Geometry of $\mathcal{I}$-extremization and black holes microstates

The entropy of a class of asymptotically AdS$_4$ magnetically charged BPS black holes can be obtained by extremizing the topologically twisted index of the dual three-dimensional field theory. This principle is known as $\mathcal{I}$-extremization. A gravitational dual of $\mathcal{I}$-extremization for a class of theories obtained by twisted compactifications of M2-branes living at a Calabi-Yau four-fold has been recently proposed. In this paper we investigate the relation between the two extremization principles. We show that the two extremization procedures are equivalent for theories without baryonic symmetries, which include ABJM and the theory dual to the non-toric Sasaki-Einstein manifold $V^{5,2}$. We then consider a class of quivers dual to M2-branes at toric Calabi-Yau four-folds for which the $\mathcal{I}$-functional can be computed in the large $N$ limit, and depends on three mesonic fluxes. We propose a gravitational dual for this construction, that we call mesonic twist, and we show that the gravitational extremization problem and $\mathcal{I}$-extremization are equivalent. We comment on more general cases.


Introduction
In this paper we are interested in theories living on M2-branes sitting at the tip of a Calabi-Yau four-fold cone and the relation of their twisted compactifications on a Riemann surface to AdS black holes physics. There are some interesting extremization problems that can be formulated for such theories. The theory on M2-branes at a Calabi-Yau cone C(Y 7 ), where Y 7 is a seven-dimensional Sasaki-Einstein manifold, is a three-dimensional N = 2 superconformal field theory (SCFT) dual to AdS 4 × Y 7 . In supersymmetric gauge theories the R-symmetry current is not necessarily unique and mixes with the global symmetry currents. It is known that the exact R-symmetry of the three-dimensional theory is obtained by extremizing the free energy on S 3 , F S 3 (∆ a ), as a function of the Rcharges of fields and, possibly, monopole operators [1]. The gravity dual of F -maximization is the volume minimization principle discovered in [2,3]. The equivalence of the two extremization principles has been checked in many examples [4][5][6][7][8][9][10], although there is no general proof, since the large N limit of the S 3 free energy is currently available only for a restricted class of theories.
On the other hand, twisted compactifications of M2-brane theories are IR dual to AdS 2 × Y 9 backgrounds in M-theory, where Y 9 is a fibration of Y 7 over a Riemann surface Σ g , depending on a choice of magnetic fluxes n a . These backgrounds can be interpreted as the horizon of asymptotically AdS 4 ×Y 7 magnetically charged BPS black holes. An extremization principle for finding the entropy of magnetically charged black holes in AdS 4 × Y 7 has been proposed in [11,12]. It involves extremizing the logarithm of the topologically twisted index, i.e. I = log Z Σg×S 1 , the partition function of the three-dimensional SCFT on Σ g × S 1 with a A-twist along Σ g [13]. This principle, dubbed I-extremization, has been successfully applied to the microscopic counting of the entropy of black holes in AdS 4 × S 7 [11,12] and in many other situations [14][15][16][17][18][19][20][21]. 1 A gravity dual for I-extremization was recently proposed in a series of interesting papers [41,42]. It is the purpose of this paper to investigate the relation between the two extremization principles. The analogous construction for D3-branes at Calabi-Yau three-fold toric cones has been recently proved to be equivalent to c-extremization for two-dimensional (0, 2) conformal field theories in full generality [41][42][43]. We already proved in [43] that the construction in [41,42] is equivalent off-shell to I-extremization for the class of magnetically charged BPS black holes in AdS 4 × S 7 studied in [11,12], where the dual field theory is ABJM [44]. In this paper, we extend this observation in various directions.
We will compare the construction of [41,42] with the field theory result in [14] which shows that, for the very same theories for which we can compute the S 3 free energy in the large N limit, the I-functional can be written as where ∆ I are the R-charges of a set of fields and monopoles, and n I the corresponding magnetic fluxes (which can be set in relation to the magnetic charges n a of the black hole).
If we choose an R-charge parameterization such that F S 3 (∆ I ) is homogeneous of degree 1 See also [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] for related works in a similar context or other dimensions. As we will review in details, the R-charges ∆ I and the fluxes n I parameterize the mixing of R-symmetry with the abelian global symmetries of the theory. It will be important for us to distinguish between mesonic and baryonic symmetries. According to the holographic dictionary, the first are associated with the isometries of the internal manifold Y 7 , while the second come from the reduction of the M-theory form on non-trivial five-cycles in Y 7 .
For manifolds Y 7 with no non-trivial five cycles, we will show that the construction in [41,42] is completely equivalent to the extremization of the I-functional given in (1.2), and that this equivalence holds off-shell. This class of manifolds is somehow limited but it contains Y 7 = S 7 , whose dual is the ABJM theory [44], and the non-toric manifold V 5,2 = SO(5)/SO(3), whose dual field theory has been found in [45]. For manifolds without five-cycles we can turn on magnetic charges only for the mesonic symmetries associated with the isometry of Y 7 . We will discuss in detail the V 5,2 example as a prototype of this class of compactifications. In particular, we obtain a prediction for the entropy of the general two-parameter family of asymptotically AdS 4 × V 5,2 black holes with arbitrary magnetic charges under the Cartan subgroup of the internal isometry SO (5). It would be interesting to find these solutions explicitly.
We then consider the general case of toric Calabi-Yau cones C(Y 7 ), for which we can use the master volume construction defined in [42]. In addition to the mesonic charges associated to the three U(1) isometries of Y 7 , we can have a baryonic charge for every non-trivial five-cycle S α ⊂ Y 7 . Here the situation is more complicated. We want to compare the former construction with the existing results for the large N limit of the topologically twisted index, which only depend on a linear combination of the available magnetic charges, corresponding to the mesonic directions only [14,15]. Correspondingly, we identify a three-parameter family of twisted compactifications, which we dub mesonic twist, where the extremization is performed along the mesonic directions only. We will show that, for such compactifications, the construction in [41,42] is equivalent off-shell to the extremization of the I-functional (1.2). This result is general and it can be proved for all toric Calabi-Yau cones C(Y 7 ) for which the equivalence between F -maximization and volume minimization is valid. In particular, all the existing field theory computations to date have a counterpart in the framework of [41,42].
Our results for toric Calabi-Yau cones are restricted to a particular class of twisted compactifications. This should be contrasted with the equivalence of the construction in [41,42] with c-extremization, which can be proved for an arbitrary toric Y 7 and an arbitrary choice of fluxes [43]. The reason for such restriction lies in our incomplete understanding of the large N limit of the S 3 free energy and the topologically twisted index for quivers with a holographic dual. It is known, for example, that the computation performed in [5] and [14,

F -maximization from geometry
We are interested in gauge theories that are holographically dual to AdS 4 ×Y 7 backgrounds in M-theory, where Y 7 is a Sasaki-Einstein manifold. The holographic dictionary relates the volume of the Sasaki-Einsten manifold to the S 3 free energy of the dual CFT [4] . (2.1) Many N = 2 quivers describing such theories have been proposed in the literature. Most of them are obtained by dimensionally reducing a parent four-dimensional quiver gauge theory with bi-fundamentals and adjoints with an AdS 5 × Y 5 dual, and then adding Chern-Simons terms (whose levels sum to zero) and flavoring with fundamentals. The value of F S 3 and the exact R-symmetry of a three-dimensional N = 2 theory can be found by extremizing the S 3 free energy as a function of the R-charges ∆ I of the chiral elementary fields and monopoles [1]. This procedure is called F -maximization. The R-charges ∆ I parameterize the mixing of the R-symmetry with the abelian global symmetries of the theory. We will distinguish between mesonic and baryonic symmetries.
For theories dual to AdS 4 × Y 7 , mesonic symmetries are associated with the isometries of Y 7 , which we take to be U(1) s , with s ≥ 1. We will be mostly interested in the toric case where s = 4, but we will also consider non-toric examples. One of the mesonic symmetries is the exact R-symmetry and the other s − 1 are global symmetries. In addition, we have a baryonic symmetry for each non-trivial five-cycle S α of Y 7 . They are holographically dual to the gauge fields that we obtain by reducing the M-theory six-form potential on the five-cycles S α [49][50][51]. In supergravity language, the corresponding vector multiplets are called Betti multiplets. 4 In all known examples, the three-dimensional N = 2 theories are quiver gauge theories that can have U(N ) G or U(1) × SU(N ) G gauge groups, where G is the number of nodes, depending on the choice of quantization [44,52]. When the gauge group is U(1) × SU(N ) G we have baryonic operators obtained by wrapping M5-branes on the five-cycles and this is the case we will be mostly interested in. 5 The gravitational dual of F -maximization is the volume minimization found in [2,3], which works as follows. One can relax the Einstein condition on the metric and write the volumes of a generic Sasaki manifold Y 7 (b i ) and of its five-cycles S α (b i ) as functions of the Reeb vector b = (b 1 , b 2 , . . . , b s ). Supersymmetry requires b 1 = 4. In general, the cone C(Y 7 ) is a Calabi-Yau three-fold. As shown in [2], the extremization of the function 4 Notice that what we call baryonic not always reflects the field theory notion of baryonic symmetry for the parent four-dimensional quiver. For example, the ABJM theory is dual to AdS 4 × S 7 and there are no nontrivial five-cycles and therefore no baryonic symmetries in our sense. The baryonic symmetry of the parent four-dimensional quiver, which is the well-known Klebanov-Witten theory [49], corresponds to an isometry of S 7 and it is a mesonic symmetry in our language. 5 The number of global symmetries is the same for both choices of gauge group, U(N ) G or U(1)×SU(N ) G [44,52]. In the case of U(N ) G , there are no baryons but we have instead monopole operators obtained by wrapping M2-branes.
Vol S (Y 7 (b i )) reproduces the Reeb vectorb = (b 1 ,b 2 , . . . ,b s ) and the volumes of the Sasaki-Einstein manifold Y 7 . Remarkably, for a large class of examples, the volume functional agrees off-shell with the S 3 free energy [5,6] with a suitable parameterization of the R-charges ∆ I (b i ) of fields and monopoles. In order to find the right parameterization, one considers all baryonic operators made with elementary fields and basic monopoles. These correspond to M5-branes wrapped on linear combinations of the five-cycles and their dimension can be computed from the corresponding volumes. 6 In all known examples, the R-charges ∆ I of fields and monopoles can be computed as linear integer combinations of basic R-charges, corresponding to M5-branes wrapped over U(1) s invariant five-cycles S a and expressed in terms of the Sasaki volumes [50,51] In the toric case there are quite explicit expressions for these volumes, which are associated with the vectors v a , a = 1, . . . , d, of the toric diagram [2,54] Vol where w a , k = 1, . . . , a , is a counterclockwise ordered sequence of vectors adjacent to v a . There is also an alternative way of computing the volumes using the Hilbert series, for which we refer to [3,8,54]. This method can be used also for non-toric manifolds. We will see an example in section 3.1. Notice that, in the large N limit, the free energy F S 3 only depends on a set of linear combinations of the ∆ I equal to the number of independent parameters b i and corresponding to the mixing of the R-symmetry with the mesonic symmetries. Indeed, as shown in [5], there are accidental flat directions at large N in F S 3 (∆ I ) and the R-charges parameterizing the baryonic symmetries disappear from the free energy functional. This should be contrasted with the case of a-maximization and its relation to volume minimization [58], where the baryonic symmetries explicitly enter in the trial a-charge. Extremizing F S 3 (∆ I ), we can only predict the exact R-charges of the mesonic operators of the theory. However, when the gauge group is U(1) × SU(N ) G , consistency of the solution allows to derive constraints expressing some of the remaining R-charges in terms of those appearing in F S 3 (∆ I ) [5]. This can be used to compute the exact R-charge of some baryonic operators in the theory. We will see examples of such constraints in section 5.

I-extremization from geometry
We are actually interested in twisted compactifications of the three-dimensional N = 2 theories discussed in the previous section on a Riemann surface Σ g of genus g. The holographic dual is an M-theory background AdS 2 ×Y 9 , where topologically Y 9 is a fibration of Y 7 over Σ g , and can be interpreted as the horizon of magnetically charged BPS AdS 4 black holes. As in the original computation in [11,12] for AdS 4 × S 7 black holes, the entropy of magnetically charged BPS black holes should be obtained by extremizing the functional where Z Σg×S 1 (∆ I , n I ) is the topologically twisted index [13], n I denote the magnetic charges and ∆ I are chemical potentials associated with elementary fields and monopoles. We refer to this principle as I-extremization. A field theory computation, valid for a large class of theories, shows that, in the large N limit, the I-functional can be parameterized in terms of the R-charges of the fields and reads [14] I(∆ I , n I ) = − 1 2 This formula (2.6) is only valid for an R-charge parameterization that makes F S 3 (∆ I ) homogeneous. 7 The same accidental symmetry that affects the large N limit of the S 3 free energy appears in the computation of the topologically twisted index at large N , and, therefore, the I-functional only depends on R-charges and fluxes along the mesonic directions [14]. The gravitational dual of I-extremization has been found in [41,42]. The authors of [41,42] considered a class of off-shell backgrounds by imposing supersymmetry but relaxing the equations of motion. More precisely, they considered M-theory backgrounds of the form where F is a closed two-form on Y 9 and ds 2 9 = η 2 + e B ds 2 . (2.8) Here, η ≡ (dz + P )/b 1 , where ρ = dP is the transverse Ricci form, and ds 2 is a Kähler metric. The Reeb vector field associated with the R-symmetry reads where ∂ ϕ i are real holomorphic vector fields generating the U(1) s action on Y 7 . When Y 7 is toric, s = 4. Finally, the closed two-form F is given by where J is the transverse Kähler form. Supersymmetry requires b 1 = 1.
For the background of interest, the transverse Kähler cohomology class decomposes as where ω is a transverse Kähler form on Y 7 and A > 0 is a constant parameterizing the Kähler class of Σ g . We normalize Σg Vol Σg = 1. The fibration of Y 7 over Σ g is specified by s integer magnetic fluxes n i . We can introduce them through s U(1) gauge fields where F i = dA i . Supersymmetry requires that we refer to as the twisting condition. As shown in [41,42], the on-shell background and the exact R-symmetry vector can be found by extremizing the supersymmetric action which is a function of the Reeb vector and the cohomology class of J, with the constraint The flux quantization conditions also require 8 16) where N is the total number of M2-branes and where n α is an integer, for all five-cycles S α ⊂ Y 7 , and α = 1, . . . , dim H 5 (Y 7 , Z). The integer numbers n i and n α are interpreted as the magnetic fluxes of the twisted compactification on Σ g . The n α are associated with the baryonic symmetries and the n i associated with the mesonic ones. Finally, the R-charge of an operator obtained by wrapping a M5-branes on a five-cycle S α ⊂ Y 7 is given by [41,59] where ω is the restriction of J to Y 7 . Since this is the R-charge of a baryonic operator, As noticed in [41,42] in the specific example of the magnetically charged AdS 4 black holes of [60], the on-shell value of S SUSY reproduces the entropy. This is also true for all magnetically charged black holes in AdS 4 × S 7 , as indirectly checked in [43] by comparing with I-extremization. Since S SUSY is related to the holographic free energy of the horizon solution, and the latter to the entropy by general arguments [17,26,27], we may expect this to be true in general. If this is the case, the construction in [41,42] gives an efficient method to write the entropy of a class of black holes from few geometrical data, even without knowing the explicit metric on Y 7 .

Theories with no baryonic symmetries
In this section we consider the case of manifolds Y 7 with no non-trivial five-cycles and therefore dual field theories with no baryonic symmetries. Examples include Y 7 = S 7 , whose dual is the ABJM theory [44], and the non-toric V 5,2 = SO(5)/SO(3), whose dual field theory has been found in [45]. 9 For the ABJM theory, we already checked in [43] that S(b i , n a ) is equal off-shell to the I-functional (2.6). In this section we will check that this is also true for the case of V 5,2 and, in general, for all manifolds Y 7 with a U(1) s action and no five-cycles.
It is convenient to first rewrite the conditions of supersymmetry as follows [41,42]. The supersymmetric action can be written as [42] where we defined the operator We can also write the constraints (2.15) and (2.16) as The derivation of (3.1) and (3.3) is given in [42] for toric Y 5 but extends with little modifications to seven dimensions and to the non-toric case. We are interested in manifolds Y 7 with no five-cycles. The supersymmetry conditions (3.1) and (3.3) only depend on the cohomology classes of ω and ρ on the foliation transverse to the Reeb vector action. Focusing, for simplicity, on the quasi regular case we can take the quotient with respect to the Reeb action, and consider the base V = Y 7 /U(1). 10 The manifold V has only one two-cycle and, therefore, ω and ρ are proportional in cohomology where λ(b) is a function to be found. Using (3.4) and the first equation in (3.3), we then obtain (3.5)

Using the last equation in (3.3) we find
Aλ and therefore, since ∇N = 0, Now we are in a position to evaluate the functional (3.1). We may write, using the plus sign in (3.5), We now notice that the expression is formally identical to the volume of a Sasakian manifold Y 7 (b i ) with Reeb vector b. Indeed, this expression can be evaluated using a fixed point theorem which only depends on the local complex geometry of C(Y 7 ) near the fixed points [41]. Hence it formally coincides with the expression for the Sasakian volume of Y 7 (b i ) computed in [2,3] and given in (2.4) in the toric case. Form now on, we will understand the explicit dependence on b i in (2.4) and just use the subscript S to indicate that the volumes are computed using the Sasaki metric on Y 7 (b i ).
We conclude that the entropy functional S(b i , n i ) is given by Notice that, since the equations involve derivatives with respect to b 1 , we can set b 1 = 1, as required by supersymmetry, only at the end of the computation. We are considering manifolds with no five-cycles. However, in analogy with the toric case that we will discuss in the next section, we can consider the set S a ⊂ Y 7 of U(1) s invariant submanifolds of dimension five and formally extend the quantization condition to these cycles. We then also impose [42] where n a are integers. Notice that, since there are no baryonic symmetries, the n a must be linear combinations of the mesonic fluxes n i . Using (3.5) and (3.7) we find (3.12) Defining normalized R-charges (see (2.18)) we can therefore rewrite (3.12) as (3.14) In many quivers, including those for ABJM and V 5,2 , the elementary fields can be associated with linear combinations of invariant five-cycles S a and their R-charges can be computed using (3.13).

Example: the manifold V 5,2
The non-toric Sasaki manifold V 5,2 = SO(5)/SO(3) possesses an SO(5) × U(1) R ⊃ U(1) 3 isometry, where U(1) R is identified with the R-symmetry on the SCFT side. The base V is the Grassmannian Gr 2 (R 5 ), which admits a complex structure, and has only one two-cycle. The dual gauge theory is given by the quiver [45] N +k We will focus on the quiver with k = 1 which is dual to AdS 4 × V 5,2 . The Calabi-Yau cone C(V 5,2 ) is described by the equation which has a manifest SO (5) invariance. This arises as the solution to the F -term equation for the adjoint fields using the variables The R-charges of the adjoint fields are constrained by the superpotential to be ∆ φ i = 2/3 while those of the bi-fundamental fields must satisfy There are five obvious U(1) 3 invariant divisors in C(V 5,2 ) that are obtained by setting φ 1 = 0, A 1 = 0, A 2 = 0, B 1 = 0 and B 2 = 0, respectively; and can be associated in a one-to-one way to the fields. The restriction to V 5,2 gives five invariant five-cycles S a . The S 3 free energy was derived in [5][6][7]15] from localization and reads In order to compute the volume (3.9) we can use the Hilbert series method [3,8,54,61]. 11 The Hilbert series is just the generating function of holomorphic functions on the cone C(V 5,2 ), graded under U(1) 3 . We assign fugacities to the fields where t corresponds to the R-symmetry, x and y are fugacities for the Cartan of SO(5), and is just a rescaling parameter that we will send to zero at the end of the computation. Since C(V 5,2 ) is a complete intersection, its Hilbert series is simply given by (3.23) 11 We refer to these papers for more details on the method.
The volume (3.9) of V 5,2 can be extracted as the coefficient of the leading pole in H(t, x, y) when → 0 [3], i.e.
Hence, we obtain Comparing (3.21) with (3.25), we see that (2.2) is satisfied, when we set b 1 = 4, as appropriate for three-dimensional computations. We thus see that volume minimization is equivalent to F -maximization off-shell. 12 The action of the Reeb vector on the fields is the linearization of the U(1) x, y), and, therefore, from (3.22) we read (3.26) where we normalized the R-charges by requiring that ∆ φ 1 = 2/3. The same expression can be derived by evaluating the volumes (3.13) as a limit of the Hilbert series for divisors, using the method discussed in [54]. The volume (3.25) then reads .
From (3.14) we also find where we associate a = 1, 2, 3, 4, 5 to (φ 1 , A 1 , A 2 , B 1 , B 2 ) in this order. At this point it is straightforward to check that where, this time, we set b 1 = 1 at the end of the computation. To obtain the correct normalization it is important to remember that, for black hole solutions, supersymmetry requires b 1 = 1, while, for volume minimization, we impose b 1 = 4. We learn that the construction in [41,42] is equivalent off-shell to I-extremization The same analysis can be applied to other Sasaki-Einsten manifolds Y 7 without fivecycles. In particular, it applies to some of the models recently discussed in [10]. We can also provide a very general argument that applies to many different theories and it is based on very few assumptions.

A general argument
On general grounds, the entropy functional (3.10) coincides with the I-functional (2.6) for all quivers with no baryonic symmetries where the off-shell equivalence of F -maximization and volume minimization (2.2) is valid. Indeed, for any quiver gauge theory with superpotential W = a W a , the R-charges of the fields must satisfy for each superpotential monomial W a , where the sum is restricted to the fields that appear in W a . Similarly, the fluxes are constrained to satisfy for each superpotential term. This is just the twisting condition (2.13). In theories with no baryonic symmetries, there is the same number of independent ∆ I and n I as the number of independent components of the Reeb vector. There must be then a linear relation among the ∆ I and the b i , that parameterizes the U(1) s action on the R-charges. By consistency, a similar linear relation exists between the n I and the n i where the normalization is determined by comparing (3.31) and (3.32), and recalling that n 1 = 2 − 2g. These expressions generalize (3.26) and (3.28). We now assume that, for our theory, F -maximization is equivalent to volume minimization, i.e.
, (3.35) when substituting (3.33) and imposing b 1 = 4. We also assume that there exists a parameterization of R-charges such that F S 3 (∆ I ) is a homogeneous function of degree two. This is generally the case. We can then write for some choice of a term W a in the superpotential and a function f (∆ I ) homogeneous of degree zero. Since Vol S (Y 7 ) is homogeneous of degree minus four in b i , we must also have for the same function f . Setting b 1 = 4 and using (3.33) and (3.31) we find indeed (3.35). It takes then a short computation to show that (3.36) and (3.37) imply using (3.34) and setting b 1 = 1 after taking derivatives. This confirms the off-shell equivalence of the entropy functional with the I-functional. In quivers where the elementary fields can be associated with linear combinations of invariant five-cycles S a , their R-charge can be computed using (3.13). By consistency, the result must reproduce (3.33). Formula (3.14) is then clearly consistent with (3.34).

Extremization for toric manifolds
In this section we consider the F -maximization and I-extremization principles in the case of toric manifolds. We will first review the construction of [42] for the toric case. We will then see that for the class of compactifications with a particular mesonic twist, the construction in [42] is equivalent to I-extremization. We will also comment about the general case.
Recall that the Sasaki-Einstein manifold Y 7 is toric when it has isometry U(1) 4 . In this case, the cone over Y 7 , C(Y 7 ) is a toric Calabi-Yau four-fold, which can be characterized in terms of combinatorial data associated with a toric diagram. Indeed, C(Y 7 ) can be described by a fan generated by d four-dimensional vectors v a = (1, v a ). The toric diagram is the three-dimensional polytope in R 3 with vertices the integer points v a . The toric four-fold C(Y 7 ) has exactly d U(1) 4 invariant divisors D a , one for each vertex v a . The restriction to the base Y 7 gives rise to U(1) 4 -invariant five-cycles S a , a = 1, . . . d. Since dimH 5 (Y 7 , Z) = dimH 2 (Y 7 , Z) = d − 4, only d − 4 of these cycles are inequivalent in cohomology.

The conditions of supersymmetry for toric manifolds
The off-shell family of supersymmetric backgrounds of [41,42] can be parameterized by the Reeb vector b = (b 1 , b 2 , b 3 , b 4 ) with b 1 = 1 and by the cohomology class of the transverse Kähler form. In the toric case, this can be specified as follows [41,42]. It is useful to restrict to the quasi regular case where the quotient with respect to the Reeb action, V = Y 7 /U(1), is a six-dimensional compact toric orbifold. The restriction of ω and ρ to V can be written in cohomology as where c a are the Poincaré duals of the restriction of the d toric divisors to V . The parameters λ a parameterize the cohomology class of the transverse Kähler form J. Since The master volume is then defined as the volume of the dual polytope associated with the Kähler parameter λ a [42]: where H(b) is the hyperplane (y, b) = 1/2 and y 0 = (1, 0, 0, 0)/(2b 1 ). The master volume (4.2) is invariant under and this leaves indeed d − 3 independent λ a since l 1 does not contribute (v 1 a = 1 , ∀a = 1, . . . , d).
The twisted compactification is specified by four mesonic fluxes n i and d − 4 baryonic ones n α . They can be conveniently parameterized by d integer magnetic fluxes n a , one for each toric divisor. The supersymmetry and flux quantization conditions for the off-shell background can be then written as [42] 13 Notice that n i and n a are not independent. Consistency of the equations above requires n i = d a=1 v i a n a . In particular, the twisting condition (2.13) becomes The total number of independent fluxes is d−1, corresponding to d−4 baryonic symmetries and three mesonic ones. With n i = d a=1 v i a n a , only d−2 equations in (4.4) are independent and allow to determine the d − 3 independent λ a and the area A as a function of n a and b i . Notice that, since the equations involve derivatives with respect to b 1 , we can set b 1 = 1 only at the end of the computation.
One can also simplify S SUSY , and define the entropy functional [42] For future reference, we also define the on-shell value of the master volume (4.2): .
We can explicitly solve the conditions of supersymmetry for a certain class of twisted compactifications. In order to simplify the exposition, we summarize the results and refer to appendix A for the proof.

The universal twist
The simplest case of twisting is when we twist the theory only along the exact R-symmetry of the theory, i.e.
Recall that n 1 = 2 − 2g. This is the so-called universal twist [47]. According to the general argument in [17], we expect that the entropy of the black holes is given by This formula indeed follows from the conditions of supersymmetry (4.4). The argument is analogous to the similar one discussed in [42] in the context of c-extremization. As shown in appendix A.2, there is a solution to the equations (4.4) where all the λ a are equal 14 Consistency of the equations (4.4) then implies that where we introduced the set of basic R-charges (2.3). Notice that d a=1 ∆ a (b i ) = 2. Note also that d a=1 n a = n 1 hence (4.5) is correctly satisfied. The entropy functional (4.6) becomes , being the Reeb vector of the Calabi-Yau four-fold [2]. In our case supersymmetry requires b 1 = 1 and we obtain b i =b i /4, and thus, using the homogeneity of the volume function, where Vol(Y 7 ) is the volume of the Sasaki-Einstein manifold. The entropy functional (4.12) at the extremum is then given by where we set b 1 = 1 and used (2.2) and (4.5). This is in agreement with the results in [14,17]. Note also that Finally notice that, since volume minimization is the dual of F -maximization, at the extremum, the R-charges ∆(b i ) becomes the exact R-charges of the fields of the threedimensional CFT. 15 Therefore, the relation (4.11) 16) tells us that all the fluxes are proportional to the corresponding exact R-charges, hence confirming that we are dealing with the universal twist. Since the fluxes n a must be integers, the universal twist is defined only for manifolds Y 7 where the exact R-charges ∆ a are rational and for certain values of g. Examples of Sasaki-Einstein manifolds with known duals admitting the universal twist are discussed in [17].

Mesonic twist
In this section we perform a particular topological twist that we dub mesonic twist. It depends on three magnetic fluxes that we can take to be the n i (recall that The condition (4.17) requires a particular choice for the fluxes n a that are specific functions of b i , n i and g, n a = n a (b i , n i , g). As for the universal twist, the quantization conditions 15 More precisely, the R-charges of the fields are integer linear combinations of the ∆(b i ).
for the fluxes give constraints on the twisted compactification. In particular, the value of b i at the extremum and g must be such that n a (b i , n i , g) is an integer. The constraint here is milder than for the universal twist, since the value of b i at the extremum can be tuned by varying n i , but still it restricts the class of solutions.
To study the mesonic twist, it is convenient to use the freedom (4.3) to fix some of the λ a and work in the gauge Furthermore, we will prove in appendix A that there exists a solution to the set of equations (4.4), compatible with (4.17), such that Notice that (4.17) is satisfied as a consequence of (4.18).
As in [43] we define the normalized R-charges inspired by (2.18). As a consequence of the first equation in (4.4) they satisfy Quite remarkably, as shown in appendix A.3, the conditions of supersymmetry (4.4) imply that the ∆ a are actually independent of the fluxes n a and are given by the Sasakian parameterization (2.3) where the volumes are given in (2.4). They satisfy the useful identity The on-shell value of the master volume can be written as where in the last step we set b 1 = 1 and used the the equivalence between volume minimization and F -maximization, see (2.2). More precisely, since Vol S (Y 7 ) is a homogeneous function of the b i of degree −4, and we can choose F S 3 (∆ a ) to be homogeneous of degree 2, we can write (4.26) for some function f . Setting b 1 = 4, as appropriate for F -maximization, we find (2.2). Setting b 1 = 1, we find (4.25). The entropy functional (4.6) is given by where we defined the operator Notice that, since ∇ explicitly takes a derivative with respect to b 1 , we can set b 1 = 1 only at the end of the computation. On the other hand, the topologically twisted index of N = 2 theories in the large N limit reads [14] I(∆ a , n a ) = − It takes a short computation to show that (4.26) implies where we used n i = d a=1 v i a n a . This confirms the off-shell equivalence of the entropy functional with the I-functional.
As anticipated, the solution is only consistent if we impose the following constraints on the fluxes n a = 1 2 leaving only the n i as independent fluxes. Notice that, as required by consistency, d a=1 v a n a = 1 2 where we used (4.24). Note that (4.27) and (4.31) for the mesonic twist are formally identical to the expressions that we found for theories with no baryonic symmetries, (3.10) and (3.14).

Interpreting the mesonic twist
The condition (4.17) and the constraints (4.31) have a nice geometrical interpretation. The condition (4.17) is equivalent to requiring that the R-charges ∆(b i ) are parameterized in terms of the Sasaki volumes as in the three-dimensional F -maximization problem. Since (4.23) only depends on the Reeb vector components b i , there are d − 4 constraints among the ∆(b i ). In all the examples that we have checked, these constraints can be written as cubic polynomials in the ∆ a . Moreover, they can be compactlly written in terms of derivatives of a single auxiliary quartic polynomial. 16 In all known examples, there exist indeed a quartic polynomial a 3d (∆) that identically coincides with F S 3 (∆ a ) 2 on the locus parameterized by (4.23) [8,62]. We have checked that the d − 4 constraints among Rcharges can be written as which is quadratic in ∆ a . The last statement is equivalent to say that the flux constraints (4.31) can be obtained by applying the operator d a=1 n a ∂ ∆a to the R-charge constraints. There is also some interesting field theory interpretation for some of R-charge constraints (4.33). In the available computation for the large N limit of the S 3 free energy [5] and the topologically twisted index of N = 2 quivers [14], they arise when one imposes that the theory has gauge group U(1) × SU(N ) G . The free energy and the index are the same for U(N ) and SU(N ) groups and, as already discussed, depend only on a linear combination of the ∆ a parameterizing the mesonic directions. However, for SU(N ) gauge groups one has to impose that the distribution of eigenvalues of the matrix model is traceless. This gives some constraints among the ∆ a that allows to fix some of R-charges of baryonic operators, as already mentioned at the end of section 2.1. In all known examples, the SU(N ) constraints -when nontrivial -coincide with a subset of the R-charge constraints (4.33). 17 We will see explicit examples in section 5.
Finally, we notice that there is a similar story for flows from four-dimensional N = 1 CFTs to (0, 2) two-dimensional CFTs induced by twisted compactifications on Σ g . In this case, volume minimization [2,3] is the dual of a-maximization [63] and the construction in [41,42] is the dual of c-extremization [47,64]. A general solution to the equations in [41,42] for arbitrary fluxes n a has been provided in [43] together with a general formula for ∆ a (n a , b i ). This solution automatically satisfies d a=1 where c 2d (∆) is the two-dimensional trial right-moving central charge, and a 4d (∆) is the four-dimensional trial a central charge. Physically, this condition guarantees that the twodimensional central charge is extremized with respect to the baryonic directions. Moreover, using the explicit solution in [43], it is not difficult to show that d a=1 and (4.35) becomes a condition that expresses the baryonic fluxes in terms of the mesonic ones, n a = n a (b i , n i ), as in the three-dimensional case. The conditions (4.37) and (4.35) are the analogue of (4.33) and (4.34). Notice that (4.37) can be interpreted as the extremization of the four-dimensional a central charge with respect to the baryonic directions.
As such, it enters as a decoupling condition for the baryonic symmetries in the proof of the the equivalence between volume minimization and a-maximization [58]. We see that, in the four-dimensional context, the mesonic twist corresponds to enforce the decoupling of baryonic symmetries before flowing from four to two dimensions. This explains the name and it was our original motivation for studying it. We expect that, similarly to the case of c-extremization discussed in [43], one can find a general solution to the equations (4.4) such that it reduces to the mesonic twist when further imposing the decoupling condition (4.17). Unfortunately, solving (4.4) in general is hard because the equations are quadratic in λ a . It would be very interesting to see if the quartic function a 3d (∆) plays some role in the general solution, as its counterpart a 4d (∆) does for c-extremization [43].

Toric examples
In this section we consider some examples of the general construction presented in section 4.3. We thus perform a mesonic twist of the three-dimensional N = 2 theories. We will use the results in [5,6,54,56] to which we refer for more details on the parameterization of R-charges and the large N limit of S 3 free energy.

The C × C geometry
Our first example is a flavored ABJM theory (with Chern-Simons levels set to zero), whose quiver is depicted below [56,65] where a circular node denotes a U(N ) gauge group and the square node denotes a U(1) flavor symmetry. There are bi-fundamental chiral multiplets between the two gauge groups labeled by (A i , B i ), i = 1, 2; fundamental and anti-fundamental chiral multiplets labeled by (q 1 ,q 1 ). The theory has the superpotential The moduli space of the quiver gauge theory (5.1) can be characterized in terms of the fields A i , B i and two monopole operators T 1 and T 2 satisfying T 1 T 2 = A 1 [56]. The mesonic spectrum is generated by the gauge invariant operators [56], subject to the relation x 1 x 2 = x 3 x 4 . Hence, the dual geometry is C × C (C the conifold). Denote the R-charges of the chiral bi-fundamental fields (A 1 , A 2 , B 1 , B 2 ) and the monopoles (T 1 , T 2 ) by (∆ A 1 , ∆ A 2 , ∆ B 1 , ∆ B 2 ) and (∆ T 1 , ∆ T 2 ), respectively. We also define the bare monopole R-charges ∆ m 1 and ∆ m 2 associated with the topological symmetries [5], satisfying That the superpotential (5.2) has R-charge two, imposes the following constraint on the R-charges We also introduce the magnetic fluxes (n A 1 , n A 2 , n B 1 , n B 2 ) and (n m 1 , n m 2 ). Supersymmetry imposes the constraint The dual geometry C × C is specified by the vectors Its toric diagram is shown below We associate R-charges ∆ a and fluxes n a , a = 1, . . . , 5, to the five vertices v a of the toric diagram (5.7). They are related to the R-charges and fluxes of the chiral fields and monopoles by [5,56] 18 8) where n m = n m 1 + n m 2 . The S 3 free energy of this theory was computed in [5, (6.9)] and it reads 19 In the SU(N ) × SU(N ) × U(1) theory one has to impose the additional constraint as discussed in [5]. This arises due the tracelessness condition on the eigenvalues distribution. Remarkably, this constraint is equivalent to where a 3d (∆ a ) is given by 12) 18 The ∆ a can be associated to a parameterization in terms of GLSM fields. See in particular [56, (6.21)]. 19 We correct a typo there. and the baryonic symmetry, characterized by (4.18), reads is homogeneous of degree two, the topologically twisted index of this theory is simply given by (2.6), i.e. (5.14) The dual polytope associated with the Kähler parameters λ a is given by (5.18) From now on, we work in the gauge (4.19), i.e. λ 1 = λ 2 = λ 3 = 0. Using (4.4) and (4.20), we may fix the remaining λ a as Notice that a B a λ a = 0, as it is required by the mesonic twist. The last equation in (4.4) is easily solved for A. Plugging the solutions for A and λ a back into (5.18) we obtain For the entropy functional (4.6) we find that

(5.21)
Using the expression (4.21) for the R-charges, we also obtain .

(5.22)
Let us emphasize again that in the case of the mesonic twist the R-charges are independent of the magnetic charges and coincide with the Sasakian parameterization (2.3). Notice that the R-charges (5.22) automatically satisfy (5.10). Finally, the second equation in (4.4) imposes the following constraint on the magnetic fluxes Notice that the constraint on fluxes can be obtained by applying the operator d a=1 n a ∂ ∆a to the R-charge constraint (5.10). This constraint is also equivalent to (4.34). This will be true for all other examples in this paper. It is easy to see that, using (5.22) and (5.23), and setting b 1 = 1, .

(5.24)
We thus proved the off-shell equivalence of the I-extremization principle and its geometric dual. It is also interesting to observe that . (5.25)

The cone over Q 1,1,1
Our second example is a gauge theory described by the quiver [56,65] N N where the notation is understood as before. The theory has the superpotential The manifold in this case is Y 7 = Q 1,1,1 that is defined by the coset The cone C(Q 1,1,1 ) determines a polytope with six vertices The toric diagram is depicted below Since dimH 2 (Q 1,1,1 , Z) = 2, there are two baryonic symmetries U(1) B 1 × U(1) B 2 that are characterized by (4.18). They are given by Thus, the symmetries of the model is SU where SU(2) 3 × U(1) R are the isometries of Q 1,1,1 (mesonic symmetries). The moduli space of the quiver gauge theory (5.26) can be characterized in terms of the fields A i , B i and two monopole operators T 1 and T 2 satisfying T 1 T 2 = A 1 A 2 [56]. The mesonic spectrum is generated by the gauge invariants B i A j and B i T j , transforming as (2, 2, 2) of SU(2) 3 . Using [56, (6.28)], and assigning R-charges ∆ T i to the monopoles, we find the relation between the chiral fields R-charges ∆ I and the basic R-charges ∆ a associated with toric geometry The dual polytope associated with the Kähler parameters λ a is given by The master volume can be easily computed from (5.33). Since the resulting expression is too long we report it in the appendix -see (B.1). As before we work in the gauge (4.19).
Using (4.4) and (4.20) we may fix λ a and A. Their explicit expressions are quite long and we shall not present them here. Employing the parameterization (4.21) for the R-charges, we obtain , , .

(5.35)
Notice that the R-charges (5.35) are independent of the fluxes and they satisfy with [8] a 3d (∆ a ) ≡ 1 24 Explicitly, we only write one of the two constraints that we will use later on, i.e.
The associated constraint on the magnetic fluxes follows from the second equation in (4.4) and it can be written as To compare with the results in [5,15], using the symmetries of the quiver (5.26), we restrict Hence, the constraint (5.38) becomes (5.42) The decoupling condition (5.39) is also simplified to The S 3 free energy of this theory was computed in [5, (6.15)] and it is given by together with constraint (5.42), that exists in the SU(N ) × SU(N ) × U(1) theory, see [5, (6.16)]. The topologically twisted index was also computed in [15, (5.47)] and it reads .

(5.46)
It is also interesting to observe that .

(5.47)
Finally, extremizing the index (5.45) with respect to ∆ m we obtain (5.48) Using (5.42) and (5.43), we also find that As can be seen from the above equations, one can find many twisted compactifications where the quantization conditions on the fluxes (n, n m ) are fulfilled. In particular, it is enough to have a rational value for ∆ m at the extremum since then (n, n m ) are also rational. We then see from (5.40) that we can make all the n a integer by taking the genus g large enough.

Flavoring N = 8 SYM
We consider the flavored N = 8 super Yang-Mills whose quiver description is given by [56] N r 1 r 2 where the loop around the circular node denotes the adjoint chiral multiplets φ i , i, 1, 2, 3, and the rest of the notation is understood as before. The theory has the superpotential The quantum corrected moduli space of vacua is a toric Calabi-Yau cone, parameterized by the complex coordinates φ i and the monopole operators T 1 , T 2 fulfilling the constraint j ), and (∆ T 1 , ∆ T 2 ) to the monopoles (T 1 , T 2 ), respectively. We also define the bare monopole R-charge (5.53) The superpotential (5.51) and (5.52) impose the constraints The toric diagram is shown below.
The baryonic symmetries are characterized by (4.18). They read It is now easy to compute the master volume, whose explicit expression is given in (B.2). We work in the gauge (4.19). Then, the remaining λ a and A can be found, using the ansatz (4.20), and solving (4.4) explicitly. In particular, ) . (5.60) Notice that the above solution satisfies (4.17). Plugging these expressions into (B.2), we find that For the entropy functional (4.6) we obtain (5.62) We assign the R-charge ∆ a and the flux n a , a = 1, . . . , 6, to each vertex of the toric diagram (5.56). Recall that supersymmetry requires 6 a=1 ∆ a = 2 , 6 a=1 n a = 2 − 2g . (5.63) The R-charges associated to the vertices of the toric diagram are mapped to those of the chiral fields and the monopoles by [5, (7.10)] Using the parameterization (4.21) for the R-charges, we also find , , , , .
The R-charges satisfy These constraints are equivalent to where a 3d (∆ a ) is given by

(5.69)
These constraints can also be obtained from (4.34). The S 3 free energy of this theory was computed in [5, (4.10)] and it can be written as .

The SPP theory
We now consider the quiver gauge theory [54] N k1 describing the dynamics of N M2-branes at a fibration over the suspended pinch point (SPP) singularity [66]. In the following, we choose Chern-Simons levels (k 1 , k 2 , k 3 ) = (2, −1, −1). The superpotential reads (5.74) Therefore, the R-charges of the chiral fields must satisfy We also introduce the magnetic fluxes (n φ , n A i , n B i , n C i ) for the chiral fields (φ, A i , B i , C i ), respectively. They satisfy The toric cone is determined by the vectors The toric diagram is depicted below.
The baryonic symmetries are characterized by (4.18) and they are given by The master volume is then easily computed and its explicit form can be found in (B.3).
As before we work in the gauge (4.19). Using (4.4) and (4.20) we may fix λ a and A. We will not report the long resulting expressions here. We assign the R-charge ∆ a and the flux n a , to each vertex of the toric diagram (5.78). Supersymmetry then requires The R-charges associated to the vertices of the toric diagram are mapped to those of the chiral fields by [54] Employing the parameterization (4.21) for the R-charges, we obtain , , (5.84) that are independent of the magnetic fluxes (as we are dealing with the mesonic twist). To compare with the results in [6,14], using the symmetries of the quiver (5.73), we restrict The S 3 free energy of this theory was computed in [6, (5.20)] and it is given by The topologically twisted index was also computed in [14, (B.19)] and it reads .

(5.89)
Notice also that where [8] a 3d (∆ a ) ≡ 1 24 Once again the constraints on the R-charges and the fluxes can be written as in (4.33) and (4.34), respectively.

5.5
The cone over M 1,1,1 Consider the cone over the Y 7 = M 1,1,1 , i.e. C(M 1,1,1 ). The gauge theory dual to AdS 4 × M 1,1,1 has a chiral quiver (in a four-dimensional sense) [55], and thus the large N limit of its partition functions on S 3 and Σ g × S 1 are not known [5,14]. However, it is interesting to evaluate the entropy functional (4.6) for M 1,1,1 and provide a prediction for the large N topologically twisted index of the dual gauge theory. The C(M 1,1,1 ) determines a polytope with six vertices where v 6 is an internal point. The toric diagram is shown below.
The baryonic symmetry is characterized by (4.18) and it is given by Having determined the dual polytope (5.95) it is now straightforward to compute the master volume, whose explicit expression can be found in (B.4). As before we work in the gauge (4.19). Using the first equation in (4.4) and (4.20) we can fix the remaining λ a as .
(5.98) The last equation of (4.4) can also be solved for A. Substituting these expressions into (B.4) we obtain .
(5.99) For the entropy functional (4.6) we find that .
(5.100) As usual, we associate the R-charge ∆ a and the flux n a to the vertex v a , a = 1, . . . , 5, of the toric diagram (5.93) [54]. They satisfy Note that we did not include the internal point v 6 . Using the parameterization (4.21) for the R-charges, we also obtain .

(5.102)
Notice that the R-charges (5.102) are independent of the fluxes and they fulfill the following relation This constraint can be obtained by looking at where a 3d (∆ a ) is given by [8] a 3d (∆ a ) ≡ 1 24 (5.105) The second equation in (4.4) also imposes the following constraint on the magnetic fluxes 106) This is equivalent to (4.34). Finally, defining the quantity, we may rewrite (5.100) as . (5.108) Also, observe that . (5.109)

Discussion and conclusions
In this paper we investigated the relation between I-extremization and its gravitational dual, recently proposed in [41,42], and, we provided many examples and a large class of twisted compactifications where the two extremizations agree off-shell. Our analysis also raises many questions and open problems.
In particular, as noticed in [14,15,17], baryonic symmetries disappear in the large N limit of the topologically twisted index for known three-dimensional quiver gauge theories with holographic duals and N 3/2 scaling. In order to match the existing field theory computation with the formalism of [41,42], we restricted to a particular class of twisted compactifications, characterized by the absence of a twist in the baryonic directions. However, there certainly exist AdS 2 solutions with baryonic fluxes [17,46,67]. It would be very interesting to understand if there is a way to introduce baryonic charges in the large N limit considered in [14,15,17] or to find more general saddle points and compare the result with the construction of [41,42].
Other obvious questions concern the interpretation of the cubic constraints (4.33). Notice that the constraints are a consequence of the Sasakian parameterization (2.3). Therefore they already show up in the three-dimensional aspects of the physics and in the discussion about the equivalence between F -maximization and volume minimization. From the physical point of view, F S 3 is only function of the mesonic directions, and its extremization leads to a prediction for the R-charges of the mesonic operators of the theory, corresponding to the KK modes of the compactification. The d−4 constraints (4.33) allow to determine the on-shell value of all the R-charges ∆ a and lead to a prediction for the Rcharges and dimensions of the baryonic operators also, since these are usually obtained by wrapping M5-branes on certain linear combinations of the cycles S α . As we saw in various examples, some of the constraints arise in the large N limit of QFT partition functions when we impose that the theory has gauge group SU(N ) and the distribution of eigenvalues is traceless. However, not all baryonic symmetries for the known quivers are related to a U(1) subgroup of a U(N ) gauge symmetry. Some of them appear as symmetries rotating the flavors. In all these cases, a field theory interpretation is still missing.
Similarly, the role of the quartic polynomial a 3d (∆) is still unclear both from the geometrical and physical point of view. It is a quartic polynomial with the property that, when restricted to the Sasakian parameterization (2.3), it coincides with F 2 S 3 as a function of b i . It has been originally found by analyzing examples in [8,62] but a general formula is still lacking. It is known to be of the form [8,62] a 3d (∆ a ) = 1 24 where the correction terms are related to internal lines in the toric diagram. Without the corrections terms, this expression would be the analogue of the well-known expression for the trial a central charge of the quiver associated to D3-branes at Calabi-Yau toric three-folds [68]. In this paper, we further noticed that, quite remarkably, the constraints among R-charges can be written in terms of a 3d (∆) using (4.33). It would be interesting to find a direct geometric interpretation for a 3d (∆). Even from the physical point of view, the analogy of a 3d (∆) with the four-dimensional trial central charge a 4d (∆) is quite intriguing. As its four-dimensional analogue [58], a 3d (∆) is automatically extremized with respect to the baryonic symmetries and it coincides off-shell with the inverse of the volume functional -which is also F 2 S 3 (∆ a ) -when imposing the Sasakian parameterization (2.3). Therefore, F -maximization is also equivalent to the extremization of a 3d (∆) with respect to all the directions, including the baryonic ones. It would be interesting to see if there is a field theory interpretation of this observation.
Similar questions arise for the twisted compactifications of the three-dimensional theories on Σ g . In particular, it would be nice to find a purely field theory interpretation of the constraints (4.31), or equivalently (4.34). It would be also interesting to see if a 3d (∆) plays some role in the solution to the equations in [41,42] for a generic choice of fluxes. In particular, in all our examples for the mesonic twist it is true that It would be interesting to see if there is a similar expression for an arbitrary twist.

A.1 Master and Sasaki volumes
The cone over a Sasaki manifold Y 7 (b i ), C(Y 7 (b i )), is a Kähler manifold but it is not in general Calabi-Yau. Considering the Reeb vector and the dual one-form η, i ξ η = 1, for a Sasaki manifold we have dη = 2ω S , where ω S is the Kähler form transverse to the Reeb foliation. For a Sasaki manifold, we also have ρ = 2b 1 ω S , where ρ is the Ricci two-form on Y 7 (b i ). The volumes of Y 7 (b i ) and of its cycles S a (b i ) are given by the explicit formulae [2,54] , where w a , k = 1, . . . , a , is a counterclockwise ordered sequence of vectors adjacent to v a . For the backgrounds in [41,42], 20 it is still true that dη = ρ/b 1 but now the restriction of ρ and ω to Y 7 are no more proportional. However, it is still true that where the subscript S indicates us that the volume are computed for the Sasaki metric on Y 7 (b i ). Indeed, the integrals in (A.3) can be computed using the Duistermaat-Heckman localization formula using a resolution of the complex cone C(Y 7 ) and dη = ρ/b 1 [3,41].
Since we can use a fixed point formula for evaluating the integrals, the result coincides with the formulae given in [2,3]. Consider now the master volume (4.2). Using the expression (4.1) for ω, we see that V is a cubic form in λ a For λ a = −1/(2b 1 ), the metric becomes Sasaki and the master volume coincides with Vol S (Y 7 ). More generally, using the expression (4.1) for ρ, we also see that the Sasakian volumes (A.3) are related to the intersection numbers by 21 (A.6)

A.2 Supersymmetry conditions: universal twist
The universal twist is defined by where n 1 = 2 − 2g. We can solve (4.4) by taking all the λ a equal Using (A.4) and (A.6) we find .
(A.9) Note that V, Vol S (Y 7 ), and Vol S (S a ) are homogeneous functions of b i of degree −1, −4, and −3, respectively. The set of equations (4.4) can then be rewritten as where we used (A.7) and Let us emphasize that the above set of equations depends on the choice of the Reeb vector through Vol S (Y 7 ) and Vol S (S a ). We thus obtain (A.11) 21 We used Y7 η ∧ c a ∧ ρ 2 = Sa η ∧ ρ 2 .
where we introduced the set of basic R-charges (2.3). Notice that d a=1 ∆ a (b i ) = 2. Note also that d a=1 n a = n 1 hence (4.5) is correctly satisfied. Evaluating the entropy functional (4.6) and using the plus sign in (A.11), we find , (A.12) thus reproducing (4.12).

A.3 Supersymmetry conditions: mesonic twist
The mesonic twist is characterized by the condition We now prove that there exists a solution to the set of equations (4.4), compatible with (A.13), such that We will use repeatedly the identity [42]  Imposing d a=1 ∆ a = 2 and using (A.6), we fix the value of the Kähler parameter λ , (A. 18) which depends on the choice of the Reeb vector through Vol S (Y 7 ). Furthermore, we discover that the ∆ a are actually independent of the fluxes n a and are given by We can also evaluate the on-shell value of the master volume (see (4.7)). It reads which is actually the simplest way of comparing the two sides of (2.2). Let us now move and solve the equations (4.4). We already used the first equation in order to find λ. The other equations can be written as where we used the identity [42]

B Explicit expressions of master volumes
In this appendix we collect the expressions for the master volume (4.2) of the toric examples discussed in section 5.