Seiberg Duality for Quiver Gauge Theories
David Berenstein, Michael R. Douglas
TL;DR
This work reframes Seiberg duality for N=1 quiver gauge theories as a change of basis in brane configurations, formalized algebraically through bound-state complexes and Ext groups. It develops a general, constructive procedure—via the reflection functor, FI-term control, and an appropriately chosen superpotential—to derive dual quivers and their spectra, including cases with adjoint matter, and connects these to D-brane realizations and derived-category tilting. By embedding these ideas in quiver categories and path-algebras, the authors show that Seiberg-like dualities are tilting equivalences of derived categories, offering a rigorous mathematical underpinning and extending the duality to toric and non-geometric settings. The framework clarifies how moduli-space matching serves as a robust classical test while highlighting the role of brane–antibrane annihilation as a generalized gauge equivalence, with potential to uncover new dualities via categorical mutations. Overall, the paper unifies physical dualities with noncommutative geometry and representation theory, linking toric duality, partial resolutions, and Calabi–Yau conditions within a single, versatile formalism.
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.