Symmetries of Toric Duality

TL;DR

This paper explores the symmetries underlying toric duality in D-brane world volume theories, revealing a multiplicity symmetry that explains known dualities and relates to geometric isometries and monodromy.

Contribution

It introduces the concept of multiplicity symmetry in toric duality, providing a new combinatorial perspective and connecting it to monodromy and Seiberg duality.

Findings

Identifies permutation symmetries as the origin of toric duality.

Reveals combinatorial properties of multiplicities in gauged linear sigma models.

Links multiplicity symmetry to geometric isometries and superpotential structures.

Abstract

This paper serves to elucidate the nature of toric duality dubbed in hep-th/0003085 in the construction for world volume theories of D-branes probing arbitrary toric singularities. This duality will be seen to be due to certain permutation symmetries of multiplicities in the gauged linear sigma model fields. To this symmetry we shall refer as ``multiplicity symmetry.'' We present beautiful combinatorial properties of these multiplicities and rederive all known cases of torically dual theories under this new light. We also initiate an understanding of why such multiplicity symmetry naturally leads to monodromy and Seiberg duality. Furthermore we discuss certain ``flavor'' and ``node'' symmetries of the quiver and superpotential and how they are intimately related to the isometry of the background geometry, as well as how in certain cases complicated superpotentials can be derived by observations of the symmetries alone.