Consistency conditions for brane tilings

TL;DR

This paper introduces a new method to analyze brane tilings on tori, proving their associated algebras are 3-Calabi-Yau and computing related invariants, with generalizations to other surfaces.

Contribution

It provides a novel proof technique using consistency conditions, introduces grading and winding number concepts, and extends results to all consistent brane tilings on K(pi,1) surfaces.

Findings

Algebras from consistent brane tilings are 3-Calabi-Yau.

Calculated Donaldson-Thomas invariants for moduli spaces.

No sphere-based brane tiling yields a 3-Calabi-Yau algebra.

Abstract

Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendroi, 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 R^2. These ideas allow us to generalise the above results to all consistent brane tilings on K(pi,1) surfaces. We also prove a converse: no consistent brane tiling on a sphere gives rise to a 3-Calabi-Yau algebra.