Gaussian and Mean Field Approximations for Reduced 4D Supersymmetric Yang-Mills Integral

TL;DR

This paper applies Gaussian and mean field approximations to a reduced 4D supersymmetric Yang-Mills integral, successfully calculating key observables and demonstrating good agreement with known results and numerical simulations.

Contribution

It introduces an improved Gaussian approximation method for supersymmetric Yang-Mills integrals and extends previous bosonic approaches to the supersymmetric case.

Findings

Accurate calculation of free energy, Polyakov loop, and Wilson loop expectations.

Results match exact and numerical data, confirming the approximation's validity.

Reproduction of known scaling behaviors and large N limit formulas.

Abstract

In this paper, we consider a reduced supersymmetric Yang-Mills integral with four supercharges by using a Gaussian approximation scheme and its improved version. We calculate the free energy and the expectation values of Polyakov loop and Wilson loop operators by extending the method employed in the bosonic case in the previous paper. Our results nicely match to the exact and the numerical results obtained before. The loop amplitudes exhibit good scaling behaviors similarly as in the bosonic case. The 't Hooft like large $N$ limit leads simple formulas for the case of the loop length smaller. Also, the Polyakov loop and the Wilson loop are computed for the case of the loop length sufficiently large, where we see that the behavior of the Wilson loop reproduces the result simulated for a few smaller values of $N$ at least qualitatively.