Wall-Crossing in Coupled 2d-4d Systems

TL;DR

This paper introduces a unified wall-crossing formula for 2d-4d supersymmetric systems, linking BPS state changes with hyperholomorphic connections and providing tools for analyzing theories of class S and Hitchin equations.

Contribution

It presents a new wall-crossing formula that generalizes existing formulas for 2d and 4d systems, applicable to coupled 2d-4d supersymmetric theories and their geometric interpretations.

Findings

Derived a combined wall-crossing formula for 2d-4d systems.

Connected wall-crossing behavior to hyperholomorphic connections on hyperkahler spaces.

Applied the framework to theories of class S and solutions to Hitchin equations.

Abstract

We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class S, that is, for those theories obtained by compactifying the six-dimensional (0,2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.