Three-sphere free energy for classical gauge groups

In this note, we calculate the $S^3$ free energy $F$ of 3-d ${\cal N}\geq 4$ supersymmetric gauge theories with $U(N)$, $O(N)$, and $USp(2N)$ gauge groups and matter hypermultiplets in the fundamental and two-index tensor representations. Supersymmetric localization reduces the computation of $F$ to a matrix model that we solve in the large $N$ limit using two different methods. The first method is a saddle point approximation first introduced in arXiv:1011.5487, which we extend to next-to-leading order in $1/N$. The second method generalizes the Fermi gas approach of arXiv:1110.4066 to theories with symplectic and orthogonal gauge groups, and yields an expression for $F$ valid to all orders in $1/N$. In developing the second method, we use a non-trivial generalization of the Cauchy determinant formula.


Introduction
In the absence of a perturbative understanding of the fundamental degrees of freedom, one can learn about M-theory only through various dualities. A promising avenue is to use the AdS/CFT correspondence [3][4][5] to extract information about M-theory that takes us beyond its leading (two-derivative) eleven-dimensional supergravity limit. Such progress is enabled by the discovery of 3-d superconformal field theories (SCFTs) dual to backgrounds of M-theory of the form AdS 4 × X [6][7][8][9][10][11][12][13], as well as the development of the technique of supersymmetric localization in these SCFTs [14][15][16] (see also [17]). For instance, computations in these SCFTs may impose constraints on the otherwise unknown higher-derivative corrections to the leading supergravity action.
In this paper we study several 3-d SCFTs, with the goal of extracting some information about M-theory on AdS 4 ×X that is not accessible from the two-derivative eleven-dimensional supergravity approximation. These theories can be engineered by placing a stack of N M2branes at the tip of a cone over the space X. A good measure of the number of degrees of freedom in these theories, and the quantity we will focus on, is the S 3 free energy F defined as minus the logarithm of the S 3 partition function, F = − log |Z S 3 | [18][19][20][21]. At large N , the F -coefficient of an SCFT dual to AdS 4 × X admits an expansion of the form [1,22] F = f 3/2 N 3/2 + f 1/2 N 1/2 + . . . . (1.1) The coefficient f 3/2 can be easily computed from two-derivative 11-d supergravity [1,22] f 3/2 = 2π 6 27 Vol(X) , (1.2) whereas the coefficient f 1/2 together with the higher-order corrections in (1.1) cannot [23,24].
In this paper we will calculate f 1/2 for various SCFTs with M-theory duals.
We focus on SCFTs with N ≥ 4 supersymmetry. In such theories, supersymmetric localization reduces the computation of Z S 3 to certain matrix models [25]. For instance, for the N = 6 ABJM theory [6], which is a U (N ) k × U (N ) −k Chern-Simons matter gauge theory, one has [14,22] where the integration variables are the eigenvalues of the auxiliary scalar fields in the two N = 2 vectormultiplets. This theory corresponds to the case where the internal space X is a freely-acting orbifold of S 7 , X = S 7 /Z k . The integral (1.3) can be computed approximately at large N by three methods: I. By mapping it to the matrix model describing Chern-Simons theory on the Lens space S 3 /Z 2 , and using standard matrix model techniques to find the eigenvalue distribution [22]. This method applies at large N and fixed N/k. To extract f 3/2 and f 1/2 in (1.1) one needs to expand the result at large 't Hooft coupling N/k.
II. By expanding Z S 3 directly at large N and fixed k [1]. In this limit, the eigenvalues λ i andλ i are uniformly distributed along straight lines in the complex plane.
III. By rewriting (1.3) as the partition function of N non-interacting fermions on the real line with a non-standard kinetic term [2]. The partition function can then be evaluated at large N and small k using statistical mechanics techniques.
Using the Fermi gas approach (III), for instance, one obtains [2] Z = A(k) Ai where A(k) is an N -independent constant. From this expression one can extract (1.5) These expressions can be reproduced from the first method mentioned above [22], and f 3/2 can also be computed using the second method [1].
While ABJM theory teaches us about M-theory on AdS 4 × (S 7 /Z k ), it would be desirable to calculate F for other SCFTs with M-theory duals, so one may wonder how general the above methods are and/or whether they can be generalized further. So far, the first method has been generalized to a class of N = 3 theories obtained by adding fundamental matter to ABJM theory [26]. 1 The second method can be applied to many N ≥ 2 theories with Mtheory duals [18,[27][28][29][30][31], but so far it can only be used to calculate f 3/2 . The third method has been generalized to certain N ≥ 2 supersymmetric theories with unitary gauge groups [32]; in all these models, Z S 3 is expressible in terms of an Airy function.
We provide two extensions of the above methods. We first extend method (II) to calculate the k 3/2 contribution to f 1/2 in (1.5), and provide a generalization to other SCFTs. We then extend the Fermi gas approach (III) to SCFTs with orthogonal and symplectic gauge groups.
This method allows us to extract f 1/2 exactly for these theories, and we find agreement with results obtained using method (II). The extension of the Fermi gas approach to theories with symplectic and orthogonal gauge groups requires a fairly non-trivial generalization of the Cauchy determinant formula that we prove in the Appendix. This formula allows us to write Z S 3 as the partition function of non-interacting fermions that can move on half of the real line and obey either Dirichlet or Neumann boundary conditions at x = 0. We find that the result for Z S 3 is again an Airy function.
The rest of this paper is organized as follows. In Section 2 we describe the field theories that we will consider in this paper. These theories are not new. They can be constructed in type IIA string theory using D2 and D6 branes, as well as O2 and O6 orientifold planes. In Section 3 we extend the large N expansion (II) to the next order. In Section 4 we extend the Fermi gas approach (III) to our theories of interest. We end with a discussion of our results in Section 5. We include several appendices. In Appendix A we determine the moduli space of vacua using field theory techniques. Appendix B provides a brief summary of the Fermi gas approach [2]. Appendix C contain some details of our computations. Lastly, in Appendix D we prove the generalization of the Cauchy determinant formula used in the Fermi gas approach. The two-index tensor representation can be the adjoint in the case of U (N ), or it can be a rank-two symmetric or anti-symmetric tensor representation in the other cases.
These SCFTs can be realized as low-energy effective theories on the intersection of various D-branes and orientifold planes in type IIA string theory as follows. In all of our constructions, we consider D2-branes stretched in the 012 directions, D6-branes stretched in the 0123456 directions, as well as O2-planes parallel to the D2-branes and O6-planes parallel to the D6-branes-See Table 1. Our constructions will have either an O2-plane or an O6plane, but not both. The gauge theory lives in the 012 directions, and the choice of gauge group and two-index tensor representation is dictated by the kind of O2 or O6-plane that is present. The role of the D6-branes is to provide the fundamental hypermultiplet flavors.
See Figure 1 for a picture of the brane configurations, and Table 2 for which gauge theories correspond to which brane/orientifold constructions.
Object 0 1 2 3 4 5 6 7 8 9 (a) Brane construction with an O6-plane.   The type IIA brane construction presented above can be straightforwardly lifted to Mtheory, where one obtains N M2-branes probing an 8-(real)-dimensional hyperkähler cone. 7 Indeed, if one ignores the D2-branes and orientifold planes for a moment, the configuration of N f separated D6-branes lifts to a configuration of N f unit mass Kaluza-Klein (KK) 2 What we mean by this is that we have N half D2-branes and their N images. The O2 − can be thought of as having a half D2-brane stuck to an O2 − plane, and hence naturally gives an O(2N + 1) gauge group. 3 In the case of orientifold planes, the D6-branes should be more correctly referred to as N f half D6-branes and their N f images under the orientifold action. 4 In a similar construction involving 2N D3-branes coincident with an O3 + or with an O3 + plane one does obtain two distinct gauge theories with symplectic gauge groups denoted by U Sp(2N ) and U Sp (2N ), respectively. These theories differ in their spectra of dyonic line operators. 5 We do not consider O6 ± planes in our brane constructions, as they require a non-zero cosmological constant [33,34] in ten dimensions. These orientifold planes therefore only exist in massive type IIA string theory and their M-theory lifts are unknown. From the effective 2 + 1-dimensional field theory perspective, an O6 − -plane would introduce an extra fundamental half hypermultiplet compared to the O6 − case. The extra half hypermultiplet introduces a parity anomaly, which can be canceled by adding a bare Chern-Simons term. This Chern-Simons term reduces the supersymmetry to N = 3 [34] and is related to the cosmological constant in ten dimensions. 6 We remind the reader that it is impossible to have a half D2-brane stuck to an O6 − -plane, because the way the orientifold projection is implemented on the Chan-Paton factors requires an even number of such branes [35]. 7 The M-theory description is valid at large N and fixed N f . When N f is also large, a more useful description is in terms of type IIA string theory. monopoles, and near every monopole core the spacetime is regular [36]. N f coincident D6branes correspond to coincident KK monopoles, whose core now has an A N f −1 singularity; in other words, the transverse space to the monopole is C 2 /Z N f in this case. The infrared limit of the field theories living on the D2-branes is captured by M2-branes probing the region close to the core of the 11d KK monopole. Let us write the transverse directions to the M2branes in complex coordinates. Let z 1 , z 2 be the directions along which the KK monopole is extended, and z 3 , z 4 be the directions transverse to it. Then the M2-branes probe the space where the Z N f action on the coordinates is given by The orbifold acts precisely in the direction of the M-theory circle, which therefore rotates (z 3 , z 4 ) by the same angle and is non-trivially fibered over the 7 directions transverse to the D2-branes. 8 Back-reacting the N M2-branes and taking the near horizon limit yields AdS where the Z N f action on S 7 is that induced from C 4 , namely (2.1). This orbifold action is not free, hence S 7 /Z N f is a singular space. Since we have not included orientifold planes yet, this AdS 4 × (S 7 /Z N f ) background of M-theory is dual to the U (N ) theory with an 8 Explicitly, the coordinates x 3 , . . . , x 9 transverse to the D2-branes can be identified with (Re z 1 , Im z 1 , Re z 2 , Im z 2 , Re(z 3 z * 4 ), Im(z 3 z * 4 ), |z 3 | 2 − |z 4 | 2 ). The M-theory circle is parameterized by ψ = adjoint and N f fundamental hypermultiplets. Note that for N f = 1 the monopole core is regular, the transverse space to the monopoles is C 2 , and the gravitational dual is M-theory on AdS 4 × S 7 . At low energies, M-theory on this background is dual to ABJM theory at Chern-Simons level k = 1 [6]; therefore, the U (N ) gauge theory with an adjoint and a flavor hypermultiplet described above is dual to ABJM theory at CS level k = 1 [25].
Introducing orientifolds in the type IIA construction corresponds to further orbifolding the 11d geometry. 9 The case of O2-planes is simpler: the orbifold in 11d is generated by the action: O2 lift: For N f = 1 the orbifold group isD 1 = Z 4 .
In M-theory, we therefore have N M2-branes probing a C 4 /D N f singularity, whereD N f is generated by (2.1) (with N f → 2N f ) and (2.2). In the near-horizon limit, the eleven dimensional geometry is AdS 4 × (S 7 /D N f ) free . The subscript "free" emphasizes that the orbifold action induced from (2.1)-(2.2) on the S 7 base of C 4 is free, and hence the corresponding eleven-dimensional background is smooth. Note that theD N f orbifolds here are not the same as those in [37] obtained from similar brane constructions. 11 The O6 case is more involved. The O6 − -plane lifts to Atiyah-Hitchin space in Mtheory [39,40]. The O6 − -plane together with 2N f coincident D6-branes away from the 9 We thank Oren Bergman and especially Ofer Aharony for helpful discussions on the lift of orientifolds to M-theory. 10 Let us denote the O2 action in (2.2) by a and the orbifold action (2.1) (with N f → 2N f ) by b. We then get the presentation of the dicyclic groupD 11 The N f = 0 case is special, because there are no D6-branes in this case. In M-theory one obtains a pair of Z 2 singularities corresponding to a pair of OM2 planes sitting at opposite points on the M-theory circle. The gauge theory is simply N = 8 SYM with O(2N ), O(2N + 1), or U Sp(2N ) gauge group, and just like N = 8 SYM with gauge group U (N ), its infrared limit is non-standard. We expect N = 8 SYM with orthogonal or symplectic gauge group to flow to an ABJ(M) theory with Chern-Simons level k = 2.
center of the Atiyah-Hitchin space can be thought of as a KK monopole with mass (−4) (as the D6-brane charge of O6 − is (−4) [41]) and a KK monopole of mass 2N f , which we discussed above. When the D6-branes coincide with the O6 − -plane, we get a KK monopole of mass 2N f − 4 (away from the center). We should therefore consider the orbifold (2.1) with In addition, the O6 plane yields an extra orbifold in 11d generated by O6 lift: As in the O2 case, this action can be derived from the fact that in type IIA an O6-plane acts by flipping the sign of all the transverse coordinates and of the R-R one-form A 1 . Together, give a D N f singularity. The corresponding orbifold group is again the dicyclic group, The M-theory lift of the O6 + plane is a peculiar kind of D 4 singularity, perhaps with extra fluxes that prevent the possibility of blowing it up [42,43]. Further adding adding 2N f D6-branes results in a D N f +4 singularity. The corresponding orbifold group isD N f +2 , so in we shift N f → N f + 4 in the O6 − case, we get the same orbifold singularity as in the O6 + case, perhaps with different torsion fluxes. As we will see, the corresponding field theories do not have the same S 3 partition functions, so they are not dual to each other.
For theories that are constructed with O6 planes, the near horizon limit of the M2-brane geometry is AdS 4 × (S 7 /D N f ±2 ), where theD N f ±2 action on S 7 is that induced from (2.1) (with N f → 2N f ± 4) and (2.3). Within C 4 , the orbifold leaves the C 2 at z 3 = z 4 = 0 fixed, hence S 7 /D N f ±2 is singular along the corresponding S 3 .
In Appendix A we provide some evidence that the field theories mentioned above are indeed dual to M-theory on the backgrounds summarized in Table 2 by computing the Coulomb branch of the moduli space. In these moduli space computations an important role is played by certain BPS monopole operators that satisfy non-trivial chiral ring relations. The Coulomb branch of the U (N ) theory with an adjoint and N f fundamental hypermultiplets is where the symmetric group S N permutes the factors in the product; this branch of moduli space is precisely what is expected for N M2-branes probing the hyperkähler space C 2 × (C 2 /Z N f ). The Coulomb branch of the theories constructed from 12 The cases N f = 0, 1, 2 are special. When N f = 0, 1, the 11-d geometry is smooth, and we therefore expect that the low-energy dynamics is the same as that of ABJM theory at level k = 1. When N f = 2, the 11-d geometry has a pair of Z 2 singularities. Near each singularity the hyperkähler space looks like Coulomb branch of the theories constructed from O6-planes is ( In all the cases, the eleven-dimensional geometry takes the form: where R is the AdS radius, vol AdS 4 is the volume form on an AdS 4 of unit radius, X is the internal seven-dimensional manifold (tri-Sasakian in this case), and p is the Planck length.
This background should be accompanied by discrete torsion flux through a torsion threecycle of X, but we do not attempt to determine this discrete torsion flux precisely. Since the volume of X is given by the volume of the unit S 7 divided by the order of the orbifold group, we predict using (1. 2) that (2.5) 13 The gauging of the charge conjugation symmetry in the SO(2N +1) gauge theory does not seem to affect the dynamics provided that 2N +1 > N f . When 2N +1 ≤ N f , the SO(2N +1) theory has baryonic operators of the form q 2N +1 , where the color indices are contracted with the anti-symmetric tensor of SO(2N + 1). These operators are odd under charge conjugation, and are therefore absent from the O(2N + 1) theory. When 2N + 1 > N f , however, the operator content of the SO(2N + 1) and O(2N + 1) gauge theories is the same. See also [44].
These results will be reproduced by the field theory calculations presented in the remainder of this paper. See Table 5.

Matrix model for the S 3 free energy
The S 3 partition function of U (N ) gauge theory with one adjoint and N f fundamental hypermultiplets can be written down using the rules summarized in [31]: The normalization includes a division by the order of the Weyl group |W| = N ! and the contributions from the N zero weights in the adjoint representations.
The S 3 partition function for the theories with orthogonal and symplectic gauge groups is given by: (2.7) The constants a, b, c, d, and C are given in Table 3 for the various theories we study. The normalization C includes a division by the order of the Weyl group W (see Table 4) and the contributions from in the zero weights the matter representations: O(2N + 1) gauge groups from SO(2N ) and SO(2N + 1), respectively. In the rest of this paper, we find it convenient to rescale Z by a factor of 2 N and calculate instead The numerator in the integrand of (2.7) comes solely from the N = 4 vectormultiplet; note that an N = 4 vector can be written as an N = 2 vector and an N = 2 chiral multiplet Table 3: The values of the constansts a, b, c, and d appearing in (2.7) for gauge group G, N f fundamental flavors, and a two-index antisymmetric (A) or symmetric (S) hypermultiplet.  (2.8) and multiply the answer by an extra factor of 1/2 coming from the gauging of the Z 2 charge conjugation symmetry, as mentioned in the main text.
with R-charge ∆ vec = 1, and only the N = 2 vector gives a non-trivial contribution to the integrand. The first factor in the denominator comes from the two-index hypermultiplet, while the additional factors come from both the two-index tensor and the N f fundamental hypermultiplets.
Note that there is a redundancy in the parameters a, b, and c. Using sinh 2λ = 2 sinh λ cosh λ,

Large N approximation
In this section we calculate the S 3 partition functions of the field theories presented above using the large N approach of [1], which we extend to include one more order in the large N expansion. Explicitly, we do three computations. In Section 3.1 we present the computation for ABJM theory, whose S 3 partition function was given in (1.3). In Section 3.2, we calculate the F -coefficient of the N = 4 U (N ) gauge theory with one adjoint and N f fundamental hypermultiplets for which we wrote down the S 3 partition function in (2.6). Lastly, in Section 3.3 we generalize this computation to theories with a symplectic or orthogonal gauge group, for which the S 3 partition function takes the form (2.7) with various values of the parameters a, b, c, and d-see Table 3.

ABJM theory
At large N one can calculate the S 3 partition function for ABJM theory (1.3) in a fairly elementary fashion using the saddle point approximation. Let us write for some function F (λ i ,λ j ) that can be easily read off from (1.3). The factor of (N !) 2 that appears in (3.1) is nothing but the order of the Weyl group W, which in this case is S N × S N , S N being the symmetric group on N elements. The saddle point equations are Since F (λ i ,λ j ) is invariant under permuting the λ i or theλ j separately, the saddle point equations have a S N × S N symmetry. For any solution of (3.2) that is not invariant under this symmetry, as will be those we find below, there are (N !) 2 − 1 other solutions that can be obtained by permuting the λ i and theλ j . That our saddle point comes with multiplicity (N !) 2 means that we can approximate where F * equals the function F (λ i ,λ j ) evaluated on any of the solutions of the saddle point equations. In other words, the multiplicity of the saddle precisely cancels the 1/(N !) 2 pref-actor in (3.1).
The saddle point equations (3.2) are invariant under interchangingλ i ↔ λ * i , and therefore one expects to find saddles whereλ i = λ * i . If one parameterizes the eigenvalues by their real part x i , the density of the real part ρ(x) = 1 N N i=1 δ(x − x i ) and λ i become continuous functions of x in the limit N → ∞. The density ρ(x) is constrained to be non-negative and to integrate to 1. Expanding F (λ i ,λ j ) to leading order in N (at fixed N/k), one obtains a continuum approximation: The corrections to this expression are suppressed by inverse powers of N . In the N → ∞ limit the saddle point approximation becomes exact, and to leading order in N one can simply evaluate F on the solution to the equations of motion following from (3.4).
At large N/k, one should further expand [1]: expanding at large N/k, we obtain Note that the double integral in (3.4) becomes a single integral in (3.6) after using the fact that, in the continuum limit (3.4), the scaling behavior (3.5) implies that the interaction forces between the eigenvalues are short-ranged. The expression in (3.6) should then be extremized order by order in k/N . To leading order, the extremum was found in [1]: This eigenvalue distribution only receives corrections from the next-to-leading term in the expansion (3.6), so it is correct to plug (3.7) into (3.6) and obtain If one wants to go to higher orders in the k/N expansion, one would have to consider corrections to the eigenvalue distribution (3.7).
The result (3.8) is in agreement with the Fermi gas approach [2], when the latter is expanded at large N/k and large N as in (3.8).

N = 4 U (N ) gauge theory with adjoint and fundamental matter
We now move on to a more complicated example, namely the N = 4 U (N ) gauge theory introduced in Section 2 whose S 3 partition function was given in (2.6). Let us denote Explicitly, we have As in the ABJM case, every saddle comes with a degeneracy equal to the order of the Weyl group (S N in this case), so we can approximate Z ≈ e −F * , where F * equals F (λ i ) evaluated on any given solution of the saddle point equations ∂F/∂λ i = 0.
In the U (N ) gauge theory the eigenvalues are real, and in the N → ∞ limit we again introduce a density of eigenvalues ρ(x). We will be interested in taking N to infinity while working in the Veneziano limit where t ≡ N/N f is held fixed and then taking the limit of large t. At large N , the free energy is a functional of ρ(x): As in the ABJM case, the appropriate scaling at large t is λ ∝ √ t, so we can define It is convenient to further introduce another parameter T and write (3.11) as (3.13) Of course, we are eventually interested in setting T = t, but it will turn out to be convenient to have two different parameters and expand both at large t and large T . (3.14) If we assume that ρ is supported on [−x * , x * ] for some x * > 0, we should extremize (3.14) order by order in N under the condition that ρ(x) ≥ 0 and that We can impose the latter condition with a Lagrange multiplier and extremize instead of (3.14).

Leading order result
To obtain the leading order free energy we can simply take the limit T → ∞ in (3.14) and ignore the 1/t 3/2 term in the first line of (3.14). The free energy takes the form The normalized ρ(x) that minimizes (3.17) is The value of F we obtain from (3.18) is After writing t = N/N f , one can check that this term reproduces the expected N 3/2 behavior of a SCFT dual to AdS 4 × S 7 /Z N f .

Subleading corrections
To obtain the t −3/2 term in (3.19) we should find the 1/T corrections to the extremum of the t −1/2 terms in (3.14), and we should evaluate the t −3/2 term in (3.14) by plugging in the leading result (3.18).
Focusing on the t −1/2 terms first, the equation of motion for ρ gives Up to exponentially small corrections (at large T ), the normalization condition (3.15) fixes . (3.21) Plugging this expression back into F [ρ] and minimizing with respect to x * , one obtains again only up to exponentially suppressed corrections.
Then (3.14) evaluates to where we included the t −3/2 term. We see now that if we had taken T → ∞ directly in (3.14) we would have missed the second term in (3.23). Setting In analogy with the ABJM case we expect that fluctuations and finite N corrections will contribute to the free energy starting at N 1/2 order. However, they will have different N f dependence then the term (3.24), and the saddle point computation can be thought of as the first term in the large N f expansion. 14 These expectations will be verified in the Fermi gas approach in Section 4.
3.3 N = 4 gauge theories with orthogonal and symplectic gauge groups As a final example, let us discuss the N = 4 theories with symplectic and orthogonal gauge groups for which the S 3 partition function was written down in (2.7). (See Table 3  We can therefore restrict ourselves to saddles for which λ i ≥ 0 for all i. If F * is the free 14 One could think of t = N/N f as the analog of the 't Hooft coupling in this case. energy of any such saddle, we have Z ≈ e −F * , up to a O(N 0 ) normalization factor coming from the constant C in (2.7) that we will henceforth ignore.
Instead of extremizing F (λ i ) with respect to the N variables λ i , i = 1, . . . , N , it is convenient to introduce 2N variables µ i , i = 1, . . . , 2N , and extremize instead under the constraint µ i+N = −µ i . In the case at hand, one can actually drop this constraint, because the extrema of the unconstrained minimization of F (µ i ) satisfy µ i+N = −µ i (after a potential relabeling of the µ i ).
If the µ i are large, then extremizing (3.25) is equivalent up to exponentially small corrections to extremizing We performed a similar extremization problem in the previous section. From comparing (3.26) with (3.10), we see that the extremum of (3.26) can be obtained after replacing (3.24) and multiplying the answer by 1/2: The first term reproduces the expected N 3/2 behavior of an SCFT dual to AdS 4 × X where X is an orbifold of S 7 of order 4 N f , in agreement with (2.5). We will reproduce (3.28) from the Fermi gas approach in the following section, where we will also be able to calculate the other terms of order N 1/2 that have a different N f dependence from the one in (3.28).
which is nothing but a slight rewriting of the Cauchy determinant formula that holds for any u i and v i , with i = 1, . . . , N . Eq. (4.1) can be obtained from (4.2) by writing u i = e 2πx i and v i = e 2πy i .
Using (4.1) in the particular case y i = x i , we can write (2.6) in the form .
Z can then be rewritten as the partition function of an ideal Fermi gas of N noninteracting particles, namely where ρ(x 1 , x 2 ) ≡ x 1 |ρ|x 2 is the one particle density matrix, and the sum is over the elements of the permutation group S N . We can read off the density matrix by comparing (4.4) with (4.3). In the position representation, ρ is given by . We then rescale x ≡ y/(2πN f ) and p ≡ k/(2π) to get where The rescaling was motivated by the following nice properties: (4.10) We identify = 2πN f , and perform a semiclassical computation of the canonical free energy of the Fermi gas. In Appendix B we give a brief review of the relevant results from [2]. These results enable us to calculate the free energy from the above ingredients. In summary, we calculate the Fermi surface area as a function of the energy for the Wigner Hamiltonian (B.11).
In the semiclassical approximation, to zeroth order the phase space volume enclosed by the Fermi surface is:  We can perform the calculation and conclude that n(E) defined in (B.1) takes the form: In (B.5) we parametrized the E dependence of n(E) as (4.14) so from (4.13) we can read off 15) and the partition function takes the form [2] .

(4.16)
A(N f ) is an N -independent constant that our approach only determines perturbatively for small N f , and we are not interested in its value. Expanding the F = − log Z we obtain: We conclude that the free energy goes as: The Fermi gas computation is in principle only valid in the semiclassical, small , i.e. small N f regime. However, because the small N f series expansions terminate, we obtain the exact answer. Then we can compare to the matrix model result (3.24) valid at large N f , and find perfect agreement to leading order in N f . 15 As discussed in Section 2, at N f = 1 the U (N ) theory is dual to ABJM theory at k = 1, and the free energy computation in both representations should give the same result [25].
Plugging k = 1 into (1.1) and (1.5) indeed gives (4.18) with N f = 1. 15 Grassi and Mariño informed us that they calculated the free energy of this theory in the large N , fixed N/N f limit using method (I) discussed in Section 1. Their result is up to exponentially small corrections in N/N f and subleading terms in 1/N . This expression agrees with the large N , fixed N/N f limit of the Fermi gas result (4.15)-(4.16) of this section. We thank Marcos Mariño for sharing these results with us.

N = 4 gauge theories with orthogonal and symplectic gauge groups
To generalize the Fermi gas approach to SCFTs with orthogonal and symplectic gauge groups, one needs the following generalization of the Cauchy determinant formula (4.2): which holds for any u i and v i , with i = 1, . . . , N . 16 Upon writing u i = e 2πx i and v i = e 2πy i , this expression becomes . (4.20) In addition to this generalization of the Cauchy determinant formula, our analysis involves an extra ingredient. The one-particle density matrix of the resulting Fermi gas will be expressible not only just in terms of the usual position and momentum operatorsx andp as before, but also in terms of a reflection operator R that we will need in order to project onto symmetric or anti-symmetric wave-functions on the real line.
Using (4.20) in the particular case y i = x i , one can rewrite (2.7) as .

(4.21)
As in (4.4) we recognize the appearence of the partition function of an ideal Fermi gas of N noninteracting particles, and can read off the one-particle density matrixρ from comparing (4.4) with (4.21). From Table 4 we see that 2 N C ≈ 1/(2 N N !), up to a O(N 0 ) pre-factor that we will henceforth ignore. In the position representation, ρ(x 1 , x 2 ) ≡ x 1 |ρ|x 2 is given by .

(4.22)
To put this expression in a more useful form, we note that if we set h = 1, we can write where R is the reflection operator that sends x → −x. For the derivation of this identity see Appendix C. Then we can writê Similarly, we could use the identity and writeρ and the same expression for T (p) as before.
To be able to use N f as a parameter analogous to k in ABJM theory, we rescale x ≡ y/(4π N f ) and p ≡ k/(2π). Under this rescaling, we havê where we used that U (x) commutes with R, and for the (+) sign  and T (k) is as above.
After rescaling, we get the following nice properties: (4.32) We then identify = 4π N f , and calculate the area of the Fermi surface as a function of energy using the Wigner Hamiltonian (B.11). It is important to bear in mind that the projector halves the density of states, as consecutive energy eigenvalues correspond to eigenfunctions of opposite parity. 17 To zeroth order, the phase space volume enclosed by the Fermi surface is again given by (4.11), and the correction is given by (4.12).
The ∞ 0 dy U (y) part of the latter formula seems to be problematic at first sight. For 17 We can also think of the projection as Neumann or Dirichlet boundary conditions at the origin.
generic a, b parameter values U (y) ∼ log |y| (y → 0) , (4.33) and the integral is divergent. Physically, this divergence would be the consequence of the careless semi-classical treatment of a Fermi gas in a singular potential (4.33). We will not have to deal with such subtleties, however, for the following reason. In the cases of interest we either have a + b = 0 or a + b = 1-see Table 3. If a + b = 0, we choose U (y) of (4.30) corresponding to the projection by (1 + R)/2, which is regular at the origin. If a + b = 1 we choose U (y) of (4.31) corresponding to the projection by (1 − R)/2, and the potential is again regular. With these choices, we can go ahead and calculate (4.12).
For the number of eigenvalues below energy E we get: where we have 4π instead of the conventional 2π due to the projection. The constants C and B from (4.16) take the values Analogously to (4.36), the free energy F has the following large N expansion:  where C and B are given in Table 5. Using the relation f 3/2 = 2/(3 √ C) from (4.17), we get agreement with the supergravity calculation (2.5).
From Table 5  More generally, the results collected in Table 5, together with the results of [2,32] for U (N ) quiver theories, represent predictions for M-theory computations that go beyond the leading two-derivative 11-d supergravity. In the case of ABJM theory, the k 3/2 contribution to f 1/2 appearing (1.1) is accounted for by the shift in the membrane charge from higher derivative corrections on the supergravity side [23]. It would be very interesting to derive the shifts in membrane charge and to take into account higher derivative corrections on the supergravity side for the other examples. Note that from the large N expansion of the Airy function (5.1), one obtains a universal logarithmic term in the free energy equal to − 1 4 log N ; this term matches a one-loop supergravity computation on AdS 4 × X [24]. 19 Perhaps one could derive the full Airy function behavior from supergravity calculations.
It would be desirable to generalize the methods in this paper to more complicated quiver theories with classical gauge groups and Chern-Simons interactions. Although at first sight it may seem straightforward to generalize the large N approximation of Section 3 to the more general setup, there are additional complications related to the non-smoothness of the eigenvalue distributions at leading order in large N and the non-exact cancellation of longrange forces between eigenvalues at subleading order. We leave such a general treatment for future work. The Fermi gas approach explored in Section 4 is very powerful, but it 18 The equivalence (2.10) does not take the two integrands into each other. See Table 3. 19 We thank Nikolay Bobev for discussions on this issue. G + matter IIA orientifold M-theory on AdS 4 × X C B U (N ) + adj no orientifold

A Quantum-corrected moduli space
As a check that the field theories presented in Table 2 are dual to M-theory on AdS 4 × X, where X is the quotient of S 7 in Table 2, one can make sure that the moduli space of these field theories does indeed match the moduli space of N M2-branes probing the 11d geometry. We will do so at the level of algebraic geometry, without explicitly constructing the full hyperkähler metric on the moduli space. In this computation, monopole operators play a crucial role, because they parameterize certain directions in the moduli space [7,8]. It is very important to include quantum corrections to their scaling dimensions, which essentially determine their OPE as in [7,8,11,12].
To define monopole operators, one should first consider monopole backgrounds. We use the convention where for a gauge theory with gauge group G, the gauge field A corresponding to a GNO monopole background centered at the origin takes the form where H is an element of the Lie algebra g. Using the gauge symmetry, one can rotate H into the Cartan {h i } subalgebra, namely where r is the rank of G. The Dirac quantization condition requires for any allowed weight w of an irreducible representation of G. These monopole backgrounds should be considered only modulo the action of the Weyl group.
The background (A.1) above breaks all supersymmetry by itself. To define a supersymmetric background, one should supplement (A.1) with a non-trivial profile for one of the three real scalars in the N = 4 vectormultiplet. Let this scalar be σ; we must take σ = H/ |x|. The choice of such a scalar breaks the SO(4) R symmetry of the N = 4 supersymmetry algebra to an SO(2) R subgroup corresponding to an N = 2 subalgebra. In this N = 2 language, one can define chiral monopole operators M q corresponding to the GNO background described above. Being chiral, one can identify their scaling dimension ∆ q with the SO(2) R charge.
As shown in [8,46], the BPS monopole operator M q acquires at one-loop the R-charge is consistent with the R-charges of X, Y , q i , and q i being equal 1/2, and that of Z being equal 1, as can be derived for instance using the F -maximization procedure [15,18,47].
In the N = 1 case, the moduli space of this theory should match precisely the eight- where in the middle equality we exhibited explicitly the contributions from the N f fundamentals, the adjoints, and the N = 4 vector, respectively. One then expects the OPE [7,8,11,12] which should be imposed as a relation in the chiral ring. Giving T , T , and Z expectation values obeying (A.7) describes the orbifold C 2 /Z N f , as can be seen from "solving" (A.7) by writing T = a N f , T = b N f , and Z = ab. The coordinates a and b parameterize C 2 /Z N f because both (a, b) and (ae 2πi/N f , be −2πi/N f ) yield the same point in (A.7). The moduli space of the U (1) theory is therefore C 2 × (C 2 /Z N f ), where the C 2 factor is parameterized by the free fields X and Y , and the C 2 /Z N f factor is really just the complex surface (A.7). Defining we obtain the description of C 2 × (C 2 /Z N f ) used around eq. (2.1).
When N > 1, the theory has a Coulomb branch where the fundamentals vanish and the adjoint fields X, Y , and Z have diagonal expectation values (to ensure vanishing of the F-term potential), thus breaking the gauge group generically to U (1) N . In addition, there are BPS monopole operators corresponding to we can denote the BPS monopole operators with q i = ±1/2 and q j = 0 with i = j by T i (for the plus sign) and T i (for the minus sign). An argument like the one in the Abelian case above shows that for every i, we have For each i we therefore have a C 2 × (C 2 /Z N f ) factor in the moduli space parameterized by The Weyl group of U (N ) acts by permuting these factors, so the Coulomb branch of the U (N ) theory is the symmetric product of N C 2 × (C 2 /Z N f ) factors. This space is precisely the expected Coulomb branch of N M2-branes probing C 2 ×(C 2 /Z N f ). In addition to the Coulomb branch, the moduli space also has a Higgs branch where the fundamental fields q and q acquire expectation values. This branch is not realized geometrically, however, and is therefore of no interest to us.
where, as in (A.6), in the middle equality we exhibited explicitly the contributions from the where the trace is taken in the fundamental representation of SU (2). The relation (A.14) should be imposed as a relation in the chiral ring. Note that ∆ in (A.13) is always an integer, so tr Z 2∆ does not vanish. Also note that if N f = 0 in the adjoint case and N f ≤ 2 in the singlet case we obtain monopole operators with R-charge ∆ ≤ 0, which signifies that one of the assumptions in our UV description of the theory must break down as we flow to the IR critical point. Such theories were called "bad" in [46], and we will not examine them. See also footnotes 11 and 12.
We are now ready to give the full description of the Coulomb branch. It is parameterized by the complex fields x, y, z, T , and T . The latter three satisfy as can be easily seen from (A.14). In addition, SU (2) has a Z 2 Weyl group, which sends A straightforward analysis shows that if X and Y are symmetric tensors, the Coulomb branch is the N th symmetric power of the space C 4 /D N f found above in the N = 1 case; if X and Y are anti-symmetric tensors, the Coulomb branch is the N th symmetric power of . These spaces are precisely the expected moduli spaces of N M2-branes probing . In addition to the Coulomb branch, the theory also has a Higgs branch where the fundamentals q i and q i acquire VEVs, but the Higgs branch is not realized geometrically in M-theory.
It is worth noting that if X and Y are anti-symmetric tensors of U Sp(2N ), one could consider imposing a symplectic tracelessness condition on these fields. When N = 1, the fields X and Y would be completely absent, because what survived in the analysis above was precisely their symplectic trace. The moduli space would therefore be only C 2 /D N f if the symplectic trace were removed from X and Y , and it would not match the elevendimensional geometry. The correct field theory that arises from the brane construction of Section 2.1 is that where X and Y are not required to be symplectic traceless.

A.3 The O(2N ) theories
The be the Hermitian generator of O(2). On the Coulomb branch, we should take Z = zJ. If X and Y are symmetric matrices, the scalar potential vanishes if these matrices commute with Z, so we should take X = xI 2 and Y = yI 2 , where I 2 is the 2 × 2 identity matrix. If X and Y are anti-symmetric matrices, the only option is X = xJ and Y = yJ for some complex numbers x and y. Denoting x = z 1 , y = z 2 , a = z 3 , and c = z 4 as in the SU (2) case we obtain the same description of the eight-dimensional hyperkähler space that appears in the eleven-dimensional geometry, as described in Section 2.1.
In the N > 1 case, one can check that the Coulomb branch is the N th symmetric In the N = 1 case, we can take the theory to the Coulomb branch by giving an expectation value to Z = zJ 12 to the complex scalar Z belonging to the N = 4 vectormultiplet. Here, is the generator of rotations in the 12-plane in color space. To ensure that the scalar potential vanishes, one should also take X = xJ 12 and Y = yJ 12 in the case where X and Y are antisymmetric tensors, and X = diag{x, x, x}, Y = diag{y, y, y} in the case where X and Y are symmetric tensors. In both cases, the vanishing of the F-term potential requires q i = q i = 0.
The relevant BPS monopole operators in this case correspond to Dirac quantization implies q ∈ Z/2, and as before we denote T = M 1/2 and T = M −1/2 .
The operators T and T are distinct on the Coulomb branch, but at the CFT fixed point they get identified. Indeed, the gauge transformations corresponding to Based on these R-charges, one can infer that T and T satisfy the OPE (A.14).
The Coulomb branch in this case is parameterized by x, y, z, T , T , as well as x and y in the symmetric tensor case. The fields z, T , and T satisfy the chiral ring relation (A.15). This discussion generalizes to N > 1. If X and Y are anti-symmetric tensors, the Coulomb branch is the N th symmetric power of C 4 /D N f +2 , as expected from N M2-branes probing C 4 /D N f . If X and Y are symmetric tensors, the Coulomb branch is C 2 times the N th symmetric power of C 2 × C 2 /D N f +2 . This moduli spaces is also as expected from N M2-branes probing C 2 × (C 2 /D N f ), together with a fractional M2-brane that is stuck at the C 2 /D N f singularity and can only explore the C 2 part of the geometry. This fractional M2brane corresponds to the half-D2-brane that is stuck to the O6 + -plane. As in the previous cases, the moduli space also has a Higgs branch where the fundamental fields q i and q i have expectation values, but this branch is not realized geometrically.
B Lightning review of the Fermi gas method of [2] In this Appendix we review briefly the approach of [2] for computing the partition function of a non-interacting Fermi gas. For such a system, the number of energy eigenvalues below some energy E is given by: where E n is the nth energy eigenvalue of the full system. The density of states is defined by In the thermodynamic limit, ρ(E) becomes a continuous function. The grand canonical potential of the non-interacting gas is given by:  The Wigner transform obeys the product rule (B.8) The trace of an operator is given by the phase space integral of the Wigner transform: TrÂ = dx dp 2π A W (x, p) . (B.9) In the Fermi gases of interest in this paper, we will encounter one particle density matrices of the form 20ρ C Derivation of (4.23) Let us note that using simple trigonometric identities 4 cosh(πx 1 ) cosh(πx 2 ) 4 cosh (π(x 1 − x 2 )) cosh (π(x 1 + x 2 )) = 1 2 cosh (π(x 1 − x 2 )) + 1 2 cosh (π(x 1 + x 2 )) . (C.1) Using the Fourier transform dx e 2πipx 1 cosh πp = 1 cosh πx , it is easy to see that 1 2 cosh (π(x 1 − x 2 )) = x 1 1 2 cosh(πp) x 2 , (C. 3) where in h = 1 unitsp = − 1 2πi ∂ x in position space. Finally, we make use of R |x 2 = |−x 2 to combine (C.1) and (C.3): 4 cosh(πx 1 ) cosh(πx 2 ) 4 cosh (π(x 1 − x 2 )) cosh (π(x 1 + x 2 )) = x 1 1 + R 2 cosh(πp) This expression is the same as (4.23), which is what we set out to show.

D Derivation of the determinant formula
The Cauchy determinant formula states that for any numbers u i and v i , with 1 ≤ i, j ≤ N , the following identity holds In this Appendix we derive a similar determinant formula 21 : . (D. 2) The proof of (D. .

(D.3)
Subtracting the last column from each column j < N , one obtains the following entries .

(D.6)
Extracting a factor of (u N − u i )(u N u i − 1) from each row i < N and a factor of 1/(u N + v j )(u N + v −1 j ) from each column j < N one obtains By induction, we can then show after rearranging the factors of v i in (D.8) we have , (D. 10) which is the same expression as (D.2).