On a singular Fredholm-type integral equation arising in N=2 super Yang-Mills theories

TL;DR

This paper analyzes a specific integral equation from N=2 supersymmetric Yang-Mills theories, solving it exactly, and deriving the instanton partition function with implications for quantization and matrix model methods.

Contribution

It provides an exact solution to a Fredholm-type integral equation in N=2 super Yang-Mills theories and connects it with matrix model techniques to derive the full instanton partition function.

Findings

Exact solution to the integral equation in the Nekrasov-Shatashvili limit

Quantization conditions for theory parameters derived from finiteness of the energy

Field theoretical expression for the instanton partition function obtained

Abstract

In this work we study the Nekrasov-Shatashvili limit of the Nekrasov instanton partition function of Yang-Mills field theories with N=2 supersymmetry and gauge group SU(n). The theories are coupled with fundamental matter. The equation that determines the density of eigenvalues at the leading order in the saddle-point approximation is exactly solved. The dominating contribution to the instanton free energy is computed. The requirement that this energy is finite imposes quantization conditions on the parameters of the theory that are in agreement with analogous conditions that have been derived in previous works. Using methods borrowed from the theory of matrix models, a field theoretical expression of the full instanton partition function is derived. It is checked that in the Nekrasov-Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations.