Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-31T04:24:13.288Z Has data issue: false hasContentIssue false
  • English
  • Français

CONSISTENCY CONDITIONS FOR DIMER MODELS

Published online by Cambridge University Press:  29 March 2012

RAF BOCKLANDT
RAF BOCKLANDT*
Affiliation:
School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK e-mail: raf.bocklandt@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

14M2514A2216S38

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 429 - 447
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra. 212 (1) (2008), 14–32.CrossRefGoogle Scholar
2.Bocklandt, R., Calabi-Yau algebras and quiver polyhedra, arXiv:0905.0232.Google Scholar
3.Bocklandt, R., Note on infinite CY-3 dimer models that are not cancellation.Google Scholar
4.Bridgeland, T., Flops and derived categories, Invent. Math. 147 (2002), 613–632.CrossRefGoogle Scholar
5.Broomhead, N., Dimer models and Calabi-Yau algebras, arXiv:0901.4662.Google Scholar
6.Butler, M. C. R. and King, A. D., Minimal resolutions of algebras, J. Algebra 212 (1) (1999), 323–362.CrossRefGoogle Scholar
7.Davison, B., Consistency conditions for brane tilings, arXiv:0812.4185.Google Scholar
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.CrossRefGoogle Scholar
9.Ginzburg, V., Calabi-Yau algebras, math/0612139.Google Scholar
10.Hanany, A., Herzog, C. P. and Vegh, D., Brane tilings and exceptional collections, J. High Energy Phys. 7 (2006), 44.Google Scholar
11.Hanany, A. and Kennaway, K. D., Dimer models and toric diagrams, hep-th/0602041.Google Scholar
12.Hanany, A. and Vegh, D., Quivers, tilings, branes and rhombi, J. High Energy Phys. 10 (2007), 35.Google Scholar
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).Google Scholar
14.Ishii, A. and Ueda, K., A note on consistency conditions on dimer models, arXiv:1012.5449.Google Scholar
15.Kennaway, K. D., Brane tilings, Int. J. Modern Phys. A 22 (18), (2007), 29773038. hep-th/0710.1660.CrossRefGoogle Scholar
16.Kenyon, R., An introduction to the dimer model (School and Conference on Probability Theory, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004).Google Scholar
17.Kenyon, R. and Schlenker, J. M., Rhombic embeddings of planar quad-graphs, Trans. AMS. 357 (9) (2004), 3443–3458.CrossRefGoogle Scholar
18.Le Bruyn, L., A cohomological interpretation of the reflexive Brauer group, J. Algebra 105 (1987), 250–254.CrossRefGoogle Scholar
19.Mozgovoy, S. and Reineke, M., On the noncommutative Donaldson-Thomas invariants arising from brane tilings, Adv. Math. 223 (5) (2010), 1521–1544.CrossRefGoogle Scholar
20.Orzech, M., Brauer Groups and Class Groups for a Krull Domain, in Brauer groups in ring theory and algebraic geometry (Springer, 1982), 68–87.Google Scholar
21.Reichstein, Z. and Vonessen, N., Polynomial identity rings as rings of functions. J. Algebra, 310 (2) (2007), 624–647.CrossRefGoogle Scholar
22.Stafford, J. T. and Van den Bergh, M., Non-commutative resolutions and rational singularities, Michigan Math. J. 57 (2008), 659–674.CrossRefGoogle Scholar
23.Van den Bergh, M., Non-commutative crepant resolutions, in The legacy of Niels Hendrik Abel (Springer, 2002), 749–770.Google Scholar