The extended tropical vertex group

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.

arXiv comment103 pages, 14 figures. PhD thesis of the author, which includes parts of arXiv:1912.09956

References in corpus (6)

• On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)
• Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
• Open Gromov-Witten invariants, mirror maps, and Seidel representations for toric manifolds
• Scattering diagrams from asymptotic analysis on Maurer-Cartan equations
• On 2d-4d motivic wall-crossing formulas
• Deformations of holomorphic pairs and 2d-4d wall-crossing