On the noncommutative Donaldson-Thomas invariants arising from brane tilings

TL;DR

This paper establishes a formula for noncommutative Donaldson-Thomas invariants derived from brane tilings, connecting algebraic, combinatorial, and geometric aspects of the associated quiver potential algebras.

Contribution

It provides a new formula for DT invariants from brane tilings and links these invariants to perfect matchings and 3-Calabi-Yau properties of the algebra.

Findings

Derived a formula for DT invariants from brane tilings

Connected invariants to perfect matchings of tilings

Proved the algebra is 3-Calabi-Yau under certain conditions

Abstract

Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi-Yau algebra. We also formulate a rationality conjecture for the generating functions of the Donaldson-Thomas type invariants.