Free energy v.s. Sasaki-Einstein volume for infinite families of M2-brane theories

We investigate infinite families of 3d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = {2} $\end{document} superconformal Chern-Simons quivers with an arbitrarily large number of gauge groups arising on M2-branes over toric CY4’s. These theories have the same matter content and superpotential of those on D3-branes probing cones over La,b,a Sasaki-Einstein manifolds. For all these infinite families, we explicitly show the correspondence between the free energy F on S3 and the volume of the 7-dimensional base of the associated CY4, even before extremization. Symmetries of the toric diagram are exploited for reducing the dimensionality of the space over which the volume of the Sasaki-Einstein manifold is extremized. Similarly, the space of trial R-charges of the gauge theory is constrained using symmetries of the quiver. Our results add to those existing in the literature, providing further support for the correspondence. We develop a lifting algorithm, based on the Type IIB realization of these theories, that takes from CY3’s to CY4’s and we use it to efficiently generate the models studied in the paper. Finally, we show that in all the infinite families we consider F2 can be expressed, even off-shell, as a quartic function in R-charges associated to certain 5-cycles. This suggests that a quartic formula on R-charges, analogous to a similar cubic function for the central charge a in 4d, exists for all toric toric CY4’s and we present some ideas regarding its general form.


JHEP09(2012)034
accumulate evidence that not only supports the application of the localization ideas to the determination of the free energy in theories with reduced SUSY, but also validates the gauge theories we consider as the correct theories on M2-brane over the corresponding CY 4 's. This paper is organized as follows. In section 2, we review the computation of the volume of Sasaki-Einstein 7-manifolds at the base of toric CY 4 cones, the calculation of the free energy of SCFT 3 's, and the correspondence between gauge theory, geometry and dimer models. Section 3 discusses the L a,b,a k infinite family of gauge theories, which are the main focus of the paper. These theories have the same quiver and superpotential of L a,b,a models in 4d [52][53][54] and, in addition, CS couplings encoded in k. Section 4 is devoted to the Type IIB realization of these theories and introduces an algorithm that lifts the cone over L a,b,a to the toric CY 4 that corresponds to the mesonic moduli space of the CS quiver. The lifting algorithm is used in section 5 to generate infinite classes of models, for which the agreement between the geometric and field theoretic determinations of the free energy is established. In section 6 we show that, in all the infinite classes of models considered in the paper, it is possible to express the free energy as a quartic function of the R-charges of extremal perfect matchings, even before extremization. We present some thoughts about a general expression for such a quartic function. In section 7, we show that the free energy is invariant for certain toric duals obtained by permuting 5-branes in the Type IIB construction of the L a,b,a k models. We conclude in section 8. We also include two appendices, which discuss the symmetries of the toric diagrams under consideration, and investigate the applicability of the proposed quartic formulas on R-charges for the free energy for some non-L a,b,a k theories.

Some background
In this section we review some topics we will later use throughout the paper.

Sasaki-Einstein volumes
We are interested in the quiver gauge theory on the worldvolume of M2-branes probing a CY 4 that is real cone over a 7-dimensional Sasaki-Einstein (SE) manifold Y 7 . The volume of Y 7 is expected to control the number of degrees of freedom of the gauge theory. For toric CY 4 's, this volume can be computed from the toric diagram in terms of the Reeb vector b = (b 1 , b 2 , b 3 , b 4 ) [56], which is a constant norm Killing vector field commuting with all the isometries of the SE manifold.
There is a one-to-one correspondence between extremal perfect matchings, i.e. corners, of the toric diagram and a basis of 5-cycles Σ i in the base over which M5-branes can be wrapped. 1 The R-charge of a single M5-brane wrapped over Σ i is given by Vol(Σ i ) Vol(Y 7 ) . (2.1)

JHEP09(2012)034
This is a function of the Reeb vector b, and the exact superconformal R-charge is obtained by extremizing the function Z MSY defined as Vol(Σ i ), (2.2) where d is the number of corners of the toric diagram. In terms of Z MSY , the volume of Y 7 and the R-charges of extremal perfect matchings are 3) The volumes Vol(Σ i ) can be calculated from the toric diagram thanks to the algorithm introduced in [56], extended to CY 4 's in [17]. Every point in the toric diagram is given by a 4-vector that, due to the Calabi-Yau condition, can be taken to the form v i = (ṽ i , 1), withṽ i a 3-vector. Considering the counterclockwise sequence w k , k = 1, . . . , n i of vectors adjacent to a given vector v i one has where · indicate column 4-vectors and ·, ·, ·, · is the determinant of the resulting 4 × 4 matrix.

Free energy
We now briefly review the calculation of the free energy in 3d, vector-like, CS quivers in the large-N limit. The free energy is computed in terms of the partition function on a 3-sphere Z S 3 as F = − log |Z S 3 | . (2.5) The partition function has been calculated in [33,34] by exploiting the localization technique [27], which reduces it to a matrix integral. For a gauge group G, with CS level k, and matter (by which we mean chiral multiplets) in the representation R of the gauge group with quantum scaling dimension ∆, one has The integral is performed over the Cartan subgroup of the gauge group. The first exponential corresponds to the CS and monopole contributions. The determinants come from the 1-loop contributions of the vector multiplet and the matter fields. For N = 2, the 1-loop determinant of matter fields is expressed in terms of the function l(z), which is defined through its derivative as follows and fixing the integration constant such that l(0) = 0.

JHEP09(2012)034
In this paper, we are interested in computing (2.6) in the large-N limit of vector-like quiver gauge theories with gauge group G = a U(N ) ka and a k a = 0. The integral is dominated by the minimum of the free energy. One can distinguish two contributions to the equations of motion, so called long and short range forces. Long range forces cancel in this class of models and only the short range ones contribute [30,41,43]. The eigenvalue λ i (a) of the a-th gauge group scales as where x and y are real [43]. The real part of (2.8) becomes dense, with density ρ(x), while the imaginary part becomes a continuous function of x, y (a) i → y a (x). The free energy follows from the saddle point equations ∂ λ F = 0 [43]. The relevant contributions for the case of vector-like theories with bifundamental and adjoint matter and a k a = 0 are where the first equation is the CS contribution, the second one is the contribution of a bifundamental-antibifundamental pair connecting the a-th and the b-th nodes, and the last one is the contribution of an adjoint field. We have defined ∆ In the partition function one should take into account the diagonal monopole charge, which is given by ∆ m = ∆(T ) − ∆(T ), where T andT are the diagonal monopole and antimonopole operators. Since vector-like models are charge conjugation invariant, ∆(T ) = ∆(T ), and we can set ∆ m = 0. The bifundamental contribution is only valid when δy ab = y a − y b is in the regime |δy ab + π∆ − ab | ≤ π∆ + ab . The leading contribution to the free energy in the large-N limit is then obtained by extremizing the free energy functional over ρ and y a while imposing the normalization of ρ.
As shown in [29,43], building on results from [28], the supergravity scaling N 3/2 [5] is recovered and the free energy matches the volume computation from AdS/CFT for theories with N > 2 SUSY. The N = 2 case is more involved, because R-charges of matter fields usually differ from the classical value ∆ = 1/2. Indeed, the exact superconformal R-charges is obtained by extremizing the free energy itself [33].
Some examples of the agreement between the field theory computation of the free energy and the geometric calculation of volumes have been presented in [41-43, 48, 49]. One of the main goals of this paper is to extend this matching to infinite classes of theories with arbitrarily large number of gauge groups, in the spirit of similar tests performed in the context of the AdS 5 /CFT 4 correspondence [51][52][53][54]. Some infinite families of models, consisting of flavored quivers with one or two gauge groups and necklace quivers with N ≥ 2 SUSY, have already been considered in the literature [30,31,43,55].

JHEP09(2012)034
The general conjecture is that the free energy of the gauge theory on S 3 is related to Vol(Y 7 ) via . (2.10) We will later see that, in an infinite number of examples, the previous expression holds even off-shell, i.e. even before maximizing the free energy or minimizing the volume.

Geometry, dimer models and R-charges
In this paper we will focus, as we will discuss in greater detail in section 3, on theories with the same quivers of 4d parents and with additional CS couplings for gauge groups. This class of theories can be encoded in terms of brane tilings [53,67], as originally studied in [14]. Their mesonic moduli space is most efficiently described in terms of perfect matchings of the tiling, which are in one-to-one correspondence with the gauged linear sigma model (GLSM) fields in the toric construction of the moduli space, i.e. they map to points in the toric diagram. 2 The mapping between chiral fields in the quiver X i and perfect matchings p α is given by where c is the total number of perfect matchings, and P iα is equal to 1 if the edge in the brane tiling associated to the chiral field X i is contained in p α and zero otherwise.
A prominent role is played by the subset of extremal perfect matchings, i.e. those corresponding to corners of the toric diagram, which we callp µ , µ = 1, . . . , d. The gauge theory contains a U(1) R × U(1) 3 F × U(1) n B B global symmetry group, where F and B indicate flavor and baryonic symmetries, and n B is a positive integer that depends on the underlying geometry. Extremal perfect matchings are the only ones with non-trivial charges under these symmetries [58]. In other words, the global U(1) symmetries of all chiral fields in the quiver are determined by theirp µ content. It is then useful to construct a reduced ma-trixP , which is simply a restriction of P to the columns associated with extremal perfect matchings. Its entries are given byP Consider any of the global U(1) symmetries, under whichp u has charge a µ . The charge of a chiral field is then given by (2.14)

JHEP09(2012)034
In the case of the R-symmetry, the charges of the extremal perfect matchings are constrained by d µ=1 a µ = 2. For other U(1) symmetries, the constraint is d µ=1 a µ = 0. In most of the theories considered in this paper, we will use the symmetries of the quivers under consideration, which will be discussed in section 3, to simplify the computation of the free energy. Having said that, we would like to emphasize that the ideas presented in this section provide an alternative way of organizing this calculation according to the following two steps: • Use (2.14) to parametrize R-charges of matter fields in terms of those of extremal perfect matchings. A corollary of this parametrization is that symmetries of the toric diagram that reduce the number of independent R-charges of extremal perfect matchings also result in a lower dimensional space of R-charges for the chiral fields in the quiver.
• Maximize the free energy over the resulting lower dimensional space.
Let us explain in more detail what we mean by using symmetries of the toric diagram to restrict a field theory computation. We do not have in mind a top-down perspective, which indeed is logically incorrect for our goal, in which we start from a toric CY 4-fold with some symmetries and assume they survive at the level of the quiver. This reasoning has a potential drawback: additional elements, such as torsion fluxes (see e.g. [62][63][64]), might be present in the M-theory construction such that the full symmetry of the original geometry does not subsist in the resulting gauge theory. On the contrary, in all the theories we consider, we have explicitly checked that we recover the corresponding toric geometry via a purely field theoretic computation: the calculation of the moduli space. We can then make the rather simple and natural assumption that the field theory exhibits the symmetries of its moduli space. We have verified in all our models that the two step procedure based on perfect matchings and symmetries of the toric diagram outlined above reproduces the free energy computation using quiver symmetries we present in section 5.
3 L a,b,a k theories In section 1, we reviewed the extent to which quiver CS theories have been tested as theories on M2-branes and mentioned the difficulties encountered when trying to do so. In order to remain on the conservative side, we will focus in this paper in theories with toric, non-chiral, 4d parents. These parents can be fully classified using toric geometry. They correspond to all toric Calabi-Yau 3-folds (CY 3 ) without compact 4-cycles, i.e. those with toric diagrams without internal points. All geometries satisfying this condition are C 3 /(Z 2 × Z 2 ) and the infinite L a,b,a family. Figure 1 shows the toric diagram for the cones over L a,b,a manifolds, consisting of two parallel lines of (a + 1) and (b + 1) points, respectively.
The corresponding gauge theory can be taken to the form given in figure 2 [52][53][54]. The superpotential is given by where X i,j indicates a bifundamental field connecting nodes i and j and X i,i corresponds to an adjoint of node i. The nodes in the quiver are identified according to a + b + 1 ≡ 1.  Marginality of the superpotential, which is necessary for conformal invariance, requires that all superpotential terms have R-charge equal to 2, restricting the space of trial R-charges entering the extremization of the free energy. This space can be reduced even further by exploiting the symmetries of the gauge theory. Since both the quiver and the superpotential for this class of theories are symmetric under the exchange X i,i+1 ↔ X i+1,i , it is natural to assume that the saddle point has ∆(X i,i+1 ) = ∆(X i+1,i ). With this further constraint, the space of trial R-charges becomes one dimensional and can be parametrized as shown in figure 2. This parametrization will be used in section 5 to deal with some of the more involved examples. The previous symmetry argument is the same one originally introduced in [43] for N = 2 and N = 3 necklace quivers and has been also implicitly used for some examples in [41]. 3 We will add to these models CS couplings for the gauge groups, which can be arranged in a vector k = (k 1 , . . . , k a+b ). We will often use the notation k = (k 1 , . . . , k b−a ||k b−a+1 , . . . , k a+b ), (3.2) where we use a double line to separate nodes with and without an adjoint field. We denote the resulting theories L a,b,a k . In section 4, we introduce an algorithm that determines how the inclusion of k lifts the CY 3 given by the real cone over L a,b,a to a CY 4 . The N = 3 necklace quivers of [30,31,55] have the same matter content of our models for a = b, but additional quartic superpotential interactions. N = 2 deformations of these JHEP09(2012)034 theories, obtained by integrating-in adjoint fields and adding polynomial superpotential interactions for them, have also been considered [43].

Lifting Calabi-Yau 3-folds to Calabi-Yau 4-folds
By now, it is well-known that candidates for 3d theories on M2-branes over toric CY 4 's can be constructed by starting from theories with the same quivers and superpotentials of 4d theories on D3-branes over toric CY 3 's, to which we refer as "parents", and adding CS terms for the gauge groups. This strategy was exploited soon after the introduction of the ABJM model for generating potential M2-brane theories with reduced SUSY [8,14,15,18,20]. The 3d toric diagram of the "uplifted" CY 4 is such that it reduces to the 2d one of the parent CY 3 when projected along a direction determined by the CS levels. From the perspective of the computation of moduli spaces, this additional projection arises from an extra D-term constraint that is imposed in the 4d theories. Models in which such a projection is not possible, and hence do not descend from a 4d parent, have also been proposed [17-19, 21, 22].
In what follows, we will focus our discussion on L a,b,a k theories. The most general uplift of figure 1 into a 3d toric diagram corresponds to the two lines turning into convex polygons living on parallel planes, as sketched in figure 3.

A lifting algorithm
In this section we introduce a general algorithm for lifting cones over L a,b,a to CY 4 's by appropriate choices of CS levels in the corresponding quivers. We will exploit this procedure in section 5 for generating interesting classes of models. The method is a specialization of the ideas in [15] to L a,b,a theories.
A useful starting point is the Type IIB brane realization of L a,b,a k theories. They can be engineered in terms of an elliptic model consisting of a stack of N D3-branes with one of their worldvolume directions compactified on a circle, suspended between a set of (b + a) (1, p i ) 5-branes. An (1, p i ) 5-brane is a bound state of one NS5-brane and p i D5-branes.  The p i integers determine the CS levels in the quiver according to the following expression We split the 5-branes, i.e. the integers p i , into two sets: Q α , α = 1, . . . , a, and P β , β = 1, . . . , b. The branes in the configuration are extended as follows Brane 0 1 2 3 4 5 6 7 8 9 D3 The SCFT lives in the (0, 1, 2) directions common to all the branes. The D3-branes are, in addition, extended along x 6 , which is compactified on a circle. The (1, Q α ) 5-brane is a bound state of the NS5 α and Q α D5 α branes and extends along (0, 1, 2, [37] θα , 4, 5). Similarly, the (1, P β ) 5-brane is a bound state of the NS5 β and P β D5 β branes and extends along (0, 1, 2, [37] θ β , 8,9). The final configuration is shown in figure 4. In order to reproduce the quiver in figure 2, we distribute the 5-branes on the circle as follows. First we put (b−a) (1, P β ) 5-branes and then we alternate the remaining a (1, P β ) and a (1, Q α ) branes. It is possible to reorder the 5-branes along the x 6 circle, which results in dual gauge theories. The D3-branes stretched between each pair of 5-branes give rise to a gauge group in the quiver. Each 5-brane is associated to a pair of bifundamental chiral fields as shown in figure 5. In addition, we have an adjoint chiral field for each consecutive pair of 5-branes of the same type.
Every point in a 2d toric diagram corresponds to a perfect matching in the associated brane tiling, or a collection of them. In order to connect geometry to perfect matchings,

JHEP09(2012)034
we recall that the 2d (3d) toric diagram describes the moduli space of the 4d (3d) gauge theory. For 4d theories defined by brane tilings, the correspondence between points in the toric diagram and sets of perfect matchings was proved in [65], to where we refer the reader for details of the proof, which involves interesting ideas such as a connection between the 4d theory and an auxiliary gauge theory living on the nodes of the bipartite graph. The computation of the moduli space for the class of 3d theories we focus on in this paper only differs from the 4d case by the fact that one linear combination of the D-term equations, determined by the CS levels, is not imposed [13,14]. The CS couplings of all infinite families considered in section 5 will be chosen in such a way that the identification between points and perfect matchings extends without changes to the 3d toric diagrams. We continue our discussion under this assumption.
The toric diagram for L a,b,a theories was given in figure 1. As we will now explain, the P β 's control how the top line of the toric diagram is lifted to a plane. Similarly, the Q α 's determine the lift of the bottom line. The degeneracies of perfect matchings associated to points in the toric diagram are b µ and a ν , where µ = 0, . . . , b and ν = 0, . . . , a run over the points on the bottom and top row respectively.
Perfect matchings correspond to certain collections of edges in the associated brane tilings, which map to sets of chiral fields in the quiver. Indeed, thinking in terms of the quiver provides a clear visualization of these multiplicities. Let us first consider the (b + 1) points in the lowest line of the toric diagram. The perfect matching for µ = 0 consists of all the fields in the quiver with the arrows pointing from right to left. The perfect matchings for the µ-th point correspond to reversing the orientation of µ of the fields, giving rise to the multiplicity described by the binomial coefficients. Repeating this procedure, we reach the µ = b point in which all the fields are arrows in the quiver point from left to right. The line with (a + 1) is constructed in the same way, by using the fields labeled by Q α , but also including the adjoint fields.
The new mesonic direction in the CY 4 is determined by the P β or Q α charges. In order to see this, it is useful to define a new integer quantity Q CS associated to every chiral multiplet. Q CS is defined such that it is equal to p i (i.e. equal to P β or Q α ) for every bifundamental field pointing from left to right in the quiver and zero otherwise.
The value of Q CS associated to a perfect matching is obtained by summing the contributions of all chiral fields it contains. The new mesonic direction in the CY 4 is then determined following a simple prescription: Every perfect matching in the 2d toric diagram gets a shift into the third dimension equal to its Q CS value.
Let us first consider the effect of this rule on the lowest line. The first point, µ = 0, has Q CS = 0 and hence does not move. The other endpoint of the line, µ = b, gets the maximum possible shift, equal b β=1 P β . The intermediate points are not only shifted but they can also be split, depending on the total Q CS of each of the perfect matchings associated to a JHEP09(2012)034 given point. The expansion of the top line of the toric diagram into the third dimension follows the same prescription, with the values of Q CS determined by Q α . Positivity of the P β and Q α charges guarantees the convexity of the resulting 3d toric diagram.
It is possible to take the theory to a conventional form in which the P β are arranged in increasing order and similarly for the Q α . In order for all perfect matchings to get different shifts, they must have different values of Q CS . Then, a necessary condition for fully lifting the degeneracy of points in the bottom and top lines of the toric diagram is that all P β and ll Q α are different, respectively. The algorithm we have just described leads to a broad range of results. For example, in the simple case in which P β ≡ P for all β and Q α = Q (with Q = P ) for all α, the two lines are lifted to lines, giving rise to the toric diagram for C 2 /Z a × C 2 /Z b . This is the situation considered in [8]. On the other end of the spectrum, we have cases in which every internal point of the lines is expanded and generates two new corners. Together with the four external points of the original 2d toric diagram, they lead to a toric diagram with 2(a+b) corners. A necessary condition for this to happen is that all the inequalities in (4.3) are strict.
Let us now discuss in further detail the lift of internal points inside the two lines in the L a,b,a toric diagram. We discuss the bottom line, the top line behaves in a similar way. The µ = 1 point expands into a segment in which, if we sort the P β 's as in (4.3), the bottom and top endpoints are shifted by P 1 and P b units, respectively. I.e., this point turns into segment of length (P b − P 1 ). 4 The µ = b − 1 point also expands into a segment, with its endpoints shifted by b−1 β=1 P β and b β=2 P β . Once again, the length of the resulting segment is (P b −P 1 ). The same phenomenon occurs for all other internal point, i.e. the µ-th and (b − µ)-th points turn into segments of equal length. The maximal length is attained for the (b − 1)/2-th and the (b + 1)/2-th points for odd b, and for the b-th point for odd b.
An example. Let us illustrate the previous ideas with an explicit example. Consider the L 2,5,2 theory and take which, following (4.1), generates the following CS levels for the quiver (4.5) Figure 6 shows the result of applying the lifting algorithm. We see the, in this case partial, lift of degeneracies of points in the toric diagram and the appearance of new corners.

Infinite families
In this section we present various infinite families of gauge theories and the associated CY 4 's obtained from L a,b,a models by the lifting algorithm introduced in section 4. In all JHEP09(2012)034 these cases we show the volume computation and the gauge theory calculation of the free energy agree. Interestingly, this agreement holds even off-shell.
We present geometries whose toric diagrams have 4, 6 and 8 extremal perfect matchings. The models presented towards the end are interesting because they give rise to non-trivial R-charges. We start our investigation of infinite classes of models by considering geometries whose toric diagrams, shown in figure 7, have four corners given by the vectors These geometries are C 2 /Z a × C 2 /Z b orbifolds and their dual gauge theories were introduced and investigated in [8]. They are obtained via the lifting algorithm by setting, for example, The resulting CS couplings are Geometric computation. The Z MSY function is obtained by summing over the volumes of the 5-cycles corresponding to the extremal points in the toric diagram. They are functions of the Reeb vector b and Z MSY , which corresponds to the sum of these contribution, becomes where we have set, as in all the examples that follow, b 4 = 4. Assigning an R-charge ∆ i to each of the four extremal points v i , these R-charges, corresponding to the charges of the extremal perfect matchings, can be expressed in terms of the Reeb vector as without loss of generality we consider ∆ 1 < ∆ 4 and ∆ 3 < ∆ 2 . The volume function becomes Under the constraint i ∆ i = 2, the volume is minimized for ∆ i = 1/2. We have included an extra k factor with respect to (2.3) in the denominator of the volume due to an additional Z k orbifold action on the moduli space, where k = gcd({k a }) [5]. This factor will also be present in the volumes of all the examples that follow.
Free energy computation. Let us now compute the free energy of this class of models.
Recall that a perfect matching is a subset of edges such that every vertex in the brane tiling is an endpoint of precisely one edge in the set. Using the dictionary between brane tilings and gauge theories [67], a perfect matching can be interpreted as a subset of the chiral fields in the quiver such that it contains exactly one field for each superpotential term. The four extremal perfect matchings can be simply represented in terms of the quiver as shown in figure 8. It is then straightforward to determine the matrix P iµ and the R-charges of chiral fields in terms of those of the extremal perfect matchings. We show the result in figure 9.   Figure 9. R-charges of chiral fields in terms of the R-charges of extremal perfect matchings. We have defined ∆ 34 The free energy is given by the sum of the CS and the matter field (bifundamentals and adjoints) contributions. As we already anticipated we are setting the monopole charge to zero even off-shell. The CS contribution to the large-N free energy is The matter contribution is where e(B) and e(W ) refer to the black and white nodes in the quiver as shown in figure 9. We moreover denote e(B) ′ the subset of the black nodes containing adjoint fields. We have defined ∆ (±) i,j = ∆ i,j + ∆ j,i . We solve the Euler-Lagrange equations with the δy variables subject to the following constraint i,i+1 = ∆ 3 − ∆ 4 . The eigenvalue distribution is reduced to a piecewise function over three connected domains as follows Integrating over the piecewise domain and imposing the normalization on ρ we obtain We consider a family with the toric diagram shown in figure 5.14, whose corners are given by the following vectors (5.14) The charges associated to the lifting algorithm are This family contains and generalizes the D3 model considered in [17,18], which corresponds to L 1,2,1 (k||−k,0) . The symmetries of the toric diagram are useful for constraining the space over which Vol(Y 7 ) is extremized. The toric diagram in figure 10 has a Z 2 symmetry that maps where we have denoted p i the perfect matching associated to the vector v i . This is not the full symmetry of the geometry. In fact, we can make its symmetry more manifest by acting with an appropriate SL(4, Z) transformation. This approach will be also exploited in the examples that follow, for which the symmetric versions of the toric diagrams and the necessary SL(4, Z) transformations are summarized in appendix A. For simplicity, let us first assume (b − a) is even and consider the following SL(4, Z) matrix whose corresponding toric diagram is shown in figure 11. We conclude that there is an additional Z 2 symmetry mapping It is also straightforward to find an SL(4, Z) transformation that makes these additional symmetries manifest for odd (b − a). Geometric computation. For the original toric data in (5.14), Z MSY can be written as As a result of the symmetries in (5.17) and (5.20), the R-charges of perfect matchings are identified as follows which implies we can parametrize the components of the Reeb vector as and the volume becomes The volume is minimized for ∆ = 1/3.
Free energy computation. Before computing the free energy, we specify the perfect matchings as collections of chiral fields in the quiver. The six extremal perfect matchings for this class of models are represented in terms of the quiver as shown in figure 12.
The CS contribution to the free energy in this case is The sum over matter fields can be organized as follows. First, we distinguish three different kinds of δy's: "red", "green" and "blue" as in figure 13. Notice that we have enforced the constraint δy = 0 by drawing δy b+a,1 in black. Moreover one can check that the equations of motion give the same value to the δy's with the same color. Using the ansatz discussed section 3, we parametrize the R-charges in terms of a single parameter ∆. Then, the matter contribution to the free energy becomes The Euler-Lagrange equations give Integrating this distribution, we have (5.28) in agreement with the geometric computation (5.24). As can be easily observed in (5.15) b has to be greater than a otherwise all the CS levels vanish and the model is not associated to a SCFT in 3d. We now consider models with toric diagram given in figure 14. The six corners of the toric diagram have coordinates given by the matrix This class of theories contains and generalizes the modified SPP model studied in [17], which corresponds in this notation to L 1,2,1 (−2||1,1) , for which the agreement between the free energy and the volume has been shown in [41].

Family
From figure 14, we see there is a symmetry that exchanges As in the previous example, we can apply an SL(4, Z) transformation to the toric diagram, that makes the additional symmetry under the interchange of manifest, as explained in appendix A.
Geometric computation. The volumes are written in terms of the components of the Reeb vector and we have (5.34) Using the symmetries in (5.32) and (5.33), we have For ∆ 2 = ∆ the R-charges of the extremal perfect matchings can be parametrized as and the volume with this parametrization becomes .

(5.39)
We see that this infinite family of theories already generates rather non-trivial R-charges. This is also the case for the families with eight corners in the toric diagram discussed in the next section.
Free energy computation. On the field theory side the six extremal perfect matchings are associated to chiral fields as in figure 15.
Parametrizing R-charges as in figure 2, we can now calculate the large-N free energy and compare it to the volume. The large-N free energy has the following contribution from the CS term while matter fields give

JHEP09(2012)034
where e(B) and e(W ) refer to white and black nodes as in figure 9. Then we impose the constraint i∈e(B) δy i + i∈e(W ) δy i = 0 and we compute the Euler-Lagrange equations for ρ and δy . We find Out of this region we have By integrating the piecewise function ρ over the domain where it is non-vanishing we obtain which agrees with the result we got from the geometry. We continue our exploration considering a more involved family of geometries with toric diagrams with eight extremal points. The 4-vectors giving the corners of the toric diagram are given in matrix form in (5.46). The corresponding toric diagram is given in figure 16.
This class of models is generated by the lifting algorithm by choosing the P (β) and the Q(α) as Type Multiplicity Value From the discussion in appendix A, we see that this model has symmetries that map Geometric computation. The Z MSY function in terms of the Reeb vector is By exploiting the symmetries of toric diagram (5.49), we can set and parametrize the components of the Reeb vector as where ∆ has a simple relation to the R-charges of fields in the quiver as it will be shown below. The volume function then becomes Extremizing it, we obtain  Figure 17. Perfect matchings associated to the eight corners of the toric diagram given in (16).
Red arrows indicate the chiral fields associated with edges in the perfect matching.
where we have defined ). (5.56) As in the previous family of theories, these models exhibit highly non-trivial values of the R-charges.
Free energy computation. We will now recover this complicated structure from the field theory computation of the free energy at large-N . The eight extremal perfect matchings are associated to chiral fields as in figure 17. Following the R-charge parametrization of figure 2, the CS contribution to the free energy is (5.57) and the matter contribution is given by the general expression (5.41). After enforcing the constraint b+1 i=1 δy i,i+2 = 0, we solve the Euler-Lagrange equations to obtain The free energy is then  In this section we study a second family with 8 extremal points. This family generates the toric diagram in figure 18. We can arrange the 4d vectors generating the diagram in matrix form as follows This geometry follows from L a,b,a by using the lifting algorithm with Type Multiplicity Value The resulting CS levels are k = (−k 1 , 0, . . . , 0, k a+b−2X , 0, . . . , 0, −k a+b−2Y , k a+b−2Y +1 , 0, . . . , 0), (5.62) where the subindices indicate the position of the non-zero entries in the vector k, which only take values ±k. We will focus on the case in which b > a and X > Y . In the notation of (3.2), we can thus distinguish two possibilities The case b > a > Y > X can be studied in a completely analogous way. The symmetries of this theory, as explained in appendix A, map

JHEP09(2012)034
Geometric computation. For this class of models, Z MSY takes the form .
The symmetries in (5.64) imply that and the volume becomes .
As in previous examples, it is straightforward to find the value of ∆ that minimizes the volume analytically. The resulting expression is not terribly illuminating, so we do not quote it here.
Free energy computation. The eight extremal perfect matchings are associated to chiral fields as in figure 19. 5 Given the parametrization in figure 2, the free energy for the gauge theory can be written by distinguishing two different δy's as where F bif and F adj are the contributions to the free energy of a couple of bifundamental anti-bifundamental and of an adjoint field. By computing the saddle point equations we find which matches the volume computation.

Free energy as a quartic function in R-charges
In this section we would like to discuss the existence of a geometrical formula capable of reproducing the free energy in terms of the charges of the perfect matchings similar to the one derived in [58] for SCFT 4 's.  Figure 19. Perfect matchings associated to the eight corners of the toric diagram given in (18), with X = Y = 1. Red arrows indicate the chiral fields associated with edges in the perfect matching.

4d preliminaries
Before continuing our study of the free energy of SCFT 3 's, it is useful discuss related questions in 4d. The number of degrees of freedom of an N = 1 SCFT in 4d is counted by the central charge a, which can be determined in terms of superconformal R-charges [35,66] as follows a = 3 32 (3TrR 3 − TrR). (6.1) Furthermore, for SCFTs on D3-branes, TrR = 0 in the large-N limit, i.e. to order N 2 , and (6.1) becomes purely cubic. In [56,58,59,61], it has been shown that a also admits a similar cubic expression based on the underlying geometry, which takes the form where the v i are the 3-dimensional vectors defining the extremal points of the two dimensional toric diagram and the R i are the R-charges of the perfect matchings associated to v i . While (6.2) is written in terms of quantities that allow a direct contact with geometry, it is important to keep in mind that perfect matchings are indeed identified with GLSM fields which, in turn, can be found in purely field theoretic terms starting from the gauge theory and computing its moduli space. Equation (6.2) can also be obtained by rewriting the inverse of the volume of the 5d Sasaki-Einstein base of the corresponding toric CY 3 , which takes the form where the v i vectors are the 3-vectors with the coordinates of extremal points in the toric diagram and b = (b 1 , b 2 , 3) is the Reeb vector. Due to the Calabi-Yau condition, we can take v i = (ṽ i , 1), withṽ i a 2-vector. ·, ·, · is the determinant of the resulting 3 × 3 matrix.

Free energy in 3d
In section 2.2, we have explained how to compute the free energy of SCFT 3 's. Furthermore, we have shown that its value agrees with the geometric computation in various infinite classes of theories. It is natural to wonder whether a simple expression for the free energy, similar in spirit to (6.1) exist in 3d. The main obstacle for going into this direction is the absence of anomalies associated to continuous symmetries in 3d. Having said this, the similarity between the volume formulas (2.4), (2.3) and (6.3) suggest that an expression in terms of R-charges of perfect matchings, i.e. of GLSM fields, analogous to (6.2) might exist. The most naive generalization of (6.2) to 3d takes the form where we have used ∆ i instead of R i to match the notation we have been using for SCFT 3 's.
Remarkably, it has been observed in [48] that this formula reproduces the free energy of several theories. Even in specific models for which (6.4) does not give the correct result, it has been possible to introduce additional terms such that the free energy is still given by a quartic formula in the R-charges of extremal perfect matchings. Interestingly, in all the theories considered in [48] the corrections to (6.4) seem to be connected to the existence of internal lines in the toric diagram, i.e. lines connecting extremal points that do not live on edges or faces. We now go over all the classes of models considered in section 5 and show that, in all of them, the free energy can be written as a quartic function of the R-charges of extremal perfect matchings. It is important to emphasize that this agreement holds off-shell, i.e. even before extremizing the free energy. For the first two families of L a,b,a (0,...,0||k,−k,...,k,−k) and L a,b,a (k,0,...,0||−k,0,...,0) theories, discussed in sections 5.1 and 5.2.1, the free energy is exactly reproduced by (6.4). Geometrically, these two families distinguish themselves from the others in that their toric diagrams do not contain internal lines, i.e. all lines connecting corners of the toric diagram live on edges or external faces.

Towards a general quartic formula
The previous examples lead us to some conjectures regarding the possible structure of a general quartic formula. It appears that there are two possible types of corrections to (6.4), which arise in the presence of internal lines in the toric diagram: 1) A correction proportional to ∆ 2 i ∆ 2 j , whenever the line connecting extremal points i and j of the toric diagram is internal.
2) A correction proportional to ∆ i ∆ j ∆ k ∆ l , whenever the lines connecting the extremal points i and j, and k and l are both internal.
Furthermore, based on the examples, it is possible to conjecture an explicit expression for the numerical coefficient multiplying the corrections of type (1). If a line connecting extremal point intersects an internal triangle, then one takes the product of the volumes of the three possible tetrahedra (V 1 , V 2 and V 3 ) whose vertices are the two endpoints of the line and a pair of vertices of the triangle, and divide it by the product of the volumes of two tetrahedra (V 4 and V 5 ) given by the triangle and each of the endpoints of the line. The corresponding correction to the free energy is of the form JHEP09(2012)034 The (−4) prefactor is universal whenever an internal line intersects a triangle. This prescription admits a nice graphical representation as shown in figure 21 More complicated situations can be obtained by triangulation. For example, if a line line (i → j) crosses a polygon formed by four extremal points of the toric diagram ({v 1 , v 2 , v 3 , v 4 }) we generate four tetrahedra with vertices being i and j and the other two on the polygon, whose volumes are: On the other hand, we construct two other volumes made out of tetrahedra with a single vertex being i or j: In appendix B, we present additional examples beyond the L a,b,a k theories, providing further support for the general ideas advocated in this section. Eventually, we expect a geometric identity that systematically re-expresses the volume function (2.4) as a quartic function in the volumes Vol(Σ i ) of elements in the basis of 5-cycles.

Toric duality and the free energy
A corollary of the lifting algorithm of section 4 is that individual permutations of the P β and Q α subsets of 5-branes lead to gauge theories with different CS couplings but the same CY 4 manifold as their mesonic moduli space. This invariance suggests that the corresponding gauge theories are dual. Theories that share the same toric moduli space, in any dimensions, go under the general denomination of toric duals [68]. In fact, some of these permutations, those that exchange a pair of adjacent P β and Q α branes, indeed correspond to the 3d version of Seiberg duality discussed in [18, 20-22, 69, 70]. It was then shown in [30,48,55,71] that the large-N free energy is preserved under this duality.
We now show that, as expected for dual theories, the free energy is invariant under permutations of P β and Q α branes. Let us start from the CS contribution which, following (2.9), is proportional to

JHEP09(2012)034
Following the separation of the p i into two sets {p i } = {P β , Q α } we also divide the δy i as {δy i } = {δy β , δy α }. The CS contribution then becomes the matter contribution is given by Next, we consider the action of two arbitrary elements S a and S b of the symmetric group acting on Q α and P β , respectively. We see that both (7.2) and (7.4) are preserved if we simultaneously act with the same permutation actions S a and S b on δy α 's and δy β 's. This shows that, as expected from the invariance of the moduli space, the large-N free energy is preserved.

Conclusions
Remarkable progress in understanding SCFT 3 's on M2-branes and in the field theoretic calculation of the number of degrees of freedom in these SCFTs has taken place in recent years. One of the main goals of this paper has been to accumulate a large body of evidence, in the form of infinite classes of theories, explicitly showing the expected agreement [41,43] between the volume of the Sasaki-Einstein horizon of the probed CY 4 cone and the free energy of the dual field theory computed on a round S 3 . The infinite families of models we investigated in section 5 belong to the L a,b,a k class, and their corresponding gauge theories have generically N = 2 SUSY and the same vector-like quivers and superpotentials of D3-branes on real cones over L a,b,a manifolds. These theories also include CS couplings, encoded in the vector k, which dictate how the parent CY 3 manifold is lifted to a CY 4 . Our results provide non-trivial checks of the AdS 4 /CFT 3 correspondence for infinite families of gauge theories and it is a step towards a general proof of the equivalence between the Z MSY -minimization and the F -maximization.
The infinite families we studied were generated with the aid of a lifting algorithm we introduced in section 4, which is based on the Type IIB realization of these theories and allows us to efficiently generate the CY 4 geometries for L a,b,a k theories. Our results are similar to the equivalence between Z MSY -minimization and amaximization in 4d [56]- [61], whose proof for toric theories relies on the existence of a geometric formula for the central charge, a geom , that is cubic in the R-charges of extremal JHEP09(2012)034 perfect matchings [56,58]. This fact follows crucially from the relation between the geometry of extremal perfect matchings and triangle anomalies in field theory [72]. Despite the absence of anomalies in 3d, the similarity of the geometric expression for the horizon volumes between the 3d and 4d cases makes it natural to expect that a geometric expression for the free energy F 2 geom , quartic in the R-charges of extremal perfect matchings, exists in 3d. In section 6, we have shown that this expression exists for all the infinite families of theories we studied. Furthermore, the correspondence is valid even before extremization. Counting with an infinite catalogue of examples has allowed us to make various conjectures regarding the general form of the quartic formula. These ideas were tested in additional, non-L a,b,a k models in appendix B, verifying that they indeed agree with the volume. We find all these results are encouraging and make us expect that it is possible to rewrite the volume formula as a quartic expression in volumes of 5-cycles. It would be very interesting to show that such a formula exists and to give a systematic prescription for writing it based on the toric data.
In the future, it is certainly desirable to prove the equivalence between Z MSYminimization and F -maximization for general toric geometries. A more modest objective is to prove the equivalence within some sub-class of theories, such as the L aba k models. Our results go a long way in this direction, but we had to be specific about the choice of CS levels in order to perform the calculations. It would be interesting to find an efficient procedure for dealing with a generic choice of CS levels. First, one should manage to find the volume of Y 7 for an arbitrary choice gauge theory data: a, b and k. Hilbert Series techniques [22,24] seem to be a promising direction for achieving this goal. On the field theory, one should compute the free energy, i.e. solve the corresponding Euler-Lagrange equations, for a generic distribution of CS levels. Computing the free energy from a Fermi gas, as proposed in [32] for N ≥ 3 theories, is perhaps a more promising approach, since no matrix model techniques are needed.
We conclude with some comments on toric duality. In section 7, we have shown that toric duals generated by permuting 5-branes in the type IIB realization of L a,b,a k theories preserve the large-N free energy without fractional branes, i.e. for all the ranks of the gauge group being equal. It is natural to expect that at finite N the precise ranks of the gauge groups might become important for the duality. This issue can be investigated by rewriting the free energy as in [73], using the formalism of [74]. It should be possible to determine the number of fractional branes required by duality by using the equivalences among the integrals discussed in [74].

A Symmetric toric diagrams
In this appendix we complete the presentation of the symmetric versions of the toric diagrams discussed in section 5, provide the SL(4, Z) transformations necessary to achieve them and discuss the resulting symmetries. Applying this transformation to (5.29), we obtain  to (5.46), we obtain We show the corresponding toric diagram in figure 23 This diagram has two manifest Z 2 symmetries that act as reflections. One of them interchanges The analysis of this family depends on the parities of (X − Y ) and (b − a). The discussion below applies to the case in which both of them are even. Other combinations of parities follow analogously and yield the same symmetries.
Starting from (5.60), we apply the following SL(4, Z) transformation and obtain The associated toric diagram is given in figure 24. It has two manifest Z 2 symmetries. One of them exchanges B Non-L a,b,a k theories and quartic formulas In this appendix, we provide additional evidence supporting our proposals of section 6.4.
To do so, we consider two families of theories that do not fit within the L a,b,a k classification.

JHEP09(2012)034
In the first class of geometries, the toric diagram is given by where X i > 0. These theories have already been studied in [48], where it has been shown that the geometrical free energy is given by (6.4) plus the following correction This correction is associated to an internal line connecting v 4 and v 5 in the toric diagram.
The numerical coefficient of this correction is in perfect agreement with our proposal (6.7).
Using it, we obtain The final set of models we would like to consider has a toric diagram given by In this case, a quartic expression for the free energy also exists, and it is given by (6.4) plus the rather non-trivial correction It is possible to check that the ∆ 2 1 ∆ 2 2 , ∆ 2 3 ∆ 2 4 and ∆ 2 5 ∆ 2 6 terms are indeed in agreement with (6.8), including its (−1) prefactor.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.