaðXmÞ2_þb _ _L_

_

RÞðe L2Rþe R2LÞ; (74) wherem_¼0;1;2;3, and the constantsaandb, are fixed by the BRST procedure. Let us start by noting that the relations

Z

eLJþþ

L JL ðe L2RÞ ¼1;

Z

eRJþþ

R JR ðe R2LÞ ¼1

(75)

render the following choice of the operator,

V_¼ 1

2½ðX

þ

þ0X 0 X0þþX0 þþ_{L R}þ _Lþþ_R Þ

ðe L2

Rþe R2Lފ; (76) a trivial deformation of the theory. To see this, one first notes that

Q_d¼4V¼

1 2½QðX

þ

þ0X 0 X0þþX0 þþ_{L R}þ _Lþþ_R ފ

ðe L2

Rþe R2LÞ 1 2ðX

þ

þ0X 0 X0þþX0

þþ

L Rþ LþþR Þ½Qðe L2Rþe R2Lފ:

(77)

Now, it can be shown that,

Q_d¼4ðXþþ0X 0 X0þþX0 þþ_{L R}þ _Lþþ_R Þ ¼0:

(78)

To check this explicitly, one can calculate how the left-invariant currents rotate various coordinates appearing in (76). Thus, underJ_L andJþþ

L , we have

_{L ¼L} ; þ

L ¼L X

þ þ0;

þ

L ¼ L X0

þ

þ; X 0 ¼ _{L R}þ; (79)

and

þþ

L ¼þþL ; þL ¼ þþL X0 ;

þ

L ¼þþL X 0 X

þ

þ0¼ þþ_L þ_R ;

X0þ

þ¼þþ_{L R}þ; (80)

and the transformations with respect toþþ

R , R can be obtained similarly. So, the first term in (77) drops out. Using the relations in (75), one gets

I¼ ðXþþ0X 0 X0þþX0 þþ_{L R}þ _Lþþ_R Þ: (81)

This is the required operator, as it also satisfiesQI_¼0, as already shown above.

The BRST exact term for the case of instantons can be written down in a similar manner. In this case, we need to do integrations over fourX_mand_ðL; _RÞzero modes in (57)

Qd¼4¼Gþd¼L4þGþd¼R4: (82)

In terms of left-invariant currents, we have

GþL d¼4¼

Z

e LJþ

L JLþ; Gþd¼R4¼

Z

e RJþ

R JRþ: (83)

The operator Q_d¼4 in Eq. (82) is also nilpotent, as it is constructed from a sum of GþL

d¼4 and Gþd¼R4, which have regular OPEs of the N_¼2algebra. The action can again be shown to be closed underQd¼4as shown for the case of anti-instantons. Thus, one now adds a term coming from

I_{¼ ð}Xþþ0X 0 X0þþX0 þþþ_{L R L} þþ_R Þ; (84)

to the path integral in (57) as

Z

d4_Xd2

Ld2Re tI: (85)

Note thatIsatisfiesQd¼4I¼0andI¼Qd¼4Vfor a suit-ableV. To proceed, one makes a Euclidean rotation to write the bosonic part ofI asXþþ0X 0þX0þþX0 . The Euclidean

AdS2S2can be taken as in [33] to be

ds2 _¼ 4dwdw ð1 ww_Þ2þ

4dzdz

ð1_þzz_Þ2; (86) and using the embedding

Xþþ0X¼0 ¼ j!j

2

ð1 _j!_j2_Þ2; X0

þ

þX0¼¼ jzj

2

ð1_{þ j}z_j2_Þ2; (87) the integral in (85) is

Z

d2_!d2_zdþþ

L dL dR dþþR

e tðððj!j2_þjzj2_ðð1þj!j2_jzj2_{Þ=ðð1 j!j}2_Þ2_ð1þjzj2_Þ2_ÞÞ þþ

L R þL þþR Þ: