The thermal instanton determinant in compact form

TL;DR

This paper simplifies the calculation of the thermal instanton determinant for SU(2) gauge theory, reducing it to a manageable form involving only two functions and a single numerical integral, with applications to free energy calculations.

Contribution

It presents a compact, simplified expression for the thermal instanton determinant in SU(2) gauge theory, accommodating various boundary conditions.

Findings

Reduced the determinant to a form with two functions

Numerical integration involves only a two-dimensional integral

Applied method to compute free energy contribution in $\

Abstract

The thermal instanton determinant for the gauge group $SU(2)$ can be reduced to a form involving two simple functions. Various boundary conditions can easily incorporated. Only a two dimensional integral has to be done numerically. As an example we compute the contribution to the free energy of $\mathcal{N}=1$ theory.