Lectures on AdS Black Holes, Holography and Localization

TL;DR

This paper reviews recent advances in counting microstates of AdS supersymmetric black holes using holography, localization, and matrix models, focusing on AdS4 and AdS5 black holes.

Contribution

It provides a detailed overview of applying localization and matrix model techniques to compute black hole microstates in AdS/CFT correspondence.

Findings

Successful microstate counting for AdS4 black holes

Application of localization techniques to dual field theories

Discussion of current methods for AdS5 black holes

Abstract

In these lectures I review some recent progresses in counting the number of microstates of AdS supersymmetric black holes in dimensions equal or greater than four using holography. The counting is obtained by applying localization and matrix model techniques to the dual field theory. I cover in details the case of dyonic AdS$_4$ black holes, corresponding to a twisted compactification of the dual field theory, and I discuss the state of the art for rotating AdS$_5$ black holes.

Tables

Table 1. Table 1: Entropy functionals for electrically charged rotating black holes. For simplicity of notations, we opted for a uniform notation for all dimensions, involving some sign redefinitions in comparison to Hosseini:2017mds ; Hosseini:2018dob ; Choi:2018fdc , to which we refer for more precise statements.

AdS${}_4{}^{}\times S^7$	$F\left({\Delta }_a\right)=\sqrt{{\Delta }_1{\Delta }_2{\Delta }_3{\Delta }_4}$	$\log 𝒵({\Delta }_a,{\omega }_i)=-\frac{4\sqrt{2}{N}^{3/2}}{3}\frac{\sqrt{{\Delta }_1{\Delta }_2{\Delta }_3{\Delta }_4}}{{\omega }_1}$
	${\Delta }_1+{\Delta }_2+{\Delta }_3+{\Delta }_4=2$	${\Delta }_1+{\Delta }_2+{\Delta }_3+{\Delta }_4-{\omega }_1=2\pi$
AdS${}_5{}^{}\times S^5$	$F\left({\Delta }_a\right)={\Delta }_1{\Delta }_2{\Delta }_3$	$\log 𝒵({\Delta }_a,{\omega }_i)=-i\frac{N^2}{2}\frac{{\Delta }_1{\Delta }_2{\Delta }_3}{{\omega }_1{\omega }_2}$
	${\Delta }_1+{\Delta }_2+{\Delta }_3=2$	${\Delta }_1+{\Delta }_2+{\Delta }_3-{\omega }_1-{\omega }_2=2\pi$
AdS${}_6{}^{}\times _W^{}{S}^4$	$F\left({\Delta }_a\right)={\left({\Delta }_1{\Delta }_2\right)}^{3/2}$	$\log 𝒵({\Delta }_a,{\omega }_i)\sim N^{5/2}\frac{{\left({\Delta }_1{\Delta }_2\right)}^{3/2}}{{\omega }_1{\omega }_2}$
	${\Delta }_1+{\Delta }_2=2$	${\Delta }_1+{\Delta }_2-{\omega }_1-{\omega }_2=2\pi$
AdS${}_7{}^{}\times S^4$	$F\left({\Delta }_a\right)={\left({\Delta }_1{\Delta }_2\right)}^3$	$\log 𝒵({\Delta }_a,{\omega }_i)=i\frac{N^3}{24}\frac{{\left({\Delta }_1{\Delta }_2\right)}^3}{{\omega }_1{\omega }_2{\omega }_3}$
	${\Delta }_1+{\Delta }_2=2$	${\Delta }_1+{\Delta }_2-{\omega }_1-{\omega }_2-{\omega }_3=2\pi$

Table 2. Table 2: Entropy functionals for magnetically charged static spherically symmetric black objects with a twist. The cases of AdS × 5 S 5 {}_{5}\times S^{5} and AdS × 7 S 4 {}_{7}\times S^{4} correspond to black strings. The Legendre transform of log ⁡ 𝒵 𝒵 \log\mathcal{Z} reproduces the entropy of the static black hole obtained by reduction on a circle, as discussed in Sect.  5 . Details and normalizations for AdS 6 and AdS 7 can be found in Hosseini:2018uzp ; Crichigno:2018adf ; Hosseini:2018usu ; Suh:2018szn .

AdS${}_4{}^{}\times S^7$	$F\left({\Delta }_a\right)=\sqrt{{\Delta }_1{\Delta }_2{\Delta }_3{\Delta }_4}$	$\log 𝒵=-\frac{2\sqrt{2}{N}^{3/2}}{3}{\sum }_{a=1}^4{𝔭}_a\frac{\partial ℱ\left(\Delta \right)}{\partial {\Delta }_a}$
	${\Delta }_1+{\Delta }_2+{\Delta }_3+{\Delta }_4=2$	${\Delta }_1+{\Delta }_2+{\Delta }_3+{\Delta }_4=2\pi$
AdS${}_5{}^{}\times S^5$	$F\left({\Delta }_a\right)={\Delta }_1{\Delta }_2{\Delta }_3$	$\log 𝒵=-\frac{N^2}{2\beta }{\sum }_{a=1}^3{𝔭}_a\frac{\partial ℱ\left(\Delta \right)}{\partial {\Delta }_a}$
	${\Delta }_1+{\Delta }_2+{\Delta }_3=2$	${\Delta }_1+{\Delta }_2+{\Delta }_3=2\pi$
AdS${}_6{}^{}\times _W^{}{S}^4$	$F\left({\Delta }_a\right)={\left({\Delta }_1{\Delta }_2\right)}^{3/2}$	$\log 𝒵\sim N^{5/2}{\sum }_{a,b=1}^2{𝔭}_a{\widetilde{𝔭}}_b\frac{{\partial }^2ℱ\left(\Delta \right)}{\partial {\Delta }_a\partial {\Delta }_b}$
	${\Delta }_1+{\Delta }_2=2$	${\Delta }_1+{\Delta }_2=2\pi$
AdS${}_7{}^{}\times S^4$	$F\left({\Delta }_a\right)={\left({\Delta }_1{\Delta }_2\right)}^2$	$\log 𝒵\sim \frac{N^3}{\beta }{\sum }_{a,b=1}^2{𝔭}_a{\widetilde{𝔭}}_b\frac{{\partial }^2ℱ\left(\Delta \right)}{\partial {\Delta }_a\partial {\Delta }_b}$
	${\Delta }_1+{\Delta }_2=2$	${\Delta }_1+{\Delta }_2=2\pi$

Equations

$$S_{\mathrm{BH}}=k_{\mathrm{B}}\frac{c^{3}A}{4\hslash G_{\mathrm{N}}},$$

$$S_{\mathrm{BH}}=k_{\mathrm{B}}\log n,$$

$$S=\frac{A}{4G_{\text{N}}},$$

$$(j_1,j_2)U(1{)}^2\subset SO(4)\subset SO(2,4),$$

$$(q_1,q_2,q_3)U(1{)}^3\subset SO(6),$$

$$(q_1,q_2,q_3,q_4)(p^1,p^2,p^3,p^4),$$

$$\mathcal{L}=\sqrt{g}\left(R+g_{ab}({\varphi }_i)F_{\mu \nu }^a{F}^{\mu \nu b}+\dots \right)$$

$$\int d^{4}x{A}_{\mu }^a{J}^{\mu a}$$

$$d{s}^2$$

$$A_{\mu }^a(x,r)$$

$$\varphi (x,r)=r^{{\alpha }_1}{\varphi }_0(x)+r^{{\alpha }_2}{\varphi }_1(x),$$

$${\mathcal{L}}_{CFT}(x)\to {\mathcal{L}}_{CFT}(x)+{\varphi }_0(x)O(x),$$

$$\frac{1}{2\pi }{\int }_{{\Sigma }_{\mathfrak{g}}}{F}^a=p^a,$$

$${\mathcal{L}}_{CFT}(x)\to {\mathcal{L}}_{CFT}(x)+{\hat{A}}_{\mu }^a(x)J^{\mu a}(x).$$

$$\delta {\psi }_{\mu }={\partial }_{\mu }ϵ+\frac{1}{4}{\omega }_{\mu }^{ab}{\Gamma }_{ab}ϵ-i{A}_{\mu }^Rϵ+\dots .$$

$$\delta {\psi }_{\mu }={\partial }_{\mu }ϵ=0.$$

$$\{{\mathcal{Q}}^{†},\mathcal{Q}\}=H-{\mu }_a{Q}_a-{\nu }_i{J}_i,$$

$$E={\mu }_a{q}_a+{\nu }_i{j}_i.$$

$${\cal Z}(\Delta_{a},\omega_{i})=\operatorname{Tr}\Big{|}_{{\cal Q}=0}e^{i(\Delta_{a}Q_{a}+\omega_{i}J_{i})}=\sum_{q_{a},j_{i}}c(q_{a},j_{i})e^{i(\Delta_{a}q_{a}+\omega_{i}j_{i})}\,,$$

$$S(q_a,j_i)=\log c(q_a,j_i),$$

$$e^{S(q_a,j_i)}=c(q_a,j_i)=\int \frac{d{\Delta }_a}{2\pi }\frac{d{\omega }_i}{2\pi }\mathcal{Z}({\Delta }_a,{\omega }_i)e^{-i({\Delta }_a{q}_a+{\omega }_i{j}_i)},$$

$$S(q_{a},j_{i})=\log\mathcal{Z}(\Delta_{a},\omega_{i})-i(\Delta_{a}q_{a}+\omega_{i}j_{i})\Big{|}_{\bar{\Delta}_{a},\bar{\omega}_{i}}\,,$$

$$\mathcal{I}({\Delta }_a,{\omega }_i)=\log \mathcal{Z}({\Delta }_a,{\omega }_i)-i({\Delta }_a{q}_a+{\omega }_i{j}_i),$$

$$\partial_{\Delta_{a}}{\cal I}(\Delta_{a},\omega_{i})=\partial_{\omega_{i}}{\cal I}(\Delta_{a},\omega_{i})=0\Big{|}_{\bar{\Delta}_{a},\bar{\omega}_{i}}\,.$$

$$\log \mathcal{Z}=-(E-TS-i{\tilde{\Delta }}_a{q}_a-i{\tilde{\omega }}_i{j}_i)/T,$$

$$\displaystyle\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})=\operatorname{Tr}\Big{|}_{{\cal Q}=0}(-1)^{F}e^{i(\Delta_{a}Q_{a}+\omega_{i}J_{i})}\,,$$

$${\mathcal{Z}}_{\mathrm{index}}({\Delta }_a,{\omega }_i)=\mathrm{Tr}(-1{)}^F{e}^{-\beta \{{\mathcal{Q}}^{†},\mathcal{Q}\}}{e}^{i({\Delta }_a{Q}_a+{\omega }_i{J}_i)}=Z_{S^1\times {\mathcal{M}}_{d-1}}^{susy}({\Delta }_a,{\omega }_i),$$

$$S(X^a,{\Omega }^i,q_a,j_i)=\mathcal{E}(X^a,{\Omega }^i)-i(X^a{q}_a+{\Omega }^i{j}_i),$$

$$S(X^a)=-\frac{2\sqrt{2}{N}^{\frac{3}{2}}}{3}\sum _{a=1}^4{\mathfrak{p}}_a\frac{\partial }{\partial X^a}\sqrt{X^1{X}^2{X}^3{X}^4}-i\sum _{a=1}{X}^a{q}_a,$$

$$S(X^a,{\Omega }^i)=-i\frac{N^2}{2}\frac{X^1{X}^2{X}^3}{{\Omega }^1{\Omega }^2}-i\left(\sum _{a=1}^3{X}^a{q}_a+\sum _{i=1}^2{\Omega }^i{j}_i\right),$$

$$\displaystyle\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})=\operatorname{Tr}\Big{|}_{{\cal Q}=0}(-1)^{F}e^{i(\Delta_{a}Q_{a}+\omega_{i}J_{i})}=\operatorname{Tr}\Big{|}_{{\cal Q}=0}e^{\pi iR(\Delta,\omega)}e^{-\operatorname{\mathbb{I}m}(\Delta_{a}Q_{a}+\omega_{i}J_{i})}\,,$$

Full text

Dated: March 16, 2024

1 Introduction

We know since the seventies that black holes in general relativity are thermodynamic objects, in particular they have a temperature and an entropy (Bekenstein:1972tm, ; Bardeen:1973gs, ; Hawking:1974sw, ). One of the most intriguing and celebrated relation in theoretical physics is the Bekenstein–Hawking formula expressing the entropy of a black hole in terms of the area $A$ of the event horizon and the natural constants $c,\hbar,k_{{\rm B}},G_{{\rm N}}$

[TABLE]

merging in a single expression gravity, relativity, statistical mechanics and quantum theory. Using a standard statistical mechanics interpretation, we are led to write the entropy as

[TABLE]

in terms of the number $n$ of microscopic degrees of freedom of the system, the microstates of the black hole. It is a main challenge for all theories of quantum gravity to give an explanation of the Bekenstein–Hawking formula and to identify the corresponding microstates.

The Bekenstein–Hawking formula suggests that the microstates are localized on the event horizon. This is an instance of the holographic principle (tHooft:1993dmi, ; Susskind:1994vu, ), which states that a volume of space in quantum gravity can be described just in terms of boundary degrees of freedom. A concrete incarnation of this general principle is the AdS/CFT correspondence (Maldacena:1997re, ), a cornerstone of modern theoretical physics. The correspondence explicitly identifies a theory of quantum gravity in Anti-de-Sitter space-time (AdS) with a dual conformal quantum field theory (CFT), which we may naively think of as living on the boundary of AdS. String theory provides many explicit examples of AdS backgrounds and dual CFTs. The most famous is the original example of dual pairs, type IIB string theory on AdS${}_{5}\times S^{5}$ and the maximally supersymmetric gauge theory in four dimensions, ${\cal N}=4$ super-Yang–Mills (SYM). Impressive checks of the correctness of this duality have been made over the last twenty years.

In this context, one of the great successes of string theory is the microscopic explanation of the entropy of certain asymptotically flat black holes. The first result was obtained in Strominger:1996sh , more than twenty years ago, and has been followed by an immense literature, which would be too long to refer to. However, quite curiously, no similar results exist for asymptotically AdS black holes in dimension four or greater until very recently. Since holography suggests that the microstates of the black hole correspond to states in a dual conformal field theory, the AdS/CFT correspondence is the natural setting where to explain the black hole entropy in terms of a microscopical theory. In the past, various attempts have been made to derive the entropy of a class of rotating black holes in AdS${}_{5}\times S^{5}$ in terms of states of the dual ${\cal N}=4$ SYM theory in the large $N$ limit, but none was completely successful. The more recent advent of localization techniques for supersymmetric quantum field theories, in the spirit of Nekrasov:2002qd ; Pestun:2007rz , opens a new perspective on this problem. In these lectures we discuss how to use localization to derive the entropy for a class of supersymmetric black holes in AdS4 and AdS5 and discuss the current status for other black holes appearing in holography. We will work in dimension equal or greater than four. AdS3 is somehow special, and well-studied in the literature, and it will not be discussed in these notes.

1.1 Content of the lectures

The first microscopic counting for AdS black holes in dimension equal or greater than four was performed in Benini:2015eyy , considering asymptotically AdS4 static supersymmetric black holes. One of the main characteristics of this class of black holes is the presence of magnetic charges that correspond to a topological twist in the dual field theory. The black holes considered in Benini:2015eyy can be embedded in M theory and are asymptotic to AdS${}_{4}\times S^{7}$. They are dual to a topologically twisted compactification of the ABJM theory in three dimensions (Aharony:2008ug, ), and their entropy scales as $N^{3/2}$ at large $N$, as familiar from three-dimensional holography. In these lectures, we will focus on this example as a prototype for many similar computations. The entropy can be extracted from the (regularized) Witten index of the quantum mechanics obtained by compactifying ABJM on a Riemann surface $\Sigma_{\mathfrak{g}}$. Holographically, the quantum mechanics describes the physics of the near horizon geometry AdS${}_{2}\times\Sigma_{\mathfrak{g}}$ of the black holes. The index can be computed, using localization, as the three-dimensional supersymmetric partition function of ABJM on $\Sigma_{\mathfrak{g}}\times S^{1}$, topologically twisted along the Riemann surface $\Sigma_{\mathfrak{g}}$. From this perspective, this computation can be generalized to more general domain wall solutions interpolating between AdSd+n and $AdS_{d}\times\mathcal{M}_{n}$, with a topological twist along the $n$-dimensional compact manifold $\mathcal{M}_{n}$, thus providing general tests of holography.

In the second part of these lectures, we also discuss the analogous problem for supersymmetric rotating electrically charged black holes in AdSd. The main difference with the previous case is the absence of a topological twist. The entropy for such black holes should be obtained by counting states with given electric charge and spin in the dual field theory and the natural observable to consider is the superconformal index, which receives contributions precisely from the BPS states of the theory. Recent results in this direction have been obtained starting with the work (Cabo-Bizet:2018ehj, ; Choi:2018hmj, ; Benini:2018ywd, ) and we will discuss these recent progresses obtained in various overlapping limits but all pointing towards a unified picture.

Many of the relevant field theory computations are performed using localization. This allows to reduce exact path integrals in quantum field theory to matrix models, which can be solved in the large $N$ limit combining standard and more recent techniques. One successful approach for the physics of black holes, that works both in four (Benini:2015eyy, ; Hosseini:2016cyf, ) and five dimensions (Benini:2018ywd, ), involves writing the matrix model partition function as a sum of Bethe vacua (Nekrasov:2009uh, ) of an auxiliary theory. Some technical aspects of this approach are discussed in Sect. 3.

Let us stress that many derivations of the entropy for asymptotically flat black holes involve the use of the Cardy formula for the asymptotic growing of states of a two-dimensional CFT. In the localization approach for AdS black holes, we directly count the number of microstates using an index.111Although some results about rotating electrically charged black holes have been obtained in a Cardy limit, which provides a generalization of the Cardy formula to higher dimensions. We will briefly make contact with the original Cardy approach based on a two-dimensional CFT in Sect. 5 where we discuss black strings.

These lectures assume some familiarity with supersymmetry and the main examples of holographic dualities in various dimensions. We assume that the reader knows that ${\cal N}=4$ SYM in four dimensions is dual to AdS${}_{5}\times S^{5}$, the ABJM theory to AdS${}_{4}\times S^{7}$ and the so-called $(2,0)$ theory in six dimensions to AdS${}_{7}\times S^{4}$. Some preliminary exposure to localization computation222We refer to Marino:2011nm for a nice introduction and Pestun:2016jze for a more comprehensive review. would be also useful although not necessary. We review instead in Sect. 4.1 the elements of gauged supergravity that are needed for these lectures.

The lectures are organized as follows. In Sect. 2 we give a general overview of the various classes of supersymmetric black holes that are relevant for holography, stressing that they fall into two main classes, distinguished by the presence or absence of magnetic charges (or more precisely of a twist). We also discuss how we should compute their entropy using field theory methods and we introduce the concepts of entropy functional and attractor mechanism that prove useful in the comparison between gravity and field theory. In Sect. 3 and 4 we discuss in details the example of dyonic black holes asymptotic to AdS${}_{4}\times S^{7}$ and dual to a twisted compactification of ABJM. In Sect. 3 we discuss the field theory aspects of the story, introducing the topologically twisted index and showing how to evaluate it using localization. In Sect. 4 we perform the large $N$ limit of the resulting matrix model and we compare with gravity. In Sect. 5 we discuss black string solutions interpolating between AdS5 and AdS${}_{3}\times\Sigma_{\mathfrak{g}}$, as a prototype of more general domain walls interpolating between AdS spaces that can be studied with these techniques. In Sect. 6 and 7 we discuss the case of rotating electrically charged black holes in AdSd. In Sect. 6 we discuss the field theory aspects of the story, introducing the superconformal index. Finally, in Sect. 7 we discuss the state of the art of the comparison between field theory and gravity for such black holes.

2 AdS black holes in $d\geq 4$

In these lectures we are interested in supersymmetric black holes that can be embedded in string theory or M-theory and are asymptotic to AdSd vacua with a known field theory dual. There are many such black holes that can be embedded in maximally supersymmetric backgrounds. For example, we can find supersymmetric black holes in AdS${}_{5}\times S^{5}$, AdS${}_{4}\times S^{7}$ and AdS${}_{7}\times S^{4}$, whose dual field theories are well known. Supersymmetric black holes are extremal and have zero temperature. They also satisfy a BPS condition that relates their mass to the other conserved charges. In the limit where gravity is weakly coupled, the entropy of a black hole can be computed with the Bekenstein–Hawking formula

[TABLE]

where $A$ is the area of the horizon and $G_{\text{N}}$ is the Newton constant. We set $c=\hbar=k_{{\rm B}}=1$.

We now discuss some general features of these black holes and their holographic interpretation.

2.1 AdS black holes and holography

For the purposes of holography, we can divide the known supersymmetric black holes in dimension $d\geq 4$ into two main classes, distinguished by how supersymmetry is realized and the holographic interpretation. In particular, they are distinguished by existence (or absence) of certain magnetic charges corresponding to a topological twist.

2.1.1 Kerr–Newman black holes and generalizations

The first class of black holes consists of supersymmetric electrically charged rotating black holes (Kerr–Newman). The most famous examples are the type IIB supergravity black holes asymptotic to AdS${}_{5}\times S^{5}$ found in Gutowski:2004yv ; Gutowski:2004ez ; Chong:2005da ; Chong:2005hr ; Kunduri:2006ek . They depend on two angular momenta corresponding to two Cartan isometries of AdS5

[TABLE]

and three electric charges under the Cartan isometries of $S^{5}$

[TABLE]

parameterizing rotations in the internal space $S^{5}$. Supersymmetry actually imposes a relation among the conserved charges, $f(j_{1},j_{2},q_{1},q_{2},q_{3})=0$, so that there are only four independent parameters.333For many different families of BPS black holes, supersymmetry imposes constraints among the charges. The reason why this happens is still unclear. In the case of AdS${}_{5}\times S^{5}$, BPS hairy black holes depending on all the charges have been found in Markeviciute:2018yal ; Markeviciute:2018cqs . These black holes preserve two real supercharges out of the original thirty-two of type IIB supergravity on AdS${}_{5}\times S^{5}$. The five-dimensional part of the metric is asymptotic to AdS5 with $\mathbb{R}\times S^{3}$ as conformal boundary. As well known, type IIB string theory on AdS${}_{5}\times S^{5}$ is dual to ${\cal N}=4$ SYM in four dimensions. It is then a natural expectation that the black holes correspond holographically to an ensemble of states of ${\cal N}=4$ SYM on $\mathbb{R}\times S^{3}$ that preserve the same supersymmetries and have the same electric charges and the same angular momenta. It is also natural to expect that, by counting all the $1/16$ BPS states of ${\cal N}=4$ SYM on $\mathbb{R}\times S^{3}$ with electric charges $(q_{1},q_{2},q_{3})$ and spin $(j_{1},j_{2})$, we should be able to reproduce the entropy of these black holes. We will work under these assumptions. We are interested in macroscopic black holes whose entropy, when expressed in terms of field theory data, scales as $O(N^{2})$, where $N$ is the number of colors of the dual field theory.

The situation is analogous in other dimensions (Chong:2004dy, ; Cvetic:2005zi, ; Chow:2007ts, ; Chow:2008ip, ; Hristov:2019mqp, ). Consider the maximally supersymmetric backgrounds AdS${}_{4}\times S^{7}$ and AdS${}_{7}\times S^{4}$ in M-theory. The isometry of AdS${}_{4}\times S^{7}$ is $SO(2,3)\times SO(8)$ and we can find electrically charged rotating black holes depending on one angular momentum $j$ and four electric charges $(q_{1},q_{2},q_{3},q_{4})$ with a constraint. They preserve two real supercharges. We expect to reproduce the entropy of such black holes by counting all $1/16$ BPS states of the dual field theory on $\mathbb{R}\times S^{2}$ with the same quantum numbers. As well-known, the dual of M-theory on AdS${}_{4}\times S^{7}$ is the three-dimensional ABJM theory at Chern–Simons level $k=1$ (Aharony:2008ug, ). The entropy of these black holes scales as $O(N^{3/2})$. Similarly, since the isometry of AdS${}_{7}\times S^{4}$ is $SO(2,6)\times SO(5)$, there are black holes depending on three angular momenta $(j_{1},j_{2},j_{3})$ and two electric charges $(q_{1},q_{2})$ with a constraint, again preserving two real supercharges.444Actually, only black holes with equal charges or equal momenta have been studied. However, we expect a family with at least four independent parameters to exist. In this case, the entropy, which scales as $O(N^{3})$, should be reproduced by counting states in the ${\cal N}=(2,0)$ theory in six dimensions (Witten:1995zh, ) on $\mathbb{R}\times S^{5}$.

Notice that all these supersymmetric black holes rotate. If we turn off the angular momenta $j_{i}$, we find singularities.

In principle, although there are not so many examples in the literature, we expect the existence of similar supersymmetric black holes in more general type II or M-theory backgrounds with an AdSd vacuum, rotating in AdSd and charged under the isometries of the compactification manifold. The holographic interpretation is similar. For example, for type IIB black holes in AdS${}_{5}\times SE_{5}$ (Klebanov:1998hh, ), where $SE_{5}$ is a five-dimensional Sasaki–Einstein manifold, we should try to match the entropy by counting $1/4$ BPS states of the dual ${\cal N}=1$ superconformal field theory on $\mathbb{R}\times S^{3}$.

2.1.2 Magnetically charged black holes with a twist

The second class of black holes are characterized by the existence of certain magnetic charges. We should more properly refer to such black holes as solutions where supersymmetry is realized with a topological twist, as we will see. Although there are examples in higher dimensions,555There are exotic $d$-dimensional solutions with horizon AdS${}_{2}\times\mathcal{M}_{d-2}$, where $\mathcal{M}_{d-2}$ is a compact manifold, with non-zero fluxes of the gauge fields on $\mathcal{M}_{d-2}$. we will focus on four dimensions, where these black holes arise naturally. Indeed we can have both magnetic and electric charges in $d=4$ and it is natural to consider dyonic black holes. There exists a family of BPS black holes in AdS${}_{4}\times S^{7}$ depending on one angular momentum $j_{1}$ in AdS4 and on electric and magnetic charges

[TABLE]

under the abelian $U(1)^{4}\subset SO(8)$ isometries of $S^{7}$ (Cacciatori:2009iz, ; DallAgata:2010ejj, ; Hristov:2010ri, ; Katmadas:2014faa, ; Halmagyi:2014qza, ; Hristov:2018spe, ). Supersymmetry requires a linear constraint among the magnetic charges $p^{a}$ and non-linear ones among the conserved charges, so that we have a six-dimensional family of rotating, dyonic black holes. For this class of solutions, we can also turn off rotation and have static supersymmetric black holes.666These BPS black holes have been found in ${\cal N}=2$ gauged supergravity in $d=4$ with vector multiplets and later uplifted to M-theory. The first static example, with a hyperbolic horizon, was found in Sabra:1999ux in minimal gauged supergravity. The first static spherically symmetric example was found in Cacciatori:2009iz , further discussed in DallAgata:2010ejj ; Hristov:2010ri and generalized to the dyonic case in Katmadas:2014faa ; Halmagyi:2014qza . The rotating case has been discussed in Hristov:2018spe (for other examples, see also (Daniele:2019rpr, )).

In general, AdS4 black holes with magnetic charges are qualitatively different from those with zero magnetic charge, as first noticed in Romans:1991nq and elaborated in Hristov:2011ye ; Hristov:2011qr . The difference is well explained using holography. Consider the black holes as solutions of an effective four-dimensional theory with a AdS4 vacuum dual to a boundary conformal field theory. For most of these lectures, the CFT will be ABJM, but the following arguments apply to black holes in general compactifications and more general CFTs with at least ${\cal N}=2$ supersymmetry. For our purposes, we need the terms of the effective theory describing the dynamics of the metric and of the vector fields $A_{\mu}^{a}$ corresponding to the abelian isometries of the internal manifold. Their dynamics is described by an Einstein-Maxwell theory

[TABLE]

where, in general, the matrix of coupling constants depends on the scalar fields $\phi_{i}$ of the theory. According to the rules of holography, gauge fields in the bulk correspond to global symmetries in the boundary CFT. For example, in the case of AdS${}_{4}\times S^{7}$ we are interested in the four fields $A_{\mu}^{a}$ corresponding to the abelian isometries $U(1)^{4}\subset SO(8)$ and they couple to the field theory conserved currents $J^{\mu a}$

[TABLE]

associated to the Cartan generators of the $SO(8)$ R-symmetry of ABJM at Chern–Simons level $k=1$. Focusing for simplicity on the static case, one finds that, near the boundary, the black hole solutions behave as

[TABLE]

where $r$ is some large radial coordinate, the ellipsis refers to terms suppressed by inverse powers of $r$, and $x$ are coordinates on the boundary manifold. For spherically symmetric black holes the boundary manifold is $\mathcal{M}_{3}=\mathbb{R}\times S^{2}$. However we can have more exotic solutions with horizon AdS${}_{2}\times\Sigma_{\mathfrak{g}}$, where $\Sigma_{\mathfrak{g}}$ is a Riemann surface of genus $g$, and, in this case, $\mathcal{M}_{3}=\mathbb{R}\times\Sigma_{\mathfrak{g}}$. Holography tells us that we should interpret (2.7) as the dual of our CFT defined on the curved manifold $\mathcal{M}_{3}$. What can we say about $A_{\mu}^{a}$? Recall again the basic rules of the AdS/CFT correspondence (Klebanov:1999tb, ). Any field $\phi(x,r)$ in AdS is associated with an operator $O(x)$ in the dual CFT. If we have an expansion

[TABLE]

of the solution of the second order equations of motion for $\phi$,777The values of $\alpha_{i}$ are related to the conformal dimension of $O$. we interpret the non-normalizable piece, $\phi_{0}(x)$, as a deformation of the original CFT with the corresponding operator $O$,

[TABLE]

while we interpret the normalizable one, $\phi_{1}(x)$, as a vacuum expectation value (vev) for $O$, $\langle O\rangle\neq 0$. More precisely, if $\phi_{0}(x)\neq 0$, we are deforming the CFT with $O$; if $\phi_{0}(x)=0$ and $\phi_{1}(x)\neq 0$ we have a state of the CFT with non zero vev for $O$. There are situations where both modes $\phi_{0}(x)$ and $\phi_{1}(x)$ are normalizable (or better have finite energy). In this case there are different possible quantizations of the same theory and we have to choose who plays the role of $\phi_{0}$. Massless vector fields in AdS4 allow for different types of quantizations, related to electric/magnetic duality in the bulk, and this leads to interesting applications, but this is not strictly the most important point. What is important is that, in the expansion (2.7) for $A_{\mu}^{a}$ both leading and sub-leading terms are turned on. The field $A_{\mu}^{a}$ has a leading contribution for $r\gg 1$ that approaches a constant value on the boundary $\mathcal{M}_{3}=\mathbb{R}\times\Sigma_{\mathfrak{g}}$, corresponding to the magnetic charge of the black hole

[TABLE]

and sub-leading terms (the ellipsis in (2.7)) that encode information about the electric charges. This means that a dyonic black hole is holographically dual to a deformation of the dual CFT. In the natural quantization of the theory, the non-zero value of $A_{\mu}^{a}$ at the boundary corresponds to the deformation

[TABLE]

This deformation in field theory is equivalent to turning on a background gauge field for a global symmetry. For example, on $S^{2}$ we would turn on a background which is just the familiar Dirac monopole $\hat{A}_{\mu}^{a}=-\frac{1}{2}p^{a}\cos\theta d\phi$. On a torus $T^{2}$ we would just turn on a background constant magnetic field. Fields satisfying (2.10) can be also written explicitly for all $\Sigma_{\mathfrak{g}}$ but their expression is not particular illuminating. We will see in Sect. 3 that the deformation (2.11) is compatible with supersymmetry.

To understand better what is going on, it is useful to have a look at how supersymmetry is preserved in the presence of a generic assignment of magnetic charges. We will be very schematic here just to convey the general idea. Consider the case where our effective theory is a certain ${\cal N}=2$ gauged supergravity in four dimensions, corresponding to a ${\cal N}=2$ three-dimensional CFT. The effective theory contains the graviphoton field, $A_{\mu}^{R}$, holographically dual to the $U(1)$ R-symmetry of the theory. In general, $A_{\mu}^{R}$ is a linear combination of the vector fields $A_{\mu}^{a}$ corresponding to the isometries of the internal manifold. For the solution to be supersymmetric, all fermion variations in the black hole background must be zero. The gravitino variation in ${\cal N}=2$ gauged supergravity is schematically given by

[TABLE]

The magnetically charged static black holes of interest in this section satisfy the BPS condition $\delta\psi_{\mu}=0$ by cancelling the spin connection with a background field for the R-symmetry. More precisely, we can regard the spin connection $\omega_{\mu}$ along $\Sigma_{\mathfrak{g}}$ as a $U(1)$ gauge field. An explicit computation shows that $\omega_{\mu}$ is just a monopole of charge $2-2\mathfrak{g}$, as the familiar relation $\frac{1}{2\pi}\int_{\Sigma_{\mathfrak{g}}}R=2-2\mathfrak{g}$, with $R=d\omega$, clearly shows. Since $A_{\mu}^{R}$ is a linear combination of the $A_{\mu}^{a}$, it is also a monopole, with charge given by a linear combination of the $p^{a}$. By appropriately choosing this linear combination888This is the linear constraint on the magnetic charges of the black hole that we mentioned before. and the spinor $\epsilon$, we can cancel the second and the third term on the right hand side in (2.12). We will come back to more precise expressions in Sect. 3. For the dyonic static black holes, restricting the index $\mu$ to lie along $\Sigma_{g}$, one discovers that the ellipsis cancels independently and we are left with the equation

[TABLE]

This equation is solved by taking $\epsilon$ constant along $\Sigma_{\mathfrak{g}}$. One also finds that the other components of the supersymmetry variations imply that $\epsilon$ is time-independent but, in general, has a non-trivial profile in $r$.

This discussion can be also applied to the dual CFT. By restricting the variations to the boundary, we see that the field theory on $\mathbb{R}\times\Sigma_{\mathfrak{g}}$ is invariant under supersymmetry transformations with a constant spinor. The very same mechanism is at work on the boundary: we are turning on a magnetic background for the R-symmetry that compensates the spin connection. In quantum field theory, this construction is well-known (Witten:1988ze, ). It is called topological twist, as we will discuss in details in Sect. 3. The conclusion is that the dual CFT is deformed by the presence of magnetic fluxes for all the global and R-symmetries, and, in particular, it is topologically twisted by the magnetic flux for the R-symmetry. Notice that our argument was based on ${\cal N}=2$ supersymmetry with a $U(1)$ R-symmetry. Theories can have a larger R-symmetry group, like ABJM, or many flavor symmetries. In these cases, the choice of a $U(1)$ R-symmetry is not unique. Each choice corresponds to a different twist. We can indeed think of the magnetic charges $p^{a}$ as parameterizing a family of inequivalent twists. It is important to remember, however, that a linear combination of the $p^{a}$ is fixed by the condition that the background for the selected $U(1)$ R-symmetry cancels the spin connection. There are only $n_{V}-1$ independent magnetic charges, where $n_{V}$ is the number of massless vectors. This number is $n_{V}=4$ for ABJM.

The interpretation of the general rotating dyonic black hole is more complicated but similar in spirit. We can have rotation only in the spherically symmetric case, where $j$ is the spin along $S^{2}$. Rotations in the bulk correspond to turning on an Omega-background (Nekrasov:2003rj, ) in the boundary theory on $S^{2}$. The theory is still topologically twisted.

All this should be contrasted with the black holes discussed in Sects. 2.1.1 where there is no cancellation between the spin connection and the R-symmetry. It can be expressed more formally in a difference between the supersymmetry algebra, as discussed in Hristov:2011ye ; Hristov:2011qr . The real discriminant among the two class of black holes is the topological twist, or equivalently a magnetic charge associated with the R-symmetry. There exist black holes with non-zero magnetic charges for the flavor symmetries only.999See Hristov:2019mqp for example of rotating dyonic black holes with flavor magnetic charges. They correspond to a CFT in a magnetic background but with no topological twist. From the point of view of micro-state counting using holography, they are more similar in spirit to the black holes discussed in Sect. 2.1.1.

Now it is clear what we should do in order to compute the entropy of magnetically charged black holes (with a twist) using field theory: enumerate all the states with electric charges $q_{i}$ and angular momentum $j$ and the right amount of supersymmetry in the twisted CFT on $\mathbb{R}\times\Sigma_{\mathfrak{g}}$. The theory is topologically twisted101010And also Omega-deformed if there is rotation. by the magnetic background for a $U(1)$ R-symmetry and possibly deformed by magnetic fluxes for all other global symmetries.

2.2 Computing the entropy

It is reasonable to expect that we can recover the entropy of the two classes of AdSd black holes by enumerating supersymmetric states in the dual field theory on $\mathbb{R}\times\mathcal{M}_{d-2}$. For all our examples, the preserved supersymmetry $\mathcal{Q}$ satisfies an algebra of the form

[TABLE]

where $H$ is the Hamiltonian and $Q_{a}$ and $J_{i}$ are the charge operators associated with the global symmetries and the angular momenta, respectively, with certain constants $\mu_{a}$ and $\nu_{i}$ that depend on the model. The explicit form of the algebra is different for different types of black holes and it will be discussed in details in the rest of these lectures. For the moment, we notice that the R-symmetry charge enters in the supersymmetry algebra for Kerr-Newmann black holes but not for topologically twisted ones. Supersymmetric states are annihilated by $\mathcal{Q}$ and their energy is determined by the BPS condition111111It follows from the algebra that all states in the theory satisfy the BPS bound $E\geq\mu_{a}q_{a}+\nu_{i}j_{i}$.

[TABLE]

2.2.1 The grand canonical partition function

Enumerating BPS states is equivalent to knowing the grand canonical partition function

[TABLE]

where the trace is taken over the Hilbert space of states on $\mathcal{M}_{d-2}$ that preserves the same amount of supersymmetry of the black hole, and $\Delta_{a}$ and $\omega_{i}$ chemical potentials conjugated to $Q_{a}$ and $J_{i}$, respectively. In practical applications, ${\cal Z}$ is a function of complex chemical potentials and converges in an appropriate domain of the complex plane for the fugacities $y_{a}=e^{i\Delta_{a}}$, $\zeta_{i}=e^{i\omega_{i}}$. In the previous formula, $c(q_{a},j_{i})$ is the number of supersymmetric states of electric charge $q_{a}$ and angular momentum $j_{i}$. Electric and magnetic charges enter in an asymmetric way in this construction. The magnetic charges $p^{a}$ enter explicitly as a set of couplings in the Lagrangian of the deformed CFT, while the electric charges $q_{a}$ are introduced through chemical potentials. In particular, for magnetically charged black holes with a twist, the trace must be taken in the topologically twisted theory.

The grand canonical partition function (2.16) should also enumerate the BPS states in the dual gravity theory. Our working assumption is that, for large charges (scaling with appropriate powers of $N$) the supersymmetric density of states is dominated by the macroscopic black holes discussed before. Under this assumption, by the very definition of entropy, the entropy of the black hole is given by

[TABLE]

where the dependence on the magnetic charges $p^{a}$, if present, is hidden in the form of the function $c$. If $\mathcal{Z}(\Delta_{a},\omega_{i})$ is known, the entropy can be extracted as a Fourier coefficient

[TABLE]

with an appropriate integration contour. In the limit of large charges, this can be evaluated by a saddle point approximation

[TABLE]

where $\bar{\Delta}_{a}$ and $\bar{\omega}_{i}$ are obtained by extremizing the functional

[TABLE]

with respect to $\Delta_{a}$ and $\omega_{i}$,

[TABLE]

We see that the entropy is just the Legendre transform of the logarithm of the partition function.

To understand better this point, recall that we are interested in extremal supersymmetric black holes. In particular, they have zero temperature. The standard thermodynamics relation for the grand canonical partition function of a system with temperature $T$,

[TABLE]

looks singular in the zero-temperature limit. However, supersymmetric states satisfy the BPS condition (2.15), namely $E=\mu_{a}q_{a}+\nu_{i}j_{i}$. When we take the zero temperature limit, we need also to scale $\tilde{\Delta}_{a}(T)=-i\mu_{a}+\Delta_{a}T$ and $\omega_{i}(T)=-i\nu_{i}+\omega_{i}T$. In this way we obtain the Legendre transform $S=\log\mathcal{Z}-i\Delta_{a}q_{a}-i\omega_{i}j_{i}$.121212 For explicit examples of this zero-temperature limit from the gravitational point of view see Silva:2006xv , and, in particular, for AdS5 black holes, see Cabo-Bizet:2018ehj ; Choi:2018hmj .

The entropy is also obtained via a Legendre transform in many other approaches, as the OSV conjecture (Ooguri:2004zv, ) and the Sen’s quantum entropy functional (Sen:2005wa, ; Sen:2008vm, ; Sen:2009vz, ) for asymptotically flat black holes.

2.2.2 The supersymmetric index

So far everything was simple. The problem is that $\mathcal{Z}(\Delta_{a},\omega_{i})$ is too hard to compute, in general. For electrically charged rotating black holes in AdS${}_{5}\times S^{5}$, computing $\mathcal{Z}(\Delta_{a},\omega_{i})$ would correspond to enumerate all the 1/16 BPS states of ${\cal N}=4$ SYM. For comparison, in a four-dimensional theory with ${\cal N}=1$ supersymmetry, this would correspond to count all the 1/4 BPS states. While almost everything is known about the counting of 1/2 BPS states in an ${\cal N}=1$ theory (Kinney:2005ej, ; Benvenuti:2006qr, ), the analogous problem for 1/4 BPS states is still open.

What we can instead compute is a supersymmetric index

[TABLE]

with the insertion of the fermionic number $(-1)^{F}$. Standard arguments tell us that we can also write

[TABLE]

if $Q_{a}$ and $J_{i}$ commute with ${\cal Q}$. In the first step of the previous identification we used the fact that states with ${\cal Q}\neq 0$ do not contribute to the trace, since bosonic and fermionic states are paired and contribute with opposite sign.131313This is the logic of the Witten index (Witten:1982df, ). For every state $|\Omega\rangle$ not annihilated by $\mathcal{Q}$ there is a state ${\cal Q}|\Omega\rangle$ of opposite statistics and the same value of $\{{\cal Q}^{\dagger}\,,{\cal Q}\}$. Since these states have opposite value of $(-1)^{F}$, their contribution cancels in the trace in (2.24). Therefore, only supersymmetric states, annihilated by ${\cal Q}$, contribute to the Witten index. See also the discussion in Sect. 3.3.1. In particular, the index is independent of $\beta$. In the second step, we identified the trace at temperature $1/\beta$ with the Euclidean path integral $Z^{susy}_{S^{1}\times\mathcal{M}_{d-1}}$ of the theory compactified on a circle of radius $\beta$.141414 It is a standard textbook result that the finite temperature partition function $\operatorname{Tr}e^{-\beta H}$ can be expressed as an Euclidean partition function with time compactified on a circle of radius $\beta$ and periodic boundary conditions for bosons and anti-periodic boundary conditions for fermions. In a supersymmetric partition functions, bosons and fermions should have the same boundary conditions and this is enforced by the fermionic number $(-1)^{F}$. The extra insertion of $e^{\beta(\mu_{a}{\cal R}_{a}-\nu_{i}{\cal J}_{i})}$ just introduces twisted boundary conditions along $S^{1}$ for the symmetries associated with ${\cal R}_{a}$ and ${\cal J}_{i}$. The latter partition function can be computed using localization techniques in quantum field theory as we discuss in Sect. 3.

In general, $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})\neq\mathcal{Z}(\Delta_{a},\omega_{i})$. First of all, the index can accommodate fugacities only for the conserved charges that commute with ${\cal Q}$ and, in general, contains less parameters than the BPS partition function. We will see that this is not a major problem for the black holes considered in this paper since they also have a constraint among charges. More importantly, the entropy should count all the BPS ground states of the theory, while the index counts bosonic ground states with a plus and fermionic ground states with a minus. However, it may happen for particular theories that the majority of ground states are of definite statistic. In this case there is no cancellation between bosonic and fermionic ground states and the index correctly reproduces the entropy in a suitable limit. This typically happens for asymptotically flat black holes in the limit of large charges, and we may hope that the same is true for asymptotically AdS black holes in the large $N$ limit. In the case of certain asymptotically flat spherically symmetric black holes, there is an extra symmetry that implies $(-1)^{F}=1$ on the relevant set of states (Sen:2009vz, ) and one can prove that $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})=\mathcal{Z}(\Delta_{a},\omega_{i})$. A similar argument for asymptotically AdS black holes is more subtle and it is discussed in Sect. 2.4. More generally, we can always rely on an explicit computation, and, as we will see in the rest of these lectures:

• •

for magnetically charged black holes in AdS4, the dual field theory is topologically twisted. $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})$ is the so-called topologically twisted index that we define and study in Sect. 3. There is no cancellation between bosonic and fermionic ground states and the index correctly reproduces the entropy at large $N$, as we will see in Sect. 4;

• •

for electrically charged rotating black holes, we are just counting states of the CFT on $\mathbb{R}\times S^{d-1}$. $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})$ is the superconformal index, whose properties we discuss in Sect. 6. The superconformal index is known to have large cancellations between bosons and fermions for real values of the fugacities (Kinney:2005ej, ), and this would suggest that we really need to compute the original BPS partition function $\mathcal{Z}(\Delta_{a},\omega_{i})$. However, it has been recently realized that introducing phases for the fugacities can obstruct the cancellation between bosonic and fermionic states and that the entropy of KN black holes is indeed correctly captured by the index for complex values of the chemical potentials. We will discuss these results in Sect. 7.

Notice also that the entropy and the index of asymptotically flat black holes coincide at leading order in the charges but are in general different when corrections are included (Dabholkar:2010rm, ). We might expect the same for asymptotically AdS black holes.

2.3 Entropy functionals and the attractor mechanism

In the limit where gravity is weakly coupled, we can compute the entropy of a black hole from the area of the horizon using the Bekenstein–Hawking formula (2.1). The area can be extracted from the explicit solution of the relevant gauged supergravity, which typically contains many scalars $X^{I}(r)$ varying with the radial distance. In principle, the area may depend on many parameters including asymptotic moduli. However, the microscopic entropy of our black holes is just a function of the conserved charges $q_{a}$ and $j_{i}$. This is explicitly realized through an attractor mechanism: independently of the asymptotic moduli, the scalar fields approach a value at the horizon, $X^{I}_{*}(q_{a},j_{i})$, that is a function of the conserved charges only. Moreover, this mechanism often allows to express the area, and therefore the entropy, in terms of the values of the scalar fields at the horizon with simple algebraic equations. This is the case of the attractor mechanism for ${\cal N}=2$ supergravity discovered in Ferrara:1996dd ; Ferrara:1995ih . It is also the idea behind Sen’s quantum entropy function (Sen:2005wa, ) that allows to find the entropy of black holes with AdS2 horizon including higher derivative corrections. In all these cases, one can define some sort of entropy functional $S(X^{a},\Omega^{i},q_{a},j_{i})$, which is a function of the conserved charges and the horizon value $X^{a},\,\Omega^{a}$ of the scalar fields and possibly other modes, and whose extremization with respect to $X^{a}$ and $\Omega^{i}$ reproduces the entropy. This extremization is expected to be the gravity analog of (2.19).

In order to make the comparison between gravity and field theory more manifest it is convenient to write the entropy functional as a Legendre transform

[TABLE]

where $X^{a}$ and $\Omega^{i}$ are now interpreted as the black hole chemical potentials and ${\cal E}(X^{a},\Omega^{i})$ as the black hole gran-canonical partition function. According to standard arguments (Gibbons:1976ue, ), ${\cal E}(X^{a},\Omega^{i})$ can be identified with the on-shell action of the Euclidean continuation of the black hole.

This description fully agrees with the field theory picture if we identify $X^{a}$ and $\Omega^{i}$ with $\Delta^{a}$ and $\omega_{i}$ and ${\cal E}(X^{a},\Omega^{i})$ with $\log\mathcal{Z}(\Delta_{a},\omega_{i})$. The latter is indeed the field theory gran-canonical partition function and, according to the rule of holography, should be also identified with the on-shell action of the gravity solution corresponding to the chemical potentials $\Delta^{a}$ and $\omega_{i}$.

The attractor mechanism has played an important role in the interpretation of the field theory result leading to the black hole entropy. For example, the attractor mechanism in N = 2 gauged supergravity (Cacciatori:2009iz, ; DallAgata:2010ejj, ) for static dyonic black holes in AdS${}_{4}\times S^{7}$ predicts

[TABLE]

with the constraint $\sum_{a=1}^{a}X^{a}=2\pi$ and this result perfectly matches with the field theory computation based on the topologically twisted index (Benini:2015eyy, ) as we will discuss in Sect. 4 (see formula (4.48)). For other black holes the attractor mechanism is not known in supergravity, and the entropy functional has been written using combined field theory and gravity intuition. This approach was successfully used in Hosseini:2017mds to write an entropy functional for KN black holes in AdS${}_{5}\times S^{5}$,

[TABLE]

with the constraint $\sum_{a=1}^{3}X^{a}-\sum_{i=1}^{2}\Omega^{i}=2\pi$. This result has been instrumental in the later developments and has been matched with field theory computations based on the superconformal index (Cabo-Bizet:2018ehj, ; Choi:2018hmj, ; Benini:2018ywd, ), as we will discuss in Sect. 7. Entropy functional for other electrically charged and rotating black holes in diverse dimensions has been later found in Hosseini:2018dob ; Choi:2018fdc and, in some cases, successfully compared to quantum field theory expectations. These entropy functionals can be also obtained by computing the zero-temperature limit of the on-shell action of a class of supersymmetric but non-extremal Euclidean black holes (Cabo-Bizet:2018ehj, ; Cassani:2019mms, ).

A general field theory inspired formula for an entropy functional that generalizes (2.26) and (2.27) and covers all existing black hole solutions in four and five dimensions has been discussed in Hosseini:2019iad .

2.4 $\mathcal{I}$-extremization

We can provide a perhaps more speculative but intriguing explanation of the extremization principle (2.19) based on the renormalization group flow. All the black holes we will consider have a (possibly warped) AdS2 factor in the near-horizon region. This suggests the existence of a superconformal quantum mechanics describing the near-horizon degrees of freedom and whose ground states correspond to the black holes microstates. The superconformal algebra associated with AdS2 is su$(1,1|1)$ and contains the bosonic factor $SL(2,\mathbb{R})\times U(1)_{R}$, where $U(1)_{R}$ is the R-symmetry. From the field theory point of view, the quantum mechanics arises from the reduction of the dual CFT on $\mathcal{M}_{d-2}$. We can see the black hole as a solution interpolating between AdSd and AdS2 with the dual interpretation of a renormalization group flow across dimensions where a $d-1$-dimensional CFT compatified on $\mathcal{M}_{d-2}$ flows to a infrared superconformal quantum mechanics. The index $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})$ is invariant under renormalization group flow and can be interpreted as the Witten index of the infrared quantum mechanics.

We can write the index as

[TABLE]

where

[TABLE]

In the examples that we will discuss the fermion number $F$ acts precisely as a particular R-symmetry, $F\equiv R_{0}$. In a general supersymmetric theory with global symmetries, the R-symmetry is not unique. If $R_{0}$ is a R-symmetry and $Q$ is a global symmetry, also $R_{0}+Q$ is a R-symmetry. We see that $R(\Delta,\omega)$ is a R-symmetry of our quantum mechanics, and given the symmetries of the problem, we expect it to be the most general R-symmetry we can write. However, when a supersymmetric theory is also conformal there exists the notion of the exact R-symmetry, which is the one singled out by the superconformal algebra. The problem of finding the exact R-symmetry is central in supersymmetric quantum field theory and it is usually associated with an extremization problem. We have indeed $c$-extremization in two dimensions (Benini:2012cz, ), $F$-maximization in three dimensions (Jafferis:2010un, ) and $a$-maximization in four dimensions (Intriligator:2003jj, ). It is tempting to propose that the extremization (2.19) selects among all $R(\Delta,\omega)$ precisely the exact R-symmetry of our quantum mechanics. This principle has been called $\mathcal{I}$-extremization in Benini:2015eyy ; Benini:2016rke . It is suggested by the fact that, in odd dimensions, the exact R-symmetry is obtained by extremizing the supersymmetric sphere partition function, which, in the case of one dimension, is just the Witten index. It would be interesting to prove or disprove this principle for generic holographic theories.

This interpretation would explain why there is no cancellation in the index between vacua of different statistic for large charges. Adapting an argument in Sen:2009vz , we might expect that the microstates are invariant under the superconformal algebra su$(1,1|1)$ and this implies that they have zero exact R-charge. This is certainly true if the quantum mechanics consists of a set of degenerate ground states with an energy gap to the first excited state. Under this assumption and using also large $N$ factorization of the correlation functions, the trace (2.28) becomes

[TABLE]

where we used that, for the values of $\Delta_{a}$ and $\omega_{i}$ that select the exact R-charge, all states have $R(\Delta,\omega)=0$. We then see that the cancelation due to $(-1)^{F}$ is balanced by the non-zero phases of the fugacities at the saddle point and all the ground states contribute with the same sign. The previous equation is consistent with (2.19). Indeed, by taking the logarithm of (2.30), we find

[TABLE]

In all our example, the extremum of (2.19) is actually real and the real part in the previous formula is superfluous.151515That the extremum of (2.19) is real is usually part of the attractor mechanism. As we will see, the fact that the extremization of (2.19) leads to a real number is equivalent to the non-linear constraint among charges imposed by supersymmetry.

Let us also notice that $c$-extremization and $F$- and $a$-maximization have a well-known gravitational dual (Martelli:2005tp, ; Couzens:2018wnk, ; Gauntlett:2018dpc, ) which allows to determine the exact R-symmetry in terms of geometrical data of the supergravity background. A gravitational dual for $I$-extremization has been also proposed in Couzens:2018wnk and further discussed in Gauntlett:2019roi ; Hosseini:2019ddy ; Kim:2019umc .

3 The topologically twisted index

As we discussed in Sect. 2, magnetically charged black holes in AdS4 are dual to topologically twisted CFT3. In this section, we discuss the topologically twisted index in three dimensions, $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})$, defined as the supersymmetric partition function $Z^{susy}_{\Sigma_{\mathfrak{g}}\times S^{1}}(\Delta_{a},\omega_{i})$ with a topological A-twist along $\Sigma_{\mathfrak{g}}$. We will discuss the case of a generic ${\cal N}=2$ Yang–Mills–Chern–Simons theory in three dimensions with an R-symmetry and we will specialize to the ABJM theory in Sect. 4.

The index can be computed in many different ways. We will discuss the localization approach here, following Benini:2015noa ; Benini:2016hjo . The index has been first derived by topological field theory arguments in various examples in Okuda:2012nx ; Okuda:2013fea and discussed in general in Nekrasov:2014xaa . In this second approach, further discussed and generalized in Gukov:2015sna ; Okuda:2015yea ; Closset:2016arn ; Gukov:2016gkn ; Closset:2017zgf ; Closset:2017bse ; Closset:2018ghr , the index is written as a sum of contributions coming from the Bethe vacua, the critical points of the twisted superpotential of the two-dimensional theory obtained by compactifying on $S^{1}$ (Nekrasov:2009uh, ; Nekrasov:2009rc, ). We will discuss the connections between the two approaches in Sect. 3.3.2. The reader that is not interested in quantum field theory properties of the index can just have a quick look at Sect. 3.1, the first part of Sect. 3.2 and Sect. 3.3.2.

3.1 The topological twist

Consider an ${\cal N}=2$ quantum field theory in three dimensions. The ${\cal N}=2$ supersymmetry multiplets are:

• •

the vector multiplet, $(A_{\mu},\lambda,\sigma,D)$, where $\lambda$ is a Dirac spinor and $\sigma$ and $D$ are real scalars. $D$ is an auxiliary field;

• •

the chiral multiplet, $(\phi,\psi,F)$, where $\psi$ is a Dirac spinor and $\phi$ and $F$ are complex scalars. $F$ is an auxiliary field.

These multiplets can be obtained by dimensional reduction from the corresponding ${\cal N}=1$ multiplets in four dimensions. We assume that the theory has an R-symmetry161616The sign of the charges is somehow unconventional (for example, $\lambda$ and $\epsilon$ have charge $-1$) but we keep it for consistency with Benini:2015noa ; Benini:2016hjo .

[TABLE]

with charges $r_{\phi}$ for the chiral fields. The charges should be integrally quantized, as we will discuss in the following. We want to define a supersymmetric theory on $\Sigma_{\mathfrak{g}}\times S^{1}$ using a non-trivial background for the R-symmetry. Let us start, for simplicity, with the case of $S^{2}\times S^{1}$ with metric

[TABLE]

and a non-trivial R-symmetry background field $A_{\mu}^{R}$.

To see if supersymmetry is preserved we can use the approach of Festuccia:2011ws . We promote the metric $g_{\mu\nu}$ and the R-symmetry background field $A_{\mu}^{R}$ to dynamical fields by coupling the theory to supergravity, using an appropriate off-shell formulation. The theory coupled to gravity is invariant under supersymmetry transformations with a local spinorial parameter $\epsilon(x)$. We can recover the rigid theory by freezing the supergravity multiplet to a background value. This can be done, for example, by sending the Planck mass to infinity. In this process we set all supergravity fermionic fields to zero, while keeping a non-trivial background for the metric and $A_{\mu}^{R}$, and possibly some auxiliary fields. The rigid theory so obtained is no more invariant under local supersymmetries. However, it is still invariant under those transformations that preserve the background fields. The supersymmetry variation of the bosonic supergravity fields is automatically zero since it is proportional to the supergravity fermions that vanish in the background. On the other hand, the vanishing of the fermionic variations gives a differential equation for $\epsilon(x)$. The solutions to this equation determine the rigid supersymmetries that are preserved by the curved background.171717These conditions impose constraints on the manifold and the choice of background fields. For examples related to our context see Festuccia:2011ws ; Klare:2012gn ; Dumitrescu:2012ha ; Closset:2012ru ; Hristov:2013spa .

In any supergravity with a R-symmetry gauge field, when all the other supergravity fields are set to zero, the fermionic variations have the universal form

[TABLE]

In three dimensions, we can choose $\gamma_{a}=\sigma_{a}$, where $\sigma_{a}$ are the Pauli matrices. The non-trivial components of the spin connection are easily computed181818We use the frame $e^{1}=Rd\theta,e^{2}=R\sin\theta d\phi$ and $e^{3}=\beta dt$. to be $\omega^{12}=-\cos\theta d\phi$. If we take $\gamma_{3}\epsilon=\epsilon$, so that $\gamma_{12}\epsilon=i\epsilon$, we see that the background field

[TABLE]

precisely cancels the spin connection. The equation reduces to

[TABLE]

which is solved by a constant spinor $\epsilon$. We thus see that the background (3.4) allows to define a supersymmetric theory on $S^{2}\times S^{1}$. The generalization of the above discussion to $\Sigma_{\mathfrak{g}}\times S^{1}$ is straightforward: we just turn on a background for the R-symmetry with $A^{R}=-\omega/2$ and everything else works in the same way.

This way of preserving supersymmetry corresponds to a topological twist along $\Sigma_{\mathfrak{g}}$ in the sense of Witten:1988ze ; Witten:1991zz . To make contact with the language used in Witten:1988ze ; Witten:1991zz , we can interpret the background field for $A^{R}$ as effectively changing the spin of the fields in the theory, thus transforming $\epsilon$ into a scalar.

A supersymmetric Lagrangian for a Yang–Mills–Chern–Simons theory with gauge group $G$ and chiral matter transforming in a representation ${\cal R}$ on $\Sigma_{\mathfrak{g}}\times S^{1}$ can be written as $\mathcal{L}=\mathcal{L}_{\text{YM}}+\mathcal{L}_{CS}+\mathcal{L}_{\text{mat}}+\mathcal{L}_{\text{W}}$ with191919This can be obtained for example by taking the rigid limit of supergravity in the background (3.2), as suggested in Festuccia:2011ws .

[TABLE]

where the superpotential $W(\phi_{i})$ is a holomorphic function of R-charge two and the fields $A_{\mu},\sigma,D$ act on the matter fields in the appropriate representation. Here the derivative $D_{\mu}$ is covariantized with respect to the spin and gauge connection and also to the R-symmetry background $A^{R}$. As usual in Euclidean signature, fields and their conjugate, $\phi$ and $\phi^{\dagger}$ for example, should be considered as independent variables. Notice that a vev for the scalar field $\sigma$ gives mass to the matter fields $\phi_{i}$ and $\psi_{i}$. This kind of coupling is typical of three dimensions and called real mass to distinguish it from the mass terms that can be introduced through the superpotential $W$. One can check that the Lagrangian is invariant under the following supersymmetry transformations

[TABLE]

for the vector multiplet fields and

[TABLE]

for the chiral multiplets. To future purposes, we also define ${\cal Q}=Q+\widetilde{Q}$.

Notice that the Lagrangian (3.6) and the transformations of supersymmetry are almost identical to the flat space ones with the further covariantization with respect to the metric and the background R-symmetry. This is not always the case in curved space, where extra terms should be included to maintain supersymmetry. In general, a Lagrangian is not invariant under flat space supersymmetry transformations when defined on a curved space because covariant derivatives do not commute anymore. With a topological twist, the spinor $\epsilon$ is covariantly constant and the problem is milder.

3.2 The localization formula

The topologically twisted index is just the $\Sigma_{\mathfrak{g}}\times S^{1}$ path integral of the theory discussed in the previous section. We can evaluate it using localization. The basic idea of localization is simple. Let us review it briefly, referring to Pestun:2016jze ; Marino:2011nm for more details.202020These references cover the developments after Pestun:2007rz . The idea of localization in physics is much older and it has been applied to many systems since Witten:1982im . In a theory with a fermionic symmetry squaring to zero (or to a bosonic symmetry of the theory)212121In our case ${\cal Q}^{2}$ is a linear combination of a gauge transformation and a rotation along $S^{1}$. we can deform the action with a ${\cal Q}$-exact term, ${\cal Q}V$, where $V$ is a fermionic functional invariant under all the symmetries. The new action is still ${\cal Q}$ invariant and the path integral independent of the deformation

[TABLE]

since ${\cal Q}$ acts as a total derivative. The path integral can be then computed for $t\rightarrow\infty$ and, if we choose $V$ cleverly and we are lucky, it reduces to a sum over saddle points of a classical contribution and a one-loop determinant,

[TABLE]

In many supersymmetric gauge theories on Euclidean manifolds, this approach successfully reduces the path integral to the evaluation of a matrix model.

In our theory, $\mathcal{L}_{\text{YM}},\mathcal{L}_{\text{mat}}$ and $\mathcal{L}_{\text{W}}$ are not only ${\cal Q}$-closed but also ${\cal Q}$-exact. For example, up to total derivatives,

[TABLE]

Therefore we can put arbitrary coefficients in front of the various ${\cal Q}$-exact terms in the Lagrangian

[TABLE]

and the path integral is independent of $g,\lambda,\eta$. We can then take the limit $g,\lambda,\eta\rightarrow 0$ and evaluate the path integral in the saddle point approximation. Notice in particular that $\mathcal{L}_{\text{W}}$ is ${\cal Q}$-exact. This means that the partition function is independent of the precise form of the interactions in the Lagrangian. The superpotential is important nevertheless for determining the global symmetries of the theory, which enter in the partition function through chemical potentials.

We now give the localization formula for the topologically twisted index. Since the computation is complicated and subtle, we just provide the final formula, referring to Pestun:2016jze ; Marino:2011nm for a general introduction to localization and to Benini:2015noa ; Benini:2016hjo for the details of this particular computation.

The path integral of the topologically twisted theory on $\Sigma_{\mathfrak{g}}\times S^{1}$ for a ${\cal N}=2$ supersymmetric gauge theory with gauge group $G$ can be written as a contour integral

[TABLE]

of a meromorphic form of Cartan-valued variables $u$, summed over a lattice $\Gamma$ of magnetic fluxes. $W$ is just the order of the Weyl group of $G$. We will explain all the ingredients in the following, referring to Benini:2015noa ; Benini:2016hjo for proofs. Notice that from now on we drop the superscript susy from the partition functions.

3.2.1 The BPS locus

We have a family of saddle points labeled by the vev of the scalar field $\sigma$, the value of the Wilson line $A_{t}$ along $S^{1}$ and a quantized magnetic flux $\mathfrak{m}$ along $\Sigma_{\mathfrak{g}}$. As standard in localization computation, these saddle points can be found as the locus where the fermionic variations vanish. The gaugino BPS equations read

[TABLE]

and are solved by setting the two terms on the right-hand side to zero. The second term in (3.12), $D_{\mu}\sigma$, vanishes for constant commuting adjoint fields $\sigma$ and $A_{t}$. With a gauge transformation, we can transform them in the Cartan subalgebra. We can combine these fields in a complex Cartan-valued quantity

[TABLE]

The Wilson line $A_{t}$ is periodic, invariant under a shift of any element $\chi$ of the co-root lattice $\Gamma$, $A_{t}\sim A_{t}+2\pi\chi$,222222The co-root lattice is defined by the requirement that $e^{2\pi i\chi}$ acts as the identity on any representation of the group $G$, and defines the weight lattice of the Langland (or S-dual) group $\check{G}$. the physical object being the holonomy $e^{iA_{t}}$. So $u$ naturally lives on a cylinder and it is convenient to define the quantity $x=e^{iu}$. For a $U(1)$ theory, $u$ lives on the cylinder $S^{1}\times\mathbb{R}$ and $x$ on the punctured plane $\mathbb{C}^{*}$. The first term in (3.12), using $\gamma_{12}\epsilon=i\epsilon$, implies that the auxiliary field $D$ is proportional to the gauge field strength along $\Sigma_{\mathfrak{g}}$, $D=iF_{12}$, and both live in the Cartan subalgebra. The curvature along $\Sigma_{\mathfrak{g}}$ is quantized

[TABLE]

where $\Gamma$ is again the co-root lattice. For a $U(1)$ theory $\mathfrak{m}$ is just an integer.

The path integral involves a sum over saddle points and is therefore given as an integral over $u$ and a sum over the magnetic fluxes $\mathfrak{m}$. Both variables live in the Cartan subalgebra and are only defined up to an action of the Weyl group, the surviving gauge symmetry. This explains the factor $1/|W|$ in (3.11). For a $U(N)$ theory the co-root lattice is just $\Gamma=\mathbb{Z}^{N}$ and the Weyl group is the permutation group of $N$ elements with $|W|=N!$

3.2.2 The integrand

The contribution to the saddle point of the classical action comes only from the Chern–Simons term232323This expression follows from a holomorphic recombination of the terms $A_{t}\wedge F_{12}$ and $\sigma D$ in the Chern–Simons action, using $D=iF_{12}$ and $u=A_{t}+i\beta\sigma$.

[TABLE]

where $x=e^{iu}$.

The one-loop determinant receives contributions from the vector multiplets and the chiral multiplets. The vector multiplet contribution is

[TABLE]

where $\alpha$ are the roots of $G$ and, for convenience, we included the integration measure $(du)^{r}$ in this expression. The chiral multiplet contribution is

[TABLE]

where $\mathfrak{R}$ is the representation under the gauge group $G$, $\rho$ are the corresponding weights and $r_{\phi}$ is the R-charge of the field. These expressions arise by taking ratios of determinants for fermionic and bosonic fields, computed by expanding in modes on $\Sigma_{\mathfrak{g}}\times S^{1}$. Due to supersymmetry, most of the modes cancel between bosons and fermions and we are left with the contribution of a set of zero-modes (indeed a convenient way to perform this computation is via an index theorem). For a chiral multiplet these zero modes contribute

[TABLE]

The term in bracket represents the mass of a chiral multiplet mode due to the coupling to $\sigma$, which acts as a real mass, to the Wilson line $A_{t}$ and to the KK momentum $n$ along the circle $S^{1}$. The exponent is the multiplicity of the zero-mode that can be easily obtained using the Riemann-Roch theorem. Notice that this multiplicity must be an integer and therefore the R-charges $r_{\phi}$ must be quantized.242424On $S^{2}\times S^{1}$ the R-charges $r_{\phi}$ must be integer. In the case of a higher genus Riemann surface it is enough to require that the quantity $(1-\mathfrak{g})(r_{\phi}-1)$ is an integer. This infinite product needs to be regularized. In (3.17) we chose a parity invariant regularization. There are other possible ones.252525Parity acts as $u\rightarrow-u$ and (3.17) is obviously invariant. A gauge invariant regularization breaking parity is used in Closset:2016arn ; Closset:2017zgf ; Closset:2017bse ; Closset:2018ghr . The latter has the advantage of clarifying subtle sign issues and simplifying the mapping of parameters between dual theories. However, even for theories with zero CS, in the gauge invariant regularization one has to introduce extra effective CS contact terms, which makes the physical interpretation less transparent.

The full integrand is

[TABLE]

where

[TABLE]

The determinant term exists only on a Riemann surface of genus $\mathfrak{g}>0$ and arise from the integration of the extra $\mathfrak{g}$ fermionic zero-modes existing on these surfaces.

3.2.3 The contour

The integrand (3.19) is a meromorphic form in the Cartan variables $u$ with poles at $x^{\rho}=1$, the points in the BPS locus where chiral multiplets become massless, and at the boundaries $x=0$ and $x=\infty$ of the moduli space. The partition function is obtained by using the residue theorem. Supersymmetry will choose the correct integration contour and tell us which poles to include. One might hope that we need to integrate over some simple contour, like the unit circle in the plane $x$, but one actually discovers that the contour is highly non-trivial and depends on the charges of the matter fields. For example, for a $U(1)$ theory with chiral fields of charge $Q_{i}$ and Chern–Simons level $k$, defining the effective Chern–Simons level262626This is actually the Chern–Simons level that one sees at one-loop after integrating out the matter fields (they have mass $\sigma$ at a generic point of the BPS locus).

[TABLE]

the rule is to take the residues of the poles created by fields with positive charge $Q_{i}>0$, the residue at the origin $x=0$ if $k_{\rm eff}(\infty)<0$ and the residue at infinity $x=\infty$ if $k_{\rm eff}(-\infty)>0$. The rule for a generic gauge group can be written in terms of the so-called Jeffrey–Kirwan (JK) residue (JeffreyKirwan, ), a prescription for dealing with poles arising from multiple intersecting hyperplane singularities. To explain it properly will lead us too far and we refer to Benini:2015noa ; Benini:2016hjo for details. The JK residue also appears in localization computations for elliptic genera in two dimensions, quantum mechanics, and various other partition functions (Benini:2013nda, ; Hori:2014tda, ; Closset:2015rna, ).

The reader may object that we are supposed to integrate over the BPS locus, which is the whole complex plane, and not to perform a contour integral in $u$. Luckily again supersymmetry comes to a rescue. On $\Sigma_{\mathfrak{g}}\times S^{1}$ there are gaugino zero-modes that contribute an extra term in the integrand in addition to the one-loop determinant. It turns out that the full integrand is a total derivative in $\bar{u}$ and we can reduce the integral over the $u$ plane to a contour integral around the singularities, as discussed in details in Benini:2015noa .

3.2.4 Adding flavor fugacities

If the theory has a flavor symmetry group $F$ acting on the chiral fields, we can introduce extra parameters in a supersymmetric way. We can just gauge the flavor symmetry and then freeze all the bosonic fields to background values that are preserved by supersymmetry. The background bosonic fields give rise to supersymmetric couplings in the Lagrangian. The analysis of fermionic variations is identical to the one performed in Sect. 3.2.1 for gauge symmetries. We need to solve (3.12) for a background multiplet, $(A^{F}_{\mu},\lambda^{F},\sigma^{F},D^{F})$. The result is that we can turn on in a supersymmetric way a constant value for $\sigma^{F}$ and $A_{t}^{F}$ which we combine into a complex quantity $u_{F}=A_{t}^{F}+i\beta\sigma^{F}$, and a background magnetic flux $\mathfrak{m}_{F}$ with $D^{F}=i\mathfrak{m}_{F}$. $\sigma^{F}$ appears in the Lagrangian as a real mass for the chiral fields. In three dimensions, any gauge theory with a $U(1)$ factor with field strength $F$ has also a topological symmetry associated with the current $J=*F$, which is automatically conserved. We can similarly introduce parameters $u_{T}$ and $\mathfrak{m}_{T}$ for the topological symmetry.

The path integral is then a function of $x_{F},x_{T}$ and $\mathfrak{m}_{F},\mathfrak{m}_{T}$. In the localization formula we just need to replace the one-loop determinant of a chiral field with

[TABLE]

where $x_{F}=e^{iu_{F}}$ and $\nu$ is the weight of the chiral field under the flavor symmetry $F$. There is no modification to the vector multiplet determinant. A $U(1)$ topological symmetry just contributes a classical term

[TABLE]

to the classical action.

3.2.5 The trace interpretation

As any path integral that involves an $S^{1}$ factor, the topologically twisted index can be written as a trace272727See footnote 14. The factor $e^{iA_{t}^{G}J^{G}}$ represents the insertion of a Wilson line $A_{t}^{G}$.

[TABLE]

where $H_{\mathfrak{g}}$ is the Hamiltonian of the topologically twisted theory on $\Sigma_{\mathfrak{g}}$, in the presence of magnetic fluxes $\mathfrak{m}_{G}=(\mathfrak{m}_{F},\mathfrak{m}_{T})$ and a supersymmetric background $x_{G}=(x_{F},x_{T})$ for the global symmetries, whose conserved charges have been denoted as $J^{G}$. The Hamiltonian $H_{\mathfrak{g}}$ explicitly depends on the magnetic fluxes $\mathfrak{m}_{G}$ and the real masses $\sigma^{G}$. If sufficiently real masses are turned on, the spectrum of $H_{\mathfrak{g}}$ is discrete and the trace is well-defined.

3.2.6 An Example: SQED

To explain all the ingredients, we can give a simple example of the final formula, using supersymmetric QED. This is a $U(1)$ theory with two chiral multiplets $Q$ and $\tilde{Q}$ of charges $\pm 1$ (electron and positron), and no Chern–Simons couplings. Since there is no superpotential, we have many possible choices of integer R-charges for the fields. We choose to assign R-charge $+1$ to both $Q$ and $\tilde{Q}$. There is an axial flavor symmetry $U(1)_{A}$ acting on $Q$ and $\tilde{Q}$ with equal charges and a topological symmetry $U(1)_{T}$. The charges of the chiral fields are

[TABLE]

The topological symmetry acts only on non-perturbative states constructed with monopole operators. We introduce a gauge variable $x$, with associated magnetic flux $\mathfrak{m}$, and flavor and topological variables $y=x_{F}$ and $\xi=x_{T}$, with associated background fluxes $\mathfrak{n}=\mathfrak{m}_{F}$ and $\mathfrak{t}=\mathfrak{m}_{T}$. According to our rules, the partition function on $S^{2}\times S^{1}$ is

[TABLE]

where we included an extra $(-1)^{\mathfrak{m}}$, which can be reabsorbed in the definition of $\xi$, for later convenience.282828For a more careful discussion of sign ambiguities see Closset:2017zgf . They will not play any important role in these lectures. Notice that gauge and flavor variables enter in a similar way in this formula. The main difference is that $x$ and $\mathfrak{m}$ are integrated and summed over, while $y,\xi$ and $\mathfrak{n},\mathfrak{t}$ are background parameters.

Our prescription instructs us to take the residues from the field $Q$ with positive gauge charge, whose pole is at $x=\frac{1}{y}$. By computing residues and resumming the result, one finds

[TABLE]

One recognizes here the product of three factors of the form (3.22) that we can associate with chiral multiplets. Indeed it is well known that the mirror theory to SQED is a Wess-Zumino model with fields $M,T,\tilde{T}$ and a cubic superpotential $W=MT\tilde{T}$ (Aharony:1997bx, ).

3.3 Interpretation of the localization formula

We can give an interpretation of the localization formula for the theory on $\Sigma_{\mathfrak{g}}\times S^{1}$ in two different ways that correspond to two different dimensional reductions of the three-dimensional theory. Compactification on $\Sigma_{\mathfrak{g}}$ gives rise to a quantum mechanics and compactification on $S^{1}$ to a two-dimensional $(2,2)$ supersymmetric theory.

3.3.1 Reduction to quantum mechanics

Compactifying on $\Sigma_{\mathfrak{g}}$, we obtain a supersymmetric quantum mechanics describing an infinite number of KK modes on $\Sigma_{\mathfrak{g}}$. These are particles living on the Riemann surface in the presence of a magnetic field for the R-symmetry and magnetic fluxes $\mathfrak{m}_{G}$ for the global symmetries. These magnetic fields create Landau levels. The trace (3.24) can be interpreted as the Witten index (Witten:1982df, ) of this particular quantum mechanics. Let us understand this concept better.

The quantum mechanics in question has ${\cal N}=2$ supersymmetry. With no background for the global symmetries, the algebra of supersymmetry is simply $\{\bar{Q},Q\}=H_{\mathfrak{g}}$, where $Q$ is a complex supercharge. The index is just

[TABLE]

and, according to standard arguments, is independent of $\beta$. Indeed, any state $\psi$ with $H_{\mathfrak{g}}\neq 0$ has a non-zero partner $Q\psi$ with the same energy and opposite statistic and their contributions cancel in the trace. Therefore the only contribution comes from ground states.292929By the algebra of supersymmetry $H_{\mathfrak{g}}\psi=0$ is equivalent to $Q\psi=0$ and, therefore, ground states not necessarily have a partner. The index is then clearly independent of $\beta$ and it is an integer counting the number of ground states with signs (plus for bosonic ones, minus for fermionic ones). When we turn on backgrounds for the global symmetries, the supersymmetry algebra is modified to $\{\bar{Q},Q\}=H_{\mathfrak{g}}-\sigma^{G}J^{G}$, where $J^{G}$ is the conserved charge associated to the global symmetry.303030See, for example, Hori:2014tda or Appendix C of Benini:2015eyy . Using (3.24), we can now write the index as follows

[TABLE]

where $g(n)$ is the number of supersymmetric states, $Q\psi=0$, with charge $n$ under the global symmetry and, as usual, $x_{G}=e^{i(A_{t}^{G}+i\beta\sigma^{G})}$. In deriving this expression, we used again the fact that states with $Q\psi\neq 0$ are paired by supersymmetry and have the same energy and charge. This time, the states that contribute to the trace are chiral states with $H_{\mathfrak{g}}=\sigma_{G}J^{G}$. In this way, we have obtained an equivariant index, where the supersymmetric states are graded according to their charge by powers of the fugacity $x_{G}$. Notice also that this argument shows that the topologically twisted index is an holomorphic function of the fugacities, as we already found using localization.

Let us also notice that the integrand of the localization formula has a simple Hamiltonian interpretation. There are two type of multiplets in the ${\cal N}=2$ quantum mechanics we are discussing: the chiral multiplet containing a complex scalar $\phi$ and a spinor as dynamical fields, and the Fermi multiplet containing only a spinor (Hori:2014tda, ). The Landau levels on $\Sigma_{\mathfrak{g}}$ give rise to zero-modes with multiplicities dictated by the Riemann–Roch theorem. One can see that the zero-modes organize themselves into $\rho(\mathfrak{m})+(\mathfrak{g}-1)(r_{\phi}-1)$ chiral multiplets if $\rho(\mathfrak{m})+(\mathfrak{g}-1)(r_{\phi}-1)>0$, and $|\rho(\mathfrak{m})+(\mathfrak{g}-1)(r_{\phi}-1)|$ Fermi multiplets if $\rho(\mathfrak{m})+(\mathfrak{g}-1)(r_{\phi}-1)<0$, where for simplicity we set the flavor fugacities to zero. We can now compute the index for Fermi and chiral multiplets. For the Fermi multiplet the Hilbert space is a fermionic Fock space, and assigning charge $-\frac{\rho}{2}$ and fermion number [math] to the vacuum, the index is

[TABLE]

For the chiral multiplet the Hilbert space is the product of a bosonic Fock space generated by $\phi,\phi^{\dagger}$ and a fermionic Fock space; assigning fermion number $1$ to the vacuum, the index is

[TABLE]

Raising these quantities to a power corresponding to the multiplicity, and taking into account the different signs for the two types of multiplets, we recover exactly the contribution (3.17) of a three-dimensional chiral multiplet to the partition function.

In particular, the localization formula for the topologically twisted index is just the sum over many topological sectors labelled by $\mathfrak{m}$ of the localization formula for the Witten index of ${\cal N}=2$ quantum mechanics found in Hori:2014tda .

3.3.2 Reduction to two dimensions

We can alternatively reduce our three-dimensional theory on $S^{1}$ and obtain a $(2,2)$ supersymmetric theory containing all the KK modes on $S^{1}$. At a generic point of the Coulomb branch where $\sigma\neq 0$, all the non-Cartan gauge bosons and the chiral multiplets are massive.313131Due to the KK mass or their coupling to $\sigma$. We can integrate them out and write a Lagrangian for the Cartan modes of the vector multiplets. In two dimensions, a vector multiplet can be described using a twisted chiral multiplet $\Sigma$ and its interactions is described by a twisted superpotential $\int d\theta_{+}d\bar{\theta}_{-}\mathcal{W}$.323232$\Sigma$ has a scalar as its lowest component and it satisfies $\bar{D}_{+}\Sigma=D_{-}\Sigma=0$. See Witten:1993yc .

It is interesting to observe that such twisted superpotential $\mathcal{W}$ enters explicitly in the integrand of the localization formula (Closset:2016arn, ; Closset:2017zgf, ). Indeed, the dependence on the gauge flux $\mathfrak{m}$ can be explicitly written as

[TABLE]

where $Q(x)$ is a meromorphic function independent of $\mathfrak{m}$. The function $\mathcal{W}$, up to an overall normalization and sign ambiguities that we fix for convenience, is given by333333Polylogarithms $\operatorname{Li}_{s}(z)$ are defined by ${\rm Li}_{s}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}$ for $|z|<1$ and by analytic continuation outside the disk. Notice, in particular, that ${\rm Li}_{1}(z)=-\log(1-z)$. For $s\geq 1$, there exists a the branch cut that we take along $[1,+\infty)$. They satisfy $\partial_{u}{\rm Li}_{s}(e^{iu})=i\,{\rm Li}_{s-1}(e^{iu})$ and, for $0<\operatorname{\mathbb{R}e}u<2\pi$, ${\rm Li}_{s}(e^{iu})+(-1)^{s}{\rm Li}_{s}(e^{-iu})=-\frac{(2i\pi)^{s}}{s!}B_{s}\left(\frac{u}{2\pi}\right)\equiv i^{s-2}g_{s}(u)$, where $B_{s}(u)$ are the Bernoulli polynomials. In this paper we need, in particular, $g_{2}(u)=\frac{u^{2}}{2}-\pi u+\frac{\pi^{2}}{3}$ and $g_{3}(u)=\frac{u^{3}}{6}-\frac{\pi}{2}u^{2}+\frac{\pi^{2}}{3}u$. One then sees that, for $0<\operatorname{\mathbb{R}e}u<2\pi$ and $\operatorname{\mathbb{I}m}u\ll 0$, ${\rm Li}_{s}(e^{iu})\sim i^{s-2}g_{s}(u)$. Notice also that $\mathcal{W}$ is a multi-valued function but, since the action is defined up to integer multiples of $2\pi i$, the path integral and all physical observables are single valued.

[TABLE]

with $g_{2}(u)=\frac{u^{2}}{2}-\pi u+\frac{\pi^{2}}{3}$. As argued in Nekrasov:2009uh ; Nekrasov:2014xaa , this can be interpreted as the effective twisted superpotential of the two-dimensional theory obtained by compactifying on $S^{1}$. The first term in (3.33) is the classical contribution coming from the CS term, and the second is the sum of all the perturbative contributions of massive fields, including the infinite tower of KK modes. Indeed, a one-loop diagram for a mode of mass $m$ contributes a term proportional to $i(\Sigma+m)(\log(\Sigma+m)/2\pi-1)$ to $\mathcal{W}$ and there are no higher order corrections (Witten:1993yc, ). The contribution of the KK modes of a chiral multiplet, whose mass depends on $\sigma$, the Wilson line $A_{t}$ and the KK momentum $n$, can be resummed to

[TABLE]

The other term in the round bracket in (3.33) is local and it is due to our choice of a parity invariant regularization.343434For a choice of a gauge invariant regularization and an extensive discussion of other issues related to definition of $\mathcal{W}$ see Closset:2017zgf . Using the asymptotic expansion of the polylogarithms, we find that the content of the bracket in (3.33) behaves, for large $\sigma$, as

[TABLE]

This can be interpreted as a one-loop effective Chern–Simons term obtained by integrating out a field of mass $\rho(\sigma)$.353535For a field of mass $m$ and charge $Q_{i}$ the one-loop effective Chern–Simons term is $k_{\rm eff}=k+\frac{1}{2}Q_{i}^{2}\operatorname{sign}(m)$.

The Jeffrey–Kirwan prescription typically selects poles in the integrand of the localization formula that are contained in a half-lattice $\mathfrak{m}_{i}\geq M$ (or $\mathfrak{m}_{i}\leq M$) for some cut-off $M$. We can then use the geometric series to resum the integrand in (3.32)

[TABLE]

and evaluate the index by taking the residues at the poles

[TABLE]

The cut-off $M$ disappears in the process. The solutions to (3.37) are the so-called Bethe vacua of the two-dimensional theory. They play an important role in the Bethe/gauge correspondence (Nekrasov:2009uh, ; Nekrasov:2009rc, ).

We then find the following general characterization of the topologically twisted index as a sum over Bethe vacua

[TABLE]

where $x^{*}$ are the solutions of (3.37). The expression for the topologically twisted index as a sum over Bethe vacua was first derived by topological field theory arguments in Okuda:2012nx ; Okuda:2013fea ; Nekrasov:2014xaa ; Okuda:2015yea . In the context of localization, this expression for the index has been derived and generalized in Closset:2016arn ; Closset:2017zgf ; Closset:2017bse ; Closset:2018ghr . The expression in (3.38) can be also written as

[TABLE]

in terms of a handle-guing operator ${\cal H}(x)=e^{\Omega(x)}\det_{ij}\partial^{2}_{u_{i}u_{j}}\mathcal{W}(x)$ and an effective dilaton $\Omega(x)$ whose complete characterization in terms of field theory data can be found in the above mentioned papers. Here we just notice that, for genus $\mathfrak{g}>0$, the Hessian of $\mathcal{W}$ enters at the power $\mathfrak{g}-1$. Indeed, the determinant in (3.19) contributes $\mathfrak{g}$ extra powers of the Hessian that combine with the denominator in (3.38).

A very interesting result of Closset:2017zgf ; Closset:2017bse ; Closset:2018ghr is the generalization of formula (3.39) to three-dimensional manifolds that are not a direct product. For example, the supersymmetric partition function on a three-manifold $\mathcal{M}_{3}$ that is an $S^{1}$ fibration of Chern class $p$ over a Riemann surface $\Sigma_{\mathfrak{g}}$ can be written as a sum over the very same set of Bethe vacua,

[TABLE]

with a fibering operator $\mathcal{F}(x)$ that can be expressed in terms of field theory data. There exists a similar result for the partition function on more general three-dimensional manifolds and also for some selected four-dimensional ones (Closset:2017zgf, ; Closset:2017bse, ; Closset:2018ghr, ). The particular case of the formula for the four-dimensional superconformal index plays a role in the physics of AdS5 black holes (Benini:2018mlo, ; Benini:2018ywd, ), as discussed in Sect. 7.2.

Let us give a couple of examples of Bethe vacua. For a pure ${\cal N}=2$ Chern–Simons theory with gauge group $SU(2)$, the expression for the partition function (3.19) is

[TABLE]

where we used $x_{i}=(x,1/x)$ and $\mathfrak{m}_{i}=(\mathfrak{m},-\mathfrak{m})$. The twisted superpotential receives contribution only from the classical action: $\mathcal{W}=\sum_{i}ku_{i}^{2}/2=ku^{2}$. The Bethe vacua (3.37) are then $x^{2k}=1$ with solutions the $2k$-roots of unity. Formula (3.38) gives, up to an ambiguous sign,

[TABLE]

where $\bar{k}=k-2$. This is the well-known Verlinde formula for the CS partition function on $\Sigma_{\mathfrak{g}}\times S^{1}$.363636Since $\sigma$ and $\lambda$ are massive and free, they can be integrated out leading to a shift in the CS coupling. An ${\cal N}=2$ Chern–Simons theory is thus equivalent to a bosonic CS theory with level $\bar{k}=k-2$. Notice that the root $x=1$ is not included in the sum: as a general rule, the Bethe vacua that are also zeros of the Vandermonde determinant are not physical.

In the presence of matter, the Bethe equations (3.37) are more complicated. In the SQED example discussed in Sect. 3.2.6, the Bethe equation is

[TABLE]

with solution $x=(1-\xi y)/(y-\xi)$. It is easy to see that (3.38) correctly reproduces (3.27). For a general theory with gauge group $G$, the Bethe equations (3.37) cannot be analytically solved.

4 The entropy of dyonic AdS4 black holes

In this section we will derive microscopically the entropy of a family of BPS static dyonic black holes in AdS${}_{4}\times S^{7}$ (Benini:2015eyy, ; Benini:2016rke, ). These solutions have been found in ${\cal N}=2$ gauged supergravity in four-dimensions (Cacciatori:2009iz, ; DallAgata:2010ejj, ; Hristov:2010ri, ; Katmadas:2014faa, ; Halmagyi:2014qza, ) and later uplifted to $M$-theory. We then start by briefly discussing the main features of ${\cal N}=2$ gauged supergravity in four dimensions. This will be also useful to write an entropy functional. We then consider the large $N$ limit of the topologically twisted index for the ABJM theory in three dimensions and show that it reproduces the entropy of the dual black holes.

We will only consider the case of static black holes in this section. A field theory derivation for the entropy of dyonic rotating black holes in AdS${}_{4}\times S^{7}$ (Hristov:2018spe, ) is still missing.

4.1 AdS4 dyonic static black holes

BPS black holes in AdS4 can be found by studying an effective four-dimensional ${\cal N}=2$ gauged supergravity. We start discussing the main features of the effective theory.

The ${\cal N}=2$ supergravity multiplets are:

• •

the graviton multiplet, whose bosonic components are the metric $g_{\mu\nu}$ and a vector field $A_{\mu}^{0}$, called graviphoton;

• •

the vector multiplet, whose bosonic components are a vector $A_{\mu}^{i}$ and a complex scalar $z$;

• •

the hypermultiplet, whose bosonic components are four real scalars $q^{\alpha}$.

For simplicity, we will restrict to ${\cal N}=2$ gauged supergravities with $n_{V}$ vector multiplets and no hypermultiplets.373737Theory with hypermultiplets have been considered for matching the entropy of black holes in massive type IIA and other models (Hosseini:2017fjo, ; Benini:2017oxt, ; Bobev:2018uxk, ). This will be enough to describe the black holes in AdS${}_{4}\times S^{7}$. The theory contains $n_{V}+1$ vector multiplets $A_{\mu}^{\Lambda}$ and $n_{V}$ complex scalar fields $z_{i}$, where $\Lambda=0,1,\ldots,n_{V}$ and $i=1,\ldots,n_{V}$. The Lagrangian can be written in terms of a holomorphic prepotential $\mathcal{F}(X^{\Lambda})$, which is a homogeneous function of degree two, and a vector of magnetic and electric Fayet–Iliopoulos (FI) parameters $(g^{\Lambda},g_{\Lambda})$. $X^{\Lambda}(z_{i})$ are a set of $n_{V}+1$ homogeneous coordinates on the scalar manifold. The theory is invariant under rescaling of the $X^{\Lambda}$ and one can identify the physical scalar fields with $z_{i}=X^{i}/X^{0}$.383838Other choices of gauge fixing for the rescaling symmetry are possible, corresponding to field redefinitions. It is also convenient to define $\mathcal{F}_{\Lambda}\equiv\partial_{\Lambda}\mathcal{F}$. The theory is fully covariant under a $Sp(2n_{V}+2)$ group of electric/magnetic dualities acting on $(X^{\Lambda},{\cal F}_{\Lambda})$ and $(g^{\Lambda},g_{\Lambda})$ as symplectic vectors.

The action of the bosonic part of the theory reads (Andrianopoli:1996cm, )

[TABLE]

The metric on the scalar manifold is given by

[TABLE]

Here, $\mathcal{K}(z,\bar{z})$ is the Kähler potential and it reads

[TABLE]

The matrix $\mathcal{N}_{\Lambda\Sigma}$ of the gauge kinetic term is a function of the vector multiplet scalars and is given by

[TABLE]

Finally, the scalar potential reads

[TABLE]

where ${\cal L}=e^{{\cal K}/2}\left(X^{\Lambda}g_{\Lambda}-\mathcal{F}_{\Lambda}g^{\Lambda}\right)$ and $D_{i}{\cal L}=\partial_{i}{\cal L}+\partial_{i}{\cal K}{\cal L}/2$.

The ansatz for a static dyonic black hole with horizon $\Sigma_{\mathfrak{g}}$ is of the form393939We normalize the metric on $\Sigma_{\mathfrak{g}}$ such that the scalar curvature is $2\kappa$, where $\kappa=1$ for $S^{2}$, $\kappa=0$ for $T^{2}$, and $\kappa=-1$ for $\Sigma_{\mathfrak{g}}$ with $\mathfrak{g}>1$. The volume is then $\mathrm{Vol}(\Sigma_{\mathfrak{g}})=2\pi\eta$ where $\eta=2|\mathfrak{g}-1|$ for $\mathfrak{g}\neq 1$ and $\eta=1$ for $\mathfrak{g}=1$.

[TABLE]

where $A_{\Sigma_{\mathfrak{g}}}$ is the gauge potential for a magnetic flux on $\Sigma_{\mathfrak{g}}$. For example, for $\Sigma_{\mathfrak{g}}=S^{2}$ we can take $A_{S^{2}}=-\cos\theta d\phi$. We assume that the scalar fields $z^{i}$ have only radial dependence. We are interested in solutions that are asymptotic to AdS4 for large values of the radial coordinate,

[TABLE]

and approach a regular horizon AdS${}_{2}\times\Sigma_{\mathfrak{g}}$ at some fixed value $r=r_{0}$,

[TABLE]

Notice that we can also interpret these black holes as domain walls interpolating between AdS4 and AdS${}_{2}\times\Sigma_{\mathfrak{g}}$. The AdS2 factor suggests the existence of a superconformal quantum mechanics describing the horizon microstates. We expect that this is the IR limit of the quantum mechanics discussed in Sect. 3.3.1.

There are two conserved quantities

[TABLE]

where $G_{\Lambda}=8\pi G_{\text{N}}\,\delta(\mathscr{L}d\mathrm{vol}_{4})/\delta F^{\Lambda}$, corresponding to the magnetic and electric charges of the black hole. Under $Sp(2n_{V}+2)$ they transform as a symplectic vector $(p^{\Lambda},q_{\Lambda})$. In a frame with purely electric gauging $g_{\Lambda}$, the magnetic and electric charges are quantized as follows

[TABLE]

not summed over $\Lambda$.

As discussed in Sect. 2.1.2, supersymmetry is realized with a topological twist. In particular, the Killing spinors $\epsilon_{A}$, $A=1,2$, only depend on the radial coordinate. The BPS equations give a set of ordinary differential equations for the functions $U,V,a_{0},a_{1},z_{i},\epsilon_{A}$ that are explicitly given in Cacciatori:2009iz ; DallAgata:2010ejj ; Hristov:2010ri ; Katmadas:2014faa ; Halmagyi:2014qza . For our purposes, the only important point is that the gravitino variation contains, among other pieces,

[TABLE]

The vanishing of this variation, when the index $\mu$ is restricted to $\Sigma_{\mathfrak{g}}$, requires that $A_{\mu}^{\Lambda}$ cancels the spin connection and one obtains

[TABLE]

where, with standard notations, $\kappa=1$ for horizon $S^{2}$, $\kappa=0$ for $T^{2}$ and $\kappa=-1$ for $\mathfrak{g}>1$. We see that a linear combination of the magnetic charges is fixed by the twist. In a general theory with also magnetic FI the previous condition is replaced by

[TABLE]

which is manifestly symplectic invariant. The previous discussion closely parallels the field theory analysis of the supersymmetries preserved by a topological twist in Sect. 3.1.404040This is not a coincidence (Klare:2012gn, ): when holography applies, solving the Killing spinor equations in bulk near the AdS boundary gives a set of constraints on the boundary theory that are equivalent to those obtained with the approach proposed in Festuccia:2011ws .

It has been noticed in DallAgata:2010ejj that the BPS equations of gauged supergravity for the near-horizon geometry can be put in the form of attractor equations.414141The attractor mechanism for AdS4 static black holes in ${\cal N}=2$ gauged supergravity is discussed in Cacciatori:2009iz ; DallAgata:2010ejj ; Chimento:2015rra . For some recent progress for dyonic rotating black holes see Hristov:2018spe . The BPS equations are indeed equivalent to the extremization of the quantity

[TABLE]

with respect to the horizon-value of the symplectic sections $X^{\Lambda}$, combined with the requirement that the value of $\mathcal{I}_{\text{sugra}}$ at the critical point $\bar{X}^{\Lambda}$ is real. In general, in gauged supergravity, $\mathcal{F}(X^{\Lambda})$ is a homogeneous function of degree two, so we can equivalently define $\hat{Y}^{\Lambda}\equiv X^{\Lambda}/(g_{\Sigma}X^{\Sigma}-g^{\Sigma}\mathcal{F}_{\Sigma})$ and extremize

[TABLE]

The extremization of (4.12) gives a set of algebraic equations for the value of the physical scalars $z^{i}$ at the horizon, and the entropy of the black hole is given by evaluating the functional (4.12) at its extremum

[TABLE]

4.1.1 Black holes in AdS${}_{4}\times S^{7}$

M-theory on AdS${}_{4}\times S^{7}$ can be consistently truncated to an ${\cal N}=2$ gauged supergravity containing the four vectors parameterizing the Cartan subgroup of the $SO(8)$ isometry of $S^{7}$. In the language of gauged supergravity, one is the graviphoton and the other three give rise to a model with three vector multiplets, $n_{V}=3$. By explicitly reducing M-theory on AdS${}_{4}\times S^{7}$, one can determine the prepotential424242See for example Hristov:2012bk .

[TABLE]

and the FI, $g_{\Lambda}\equiv g,\,g^{\Lambda}=0$, that are purely electric. With these notations the AdS4 vacuum has radius $L^{2}=1/2g^{2}$.

We can introduce four magnetic and electric charges $(p^{\Lambda},q_{\Lambda})$. However, two of these charges are determined by supersymmetry. Indeed, one is fixed by the twisting condition (4.10) that gives a linear constraint among the magnetic charges. For purely magnetic black holes (Cacciatori:2009iz, ), this is the only constraint and we find a three-dimensional family of solutions. They are particularly simple since all the scalars $z_{i}$ are real. We can write, for example, the solution for a black hole with $S^{2}$ horizon

[TABLE]

where the real sections are given by $X^{\Lambda}=\frac{1}{4}-\frac{\beta_{\Lambda}}{r}$ and the parameters $\beta^{\Lambda}$ and $c$ are determined in terms of the magnetic charges by

[TABLE]

where we also set $L=1$ or, equivalently, $g=1/\sqrt{2}$. The generic dyonic black holes found in Katmadas:2014faa ; Halmagyi:2014qza are more complicated and we will not report here the form of the solution. For dyonic black holes there is an extra constraint on the charges that follows from the requirement that the entropy computed through the attractor mechanism (4.14) is a real number. This constraint is highly non linear in the charges and leaves a six-dimensional family of black holes.

The entropy can be written by using (4.14), even without knowing the explicit form of the metric. The final expression can be written in a symplectic invariant form

[TABLE]

where $\Gamma=(p^{\Lambda},q_{\Lambda})$ and $G=(g^{\Lambda},g_{\Lambda})$ are symplectic vectors containing the charges and the FI parameters, and $I_{4}$ is a quartic polynomial, known as the quartic invariant, whose explicit expression can be found in Katmadas:2014faa ; Halmagyi:2014qza . In the simplest case of a purely magnetic black hole with $p^{1}=p^{2}=p^{3}\equiv-p/(2g)$, $p^{0}=(3p-2)/(2g)$ and horizon $S^{2}$, we find an expression of the form

[TABLE]

Notice that this expression is quite complicated, especially if compared with simple forms of the entropy as a function of charges that one can find for some asymptotically flat black holes.

The solutions with spherical horizon can be generalized by adding rotation (Hristov:2018spe, ). For completeness we report the form of the entropy

[TABLE]

where $j$ is the angular momentum. This time supersymmetry imposes three constraints on the charges and leaves again a six-dimensional family of rotating black holes.

4.2 The dual field theory

The CFT dual to AdS${}_{4}\times S^{7}$ is the so-called ABJM theory (Aharony:2008ug, ). We briefly discuss its properties and then we write the corresponding topological twisted index using the rules discussed in Sect. 3.

The ABJM theory describes the low-energy dynamics of $N$ M2-branes on $\mathbb{C}^{4}/\mathbb{Z}_{k}$ (Aharony:2008ug, ). It is a three-dimensional supersymmetric Chern–Simons-matter theory with gauge group $U(N)_{k}\times U(N)_{-k}$ (the subscripts are the CS levels) and matter in bifundamental representation. The matter content, in ${\cal N}=2$ notations, is described by the quiver diagram

N$$k$$N$$-k$$A_{i}$$B_{j}

where $i,j=1,2$ and nodes represent gauge groups and arrows represent bifundamental chiral multiplets. The theory has a quartic superpotential

[TABLE]

The ABJM theory has a number of interesting properties:

• •

the theory has $\mathcal{N}=6$ superconformal symmetry, non-perturbatively enhanced to $\mathcal{N}=8$ for $k=1,2$;

• •

it has an $SU(4)$ R-symmetry, enhanced to $SO(8)$ for $k=1,2$;

• •

for $N\gg k^{5}$ the theory is well-described by a weakly coupled M-theory background, AdS${}_{4}\times S^{7}/\mathbb{Z}_{k}$;

• •

the free energy on $S^{3}$ can be computed using localization and scales as $O(N^{3/2})$ in the M-theory limit (Drukker:2010nc, )

[TABLE]

For a review of these properties see Klebanov:2009sg ; Marino:2011nm .

We will consider the case $k=1$ where the theory has maximal supersymmetry, $SO(8)$ R-symmetry and is dual to AdS${}_{4}\times S^{7}$. The four abelian symmetries of the theory, $U(1)^{4}\subset SO(8)$ correspond, in ${\cal N}=2$ notation, to an R-symmetry and three global symmetries. There are many choices of $U(1)$ R-symmetry corresponding to different decompositions $SO(8)\rightarrow U(1)_{R}\times U(1)^{3}$. In order to write the index we need to select one with integer charges. Introducing a natural basis of $U(1)$ R-symmetries,

[TABLE]

we can for example choose the R-symmetry $\sum_{a}R_{a}/2$ that has integer charges. The remaining three $U(1)$s combine to give three flavor symmetries, say $(R_{a}-R_{4})/2$ for $a=1,2,3$.434343We are cheating a little bit here. The ABJM theory has also two topological symmetries, $T_{1}$ and $T_{2}$, associated with the two $U(1)$ gauge groups. This apparently makes a total of five $U(1)$ global symmetries. However $T_{1}+T_{2}$ is decoupled, and the baryonic symmetry that rotates $A_{i}$ and $B_{i}$ with opposite charges is actually gauged. More precisely, due the CS term, a linear combination of $T_{1}-T_{2}$ and the baryonic symmetry differ by a gauge transformation and are therefore equivalent. In the index we could introduce extra fluxes and fugacities for $T_{1}$ and $T_{2}$ but these can be re-absorbed by a shift of the fluxes and a rescaling of the integration variables. See Benini:2015eyy for more details.

Our general rules for the index allow to introduce a number of independent fluxes and fugacities equal to the number of global symmetries. We then introduce three magnetic fluxes $\mathfrak{p}$ and three fugacities $y$ for the three flavor symmetries of ABJM. It will be convenient to choose a redundant but democratic parameterization of these quantities. We assign a flux and a fugacity, $\mathfrak{p}_{a}$ and $y_{a}$ with $a=1,2,3,4$, to each of the fields $A_{1},A_{2},B_{1},B_{2}$ in the order indicated. The index is given by

[TABLE]

where we used the rules of Sect. 3.2.2. Notice that the Hessian of $\mathcal{W}$ should be computed using the $2N$ variables $u_{A}=(u_{i},\tilde{u}_{i})$. We have also included the CS term $k$ in order to make clear where the terms come from but soon we will set $k=1$. As already said, the fugacities are not independent. Since the superpotential (4.18) must have charge zero under a global symmetry we must set

[TABLE]

This translates into a constraint for the corresponding (complexified) chemical potentials $\Delta_{a}$, $y_{a}=e^{i\Delta_{a}}$,

[TABLE]

since the $\Delta_{a}$ are only defined modulo $2\pi$.444444In the notation of section 3.2.4, $\Delta_{a}=A_{t\,a}^{F}+i\beta\sigma_{a}^{F}$, where $A_{t\,a}^{F}$ and $\sigma_{a}^{F}$ are the backgrounds for the $a$-th symmetry. The periodicity of $\Delta_{a}$ is due to the periodicity of the Wilson line $A_{t\,a}^{F}$. Similarly, the four fluxes $\mathfrak{p}_{a}$ are not independent. To understand our parameterization, let us compare the chiral fields contributions in (4.21) with (3.22). Identifying exponents we have

[TABLE]

where $\mathfrak{m}_{a}^{F}$ is an assignment of background fluxes for the global symmetry and $r_{a}$ the R-charge of the $a$-th field. Since $W$ has charge zero under global symmetries and charge two under R-symmetries, we have $\sum_{a=1}^{4}\mathfrak{m}_{a}^{F}=0$ and $\sum_{a=1}^{4}r_{a}=2$, so that

[TABLE]

The dependence of our index on three magnetic fluxes and three fugacities fits well with the family of black holes discussed in Sect. 4.1.1 that have three magnetic and three electric charges. (4.25) is clearly the analog of (4.10) and already suggests the following identification between parameters $p^{\Lambda}\rightarrow-\kappa\mathfrak{p}_{a}/(2g(1-\mathfrak{g}))$.

The index can be written as a sum over Bethe vacua (3.38). The twisted superpotential (3.33) reads454545We used the of polylogarithm identities given in footnote 33 in order to recombine the terms in (3.33), and discarded terms that do not contribute to the Bethe equations (3.37). We also introduced an extra minus sign in the definition of $\mathcal{W}$ in order to match the original conventions in Benini:2015eyy ; Hosseini:2016tor . It is easy to check directly that the equations (4.27) give the position of the poles of the integrand after we sum the geometric series in $m_{i}$ and $\tilde{m}_{i}$ and that, with the given definition of $\mathcal{W}$, (4.27) are equivalent to (3.37).

[TABLE]

and the Bethe vacua equations are

[TABLE]

In the large $N$ limit we expect that just one Bethe vacuum dominates the partition function.

4.3 The ABJM Bethe vacua in the large $N$ limit

We want to study the solutions of (4.27) in the large $N$ limit (Benini:2015eyy, ). By running numerics, one discovers that the imaginary parts of the solutions $u_{i}$ and $\tilde{u}_{i}$ grow with $N$,

[TABLE]

and are equal for the two sets, while the real parts remain bounded. As usual, in the large $N$ limit, the distributions of $u_{i}$ and $\tilde{u}_{i}$ become almost continuous and we introduce a parameter $t(i/N)=t_{i}$, defined in an interval $[t_{-},t_{+}]$. We also introduce two functions of $t$, $v(t)$ and $\tilde{v}(t)$, defined implicitly by $v(i/N)=v_{i},\,\tilde{v}(i/N)=\tilde{v}_{i}$, and a normalized density

[TABLE]

The interesting feature of this model is that $\mathcal{W}$ becomes a local functional,464646This is similar to other matrix models solved using localization in three and five dimensions (Herzog:2010hf, ; Jafferis:2011zi, ; Jafferis:2012iv, ; Minahan:2013jwa, ).

[TABLE]

where $\delta v(t)=\tilde{v}(t)-v(t)$, $g_{3}(u)=\frac{1}{6}u^{3}-\frac{1}{2}\pi u^{2}+\frac{\pi^{2}}{3}u$ and $\epsilon_{a}=1$ for $a=1,2$ and $\epsilon_{a}=-1$ for $a=3,4$. We also assumed that $\Delta_{a}$ are real and

[TABLE]

for all $a$. The first term in $\mathcal{W}$ comes from the Chern–Simons interaction and the second is the contribution of matter fields. The derivation of (4.30) is given in Benini:2015eyy . Here we just mention few facts.

• •

$\mathcal{W}$ is local because of the exponential terms $e^{i(\tilde{u}_{j}-u_{i}\pm\Delta_{a})}$ in the arguments of polylogs in (4.26). Due to (4.28), for $j>i$ the polylogs are exponentially suppressed in the large $N$ limit. For $i>j$ the exponential is large but we can use the identity $\operatorname{Li}_{2}(e^{iu})+\operatorname{Li}_{2}(e^{-iu})=\frac{1}{2}u^{2}-\pi u+\frac{\pi^{2}}{3}$, valid for $\operatorname{\mathbb{R}e}u\in[0,2\pi]$, (see footnote 33) to transform it into a polynomial plus exponentially suppressed terms. As a consequence, up to polynomial terms, the main contribution comes for values of the indices $i\sim j$ and makes the functional local.

• •

Terms with higher powers of $N$ cancel. For more general ${\cal N}=2$ theories this is not automatic and imposes conditions on the matter content of the theories for which this method works (Hosseini:2016tor, ).

• •

Polynomial terms coming from this manipulation or Chern–Simons terms that are not in (4.30) happily combine into a contribution $\sum_{i=1}^{N}2\pi n_{i}u_{i}+2\pi\tilde{n}_{i}\tilde{u}_{i}$ to $\mathcal{W}$, where $n_{i}$ and $\tilde{n}_{i}$ are integers. These angular ambiguities disappear in the Bethe equations (3.37).

In general, the two contributions in (4.30) have different powers of $N$. They compete and give a sensible functional with a minimum only for $\alpha=1/2$. We then see that $\mathcal{W}$ scales as $N^{3/2}$ as predicted by holography for AdS4 black holes. We will then set $\alpha=1/2$ from now on.

In order to extremize (4.30) we add to $\mathcal{W}$ a Lagrange multiplier term

[TABLE]

that enforces the normalization condition (4.29). Differentiating $\mathcal{W}$ with respect to $\delta v(t)$ and $\rho(t)$, we obtain a pair of algebraic equations

[TABLE]

which can be easily solved in terms of rational functions of $t$. The solution is depicted in Fig. 1 together with the numerical solution for large $N$.

From the figure we see that $\rho(t)$ and $\delta v(t)$ are piece-wise continuous functions of $t$. The solution in (4.33) and (4.34) only covers the central part of these functions. Numerics suggest that there are two external intervals, which we dub tails, where $\delta v(t)$ is actually constant in the large $N$ limit. It turns out that such constant value corresponds to the saturation of the inequality (4.31) for some value of $a$, $\delta v(t)=\epsilon_{a}\Delta_{a}$. The inequalities (4.31) are necessary to restrict to a particular determination of the multi-valued polylog functions and the saturation corresponds to the position of the cuts. The numerics suggest that, once $v_{i}$ and $\tilde{v}_{i}$ hit the cut, their value is frozen. The value for $\rho(t)$ in the tails can be obtained by solving its equation of motion, (4.34), setting $\delta v(t)$ to the constant value $\epsilon_{a}\Delta_{a}$ and ignoring the equation of motion for $\delta v(t)$, (4.33), which would be inconsistent. The end-points of the interval, $t_{-}$ and $t_{+}$, are finally determined by $\rho(t_{\pm})=0$.

Obviously, the equation of motion for $\delta v(t)$, (4.33), must be satisfied at finite $N$. The main correction to $\delta v(t)$ and to its equation comes from the terms with $i=j$ in (4.26). Such terms contribute

[TABLE]

Notice that these terms are suppressed compared to $\mathcal{W}$.474747This is a standard argument in the context of matrix models: $\sum_{i\neq j}^{N}=O(N^{2})$ while $\sum_{i=1}^{N}=O(N)$. Notice that the contribution to $\mathcal{W}$ comes from terms with $i$ almost equal to $j$ and contributions to $\delta\mathcal{W}$ from terms with $i=j$. They contribute a term

[TABLE]

to the right-hand side of the equation (4.33) for $\delta v(t)$. Such a correction is generically of order $1/\sqrt{N}$. However, on the tails, since $\delta v(t)=\epsilon_{b}\Delta_{b}$ for some $b$, one of the logarithms blows up and the correction can be effectively of order one. Indeed, the equation of motion for $\delta v(t)$ can be satisfied if

[TABLE]

where $Y_{b}(t)$ is a quantity of order one. In this case, on the tail, the equation becomes

[TABLE]

not summed over $b$, which determines the value of $Y_{b}(t)$. Notice that, quite remarkably, the correction to $\delta v(t)$ is not power-like but exponentially small. The equations for $\rho(t)$ and the value of $\mathcal{W}$ on the solution are not affected by these corrections in the large $N$ limit since $\operatorname{Li}_{2}(z)$ is finite for $z\rightarrow 1$. However, these corrections are important for evaluating the index.

The explicit solution is as follows (Benini:2015eyy, ). Let us first take $\Delta_{a}$ real. Using the periodicity of $\Delta_{a}$, we can always restrict to the case where $0\leq\Delta_{a}\leq 2\pi$. We will also assume that $\Delta_{1}\leq\Delta_{2}$, $\Delta_{3}\leq\Delta_{4}$. The constraint (4.23) can be satisfied only for $\sum_{a}\Delta_{a}=0,2\pi,4\pi,6\pi,8\pi$ and we need to consider all possible cases. We find a solution for $\sum_{a}\Delta_{a}=2\pi$. We have a central region where

[TABLE]

When $\delta v$ hits $-\Delta_{3}$ on the left the solution becomes

[TABLE]

with the exponentially small correction $Y_{3}=(-t\Delta_{4}-\mu)/(\Delta_{4}-\Delta_{3})$, while when $\delta v$ hits $\Delta_{1}$ on the right the solution becomes

[TABLE]

with $Y_{1}=(t\Delta_{2}-\mu)/(\Delta_{2}-\Delta_{1})$. It turns out that, for $\sum_{a=1}^{4}\Delta_{a}=0,4\pi,8\pi$, equations (4.33) and (4.34) have no regular solutions. There is also a solution for $\sum_{a}\Delta_{a}=6\pi$ which, however, is obtained by the previous one by a discrete symmetry of the index: $\Delta_{a}\rightarrow 2\pi-\Delta_{a}$ $\left(y_{a}\rightarrow y_{a}^{-1}\right)$.

We can also evaluate the twisted superpotential on the solution and find

[TABLE]

The result for a different determination of the $\Delta_{a}$ is obtained by periodicity: just replace $\Delta_{a}$ with $[\Delta_{a}]=(\Delta_{a}\,{\rm mod}\,2\pi)$. We can also extend by holomorphicity the result to complex $\Delta_{a}$.

It is interesting to observe that $\widetilde{\mathcal{W}}(\Delta)$ has the same functional dependence of two important physical quantities appearing in the study of ABJM and its dual AdS${}_{4}\times S^{7}$. One is of purely field theory origin. It is known that, for any ${\cal N}=2$ theory, there exists a family of supersymmetric Lagrangians on $S^{3}$ parameterized by arbitrary R-charges of the chiral fields (Hama:2011ea, ; Jafferis:2010un, ). When the theory is superconformal, the resulting partition function has an extremum precisely at the exact R-symmetry of the theory (Jafferis:2010un, ). For ABJM at $k=1$, the $S^{3}$ free energy reads (Jafferis:2011zi, )

[TABLE]

where $r_{a}$ are a general assignment of R-charges for the fields $A_{1},A_{2},B_{1},B_{2}$ satisfying $\sum_{a=1}^{4}r_{a}=2$. We see that (Hosseini:2016tor, )

[TABLE]

The second quantity, of supergravity origin, is the prepotential of the ${\cal N}=2$ gauged supergravity describing the low energy theory of the holographic dual. Restricted to the $U(1)^{4}$ gauge sector, the prepotential is given by (4.15) and we see that

[TABLE]

This will be important for comparison with the attractor mechanism.

4.4 The large $N$ limit of the ABJM index

Using (3.38), up to an irrelevant overall factor, the index is given by

[TABLE]

where $x_{i}^{*}$ and $\tilde{x}_{i}^{*}$ is the large $N$ Bethe vacuum found in the previous section. By taking the large $N$ limit of (LABEL:final_Z), after some manipulations, one finds

[TABLE]

up to corrections of order $N\log N$. The first contribution in the first line of (4.4) comes from the Vandermonde determinant and the second from the matter contribution. The second line in (4.4) comes from the tails. Since the logarithm of the one-loop determinant of the chiral fields is singular on such regions, we need to take into account the exponentially small corrections $Y_{a}$ to the tails. The exponent $-\mathfrak{p}_{a}+1-\mathfrak{g}$ of the one-loop determinant is corrected to $-\mathfrak{p}_{a}$ by an analogous and subtle contribution from the determinant of the Hessian of $\mathcal{W}$.484848See Benini:2015eyy for details.

By plugging in the explicit solution for $\rho(t),\delta v(t)$ and $Y_{a}(t)$ we find

[TABLE]

Notice that

[TABLE]

One can show that this remarkable identity can be extended to other ${\cal N}=2$ quiver theories. Indeed it can be proved using only the equations of motion for $\rho$ and $\delta v$, and taking into account the extra terms on the tails (Hosseini:2016tor, ).

4.5 Matching index and entropy for ABJM

We can now extract the degeneracy of states from the index using (2.19). We introduce four electric charges $\mathfrak{q}_{a}$, adapted to the democratic basis of charges we are using, and we set $j_{i}=0$ since we are interested in static black holes. The degeneracy of states with given electric charge is then obtain by extremizing the functional

[TABLE]

which implicitly depends on the magnetic charges $\mathfrak{p}_{a}$ through $Z$. This is the $\mathcal{I}$- extremization principle introduced in Benini:2015eyy ; Benini:2016rke . For purely magnetic black holes we just extremize $\log Z$. For a generic dyonic static black hole, we obtain

[TABLE]

evaluated at its critical point in $\Delta_{a}$ with the constraints $\sum_{a=1}^{4}\Delta_{a}=2\pi$ and $\operatorname{\mathbb{R}e}\Delta_{a}\in[0,2\pi]$.

By an explicit computation one can see that this expression correctly reproduces the entropy of the black holes in Cacciatori:2009iz ; DallAgata:2010ejj ; Hristov:2010ri ; Katmadas:2014faa ; Halmagyi:2014qza . This can be checked more easily by comparing with the attractor equations (4.14). Using (4.12) we find

[TABLE]

Here we used the data of the relevant gauged supergravity (see (4.15))

[TABLE]

we defined adapted scalar fields

[TABLE]

satisfying $\sum_{\Lambda=0}^{3}\hat{X}^{\Lambda}=2\pi$, and enforced the Dirac quantization condition of fluxes (4.8) by defining the integers $\hat{\mathfrak{p}}^{\Lambda}$ and $\hat{\mathfrak{q}}_{\Lambda}$,

[TABLE]

Finally, we used the known holographic dictionary (see for example Marino:2011nm )

[TABLE]

Using the identification494949Notice that the identification is consistent with the twisting condition (4.10) and the constraint $\sum_{a=1}^{4}\Delta_{a}=2\pi$.

[TABLE]

we see that $S_{{\rm micro}}(\mathfrak{p}_{a},\mathfrak{q}_{a})=S_{{\rm BH}}(\mathfrak{p}^{\Lambda},\mathfrak{q}_{\Lambda})$. The extremization of (4.51) is thus completely equivalent to the attractor mechanism in gauged supergravity. The correspondence, up to constants, is

[TABLE]

Notice that the functional $\mathcal{I}(\Delta)$ is only defined up to integer multiples of $2\pi i$, due to the presence of $\log Z$. So the microscopical entropy should be properly defined as $S_{{\rm micro}}(\mathfrak{p}_{a},\mathfrak{q}_{a})=\mathcal{I}\,({\rm mod}\,2\pi i)$.

In field theory, $\mathcal{I}$ only depends on three independent chemical potentials, which we can choose to be $\Delta_{a}$ with $a=1,2,3$. The extremization of $\mathcal{I}$ determines them in terms of three electric charges, $q_{a}-q_{4}$,

[TABLE]

The entropy can be then expressed in terms of $\mathfrak{q}_{a}-\mathfrak{q}_{4}$ as

[TABLE]

In these derivations we used that $\mathfrak{q}_{a}$ are integers and $\sum_{a=1}^{4}\Delta_{a}=2\pi$.

Correspondingly, in gravity, the family of black holes depends only on three independent electric charges, as discussed in Sect. 2.3. Indeed, the requirement that $S_{{\rm BH}}(\mathfrak{p}^{\Lambda},\mathfrak{q}_{\Lambda})$ is real gives a constraints on the charges. For given magnetic charges $\mathfrak{p}_{a}$ and flavor electric charges $\mathfrak{q}_{a}-\mathfrak{q}_{4}$, there is at most one value of $\mathfrak{q}_{4}$ that leads to black hole with regular horizon.

We can see the left hand side of (4.59) as determining the average electric charge $\mathfrak{q}_{a}-\mathfrak{q}_{4}$ in our statistical ensemble at large $N$. The index only depends on the global symmetries of the theory and there are three of them. Correspondingly, with our method, we cannot determine the average electric charge associated with the R-symmetry. However, this value is fixed by the BPS equations in gravity. It would be interesting to find a purely field theoretical method for testing this prediction for the fourth charge. Notice that, with the permutationally invariant definition of $\mathcal{I}$ given in (4.50), the value of $q_{4}$ predicted by gravity is precisely the one that makes the critical value of $\mathcal{I}$ real.

Let us conclude this section with a comment. We computed the entropy as a Legendre transform of the grand canonical partition function, according to a natural microscopic point of view. However, in holography, the partition function of the boundary field theory can be also identified with the (holographically renormalized) on-shell action of the bulk theory. By consistency, we should be able to extract the same information from the gravitational on-shell action. The agreement of these points of view has been discussed in Azzurli:2017kxo ; Halmagyi:2017hmw ; Cabo-Bizet:2017xdr . In particular, the authors of Bobev:2020pjk found a family of smooth Euclidean solutions with boundary $\Sigma_{g}\times S^{1}$ whose on-shell action coincides with the topologically twisted index of ABJM for general values of the fugacities and that contains one element with an AdS2 geometry that can be wick-rotated to the original Lorentzian black hole.

4.6 Other examples and generalizations

The large $N$ limit of the topologically twisted index can be evaluated for other ${\cal N}=2$ quiver Chern–Simons theories with an AdS4 dual. There is a large class of Yang–Mills–Chern–Simons theories with fundamental and bi-fundamental chiral fields that have been proposed as duals of M-theory and massive type IIA compactifications. Most of these are obtained by dimensionally reducing a parent four-dimensional quiver gauge theory with an AdS${}_{5}\times SE_{5}$ dual, where $SE_{5}$ is a five-dimensional Sasaki–Einstein manifold, adding Chern–Simons terms and flavoring with fundamentals. Holography predicts that the twisted index scales as $N^{3/2}$ for theories dual to M-theory on seven-dimensional Sasaki–Einstein manifolds, and as $N^{5/3}$ for a class of theories dual to massive type IIA on warped six-manifolds.505050See Hanany:2008fj ; Hanany:2008cd ; Martelli:2008si ; Gaiotto:2009tk ; Benini:2009qs ; Jafferis:2008qz ; Herzog:2010hf ; Gulotta:2011vp ; Crichigno:2012sk ; Martelli:2009ga for examples of theories with M-theory dual and Guarino:2015jca ; Fluder:2015eoa for theories with massive type IIA dual. For a large class of these theories the large $N$ method discussed in Sect. 4.3 applies and the twisted index has been evaluated in Hosseini:2016tor ; Hosseini:2016ume ; Jain:2019lqb ; Jain:2019euv . Quite remarkably, in all these theories the on-shell twisted superpotential coincides with the corresponding $S^{3}$ free energy (Hosseini:2016tor, )

[TABLE]

where $d=2$ for M-theory examples and $d=3$ in massive type IIA ones.515151The twisted index depends on chemical potentials for the global symmetries. As for ABJM, we can use a redundant basis where we assign a chemical potential $\Delta_{a}$ to each field $\phi_{a}$. For each term $W_{I}$ in the superpotential we must require $\sum_{a\in W_{I}}\Delta_{a}=2\pi p$ with $p\in\mathbb{Z}$ for all the fields $\phi_{a}$ appearing in $W_{I}$. It turns out that, as for ABJM, a large $N$ solution exists only for $p=1$ (up to equivalent solutions). On the other hand the $S^{3}$ free energy is a function of the R-charges of the fields, and these are constrained to satisfy $\sum_{a\in W_{I}}r_{a}=2$. The identity (4.61) makes sense because $p=1$. The partition function is given by the following index theorem (Hosseini:2016tor, )525252See Hosseini:2018qsx for more details and applications.

[TABLE]

It is often possible, as for ABJM, to choose a convenient redundant parameterization for the chemical potentials such that $\widetilde{\mathcal{W}}$ becomes a homogeneous functions of degree $d$. For this choice the twisted index is given again by

[TABLE]

The entropy of dyonic static black holes in such theories is obtained by taking the Legendre transform of (4.63). Comparison with the attractor mechanism in the form (4.13) then requires that

[TABLE]

under some map $\Delta_{a}\rightarrow X^{\Lambda}$. (4.64) is however oversimplified. The effective low energy theory for M or type IIA compactifications on general manifolds is an ${\cal N}=2$ gauged supergravity containing massless vector multiplets associated with the global symmetries of the CFT. In general, such gauged supergravity contains also hypermultiplets and a number of vector multiplet that is larger than the number of global symmetries. Therefore, in (4.64), the number of $\Delta_{a}$ does not match the number of sections $X^{\Lambda}$. In general, the hypermultiplets have a VEV at the horizon and give a mass to some of the vector fields. If $n_{V}$ is the number of vector multiplets and $n_{H}$ is the number of hypermultiplets, $n_{V}-n_{H}$ is the number of massless vectors, corresponding to the number of global symmetries and therefore to the number of independent $\Delta_{a}$. The BPS equations for the hyperinos are typically algebraic and give linear constraints among the sections $X^{\Lambda}$. By solving these constraints we can obtain an effective prepotential $\mathcal{F}_{{\rm eff}}(X^{\Lambda})$ for the massless vector multiplets only. If we believe that this further truncated theory correctly describes the horizon of the black hole, (4.64) should hold with $\mathcal{F}(X^{\Lambda})$ replaced by $\mathcal{F}_{{\rm eff}}(X^{\Lambda})$.535353This happens for example for the mass deformed ABJM model (Bobev:2018uxk, ), where the value of one $\Delta_{a}$ is fixed by the mass deformation and, in gravity, one $X^{\Lambda}$ is fixed by the hyperino conditions.

Two notable applications where this works perfectly are the following. A well-known example of massive type IIA background is the warped AdS${}_{4}\times Y_{6}$ flux vacua of massive type IIA dual to the $\mathcal{N}=2$ $U(N)$ gauge theory with three adjoint multiplets and a Chern–Simons coupling $k$ (Guarino:2015jca, ). It corresponds to an internal manifold $Y_{6}$ with the topology of $S^{6}$. The theory has two global symmetries and there exists a family of black holes depending on two magnetic charges (Guarino:2017pkw, ). The entropy of such black holes has been matched with the prediction of the twisted index in Hosseini:2017fjo ; Benini:2017oxt . For this example

[TABLE]

The second example involves the so-called universal black hole (Azzurli:2017kxo, ). This is dual to the universal twist (Benini:2015bwz, ; Bobev:2017uzs, ) defined by a set of magnetic fluxes $\mathfrak{p}_{a}$ proportional to the exact R-symmetry of the CFT. This black hole is a solution of minimal gauged supergravity and, as such, can be embedded in all M-theory and massive type IIA compactifications, thus explaining the name universal. Hence it provides an infinite family of examples. Since $\widetilde{\mathcal{W}}$ is proportional to the $S^{3}$ free energy (see (4.61)) and the latter is extremized at the exact R-symmetry, it follows easily from (4.62) that

[TABLE]

This relation agrees with the gravitational prediction based on minimal gauged supergravity (Azzurli:2017kxo, ).

There have been progresses in various directions and also open problems:

• •

For theories with M-theory dual the large $N$ method discussed above works only when the bi-fundamental fields transform in a real representation of the gauge group and the total number of fundamentals is equal to the total number of anti-fundamentals. The same restriction has been found in Jafferis:2011zi for the large $N$ evaluation of the $S^{3}$ free energy. It is not known yet how to take the large $N$ limit for the $S^{3}$ free energy or the twisted index in the case of chiral quivers. Also in the case of vector-like quivers, where the method works, there is an extra complication since baryonic symmetries are invisible in the large $N$ limit (accidental flat directions of the matrix model). It is still not clear how to introduce and study baryonic fluxes in the large $N$ limit of the twisted index.

• •

A gravitational dual of $\mathcal{I}$-extremization was proposed in Couzens:2018wnk , generalizing similar results for $a$- and $c$-extremization (Martelli:2005tp, ; Gauntlett:2018dpc, ). In this approach, the entropy of an M theory black hole with AdS2 horizon is obtained by extremizing a geometric quantity associated with a supersymmetric background where some of the equations of motion have been relaxed. As a result, the entropy can be expressed in terms of simple geometrical data of the internal manifold and can be computed also in the cases where the explicit solution is not known. In particular, this provides an explicit formula for the entropy and the exact R-symmetry of the associated superconformal mechanics for an infinite class of models based on toric Sasaki-Einstein manifolds. The agreement of the gravitational picture with the available field theory results at large $N$ has been discussed in Gauntlett:2019roi ; Hosseini:2019ddy ; Kim:2019umc .

• •

The microstate counting for dyonic rotating black holes (Hristov:2018spe, ) is still an open problem. The chemical potential dual to rotation in the bulk is associated with an Omega-background for the boundary theory on $S^{2}\times S^{1}$. The topologically twisted index in an Omega-background is known (Benini:2015noa, ) but it gives rise to a complicated matrix model. For particular values of the $\Omega$-deformation, it has been written as a sum over Bethe vacua (Closset:2018ghr, ). This could be useful to solve the matrix model in the large $N$ limit. The result is expected to match the entropy functional proposed in Hosseini:2019iad by gluing gravitational blocks.

• •

For asymptotically flat black holes, there is a huge literature including higher derivative corrections and highly detailed precision tests. For asymptotically AdS black holes, the story is just begun. At the moment, $1/N$ corrections to the twisted index for ABJM and the above massive type IIA background have been computed numerically focusing on the universal logarithmic correction (Liu:2017vbl, ; Liu:2017vll, ; Jeon:2017aif, ; Liu:2018bac, ; PandoZayas:2019hdb, ). The ABJM matrix model provides a quantum corrected entropy functional that would be interesting to study further. In particular, it would be interesting to find the analog of standard results and conjectures about the quantum entropy of asymptotically flat back holes, like the OSV conjecture (Ooguri:2004zv, ), the Sen’s quantum entropy functional (Sen:2008vm, ; Sen:2009vz, ) and localization in supergravity (Dabholkar:2010uh, ; Dabholkar:2011ec, ),545454See Iqbal:2003ds for an earlier localization computation in gravity. to cite only few of them. In particular, some interesting results about localization in AdS${}_{2}\times S^{2}$ have been already found in Hristov:2018lod ; Hristov:2019xku .

• •

The topologically twisted index has been extended to five-dimensions (Hosseini:2018uzp, ; Crichigno:2018adf, ) and successfully compared with the entropy of AdS6 black holes with horizon AdS${}_{2}\times\Sigma_{\mathfrak{g}_{1}}\times\Sigma_{\mathfrak{g}_{2}}$ (Suh:2018tul, ; Hosseini:2018usu, ; Suh:2018szn, ; Suh:2018qyv, ; Fluder:2019szh, ). The expression for the entropy is given by a generalization of the index theorem (4.63) and (4.62).

Other interesting related results and solutions can be found in Cabo-Bizet:2017jsl ; Toldo:2017qsh ; Gang:2018hjd ; Gang:2019uay ; Hosseini:2019and ; Bae:2019poj .

5 Black strings in AdS5

The methods introduced in Sect. 4 can be generalized to other dimensions and can be used to provide general tests of holography. In particular, they can be applied to domain wall solutions interpolating between AdSd+n and AdS${}_{d}\times\mathcal{M}_{n}$, with a topological twist along the $n$-dimensional compact manifold $\mathcal{M}_{n}$. In this section, making a brief digression from the main subject of these notes, we discuss the example of black strings in AdS5.

More precisely, we consider a family of black strings in AdS5 with near horizon geometry AdS${}_{3}\times\Sigma_{\mathfrak{g}}$. They correspond holographically to a twisted compactification of a four-dimensional theory flowing to a two-dimensional CFT in the IR. The properties of this CFT, and, in particular, its elliptic genus, can be computed using a topologically twisted index for four-dimensional theories on $\Sigma_{\mathfrak{g}}\times T^{2}$. We can make contact with the physics of black holes by compactifying the black string on a circle, as in the standard example (Strominger:1996sh, ). In this way we can also make contact with a Cardy formula approach to the microstate counting. Other examples of similar tests of holography related to twisted compactifications can be found in Toldo:2017qsh ; Crichigno:2018adf ; Hosseini:2018uzp .

5.1 Black string solutions in AdS${}_{5}\times S^{5}$

We consider solutions of a five-dimensional ${\cal N}=2$ effective gauged supergravity with abelian vector multiplets of the form

[TABLE]

where $A_{\Sigma_{\mathfrak{g}}}$ is the gauge potential for a magnetic flux on $\Sigma_{\mathfrak{g}}$ and supersymmetry is preserved with a twist along $\Sigma_{\mathfrak{g}}$. We are interested in solutions that are asymptotic to AdS5 for large values of the radial coordinate,

[TABLE]

and approach a regular horizon AdS${}_{3}\times\Sigma_{\mathfrak{g}}$ at some fixed value $r=r_{0}$,

[TABLE]

These solutions can be interpreted as black strings extended in the direction $z$ or, equivalently, as domain walls interpolating between AdS5 and AdS${}_{3}\times\Sigma_{\mathfrak{g}}$. They are holographically dual to a twisted compactification of a SCFT4 on $\Sigma_{\mathfrak{g}}$ that flows in the IR to a two-dimensional $(0,2)$ SCFT2 associated with the horizon factor AdS3.

Solutions that can be embedded in AdS${}_{5}\times S^{5}$ have been found in Benini:2013cda , using a five-dimensional gauged supergravity with three abelian gauge fields associated with the isometries $U(1)^{3}\subset SO(6)$ of $S^{5}$. The solution depends on three fluxes $\mathfrak{p}_{a}$ constrained by the twisting condition

[TABLE]

and corresponds to a twisted compactification of ${\cal N}=4$ SYM on $\Sigma_{\mathfrak{g}}$. Holography suggests that this theory flows to an IR two-dimensional SCFT. The value of the central charge of the IR SCFT2 in the large $N$ limit can be extracted from the solution using standard arguments (Brown:1986nw, ; Henningson:1998gx, ) and reads (Benini:2013cda, )

[TABLE]

This result can be successfully compared with field theory (Benini:2013cda, ), where central charges can be computed using extremization techniques. Let us briefly explain how this works. Consider first the case of ${\cal N}=1$ four-dimensional SCFTs and ${\cal N}=4$ SYM as an example. In ${\cal N}=1$ language, the theory contains three chiral multiplets $\phi_{i}$ with the superpotential

[TABLE]

We introduce a generic R-charge assignment $\Delta_{a}$ for the three chiral fields $\phi_{i}$. Since the superpotential has R-charge $2$, we must have $\Delta_{1}+\Delta_{2}+\Delta_{3}=2$. The exact R-symmetry of an ${\cal N}=1$ superconformal theory can be obtained by extremizing a trial $a$-charge

[TABLE]

where $R(\Delta_{a})$ is the matrix of R-charges of the fermionic fields. This construction is known as $a$-maximization (Intriligator:2003jj, ). For ${\cal N}=4$ SYM at large $N$ we find555555For ${\cal N}=4$ SYM $\operatorname{Tr}R$ is identically zero. For theories with an AdS dual, $\operatorname{Tr}R=0$ in the large $N$ limit.

[TABLE]

where the first contribution in the bracket comes from the gauginos and the second from the fermions in the chiral multiplets. This expression is trivially extremized for $\Delta_{1}=\Delta_{2}=\Delta_{3}=2/3$, the exact R-charges of the fields $\phi_{i}$. The critical value of $a(\Delta)$ is the central charge $a$ of the SCFT4. Similarly, the exact R-symmetry and the right-moving central charge of a two-dimensional $(0,2)$ SCFT are obtained by extremizing the trial quantity (Benini:2012cz, )

[TABLE]

where $\gamma_{3}$ is the two-dimensional chirality operator and $R(\Delta_{a})$ the matrix of R-charges of the massless fermionic fields. This construction is known as $c$-extremization (Benini:2012cz, ). For the twisted compactification of ${\cal N}=4$ SYM, the quantity $c_{r}$ can be computed using topological arguments. We just need to know the difference between the number of fermionic zero modes with positive and negative chirality. This is easily computed from the Riemann-Roch theorem as in section 3.3.1. We then find at large $N$ (Benini:2013cda, )

[TABLE]

where again the first contribution in the first line bracket comes from gauginos and the second from matter fields. One can easily check that the extremization of (5.9) with respect to $\Delta_{a}$ reproduces (5.4). The agreement is valid in the large $N$ limit where $c=c_{l}=c_{r}$.565656$c_{r}-c_{l}$ is equal to the gravitational anomaly $k={\rm Tr}\gamma_{3}$ which is of order one in the large $N$ limit (Benini:2012cz, ).

Notice that, in the large $N$ limit, the $c_{r}$ trial central charge can be written as (Hosseini:2016cyf, )

[TABLE]

It is interesting to observe that formula (5.10), with a suitable parameterization for the fluxes and the R-charges, holds for all the twisted compactifications of ${\cal N}=1$ SCFT4 dual to AdS${}_{5}\times SE_{5}$, where $SE_{5}$ is a toric Sasaki–Einstein manifold (Hosseini:2016cyf, ).575757We refer to Hosseini:2016cyf for a proof, to Amariti:2017iuz for more detailed expressions, and to the appendices of Hosseini:2018uzp ; Hosseini:2019use for an alternative derivation based on the anomaly polynomial.

In the case of black strings the information about states is encoded in the elliptic genus of the two-dimensional $(0,2)$ CFT

[TABLE]

where $q=e^{2\pi i\tau}$, with $\tau$ the modular parameter of $T^{2}$, $y_{I}$ are fugacities for the global symmetries and $L_{0}$ the left-moving Virasoro generator. Since the index is independent of the scale, we can evaluate it in the UV, where it becomes the topologically twisted index, defined as the partition function on $\Sigma_{\mathfrak{g}}\times T^{2}$, with a topological twist along $\Sigma_{\mathfrak{g}}$.

5.2 The topologically twisted index on $T^{2}\times\Sigma_{\mathfrak{g}}$

For an $\mathcal{N}=1$ four-dimensional gauge theory with a non-anomalous $\mathrm{U}(1)$ R-symmetry, the topologically twisted index is a function of $q=e^{2\pi i\tau}$, where $\tau$ is the modular parameter of $T^{2}$, fugacities $y_{I}$ for the global symmetries and flavor magnetic fluxes $\mathfrak{m}^{F}_{I}$ on $\Sigma_{\mathfrak{g}}$ parameterizing the twist. It can be computed using localization and it is given by an elliptic generalization of the formulae in Sect. 3.2 (Closset:2013sxa, ; Benini:2015noa, ). Explicitly, for a theory with gauge group $G$ and a set of chiral multiplets transforming in representations $\mathfrak{R}_{I}$ of $G$ with R-charge $r_{I}$, the topologically twisted index is given by a contour integral of a meromorphic form

[TABLE]

where $\alpha$ are the roots of $G$, $\rho_{I}$ the weights of the representation $\mathfrak{R}_{I}$ and $|W|$ denotes the order of the Weyl group. In this formula, $\theta_{1}(x;q)$ is a Jacobi theta function and $\eta(q)$ is the Dedekind eta function. The zero-mode gauge variables $x=e^{iu}$ parameterize the Wilson lines on the two directions of the torus

[TABLE]

and are defined modulo

[TABLE]

As in three dimensions, the result is summed over a lattice of gauge magnetic fluxes $\mathfrak{m}$ living in the co-root lattice $\Gamma_{\mathfrak{h}}$ of the gauge group $G$ and the contour of integration selects the Jeffrey–Kirwan prescription for taking the residues. In four dimensions there is a one-loop contribution from the Cartan components of the vector multiplets. One can show that the integrand in (5.12) is a well-defined meromorphic function of $x$ on the torus provided that the gauge and the gauge-flavor anomalies vanish. The index has a trace interpretation as a sum over a Hilbert space of states on $\Sigma_{\mathfrak{g}}\times S^{1}$

[TABLE]

This trace reduces to the elliptic genus of the two-dimensional theory obtained by the twisted compactification on $\Sigma_{\mathfrak{g}}$.

We now consider the index for ${\cal N}=4$ SYM. The superpotential (5.5) imposes the following constraints on the chemical potentials $\Delta_{a}$ and the flavor magnetic fluxes $-\mathfrak{p}_{a}+1-\mathfrak{g}=(\mathfrak{g}-1)(r_{a}-1)+\mathfrak{m}^{F}_{a}$ associated with the fields $\phi_{a}$,

[TABLE]

The topologically twisted index for the SYM theory with gauge group $SU(N)$ is then given by

[TABLE]

with $y_{a}=e^{i\Delta_{a}}$. The Bethe vacua are determined by

[TABLE]

The Bethe equations (5.18) have a remarkably simple solution (Hosseini:2016cyf, ; Hong:2018viz, )585858This solution was found in the high temperature limit in Hosseini:2016cyf and proved to be exact for all $\tau$ in Hong:2018viz . Using $SL(2,\mathbb{Z})$, the authors of Hong:2018viz have found many other solutions of the Bethe equations for ${\cal N}=4$ SYM at finite $\tau$. Non-standard solutions corresponding to continua of Bethe vacua have been found in ArabiArdehali:2019orz . The solution (5.19) extends to more general quivers (Hosseini:2016cyf, ; Lezcano:2019pae, ; Lanir:2019abx, ).

[TABLE]

These Bethe vacua will play a role also in the physics of AdS5 black holes (Benini:2018mlo, ; Benini:2018ywd, ), as discussed in Sect. 7.2.

This time the index is too hard to solve in the large $N$ limit. We can study instead the high temperature limit corresponding to a shrinking of the torus given by $\tau=i\beta/(2\pi)$ with $\beta\rightarrow 0^{+}$ (Hosseini:2016cyf, ).595959Notice that $\beta$ is not really a temperature, but rather a parameterization of the modular parameter of the torus. This limit is also particularly interesting from the field theory point of view because it controls the asymptotic growing of the number of states with the dimension (Cardy:1986ie, ). Using (5.19) it is easy to compute, at leading order in $\beta$, (Hosseini:2016cyf, )

[TABLE]

where $\widetilde{\mathcal{W}}(\Delta)$ is the on-shell value of the twisted superpotential. As in the three-dimensional case, the result is valid for $\sum_{a}\Delta=2\pi$. We see a striking similarity with the black hole case, in particular equation (4.63). Moreover, the twisted superpotential is proportional to the trial $a$-central charge (5.7) of the four-dimensional SCFT. This statement is the four-dimensional analog of (4.61).

Comparing with (5.10) we find the Cardy formula

[TABLE]

valid at leading order in $\beta$ and at large $N$.

This result can be extended to all ${\cal N}=1$ SCFT4 dual to AdS${}_{5}\times SE_{5}$ vacua (Hosseini:2016cyf, ). It can be also generalized to the case of the refined topologically twisted index on $S^{2}\times T^{2}$

[TABLE]

where $\zeta=e^{i\omega/2}$ is a fugacity for the angular momentum. The result is (Hosseini:2019lkt, )

[TABLE]

valid at leading order in $\beta$ and at large $N$.

The example discussed in this section can be actually generalized to many other flows interpolating between AdS3+n and AdS${}_{3}\times\mathcal{M}_{n}$ where supersymmetry is preserved along the compact manifold $\mathcal{M}_{n}$ with a topological twist. Examples of compactifications of the $(2,0)$ theory in six dimensions on the product of two Riemann surfaces $\Sigma_{\mathfrak{g}_{1}}\times\Sigma_{\mathfrak{g}_{2}}$ are discussed for example in Hosseini:2018uzp .

5.3 Back to Cardy

Similarly to what is done for generic CFT2 (Cardy:1986ie, ), we can extract information on the growing of the number of supersymmetric states with the energy from the asymptotic behavior (5.21) of the elliptic genus.

From the definition of the index as a trace (5.22), we see that the number of supersymmetric states with momentum $n$, electric charge $\mathfrak{q}_{a}$ under the Cartan subgroup of $SO(6)$ and angular momentum $j$ can be extracted as a Fourier coefficient

[TABLE]

where $\beta=-2\pi i\tau$ and the corresponding integration is over the imaginary axis.

In the limit of large charges, we can use the saddle point approximation. Consider first $\mathfrak{q}=0$ and $j=0$. The number of supersymmetric states with charges $(\mathfrak{p},n)$ can be obtained by extremizing

[TABLE]

with respect to $\Delta$ and $\beta$. Given (5.21), we see that the extremization with respect to $\Delta$ is the $c$-extremization principle (Benini:2012cz, ; Benini:2013cda, ) and sets the trial right-moving central charge $c_{r}(\Delta)$ to its exact value $c_{{\rm CFT}}(\mathfrak{p})$ given in (5.4). Extremizing $\mathcal{I}(\beta,\Delta)$ with respect to $\beta$ yields

[TABLE]

Plugging back (5.26) into $\mathcal{I}(\beta,\Delta)$, we find for the entropy of states

[TABLE]

This is obviously nothing else than Cardy formula (Cardy:1986ie, ).606060For a further discussion about the asymptotic behavior see Hosseini:2018qsx .

One can generalize the previous computation to the case $\mathfrak{q}_{a}\neq 0$ and $j\neq 0$ by extremizing

[TABLE]

After some manipulations, the result can be expressed in the form (Hosseini:2019lkt, ; Hosseini:2020vgl, )

[TABLE]

where the index $I$ runs over a set of independent global symmetries $J_{K}$ and $A_{IK}=\operatorname{Tr}\gamma_{3}J_{I}J_{K}$ is the ’t Hooft anomaly matrix of the two-dimensional theory. This result holds for a generic ${\cal N}=1$ quiver with a Sasaki-Einstein dual. The particular combination of $n$, electric charges and angular momentum appearing in (5.29) is related to the properties of the elliptic genus and is familiar from the physics of asymptotically flat black holes (Dijkgraaf:2000fq, ; Kraus:2006nb, ; Manschot:2007ha, ).616161Our elliptic genus is actually a meromorphic function of $\Delta_{a}$ and this leads to a more complicated structure of the corresponding modular forms, related to wall-crossing phenomena. For asymptotically flat black holes this leads to interesting behaviours (see for example Dabholkar:2012nd ). It would be interesting to see if the same happens for the black holes discussed in Sect. 5.4 obtained by compactifying the black string.

5.4 Cardy formula and black holes

There is standard argument to obtain a black holes from a black string: compactify along the circle inside AdS3 and add a momentum $n$ along it. More precisely, one replaces the near-horizon geometry of the five-dimensional black string with $\text{BTZ}\times\Sigma_{\mathfrak{g}_{1}}$, where the metric for the extremal BTZ reads (Banados:1992wn, )

[TABLE]

Here, the parameter $\rho$ is related to the electric charge $n$. This solution is locally equivalent to AdS3, since there exists locally only one constant curvature metric in three dimensions, and solves the same BPS equations; however, BTZ and AdS3 are inequivalent globally. Compactifying the full five-dimensional black string of Benini:2013cda on the circle with the extra momentum we obtain a static BPS black hole in four dimensions, with magnetic charges $\mathfrak{p}_{a}$ and electric charge $n$. This can be thought as a domain wall that interpolates between an AdS${}_{2}\times\Sigma_{\mathfrak{g}_{1}}$ near-horizon region and a complicated asymptotic non-AdS4 vacuum (Hristov:2014eza, ). By another standard argument, the entropy of such a black hole is given by the number of states of the CFT with momentum $n$, and is therefore given by the Cardy formula (5.27)

[TABLE]

One can see that this prediction matches the black holes entropy computed from supergravity (Hristov:2014eza, ). Moreover, a family of rotating dyonic black strings in AdS${}_{5}\times S^{5}$ was found in Hosseini:2019lkt and the entropy of the compactified black hole successfully compared with (5.29). An analogous matching for black strings in AdS${}_{7}\times S^{4}$ was discussed in Hosseini:2020vgl .

It is also interesting to observe that, for static black holes, the field theory extremization of $\mathcal{I}$ becomes again equivalent to the attractor mechanism in the reduced four-dimensional gauged supergravity. The dimensional reduced supergravity has one more vector coming from the reduction on the circle, prepotential $\mathcal{F}\sim\frac{X^{1}X^{2}X^{3}}{X^{0}}$, and purely electric FI parameters $g_{0}=0$ and $g_{i}=g$ (Hristov:2014eza, ). Under the identifications $\hat{X}^{0}\rightarrow\beta\,,\hat{X}^{i}\rightarrow i\Delta_{a}\,,(\mathfrak{q}_{\Lambda},\mathfrak{p}^{\Lambda})\rightarrow(-n,\mathfrak{q}_{1},\mathfrak{q}_{2},\mathfrak{q}_{3},0,\mathfrak{p}_{1},\mathfrak{p}_{2},\mathfrak{p}_{3})$ and $\mathcal{F}(\hat{X})\rightarrow\widetilde{\mathcal{W}}(\Delta)$, the attractor functional (4.13) can be identified with the $\mathcal{I}$-functional (5.25).

6 The superconformal index

As we discussed in Sect. 2, we should be able to obtain the entropy of supersymmetric Kerr–Newman black holes in AdSd+1 by counting states in the dual CFT on $S^{d}\times\mathbb{R}$. The relevant supersymmetric quantity to consider is the superconformal index, which enumerates BPS states on $S^{d}\times\mathbb{R}$ (Romelsberger:2005eg, ; Kinney:2005ej, ).

We start by considering ${\cal N}=1$ supersymmetric field theories on $S^{3}\times\mathbb{R}$. We refer to Festuccia:2011ws for an explicit description of the supersymmetric Lagrangian on $S^{3}\times\mathbb{R}$ and the corresponding supersymmetry transformations. The superconformal index is defined as the trace

[TABLE]

on the Hilbert space of states on $S^{3}$. Here $J_{1}$ and $J_{2}$ generate rotations on $S^{3}$, $R$ is the R-symmetry generator, and $J$ denotes collectively the global symmetries. $\mathcal{Q}$ is a supercharge with $J_{1}=J_{2}=-1/2$ and $R=1$ and satisfies the algebra

[TABLE]

where $\Delta$ generates translation along $\mathbb{R}$. We introduced fugacities, $p$, $q$, and $u$, for all the generators that commute with $\mathcal{Q}$, which are $J_{1}+R/2$, $J_{2}+R/2$ and the global symmetries $J$. The quantity (6.1) is a Witten index in the sense discussed in Sect. 2.2.2 and it is therefore independent of $\beta$ and invariant under continuous small deformations of the Lagrangian. Despite the name, the supersymmetric index (6.1) makes sense also for non-conformal theories. Since we are interested in holography, we will just consider the case of conformal theories where $\Delta$ can be identified with the dilatation operator.

The index (6.1) can be also written as the Euclidean supersymmetric partition function on $S^{3}\times S^{1}$, as discussed in Sect. 2.2.2, and computed using localization. The precise relation among the two quantities involves a prefactor

[TABLE]

where the quantity $E_{c.e}$, called supersymmetric Casimir energy (Assel:2014paa, ; Assel:2015nca, ; Lorenzen:2014pna, ; Genolini:2016sxe, ; Martelli:2015kuk, ; Closset:2019ucb, ), is due to the regularization of the one-loop determinants and it can be interpreted as the vacuum expectation value of the Hamiltonian.

The superconformal index can be computed explicitly through localization or, being invariant under continuous deformations, just by enumerating the gauge invariant states annihilated by $\mathcal{Q}$ and $\mathcal{Q}^{\dagger}$ in the weakly coupled UV theory. The result for an ${\cal N}=1$ theory with gauge group $G$ and chiral matter in the representation $\mathcal{R}_{a}$ and R-charge $r_{a}$ is (Romelsberger:2005eg, ; Kinney:2005ej, ; Dolan:2008qi, )

[TABLE]

where $r$ is the rank of the gauge group, $\alpha$ the roots, $\rho_{a}$ are the weights of the representation $\mathcal{R}_{a}$, $\nu_{a}$ the flavor weights, and $|W|$ the order of the gauge group. The integration is over the Cartan subgroup of $G$ and is taken over the unit circle for all variables $z_{i}$. The special functions appearing in the previous formula are the elliptic Gamma function (Felder_2000, )

[TABLE]

and the q-Pochhammer symbol

[TABLE]

In the localization approach, numerator and denominator of (6.4) arise as one-loop determinants for chiral matter multiplets and vector multiplets, respectively. The reader can compare the formula for the superconformal index (6.4) with the analogous one for the topologically twisted one (3.11) and look for analogies and differences.626262One main difference is the sum over magnetic fluxes in the (3.11). From a more technical point of view, another big difference is the integration contour.

The superconformal index can be defined and computed through localization also in other dimensions (Bhattacharya:2008zy, ; Pestun:2016jze, ). For three-dimensional theories the elliptic Gamma functions are replaced by hyperbolic ones and there is an extra sum over magnetic fluxes, as in (3.11), due to the existence of local BPS monopole operators in three dimensions (Kim:2009wb, ; Imamura:2011su, ; Kapustin:2011jm, ; Dimofte:2011py, ).

7 Electrically charged rotating black holes

In this section we discuss the case of Kerr–Newman black holes in various dimensions. These are electrically charged rotating black holes without a twist. As argued in Sect. 2, they are qualitatively different from the magnetically charged black holes discussed in Sect. 4.

In general, we should be able to recover the entropy of the electrically charged rotating black holes in AdSd from the BPS partition function (2.16) that counts supersymmetric states of the dual CFT on $S^{d-2}\times\mathbb{R}$. Since the black holes preserve just two real supercharges, we need to count 1/16 BPS states and this is a hard problem. Old attempts to evaluate the BPS partition function of ${\cal N}=4$ SYM (Grant:2008sk, ; Chang:2013fba, ; Yokoyama:2014qwa, ) reached the somehow disappointing conclusion that there is a subset of states whose number grows with $N$ but slower than the entropy of the black holes. Alternatively, one may try to replace the BPS partition function with the corresponding index (2.23). The appropriate index is the superconformal one, defined in the previous section, that can be expressed and computed as a supersymmetric Euclidean path integral on $S^{d-2}\times S^{1}$. It is known that, for generic real fugacities, the superconformal index is a quantity of order one in the large $N$ limit (Kinney:2005ej, ) and, as such, it does not reproduce the entropy which grows with powers of $N$. As a difference with the twisted index, already in the large $N$ limit, there is a large cancellation between bosonic and fermionic supersymmetric states and $\mathcal{Z}_{\rm index}(\Delta_{a},\omega_{i})\neq\mathcal{Z}(\Delta_{a},\omega_{i})$. All these (partial) results have stood for long time as puzzles about supersymmetric electrically charged rotating black holes.

However, the entropy functionals for many Kerr–Newman black holes in different dimensions can be expressed in terms of quantities with a clear field theory interpretation (Hosseini:2017mds, ; Hosseini:2018dob, ; Choi:2018fdc, ), thus suggesting that the entropy can be always reproduced by a field theory computation. These entropy functionals also suggest complex value for the chemical potentials $\Delta_{a}$ and $\omega_{i}$. As stressed in Choi:2018hmj ; Benini:2018ywd , the computation in Kinney:2005ej is valid only for real fugacities and the introduction of phases in the fugacities can obstruct the cancellation at large $N$ and lead to an enhancement of the entropy. This is also in the spirit of the $\mathcal{I}$-extremization principle discussed in Sect. 2.4. Recent results for AdS5 and other dimensions, started with the work of Cabo-Bizet:2018ehj ; Choi:2018hmj ; Benini:2018ywd , confirm this point of view and lead to various derivations of the entropy using the index, as we discuss in this section.

7.1 The entropy functional for electrically charged rotating black holes

For many Kerr–Newman black holes, the entropy, as a function of electric charges $q_{a}$ and angular momenta $j_{i}$, can be written as a Legendre transform

[TABLE]

of a quantity $\log\mathcal{Z}(\Delta_{a},\omega_{i})$ related to anomalies or free energies of the dual CFT. This was derived in Hosseini:2017mds for five-dimensional black holes and generalized to other dimensions in Hosseini:2018dob ; Choi:2018fdc .636363A proposal for non BPS black holes can be found in Larsen:2019oll . We consider first the example of AdS${}_{5}\times S^{5}$ and we come back later to the general case.

7.1.1 The entropy functional for Kerr–Newman black holes in AdS${}_{5}\times S^{5}$

As discussed in Sect. 2.1.1, the Kerr–Newman black holes in AdS${}_{5}\times S^{5}$ depend on three charges $q_{1},q_{2},q_{3}$ associated with $U(1)^{3}\subset SO(6)$, the Cartan subgroup of the isometry of $S^{5}$, and two angular momenta $j_{1}$ and $j_{2}$ in AdS5. Supersymmetry requires a non-linear constraint among these conserved charges, whose form we discuss below, and leaves a four-dimensional family of BPS black holes (Gutowski:2004yv, ; Gutowski:2004ez, ; Chong:2005da, ; Chong:2005hr, ; Kunduri:2006ek, ). The entropy can be compactly written as (Kim:2006he, )

[TABLE]

where $G_{{\text{N}}}^{(5)}$ is the five-dimensional Newton constant and $g$ the gauge coupling of the five-dimensional effective supergravity. Holography relates these quantities to the number of colors of the dual field theory, ${\cal N}=4$ SYM in four dimensions, through $N^{2}=\pi/(2G_{{\text{N}}}^{(5)}g^{3})$. We see that black holes with charges and angular momenta of order $O(N^{2})$ have an entropy of order $O(N^{2})$.

Quite remarkably, the entropy (7.2) can be written as the Legendre transform of a very simple quantity (Hosseini:2017mds, )

[TABLE]

where the chemical potentials are constrained by

[TABLE]

This constraint resembles the analogous one, (4.42), for black holes in ABJM. The quantity in (7.3) is extremized for complex values of the chemical potentials $\Delta_{a}$ and $\omega_{i}$. However, quite remarkably, the on-shell value (7.3) becomes real once we impose the non-linear constraint among charges imposed by supersymmetry. In fact, a simple way of characterize the constraint on charges is to identify it with the imaginary part of entropy functional (7.3) at its extremum. The fact that the critical value for $\Delta_{a}$ and $\omega_{i}$ are complex will also play an important for the field theory interpretation of the result. The two choice of signs in (7.4) lead to the same final result. Formally, this is due to the fact that (7.3) is a holomorphic homogeneous function of the chemical potentials of degree one.646464Consider the two functionals $S_{\pm}=S_{\text{BH}}-i\Lambda(\sum_{a=1}^{3}\Delta_{a}-\sum_{i=1}^{2}\omega_{i}\mp 2\pi)$, where the constraint is enforced by the Lagrange multiplier $\Lambda$. Given the homogeneity of $S_{\text{BH}}$, the extremal value is $S_{\pm}=\pm 2\pi i\Lambda$. Since $q_{a}$ and $j_{i}$ are real, it is immediate to see that, if $(\Delta_{a},\omega_{i},\Lambda)$ is an extremum of $S_{+}$, then $(-\bar{\Delta}_{a},-\bar{\omega}_{i},\bar{\Lambda})$ is an extremum of $S_{-}$ and the extremal values are related by $S_{+}=\bar{S}_{-}$. Therefore the extremization with different constraints (7.4) give the same results for the real part of the entropy.

The quantity

[TABLE]

must be interpreted as a grand canonical partition function, and, as discussed in 2.3, should be related to the on-shell action of the Euclidean black hole. This has been proved in Cabo-Bizet:2018ehj by taking the zero-temperature limit of the on-shell action of a family of supersymmetric complexified non-extremal Euclidean solutions. In this approach, one can also derive the complex value for the chemical potentials at the saddle point.656565The chemical potentials are extracted following the logic discussed at the end of Sect. 2.2.1. The constraints $\sum_{a}\Delta_{a}-\sum_{i}\omega_{i}=\pm 2\pi$ arise due to regularity conditions to be imposed on the Killing spinors.

7.1.2 The entropy functional for Kerr–Newman black holes in diverse dimensions

The expression for the entropy functionals for Kerr–Newman black holes in diverse dimensions is schematically given in Table 1.

The content of the table refers to electrically charged rotating black holes in each dimension that can be embedded in a maximally supersymmetric string theory or M-theory background. In addition to the AdS${}_{5}\times S^{5}$ black holes in type IIB, there are analogous AdS${}_{4}\times S^{7}$ and AdS${}_{7}\times S^{4}$ black holes in M-theory (Chow:2007ts, ; Cvetic:2005zi, ; Chong:2004dy, ). The dual field theories are well known: the ABJM theory in three dimensions and the $(2,0)$ theory in six dimensions. In addition, there are black holes in the warped AdS${}_{6}\times_{W}S^{4}$ background of massive type IIA (Brandhuber:1999np, ; Chow:2008ip, ). The dual CFT is the ${\cal N}=1$ five-dimensional fixed point associated with D4-D8-O8 branes in type IIA found in Seiberg:1996bd . Notice that, in six dimensions, the maximal superconformal algebra has only sixteen supercharges instead of thirty-two and the superconformal theory with such an algebra is not unique. Among the theories with an AdS dual, the D4-D8-O8 system is somehow the simplest and most studied.

The chemical potentials in the table refer to the isometries of the internal manifold.666666In the case of the D4-D8-O8 system, the field theory has $SU(2)_{R}\times SU(2)\times E_{n_{f}+1}$ symmetry, where $SU(2)_{R}\times SU(2)$ is realized by the isometry of the warped $S^{4}$ and $E_{n_{f}+1}$ by the theory on the $N_{f}$ physical D8 branes. We introduced a symmetric notation for the chemical potentials associated with $SU(2)_{R}\times SU(2)$ following the notations of Hosseini:2018uzp . The entropy functional discussed in Choi:2018fdc refers to the case $\Delta_{1}=\Delta_{2}$. Notice that the chemical potentials are always subject to a constraint. Indeed, as already discussed in Sect. 2, in all dimensions, we expect the existence of a family of supersymmetric black holes depending on the possible electric charges and spins with a constraint among them.676767Supersymmetric hairy AdS5 black holes depending on all the charges has been recently found in Markeviciute:2018yal ; Markeviciute:2018cqs . Their entropy seems to be subleading compared to the Kerr–Newman black hole. This explains the constraint among chemical potentials. The family of black holes with generic charges and spins allowed by the constraint is known only for AdS${}_{5}\times S^{5}$ (and AdS${}_{4}\times S^{7}$) and (7.1) has been fully checked only in these cases (Hosseini:2017mds, ). In all other dimensions the relation (7.1) has been checked for the solutions available in the literature (Hosseini:2018dob, ; Choi:2018fdc, ).

In the second column of the table, there is a quantity, $F(\Delta_{a})$, with a clear field theory interpretation. The reader can recognize the $S^{3}$ free energy of ABJM in the first row and the trial $a$-charge of ${\cal N}=4$ SYM in the second row, see (4.43) and (5.7), which are both functions of trial R-charges satisfying

[TABLE]

In general, $F(\Delta_{a})$ is the sphere free energy for odd-dimensional CFTs, and a particular combination of t’Hooft anomaly coefficients in even dimensions.686868 It is also curious to observe that $F(\Delta)$ is the on-shell value of the twisted superpotential of the CFT in three and four-dimensions, as discussed in Sects. 4 and 5, and the on-shell Seiberg–Witten prepotential in five- and six-dimensional computations (Hosseini:2018uzp, ). In all cases, the quantity $\log\mathcal{Z}(\Delta_{a},\omega_{i})$ can be obtained by taking the quotient of $F(\Delta)$ by the product of all angular momentum chemical potentials and by replacing the R-charge constraint (7.6) with

[TABLE]

This constraint is strongly reminiscent of the analogous constraint (4.42) for magnetically charged black holes. Notice that $\log\mathcal{Z}(\Delta_{a},\omega_{i})$ is a sort of equivariant generalization of $F(\Delta)$ with respect to rotations. Indeed, the expression for $\log\mathcal{Z}(\Delta_{a},\omega_{i})$ for AdS5 and AdS7 can be also directly obtained by an equivariant integration of the six-dimensional and eight-dimensional anomaly polynomial for ${\cal N}=4$ SYM and the $(2,0)$ theory in six dimensions, respectively (Bobev:2015kza, ).696969The expression for $\log\mathcal{Z}(\Delta_{a},\omega_{i})$ for AdS4 and AdS6 can be instead related to a small $\epsilon$ limit of the partition functions on $\mathbb{R}^{2}_{\epsilon_{1}}\times S^{1}$ and $\mathbb{R}^{2}_{\epsilon_{1}}\times\mathbb{R}^{2}_{\epsilon_{2}}\times S^{1}$, respectively, where $\epsilon_{i}\propto\omega_{i}$ are equivariant parameters for rotations in the $\mathbb{R}^{2}$ planes. The sphere and twisted partitions functions in three and five dimensions can be obtained by gluing together these basic building blocks in the spirit of Pasquetti:2011fj ; Beem:2012mb ; Nieri:2013yra ; Gukov:2016gkn ; Pasquetti:2016dyl ; Nekrasov:2003vi ; Bershtein:2016mxz ; Kim:2013nva ; Hosseini:2018uzp ; Qiu:2016dyj ; Festuccia:2018rew . This point of view has been applied to the physics of black holes in Hosseini:2018uzp ; Hosseini:2019iad and Choi:2019zpz ; Choi:2019dfu .

Various observations made for AdS${}_{5}\times S^{5}$ generalize to the other dimensions. First, the quantity in (7.1) is extremized for complex values of the chemical potentials $\Delta_{a}$ and $\omega_{i}$ but the on-shell value (7.1) is real, as an entropy must be. Secondly, there is a sign ambiguity in the constraint (7.7) that can be replaced by $\sum_{a}\Delta_{a}+\sum_{i}\omega_{i}=-2\pi$ with no differences. Thirdly, the expression of $\log\mathcal{Z}(\Delta_{a},\omega_{i})$ can be explicitly derived in gravity by taking the zero-temperature limit of the on-shell action of a family of supersymmetric Euclidean solutions (Cassani:2019mms, ).

Finally, we expect that (7.1) corresponds to the attractor mechanism in the relevant gauged supergravity. Unfortunately, the attractor mechanism in generic dimensions and, specifically, for electrically charged rotating black holes is not known. AdS5 black holes with equal angular momenta can be dimensionally reduced to static black holes in four dimensions and, in this case, one can show that (7.1) corresponds to the attractor mechanism in four-dimensional gauged supergravity (Hosseini:2017mds, ). This was actually the observation that led to write the entropy functional (7.1).

7.2 Results on the quantum field theory side

Various field theory derivations of the extremization principles (7.3) have been recently proposed for AdS5. All these results are valid in overlapping limits and the connection between different approaches still to be understood, but all seems to indicate that the entropy is correctly accounted by the large $N$ limit of the superconformal index. Partial results in other dimensions also confirm the content of table 1.

Consider first the case of AdS${}_{5}\times S^{5}$. The dual field theory is ${\cal N}=4$ SYM. In the language of ${\cal N}=1$ supersymmetry, it contains a vector multiplet $W_{\alpha}$ and three chiral multiplets $\phi_{a}$ subject to the superpotential (5.5). We introduce three R-symmetries $\mathcal{R}_{a}$ associated with $U(1)^{3}\subset SO(6)$. $\mathcal{R}_{a}$ assign charge $2$ to $\phi_{a}$ and zero to the $\phi_{b}$ with $b\neq a$. The exact R-symmetry is $R=(\mathcal{R}_{1}+\mathcal{R}_{2}+\mathcal{R}_{3})/3$ and the global symmetries are $q_{a}=(\mathcal{R}_{a}-R)/2$, with associated fugacities $u_{a}$. Only two global symmetries are independent since $\sum_{a=1}^{3}q_{a}=0$ and, as a consequence, $\prod_{a=1}^{3}u_{a}=1$. Defining $y_{a}=(pq)^{1/3}u_{a}$ we can write the superconformal index (6.1) as

[TABLE]

with $y_{a}=e^{i\Delta_{a}}$, $p=e^{i\omega_{1}}$, $q=e^{i\omega_{2}}$ and $Q_{a}=\mathcal{R}_{a}/2$. The fugacities are constrained by $\prod_{a=1}^{3}y_{a}=pq$, and the index depends only on four independent parameters, as the family of BPS black holes.

Due to cancellations between bosonic and fermionic supersymmetric states, the result obtained from the index can only be a lower bound on the number of BPS states. However, we may expect that, as for magnetically charged black holes, for large $N$, the result saturates the entropy. Most of the computations for the superconformal index in the old literature has been performed for real fugacities and give results of order $O(1)$ for large $N$. However, the extremization principle discussed above strongly suggests that we should look at the behavior of the index as a function of complex chemical potentials.

Agreement with the gravity result (7.5)

[TABLE]

with $\Delta_{1}+\Delta_{2}+\Delta_{3}-\omega_{1}-\omega_{2}=\pm 2\pi$ has been obtained analytically, up to now, in two partially overlapping limits:

• •

Large $N$ and equal angular momenta (Benini:2018mlo, ; Benini:2018ywd, ). In the large $N$ limit, the index has a Stokes behavior as a functions of the chemical potentials, and it can give a contribution to the entropy of order $O(N^{2})$ along the right direction in the complex plane. A crucial technical ingredient in this approach involves writing the superconformal index as a sum over Bethe vacua. As shown in Closset:2017zgf ; Closset:2017bse ; Closset:2018ghr , the supersymmetric partition function of many three- and four-dimensional manifolds can be expressed as a sum over two-dimensional Bethe vacua using the formalism that we briefly discussed in Sect. 3.3.2. A formula for the superconformal index was obtained in Closset:2017bse and generalized to unequal fugacities for the angular momenta in Benini:2018mlo . Schematically, it allows to write the index as in (3.38)

[TABLE]

where $x^{*}$ are the Bethe vacua of the two-dimensional theory obtained by reduction on $T^{2}$, and $Q(x)$ is a suitable function whose expression can be found in Closset:2017bse ; Benini:2018mlo .707070The four-manifold $S^{1}\times S^{3}$ can be considered as a torus fibration over a two-dimensional manifold using $S^{1}$ and a circle inside $S^{3}$ for the $T^{2}$ fiber. For details see Closset:2017bse ; Benini:2018mlo . The Bethe vacua of ${\cal N}=4$ SYM on $T^{2}$ have been already discussed in Sect. 5.2 for a different purpose and are explicitly given by solutions of (5.18).717171As argued in ArabiArdehali:2019orz , there can exist non-standard solutions corresponding to continua of Bethe vacua and the formula (7.10) must be accordingly generalized. It is argued in Benini:2018ywd that, in the large $N$ limit, the particular Bethe vacuum (5.19) dominates for sufficiently large charges and reproduces the entropy of the AdS5 black holes. The same result has been reproduced by directly analysing the saddle point of the integrand (6.4) in Cabo-Bizet:2019eaf . The extension to the case of unequal angular momenta is discussed in Benini:2020gjh . Although it is difficult to evaluate and compare the contribution of all the Bethe vacua, one can show that there is a natural choice that leads precisely to the gravity result (7.5).

• •

The Cardy limit, corresponding to $\omega_{i}\ll 1$ at fixed complex valued of $\Delta_{a}$ (Choi:2018hmj, ; Choi:2018vbz, ). This limit corresponds to large black holes with electric charges and angular momenta scaling as

[TABLE]

It is crucial that the chemical potentials are complex. As argued in Choi:2018hmj ; Choi:2018vbz , the imaginary parts of the fugacities at the saddle point introduce phases that optimally obstruct the cancellation between bosonic and fermionic states. The number of states accounted by the index in the Cardy limit correctly reproduces the entropy of large AdS5 black holes and the $\omega_{i}\ll 1$ limit of the extremization formula (7.3). This approach has been further refined and generalized in Honda:2019cio ; ArabiArdehali:2019tdm ; ArabiArdehali:2019orz .

Numerical analysis confirming the $O(N^{2})$ behavior of the index for complex chemical potential has been performed in Murthy:2020rbd ; Agarwal:2020zwm .

In all these approaches, there seems to exist instabilities when decreasing the charges, which might suggest the contribution of other types of black holes. Given also the recently found supersymmetric hairy black holes in AdS5 (Markeviciute:2018yal, ; Markeviciute:2018cqs, ), we may expect a rich structure of the index/partition function still to be uncovered.

It was observed in Hosseini:2017mds ; Hosseini:2018dob that the quantity (7.9) and its analogous for AdS7 given in table 1 matches the expression for the supersymmetric Casimir energy $E_{c.e}$ quoted in the literature with the precise coefficient for both ${\cal N}=4$ SYM and the $(2,0)$ theory (see for example Bobev:2015kza ). This observation was strengthened in Cabo-Bizet:2018ehj by considering a modified supersymmetric partition function on $S^{3}\times S^{1}$ implementing the constraint (7.4) and showing that the corresponding supersymmetric Casimir energy $E_{c.e}$ has still the expression (7.9).727272The motivation for using this partition function comes from holography, since the constraint (7.4) explicitly arises in the Euclidean description of the AdS5 black holes (Cabo-Bizet:2018ehj, ). It is not completely clear why the supersymmetric Casimir energy, which corresponds to the energy of the vacuum of the CFT (Assel:2015nca, ), should be related to the entropy of the black hole. It would be intriguing if this a consequence of some modular properties of the partition function, still to be understood.

The large $N$ holographic expectation for theories with a Sasaki–Einstein dual is (Hosseini:2018dob, )

[TABLE]

with the constraint $\sum_{a=1}^{d}\Delta-\omega_{1}-\omega_{2}=\pm 2\pi$, where $a(\Delta)$ is the trial central charge defined in section 5.1 and $f_{abc}$ are the cubic t’Hooft anomaly coefficients for a basis of $d$ independent R-charges. The coefficients $f_{abc}$ have a natural dual gravitational interpretation. They arise as intersection numbers of cycles in the internal manifold (Benvenuti:2006xg, ), and, from an effective field theory perspective, as Chern–Simons terms in the corresponding gauged supergravity in five dimensions. In particular, they determine completely the structure of ${\cal N}=1$ gauged supergravity including the prepotential. The field theory computation of the index has been extended to other ${\cal N}=1$ superconformal theories and gives results consistent with (7.12). In particular, the Cardy limit for a generic ${\cal N}=1$ superconformal theory has been studied in Kim:2019yrz ; Cabo-Bizet:2019osg with the result737373The authors use a slightly modified index where $(-1)^{F}$ is replaced by $(-1)^{R}$. This replacement has the same effect as introducing complex fugacities for the flavor symmetries. The prescription is the same introduced in Cabo-Bizet:2018ehj for the modified supersymmetric partition function on $S^{3}\times S^{1}$.

[TABLE]

when all flavor symmetries are turned off. This formula generalize a previous result by DiPietro:2014bca for the standard index with real fugacities. Flavor fugacities have been introduced in Amariti:2019mgp and consistency with (7.12) in the large $N$ limit, where $c=a$, checked for many toric models. The large $N$ limit of the index for equal angular momenta has been studied in Lanir:2019abx ; Cabo-Bizet:2020nkr with results again consistent with (7.12).747474See also Lezcano:2019pae . The case of unequal angular momenta is discussed in Benini:2020gjh .

These results have been generalized and extended to other dimensions. In particular, the entropy of Kerr–Newman black holes in AdS4 has been reproduced in the Cardy limit in Choi:2019zpz ; Nian:2019pxj ; Choi:2019dfu , one of the method involving factorization of the partition function. The case of AdS6 and AdS7 have been analysed in Choi:2019miv ; Kantor:2019lfo ; Nahmgoong:2019hko . Other interesting developments can be found in Benini:2019dyp ; Bobev:2019zmz ; Goldstein:2019gpz .

8 Conclusions and comments

At the end of our journey, it is time to recapitulate. We have seen that, using holography, the entropy of supersymmetric AdS black holes and black objects with large charges can be correctly accounted by the evaluation of the relevant index in the dual conformal field theory, which just enumerates the corresponding microstates. This solves a long standing puzzle about supersymmetric black holes in AdS and their holographic interpretation. Many results have been obtained for most of the existing black objects in maximally supersymmetric string theory backgrounds in diverse dimensions. One can reasonably expect that the agreement will persist for the more technically involved case of black objects with arbitrary rotations and magnetic charges and of black objects in string backgrounds with reduced supersymmetry.

It is interesting to observe that, in all dimensions, a single function $F(\Delta)$ controls the entropy of most of the black holes and black objects asymptotic to a maximally supersymmetric AdS vacuum, with or without magnetic charges or rotation. For comparison, the partition function for Kerr–Newman black holes and magnetically charged black objects with a twist in various dimensions is reported in Table 1 and Table 2, respectively.

We see that the function $F(\Delta)$ determines the entropy of all such black holes. A general entropy functional built out of $F(\Delta)$ that covers all existing black holes and generalises the content of table 1 and 2 to the case of arbitrary rotations and magnetic charges has been discussed in Hosseini:2019iad in analogy with the factorization properties of supersymmetric partition functions.

The function $F(\Delta)$ has various interpretations. In gravity, it determines the effective gauged supergravity action for the massless vectors, being the prepotential in four dimensions. In field theory, it is related to the anomalies of the dual CFT in even dimensions, and the round sphere partition function in odd dimensions. On a more technical side, there is a third interpretation in terms of the twisted superpotential of the two-dimensional theory obtained by reducing the CFT on circle or tori, as discussed at length in sections 3 and 5.757575For five and six-dimensional CFTs, this should be replaced with the Seiberg–Witten prepotential (Hosseini:2018uzp, ; Crichigno:2018adf, ).

In these lectures we discussed black holes in the supergravity approximation, where the charges are large in units of the number of colors of the dual CFT. For asymptotically flat black holes impressive computations and precision tests have been made beyond the supergravity limit. The story for AdS black hole has just begun.

Acknowledgements

These notes grow out of lectures given at the CERN Winter School on Strings and Fields 2017 and the School on Supersymmetric Localization, Holography and Related Topics at ICTP in 2018, for which I heartily thank the organizers. I would also like to thank Francesco Azzurli, Nicolay Bobev, Marcos Crichigno, Noppadol Mekareeya, Vincent Min, Anton Nedelin, Achilleas Passias, Alessandro Tomasiello and especially Francesco Benini, Seyed Morteza Hosseini and Kiril Hristov, for collaboration and discussions on related topics. I am partially supported by the INFN, the ERC-STG grant 637844-HBQFTNCER and the MIUR-PRIN contract 2017CC72MK003.