Four-dimensional wall-crossing via three-dimensional field theory

TL;DR

This paper provides a physical explanation for the Kontsevich-Soibelman wall-crossing formula in Seiberg-Witten theories, describing BPS spectrum changes and instanton corrections to the hyperkahler metric on R^3 x S^1.

Contribution

It introduces a four-dimensional analogue of tt* equations to describe BPS instanton corrections and explains the wall-crossing formula as a statement of metric continuity.

Findings

Derived an exact description of BPS instanton corrections

Connected wall-crossing to metric continuity in hyperkahler geometry

Extended tt* equations to four-dimensional theories

Abstract

We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on R^3 x S^1. The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.