Proving the equivalence of $c$-extremization and its gravitational dual for all toric quivers

TL;DR

This paper proves the general equivalence between $c$-extremization and its gravitational dual for all toric Calabi-Yau three-folds, extending previous specific examples and connecting to M2-brane theories and black hole entropy counting.

Contribution

It generalizes the proof of the equivalence between $c$-extremization and gravity duals to all toric Calabi-Yau three-folds, unifying previous case-by-case checks.

Findings

Proves off-shell equivalence for all toric Calabi-Yau cases.

Maps trial R-charges to gravitational extremization functional.

Connects $c$-extremization with $ ext{I}$-extremization for M2-branes.

Abstract

The gravitational dual of $c$-extremization for a class of $(0,2)$ two-dimensional theories obtained by twisted compactifications of D3-brane gauge theories living at a toric Calabi-Yau three-fold has been recently proposed. The equivalence of this construction with $c$-extremization has been checked in various examples and holds also off-shell. In this note we prove that such equivalence holds for an arbitrary toric Calabi-Yau. We do it by generalizing the proof of the equivalence between $a$-maximization and volume minimization for four-dimensional toric quivers. By an explicit parameterization of the R-charges we map the trial right-moving central charge $c_r$ into the off-shell functional to be extremized in gravity. We also observe that the similar construction for M2-branes on $\mathbb{C}^4$ is equivalent to the $\mathcal{I}$-extremization principle that leads to the microscopic counting for the entropy of magnetically charged black holes in AdS$_4\times S^7$. Also this equivalence holds off-shell.

Equations

$${\Delta }_a(b_i)=\frac{\pi \text{Vol}(S_a(b_i))}{b_1\text{Vol}(Y_5(b_i))},$$

$$\displaystyle\frac{\pi^{3}N^{2}}{4\text{Vol}(Y_{5}(b_{i}))}\equiv a(\Delta_{a})\Big{|}_{\Delta_{a}(b_{i})}\,,$$

$$\displaystyle\sum_{a}B_{a}\frac{\partial a(\Delta_{a})}{\partial\Delta_{a}}\Big{|}_{\Delta_{a}(b)}\equiv 0\,,$$

$$c_r({\Delta }_a,{\mathfrak{n}}_a)$$

$$\displaystyle c(b_{i},\mathfrak{n}_{a})\equiv c_{r}(\Delta_{a},\mathfrak{n}_{a})\Big{|}_{\Delta_{a}(b,\mathfrak{n})}\equiv-\frac{32}{9}\sum_{a=1}^{d}\mathfrak{n}_{a}\frac{\partial a(\Delta_{a})}{\partial\Delta_{a}}\Big{|}_{\Delta_{a}(b,\mathfrak{n})}\,.$$

$$\displaystyle\sum_{a}B_{a}\frac{\partial c_{r}(\Delta_{a},\mathfrak{n}_{a})}{\partial\Delta_{a}}\Big{|}_{\Delta_{a}(b,\mathfrak{n})}\equiv 0\,.$$

$$\mathcal{I}({\Delta }_a,{\mathfrak{n}}_a)=-\frac{1}{2}\sum _{a=1}^d{\mathfrak{n}}_a\frac{\partial F_{S^3}({\Delta }_a)}{\partial {\Delta }_a}.$$

$$\sum _{a=1}^d{F}_a=0.$$

$$\sum _{a=1}^d{B}_a{v}_a=0.$$

$$\sum _{a=1}^d{\Delta }_a=2.$$

$$a({\Delta }_a)=\frac{9}{32}\mathrm{Tr}R({\Delta }_a{)}^3,$$

$$\displaystyle a(\Delta_{a})=\frac{9}{32}N^{2}\bigg{(}|G|+\sum_{\Phi_{ab}}\text{ mult}(\Phi_{ab})\left(\Delta_{\Phi_{ab}}-1\right)^{3}\bigg{)}\,,$$

$$a({\Delta }_a)=\frac{9}{32}\sum _{a,b,c=1}^d{c}_{abc}{\Delta }_a{\Delta }_b{\Delta }_c,$$

$$c_{abc}=\frac{N^2}{2}|(v_a,v_b,v_c)|.$$

$$a({\Delta }_a)=\frac{27}{32}{N}^2\sum _{1\le a<b<c\le d}(v_a,v_b,v_c){\Delta }_a{\Delta }_b{\Delta }_c.$$

$${\bar{\Delta }}_a=\frac{\pi \text{Vol}(S_a)}{3\text{Vol}(Y_5)}.$$

$$a({\bar{\Delta }}_a)=\frac{{\pi }^3{N}^2}{4\text{Vol}(Y_5)}.$$

$$\sum _{a=1}^d{\mathfrak{n}}_a=2-2\mathfrak{g}.$$

$$c_r({\Delta }_a,{\mathfrak{n}}_a)=3\mathrm{Tr}{\gamma }_{3}R({\Delta }_a{)}^2,$$

$$c_r(\Delta ,\mathfrak{n})$$

$$=3N^2\sum _{1\le a<b<c\le d}(v_a,v_b,v_c)({\mathfrak{n}}_a{\Delta }_b{\Delta }_c+{\mathfrak{n}}_b{\Delta }_a{\Delta }_c+{\mathfrak{n}}_c{\Delta }_a{\Delta }_b).$$

$$\zeta =\sum _{i=1}^3{b}_i{\partial }_{{\varphi }_i},$$

$$\text{Vol}(Y_5)=\frac{{\pi }^3}{b_1}\sum _{a=1}^d\frac{(v_{a-1},v_a,v_{a+1})}{(v_{a-1},v_a,b)(v_a,v_{a+1},b)},$$

$$\text{Vol}(S_a)=2{\pi }^2\frac{(v_{a-1},v_a,v_{a+1})}{(v_{a-1},v_a,b)(v_a,v_{a+1},b)}.$$

$$a(b_i)=\frac{{\pi }^3{N}^2}{4\text{Vol}(Y_5)},$$

$${\Delta }_a(b_i)=\frac{\pi \text{Vol}(S_a)}{b_1\text{Vol}(Y_5)},$$

$$\displaystyle a(b_{i})\equiv a(\Delta_{a})\Big{|}_{\Delta_{a}(b_{i})}\equiv\frac{27}{32}N^{2}\sum_{1\leq a<b<c\leq d}(v_{a},v_{b},v_{c})\Delta_{a}\Delta_{b}\Delta_{c}\Big{|}_{\Delta_{a}(b_{i})}\,.$$

$$\displaystyle\sum_{a}B_{a}\frac{\partial a(\Delta_{a})}{\partial\Delta_{a}}\Big{|}_{\Delta_{a}(b)}\equiv 0\,,$$

$$\zeta =\sum _{i=1}^3{b}_i{\partial }_{{\varphi }_i},$$

$$\omega =-2\pi \sum _{a=1}^d{\lambda }_a{c}_a,$$

$$\sum _{a=1}^d{\mathfrak{n}}_a=2-2\mathfrak{g}.$$

Full text

††preprint: IPMU19-0006

Dated: March 2, 2024

1 Introduction

Central charges play an important role in the study of superconformal field theories (SCFTs) in even dimensions. In supersymmetric gauge theories the R-symmetry current is not necessarily unique and mixes with the global symmetry currents. This happens in particular in most models with a holographic dual. It is well known that, for ${\cal N}=1$ supersymmetric theories in four dimensions, the extremization of a trial central charge $a$ with respect to a varying R-symmetry allows to identify the exact R-symmetry of the superconformal theory Intriligator:2003jj .111The exact R-symmetry is the one appearing in the superconformal algebra. Similarly, for $\mathcal{N}=(0,2)$ supersymmetric theories in two dimensions, the extremization of a right-moving trial central charge $c_{r}$ allows to identify the exact R-symmetry Benini:2012cz ; Benini:2013cda . The gravity dual of $a$-maximization is the volume minimization principle discovered in Martelli:2005tp ; Martelli:2006yb .222See also Tachikawa:2005tq ; Szepietowski:2012tb for a different approach based on five-dimensional supergravity. See also a similar approach for $c$-extremization in Karndumri:2013iqa . The equivalence of $a$-maximization and volume minimization has been proven in Butti:2005vn for all quivers associated with D3-branes at toric Calabi-Yau three-fold singularities and generalized in Lee:2006ru ; Eager:2010yu . On the other hand, the gravity dual of $c$-extremization has been recently found in a series of very interesting papers Couzens:2018wnk ; Gauntlett:2018dpc . The authors of Couzens:2018wnk ; Gauntlett:2018dpc have checked the equivalence of their formalism with $c$-extremization in various explicit examples. It is the purpose of this note to prove this equivalence for all theories obtained by twisted compactifications of D3-branes sitting at an arbitrary toric Calabi-Yau three-fold, just by generalizing the arguments of Butti:2005vn .

The main focus in this note are theories that are obtained by a twisted compactification of four-dimensional ${\cal N}=1$ superconformal theories living on D3-branes sitting at the tip of a toric Calabi-Yau cone $C(Y_{5})$ over a Sasaki-Einstein manifold $Y_{5}$. These four-dimensional theories are well known and classified in terms of the toric data Hanany:2005ve ; Franco:2005rj ; Feng:2005gw . The gravitational dual is AdS${}_{5}\times Y_{5}$. When compactified on a Riemann surface $\Sigma_{\mathfrak{g}}$ with a topological twist parameterized by magnetic fluxes $\mathfrak{n}_{a}$, the theory can flow in the infrared (IR) to a $\mathcal{N}=(0,2)$ CFT. The gravity solution dual to such CFT is a warped background AdS${}_{3}\times_{W}Y_{7}$, where $Y_{7}$ is topologically a fibration of $Y_{5}$ over $\Sigma_{\mathfrak{g}}$, with a five-form flux.

Given the close similarity between the gravitational dual of $a$- and $c$-extremization, let us start by first reviewing the story for $a$-maximization in four-dimensions. By relaxing the equations of motion but still imposing the conditions for supersymmetry, the authors of Martelli:2005tp ; Martelli:2006yb defined an off-shell class of supersymmetric backgrounds obtained by replacing $Y_{5}$ with a general Sasaki manifold. The background depends on a Reeb vector, $b=(b_{1},b_{2},b_{3})$, which specifies the direction of the R-symmetry inside the three isometries of $Y_{5}$. It has been shown in Martelli:2005tp ; Martelli:2006yb that the extremization of the volume of the Sasaki manifold identifies the exact R-symmetry of the CFT and allows to compute its central charge. The proof that this procedure is equivalent to $a$-maximization involves choosing a convenient parameterization of the R-charges of the toric quiver in terms of the toric data and define a natural parameterization of the R-charges in terms of the Reeb vector Butti:2005vn

[TABLE]

where $S_{a}$ are toric three-cycles in $Y_{5}$. One then shows that

[TABLE]

thus demonstrating the equivalence of $a$-maximization and volume minimization. Notice that the equivalence holds not only for the extremal value but is valid off-shell, since the two expressions in (1.2) are equal for generic values of $b_{i}$. There is an important difference between the two extremization principles. $a$-maximization is performed on the space of all R-symmetries. This spans the three mesonic symmetries, associated with the isometries of $Y_{5}$ and a number (in principle large) of baryonic symmetries, associated with the non-trivial three-cycles of $Y_{5}$. On the other hand, volume minimization is performed on the direction of the Reeb vector, spanned by $b_{i}$ and corresponding to the mesonic symmetries only. The consistency of the two extremizations is a consequence of the automatic decoupling of the baryonic symmetries from the $a$-maximization procedure in the given parameterization. This follows from the identity proved in Butti:2005vn

[TABLE]

where $B_{a}$ is a baryonic symmetry.

After compactification on $\Sigma_{\mathfrak{g}}$ we obtain a two-dimensional theory depending on magnetic fluxes $\mathfrak{n}_{a}$ for all the symmetries of the original theory, including the baryonic ones. The exact R-symmetry can be found by extremizing the trial right-moving central charge with respect to the mesonic and baryonic symmetries Benini:2012cz ; Benini:2013cda . There is a simple formula for the trial central charge of the $(0,2)$ CFT at large $N$, that, in the basis for R-charges of Butti:2005vn , reads Hosseini:2016cyf

[TABLE]

In order to study the gravitational dual of $c$-extremization, the authors of Couzens:2018wnk ; Gauntlett:2018dpc defined a family of off-shell backgrounds, again depending on the Reeb vector. They also defined a functional $c(b_{i},\mathfrak{n}_{a})$ of the Reeb vector and fluxes whose extremization selects the on-shell R-symmetry. It has been explicitly checked in many examples in Couzens:2018wnk ; Gauntlett:2018dpc that this procedure is equivalent to $c$-extremization, and the equivalence holds off-shell. We will prove in this note that this is true in general for all toric quivers and that the proof Butti:2005vn extends very nicely to the two-dimensional case. Indeed, the ingredients are exactly the same. We will define a natural parameterization of the R-charges in terms of the Reeb vector and magnetic fluxes, $\Delta_{a}(b_{i},\mathfrak{n}_{a})$, just by generalizing the logic behind (1.1). Then we will show that for an arbitrary toric quiver

[TABLE]

Moreover, as in four dimensions, the baryonic symmetries explicitly decouple from the extremization process in this parameterization

[TABLE]

In particular, we do not see any particular difference in the role of baryonic symmetries in two dimensions compared to four.

It is also interesting to study the theories living on M2-branes at a toric Calabi-Yau four-fold $C(Y_{7})$ and their twisted compactifications on a Riemann surface. In this case, the exact R-symmetry of the three-dimensional theory is obtained by extremizing the free energy on $S^{3}$, $F_{S^{3}}(\Delta_{a})$. The equivalence of volume minimization for four-folds Martelli:2005tp ; Martelli:2006yb and the extremization of $F_{S^{3}}(\Delta_{a})$ has been checked in many examples in Herzog:2010hf ; Jafferis:2011zi . Given the complications of three dimensions and the absence of a complete classification of quiver duals to Calabi-Yau four-folds, there is no general proof. The twisted compactifications of M2-brane theories are dual in the IR to AdS${}_{2}\times Y_{9}$ backgrounds, where $Y_{9}$ is topologically a fibration of $Y_{7}$ over $\Sigma_{\mathfrak{g}}$. These backgrounds can be interpreted as the horizon of magnetically charged AdS4 black holes. The construction in Couzens:2018wnk also applies to these solutions and the authors of Couzens:2018wnk identified the quantity to extremize with the entropy of the black hole in various cases. Interestingly, it is suggested by a field theory computation Hosseini:2016tor that the entropy of magnetically charged black holes in AdS${}_{4}\times Y_{7}$ should be obtained by extremizing the functional

[TABLE]

This is certainly true for the theory with $Y_{7}=S^{7}$ as shown in Benini:2015eyy ; Benini:2016rke , where a microscopic counting for the entropy of magnetically charged black holes in AdS${}_{4}\times S^{7}$ has been performed. We then expect that, also off-shell, the construction of Couzens:2018wnk is dual to $\mathcal{I}$-extremization. In this note we just verify this statement for $Y_{7}=S^{7}$, reproducing the extremization of Benini:2015eyy ; Benini:2016rke also off-shell. We leave the investigation of more general Sasaki-Einstein manifold $Y_{7}$, where the computation is more complicated, to the future. The microscopic computation of the entropy of black holes in AdS${}_{4}\times Y_{7}$ for generic $Y_{7}$ is still an open problem. In particular, baryonic symmetries enter in a puzzling way in the large $N$ limit, as noticed in Hosseini:2016tor ; Hosseini:2016ume ; Azzurli:2017kxo . The formalism of Couzens:2018wnk ; Gauntlett:2018dpc seems well suited to address these problems and we hope to come back to these questions in the future. Finally, notice the analogy of (1.7) with (1.4). In the context of the large $N$ limit of topologically twisted theories these identities arise as special cases of the index theorem discussed in Hosseini:2016tor ; Hosseini:2016cyf .

The note is organized as follows. In section 2 we discuss general features of four-dimensional toric quivers and their twisted compactifications. In section 3 we first review the proof of the equivalence between $a$-maximization and volume minimization for all four-dimensional toric theories and then we extend it to the equivalence between $c$-extremization and the construction in Couzens:2018wnk ; Gauntlett:2018dpc . For the convenience of the reader, the technical aspects of the proof are deferred to appendix B. In section 4 we give explicit formulae for the R-charge parameterization and we present few examples. In section 5 we show that the formalism Couzens:2018wnk ; Gauntlett:2018dpc for $Y_{7}=S^{7}$ is equivalent off-shell to the $\mathcal{I}$-extremization principle for black holes in AdS${}_{4}\times S^{7}$. Finally, in appendix A we review the proof of (1.4) for the right-moving central charge $c_{r}$.

2 Introducing the field theory

In this section we review some general aspects of the quiver gauge theories living on D3-branes at toric Calabi-Yau singularities and of their twisted compactifications on Riemann surfaces.

2.1 ${\cal N}=1$ superconformal field theories

We first discuss the four-dimensional aspect of the story. Consider the type IIB background AdS${}_{5}\times Y_{5}$, where $Y_{5}$ is a five-dimensional Sasaki-Einstein manifold. In the AdS/CFT correspondence, this is dual to the ${\cal N}=1$ superconformal theory living on $N$ D3-branes sitting at the tip of the Calabi-Yau cone CY${}_{3}=C(Y_{5})$ with base $Y_{5}$ Klebanov:1998hh ; Acharya:1998db ; Morrison:1998cs . Familiar examples of Sasaki-Einstein manifolds include $T^{1,1}$, whose dual is the Klebanov-Witten theory Klebanov:1998hh , and the Yp,q and Lp,q,r spaces Gauntlett:2004yd ; Gauntlett:2004hh ; Cvetic:2005ft , whose dual field theories have been identified in Benvenuti:2004dy and Benvenuti:2005ja ; Butti:2005sw ; Franco:2005sm , respectively. When the CY3 is toric, there is a general prescription for constructing the gauge theory associated with the D3-branes Hanany:2005ve ; Franco:2005rj ; Feng:2005gw based on dimer models and tilings. For our purposes, we will need just some general information about the quiver, that we review following Butti:2005vn .

A toric affine CY3 is specified by its fan, a collections of vectors $v_{a}$ in $\mathbb{R}^{3}$ with integer entries. The Calabi-Yau condition requires that all the $v_{a}$ lie on a plane that we will take to be the plane orthogonal to the vector $e_{1}=(1,0,0)$. The toric cone is then specified by $d$ vectors $v_{a}=(1,{\vec{v}}_{a})$ for $a=1,\cdots\,,d$. The restriction to the plane of these vectors define a regular polygon with integer vertices called the toric diagram. There is a toric divisor $D_{a}$ for each vertex. Each $D_{a}$ is a cone over a three cycle $S_{a}$ in $Y_{5}$. There are $d$ such cycles but only $d-3$ are independent in cohomology. All the data and symmetries of the gauge theory can be extracted from the geometry Hanany:2005ve ; Franco:2005rj ; Feng:2005gw . The theory has an R-symmetry and $d-1$ $\mathrm{U}(1)$ global symmetries that can mix with it. A particularly important role is played by the baryonic symmetries. There are precisely $d-3$ of them, corresponding to the inequivalent non-trivial three-cycles $S_{a}$ of $Y_{5}$. They are holographically dual to the $d-3$ gauge fields that we obtain by reducing the type IIB four-form potential on the three-cycles $S_{a}$. The remaining three symmetries are called mesonic and are holographically dual to the three gauge fields associated with the isometries of the toric $Y_{5}$. One is an R-symmetry and the other two are global symmetries. A convenient way to parameterize the global and R-symmetry comes from the prescription in Butti:2005vn or, equivalently, from the folded quiver formalism of Benvenuti:2005ja . The $d-1$ global symmetries can be parameterized by assigning a real number $F_{a}$ to each vertex with the constraint

[TABLE]

In the minimal toric phase,333There are many different quivers that describe the same IR SCFT. They are related by Seiberg dualities. The toric phases have the same number of gauge groups but different matter content. The minimal phase corresponds to the quiver with the smallest number of chiral fields. the theory contains a number $|G|$ of gauge group factors $\mathrm{SU}(N)$ equal to twice the area of the toric diagram. Moreover, defining the vectors $w_{a}=v_{a+1}-v_{a}$ lying in the plane, there are precisely $|(e_{1},w_{a},w_{b})|$ bi-fundamental chiral fields $\Phi_{ab}$ with charge $F_{a+1}+F_{a+2}+\ldots+F_{b}$ for each pair $(a,b)$ such that the outgoing normal of $w_{a}$ can be rotated counter-clockwise into that of $w_{b}$ in the plane with an angle smaller than $\pi$.444In this note, we will use the following notation for determinants of vectors $(v_{a},v_{b},v_{c})\equiv\det(v_{a},v_{b},v_{c})$. We also identify indices modulo $d$, so that, for example, $v_{d+1}=v_{1}$.

The baryonic symmetries, which we will denote by $B_{a}$, are further characterized by the vector identity

[TABLE]

Similarly, we can parameterize the R-charges of all fields in the quiver by assigning a number $\Delta_{a}$ to each vertex with the constraint Butti:2005vn

[TABLE]

The chiral fields $\Phi_{ab}$ have R-charge $\Delta_{a+1}+\Delta_{a+2}+\ldots+\Delta_{b}$.

The quiver and all interactions can be written explicitly but we will not need the explicit matter content in the following. The reader can find many examples in Butti:2005vn ; Butti:2005ps . The only important information is that there is a very simple formula for the central charge of the CFT in the large $N$ limit. According to $a$-maximization Intriligator:2003jj , the exact central charge $a$ of the SCFT can be obtained by extremizing the trial central charge555In general $a=\frac{9}{32}\operatorname{Tr}R^{3}-\frac{3}{32}\operatorname{Tr}R^{3}$, but we work in the large $N$ limit where $c=a$ Henningson:1998gx . In particular, for all our quivers, in the large $N$ limit, $\operatorname{Tr}R=-16(c-a)=0$.

[TABLE]

where the trace runs over all the fermions of the theory and $R(\Delta)$ denotes their R-charges as a function of $\Delta_{a}$. Explicitly, we have

[TABLE]

where $\text{mult}(\Phi_{ab})=|(e_{1},v_{a},v_{b})|$. It has been shown in Benvenuti:2006xg that the trial central charge of the theory can be written in the large $N$ limit as

[TABLE]

where the t’Hooft anomaly coefficients are given by

[TABLE]

We can remove the absolute value if we assume an order in the toric diagram. Assuming that the vertices of the toric diagram are numerated in counter-clockwise direction, if $1\leq a<b<c\leq d$ we can write $c_{abc}=N^{2}(v_{a},v_{b},v_{c})/2>0$. The trial central charge can be then written as

[TABLE]

All our formulae are strictly valid in the large $N$ limit where $c=a$.

The extremization of (2.8) gives the exact R-charges $\bar{\Delta}_{a}$ of the fields in the SCFT. They can be compared with the predictions for the dimension of baryonic operators in the gravity dual. The baryonic operator $\det\Phi_{a-1,a}$ is obtained by wrapping a D3-branes on the three-cycle $S_{a}$ and the R-charge of $\Phi_{a-1,a}$ can be computed by the standard formula Gubser:1998fp

[TABLE]

The value of the exact central charge of the CFT is also given by Gubser:1998vd

[TABLE]

That $a$-maximization reproduces these formulae has been tested in many examples and it can be proved in general for all toric quivers Butti:2005vn .

2.2 Twisted compactification to two dimensions

Let us now consider the theory compactified on a Riemann surface $\Sigma_{\mathfrak{g}}$ with a topological twist and assume that it flows to a two-dimensional $\mathcal{N}=(0,2)$ CFT at low energies. The gravitational dual is a type IIB solution interpolating between AdS${}_{5}\times Y_{5}$ and a warped compactification AdS${}_{3}\times_{W}Y_{7}$ where $Y_{7}$ is topologically a fibration of $Y_{5}$ over $\Sigma_{\mathfrak{g}}$ Benini:2012cz ; Benini:2013cda . In general, we have a family of such two-dimensional CFTs labeled by the magnetic flux of the R-symmetry on $\Sigma_{\mathfrak{g}}$. Once again, we can parameterize the R-symmetry flux with integers $\mathfrak{n}_{a}$ associated with the vertices of the toric diagram and satisfying

[TABLE]

We will refer to this constraint as the twisting condition. It is equivalent to the requirement that the background for the R-symmetry cancels the spin connection. As shown in Benini:2012cz ; Benini:2013cda the right-moving central charge of the two-dimensional theory can be found by extremizing the trial right-moving central charge

[TABLE]

where $\gamma_{3}$ is the chirality operator in two dimensions and the trace runs over all the two-dimensional fermions. As shown in Hosseini:2016cyf , the trial right-moving central charge of the theory compactified on $\Sigma_{\mathfrak{g}}$ in the large $N$ limit can be compactly written in terms of the four-dimensional trial $a$ charge as666Notice that in (2.13) we impose the constraint $\sum_{a}\Delta_{a}=2$ after differentiation. For the logic behind it see appendix A.1.

[TABLE]

This relation between $c_{r}$ and $a$ has been proven in Hosseini:2016cyf for a large class of quivers, including the toric ones, by comparing the four-dimensional and two-dimensional central charges. It has been also verified in many toric examples in Amariti:2017iuz . It can be also obtained in a simple way by integrating the four-dimensional anomaly polynomial on $\Sigma_{\mathfrak{g}}$, following the logic in appendix C of Hosseini:2018uzp . We review the derivation in appendix A. The formula is valid in the large $N$ limit where $c_{r}=c_{l}=c$.

That $c$-extremization correctly reproduces the central charge predicted by the gravitational dual has been tested in many examples Benini:2012cz ; Benini:2013cda ; Benini:2015bwz ; Amariti:2016mnz ; Amariti:2017cyd ; Amariti:2017iuz .

3 $c$-extremization equals its gravity dual for all toric quivers

In this section we first briefly review the equivalence of $a$-maximization with the volume minimization proposed in Martelli:2005tp ; Martelli:2006yb and then we extend it to the equivalence of $c$-extremization with the construction proposed in Couzens:2018wnk ; Gauntlett:2018dpc for all toric quiver. The technical parts of the proof are discussed in appendix B.

3.1 $a$-maximization is volume minimization

The gravity dual of $a$-maximization has been found in Martelli:2005tp ; Martelli:2006yb by defining a class of off-shell backgrounds that solve the conditions for supersymmetry but relax the equations of motion. In particular, the authors of Martelli:2005tp ; Martelli:2006yb replace the Sasaki-Einsten metric on $Y_{5}$ with a general Sasaki metric. The metric depends on a Reeb vector which is a linear combinations of the vector fields $\partial_{\phi_{i}}$ generating the toric $\mathrm{U}(1)^{3}$ action

[TABLE]

and specifies the direction of the R-symmetry vector field inside the isometries of $Y_{5}$. Supersymmetry requires $b_{1}=3$. The volumes of Sasaki manifold $Y_{5}$ and of its three-cycles $S_{a}$ are now functions of the Reeb vector $b=(b_{1},b_{2},b_{3})$

[TABLE]

As shown in Martelli:2005tp ; Martelli:2006yb , the extremization of the function

[TABLE]

reproduces the Reeb vector $\bar{b}=(\bar{b}_{1},\bar{b}_{2},\bar{b}_{3})$ and the volumes of the Sasaki-Einstein manifold. By construction, $a(\bar{b}_{i})$ reproduces the gravitational prediction (2.10) for the exact central charge of the CFT.

The equivalence of $a$-maximization with volume minimization has been proved for all toric quivers in Butti:2005vn . The proof has been simplified in Lee:2006ru and generalized to other quivers in Eager:2010yu . Following Butti:2005vn , we define a natural parameterization for the R-charges in terms of the Reeb vector inspired by (2.9)

[TABLE]

where now the volumes are the functions of $b_{i}$ given in (LABEL:volumeSasaki). Notice that $\sum_{a=1}^{d}\Delta_{a}(b_{i})=2$. One then proves that Butti:2005vn ; Lee:2006ru

[TABLE]

One might be puzzled by the fact that $a$-extremization is performed on $d-1$ independent parameters, while volume minimization is an extremization with respect to the Reeb vector that depends on two independent parameters only. The Reeb vector in a sense only sees the mixing of the R-symmetry with the mesonic symmetries. The point is that, as proved in Butti:2005vn , the trial $a$-function $a(\Delta_{a})$ is automatically extremized

[TABLE]

with respect to the baryonic directions, defined by (2.2).

3.2 $c$-extremization is equivalent to its gravity dual

The gravity dual of $c$-extremization has been found in Couzens:2018wnk ; Gauntlett:2018dpc . The solution associated with a twisted compactification of the four-dimensional CFT on $\Sigma_{\mathfrak{g}}$ is a warped background AdS${}_{3}\times_{W}Y_{7}$ where $Y_{7}$ is topologically a fibration of $Y_{5}$ over $\Sigma_{\mathfrak{g}}$ with a five-form flux. The authors Couzens:2018wnk ; Gauntlett:2018dpc define a family of off-shell backgrounds, depending on the Reeb vector, that solve the conditions for supersymmetry but relax the equations of motion for the five-form. They also define a functional of the Reeb vector that, upon extremization, selects the on-shell R-symmetry and it becomes equal to the exact two-dimensional central charge. Here we describe the basic ingredients of the construction and we refer to Couzens:2018wnk ; Gauntlett:2018dpc for details. We will work under the assumption that the gravity background associated with $Y_{5}$ exists. This is not always the case, as discussed in Couzens:2018wnk ; Gauntlett:2018dpc .

The off-shell backgrounds depend on a Reeb vector $b=(b_{1},b_{2},b_{3})$, and on $d$ parameters $\lambda_{a}$ and $d$ fluxes $\mathfrak{n}_{a}$. The Reeb vector is again given by

[TABLE]

and specifies the direction of the R-symmetry vector field inside the isometries of $Y_{5}$. This time supersymmetry requires $b_{1}=2$. The parameters $\lambda_{a}$ are associated with the toric divisors $D_{a}$ and determine the Kähler class of a four-dimensional transverse slice. For simplicity, we restrict to the quasi-regular case where the quotient with respect to the Reeb action, $V=Y_{5}/\mathrm{U}(1)$, is a four-dimensional compact toric orbifold. Then the Kähler class of $V$ is given by

[TABLE]

where $c_{a}$ are the Poincaré dual of the restriction of $D_{a}$ to $V$. Only $d-2$ parameters $\lambda_{a}$ are independent, since there are only $d-2$ independent two-cycles in $V$ (one more than in $Y_{5}$). We can recover the Sasaki geometry for $\lambda_{a}=-1/2b_{1}$.777In the Sasaki case, the contact form $\eta$ associated with the Reeb vector $b$ satisfies $\mathrm{d}\eta=2\omega$ and $b_{1}=3$ Martelli:2005tp ; Martelli:2006yb . Here instead $[\mathrm{d}\eta]=[\rho]/b_{1}$ in cohomology, where the Ricci form is given by $\rho=2\pi\sum_{a=1}^{d}c_{a}$ and $b_{1}=2$. The fluxes $\mathfrak{n}_{a}$ are also associated with the divisors $D_{a}$ and satisfy the twisting condition

[TABLE]

The $\mathfrak{n}_{a}$ are magnetic fluxes for both the three gauge fields associated with the isometries of $Y_{5}$ and the $d-3$ gauge fields coming from the reduction of the four-form potential on the $d-3$ independent three-cycles $S_{a}$. The fluxes associated with the isometries enter explicitly in the fibration of SE5 over $\Sigma_{\mathfrak{g}}$ and they can be parameterized by the integers $n^{i}=\sum_{a=1}^{d}v_{a}^{i}\mathfrak{n}_{a}$. They are associated with the mesonic symmetries of the quiver. The other $d-3$ fluxes enter in the supergravity five-form and are associated with the baryonic symmetries. The relation with the fluxes defined in Gauntlett:2018dpc is $M_{a}=-\mathfrak{n}_{a}N$.

Following Gauntlett:2018dpc , we define the master volume of the five-manifold with Kähler class (3.8)

[TABLE]

Notice that we identify indices modulo $d$ so that $v_{d+1}=v_{1}$ and $\lambda_{d+1}=\lambda_{1}$. The supersymmetry and flux quantization conditions for the off-shell background can be then summarized by Gauntlett:2018dpc 888In the notation of Gauntlett:2018dpc , we set $L^{4}=2(2\pi l_{s})^{4}g_{s}$. In order to compare with Gauntlett:2018dpc one must also set $\Delta_{a}=\frac{R_{a}}{N}$ and $\mathfrak{n}_{a}=-\frac{M_{a}}{N}$.

[TABLE]

where $n^{i}=\sum_{a=1}^{d}v_{a}^{i}\mathfrak{n}_{a}$. As shown in Gauntlett:2018dpc and reviewed in section B.1, $\mathcal{V}$ is a function of only $d-2$ independent parameters $\lambda_{a}$ and only $d-1$ equations in (LABEL:constraint) are independent. We can use the constraints (LABEL:constraint) to eliminate the $d-2$ independent $\lambda_{a}$ and $A$ and write them as functions of $b_{i}$ and $\mathfrak{n}_{a}$.999The dependence on the remaining two variables $\lambda_{a}$ drops out from every physical quantity. To simplify the computation, one can also choose a gauge like in appendix B.2. We then obtain the $c$-functional Gauntlett:2018dpc

[TABLE]

For further reference, we also define the on-shell value of the master volume

[TABLE]

The authors of Gauntlett:2018dpc checked that the extremization of $c$ with respect to $b_{i}$ (with $b_{1}=2$) correctly reproduces the central charge of the two-dimensional CFT in various examples, including the Yp,q and Xp,q manifolds. By an explicit computation along the lines of Benini:2012cz ; Benini:2013cda ; Benini:2015bwz , they also show that the identification holds off-shell and $c(b_{i},\mathfrak{n}_{a})$ can be identified with the trial right-moving central charge. We want now to show that this holds for all toric quiver, using the expression (2.13) derived in Hosseini:2016cyf and a natural parameterization of the R-charges based on the toric data.

In order to prove this, in analogy with Butti:2005vn and the four-dimensional case, we define

[TABLE]

satisfying $\sum_{a}\Delta_{a}=2$. This expression is indeed the holographic prediction for the R-charges of baryonic operators obtained by wrapping D3-branes on the cycles associated with the toric divisors $D_{a}$ Couzens:2017nnr ; Gauntlett:2018dpc . As such this is the natural generalization of the four-dimensional parameterization (3.4). It is important to observe that the $\Delta_{a}$ satisfy Gauntlett:2018dpc

[TABLE]

We will show how to obtain an explicit expression for $\Delta_{a}(b_{i},\mathfrak{n}_{a})$ in section 4.

With these definitions, we will show that101010In this and the following equations we set $b_{1}=2$. Reinstating $b_{1}$ we have $\mathcal{V}=\frac{b_{1}}{108\pi^{3}}a$ and $c=\frac{b_{1}}{2}c_{r}$.

[TABLE]

thus proving the off-shell equivalence of $c$-extremization and the formalism of Gauntlett:2018dpc for all toric quivers.

As in four dimensions, one might be puzzled by the fact that $c$-extremization is performed on $d-1$ independent parameters, while the construction in Gauntlett:2018dpc is an extremization with respect to the Reeb vector that depends on only two independent parameters. The point is again that the trial $c$-function $c_{r}(\Delta_{a},\mathfrak{n}_{a})$ is automatically extremized

[TABLE]

with respect to the baryonic directions, defined by (2.2), as we will show. Again, this is completely analogous to Butti:2005vn .

Indeed, (3.16) and (3.17) follow at once from the result in appendix B, where we will prove that there exists a vector $t$ such that

[TABLE]

where $r_{a}=v_{a}-b/b_{1}$. (3.16) follows by multiplying (3.18) by $\Delta_{a}$ and summing over $a$. The term with the vector $t$ cancels since $\sum_{a}\Delta_{a}r_{a}=0$, as a consequence of (3.15). Similarly, (3.17) follows by multiplying (3.18) by $B_{a}$ and summing over $a$. The term with $c_{r}$ on the right hand side vanishes because $\sum_{a}B_{a}=0$ and the term with $t$ because $\sum_{a}B_{a}r_{a}=0$, where we used (2.2).

Let us also observe that, quite interestingly, the on-shell value of the master volume coincides with the four-dimensional trial central charge

[TABLE]

A word of caution is in order. To find the exact right-moving central charge of the two-dimensional CFT we need to extremize $c(b_{i},\mathfrak{n}_{a})$ with respect to $b_{2}$ and $b_{3}$ after setting $b_{1}=2$, or equivalently, $c(\Delta_{a},\mathfrak{n}_{a})$ with respect to $\Delta_{a}$ with the constraint $\sum_{a}\Delta_{a}=2$. Our results guarantee that the two procedures are equivalent for all toric quivers. However they do not guarantee that the exact central charge found in this way really corresponds to an IR CFT. Similarly, in gravity, nothing guarantee that the family of backgrounds discussed in Couzens:2018wnk ; Gauntlett:2018dpc contains an actual solution of the equations of motion of type IIB. Explicit examples of possible obstructions are discussed in Couzens:2018wnk .

4 Formulae for the R-charges and examples

In this section we discuss how to solve equations (LABEL:constraint). Fortunately, there is no need of solving explicitly (LABEL:constraint) in order to write the R-charges $\Delta_{a}(b_{i},\mathfrak{n}_{a})$. Indeed there is an explicit expression for $\Delta_{a}(b_{i},\mathfrak{n}_{a})$ in terms of the toric data and the fluxes $\mathfrak{n}_{a}$. Moreover, in a convenient gauge, we can also write a general expression for the solutions $\lambda_{a}$ and $A$ that allows to write $c(b_{i},\mathfrak{n}_{a})$ and $\mathcal{V}(b_{i},\mathfrak{n}_{a})$. We summarize here the result referring to appendix B for the proof. We also discuss some explicit examples.

We first show how to find the R-charges $\Delta_{a}(b_{i},\mathfrak{n}_{a})$ in terms of the toric data and the fluxes $\mathfrak{n}_{a}$. A consequence of (LABEL:constraint) is the set of equations

[TABLE]

where, as usual, we identify the indices modulo $d$. These equations allow to find explicitly $\Delta_{a}(b_{i},\mathfrak{n}_{a})$ by recursion. We can use them to express $\Delta_{a}$ in terms of $\Delta_{1}$, and, finally, determine $\Delta_{1}$ using the constraint $\sum_{a=1}^{d}\Delta_{a}=2$.

In order to write $c(b_{i},\mathfrak{n}_{a})$ and $\mathcal{V}(b_{i},\mathfrak{n}_{a})$ we also need to solve (LABEL:constraint) for the variables $\lambda_{a}$ and $A$. As already mentioned, only $d-2$ variables $\lambda_{a}$ are independent. Indeed, the master volume (3.10) is a quadratic form in $\lambda_{a}$ invariant under

[TABLE]

for arbitrary functions $l_{2}$ and $l_{2}$. We can use this freedom to choose a gauge, for example $\lambda_{1}=\lambda_{2}=0$. In this gauge, we can explicitly invert the relation (3.14) and write

[TABLE]

In this gauge, equations (LABEL:constraint) also imply

[TABLE]

Finally, the $c$-functional can be simplified to

[TABLE]

As clear from the previous formulae and manifest in the following examples, the gauge invariant quantities $\Delta_{a}(b_{i},\mathfrak{n}_{a})$, $\mathcal{V}(b_{i},\mathfrak{n}_{a})$ and $c(b_{i},\mathfrak{n}_{a})$ are homogeneous polynomials of $b_{i}/b_{1}$. In particular, $\Delta_{a}$ is a linear homogeneous polynomial in $b_{i}/b_{1}$, $c_{r}/b_{1}$ is quadratic and $\mathcal{V}/b_{1}$ is cubic. In the gauge where two $\lambda_{a}$ are set to zero, also $\lambda_{a}/b_{1}$ and $A/b_{1}$ are quadratic homogeneous polynomials of $b_{i}/b_{1}$. Setting $b_{1}=2$, as required by supersymmetry, $\Delta_{a}$ becomes a linear function of $b_{2}$ and $b_{3}$ with rational functions of the fluxes as coefficients. This should be contrasted with the case of $a$-maximization Butti:2005vn , where the R-charges $\Delta_{a}(b_{i})$ are rational functions of $b$ with poles on the sides of the toric diagram, as one can see from (3.4) and (LABEL:volumeSasaki).

We now present few examples.

4.1 $\mathcal{N}=4$ SYM

Our first example is the $\mathcal{N}=4$ super Yang-Mills (SYM) theory compactified on $\Sigma_{\mathfrak{g}}$. The holographic dual has been found in Benini:2013cda . The manifold is $Y_{5}=S^{5}$ and the toric cone is specified by the vectors

[TABLE]

In $\mathcal{N}=1$ notation, the four-dimensional theory contains three adjoint chiral fields $\Phi_{a}$, $a=1,2,3$, with superpotential

[TABLE]

In this example, the vertices are in one-to-one correspondence with the fields and fluxes. The vertex $v_{a}$ is associated with the field $\Phi_{a}$ with R-charge $\Delta_{a}$ and the flux $\mathfrak{n}_{a}$. They satisfy

[TABLE]

which just express the fact that the superpotential (4.7) has R-charge two. Since $d=3$, there are no baryonic symmetries. The trial central charge $a$, at large $N$, reads (cf. (2.5) and (2.8))

[TABLE]

The trial central charge $c_{r}$ is given by (2.13)

[TABLE]

Solving explicitly (LABEL:constraint) or using the recursion relations (LABEL:fundamentalidentitytext) we find

[TABLE]

Notice that $\Delta_{a}(b_{i})$ are independent of the fluxes $\mathfrak{n}_{a}$. This is due to the absence of baryonic symmetries. Moreover, comparing with (LABEL:volumeSasaki) we see that

[TABLE]

Therefore, for ${\cal N}=4$ SYM the parameterization (3.14) coincides with the one used in Butti:2005vn for $a$-maximization. Moreover, the on-shell value of the master volume is given by

[TABLE]

and again is independent of $\mathfrak{n}_{a}$. Setting $b_{1}=2$ we find the very simple identification

[TABLE]

One can easily verify that (3.16) and (LABEL:id3) are satisfied. For ${\cal N}=4$ SYM, (LABEL:id3) is just equivalent to the equivalence of $a$-maximization and volume minimization found in Butti:2005vn , since (4.13) holds and the parameterization of R-charges is the same in two and four dimensions.

4.2 Klebanov-Witten theory

Our second example is the twisted compactification of the Klebanov-Witten theory Klebanov:1998hh on $\Sigma_{\mathfrak{g}}$, discussed e.g. in Benini:2015bwz ; Hosseini:2016cyf . The manifold in this case is $Y_{5}=T^{1,1}$. The toric cone $C(T^{1,1})$ is determined by the vectors

[TABLE]

This theory has $\mathcal{N}=1$ supersymmetry. The quiver contains two $\mathrm{SU}(N)$ gauge groups with two bi-fundamental chiral fields $A_{i}$ in the representation $({\bf N},\overline{{\bf N}})$ and two bi-fundamental chiral fields $B_{i}$ in the representation $(\overline{{\bf N}},{\bf N})$. The theory has a quartic superpotential

[TABLE]

We introduce four chemical potentials $\Delta_{a}$ and fluxes $\mathfrak{n}_{a}$, one for each of the four fields $\{A_{i},B_{i}\}$, associated with the four vertices $v_{a}$, with the constraints

[TABLE]

The trial $a$ central charge can be computed from either (2.5) or (2.8) and it reads

[TABLE]

The trial central charge $c_{r}$ is given by (2.13)

[TABLE]

Solving explicitly (LABEL:constraint) or using the recursion relations (LABEL:fundamentalidentitytext) we find

[TABLE]

Notice that these are linear polynomials in $b_{i}/b_{1}$.

The baryonic symmetry $\mathrm{U}(1)_{B}$ is characterized by (2.2) and it is given by

[TABLE]

Thus, the decoupling condition (3.17) can be explicitly written as

[TABLE]

One can check that (4.22) is automatically satisfied by the solution (4.20).

Finally, $c(b_{i},\mathfrak{n}_{a})$ and $\mathcal{V}_{\text{on-shell}}(b_{i},\mathfrak{n}_{a})$ read

[TABLE]

One can explicitly verify that (3.16) and (LABEL:id3) are satisfied. Recall that the central charge is obtained by extremizing $c(b_{i},\mathfrak{n}_{a})$ with respect to $b_{2}$ and $b_{3}$ after setting $b_{2}=2$.

4.3 $Y^{p,q}$ quiver gauge theory

Our third example is the $Y^{p,q}$ ($p>0$ and $p\geq q\geq 0$) quiver gauge theory Benvenuti:2004dy . The dual of the twisted compactification on $\Sigma_{\mathfrak{g}}$ is discussed in Benini:2015bwz . The cone $C(Y^{p,q})$ determines a polytope with four vertices Martelli:2004wu

[TABLE]

The $Y^{p,q}$ quiver has $2p$ $\mathrm{SU}(N)$ gauge groups with $4p+2q$ chiral fields $\{Y,Z,U^{\alpha},V^{\alpha}\}$, $\alpha=1,2$, in bi-fundamental representations of pairs of gauge groups. In (4.25) we present the R-charges and their multiplicity.

[TABLE]

Supersymmetry imposes the constraints

[TABLE]

The $a$ central charge can be computed from either (2.5) or (2.8) and it is given by

[TABLE]

Solving explicitly (LABEL:constraint) or using the recursion relations (LABEL:fundamentalidentitytext) we find

[TABLE]

Notice that these are linear polynomials in $b_{i}/b_{1}$. At the end of the computation we can set $b_{1}=2$ as required by supersymmetry.

The baryonic symmetry $\mathrm{U}(1)_{B}$ is characterized by (2.2). It reads

[TABLE]

Hence, the decoupling condition (3.17) can be explicitly written as

[TABLE]

and one can see that it is automatically satisfied by the solution (4.28).

The expressions for $c(b_{i},\mathfrak{n}_{a})$ and $\mathcal{V}_{\text{on-shell}}(b_{i},\mathfrak{n}_{a})$ are too long to be reported here. One can explicitly verify that (3.16) and (LABEL:id3) are satisfied.

5 $\mathcal{I}$-extremization and black hole entropy

As discussed in Couzens:2018wnk , the construction behind the gravity dual of $c$-extremization can be extended to twisted compactifications of three-dimensional CFTs on $\Sigma_{\mathfrak{g}}$. The gravity dual of the IR physics is a warped background AdS${}_{2}\times_{W}Y_{9}$ where $Y_{9}$ is topologically a fibration of a Sasaki-Einstein $Y_{7}$ over $\Sigma_{\mathfrak{g}}$. The gravity dual then describes the horizon of magnetically charged black holes in AdS${}_{4}\times Y_{7}$. As shown in Couzens:2018wnk for the case of solutions of minimal gauged supergravity in four dimensions, the extremization of the analogue of the $c$-functional (3.12) reproduces the entropy of the black hole. It is then natural to conjecture that the construction in Couzens:2018wnk is the dual of $\mathcal{I}$-extremization Benini:2015eyy ; Benini:2016rke , that it has been successfully used to perform a microscopic counting for AdS4 black holes. The $\mathcal{I}$-extremization principle states that the entropy of magnetically charged static black holes can be obtained by extremizing the logarithm of the supersymmetric partition function on $\Sigma_{\mathfrak{g}}\times S^{1}$ — also known as topologically twisted index. The index is a function of chemical potentials and magnetic fluxes for the global symmetries of the theory Benini:2015noa ; Benini:2016hjo ; Closset:2016arn . In the case of the ABJM theory Aharony:2008ug , where $Y_{7}=S^{7}$, the $\mathcal{I}$-extremization principle states that the entropy of magnetically charged static black holes in AdS${}_{4}\times S^{7}$ is the extremum of the function Benini:2015eyy ; Benini:2016rke

[TABLE]

where

[TABLE]

and the chemical potentials and the fluxes satisfy the constraints

[TABLE]

As noticed in Hosseini:2016tor , the function (5.2) is the free energy on $S^{3}$ of ABJM.

In this note we show that the construction of Gauntlett:2018dpc adapted to three-dimensional $\mathcal{N}=2$ theories, exactly reproduces (5.1) for ABJM and the identification is valid off-shell, when we use a natural parameterization of the R-charges in terms of the Reeb vector. The parallel with four dimensions is complete. As shown in Herzog:2010hf ; Jafferis:2011zi , the extremization of $F_{S^{3}}$ is equivalent to volume minimization for Calabi-Yau eight-folds Martelli:2005tp ; Martelli:2006yb . The equivalence of the construction of Gauntlett:2018dpc with (5.1) is the analogue of (3.16).

It was shown in Hosseini:2016tor that (5.1), where $\mathcal{I}$ is the logarithm of the topologically twisted index and $F_{S^{3}}(\Delta_{a})$ the $S^{3}$ free energy, can be extended to many quivers dual to AdS${}_{4}\times Y_{7}$. (5.1) has been called the index theorem in Hosseini:2016tor . This may suggest that the equivalence between the construction in Gauntlett:2018dpc and $\mathcal{I}$-extremization extends to other Sasaki-Einstein manifolds. Since the equations are more complicated to solve for Calabi-Yau eight-folds we leave this very interesting investigation to future work. This might shed light on some puzzles about baryonic symmetries raised in Hosseini:2016tor ; Hosseini:2016ume ; Azzurli:2017kxo .

5.1 The ABJM theory

Let us consider the twisted compactification of ABJM on $\Sigma_{\mathfrak{g}}$. The ABJM theory Aharony:2008ug is a three-dimensional supersymmetric Chern-Simons-matter theory with gauge group $\mathrm{U}(N)_{k}\times\mathrm{U}(N)_{-k}$ (the subscripts denote the CS levels) with two bi-fundamental chiral fields $A_{i}$ in the representation $({\bf N},\overline{{\bf N}})$ and two bi-fundamental chiral fields $B_{i}$ in the representation $(\overline{{\bf N}},{\bf N})$. The theory has a quartic superpotential

[TABLE]

We introduce four chemical potentials $\Delta_{a}$, one for each of the four fields $\{A_{i},B_{i}\}$, and four fluxes $\mathfrak{n}_{a}$ on $\Sigma_{\mathfrak{g}}$ satisfying

[TABLE]

The ABJM theory in three dimensions is dual to AdS${}_{4}\times S^{7}/\mathbb{Z}_{k}$. We are interested in $k=1$, which corresponds to the toric Calabi-Yau four-fold $\mathbb{C}^{4}$. The toric data are

[TABLE]

We can associate each vertex to one of the fields. The supergravity background corresponding to the twisted compactification is then a warped background AdS${}_{2}\times_{W}Y_{9}$ where $Y_{9}$ is topologically a fibration of $S^{7}$ over $\Sigma_{\mathfrak{g}}$. It corresponds to the horizon geometry of the black holes found and studied in Cacciatori:2009iz ; DallAgata:2010ejj ; Hristov:2010ri .

The master volume in Gauntlett:2018dpc is defined as the volume of a dual polytope associated with the Kähler parameter $\lambda_{a}$:

[TABLE]

where $H(b)$ is the hyperplane $(y,b)=1/2$ and $y_{0}=(1,0,0,0)/(2b_{1})$. Supersymmetry now requires $b_{1}=1$ Couzens:2018wnk . For $\mathbb{C}^{4}$ the dual polytope is a tetrahedron lying on $H(b)$. Its four vertices can be found by solving for every distinct triple $v_{a},v_{b},v_{c}$ the equations

[TABLE]

and the volume can be easily computed. We then find that

[TABLE]

By adapting the arguments in Gauntlett:2018dpc , it is easy to see that equations (LABEL:constraint) have the same form with the index $i$ running from $1$ to $4$. Following Couzens:2018wnk ; Gauntlett:2018dpc we also define the entropy functional111111In the notation of Couzens:2018wnk , we set $L^{6}=(2\pi l_{P})^{6}$.

[TABLE]

The equations (LABEL:constraint) are easily solved for the independent $\lambda_{a}$ and $A$. By substituting the result into (5.9) we find that

[TABLE]

For the entropy functional we obtain

[TABLE]

Similarly to what we did for $c$-extremization, we use the following parameterization for the R-charges

[TABLE]

Plugging the solution to (LABEL:constraint) into (5.13) we obtain

[TABLE]

As in ${\cal N}=4$ SYM in one dimension more (see (4.11)), the R-charges are functions of $b_{i}$ only. Moreover, one can also check that they are expressed in terms of the Sasaki volumes

[TABLE]

of toric divisors (see Martelli:2005tp ; Martelli:2006yb ; Hanany:2008fj ; Jafferis:2011zi for explicit expressions).

We now easily see that121212We set $b_{1}=1$ in the following formulae. Reinstating $b_{1}$ we have $S=\sqrt{b_{1}}\mathcal{I}$ and $\mathcal{V}=\frac{\sqrt{b_{1}}}{64\pi^{3}}F_{S^{3}}$.

[TABLE]

thus proving that the functional $S(b_{i},\mathfrak{n}_{a})$ is equivalent to the $\mathcal{I}$-functional defined in Benini:2015eyy ; Benini:2016rke . It is also interesting to observe that

[TABLE]

(5.16) and (5.17) are the direct analogues of (3.16) and (LABEL:id3).

The entropy of the black holes in Cacciatori:2009iz ; DallAgata:2010ejj ; Hristov:2010ri can be found equivalently by extremizing $S(b_{i},\mathfrak{n}_{a})$ with respect to $b_{2},b_{3},b_{4}$ after setting $b_{1}=1$ or by extremizing $\mathcal{I}(\Delta_{a},\mathfrak{n}_{a})$ with respect to $\Delta_{a}$ with the constraint $\sum_{a}\Delta_{a}=2$. More interestingly, the equivalence between the construction of Gauntlett:2018dpc and the $\mathcal{I}$-extremization principle holds also off-shell. It would be interesting to see if this result extends to more general Sasaki-Einstein manifolds $Y_{7}$.

Acknowledgements

We would like to thank Noppadol Mekareeya for useful discussions. The work of SMH was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. AZ is partially supported by the INFN and ERC-STG grant 637844-HBQFTNCER.

Appendix A The relation between $c_{r}(\Delta_{a},\mathfrak{n}_{a})$ and $a(\Delta_{a})$

In this appendix we briefly review the derivation of (2.13) given in Hosseini:2016cyf and give an alternative one using the integration of the anomaly polynomial, in the spirit of appendix C of Hosseini:2018uzp .

A.1 Direct evaluation

The trial $a$ central charge of a four-dimensional $\mathcal{N}=1$ field theory with gauge group $G$, at large $N$, reads

[TABLE]

where the trace is taken over all the bi-fundamental fermions and gauginos and $\text{dim }\mathfrak{R}_{I}$ is the dimension of the respective matter representation with R-charge $\Delta_{I}$. On the other hand, the trial right-moving central charge of the IR two-dimensional $\mathcal{N}=(0,2)$ SCFT can be computed from the spectrum of massless fermions Benini:2012cz ; Benini:2013cda ; Benini:2015bwz . These are gauginos for all the gauge groups and fermionic zero modes for each chiral field. The difference between the number of fermions of opposite chiralities is predicted by the Riemann-Roch theorem and equals $\mathfrak{g}-1$ for gauginos and $-\mathfrak{n}_{I}-\mathfrak{g}+1$ for chiral fields Benini:2012cz ; Benini:2013cda ; Benini:2015bwz . The trial right-moving central charge is then given by

[TABLE]

Using (A.1), it is easy to see that we can write

[TABLE]

When $a(\Delta_{I})$ is a homogeneous function of degree three of the variables $\Delta_{I}$, (A.3) simplifies to

[TABLE]

which is precisely (2.13).

In evaluating the right hand side of (A.3), we have considered the R-charges $\Delta_{I}$ of all the chiral fields as independent variables. However, the R-charges satisfy the constraint $\sum_{I\in W}\Delta_{I}=2$ for each term $W$ in the superpotential. Similarly $\sum_{I\in W}\mathfrak{n}_{I}=2-2\mathfrak{g}$. Fortunately, the differential operator in (A.3) is such that we can impose the constraints equivalently before or after differentiation. (A.3) is indeed valid for all parameterizations of the R-charges and fluxes (even redundant ones) provided that, if we impose a constraint coming from a superpotential term $W$, $\sum_{I\in W}\Delta_{I}=2$, a similar constraint is imposed on $\mathfrak{n}_{I}$, $\sum_{I\in W}\mathfrak{n}_{I}=2-2\mathfrak{g}$. In particular, it is valid for the parameterization used in this note, where we express $d-1$ independent R-charges in terms of $d$ parameters $\Delta_{a}$ with a constraint $\sum_{a=1}^{d}\Delta_{a}=2$. We can apply (A.3) and (A.4) considering $a$ as a function of $d$ independent variables $\Delta_{a}$ and impose the constraint $\sum_{a=1}^{d}\Delta_{a}=2$ after differentiation.

A.2 Integrating the anomaly polynomial

The trial ’t Hooft anomaly coefficients of two-dimensional $\mathcal{N}=(0,2)$ CFT can be extracted by integrating the six-form anomaly polynomial $I_{6}$ of the four-dimensional $\mathcal{N}=1$ field theory over $\Sigma_{\mathfrak{g}}$ Alday:2009qq ; Bah:2011vv ; Bah:2012dg ; Benini:2013cda . The six-form anomaly polynomial reads

[TABLE]

where $p_{1}(T\mathcal{M})$ is the first Poyntryagin class of tangent bundle, $F$ is the curvature of the R- and global symmetry bundle $K$ and the trace runs over all the fermions in the theory. We choose a basis of generators $T^{a}$ adapted to the parameterization discussed in section 2.1 and write $F=\sum_{a}\Delta_{a}T^{a}c_{1}(F)$, where $c_{1}(F)$ is a flux coupled to the $\mathrm{U}(1)$ R-symmetry and $\sum_{a}\Delta_{a}=2$. We can extract the trial $a$ central charge, at large $N$, from

[TABLE]

and we find

[TABLE]

where $c_{abc}=\operatorname{Tr}(T^{a}T^{b}T^{c})$ are the t’Hooft anomaly coefficients.

Consider now the compactification of the four-dimensional theories on a Riemann surface $\Sigma_{\mathfrak{g}}$ with fluxes $\mathfrak{n}_{a}$. The prescription in Alday:2009qq ; Bah:2011vv ; Bah:2012dg ; Benini:2013cda for computing the anomaly coefficient $c_{r}$ of the two-dimensional SCFT amounts to first replace $F$ in (A.5) with

[TABLE]

implementing the topological twist along $\Sigma_{\mathfrak{g}}$, and then integrate the $I_{6}$ on $\Sigma_{\mathfrak{g}}$:

[TABLE]

Here $x$ denotes the Chern root of the tangent bundle to $\Sigma_{\mathfrak{g}}$, $\Delta_{a}$ parameterize the trial R-symmetry, and $\mathfrak{n}_{a}$ are the fluxes parameterizing the twist, satisfying

[TABLE]

Then we integrate $I_{6}$ over $\Sigma_{\mathfrak{g}}$ using $\int_{\Sigma_{\mathfrak{g}}}x=2-2\mathfrak{g}$. The result should be compared with the four-form anomaly polynomial of the two-dimensional SCFT that, in the large $N$ limit, where $c_{l}=c_{r}$, reads

[TABLE]

We see immediately that, since in our basis $a(\Delta_{a})$ is homogeneous,

[TABLE]

which is precisely (2.13).

Appendix B Proof of the equality between $c(b_{i},\mathfrak{n}_{a})$ and $c_{r}(\Delta_{a},\mathfrak{n}_{a})$

In this appendix we prove (3.18). As discussed in the text, (3.16) and (3.17) are simple consequences of this equation. We will also solve explicitly the equations (LABEL:constraint) in a particular gauge. We first review in appendix B.1 some technical results of Gauntlett:2018dpc that will be used in the rest of the proof. For simplicity of notations, in this appendix $\Delta_{a}$ will always refer to the quantities $\Delta_{a}(b_{i},\mathfrak{n}_{a})$ defined in (3.14), unless otherwise stated.

B.1 Some simplifications

The master volume is a quadratic form in $\lambda_{a}$,131313To compare with Gauntlett:2018dpc : $J_{ab}=(2\pi)^{2}I_{ab}$.

[TABLE]

with a symmetric matrix $J_{ab}$, of rank $d-2$. Indeed, it is invariant under

[TABLE]

since, using (Gauntlett:2018dpc, , (3.41)), we find that

[TABLE]

Notice that this leaves $d-2$ independent $\lambda_{a}$ since $l_{1}$ does not contribute ($v_{a}^{1}=1$ for all $a$). Correspondingly, the matrix $J_{a,a}$ has rank $d-2$. Since (B.3) is valid for all $b_{i}$ and all $\lambda_{a}$ we obtain

[TABLE]

The equations (LABEL:constraint) can be written as

[TABLE]

For given fluxes $\mathfrak{n}_{a}$ and number of colors $N$, these are, in principle, $d+2$ equations for $d-1=(d-2)+1$ variables $\lambda_{a}$ and $A$. But, fortunately, three equations are redundant Gauntlett:2018dpc . Indeed the three linear combinations, $k=1,2,3$, of the equations for $\mathfrak{n}_{a}$

[TABLE]

reproduce the relation between $\mathfrak{n}_{a}$ and $n^{k}$. We used the first and third equations in (LABEL:constraint), the constraint (B.4) and its derivative with respect to $b_{i}$

[TABLE]

We can also rewrite the functional (3.12) as

[TABLE]

where, in the second step, we computed $\sum_{a}\lambda_{a}\mathfrak{n}_{a}$ from the second equation in (LABEL:eqs).

The R-charges (3.14) read

[TABLE]

Multiplying (B.4) by $-\frac{2}{N}\lambda_{b}$ and summing over $b$ we obtain

[TABLE]

Notice, in particular, that $\sum_{a}\Delta_{a}=2$. Introducing the vectors $r_{a}=v_{a}-b/b_{1}$ we can also write

[TABLE]

an identity that we will use repeatedly in the following.

B.2 A convenient gauge

We can simplify the equations choosing a gauge. Using (B.2) we can set two $\lambda_{a}$ to zero, say $\lambda_{1}=\lambda_{2}=0$. From (3.10), we see that the non-zero components of the matrix $J_{ab}$ are

[TABLE]

We also know that

[TABLE]

In our gauge, $\lambda_{1}=\lambda_{2}=0$, we find that

[TABLE]

that we can solve recursively. We obtain

[TABLE]

where in the last step we used the identity

[TABLE]

valid for arbitrary vectors $A,B,C,D$. Altogether we can write the inversion formula

[TABLE]

As a consistency check, note that we can also extract $\lambda_{d}$ from the equation $\Delta_{1}=-\frac{2}{N}J_{d1}\lambda_{d}$, obtaining

[TABLE]

where in the second step we used $\sum_{c}v_{c}\Delta_{c}=2b/b_{1}$.

We can now analyze the equations (LABEL:eqs) in the gauge $\lambda_{1}=\lambda_{2}=0$. Introduce the notation $\nabla\equiv\sum_{i}n^{i}\partial_{b_{i}}$. We have

[TABLE]

where, for the second identity, we used141414The geometrical meaning of this identity is clear from figure B.26: the sum of the areas of the plane triangles of vertices $(v_{a-1},v_{a},B)$ and $(v_{a},v_{a+1},B)$ minus the area of $(v_{a-1},v_{a+1},B)$ is the area of $(v_{a-1},v_{a},v_{a+1})$. The factor of $b_{1}$ takes into account the normalization in converting three-dimensional determinants into areas in the plane.

[TABLE]

Writing the equations (LABEL:eqs) for $a=1$ and $a=2$

[TABLE]

and using (B.15) and (B.18) we find

[TABLE]

Multiplying the first by $(v_{d},v_{1},v_{2})$ and the second by $(v_{1},v_{2},v_{3})$ and subtracting we obtain

[TABLE]

This has been proved for $a=1$ and $a=2$ but should hold for all adjacent pairs $(a,a+1)$ because it is an identity for gauge invariant quantities and we can always use an adapted gauge where $\lambda_{a}=\lambda_{a+1}=0$. Therefore, we find

[TABLE]

where we identify $\mathfrak{n}_{a+d}=\mathfrak{n}_{a}$, $\Delta_{a+d}=\Delta_{a}$ and $v_{a+d}=v_{a}$, so that for example $v_{d+1}=v_{1}$ and $v_{0}=v_{d}$. This is a set of equations that allow to find an explicit expression for $\Delta_{a}$ using recursion to obtain $\Delta_{a+1}$ from $\Delta_{a}$ and enforcing $\sum_{a}\Delta_{a}=2$ in order to find the value of $\Delta_{1}$. Notice that, at each step of the recursion, $b$ only appears linearly so $\Delta_{a}$ is a regular function of $b_{i}$. $\Delta_{a}$ is actually a linear polynomial in $b/b_{1}$.

For further reference let us also quote the value of $A$:

[TABLE]

This equation and (B.17) guarantee that, in the gauge $\lambda_{1}=\lambda_{2}=0$, $\lambda_{a}/b_{1}$ and $A/b_{1}$ are quadratic polynomials in $b/b_{1}$.

B.3 Completing the proof

We now prove (3.18) using the logic of Butti:2005vn and Lee:2006ru . In figure B.26 we draw the plane orthogonal to the vector $e_{1}=(1,0,0)$, where all the endpoints of the vectors $v_{a}$ lie. The vectors $w_{a}=v_{a+1}-v_{a}$ lie entirely on the plane and correspond to the sides of the toric diagram. We also define the vectors $r_{a}=v_{a}-b/b_{1}$. They also lie entirely on the plane and connect the point $B$ with coordinates $(b_{2}/b_{1},b_{3}/b_{2})$ to the vertices of the toric diagram. All the vectors in the following are three-dimensional. We use $\langle C,D\rangle=(e_{1},C,D)$ to compute areas in the plane. Indeed, when $C$ and $D$ are vectors lying on the plane, $|\langle C,D\rangle|$ is twice the area of the triangle with sides $C$ and $D$. We also assume that the vertices are labeled in the counterclockwise direction.

[TABLE]

Define the quantities

[TABLE]

where the symbol $b,c\in[A,B]$ means a sum over all pairs $A\leq b<c\leq B+d$ with the identification $\mathfrak{n}_{a+d}=\mathfrak{n}_{a}$, $\Delta_{a+d}=\Delta_{a}$ and $v_{a+d}=v_{a}$. If we select an order, we can drop the absolute value from $|(v_{a},v_{b},v_{c})|$ but note that we need to keep the symmetrized product of $\mathfrak{n}$ and $\Delta$. We want to prove that there exists a function $S$ and a vector $u$ such that

[TABLE]

and $S$ is proportional to $c(b_{i},\mathfrak{n}_{a})$. Using repeatedly the identity $(v_{a},v_{b},v_{c})=\langle r_{a},r_{b}\rangle+\langle r_{b},r_{c}\rangle-\langle r_{a},r_{c}\rangle$,151515The geometrical interpretation of this identity is similar to the one discussed in footnote 14. we can compute the difference

[TABLE]

where we added an arbitrary term $d_{1}w_{1}$ since it gives a vanishing contribution. More generally we find

[TABLE]

where again $d_{a}$ is arbitrary. We will show now that there exists a choice of $d_{a}$ such that all the $u_{a}$ are equal. In order to show it, we compute the difference

[TABLE]

where we used $\sum_{b=1}^{d}\Delta_{b}=2$ and $\sum_{a=1}^{d}r_{a}\Delta_{a}=0$. More generally,

[TABLE]

Requiring that (B.32) is zero gives an expression for $d_{a}$. Indeed, if $u_{a+1}=u_{a}$, contracting (B.32) with $w_{a}$ and $w_{a+1}$, respectively, we obtain two different expressions for $d_{a}$:

[TABLE]

The two expressions should coincide. By shifting $a\rightarrow a+1$ in the second equation and comparing with the first one, we get the consistency condition

[TABLE]

After a short computation using $\sum_{a}\mathfrak{n}_{a}v_{a}=n$ and some geometrical identities to convert areas in the plane to three-dimensional determinants, like

[TABLE]

and generalizations, we obtain

[TABLE]

which is precisely the identity (LABEL:fundamentalidentity). This proves that $d_{a}$ such that all the $u_{a}$ are equal can be actually found. We can also write an explicit expression

[TABLE]

Notice also that, comparing with (B.25), we obtain

[TABLE]

Since all the vectors $u_{a}$ are equal we call them $u$. We have

[TABLE]

for some function $S$. Using (B.27) and (B.30), we find that

[TABLE]

We can manipulate the sum in the previous expression by using repeatedly $\sum_{c}\Delta_{c}r_{c}=0$:

[TABLE]

where in the first step we change variables in the first sum and notice that the term $c=1$ in the second sum is zero because of $\sum_{c}r_{c}\Delta_{c}=0$. In the second step, for each fixed $c$, we transform $\sum_{c<b\leq 1}r_{b}\Delta_{b}=-\sum_{2\leq b\leq c}r_{b}\Delta_{b}$ and notice that $c\leq d$. In conclusion we have

[TABLE]

This has to be compared with (B.8):

[TABLE]

where we worked in the gauge $\lambda_{1}=\lambda_{2}=0$ and we used (B.17) and the identity $(v_{c},v_{a},b)=b_{1}\langle r_{c},r_{a}\rangle$. We see that

[TABLE]

Multiplying (B.28) by $-3N^{2}$, and using the definition (B.27) we finally obtain

[TABLE]

where we set $b_{1}=2$ and $t=-3N^{2}u$. This is exactly (3.18) .

B.4 Proving that $\mathcal{V}_{\text{on-shell}}(b_{i},\mathfrak{n}_{a})=a(\Delta_{a})$

In the gauge $\lambda_{1}=\lambda_{2}=0$, the master volume (3.10) can be written as

[TABLE]

where we used (B.17) and $b=\frac{b_{1}}{2}\sum_{a}v_{a}\Delta_{a}$.161616In the last step of the proof we reorganized the sum by noticing that the term $(v_{1},v_{e},v_{f})\Delta_{1}\Delta_{e}\Delta_{f}$ with $2\leq e<f\leq d$ appears just once while $(v_{d},v_{e},v_{f})\Delta_{d}\Delta_{e}\Delta_{f}$ with $2\leq d<e<f\leq d$ appears three times, two with the sign plus and one with sign minus. Comparing with (2.8), we see that

[TABLE]