Deformations of Toric Quiver Gauge Theories

Abstract

We explore various aspects of toric quiver gauge theories, with a particular focus on their deformations and the associated geometric and algebraic structures. We review fundamental concepts in algebraic geometry, toric geometry, and quiver representation theory, delving into the construction of toric varieties and quiver representations, examining their roles in the moduli of toric quiver gauge theories, particularly through Geometric Invariant Theory (GIT) and symplectic reduction.
We study 4d supersymmetric quiver gauge theories, emphasizing their representation via quiver diagrams and brane tilings, and analyzing dualities that link different quiver theories. Central to this thesis is the study of one-parameter families of -preserving deformations, defined by zig-zag paths in the brane tiling.
These deformations, which correspond to Hanany-Witten moves in the dual -web, are explored in the context of their impact on moduli spaces and RG flows between SCFTs on D3-branes probing local toric (pseudo) del Pezzo surfaces.
Finally, we extend our analysis to non-reflexive polytopes, applying algorithms to uncover flows between toric phases of quiver gauge theories, with particular attention to those described by toric diagrams with 2 internal points.