A Review of 4D \mathcal{N}=2 Off-Shell Gauged Supergravity

# Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole

## Abstract

In this paper we discuss the supersymmetric localization of the 4D off-shell gauged supergravity in the background of the neutral topological black hole, which is the gravity dual of the ABJM theory defined on the boundary . We compute the large- expansion of the supergravity partition function. The result gives the black hole entropy with the logarithmic correction, which matches the previous result of the entanglement entropy of the ABJM theory up to some stringy effects. Our result is consistent with the previous on-shell one-loop computation of the logarithmic correction to black hole entropy. It provides an explicit example of the identification of the entanglement entropy of the boundary conformal field theory with the bulk black hole entropy beyond the leading order given by the classical Bekenstein-Hawking formula, which consequently tests the AdS/CFT correspondence at the subleading order.

\preprint

YITP-SB-17-17 a]Jun Nian \emailAddnian@ihes.fr b]Xinyu Zhang \emailAddzhangxinyuphysics@gmail.com \affiliation[a]Institut des Hautes Études Scientifiques
Le Bois-Marie, 35 route de Chartres
91440 Bures-sur-Yvette, France
\affiliation[b]C.N. Yang Institute for Theoretical Physics
Stony Brook University
Stony Brook, NY 11794-3840, USA
\keywordssupergravity, supersymmetric localization, ABJM, entanglement entropy, topological black hole, black hole entropy, AdS/CFT \arxivnumber

## 1 Introduction

The interpretation of the black hole entropy is one of the central problems in theoretical physics. The celebrated AdS/CFT correspondence [1] provides us with a new insight into the problem of the black hole entropy. Based on this principle, the conformal field theory defined on the boundary of an AdS space should capture all features of the gravity theory in the bulk. Hence, it is really tempting to identify the black hole entropy in the bulk and the entanglement entropy of the conformal field theory on the boundary [2]. When the boundary conformal field theory and the bulk gravity both have certain amount of supersymmetries, the technique of supersymmetric localization allows to compute the entropy on both sides and test the identification precisely. In this paper, we would like to study a concrete example towards this direction, i.e. the ABJM theory via the supergravity localization, to test this proposal.

As a generalization of entanglement entropy, supersymmetric Rényi entropy was first defined on a -branched three-sphere [3]. It can be computed exactly using the technique of supersymmetric localization, and the result can be expressed in terms of the partition function of the 3D superconformal field theory defined on a squashed sphere with the squashing parameter [3]. Using the technique of supersymmetric localization on curved manifolds [4], one can further express this partition function into a matrix integral [5, 6, 7, 8, 9]. In some cases, one can even evaluate the matrix integral to obtain a relatively simple result. For instance, neglecting the nonperturbative effects at large , the matrix integral for the ABJM theory on some compact manifolds can be evaluated using the Fermi gas approach [10, 11].

Since it can be computed exactly, the supersymmetric Rényi entropy provides a new quantitiy to test various dualities. For instance, one can use it to test the AdS/CFT correspondence more precisely. Before doing it, one has to first find the holographic way of computing the supersymmetric Rényi entropy. As explained in Ref. [3], the technical problem is the conical singularity caused by the branched sphere. To resolve the conical singularity, it was proposed in Refs. [12, 13] to perform a conformal transformation, which maps the branched three-sphere into , i.e.,

 ds2 =dθ2+q2sin2θdτ2+cos% 2θdϕ2 =sin2θ[d~τ2+du2+% sinh2udϕ2], (1)

where , , and

 sinhu=−cotθ. (2)

After the conformal transformation, one finds that can be viewed as the boundary of the topological black hole (TBH). Hence, in principle the supersymmetric Rényi entropy can be computed in the topological black hole holographically. The free energy and the Killing spinor equations can also be evaluated in the bulk gravity theory, which supports the holographic interpretation [12, 13]. In particular, the parameter coming from the branched sphere can be viewed as a deformation parameter of the original theory, related to the mass and the charge of the topological black hole. This relation was called the TBH/qSCFT correspondence [12, 14], where it provides another precise test of the AdS/CFT correspondence. Later, these works were generalized to other dimensions, and similar results were found [14, 15, 16, 17, 18, 19, 20, 21].

The partition functions and consequently the supersymmetric Rényi entropies of some superconformal field theories can be computed exactly using the technique of supersymmetric localization. In fact, this technique can also be applied to some supergravity theories. Different backgrounds (, ) have been studied [22, 23, 24]. In particular, the localization of the 4D off-shell supergravity on corresponds to the ABJM theory on the boundary [24], and the partition functions of both theories can be expressed in terms of Airy function. From the partition function, we can compute the entanglement entropy of the ABJM theory across a circle on the boundary, which matches the previous results [3].

It is then natural to consider the supergravity localization on 4D topological black holes, whose boundaries are . From the supergravity localization we should be able to compute holographically the supersymmetric Rényi entropies of the corresponding superconformal field theories on the boundary. Comparing the results from the bulk and the known results from the boundary provides an exact test of the AdS/CFT correspondence, and at the same time one can also check the proposal of identifying these entropies as the black hole entropies in this framework concretely.

As a starting point, in this paper we study the localization of the 4D off-shell gauged supergravity on the neutral topological black hole, which corresponds to the entanglement entropy of the superconformal ABJM theory across a circle on the boundary. The logic of our computation is as follows. The gravity dual of the ABJM theory is the 11-dimensional M-theory on [25]. We neglect all the stringy effects and consistently truncate the 11-dimensional supergravity to a 4-dimensional gauged supergravity theory, which has an off-shell formalism using superconformal gauged supergravity. We fix the values of fields in the Weyl multiplet, and apply the localization method to evaluate the supersymmetric partition function by integrating over the vector multiplets and the hypermultiplets. Our localization calculation is similar to the standard field theory localization, except that the background spacetime is noncompact. We find that the entropy of the neutral topological black hole and the entanglement entropy of the ABJM theory on the boundary coincide in the large- expansion up to some stringy effects. More precisely,

 SABJMEE=SBH=−√2π3k1/2N3/2−14log(N)+O(N0). (3)

Meanwhile, using the supergravity localization we obtain the logarithmic correction to the classical result of the black hole entropy given by the Bekenstein-Hawking formula [26, 27, 28], and this correction is consistent with the on-shell 1-loop computation from the Euclidean 11-dimensional supergravity on [29].

This paper is organized as follows. In Section 2 we review some facts about the supersymmetric Rényi entropy and the ABJM theory. The gravity dual of the supersymmetric Rényi entropy will be reviewed in Section 3. In Section 4 we discuss the localization of the 4D off-shell supergravity on the neutral topological black hole with the boundary . The bulk black hole entropy and the boundary entanglement entropy of the ABJM theory can be read off from the results of the supergravity localization, which is presented in Section 5. Some further discussions will be made in Section 6. We also present some details of the calculations in a few appendices. In Appendix A we review the 4D off-shell supergravity, while in Appendix B the Killing spinors and the convenction of the Gamma matrices are discussed. For the supergravity localization, the explicit form of the localization action will be presented in Appendix C, and we will evaluate the action along the localization locus in Appendix D.

## 2 Supersymmetric Rényi Entropy of ABJM Theory

### 2.1 Supersymmetric Rényi Entropy

We start with the well-known definitions of entanglement entropy and Rényi entropy. Suppose the space on which the theory is defined can be divided into a piece and its complement , and correspondingly the Hilbert space factorizes into a tensor product . If the density matrix over the whole Hilbert space is , then the reduced density matrix is defined as

 ρA≡trBρ. (4)

The entanglement entropy is the von Neumann entropy of ,

 SE≡−trρAlogρA, (5)

while the Rényi entropies are defined to be

 Sn≡11−nlogtr(ρA)n. (6)

Assuming an analytic continuation of can be obtained, the entanglement entropy can alternately be expressed as a limit of the Rényi entropy:

 limn→1Sn=SE. (7)

The Rényi entropy can be calculated using the so-called “replica trick”:

 Sn=11−nlog(Zn(Z1)n), (8)

where is the Euclidean partition function on a -covering space branched along .

The concept of the supersymmetric Rényi entropy is a generalization of Rényi entropy. It was first introduced in Ref. [3] for the 3-dimensional supersymmetric field theories as follows:

 SSUSYq≡11−q[log(Zsingular space(q)(ZS3)q)], (9)

where is the partition function of a supersymmetric theory on a three-sphere , while is the partition function on the q-covering of a three-sphere, , which is also called the q-branched sphere given by the metric

 ds2=L2(dθ2+q2sin2θdτ2+cos2θdϕ2) (10)

with , and . In the limit , the -branched sphere returns to the round sphere, and the supersymmetric Rényi entropy becomes the entanglement entropy. Initially, the supersymmetric Rényi entropy was defined for 3D superconformal field theories [3], and later it was generalized to other dimensions [14, 15, 16, 17, 18, 19, 20, 21].

Using the supersymmetric localization, it was derived explicitly in Ref. [3] that in the definition of the supersymmetric Rényi entropy of 3D superconformal field theories can be written as:

 Zsingular space(q)=1|W|∫rankG∏i=1dσieπikTr(σ2)∏α1Γh(α(σ))∏I∏ρ∈RIΓI(ρ(σ)+iωΔI) (11)

with

 ω=ω1+ω22,ω1=√q,ω2=1√q, (12)

and

 Γh(z)≡Γh(z;iω1,iω2), (13)

where

 Γh(z;ω1,ω2) =∏n1,n2≥0(n1+1)ω1+(n2+1)ω2−zn1ω1+n2ω2+z =exp[i∫∞0dxx(z−ωω1ω2x−sin(2x(z−ω))2% sin(ω1x)sin(ω2x))] (14)

defined for

 0

is a hyperbolic gamma function [30]. parametrize the localization locus of the Coulomb branch. stands for the Chern-Simons level, and is the index for the chiral multiplets. and denote the root of the adjoint representation and the weight of the representation of the gauge group respectively. is the R-charge of the scalar in the chiral multiplet. It turns out that the partition function equals the partition function of the same theory on a squashed three-sphere with .

### 2.2 Results for ABJM Theory

As an example of the 3D superconformal field theory, the ABJM theory has been intensively studied. As first discussed by Aharony, Bergman, Jafferis and Maldacena in Ref. [25], the ABJM theory is a 3D supersymmetric Chern-Simons-matter theory with the gauge group , where stands for the Chern-Simons level. The theory describes the low-energy dynamics of M2-branes on , and it has 4 bi-fundamental chiral multiplets, two of them in the representation and the other two in the representation. The matter content of the ABJM theory can be illustrated using the quiver diagram in Fig. 1.

With the development of the supersymmetric localization on curved manifolds [4], the partition functions of some 3D supersymmetric gauge theories including the ABJM theory were studied in Ref. [31], and they can be expressed as matrix integrals. In particular, the partition function of the ABJM theory is reduced to the following matrix model:

 Missing or unrecognized delimiter for \right (16)

where and are the roots of and respectively, and the weights in the representations and are

 ρ(N,¯¯¯¯N)i,j =σi−˜σj, ρ(¯¯¯¯N,N)i,j =−σi+˜σj. (17)

To evaluate these integrals, one still has to solve the matrix model, which sometimes can be nontrivial. To proceed the computation, on the one hand the matrix integral were evaluated directly under some approximations for the 3D superconformal field theories [32, 33], while on the other hand using the Fermi gas approach one can obtain the final results of the perturbative contributions, which was done in Ref. [10]. The result for the partition function of the ABJM theory from the Fermi gas approach was obtained by Mariño and Putrov, which can be written in terms of the Airy function [10]:

 ZABJM∝Ai[(π2k2)1/3(N−k24−13k)]. (18)

As discussed above, the supersymmetric Rényi entropies of some 3D superconformal field theories can be expressed in terms of partition functions of these theories on squashed three sphere , and these partition functions can still be written as matrix models using the technique of localization. For the ABJM theory, the partition function on a squashed three-sphere can be written as

 ZABJMb2=1(N!)2∫∏idσid˜σie−ikπ(σ2i−˜σ2i)Zvecb2Zbi-fundb2, (19)

where

 Zvecb2 =∏i

where , and is the double sine function. In the limit ,

 Zvecb2=1 =∏i

which reproduce the partition function of the ABJM theory on the round three-sphere found in Ref. [31].

Recently, Hatsuda studied the partition function of the ABJM theory on a squashed three-spheres [11], and found that for some cases the matrix model can be greatly simplified and evaluated analytically at large using the Fermi gas approach. For instance, when and , the leading contribution to the partition function is

 ZABJMb2=3=C−1/33eA3Ai[C−1/33(N−B3)]+⋯, (22)

where

 A3=−ζ(3)3π2+log36,B3=18,C3=98π2. (23)

With these results, one can study the supersymmetric Rényi entropy at large beyond the leading order.

## 3 Gravity Dual of Supersymmetric Rényi Entropy

The gravity dual of the supersymmetric Rényi entropies of 3D superconformal field theories (including the ABJM theory) has been constructed in Refs. [12, 13]. Later, it was generalized to other dimensions [14, 15, 16, 17, 18, 19, 20, 21]. In this section, we briefly review the gravity dual theory found in Refs. [12, 13].

As discussed in Ref. [3], due to the conical singularity one has to turn on a R-symmetry gauge field in order to preserve supersymmetry. In the spirit of the AdS/CFT correspondence, instead of finding an AdS space with the branched sphere as the boundary, one can first perform the conformal transformation introduced in Section 1 to the branched sphere, which maps the branched three-sphere into , i.e.

 ds2 =dθ2+q2sin2θdτ2+cos% 2θdϕ2 =sin2θ[d~τ2+du2+% sinh2udϕ2], (24)

where , , and

 sinhu=−cotθ. (25)

Next, one can find an topological black hole with the metric [34]:

 ds2=−f(r)dt2+1f(r)dr2+r2dΣ(H2), (26)

whose boundary is , where

 f(r)=r2L2−1−2mr+Q2r2, (27)

and

 dΣ(H2)=du2+sinh2udϕ2. (28)

This metric can be viewed as solutions to the 4D gauged supergravity given by the effective action [35], whose bosonic part is

 I=−12ℓ2P∫d4x√−g(2Λ+R−1g2FμνFμν). (29)

The gauge field is given by

 A=(Qr−Qrh)dt, (30)

where is the horizon radius of the black hole determined by .

As explained in Refs. [36, 12], to preserve the supersymmetry, the condition

 m2+Q2=0 (31)

holds for both the charged case () and the neutral case ().

For ,

 f(r)=r2L2−(1+mr)2, (32)

the metric (26) corresponds to a charged topological black hole. As shown in Refs. [12, 13], the Bekenstein-Hawking entropy of the charged topological black hole equals the Rényi entropy of the superconformal field theory on the boundary. In particular, the result from the gravity dual recovers the relation between the Rényi entropy and the entanglement entropy for the 3D superconformal field theories:

 Sq=3q+14qS1. (33)

For ,

 f(r)=r2L2−1, (34)

and . The gravity solution is dual to a 3D superconformal field theory on with . Correspondingly, the black hole entropy in this case equals the entanglement entropy of the boundary superconformal field theory. The evaluation of the gravity free energy at classical level supports this identification.

As we know, the Bekenstein-Hawking entropy corresponds to the classical result of the gravity. Using supersymmetric localization, we can obtain more precise result and go beyond the classical result. Hence, in this way we can test the gravity dual and the AdS/CFT correspondence more precisely.

In this paper, we consider the neutral topological black hole, whose entropy gives the entanglement entropy of the superconformal field theory on the boundary. For this case, the branching parameter , which corresponds to the round three sphere. One can nevertheless perform the conformal transformation (25). The hyperbolic space becomes an neutral topological black hole. The entanglement entropy of the superconformal field theory on the boundary is supposed to be equal to the bulk black hole entropy. The equality can be tested more precisely using the results of the localization of supergravity.

## 4 4d N=2 Off-Shell Gauged Supergravity and Its Localization

In this section, we discuss the localization of the 4D off-shell supergravity on topological black hole with the boundary . The steps are similar to the ones in Ref. [24], however, there are some subtle differences which consequently lead to different final results.

### 4.1 4d N=2 Off-Shell Gauged Supergravity

The 4D off-shell supergravity theory can be obtained as a consistent truncation of M-theory on a Sasaki-Einstein manifold . The theory was originally constructed in Ref. [37] and also reviwed in Ref. [24]. We also briefly summarize the theory in Appendix A.

The superconformal algebra has the generators:

 Pa,Mab,D,Ka,Qi,Si,Uij, (35)

which correspond to the generators of translations, Lorentz rotations, dilatations, special conformal transformations, usual supersymmetry transformations, special conformal supersymmetry transformations and the 4D R-symmetry respectively. The gauge fields corresponding to these generators are

 eaμ,ωabμ,bμ,faμ,ψiμ,ϕiμ,Vijμ (36)

respectively.

To construct a 4D off-shell supergravity, one needs the Weyl multiplet :

 W=(eaμ,ψiμ,bμ,Aμ,Viμj,Tijab,χi,D), (37)

the vector multiplet :

 XI=(XI,ΩIi,WIμ,YIij), (38)

and the hypermultiplet . More details about these multiplets and their supersymmetric transformations can be found in Appendix A.

Given a prepotential , the two-derivative off-shell action for the bosonic fields is given by

 S =∫d4x√g[NIJ¯¯¯¯¯XIXJ(R6+D)+NIJ∂¯¯¯¯¯XI∂XJ−18NIJYijIYJij +(−∇Aiβ∇Aiα−(R6−D2)AiβAiα+FiβFiα+4g2Aiβ¯¯¯¯¯XαγXγδAiδ +gAiβ(Yjk)γαAkγϵij)dαβ], (39)

where

 NIJ≡12i(FIJ−¯¯¯¯FIJ),FIJ≡∂I∂JF(X), (40)

and is related to the field discussed in Appendix A in the following way:

 Fiα=aAiα(z). (41)

The term provides a negative cosmological constant for the space.

### 4.2 Localization of Supergravity

As discussed before, to find the gravity dual of the supersymmetric Rényi entropy, one can perform a conformal transformation (25) on the boundary, which maps the branched three-sphere into . Correspondingly, the metric in the bulk now should be the topological black hole given by the metric (26) to match the boundary.

In this section, we discuss the localization of the 4D off-shell gauged supergravity on the background of the neutral topological black hole (26). In other words, we focus on the case with the branching parameter , which corresponds to the entanglement entropy of the ABJM theory across a circle on the boundary. The discussions are similar to the supergravity localization on the hyperbolic in Ref. [24].

#### BPS Equations

Let us first consider the BPS equations for various supergravity multiplets. For the Weyl multiplet, by setting one obtains the Killing spinor equation:

 2∇μϵi+iAμϵi−18Tabijγabγμϵj=γμηi, (42)

where we set the background , and

 ∇μϵi≡∂μϵi+14ωμabγabϵi. (43)

For , one can further set in Eq. (42), and the Killing spinor equation becomes

 ∇μϵi=12γμηi. (44)

The upper and the lower indices for denote the positive and the negative chirality respectively, and the opposite for . We can use the Dirac notation to combine different components into Dirac spinors

 ξ=(ξi+,ξi−),η=(ηi+,ηi−), (45)

where

 ξi+≡ϵi,ϵi≡iϵijξj−,ηi≡−ϵijηj+,ηi≡iηi−. (46)

Using these notations, we can rewrite the Killing spinor equation (44) for as follows:

 ∇μξi=i2γμηi. (47)

We would like to recover the Killing spinor equation for the topological black hole discussed in Ref. [12] for . To do so, let us first consider the general Killing spinor equation for the topological black hole [12]:

 ∇μϵ−igAμϵ+i4Fabγabγμϵ=−12gγμϵ, (48)

where

 A=(Qr−Qrh)dt (49)

with denoting the position of the horizon, and consequently only the components and are nonvanishing. For the branching paramter considered in this paper, the black hole is neutral, i.e. , hence both and vanish for this case. As discussed in Appendix B, using the charge conjugation matrix one can define the charge conjugate spinor satisfying another Killing spinor equation:

 ∇μϵc+igAμϵc+i4Fabγabγμϵc=12gγμϵc. (50)

 L=1g. (51)

For , the two Killing spinor equations are

 ∇μϵ=−12Lγμϵ,∇μϵc=12Lγμϵc, (52)

which can be written into a more compact form using the Dirac notation:

 ∇μ˜ξi=12Lγμ(σ3)ij˜ξj, (53)

with

 ˜ξ1≡ϵc,˜ξ2≡ϵ. (54)

Defining

 ξi≡1+iγ52˜ξi, (55)

one can further obtain an equivalent expression for the Killing spinor equation at :

 ∇μξi=i2Lγ5γμ(σ3)ijξj. (56)

Comparing this equation with Eq. (47), we should identify

 ηi=−1Lγ5(σ3)ijξj. (57)

Eq. (56) will be the Killing spinor equation used throughout the rest of this paper.

Next, for the vector multiplet, the BPS equations are obtained from . Setting and distinguishing different chiralities, we obtain

 δΩi+ =−i⧸∂Xξi−−12Yijξj++Xηi+=0, δΩi− =−i⧸∂¯¯¯¯¯Xξi+−12Yijξj−+¯¯¯¯¯Xηi−=0, (58)

which can be combined into

 −i⧸∂(H−iγ5J)ξi−12Yijξj−1L(H+iγ5J)γ5(σ3)ijξj=0, (59)

where we have parametrized and used the expression for given above. For constant and , the BPS equations above have the solution:

 H=0,Y11=−Y22=−2iLJ,Y12=−Y21=0. (60)

The BPS equation for the hypermultiplet can be obtained by setting the modified supersymmetric transformation (see Appendix A), which leads to

 δζα+ =i⧸∇Aiαϵijξj−+2g¯¯¯¯¯XαβAiβϵijξj+−Aiαϵijηj++Fiαϵijξj+=0, δζα− =⧸∇Aiαξi+−2giXαβAiβξi−+iAiαηi−−iFiαξi−=0, (61)

where satisfying . One can combine these two equations using the Dirac notation in the following way:

 ⧸∇Aiαξi−2gi(HI−iγ5JI)(tI)αβAiβξi+iAiαηi−iFiαξi=0. (62)

We consider the model with the charges

 tIAiα=PI(iσ3)αβAiβ, (63)

where are moment maps on the hyperkähler manifold with the scalars in the hypermultiplet as sections. In the gauge , using the relation (57) one can express the BPS equation for the hypermultiplet as

 [2g(H⋅P)−2giγ5(J⋅P)−iLγ5]Aiα(σ3)ijξj−iFiαξi=0, (64)

 Fjα=−2igAiα(σ3)ij(H⋅P),2g(J⋅P)=−1L. (65)

#### Attractor Solution

As we discussed before, given a prepotential , the two-derivative off-shell action for the bosonic fields is given by Eq. (39). Now let us take a closer look at the theory and analyze its attractor solution. Later in the localization procedure, the localization locus will fluctuate around the attractor solution discussed in this subsection.

First, the field plays the role of a Lagrange multiplier, which imposes the condition:

 NIJ¯¯¯¯¯XIXJ+12AiβAiαdαβ=0. (66)

By requiring that the terms containing the Ricci scalar reproduce the Einstein-Hilbert action, we obtain

 16NIJ¯¯¯¯¯XIXJ−16AiβAiαdαβ=116πG, (67)

where is the Newton’s constant. The equations (66) (67) lead to

 NIJ¯¯¯¯¯XIXJ=18πG,AiβAiαdαβ=−14πG. (68)

In the gauge , the second equation above implies

 Aiα=1√8πGδαi, (69)

where we have used discussed in Appendix A.

By analyzing the field equations of various fields in the action (39), we arrive at the same solution that we found before from the BPS equations (65):

 2g(J⋅P)=−1L, (70)

more precisely,

 8gJ0P0=−1L,8gJ1P1=−3L. (71)

For the prepotential , the first one of Eq. (68) becomes

 14i|X0|2⎛⎜⎝√X1X0−¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯√X1X0⎞⎟⎠3=18πG, (72)

 (J0)1/2(J1)3/2=i16πG, (73)

where we choose .

#### Localization Action

As in the standard localization procedure, we can add a SUSY-exact term to the action without changing the partition function of the theory. The SUSY-exact term is called the localization action. In our case, we choose the following localization action for the vector multiplet:

 ΔS=δ((δΩ)†Ω), (74)

where denotes the gaugino field in the vector multiplet. The bosonic part of the localization action is

 (ΔS)bos=(δΩ)†δΩ. (75)

We can solve to find the localization locus. Some details are presented in Appendix C.

When expanding the localization action, we choose the Killing spinor found in Ref. [12] for the topological black hole with :

 ϵ=e−i2qLτEeiu2γ4γ1γ2eϕ2γ23~ϵ(r) (76)

with

 ~ϵ(r)=(√rL+√f(r)−iγ4√rL−√f(r))(1−γ12)ϵ′0, (77)

where is an arbitrary constant spinor, and is the factor appearing in the metric of the topological black hole (26):

 ds2=−f(r)dt2+1f(r)dr2+r2dΣ(H2).

In principle, there are 8 independent Killing spinors . Moreover, and satisfy the Killing spinor equations (52):

 ∇μϵ=−12Lγμϵ,∇μϵc=12Lγμϵc.

As we discussed in Subsection 4.2.1, it is more convenient to work with the Killing spinors

 ξi≡1+iγ52˜ξi, (78)

with

 ˜ξ1≡ϵc,˜ξ2≡ϵ. (79)

and they satisfy the equivalent Killing spinor equation (56):

 ∇μξi=i2Lγ5γμ(σ3)ijξj.

The Killing spinors generate the Killing vector

 v=ξ†γμξ∂μ=LU(1), (80)

which is a linear combination of the compact ’s along the compact directions and in the metric (147).

Using the Killing spinor discussed above with the special choice of the constant spinor , we can compute various Killing spinor bilinears, and expand the localization action (75) explicitly. The localization action (75) can be expressed as a sum of some squares (161). By requiring these squares vanish, we obtain the following solutions:

 H=Ccosh(η),Y11=2Ccosh2(η), for u=0; (81)
 J=const,FabVb=0, (82)

where is an arbitrary constant, and the constant value of is fixed by the attractor solutions (71) (73). Together with the BPS solutions found in Subsection 4.2.1, these form the localization locus:

 XI =HI+iJI=CIcosh(η)+iJI=JIhIcosh(η)+iJI, (YI)11 =−(YI)22=2CIcosh2(η)−2iLJI=2JIhIcosh2(η)−2iLJI, (83)

where we have written the gauge index explicitly and used the parametrization , and again the values of are fixed to be the attractor solutions given by Eq. (71) and Eq. (73).

For the hypermultiplet, as discussed in Appendix C, we require for all 8 Killing spinors, which leads to the solutions

 Fiα=−2ig√8πG(σ3)αj(H⋅P),2g(J⋅P)=−1L (84)

with and given by

 Fiα=aAiα(z),tIAiα=PI(iσ3)αβAiβ. (85)

These solutions coincide with the solutions (65) to the BPS equations under the attractor solution (69).

#### Action on Localization Locus

Now we would like to evaluate the action (39) at the localization locus (83) obtained in the previous subsection. We distinguish the action for the vector multiplet and the action for the hypermultiplet:

 Svec =∫d4x√g[NIJ¯¯¯¯¯XIXJR6+NIJ∂¯¯¯¯¯XI∂XJ−18NIJYijIYJij], (86) Shyp =∫d4x√g[(−R6AiβAiα+FiβF