Mathematical Aspects of Higgs & Coulomb Branches

Abstract

This thesis contains results pertaining to different aspects of the 3d mirror symmetry between Higgs and Coulomb branches.

In Chapter 2 we verify the 3d A-model Higgs branch conjecture formulated in [BF23] for SQED with n > 3 hypermultiplets. The conjecture claims that the associated variety of the boundary VOA for the 3d A-model is isomorphic to the Higgs branch of the physical theory. We demonstrate that the boundary VOA is L1(psl(n|n)) and show that its associated variety is the closure of the minimal nilpotent orbit, verifying the conjecture.

In Chapter 3 we build on the work of [Web19a; Web22] by explicitly constructing a tilting generator for the derived category of coherent sheaves on T∗Gr(2, 4). This variety is the Coulomb branch for a quiver gauge theory and has functions described by a KRLW algebra. We achieve this result by constructing generators for modules over this diagrammatic algebra and identifying the coherent sheaves these correspond to.