On the noncommutative Donaldson-Thomas invariants arising from brane tilings

TL;DR

This work addresses the computation of noncommutative Donaldson-Thomas invariants for moduli spaces of framed cyclic modules arising from quiver potentials associated to brane tilings. It introduces a canonical $ oldsymbol{ abla}$-grading producing a torus action with finitely many fixed points, enabling a purely combinatorial DT count via ideals in a path poset and perfect matchings of the planar tiling; under two consistency conditions the quiver potential algebra is shown to be a $3$-Calabi–Yau algebra and a DT generating function formula is derived, with a rationality conjecture for the generating functions. The approach recovers known results for $C^3$, the conifold, and certain orbifolds, and provides a computational framework based on path posets and perfect matchings for broader brane tilings. The work also links to potential wall-crossing methods and suggests a universal rationality structure in the diagonal specialization, supported by several computed examples.

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.

Figures (4)

• Figure 1: The periodic quiver and a fundamental domain for $\mathbb{C}\xspace^3$.
• Figure 2: The periodic quiver and a fundamental domain for the conifold.
• Figure 3: The periodic quiver and a fundamental domain for $\mathbb{C}\xspace^3/\mathbb{Z}\xspace_4$.
• Figure 4: The right picture is obtained from the left by drawing the shortest diagonals of the small parallelograms in green.

Theorems & Definitions (57)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Lemma 2.5
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof