Oscillatory matrix model in Chern-Simons theory and Jacobi-theta determinantal point process
The partition function of the Chern-Simons theory on the three-sphere with the unitary group provides a one-matrix model. The corresponding -particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as -extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the -particle system to an infinite-dimensional determinantal point process in , in which the correlation kernel is expressed by Jacobi’s theta functions. We show that the matrix model obtained by this limit realizes the oscillatory matrix model in Chern-Simons theory discussed by de Haro and Tierz.
Chern-Simons theory on a three-manifold with a simply-laced gauge group is specified by the action
where is a -connection on and is an integer. The partition function of the Chern-Simons theory is then given by
with . Based on [31, 32], Mariño showed in  that the partition function of Chern-Simons theory on Seifert spaces can be calculated in a combinatorial way and expressed by multiple integrals. In particular, when the gauge group is chosen as the unitary group , , the Chern-Simons partition function on the three-sphere is expressed by 
where the string coupling constant is given by
and regarded (1.3) as the partition function of matrix model associated with the Stieltjes-Wigert polynomials , , which are -extensions of Hermite polynomials (see Section 2.1). He performed the integral (1.3) by using orthonormality of ’s, which is generally valid for , and obtained the exact and explicit expression for (1.3)
The above fact leads us to consider the Chern-Simons partition function (1.3) in the way that the constant is a free parameter and is not restricted by (1.4). In the present paper, we consider the case that the parameter is positive and regard (1.3) as the partition function of a statistical mechanics system of particles. The variables , in the integral (1.3) are considered to be realizations of random variables on whose probability law is given by the probability density function
Here is the normalization constant which will be explicitly given in Section 2.1. From it, the partition function (1.3) is obtained by
Let . By the mapping
(1.7) is transformed to the probability density function
where is given by
In the Gaussian unitary ensemble (GUE) with variance , the eigenvalues of random Hermitian matrices obey the probability density
where [17, 6]. In the present ensemble (1.10), individual point follows the log-normal distribution on instead of the Gaussian distribution on , while the repulsive interactions represented by are common.
Let , be random variables having the probability density function (1.10) on . Since as well as are symmetric functions of , , we shall represent a configuration as unlabeled. Let be the space of nonnegative integer-valued Radon measure on . Any element is represented as with a countable index set , where denotes a point mass (the delta measure) on . There a sequence of points in , , satisfies for any compact subset . Then we regard the present particle system as -valued and write it as with
Let be the set of all continuous real-valued function with compact support on . For , the moment generating function of the system is given by the following generalized Laplace transform of the distribution (1.10),
where and denotes the expectation with respect to . It is expanded with respect to the ‘test function’ as
where denotes , . Here gives the -point correlation function for , which is a symmetric function, . Given an integral kernel , , a Fredholm determinant with is defined as
If the system has an integral kernel such that any moment generating function (1.14) is given by a Fredholm determinant (1.16), is said to be a determinantal point process with the correlation kernel [22, 24]. By definition, we have
, . Note that the terminology ‘point process’ does not mean any stochastic process but does a spatial distribution of points as usually used in probability theory (e.g. Poisson processes). Determinantal point process is also called fermion point process [24, 22].
where , , are the Stieltjes-Wigert polynomials, , are their derivatives, and is their weight function for orthogonality, which will be explicitly given in Section 2.1. The second equality in (1.19) is given by the Christoffel-Darboux formula .
This fact implies that the original system
The purpose of the present paper is to consider an limit of the systems and . For , let
and for , let be the least integer not less than . Then for and , we will prove
(Proposition 3), where
with the weight function , and the conditional limit depending on , denoted by , is defined at the beginning of Section 3.1. Here is a version of Jacobi’s theta function defined by (see, for instance, ),
for and , and is its derivative
Thus we call the correlation kernel given by (1.25) and (1.26) the Jacobi-theta kernel. The convergence (1.24) of correlation kernel in implies that of moment generating function to the Fredholm determinant associated with the Jacobi-theta kernel (1.25), (1.26),
. Then all correlation functions , are determined and expressed by determinants. In this sense, as the limit of and , determinantal point processes with infinite numbers of particles are obtained (Theorem 4 and Corollary 5).
In  de Haro and Tierz discussed an oscillatory matrix model, which seems to appear in sufficiently large but finite in the Chern-Simons theory with the gauge. With the restriction (1.4), the limit is identified with the ’t Hooft limit; with . Since it corresponds to by (1.5), the oscillatory behavior will vanish and only classical matrix model is obtained in the limit. The oscillatory matrix model of de Haro and Tierz is realized as a crossover phenomenon in . In the present paper, we fix so that and take limit . The infinite-dimensional determinantal point process obtained by this limit of will be a stationary realization of the oscillatory matrix model observed by de Haro and Tierz . The oscillatory behavior will be demonstrated in Section 4.1 with figures. In Section 4.2 we will confirm that if we take the further limit (i.e. ), the system becomes classical with the sine-kernel as it should. In other words, in the context of random matrix theory [17, 6] the present paper reports a -extension of the bulk scaling limit of the Hermite kernel in GUE. The -extension of the edge scaling limit described by the Airy kernel will be reported in a forthcoming paper .
The paper is organized as follows. In Section 2, we define the Stieltjes-Wigert polynomials and give their asymptotic expansions as the degree of polynomials . In Section 3, we present asymptotic form of the Stieltjes-Wigert kernel described by (1.19) and (1.20) and explain its connection with the oscillatory matrix model. Section 4 is devoted to showing the oscillatory behaviors of the infinite-particle systems. Proofs of Lemma 1 and Proposition 3 are given in Section 5. In Appendix A, we rewrite the correlation kernel of the oscillatory matrix model in standard notations of theta functions  and in terms of Gosper’s -trigonometric functions . Appendix B complements the proof of Lemma 1.
2.1 Some -special functions
For and , we introduce the -Pochhammer symbol
The following identity follows from the -binomial theorem ,
For and , the orthonormal Stieltjes-Wigert polynomials are defined by 
They satisfy the orthonormality relations
with respect to the weight function
This gives a density for a log-normal distribution and solves the functional equation
We have the identity
The theta function (1.28) can be written as
which is called Jacobi’s triple product identity. One can prove the functional equation
directly from the definition (1.27).
A -exponential function is defined as
2.2 Asymptotic expansions for the Stieltjes-Wigert polynomials
For , , let
where denotes the integer part of . We also introduce the indicator function of a set such that if and otherwise.
Lemma 1 Let and . Then the orthonormal Stieltjes-Wigert polynomials (2.5) have the following asymptotic expansions as the degree of polynomials ,
The proof is given in Section 5.1 with Appendix B. Since the Stieltjes-Wigert polynomial is a -extension of the Hermite polynomials , this result can be regarded as a -analogue of the celebrated Plancherel-Rotach asymptotic formula . We note that the leading term given by the first term in the parenthesis in the RHS was given by Ismail and Zhang as equations (2.19) and (2.23) in  (and (16) and (19) in ). This lemma improves their estimate. In order to obtain the Jacobi-theta kernel given by (1.25) and (1.26) as a limit of the Stieltjes-Wigert kernel expressed by (1.19) and (1.20), the correction term given by the second term in the parenthesis is necessary (Proposition 3). Owing to the factor with (2.15) in the formula, asymptotic behavior in will depend on whether is rational or irrational as discussed in .
Lemma 1 gives the following asymptotic expansions for multiplied by the weight function (2.7); as ,
Then, we can find the following.
Lemma 2 For , ,
3 Main theorems
3.1 Jacobi-theta determinantal point process
Since we obtained the asymptotic forms of the Stieltjes-Wigert polynomials as Lemmas 1 and 2, we will be able to determine the asymptotics of the Stieltjes-Wigert kernel given by (1.19) and (1.20) in . For a technical reason, here we assume . We consider a monotonically increasing series of integers such that the equalities
holds. Note that for . Then and are defined as subsequences of such that are even and are odd, respectively, . For a given , we take the limit following the subsequences and . We write this conditional limit as .
We have the following result.
Proposition 3 Let and . Then (1.24) holds.
We expect that the statement will be extended for , but we need further improvement of Lemmas 1 and 2 to prove it.
Proposition 3 means the convergence of integral operators
. The convergence of integral operators implies that of Fredholm determinants to (1.30) for . Since the Fredholm determinants are identified with the moment generating functions in determinantal point process, we can conclude the following.
3.2 Mapping to the matrix model
We obtained an infinite-particle system on in the previous subsection. Here, we explain that the particle system is then mapped to an infinite-particle system on , which will be regarded as a stationary realization of the oscillatory matrix model considered in .
First, we remind that the -particle systems and were related by the mapping (1.9). On the other hand, when we take the limit in Theorem 4, we performed the scaling of variables as
This suggests that, only in the case of , the -dependent factor vanishes. (The value of the factor is fixed to be in the series and in by the definition of , (1.23), and of .) Hence, in the case , if we take an infinite-particle limit of the scaled version of by (3.4), we can obtain an infinite-dimensional model in Chern-Simons theory by simply putting .
Then, for with correlation kernel (1.22), we have the following corollary from Proposition 3 and Theorem 4.
Corollary 5 Set . Then
Then, the system converges to an infinite-dimensional determinantal point process on with the correlation kernel in . The correlation functions are given by
for the limit system, where denotes