Consistency conditions for brane tilings

Abstract

Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendrői, Mozgovoy and Reineke regarding the Donaldson–Thomas theory of the moduli space of framed cyclic representations of the associated algebra. Using only a natural cancellation-type consistency condition, we show that the algebras are 3-Calabi–Yau, and calculate Donaldson–Thomas type invariants of the moduli spaces. Two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations, and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to the universal cover. These ideas allow us to generalise the above results to all consistent brane tilings on K ( π , 1 ) surfaces. We also prove a converse: no consistent brane tiling on a sphere gives rise to a 3-Calabi–Yau algebra.