Wall Crossing from Dirac Zeromodes

TL;DR

This paper investigates wall crossing phenomena in BPS states within 4D N=2 super-Yang-Mills theories, using semiclassical analysis to derive explicit solutions and formulas, and explores the structure of monopole moduli spaces.

Contribution

It provides a semiclassical framework for understanding wall crossing, derives the primitive wall-crossing formula from Dirac operators, and introduces an asymptotic metric for singular monopole moduli spaces.

Findings

Recovered the primitive wall-crossing formula from asymptotic analysis.

Constructed explicit solutions illustrating wall crossing features.

Proposed evidence for stable non-BPS bound states.

Abstract

We explore the physics of two-body decay of BPS states using semiclassical analysis to construct explicit solutions that illustrate the main features of wall crossing, for both ordinary and framed BPS states. In particular we recover the primitive wall-crossing formula from an asymptotic analysis of certain Dirac-type operators on monopole moduli spaces. Along the way we give an asymptotic metric for the moduli space of singular monopoles, analogous to the Gibbons-Manton and Lee-Weinberg-Yi metrics for the moduli space of smooth monopoles, and we find evidence for the existence of stable non-BPS boundstates. Our discussion applies to four-dimensional N = 2 super-Yang-Mills theories with general gauge group and general 't Hooft defects.