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. CONTENTS 1. Introduction 1.1. Plan of the paper 1.2. Acknowledgements 2. Background 2.1. Setting 2.2. Deformations of holomorphic pairs 2.3. Scattering diagrams 2.4. Fourier transform 2.5. Symplectic dgLa 3. Deformations associated to a single wall diagram 3.1. Ansatz for a wall 3.2. Asymptotic behaviour of the gauge ϕ 4. Scattering diagrams from solutions of Maurer-Cartan 4.1. From scattering diagram to solution of Maurer-Cartan 4.2. From solution of Maurer-Cartan to the saturated scattering diagram D ∞ 4.3. Consistency of D ∞ 5. Relation with the wall-crossing formulas in coupled 2d -4d systems Appendix A. Computations MR 2483955

No abstract

No abstract