The extended tropical vertex group

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 [CCLM17a]. 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 byGaiotto–Moore–Neitzke [GMN12]. On the other hand, Gromov–Witten invariants of toricsurfaces relative to their boundary divisor appear in the commutator formulas, along with certain absolute invariants due to Gross–Pandharipande–Siebert [GPS10], which suggests a possible connection to open/closed theories in geometry and mathematical physics.