Seiberg Duality for Quiver Gauge Theories

TL;DR

This paper provides an algebraic and mathematical framework for understanding Seiberg duality in N=1 supersymmetric quiver gauge theories, connecting physical dualities with derived category tilting equivalences.

Contribution

It introduces an algebraic approach to Seiberg duality, defining brane concepts purely mathematically and linking duality to tilting equivalences in derived categories.

Findings

Matching moduli spaces confirm duality

Spectrum and superpotential computed from first principles

Connects physical duality with mathematical tilting theory

Abstract

A popular way to study N=1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized. We abstract from these arguments the idea that Seiberg duality arises because a configuration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theories obtained this way have matching moduli spaces, the primary test of Seiberg duality. Furthermore, we do this by defining ``brane'' and all the other ingredients of such arguments purely algebraically, for a very large class of N=1 quiver supersymmetric gauge theories, making physical intuitions about brane-antibrane systems and tachyon condensation precise using the tools of homological algebra. These techniques allow us to compute the spectrum and superpotential of the dual theory from first principles, and to make contact with geometry and topological string theory when this is appropriate, but in general provide a more abstract notion of ``noncommutative geometry'' which is better suited to these problems. This makes contact with mathematical results in the representation theory of algebras; in this language, Seiberg duality is a tilting equivalence between the derived categories of the quiver algebras of the dual theories.