Notes on Wall Crossing and Instanton in Compactified Gauge Theory with Matter

TL;DR

This paper investigates quantum corrections to the Coulomb branch of N=2 SU(2) supersymmetric Yang-Mills with matter on R^3 x S^1, confirming wall-crossing predictions through explicit instanton calculations and geometric interpretations.

Contribution

It provides explicit perturbative and non-perturbative corrections to the moduli space metric, verifying wall-crossing predictions with field theory computations and geometric insights.

Findings

Confirmed the predicted metric from wall-crossing theory.

Computed leading instanton contributions matching predictions.

Connected 3D quantities with brane geometric pictures.

Abstract

We study the quantum effects on the Coulomb branch of N=2 SU(2) supersymmetric Yang-Mills with fundamental matters compactified on R^3 x S^1, and extract the explicit perturbative and leading non-perturbative corrections to the moduli space metric predicted from the recent work of Gaiotto, Moore and Neitzke on wall-crossing [1]. We verify the predicted metric by computing the leading weak coupling instanton contribution to the four fermion correlation using standard field theory techniques, and demonstrate perfect agreement. We also demonstrate how previously known three dimensional quantities can be recovered in appropriate small radius limit, and provide a simple geometric picture from brane construction.