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

TL;DR

This work provides a physical and geometric realization of the Kontsevich-Soibelman wall-crossing formula for 4d ${\cal N}=2$ theories by studying the gauge theory on ${\mathbb{R}}^3\times S^1$ and its hyperkähler moduli space ${\cal M}$. The authors develop a twistorial (tt$^*$-like) framework in which BPS instanton corrections are encoded in holomorphic Darboux coordinates ${\cal X}_{\gamma}(u,\theta;\zeta)$ with jumps along BPS rays implementing KS symplectomorphisms; the resulting Riemann-Hilbert problem yields a continuously varying metric $g$ across walls. They distinguish mutually local and non-local contributions, derive exact single-particle and higher-rank corrections, and show that the full metric is smooth provided the KS wall-crossing formula holds, effectively proving the WCF from physical consistency. The work also connects these structures to integrable systems through a Thermodynamic Bethe Ansatz–like formulation and situates the construction within a broader twistor/string-theory perspective, with implications for including masses and flavor symmetries.

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.