The extended tropical vertex group

TL;DR

This thesis introduces the extended tropical vertex group, linking scattering diagrams with deformations of holomorphic pairs, and explores their applications in wall-crossing formulas and Gromov--Witten invariants.

Contribution

It develops the extended tropical vertex group and connects it to geometric wall-crossing formulas and Gromov--Witten invariants, expanding the theoretical framework.

Findings

Geometric interpretation of wall-crossing formulas for 2d-4d systems.

Appearance of Gromov--Witten invariants in commutator formulas.

Potential links to open/closed theories in geometry and physics.

Abstract

In this thesis we study the relation between scattering diagrams and deformations of holomorphic pairs, building on a recent work of Chan--Conan Leung--Ma. The new feature is the extended tropical vertex group where the scattering diagrams are defined. In addition, the extended tropical vertex provides interesting applications: on one hand we get a geometric interpretation of the wall-crossing formulas for coupled $2d$-$4d$ systems, previously introduced by Gaiotto--Moore--Neitzke. On the other hand, Gromov--Witten invariants of toric surfaces relative to their boundary divisor appear in the commutator formulas, along with certain absolute invariants due to Gross--Pandharipande--Siebert, which suggests a possible connection to open/closed theories in geometry and mathematical physics.