Quantization conditions and functional equations in ABJ(M) theories
The partition function of ABJ(M) theories on the three-sphere can be regarded as the canonical partition function of an ideal Fermi gas with a non-trivial Hamiltonian. We propose an exact expression for the spectral determinant of this Hamiltonian, which generalizes recent results obtained in the maximally supersymmetric case. As a consequence, we find an exact WKB quantization condition determining the spectrum which is in agreement with numerical results. In addition, we investigate the factorization properties and functional equations for our conjectured spectral determinants. These functional equations relate the spectral determinants of ABJ theories with consecutive ranks of gauge groups but the same Chern-Simons coupling.
Département de Physique Théorique et Section de Mathématiques,
Université de Genève, Genève, CH-1211 Switzerland
DESY Theory Group, DESY Hamburg,
Notkestrasse 85, D-22603 Hamburg, Germany
\emailAddalba.firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
In the last years, there has been a lot of progress in understanding ABJ(M) theory [1, 2]. In  the partition function of ABJ(M) theory on the three-sphere was reduced to a matrix integral which turned out to be closely related to topological strings on local . In  the connection with topological strings was used to compute recursively the full ’t Hooft expansion, which by the AdS/CFT correspondence corresponds to the genus expansion of a dual type IIA superstring theory. In order to understand the M-theory lifting of this result, one has to study ABJM theory in a different regime, usually called the M-theory regime or M-theory expansion, in which is large but the coupling constant is fixed. The study of the matrix models computing partition functions of Chern–Simons–matter theories in the M-theory regime was initiated in , where the strict large limit was solved for a large class of theories.
In , a different approach was proposed to study the M-theory regime of ABJM theory and related models. In this approach, the partition function of ABJ(M) is interpreted as the partition function of an ideal Fermi gas. The M-theory limit corresponds to the thermodynamic limit of this gas, and the coupling constant of ABJM theory becomes Planck’s constant. The Fermi gas formulation of ABJM theory has been intensively studied in [7, 8, 9, 10, 11, 12, 13], leading to an exact expression for the the partition function of ABJ(M) theory which resums the ’t Hooft expansion and includes as well non-perturbative, large instanton corrections .
An important aspect of the Fermi gas approach is that, since we are dealing with an ideal gas, all the physics of the problem is encoded in the spectrum of the one-particle Hamiltonian. Therefore one should be able to reproduce the results of  by studying the spectral problem associated to the Fermi gas of ABJ(M) theory. Conversely, the exact expression for the partition function should encode all the information about the spectrum of the Fermi gas. In [15, 16], a WKB quantization condition for ABJ(M) theory has been proposed by studying the relation between the spectrum of the Hamiltonian and the thermodynamics of the Fermi gas. This condition turns out to be exact in the cases in which the theory has maximal supersymmetry , but it needs additional corrections in the general case, as it was recently pointed out in  (see also ) by a detailed numerical analysis.
One of the key results of  is that, in the maximally supersymmetric cases, one can write an explicit expression for the grand canonical partition function of the Fermi gas, which is nothing but the spectral determinant of the Hamiltonian. This expression involves a Jacobi theta function, and the spectrum can be read from the vanishing locus of this theta function. In this paper we generalize the results of  to ABJ(M) theories with supersymmetry. We write a general formula for the spectral determinant of these theories, which involves now a generalization of the theta function. In particular, we derive an exact WKB quantization condition for the spectrum. The quantization condition proposed in [15, 16] is only an approximation to the exact quantization condition, and in general it receives corrections that we can compute analytically in this paper. Our general result explains why the quantization conditions of [15, 16] are valid in the maximally supersymmetric cases. It reproduces the corrections found numerically in the case of ABJM theory in , and we also test these corrections in detail against explicit calculations of the spectrum in both ABJM theory and ABJ theory.
As it was emphasized in the companion paper , where we studied the implications of these ideas for topological string theory, the quantization condition is obtained as a corollary of a stronger result, namely a conjectural exact expression for the spectral determinant. This expression was tested in detail in  in the maximally supersymmetric case, where it was shown that it reproduces the values for the partition functions calculated in [8, 9, 22, 23]. In the general case with supersymmetry our conjecture for the spectral determinant is more difficult to verify, since this involves a resummation of the Gopakumar–Vafa expansion of the topological string free energy . However, we are able to perform this resummation in one special case (ABJM theory with ), and we obtain a generating functional for the partition functions of this theory in full agreement with existing calculations .
In exactly solvable cases, spectral determinants enjoy very interesting properties. They can be factorized according to the parity of the eigenfunctions, and they satisfy functional equations (see for example [24, 25]). In this paper, we initiate the study of such properties in the spectral problem of ABJ(M) theory. We find for example an explicit factorization of the spectral determinant in the maximally supersymmetric case , as well as conjectural functional equations akin to those found in [24, 25] in Quantum Mechanics. These functional equations relate the spectral determinants of ABJ theories with consecutive ranks of gauge groups. In particular, if the Chern-Simons levels are odd, these equations determine all the ABJ spectral determinants from the ABJM ones via the Seiberg-like duality of ABJ theory .
This paper is organized as follows. In section 2 we review some general aspects of the Fermi gas formalism. In section 3 we introduce the spectral determinant and the generalized theta function associated to the ABJ(M) grand potential, and we deduce the exact quantization condition for the energy levels by looking at the zeros of this generalized theta function. We also give strong numerical evidence in support of our computations. In section 4 we study an example with supersymmetry and we show that the full genus expansion can be resummed into an explicit function on the moduli space. In section 5 we discuss the factorization of the spectral determinant according to the parity of the energy levels and in section 6 we give evidence for some functional identities. In section 7 we draw some conclusions. There are also three appendices. In appendix A and B we give some details for computations appearing in section 4 and in section 5. In appendix C we explain the numerical technique used to compute the spectrum.
2 The Fermi gas approach to ABJ(M) Theory
The ABJ(M) theory [1, 2] is an superconformal Chern-Simons-matter theory with gauge group . It consists in two Chern-Simons nodes, with couplings and , respectively, together with four hypermultiplets in the bifundamental representation. By using localization techniques it is possible to reduce the ABJ(M) partition function on to the following matrix integral :
These matrix integrals can be studied in the conventional ’t Hooft expansion [4, 5]. In  it was pointed out that, to fully understand the non-perturbative effects, one has to go beyond the ’t Hooft expansion and study the M-theory regime of (2.1). In this regime, the ranks of the gauge groups are large but the coupling is fixed. To study this regime it is convenient to use the Fermi gas approach  in which we rewrite the matrix integral as the canonical partition function of a one-dimensional ideal Fermi gas. In this approach, the Chern-Simons coupling plays the role of the Planck constant:
and we used the notation
In the following we will suppose that
The phase factor appearing in (2.4) is given by
and is the Chern-Simon partition function on :
The factor has the form,
The function is given as
where the function is given by
One can verify that this function is real and positive. When , the ABJ partition function becomes the partition function of ABJM theory given in (2.2):
The function (2.10) can be interpreted as a canonical density matrix,
where is the one-particle Hamiltonian. In this picture, (2.9) is then the canonical partition function of an ideal Fermi gas of particles with Hamiltonian . Mathematically, the density matrix (2.10) is given by a positive-definite, Hilbert-Schmidt integral kernel. The spectrum of the associated Hamiltonian is then determined by the integral equation
where we have ordered the eigenvalues as
As it is well-known, ideal quantum gases are better studied in the grand canonical ensemble. The grand canonical partition function is defined by
is the fugacity. We will use the notation and interchangably. When we will write
The grand canonical partition function can be also interpreted as the spectral determinant (or Fredholm determinant)
of the operator . Since this operator is positive-definite and Hilbert–Schmidt, it is of trace class and therefore its spectral determinant
has simple zeros for
The spectral determinant can be also regarded as a generating function for the partition functions :
The grand potential is defined by the usual formula,
Its power series expansion around
involves the spectral traces of the canonical density matrix,
In the context of ABJ(M) theory it is convenient to use the modified grand potential , which was introduced in . It is related to the partition function by
where is the standard Airy contour shown in Figure 1. The standard and the modified grand potentials are related via
The modified grand potential of ABJM theory was determined in a series of works [7, 8, 10, 11, 12, 13], and it can be written down in terms of the standard and refined topological strings on local . This result was extended to ABJ theory in [23, 22]. One has the following result:
The perturbative piece is a cubic polynomial in :
The constant term is given by
where is the same as in (2.8). In particular for we have
and we recover the constant map contribution of ABJM theory [32, 33]. The exact values of this constant map contribution for arbitrary integral are found in . The effective chemical potential was introduced in  to take into account bound states of worldsheet instantons and membrane instantons. It is given by
and we denoted
When is an integer, the effective chemical potential can be written in closed form 
The membrane part of the grand potential consists of two functions and , whose large expansion reads:
The particular combination of invariants appearing here involves only the so-called Nekrasov–Shatashvili limit  of the topological string free energy. The coefficients can be computed from the by using the relation conjectured in 
The worldsheet part of the grand potential takes the following form
These coefficients can also be expressed in terms of BPS invariants of local
Notice that both the worldsheet instanton contributions (2.39) and the membrane instanton contributions (2.36) have poles at rational value of . However, as noted in , the HMO cancellation mechanism of ABJM theory [10, 13] extends to ABJ theory, and these poles cancel in the total sum. As a result, the modified grand potential is a well defined and finite quantity for any value of .
3 Spectral determinant and quantization conditions
The physical information on the ABJ(M) Fermi gas can be encoded in many different ways: in the spectrum of the Hamiltonian, in the spectral determinant, and in the modified grand potential. It is clear that these three objects are equivalent, but their relationship is not trivial. The purpose of this paper is to use the explicit answer for the modified grand potential of ABJ(M) theory in order to find a useful expression for the spectral determinant and to solve the spectral problem of the Hamiltonian.
A natural starting point to find the spectrum is to use the Bohr–Sommerfeld approximation, which in this generalized setting goes as follows. Let be the classical phase space volume, i.e. the volume of the region
Here we denoted by the classical limit
The Bohr–Sommerfeld quantization condition reads
For large values of the energy one finds
as shown for instance in Figure 2. Notice that the region has a finite volume, corresponding to the fact that the operator has a discrete spectrum. In  it was pointed out that the classical volume receives two types of quantum corrections, perturbative and non perturbative in , and there should be a fully “quantum” version of the classical function incorporating these corrections, which we will denote by . It is convenient to write the quantum volume as in ,
where contains the full series of perturbative corrections in , while contains the non-perturbative corrections in . The Bohr–Sommerfeld quantization condition should be promoted to an exact WKB quantization condition of the form
similar to what happens in some problems in ordinary Quantum Mechanics .
Our goal is to extract the exact quantum volume from our knowledge of the
exact grand potential. A first attempt to do this was presented in . Although the strategy of  leads to the correct
result in the case of , it involves many technical difficulties in the study of the non-perturbative sector.
Here we overcome these difficulties by using the approach of , where the quantum volume and the spectrum are computed by studying the zeros of the spectral determinant
In the case of maximally supersymmetric theories, it was shown in  that the sum appearing in (2.26) can be written in terms of Jacobi theta functions. It is easy to see, by a computation similar to the one presented in , that the spectral determinant for ABJ(M) theory (2.26) is given by
We will call this function a generalized theta function.
As it was found in [17, 20], this involves looking at the zeros of the (generalized) theta function, and leads to a quantization condition of the form (3.7) which incorporates all the perturbative and non-perturbative corrections to the Bohr–Sommerfeld condition (3.4). It is easy to see that
In this equation we have introduced, in analogy with (2.32), the “effective” energy
In addition, we have
We also have that
Note that, if we just take into account in the generalized theta function (3.11) the terms with , we obtain the quantization condition
This is precisely the quantization condition proposed in [15, 16]. However, there will be in general corrections to this condition, due to the remaining terms in (3.11). In order to take them into account systematically, let us call this correction and write the exact quantization condition as
A straightforward calculation shows that , which is non-perturbative in (i.e. in ), is determined by the following equation 
We also conclude from this analysis that the exact quantum volume is given by
Note that the perturbative part of this quantum volume is given precisely by the all-orders WKB perturbative contribution, encoded in the quantum B-period,
while the non-perturbative contribution is given by
The perturbative and the non-perturbative part are separately divergent when is rational, as noted in , but the total quantum volume (3.21) is smooth, since the singularities cancel as a consequence of the HMO mechanism (indeed, the quantum volume is obtained from the modified grand potential, which is singularity-free). The non-perturbative part is then needed to cancel the singularities in the WKB perturbative expansion, and it contains crucial information on the spectrum. For example, as shown in appendix A, when is an integer, the finite part of vanishes and the quantum volume is largely determined by the worldsheet instanton contribution.
The correction can be computed analytically, in a series expansion in . It is easy to see that is of the form
where the first few terms are explicitly given by
and we have denoted
Note that, when , we have that , and the first term in the argument of the sine in (3.19) is always a multiple of . Therefore, the solution to (3.19) is , i.e. the correction vanishes and the quantization condition of [15, 16] is exact.
|Energy levels for|
|Energy levels for|
|Energy levels for|
|Energy levels for|
|Energy levels for|
Let us now give some concrete results for the correction series (3.24). In the case of ABJM theory, with , the first few corrections read
The results for and are in perfect agreement with the results of  based on numerical fitting. We can check as well the higher order corrections by performing a detailed numerical analysis. First of all, we compute the first two energy levels in (2.14), with high precision. To do this, we follow a procedure inspired by the analysis of  and summarized in Appendix C. On the other hand, we use (3.19) to compute the corrections to the quantum volume up to
The results are shown in tables 1 and 2, for . As expected, the more instanton corrections we include in the analytic computation, the better we approach the numerical value. This can be seen in detail by considering the quantity
where is the numerical value of the ground state energy, and is the value computed from (3.19) by including the first instanton corrections. As shown in Fig. 3, converges to as grows. However, it does not converge monotonically, in contrast to what happened in the numerical analysis of  for and . In tables 3, 4 we give additional numerical evidence for the validity of the quantization condition in the ABJ case with .
4 A case study with supersymmetry
As emphasized in , the quantization condition studied in the previous section is a consequence of a stronger result, namely our explicit formula (3.8) for the spectral determinant. In principle, using this formula, one can compute the canonical partition functions by performing an expansion around as in (2.21). In  this was checked in detail in the maximally supersymmetric cases, by using the computations of the partition functions in [9, 8, 10, 23, 22]. In the case with maximal supersymmetry, the generalized theta function becomes a standard Jacobi theta function, and the higher genus contribution to the modified grand potential vanish, so the analysis of the spectral determinant is relatively straightforward.
In this section we analyze in detail a case with supersymmetry, namely ABJM theory with . This case is slightly richer than the maximally supersymmetric cases because the grand potential involves the all genus worldsheet instantons. However, the generalized theta function of this theory becomes a conventional theta function, as in the maximally supersymmetric cases. Therefore this case is not the most generic one, but it is a good starting point to start exploring the spectral determinants of theories with supersymmetry.
It follows from (2.27) that the grand potential of ABJM theory with is given by
To calculate this quantity we have to take the limit in the general expression and be careful with the poles, as in . These however cancel, as we recalled above, so we can compute (4.1) by considering only its finite part. In particular has no finite part, as shown in appendix A, while the finite part of is
Let us now look at the worldsheet instanton part. For general and the expression (2.39) reads
It is convenient to split the sum over into even and odd part. This leads to