Exact partition functions for the $\Omega$-deformed $\mathcal N=2^{*}$ $SU(2)$ gauge theory

TL;DR

This paper derives exact expressions for the partition functions of the $ abla$-deformed $ abla$-N=2* $SU(2)$ gauge theory, revealing special points where the prepotential simplifies and can be expressed in closed form, with implications for the AGT correspondence.

Contribution

It identifies special points in the deformation parameter space where the instanton prepotential simplifies and can be expressed explicitly using modular functions, advancing understanding of the theory's structure.

Findings

Exact closed-form expressions for the partition function at special points.

Identification of finite sets of points with pole structures independent of instanton number.

Verification of the modular anomaly equation at all orders at these points.

Abstract

We study the low energy effective action of the $\Omega$-deformed $\mathcal N =2^{*}$ $SU(2) $ gauge theory. It depends on the deformation parameters $\epsilon_{1},\epsilon_{2}$, the scalar field expectation value $a$, and the hypermultiplet mass $m$. We explore the plane $(\frac{m}{\epsilon_{1}}, \frac{\epsilon_{2}}{\epsilon_{1}})$ looking for special features in the multi-instanton contributions to the prepotential, motivated by what happens in the Nekrasov-Shatashvili limit $\epsilon_{2}\to 0$. We propose a simple condition on the structure of poles of the $k$-instanton prepotential and show that it is admissible at a finite set of points in the above plane. At these special points, the prepotential has poles at fixed positions independent on the instanton number. Besides and remarkably, both the instanton partition function and the full prepotential, including the perturbative contribution, may be given in closed form as functions of the scalar expectation value $a$ and the modular parameter $q$ appearing in special combinations of Eisenstein series and Dedekind $\eta$ function. As a byproduct, the modular anomaly equation can be tested at all orders at these points. We discuss these special features from the point of view of the AGT correspondence and provide explicit toroidal 1-blocks in non-trivial closed form. The full list of solutions with 1, 2, 3, and 4 poles is determined and described in details.