Exact results in N=2 gauge theories

TL;DR

This paper derives exact formulas for partition functions and Wilson/'t Hooft loop expectation values in specific N=2 gauge theories on S^4, confirming S-duality and simplifying calculations via localization and hypergeometric functions.

Contribution

It provides explicit exact results for N=2 gauge theories with fundamental matter, including localization around critical points and connections to AGT correspondence.

Findings

Partition function expressed in terms of hypergeometric functions.

Localization at finite critical points simplifies instanton contributions.

Verification of S-duality transformations for Wilson/'t Hooft loops.

Abstract

We derive exact formulae for the partition function and the expectation values of Wilson/'t Hooft loops, thus directly checking their S-duality transformations. We focus on a special class of N=2 gauge theories on S^4 with fundamental matter. In particular we show that, for a specific choice of the masses, the matrix model integral defining the gauge theory partition function localizes around a finite set of critical points where it can be explicitly evaluated and written in terms of generalized hypergeometric functions. From the AGT perspective the gauge theory partition function, evaluated with this choice of masses, is viewed as a four point correlator involving the insertion of a degenerated field. The well known simplicity of the degenerated correlator reflects the fact that for these choices of masses only a very restrictive type of instanton configurations contributes to the gauge theory partition function.