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^{3}

We show that

The study of supersymmetric gauge theories in compact manifolds has been considerably pushed forward in recent years, after the development of the localization method [

We study in particular a 3D

A more general case, also including adjoint hypermultiplets, has also been studied, e.g., in Refs. [

Other recent exact analytical evaluations of free energies of 3D supersymmetric gauge theories can be found in Refs. [

In addition, notice that, in spite of the apparent simplicity of this model, even simpler models in 3D, like the Abelian gauge theory studied in Ref. [

The case where all the

Notice that in the Abelian case, the theory is exactly solvable, since the observables then reduce to the evaluation of a very well-known Fourier transform

The following exact evaluations, given in terms of a number of Barnes

Therefore, the large-

We advance that the issue of Stokes line crossing will not appear while moving the physical parameters (which is essentially here the FI parameter) of a given theory, unless we take

The paper is organized as follows. In the next section, we show how Eqs. (

In Sect. 3, we focus exclusively on the partition function and show that the matrix model (

In the last section we study the asymptotics of the free energy, with FI parameter, discussing the crossing of Stokes lines and the ensuing appearance of exponentially small contributions, coming from expansions of Gamma functions of complex arguments, in the asymptotic expansion of the partition functions. Explicit duality tests are carried out for the analytical expressions obtained for the partition functions. We conclude with some open directions for further work.

We want the analytical evaluation (^{1}

The mass dependence is accounted for in the prefactor

In contrast to the case of Chern–Simons theory, the absence of a Gaussian factor in the matrix model representation (^{2}

The implication of this solution for the gauge theories is what we discuss here. The central result in Ref. [^{3}

The latter expression follows from Selberg’s integral (see Refs. [^{4}

As we see from the expressions below, involving Schur polynomials, in principle it seems that we need to consider the case of an even number of flavors and take, at least for the moment, the restricted view, shown above, for the FI parameter. For definiteness, we now take

Looking for potential poles or zeroes of the expressions, recall that the Barnes

For

Regarding Wilson loops, it seems that the required specialization of Schur polynomials puts some restriction on the parameters of our model but actually these admit an expression in terms of Beta functions and this provides an extension to the whole complex plane. We show this explicitly now.

Thus, we focus now on two interesting specific instances of Eq. (

In contrast, in this case, the solvability of the model, which is in general that of a multidimensional Beta function (Selberg integral), reduces to that of the ordinary Beta integral. This leads to compact exact expressions valid for all

Then, with

The general Wilson loop expression (

Notice that these expressions do not restrict

Because of Eq. (^{5}

Note that in these expressions the Gamma functions can be traded for Pochhammer symbols, using

Likewise, as for the partition function (but this time due to the Gamma functions and not the exponential prefactor, which is real), we have that

The case of antisymmetric representation with

Notice that this is the same as the result for the fundamental but with the complex conjugate in the denominator.

There are more general representations that can also be studied very explicitly. In particular, the case of rectangular partitions is specially interesting. Recall the statement made above, before Eq. (

Likewise, from Eq. (

Thus, this case is equivalent to that of a partition function with a complex FI parameter. This setting will be briefly discussed again at the end, when considering the asymptotics of Barnes

The last explicit case that we analyze is the one corresponding to partitions represented by hooks,

The asymptotics of these Wilson loop expressions is particularly rich because of having complex arguments in the Gamma functions, then the crossing of Stokes lines (and related phenomena, like Berry smoothing transitions across lines [

In this section, we show that the partition function of the matrix models also follows, after some change of variables, from the famous evaluation of the Selberg integral [

The evaluation of this integral is valid for complex parameters

Notice that, with only a very minor modification of the change of variables proposed in Ex. 4.1.3 in Ref. [

Once again, the partition function can be written in terms of Barnes

In the

For ^{6}

The final term that comes from the

Note that for the particular case of

Recall that the free energy of the larger family of 3D

Now, setting the mass and the FI parameter

Recall that the Vandermonde determinant of the matrix model in the case of the orthogonal and symplectic gauge groups is the corresponding hyperbolic version of the Haar measure, explicitly given by:

Thus, we find that, in terms of the

To conclude this section, notice that the

Notice that the expression diverges for

From the expression (

The presence of the FI parameter has an interesting implication: the expressions for the free energies and the Wilson loops, given above, are in terms of Barnes

Let us remind ourselves first that

We focus on the

The four

The Stokes lines are located at

We need the asymptotics of the Gamma function too, whose Stokes phenomenon is similar to that of the Barnes

The expansion of the logarithm gives the asymptotics in the same form as above:
^{7}

Taking into account the specific form of the Stokes multiplier parts in Eqs. (

There is a small difference between the cases corresponding to

For example, in the

Notice that this result and the one below also holds for finite

In the

In addition, this is the last case where the Gamma function asymptotics would not contribute at large but finite

For bad theories, we have

Now, localization on

Besides, we saw above that this case is equal to an unnormalized (not divided by the partition function) Wilson loop with a rectangular representation of the type

Some simple tests of Seiberg duality can be quickly carried out. This is an additional test of the analytical formula (

If we take

Thus, this is the duality between a good theory on the l.h.s. of Eq. (

We expect to study further the asymptotics together with the duality, including further discussion on the case of Wilson loops and the setting where a gauge-R Chern–Simons term is present, characterized by an additional imaginary part in the FI parameter. The asymptotics in this case will admit more possibilities and it should be possible to also look at it from the point of view of Borel transforms. The Mellin–Barnes type of integral given for the

We conclude by briefly commenting on the fact that the matrix model studied here is not only related to an extended Selberg integral [

The author is indebted to Masazumi Honda for many discussions and very valuable comments and questions. Thanks also to Jorge Russo and David Garc ía for comments on a preliminary version and to a referee for useful observations. This work is supported by the Fundãçao para a Ciência e Tecnologia (program Investigador FCT IF2014), under Contract No. IF/01767/2014.

Open Access funding: SCOAP

^{1}Precisely, we use

^{2}This leads to a discussion of the FI term, as this argument seemingly requires that the FI is either taken to be

^{3}We change their notation slightly, since their

^{4}Recall that

^{5}A way to prove Eq. (

^{6}

^{7}Note that, with regard to the location of the Stokes lines, the asymptotics of the Gamma function is with variable