^{3}

A class of matrix models that arises as a partition function in U(

Recent progress in supersymmetric Chern–Simons matter theories has been made on the basis of the exact result by means of supersymmetric localization. This technique allows one to compute the partition function of supersymmetric theories exactly by the steepest descent method, which reduces the path integral calculation to that of a certain matrix model [

On the other hand, progress in non-supersymmetric Chern–Simons matter theories has also been made not relying on the localization technique but on the

In contrast, the exact large-

In this situation we change gears to study a class of matrix models that is to be obtained as the three-sphere partition function of Chern–Simons matter theories in order to capture a generic feature of such a class of matrix models toward the bigger goal of showing bosonization duality on the three-sphere. We are also interested in analyzing such a class of matrix models beyond the planar limit because the bosonization duality is expected to hold at the subleading order in the

To illustrate the kind of matrix models that are to be studied, let us consider the partition function of generic U

We take the Lorenz gauge

Then, by denoting the correction of the matter fields to the effective action by

In this note we analyze this class of matrix models by restricting the form of the potential to consist of single trace operators: ^{1}

The rest of this paper is written as follows. In

Throughout this paper we investigate a class of matrix models such that

See Refs. [

For explicit computation we write the form of the potential in Eq. (

See Eq. (^{2}

The matrix model of Eq. (

As a result, the partition function can be written by using a positive definite Hermitian matrix

There is a well-known method to determine the free energy in matrix models by using the so-called resolvent, which is in the current situation defined by the vacuum expectation value of the generating function of regular single trace operators:

We remark that the resolvent is well defined around infinity and formally behaves as

Once the resolvent is determined, the coupling dependence on

In order to determine the resolvent systematically, we first derive the Schwinger–Dyson equation for a generic operator

Consider a one-to-one transformation on

Suppose the (infinitesimal) transformation

To derive the loop equation from the Schwinger–Dyson equation let us choose

The first two terms are computed as

The third term is

Hereafter we assume that

By using this relation, Eq. (

Collecting these, we obtain the loop equation

Note that this is of the same form as that of the ordinary matrix models except for the integration region.

Once the resolvent is determined by the loop equation, so are the density function by Eq. (

Similarly, the

Acting

We remark that it is not guaranteed and has to be confirmed that correlators computed in this way agree with those computed from the original theory by using the path integral.^{3}

In this section we solve the loop equation in Eq. (

In the examples of supersymmetric Chern–Simons theory in

Plugging this into the loop equation and expanding with respect to

From these equations,

Let us solve the planar loop equation of Eq. (

Therefore we obtain

Suppose that the support of the density function consists of

As mentioned in the previous section, we solve the loop equation so that the resolvent behaves as ^{4}

Then the planar density function is computed from Eq. (

The endpoints of the cuts are determined in the following way. Assume that the solution obtained above behaves asymptotically as

Differentiating this with respect to ^{5}^{6}

There is a comment on the solution in Eq. (

The free energy in the leading order of the

The hole correction of the resolvent is determined by the genus-half loop equation in Eq. (

As in the planar case, we compute the discontinuity of the left-hand side between

That of the second term is

Therefore we obtain

Let us determine the genus-one correction of the resolvent. For simplicity we first study the case where

This can be solved in the same manner as in the ordinary Hermitian matrix model [

Then the second term is computed as

The third term is

Therefore we obtain

In order to compute

Substituting Eq. (

This can be rewritten as the image of the linear operator

Equivalently,

The case with

Hence, if we define ^{7}

Next we consider the case where the matrix model potential contains the

Since

This equation is of the same form as the one without the hole correction by replacing

In this section we apply the presented formulation developed in the previous section to a few examples. First we apply it to the three-sphere partition function in U

The matrix model potential for pure Chern–Simons theory is given by Eq. (

Let us first determine the planar resolvent. For

The edges of the cut,

The planar density function is computed as

This solution matches the one given in Ref. [

(a) The blue and yellow curves depict the matrix model potential and the planar density function, respectively, when

The planar free energy is given by Eq. (

We could not perform the integration on the right-hand side analytically, so instead we evaluated it numerically. The numerical result is in good agreement with the past exact result of Eq. (

Next we study the genus-one correction. Now we consider the one-cut solution, so the genus-one correction of the resolvent is given by

By inflating the contour to infinity, the first term is computed as

The genus-one correction of the free energy is computed by using Eq. (

This time we could perform the integral analytically:

This result is in precise agreement with the past exact result of Eq. (

Next we consider the case where the matrix model potential has a hole correction by the “FI parameter”:^{8}^{9}

The edges of the cut are also determined in the same way. The result is

The third term is

As a result, we obtain

This is in perfect agreement with the past exact result of Eq. (

As another example we consider the matrix model of

Without losing generality, we can assume that

Its derivative is

In order to determine the number of cuts in the resolvent by identifying that of the potential minimum, we regard the matrix model potential as an analytic function with all the parameters. When

We consider the large-

We compute the resolvent up to the hole correction by Eq. (

Then the resolvent becomes

The edges of the cut

From these equations the edges of the cut are determined order by order in

As argued in

Then, as argued in

It is known that this system has the dual description known as Seiberg-like duality [

In this paper we have performed a general analysis of a class of matrix models describing Chern–Simons matter theories on the three-sphere incorporating the standard technique of

This paper mainly focused on the construction of the framework to solve a class of matrix models. We are very much interested in applying the formula obtained in this paper to a duality pair of Chern–Simons matter systems and testing that the bosonization duality holds at the next leading order in the

In this paper, in order to study beyond the planar limit we adopted the iterative procedure given in Refs. [

Another interesting question is whether this class of matrix models has the equivalent description of some two-dimensional CFTs as ordinary Hermitian matrix models [

There is a straightforward generalization of the presented formulation to a different gauge group [

We hope to come back to these issues in the near future.

The author would like to thank S. Sugimoto and T. Takayanagi for valuable discussions and comments on the draft. The author would also like to thank Y. Imamura for a helpful comment on the first version of this paper.

Open Access funding: SCOAP

In this appendix we give a brief overview of the three-sphere partition function in U

The partition function is defined formally by a path integral over the gauge field on ^{10}

The classic paper Ref. [

After this exact result was studied in terms of the

It was subsequently pointed out that the partition function in Eq. (

This matrix model was extensively studied in relation to topological string theory [

Let us compute the matrix model in a Fermi-gas-like approach [

For this purpose we rewrite the partition function as a determinant by using the Weyl denominator formula

From this formula we can show that

The inside of the determinant is computed by Gaussian integration as follows:

Plugging this back in gives

Then the free energy is computed as

The

By using

Here, ^{11}

The summation in the last term in Eq. (^{12}

Plugging these back in, we obtain the free energy as

As a result, the coefficients in the

A few comments are in order. The leading term in the

^{1} This restriction corresponds to the representation of the matter fields excluding higher-dimensional representations such as the adjoint one.

^{2} This expansion may not be useful for practical computation, though.

^{3} The author would like to thank S. Sugimoto for discussion on this point.

^{4} One may more generally conclude that, for example,

^{5} It is possible to consider a case where the Lagrange multiplier in Eq. (

^{6} The chemical potentials mentioned in footnote 5 can be added such that

^{7} The differential equation in Eq. (

^{8} The usage of this terminology can be justified by adding some auxiliary fields into the pure Chern–Simons theory so that the theory has

^{9} The planar resolvent and the hole one are determined from this by

^{10} Here,

^{11} Our definition of the Bernoulli number is different from the one adopted in several past works such as Refs. [

^{12} This can be proved by using