Brane Tilings and Reflexive Polygons

TL;DR

This paper explores the relationship between reflexive polygons and quiver gauge theories, identifying all theories associated with these polygons and revealing dualities in their moduli space structures.

Contribution

It systematically classifies 30 quiver gauge theories linked to 16 reflexive polygons and uncovers duality relations based on the duality of reflexive polygons.

Findings

Identified all theories corresponding to reflexive polygons.

Established duality pairs of quiver gauge theories.

Linked the lattice of generators to the dual reflexive polygon.

Abstract

Reflexive polygons have attracted great interest both in mathematics and in physics. This paper discusses a new aspect of the existing study in the context of quiver gauge theories. These theories are 4d supersymmetric worldvolume theories of D3 branes with toric Calabi-Yau moduli spaces that are conveniently described with brane tilings. We find all 30 theories corresponding to the 16 reflexive polygons, some of the theories being toric (Seiberg) dual to each other. The mesonic generators of the moduli spaces are identified through the Hilbert series. It is shown that the lattice of generators is the dual reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver gauge theories with the lattice of generators being the toric diagram of the dual and vice versa.