ABJM theory as a Fermi gas

# ABJM theory as a Fermi gas

Marcos Mariño and Pavel Putrov
Département de Physique Théorique et Section de Mathématiques,
Université de Genève, Genève, CH-1211 Switzerland

,
###### Abstract:

The partition function on the three-sphere of many supersymmetric Chern–Simons–matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the behavior for ABJM theory and quiver Chern–Simons–matter theories. In addition, the full series of corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern–Simons–matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.

## 1 Introduction

One of the most interesting aspects of the AdS/CFT correspondence is that, in principle, one can use gauge theories to learn about elusive aspects of string theory and quantum gravity. For example, ABJM theory [1], as well as other supersymmetric Chern–Simons–matter (CSM) theories, are conjecturally dual to M-theory on spaces of the type AdS. Therefore, computations in the gauge theory side might give interesting insights on M-theory on these backgrounds.

It was shown in [2] that the partition function on the three-sphere of CSM theories with supersymmetry reduces, via localization, to a matrix model. This result was extended to theories with supersymmetry in [3, 4]. In the case of ABJM theory, the corresponding matrix model was solved for arbitrary ’t Hooft coupling and at all orders in in [5], providing the first gauge theory derivation of the famous growth of the number of degrees of freedom of M2 branes [6]. In the last year, many results have been obtained for these matrix models, providing precision tests of the AdS/CFT correspondence as well as beautiful field-theoretical results on supersymmetric CSM theories (see [7] for a review and a list of references).

Up to now, the “stringy” side of these matrix models has been studied less intensively, but there have been already some interesting results in this direction for the ABJM theory. In [5] it was found that the genus free energies of the matrix model contain very rich information about worldsheet instantons in type IIA theory (i.e. they include non-perturbative effects in ). In [8], by studying the large order behavior of the genus expansion, it was possible to identify as well non-perturbative corrections in the string coupling constant, conjecturally associated to D2-branes, or to membrane instantons in M-theory. Finally, building on the results of [5, 8], it was shown in [9] that, once worldsheet instantons are discarded, the full free energy of the ABJM matrix model is given by an Airy function. Schematically, we have

 ZABJM∝Ai[f(k)(N−g(k))], (1.1)

where is the CS level (i.e. the inverse coupling), is a function of determined by the large limit, and is a function of which shifts (see (2.14) below for a precise formula). This is a beautiful result which provides the all-orders expansion of the partition function in powers of the string length, and therefore resums the perturbative long-distance expansion for quantum superstrings.

This result raises an interesting possibility. The behavior of the free energy characterizes a large class of [10, 11] and [12, 13, 14, 15] Chern–Simons–matter theories. On the other hand, the result (1.1) says that, for ABJM theory, this behavior is just the leading term in the logarithm of the Airy function. It is then natural to propose the following

###### Conjecture 1.1.

In parity-invariant supersymmetric CSM theories that display an growth in the number of degrees of freedom, the leading large limit and the corrections to the partition function on the three-sphere add up to an Airy function, i.e. we have schematically

 ZCSM∝Ai[f(ka)(N−g(ka))], (1.2)

where are the different CS levels involved in the theory, is a function which is determined by the leading, large limit, and is a shift which also depends on the details of the theory (but cannot be determined by the large limit alone).

If this conjecture is true, the corresponding matrix models display a universal behavior characterized by the Airy function. For theories which are not parity-invariant, we expect the Airy function to be the crucial ingredient of the answer. It is interesting to point out that the universal role of the Airy function in summing up corrections was already proposed in [16], in the different but related context of five-dimensional black holes made out of M2 branes in certain Calabi–Yau compactifications111We would like to thank C. Vafa for discussions on this.. Unfortunately, the techniques used to derive (1.1) for ABJM theory rely heavily on the calculation of corrections based on the holomorphic anomaly equation (see [5, 9] for details and references). For other models it is not clear how to generalize these techniques, even in cases (like the one studied in [11]) where the planar resolvent is known explicitly.

All the results mentioned so far have been obtained in what we will call the ’t Hooft expansion of the matrix model, which corresponds to the genus expansion in type IIA superstring theory. The ’t Hooft expansion is the asymptotic expansion as goes to infinity and the ’t Hooft parameter of the model, , is kept fixed at large ,

 N→∞,λ=Nkfixed. (1.3)

However, in order to make contact with M-theory, we have to consider the M-theory expansion, i.e. the asymptotic expansion as goes to infinity in which is fixed,

 N→∞,kfixed. (1.4)

The ’t Hooft expansion of the matrix model gives some information about the M-theory expansion. For example, the all-orders result (1.1) presumably captures the all-orders expansion of the M-theory partition function in powers of the Planck length, at finite . However, important contributions to the partition function in the M-theory expansion (like membrane instantons and Kaluza–Klein modes) are not directly captured in the ’t Hooft expansion, and in order to use the ABJM matrix model as a tool to explore M-theory, one should study the regime (1.4) directly. A step in this direction was taken in [10], who found a very simple method to extract the large , fixed behavior of general CSM matrix models. However, in the approach of [10] it is not obvious how to compute systematically corrections to the leading large behavior, not to speak about exponentially small corrections in .

It would then be very interesting to find a method to analyze the M-theory expansion of CSM theories, beyond the leading large contribution considered in [10]. Such a method would allow us to address the above conjecture about the universal role of the Airy function in resumming the corrections, and eventually could give us information about M-theoretic features of the matrix models which are not manifest in the ’t Hooft expansion.

In this paper, we make a first step in this direction, and we propose a new method to analyze the matrix model of some Chern–Simons–matter theories which fulfills some of the expectations that we have just listed. The method consists of writing the partition function on the three-sphere as the partition function of an ideal Fermi gas with a non-trivial one-particle, quantum Hamiltonian. In this reformulation of the problem, the Chern–Simons level becomes the Planck’s constant of the quantum-mechanical problem. Since corresponds to the inverse string coupling, the semiclassical expansion of the Fermi gas is a strong coupling expansion in the type IIA theory large dual. As usual, the large limit is simply the thermodynamic limit of the gas. Our approach has the following features:

1) The large limit of the free energy at finite is governed by the thermodynamic limit of the Fermi gas, which can be determined by a semiclassical calculation. In particular, we find a completely elementary derivation of the behavior of the free energy of ABJM theory, including the correct coefficient. This can be extended in a straightforward way to necklace quivers, and the result for the free energy is in full agreement with the calculation in [17, 18] based on the matrix model analysis of [10]. Interestingly, the relevant Fermi surface describing the thermodynamic limit of the Fermi gas is a two-dimensional polytope which characterizes the geometry of the dual tri-Sasakian spaces. In the case of ABJM theory, this Fermi surface is also a real version of the tropical curve obtained in [11].

2) The full expansion of the free energy of the matrix model is determined by the first quantum correction to the semiclassical limit. In this way we reproduce the result (1.1) for ABJM theory at finite , using again elementary methods in quantum Statistical Mechanics. Moreover, we prove our conjecture (1.1) for a large class of supersymmetric CSM theories.

3) One can also compute exponentially suppressed effects at large which are clearly M-theoretic. In particular, we find a systematic method to determine the contribution of membrane instantons. These non-perturbative effects receive however corrections at all orders in the expansion, and so far we can only determine them order by order in (but to all orders in the membrane winding). The strength of these corrections (i.e. the minimal membrane action) agrees with the instanton analysis of [8].

It follows from the last point above that, at the level of non-perturbative corrections, our method does not fully capture the M-theory expansion, since so far we are only able to determine these corrections in an expansion in around . In order to make contact with the true M-theory expansion one should resum the resulting series. In spite of this limitation, the Fermi gas picture gives a concrete computational method to address non-perturbative effects in these superstring theory backgrounds. In fact, the semiclassical expansion of the Fermi gas is dual to the conventional genus expansion captured by the ’t Hooft expansion of the matrix model. For example, in the ’t Hooft expansion, worldsheet instantons appear as exponential corrections in the ’t Hooft coupling, order by order in the expansion, while membrane corrections appear as large instantons, of order . In the Fermi gas approach developed in this paper, membrane instantons appear as exponential corrections in the chemical potential of the gas, order by order in the expansion, while worldsheet instantons appear as quantum-mechanical instanton effects of order .

Finally, we would like to point out that the use of Fermi gas techniques in the analysis of matrix models goes back to the solution of matrix quantum mechanics in [19], and should be familiar from the study of the string (see for example [20]). The idea of studying the matrix integral partition function in the grand-canonical ensemble appeared in [21], and was developed in detail in [22, 23, 24]. In particular, the semiclassical limit of the Fermi gas was already used in Appendix A of [23]. The systematic application of semiclassical techniques of many-body physics in the study of these matrix integrals, which we develop in this paper, seems however to be new.

The paper is organized as follows. In section 2 we review previous results on the expansion of the ABJM matrix model in the ’t Hooft expansion, focusing on [5, 8, 9]. In section 3 we show that the matrix integral of a general class of CSM theories (necklace quivers with fundamental matter) can be written as the partition function of an ideal Fermi gas with a non-trivial one-particle Hamiltonian. In sections 4 and 5 we present the tools to analyze the Fermi gas, and we illustrate them in ABJM theory. More precisely, in section 4 we study the Fermi gas in the thermodynamic limit, by passing to the grand canonical ensemble. This makes it possible to derive the leading behavior of the free energy of ABJM theory, by using elementary tools in Statistical Mechanics. We also compute exponentially suppressed corrections to the grand canonical potential, which are interpreted as membrane instantons in M-theory. In section 5 we study the quantum corrections to the grand canonical potential. We show that, up to non-perturbative terms, a next-to-leading WKB calculation is enough to determine the full expansion of the canonical free energy. This provides a simple derivation of the Airy function resummation of [9]. In section 6 we extend our techniques to more general CSM theories, including necklace quivers and theories with fundamental matter. We show that, when the free energy on the three-sphere is real, the expansion at fixed gets resummed by an Airy function, thus proving conjecture 1.1 for this family of examples. We also consider the “massive” theory of [53], where a different scaling has been found for the free energy, and we rederive it with our techniques. Finally, in section 8 we conclude with some prospects for future work. In an Appendix we collect some results for the grand canonical potential of ABJM theory at order .

As this paper was being prepared for submission, the paper [25] appear which also considers ABJM theory in the grand canonical ensemble.

## 2 The ABJM matrix model in the ’t Hooft expansion

### 2.1 1/n expansion and non-perturbative effects

The matrix integral describing the partition function of ABJM theory on is given by [2]

 ZABJM(N) (2.1) =1N!2∫dNμ(2π)NdNν(2π)N∏i

This matrix integral can be solved in the ’t Hooft expansion (1.3) by using techniques of matrix model theory and topological string theory [26, 5]. In particular, one can obtain explicit formulae for the genus free energies appearing in the expansion

 F(λ,gs)=∞∑g=0g2g−2sFg(λ), (2.2)

where the ’t Hooft coupling is defined in (1.3), and

 gs=2πik. (2.3)

The genus free energies obtained in this way are exact interpolating functions, and they can be studied in various regimes of the ’t Hooft coupling. When they reproduce the perturbation theory of the matrix model around the Gaussian point. They can be also studied in the strong coupling regime , where one can make contact with the AdS dual. In this regime it is more convenient to use the shifted variable

 ^λ=λ−124. (2.4)

As explained in [5], this shift is expected from type IIA and M-theory arguments [27, 28]. It turns out that, when expanded at strong coupling, the genus free energies have the structure

 Fg(^λ)=Fpg(^λ)+Fnpg(^λ). (2.5)

The first term represents the perturbative contribution in , while the second term is non-perturbative in ,

 Fnpg(^λ)∼O(e−2π√2^λ) (2.6)

and it was interpreted in [5] as the contribution of worldsheet instantons in the type IIA dual. For , the perturbative part is of the form,

 Fp0(^λ) =4π3√23^λ3/2, (2.7) Fp1(^λ) =π6√2^λ−12log[2√2^λ],

while for one has

 Fpg(^λ)=fg(1√^λ), (2.8)

where

 fg(x)=g∑j=0c(g)jx2g−3+j (2.9)

is a polynomial.

Besides the non-perturbative effects in , one can use the connection between the large-order behavior of perturbation theory and instantons to deduce the structure of non-perturbative effects in the string coupling constant. In [8] a detailed analysis showed that these effects would have the form

 exp(−kπ√2λ) (2.10)

at large . These were interpreted as D2-branes wrapped around generalized Lagrangian cycles of the target geometry. We will refer to these non-perturbative effects as membrane instanton effects, since they can be interpreted as M2 instantons in M-theory [29] but they are invisible in ordinary string perturbation theory.

### 2.2 The partition function as an Airy function

It was shown in [9] that the genus expansion of the perturbative free energies can be resummed. In order to do that, one has to use the variable [8]

 λren=λ−124−13k2 (2.11)

rather than (2.4). If we define the perturbative partition function as

 ZpABJM=exp[∞∑g=0Fpg(^λ)g2g−2s] (2.12)

then

 ZpABJM∝Ai[(π2k42)1/3λren], (2.13)

where is the Airy function. This can be also written in terms of as

 ZpABJM∝Ai[C−1/3(N−k24−13k)], (2.14)

where

 C=2π2k. (2.15)

As noticed in [8], the expansion resummed in (2.14) makes perfect sense for finite . Therefore, even if (2.14) was obtained from a calculation in the ’t Hooft expansion, it should be part of the M-theory answer. Indeed, one of our goals in this paper is to verify this by computing directly in the M-theory expansion.

The Airy function appearing in (2.14) gives an exact resummation of the long-distance expansion in M-theory. To see this, one has to use the dictionary relating gauge theory quantities to gravity quantities. In particular, one has to take into account the anomalous shifts relating the rank of the gauge group to the Maxwell charge , which in turn determines the compactification radius [27, 28]. The relation is

 Q=N−124(k−1k). (2.16)

The charge determines the compactification radius in M-theory according to

 (Lℓp)6=32π2Qk, (2.17)

where is the Planck length. The shift (2.11) was interpreted in [8] as a renormalization of the expansion parameter , since it means that the natural variable is

 ˆℓpL=ℓp/L[1−12π2(ℓp/L)6]1/6, (2.18)

and then the argument of the Airy function (2.14) is given by

 (256kπ2)−2/3(Lˆℓp)6. (2.19)

The expansion of the ABJM matrix model was derived in [5] by using the holomorphic anomaly equations [30] of topological string theory. The result (2.14) was obtained in [9] by looking at the recursive structure of these equations. There is however a much simpler method to obtain (2.14) which exploits the wavefunction behavior of the topological string partition function222We would like to thank C. Vafa for reminding us this.. Our derivation of (2.14) in this paper does not depend at all on ideas from topological string theory, but since it is formally very similar, we will now present this simpler argument. We will rely on results and notations of [5]. Readers who are not familiar with topological string theory can skip the rest of this section and proceed to the next one.

As shown in [31], it follows from the holomorphic anomaly equations that the topological string partition function is a wavefunction on moduli space. In particular, its transformation from one symplectic frame to the other is given by a Fourier transform. This property was spelled out in detail and exploited in [32]. The main result is summarized as follows. Let

 Γ=(αβγδ)∈SL(2,Z) (2.20)

be a symplectic transformation relating two different frames (we assume for simplicity that there is a single modulus in the problem). This means that the periods transform as

 (∂aΓFΓ0aΓ)=Γ(∂aF0a). (2.21)

Then, the full topological string partition function

 Z(a)=exp[∞∑g=0Fg(a)g2g−2s] (2.22)

transforms as

 ZΓ(aΓ)=∫dae−S(a,aΓ)/g2sZ(a), (2.23)

where

 S(a,aΓ)=−12δγ−1a2+γ−1aaΓ−12αγ−1(aΓ)2. (2.24)

In the context of ABJM theory, as explained in detail in [26, 5], the relevant quantities correspond to topological string computations in the so-called orbifold frame, where the natural periods are (the ’t Hooft coupling of the gauge theory) and the derivative . On the other hand, the most familiar frame in topological string theory is the large radius or Gromov–Witten frame, where the natural periods are (the Kähler modulus) and the derivative . The genus free energies in the large radius frame are given by the standard formulae,

 FGW0 =T36+∑k>0N0,ke−kT, (2.25) FGW1 =T12+∑k>0N1,ke−kT, FGWg>1 =∑k>0Ng,ke−kT,

where are Gromov–Witten invariants in the local geometry (there is no constant term contribution at higher genus). The fact that the total free energy is at most cubic in , up to exponentially small corrections, is a well known fact in topological string theory.

In [5], the periods in the orbifold frame were written in terms of periods in the large radius frame in order to perform analytic continuations to strong coupling. By general principles, this relation must be a symplectic transformation like (2.20). In fact, it is easy to see that the results of [5] relating the periods can be written as the following symplectic transformation:

 (∂~λ˜F0~λ)=(01−12)(∂˜T˜FGW0˜T) (2.26)

where

 ~λ=4π2cλ,˜T=πi2cT,c2=2πi, (2.27)

and

 ˜F0 =F0−π3iλ, (2.28) ˜FGWg =(−4)g−1(FGWg−δg,0π2T3).

Then, according to (2.23), (2.24), the total partition functions are related by the following formula:

 exp[F(λ)−π3iλ/g2s]∝∫d˜Texp[−˜T2/g2s+˜T~λ/g2s+˜FGW(˜T)]. (2.29)

Notice that, up to nonperturbative terms in , this is the integral of the exponential of a cubic polynomial, therefore we will indeed get an Airy function. Let us introduce the new variable through

 T=4μk−πi. (2.30)

Then, one finds the expression

 expF(λ) ∝∫dμexp{2μ33kπ2−μN+k24μ+13kμ+O(e−4μk)} (2.31) ∝Ai[C−1/3(N−B)](1+O(e−2π√2λ)),

where we used the following integral representation of the Airy function,

 Ai(z)=12πi∫Cdtexp(t33−zt), (2.32)

and is a contour in the complex plane from to . In (2.31), is given in (2.15) and

 B=k24+13k. (2.33)

The result of (2.31) is of course the expression obtained in (2.14). Notice that the first term in the shift comes from in (2.28), while the second term is due to the first, perturbative term in . The exponentially small corrections in in (2.31), which are due to the worldsheet instantons at large radius of the topological string, become, after Fourier transform, the worldsheet instantons (2.6) of the type IIA superstring.

This derivation is nice, but it seems difficult to generalize it in its current form to other Chern–Simons–matter theories, and prove in this way the conjecture (1.2) for other cases. In this paper we will find a completely different approach to the derivation of the Airy function which turns out to formally equivalent to the one based on topological string theory. However, this approach can be extended to many CSM theories and makes it possible to verify the conjecture 1.1 for many of them.

## 3 Chern–Simons–matter theories as Fermi gases

### 3.1 ABJM theory as a Fermi gas

Our Fermi gas approach is based on the following observation. The interaction term between the eigenvalues in (2.1) can be written in a different way by using the Cauchy identity:

 ∏i

In this equation, is the permutation group of elements, and is the signature of the permutation . This identity has been used in other matrix models in [21, 22, 23] in order to study them in the grand canonical ensemble, as we will do here. In the context of ABJM theory, it was used in [33] in order to prove the equivalence of (2.1) and the matrix integral for super Yang–Mills theory in three dimensions, when . The manipulations in [33] can be easily generalized to arbitrary , and one obtains the following expression for the ABJM matrix model,

 (3.2)

We will derive this expression below with a different technique, which can be used for more general Chern–Simons–matter theories. The main property of (3.2) is that it makes contact with the standard formalism to study partition functions of ideal Fermi gases. Indeed, let us introduce the function

 ρ(x1,x2)=12πk1(2coshx12)1/21(2coshx22)1/212cosh(x1−x22k). (3.3)

If we interpret it as a one-particle density matrix in the position representation

 ρ(x1,x2)=⟨x1|^ρ|x2⟩, (3.4)

the matrix integral (2.1) can be written as the partition function of an ideal Fermi gas with particles

 Z(N)=1N!∑σ∈SN(−1)ϵ(σ)∫dNx∏iρ(xi,xσ(i)). (3.5)

It is well-known that the sum over permutations appearing in the canonical free energy of an ideal quantum gas can be written as a sum over conjugacy classes of the permutation group (see for example [34]). A conjugacy class is specified by a set of integers satisfying

 ∑ℓℓmℓ=N. (3.6)

Let us define

 Zℓ=∫dx1⋯dxℓρ(x1,x2)ρ(x2,x3)⋯ρ(xℓ−1,xℓ)ρ(xℓ,x1). (3.7)

Then, the partition function is given by,

 Z(N)=∑{mℓ}′∏ℓη(ℓ−1)mℓZmℓℓmℓ!ℓmℓ (3.8)

where the means that we only sum over the integers satisfying the constraint (3.6).

Due to the constrained sum, the canonical partition function is not easy to handle for large . As usual, the remedy is to consider the grand partition function

 Ξ=1+∞∑N=1Z(N)zN, (3.9)

where

 z=eμ (3.10)

plays the rôle of the fugacity and is the chemical potential. The grand-canonical potential is

 J(μ)=logΞ. (3.11)

Notice that this potential (like the free energy) has the opposite sign to the usual conventions in Statistical Mechanics. A standard argument (presented for example in [34]) tells us that the sum over conjugacy classes in (3.8) can be written as

 J(μ)=−∑ℓ≥1Zℓ(−z)ℓℓ. (3.12)

The canonical partition function is recovered from the grand-canonical potential as

 Z(N)=∮dz2πiΞzN+1. (3.13)

At large , this integral can be computed by applying the saddle-point method to

 Z(N)=12πi∫dμexp[J(μ)−μN]. (3.14)

The saddle point occurs at

 N=∂J∂μ=−∑ℓ≥1Zℓ(−z)ℓ, (3.15)

and defines a function . The free energy is given, at leading order as , by

 F(N)=J(μ∗)−μ∗N. (3.16)

However, it is possible to compute the corrections to this relation by simply computing the corrections to the full integral in (3.14). This is what we will eventually do. Notice the similarity between the traditional inverse transform (3.14) and the Fourier transform (2.31) in topological string theory.

We have then shown that the original ABJM matrix integral can be computed as the canonical partition function of a system of non-interacting fermions, where the one-particle density matrix is given by (3.3). We just have to solve the corresponding one-body problem in order to compute the relevant thermodynamic quantities of the system. Equivalently, one should compute the quantity introduced in (3.7). This quantity can be regarded as the partition function of a classical lattice gas with particles in a periodic lattice with nearest-neighbour interactions, as shown in Fig. 1. The density matrix plays the rôle of the classical transfer matrix of the system (see for example chapter 12 of [35]). It defines a symmetric kernel

 ⟨x|^ρ|ϕ⟩=∫dx′ρ(x,x′)ϕ(x′), (3.17)

so that

 Zℓ=Tr^ρℓ. (3.18)

It is easy to see that this kernel is a non-negative Hilbert–Schmidt operator, therefore it has a discrete, positive spectrum

 ^ρ|ϕn⟩=λn|ϕn⟩,n=0,1,⋯, (3.19)

where are orthonormal eigenfunctions and we assume that

 λ0≥λ1≥λ2≥⋯. (3.20)

We can then write the density matrix as

 ^ρ=∑n≥0λn|ϕn⟩⟨ϕn|. (3.21)

In terms of these eigenvalues we have,

 Zℓ=∑n≥0λℓn. (3.22)

When is large, this sum is dominated by the largest eigenvalue ,

 Zℓ≈λℓ0,ℓ≫1. (3.23)

It also follows from this representation that the grand-canonical partition function is given by a Fredholm determinant,

 Ξ=det(1+z^ρ)=∏n≥0(1+zλn). (3.24)

Instead of using the formulation of the lattice problem in terms of the density matrix operator, we can introduce a quantum Hamiltonian in the standard way,

 ^ρ=e−^H. (3.25)

This leads to the well-known equivalence between the partition function of a classical lattice gas (3.18) and the propagator of a quantum particle in units of discretized time (see for example [35, 36]). We can then write

 Zℓ=Tre−ℓ^H. (3.26)

To find the Hamiltonian corresponding to the ABJM matrix model, we first write the density matrix (3.3) as

 ^ρ=e−12U(^q)e−T(^p)e−12U(^q). (3.27)

In this equation, are canonically conjugate operators,

 [^q,^p]=iℏ, (3.28)

and

 ℏ=2πk. (3.29)

This is a key aspect of this formalism: is the inverse coupling constant of the gauge theory/string theory, therefore semiclassical or WKB expansions in the Fermi gas correspond to strong coupling expansions in gauge theory/string theory. The potential in (3.27) is given by

 U(q)=log(2coshq2), (3.30)

and the kinetic term is given by the same function,

 T(p)=log(2coshp2). (3.31)

The peculiar kinetic term (3.31) can be regarded as a non-trivial dispersion relation interpolating between the quadratic behavior of a non-relativistic particle at small ,

 T(p)∼log(2)+p28,p→0, (3.32)

and the linear behavior of an ultra-relativistic particle at large ,

 T(p)∼|p|2,|p|→∞. (3.33)

Notice that, as it is standard for Hamiltonians defined by transfer matrices at finite lattice spacing [35, 36], the quantum operator defined by (3.25) and (3.27) differs from

 T(^p)+U(^q) (3.34)

in corrections. There is a very elegant method to obtain these corrections based on the phase-space or Wigner approach to quantization. This method will be also extremely useful in setting the semiclassical or WKB expansion of our thermodynamic problem. We first recall that the Wigner transform of an operator is given by (see [37] for a detailed exposition of phase-space quantization)

 AW(q,p)=∫dq′⟨q−q′2∣∣∣^A∣∣∣q+q′2⟩eipq′/ℏ. (3.35)

The Wigner transform of a product is given by the -product of their Wigner transforms,

 (^A^B)W=AW⋆BW (3.36)

where the star operator is given as usual by

 ⋆=exp[iℏ2(←∂q→∂p−←∂p→∂q)], (3.37)

and is invariant under linear canonical transformations. Another useful property is that

 Tr^A=∫dpdq2πℏAW(q,p). (3.38)

In order to calculate the corrections to the Hamiltonian, we consider the Wigner transform of the density matrix (3.27). By using (3.36) we find,

 ρW=e−12U(q)⋆e−T(p)⋆e−12U(q). (3.39)

Let us note that the partition function depends only on the eigenvalues of (or, equivalently, on the traces ). Therefore there is the following freedom in the choice of :

 ^ρ→^V^ρ^V−1 (3.40)

which translates into

 ρW(q,p)→VW(q,p)⋆ρW(q,p)⋆(V−1)W(q,p) (3.41)

after the Wigner transform. Equation (3.39) defines the Wigner transform of our Hamiltonian through

 ρW=e−HW⋆, (3.42)

where the -exponential is defined by

 exp⋆(A)=1+A+12A⋆A+⋯. (3.43)

The quantum Hamiltonian can be computed by using the Baker–Campbell–Hausdorff formula, as applied to the -product. One finds,

 HW(q,p) =T+U+112[T,[T,U]⋆]⋆+124[U,[T,U]⋆]⋆+… (3.44) =T(p)+U(q)−ℏ212(T′(p))2U′′(q)+ℏ224(U′(q))2T′′(p)+O(ℏ4),

where we have used the fact that, at leading order in , the Moyal bracket is the Poisson bracket

 [A,B]⋆≡A⋆B−B⋆A=iℏ{A,B}+O(ℏ2). (3.45)

Further corrections to (3.44) can be computed to any desired order, see (A.1) for the result at order .

### 3.2 More general Chern–Simons–matter theories

The identification of the matrix model of ABJM theory as the partition function of a Fermi gas can be also made for more general Chern–Simons–matter theories. We will set up the formalism for the necklace quivers with nodes considered in [38, 39], and with fundamental matter in each node (see Fig. 2). These theories are given by a

 U(N)k1×U(N)k2×⋯U(N)kr (3.46)

Chern–Simons quiver. Each node will be labelled with the letter . There are bifundamental chiral superfields , connecting adjacent nodes, and in addition we will suppose that there are matter superfields in each node, in the fundamental representation. We will write

 ka=nak, (3.47)

and we will assume that

 r∑a=1na=0. (3.48)

According to the general localization computation in [2], the matrix model computing the partition function of a necklace quiver is given by

 (3.49)

This matrix model is very similar to the models considered in for example [22, 40], and one can use a very similar strategy in order to rewrite them as Fermi gases. First of all, we define a kernel corresponding to a pair of connected nodes by,

 Kab(x′,x)=12πkexp{inbx24πk}2cosh(x′−x2k)[2coshx2k]−Nfb, (3.50)

where we set . The grand canonical partition function corresponding to the above matrix model is defined as in (3.9). Then, if we use the Cauchy identity (3.1), a simple generalization of the above arguments makes it possible to write it again as a Fredholm determinant (3.24), where now [22]

 ^ρ=^Kr1^K12⋯^Kr−1,r (3.51)

is the product of the kernels (3.50) around the quiver. Therefore, we can apply exactly the same techniques that we used before in ABJM theory. In a sense, we are “integrating out” nodes of the quiver in order to define an effective theory in the -th node, but with a complicated Hamiltonian which takes into account the other nodes.

This idea can be made very concrete by looking at the Wigner transform of the operator in (3.51). We first compute the Wigner transform of the kernel (3.50),

 KWab(q,p)=12coshp2⋆einbq22ℏ[2coshq2k]Nfb (3.52)

where the in the product is given again by (3.29). Let us note that

where we used that

 [q2,p]⋆=2iℏq. (3.54)

We obtain then, for the Wigner transform of the density operator (3.51)

 ρW(q,p) =12coshp2⋆1[2coshq2k]Nf1⋆12coshp−n1q2⋆ (3.55) 1[2coshq2k]Nf2⋆12coshp−(n1+n2)q2⋆1[2coshq2k]Nf3⋆ ⋯⋆12coshp−(n1+⋯+nr−1)q2⋆1[2coshq2k]Nfr

where we used (3.48). For necklace theories without fundamental matter this is simply

 ρW(q,p)=12coshp2⋆12coshp−n1q2⋆12coshp−(n1+n2)q2⋆⋯⋆12coshp−(n1+⋯+nr−1)q2. (3.56)

In particular, for the ABJM necklace with fundamental matter first considered in [41, 42, 43], we have

 ρW(q,p)=12coshp2⋆1[2coshq2k]Nf⋆12coshp+q2⋆1[2coshq2k]Nf. (3.57)

If we perform a canonical transformation

 p→−q,q→p+q (3.58)

and we conjugate by to obtain a symmetric kernel, we get the equivalent representation,

 ρW(q,p)=1[2coshq2]1/2⋆1[2coshp+q2k]Nf⋆12coshp2⋆1[2coshp+q2k]Nf⋆1[2coshq2]1/2 (3.59)

which, for , agrees with the result (3.39). In this way, we have reduced the general necklace quiver theory to an ideal Fermi gas whose one-particle quantum Hamiltonian is defined by the above density matrices through (3.42).

Notice that, in general, the density operators are not Hermitian, and correspondingly is generally not real. This reflects the fact that the free energy on the three-sphere of these CSM theories is in general complex.

## 4 Thermodynamic limit

It is well-known that the thermodynamic limit of an ideal quantum gas can be evaluated by treating the one-particle problem in the semiclassical or WKB approximation. Moreover, the corrections to the thermodynamic limit can be obtained by studying the quantum corrections to the semiclassical limit. In this section we will present general results about the thermodynamic limit and we will illustrate them in ABJM theory. More general theories will be considered in section 6.

### 4.1 The thermodynamic limit of ideal Fermi gases

In the following we will need several standard results in the analysis of ideal quantum gases. The distribution operator at zero temperature is given by,

 ^n(E)=θ(E−^H) (4.1)

where is the Heaviside step function. The trace of this operator gives the function , counting the number of eigenstates whose energy is less than :

 n(E)=Tr^n(E)=∑nθ(E−En). (4.2)

Notice that

 En=−logλn, (4.3)

where are the eigenvalues (3.19) of the density matrix. The density of eigenstates is defined by

 ρ(E)=dn(E)dE=∑nδ(E−En). (4.4)

The one-particle canonical partition function is then given by the standard formula,

 Zℓ=∫∞0dEρ(E)e−ℓE, (4.5)

while the grand-canonical potential of the particle system is given by

 J(μ)=∫∞0dEρ(E)log(1+ze−E). (4.6)

Let us now consider the thermodynamic limit of the system, when . In this regime, the behavior of the system is semiclassical and the spectrum of the one-particle Hamiltonian is encoded in the functions , . The thermodynamic limit is governed by the behavior of these functions as . We notice that, if

 n(E)≈CEs,E≫1, (4.7)

then the grand-canonical potential is given by

 J(μ)≈sC∞∫0log(1+ze−E)Es−1dE=−CΓ(s+1)Lis+1(−eμ), (4.8)

where is the usual polylogarithm function. The number of particles is related to the chemical potential by

 N(μ)≈CΓ(s+1)Lis(−eμ), (4.9)

and large corresponds to large . In this regime, we have

 J(μ)≈Cs+1μs+1,N(μ)≈Cμs. (4.10)

The second equation defines as function of , and we deduce from (3.16) that the canonical free energy is given, as , by

 F(N)≈−ss+1C−1/sNs+1s. (4.11)

These formulae should be familiar from the elementary theory of ideal quantum gases. For example, the textbook ideal Fermi gas in three dimensions has .

To determine the value of for a given system we notice that, in the semiclassical limit, the trace is replaced by an integral over phase space

 Tr→∫dqdp2πℏ (4.12)

which gives the standard semiclassical formula

 n(E)≈∫dpdq2πkθ(E−H(q,p))=Vol(E)2πℏ, (4.13)

i.e. the number of eigenstates is just given by the volume of phase space. The surface

 H(q,p)=E (4.14)

is just the Fermi surface of the system. For a one-dimensional ideal gas whose one-particle Hamiltonian is of the form

 H∼A|p|α+B|q|β (4.15)

we have

 s=1α+1β. (4.16)

This will be useful later on.

### 4.2 A simple derivation of the N3/2 behavior in ABJM theory

We can now study the thermodynamic limit of the partition function of ABJM theory. In this case, the Hamiltonian appearing in the semiclassical formula (4.13) is just given by the classical counterpart of (3.34),

 Hcl(q,p)=T(p)+U(q)=log(2coshp2)+log(2coshq2). (4.17)

Here we have neglected the corrections appearing in . It is easy to show that the minimum energy is

 E0=2log2, (4.18)

which corresponds to the maximal eigenvalue of the density matrix

 λ0=14. (4.19)

This is the semiclassical value given by the leading WKB approximation, and it will be corrected quantum-mechanically. In the large regime, the discrete spectrum “condenses” along a cut in the complex plane, and signals the endpoint of the cut.

In order to proceed with the analysis of the thermodynamic limit, we should determine the Fermi surface

 Hcl(q,p)=E (4.20)

controling the density of eigenvalues. We show the shape of this surface in Fig. 3 for (left) and (right). It is clear that in the thermodynamic limit, when is large, the surface can be approximated by considering the values of , for large. In this regime we have

 U(q)≈|q|2,|q|→∞,T(p)≈|p|2,|p|→∞, (4.21)

so that (4.20) is approximately given by

 |q|+|p|=2E, (4.22)

as it is manifest in the graphic on the right in Fig. 3. From (4.15) and (4.16) we deduce that

 s=2. (4.23)

Since

 Vol(E)≈8E2, (4.24)

the number of states is given by

 n(E)≈2π2kE2. (4.25)

By comparing with (4.7), we find

 C=2π2k. (4.26)

The equation (4.11) gives immediately

 F(N)≈−π√2k3N3/2. (4.27)

This is exactly the result first found in [5] using the ’t Hooft expansion of the matrix model. The derivation presented here is however completely elementary, and relies on basic notions of quantum Statistical Mechanics: the scaling of the number of degrees of freedom is nothing but the scaling of the free energy of an ultrarelativistic gas of one-dimensional fermions in a linearly confining potential. No matrix model techniques are needed. In this sense, our derivation is even simpler than the one presented in [10], which required some detailed analysis of the eigenvalue interaction in the matrix integral.

We would like to emphasize that the above result (4.27) provides the right large behavior of the system at finite . This is because the true expansion parameter in the semiclassical expansion is , which is small for large even at finite . This can be proved rigorously for some spectral problems defined by kernels of the form (3.3) [44], and we will verify it in section 5 by a detailed analysis of the WKB expansion.

### 4.3 Large N corrections

One advantage of the statistical-mechanical framework presented here is that it makes it possible to compute corrections to the thermodynamic limit in a systematic way. To start the study of these corrections, we now look at the thermodynamics of the Fermi gas of ABJM theory in the semiclassical approximation, but taking into account the exact value of the volume of phase space (i.e. we go beyond the polygonal approximation in (4.22)). As expected, this gives sub-leading and exponentially suppressed corrections at large .

The computation of the exact volume is equivalent to computing all the exactly in the semiclassical approximation, and resumming the resulting series (3.12). Using that

 ∫∞−∞dξ(2coshξ2)ℓ=Γ2(ℓ/2)Γ(ℓ) (4.28)

we find

 Zℓ≈1ℏZ(0)ℓ, (4.29)

where

 Z(0)ℓ=∫dpdq2πe−ℓHcl(q,p)=12πΓ4(ℓ/2)Γ2(ℓ). (4.30)

Therefore,

 J(μ)≈1kJ0(μ) (4.31)

where

 J0(μ) =−∞∑ℓ=1(−z)ℓ4π2Γ4