Moduli Space and Wall-Crossing Formulae in Higher-Rank Gauge Theories

TL;DR

This paper investigates wall-crossing phenomena and instanton effects in four-dimensional  gauge theories with  gauge group SU(n), verifying the Kontsevich-Soibelman formula and smoothness of the moduli space metric.

Contribution

It provides a detailed check of the Kontsevich-Soibelman wall-crossing formula and instanton contributions in higher-rank  gauge theories on  a  , extending previous results.

Findings

Verification of the Kontsevich-Soibelman formula at weak coupling.

Confirmation of the smoothness of the moduli space metric across walls.

Explicit calculation of the one-instanton contribution using nonlinear integral equations.

Abstract

We study the interplay between wall-crossing in four-dimensional gauge theory and instanton contributions to the moduli space metric of the same theory on $\mathbb{R}^{3}\times S^{1}$. We consider $\mathcal{N}=2$ SUSY Yang--Mills with gauge group SU(n) and focus on walls of marginal stability which extend to weak coupling. By comparison with explicit field theory results we verify the Kontsevich--Soibelman formula for the change in the BPS spectrum at these walls and check the smoothness of the metric in the corresponding compactified theory. We also verify in detail the predictions for the one instanton contribution to the metric coming from the non-linear integral equations of Gaiotto, Moore and Nietzke.