Seiberg duality as derived equivalence for some quiver gauge theories

TL;DR

This paper demonstrates that Seiberg duality in certain quiver gauge theories can be understood as a derived equivalence between their module categories, providing a new algebraic perspective on gauge theory dualities.

Contribution

It establishes a connection between Seiberg duality and derived category equivalences, offering a systematic method to construct dual quivers via tilting complexes.

Findings

Seiberg duality corresponds to derived equivalence of path algebra categories.

Construction of tilting complexes yields dual quivers with equivalent derived categories.

A general scheme for obtaining dual quivers through derived equivalences is presented.

Abstract

We study Seiberg duality of quiver gauge theories associated to the complex cone over the second del Pezzo surface. Homomorphisms in the path algebra of the quivers in each of these cases satisfy relations which follow from a superpotential of the corresponding gauge theory as F-flatness conditions. We verify that Seiberg duality between each pair of these theories can be understood as a derived equivalence between the categories of modules of representation of the path algebras of the quivers. Starting from the projective modules of one quiver we construct tilting complexes whose endomorphism algebra yields the path algebra of the dual quiver. Finally, we present a general scheme for obtaining Seiberg dual quiver theories by constructing quivers whose path algebras are derived equivalent. We also discuss some combinatorial relations between this approach and some of the other approaches which has been used to study such dualities.