Deformations of holomorphic pairs and 2d-4d wall-crossing

TL;DR

This paper explores the geometric interpretation of wall-crossing formulas in 2d-4d systems through the deformation theory of holomorphic pairs, linking scattering diagrams with complex geometric structures.

Contribution

It establishes a novel connection between wall-crossing phenomena and the deformation theory of holomorphic pairs, expanding the geometric understanding of these formulas.

Findings

Wall-crossing formulas can be interpreted geometrically via holomorphic pair deformations.

A relation between scattering diagrams and deformations of holomorphic pairs is developed.

The work builds on recent advances by Chan, Conan Leung, and Ma.

Abstract

We show how wall-crossing formulas in coupled 2d-4d systems, introduced by Gaiotto, Moore and Neitzke, can be interpreted geometrically in terms of the deformation theory of holomorphic pairs, given by a complex manifold together with a holomorphic vector bundle. The main part of the paper studies the relation between scattering diagrams and deformations of holomorphic pairs, building on recent work by Chan, Conan Leung and Ma.