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CONSISTENCY CONDITIONS FOR DIMER MODELS

  • RAF BOCKLANDT (a1)
Abstract

Dimer models are a combinatorial tool to describe certain algebras that appear as noncommutative crepant resolutions of toric Gorenstein singularities. Unfortunately, not every dimer model gives rise to a noncommutative crepant resolution. Several notions of consistency have been introduced to deal with this problem. In this paper, we study the major different notions in detail and show that for dimer models on a torus, they are all equivalent.

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References
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1.Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra. 212 (1) (2008), 1432.
2.Bocklandt, R., Calabi-Yau algebras and quiver polyhedra, arXiv:0905.0232.
3.Bocklandt, R., Note on infinite CY-3 dimer models that are not cancellation.
4.Bridgeland, T., Flops and derived categories, Invent. Math. 147 (2002), 613632.
5.Broomhead, N., Dimer models and Calabi-Yau algebras, arXiv:0901.4662.
6.Butler, M. C. R. and King, A. D., Minimal resolutions of algebras, J. Algebra 212 (1) (1999), 323362.
7.Davison, B., Consistency conditions for brane tilings, arXiv:0812.4185.
8.Franco, S., Hanany, A., Kennaway, K. D., Vegh, D. and Wecht, B., Brane dimers and quiver gauge theories, JHEP 0601 (2006), 096, hep-th/0504110.
9.Ginzburg, V., Calabi-Yau algebras, math/0612139.
10.Hanany, A., Herzog, C. P. and Vegh, D., Brane tilings and exceptional collections, J. High Energy Phys. 7 (2006), 44.
11.Hanany, A. and Kennaway, K. D., Dimer models and toric diagrams, hep-th/0602041.
12.Hanany, A. and Vegh, D., Quivers, tilings, branes and rhombi, J. High Energy Phys. 10 (2007), 35.
13.Ishii, A. and Ueda, K., On moduli spaces of quiver representations associated with brane tilings, in higher dimensional algebraic varieties and vector bundles, (RIMS Kroku Bessatsu, Kyoto, 2008).
14.Ishii, A. and Ueda, K., A note on consistency conditions on dimer models, arXiv:1012.5449.
15.Kennaway, K. D., Brane tilings, Int. J. Modern Phys. A 22 (18), (2007), 29773038. hep-th/0710.1660.
16.Kenyon, R., An introduction to the dimer model (School and Conference on Probability Theory, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004).
17.Kenyon, R. and Schlenker, J. M., Rhombic embeddings of planar quad-graphs, Trans. AMS. 357 (9) (2004), 34433458.
18.Le Bruyn, L., A cohomological interpretation of the reflexive Brauer group, J. Algebra 105 (1987), 250254.
19.Mozgovoy, S. and Reineke, M., On the noncommutative Donaldson-Thomas invariants arising from brane tilings, Adv. Math. 223 (5) (2010), 15211544.
20.Orzech, M., Brauer Groups and Class Groups for a Krull Domain, in Brauer groups in ring theory and algebraic geometry (Springer, 1982), 6887.
21.Reichstein, Z. and Vonessen, N., Polynomial identity rings as rings of functions. J. Algebra, 310 (2) (2007), 624647.
22.Stafford, J. T. and Van den Bergh, M., Non-commutative resolutions and rational singularities, Michigan Math. J. 57 (2008), 659674.
23.Van den Bergh, M., Non-commutative crepant resolutions, in The legacy of Niels Hendrik Abel (Springer, 2002), 749770.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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