On the noncommutative Donaldson-Thomas invariants arising from brane tilings

By S. Mozgovoy, Markus Reineke. 2008

138 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8d558f36a13cb31539a67f1b5641eff916c70a02

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Kazushi Ueda
2. Yukinobu Toda
3. A. Ishii
4. Sergei Gukov
5. Raf Bocklandt
6. D. Joyce
7. Yinan Song
8. Tudor Dimofte

In Geometry and Topology

1. Yukinobu Toda
2. Yinan Song
3. Tudor Dimofte
4. D. Joyce
5. Sergei Gukov
6. Kazushi Ueda
7. T. Bridgeland
8. A. Ishii

In Nuclear and High Energy Physics

1. Tudor Dimofte
2. Sergei Gukov
3. Yukinobu Toda
4. A. Ishii
5. Raf Bocklandt
6. D. Joyce
7. Yinan Song
8. Kazushi Ueda

Citations in the corpus

• A theory of generalized Donaldson–Thomas invariants (2008, background): “These quiver analogues of Mα,nstp (τ ′), P Iα,n(τ ′) are not new, similar things have been studied in quiver theory by Nakajima, Reineke, Szendrői and other authors for some years [24, 82–85, 88, 89, 99].”
• Dimer models and the special McKay correspondence (2009, background): “Mozgovoy and Reineke [29] proved that the path algebra of the quiver with relations coming from a dimer model is a Calabi–Yau–3 algebra in the sense of Ginzburg [15] if the dimer model is cancellative and one extra condition which they call the second consistency condition.”
• CONSISTENCY CONDITIONS FOR DIMER MODELS (2011, methodology): “in [7]. Theorem 5.2 (Davison). The Jacobi algebra of a dimer model Qwith χQ ≤0 is CY-3 if it is a cancellation algebra. Remark 5.3. Davison’s work was a generalization of work by Mozgovoy and Reineke [20] which used an extra consistency condition. This extra condition turned out to be a consequence ”
• Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, methodology): “The purpose of this paper is to compare curve counting theories on X and X+ via categorical method, i.e. effectively use an equivalence of bounded derived categories of coherent sheaves by Bridgeland [4], Φ: Db(Coh(X+)) ∼ −→ Db(Coh(X)).”
• Refined, Motivic, and Quantum (2009, methodology): “(See, for example, [8,10,11] for explicit examples of this relation in the Szendröi region of moduli space.”