Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 18 (2022), 030, 53 pages      arXiv:1903.01636      https://doi.org/10.3842/SIGMA.2022.030

Deformations of Dimer Models

Akihiro Higashitani a and Yusuke Nakajima b
a) Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Osaka 565-0871, Japan
b) Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, 603-8555, Japan

Received August 06, 2021, in final form April 10, 2022; Published online April 16, 2022

Abstract
The combinatorial mutation of polygons, which transforms a given lattice polygon into another one, is an important operation to understand mirror partners for two-dimensional Fano manifolds, and the mutation-equivalent polygons give ${\mathbb Q}$-Gorenstein deformation-equivalent toric varieties. On the other hand, for a dimer model, which is a bipartite graph described on the real two-torus, one can assign a lattice polygon called the perfect matching polygon. It is known that for each lattice polygon $P$ there exists a dimer model having $P$ as the perfect matching polygon and satisfying certain consistency conditions. Moreover, a dimer model has rich information regarding toric geometry associated with the perfect matching polygon. In this paper, we introduce a set of operations which we call deformations of consistent dimer models, and show that the deformations of consistent dimer models realize the combinatorial mutations of the associated perfect matching polygons.

Key words: dimer models; combinatorial mutation of polygons; mirror symmetry.

pdf (901 kb)   tex (79 kb)  

References

  1. Akhtar M., Coates T., Corti A., Heuberger L., Kasprzyk A., Oneto A., Petracci A., Prince T., Tveiten K., Mirror symmetry and the classification of orbifold del Pezzo surfaces, Proc. Amer. Math. Soc. 144 (2016), 513-527, arXiv:1501.05334.
  2. Akhtar M., Coates T., Galkin S., Kasprzyk A.M., Minkowski polynomials and mutations, SIGMA 8 (2012), 094, 707 pages, arXiv:1212.1785.
  3. Beil C., Ishii A., Ueda K., Cancellativization of dimer models, arXiv:1301.5410.
  4. Bocklandt R., Consistency conditions for dimer models, Glasg. Math. J. 54 (2012), 429-447, arXiv:1104.1592.
  5. Bocklandt R., Generating toric noncommutative crepant resolutions, J. Algebra 364 (2012), 119-147, arXiv:1104.1597.
  6. Bocklandt R., A dimer ABC, Bull. Lond. Math. Soc. 48 (2016), 387-451, arXiv:1510.04242.
  7. Broomhead N., Dimer models and Calabi-Yau algebras, Mem. Amer. Math. Soc. 215 (2012), viii+86 pages, arXiv:0901.4662.
  8. Coates T., Corti A., Galkin S., Golyshev V., Kasprzyk A., Mirror Symmetry and Fano Manifolds, in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, 285-300, arXiv:1212.1722.
  9. Derksen H., Weyman J., Zelevinsky A., Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), 59-119, arXiv:0704.0649.
  10. Duffin R.J., Potential theory on a rhombic lattice, J. Combinatorial Theory 5 (1968), 258-272.
  11. Franco S., Hanany A., Vegh D., Wecht B., Kennaway K.D., Brane dimers and quiver gauge theories, J. High Energy Phys. 2006 (2006), no. 1, 096, 48 pages, arXiv:hep-th/0504110.
  12. Galkin S., Usnich A., Mutations of potentials, Preprint IPMU 10-0100, 2010.
  13. Goncharov A.B., Kenyon R., Dimers and cluster integrable systems, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), 747-813, arXiv:1107.5588.
  14. Gulotta D.R., Properly ordered dimers, $R$-charges, and an efficient inverse algorithm, J. High Energy Phys. 2008 (2008), no. 10, 014, 31 pages, arXiv:0807.3012.
  15. Hanany A., Seong R.K., Brane tilings and reflexive polygons, Fortschr. Phys. 60 (2012), 695-803, arXiv:1201.2614.
  16. Hanany A., Vegh D., Quivers, tilings, branes and rhombi, J. High Energy Phys. 2007 (2007), no. 10, 029, 35 pages, arXiv:hep-th/0511063.
  17. Higashitani A., Nakajima Y., Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs, Adv. Stud. Pure Math., to appear, arXiv:2107.04264.
  18. Ilten N.O., Mutations of Laurent polynomials and flat families with toric fibers, SIGMA 8 (2012), 047, 7 pages, arXiv:1205.4664.
  19. Ishii A., Ueda K., A note on consistency conditions on dimer models, in Higher Dimensional Algebraic Geometry, RIMS Kôky^uroku Bessatsu, Vol. B24, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, 143-164, arXiv:1012.5449.
  20. Ishii A., Ueda K., Dimer models and the special McKay correspondence, Geom. Topol. 19 (2015), 3405-3466, arXiv:0905.0059.
  21. Iyama O., Nakajima Y., On steady non-commutative crepant resolutions, J. Noncommut. Geom. 12 (2018), 457-471, arXiv:1509.09031.
  22. Kasprzyk A., Nill B., Prince T., Minimality and mutation-equivalence of polygons, Forum Math. Sigma 5 (2017), e18, 48 pages, arXiv:1501.05335.
  23. Kennaway K.D., Brane tilings, Internat. J. Modern Phys. A 22 (2007), 2977-3038, arXiv:0706.1660.
  24. Kenyon R., An introduction to the dimer model, in School and Conference on Probability Theory, ICTP Lect. Notes, Vol. 17, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, 267-304, arXiv:math.CO/0310326.
  25. Kenyon R., Schlenker J.M., Rhombic embeddings of planar quad-graphs, Trans. Amer. Math. Soc. 357 (2005), 3443-3458, arXiv:math-ph/0305057.
  26. Mercat C., Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), 177-216, arXiv:0909.3600.
  27. Nakajima Y., Mutations of splitting maximal modifying modules: the case of reflexive polygons, Int. Math. Res. Not. 2019 (2019), 470-550, arXiv:1601.05203.
  28. Nakajima Y., Semi-steady non-commutative crepant resolutions via regular dimer models, Algebr. Comb. 2 (2019), 173-195, arXiv:1608.05162.
  29. Rietsch K., Williams L., Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437-3527, arXiv:1712.00447.
  30. Schrijver A., Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986.
  31. Ueda K., Yamazaki M., A note on dimer models and McKay quivers, Comm. Math. Phys. 301 (2011), 723-747, arXiv:math.AG/0605780.

Previous article  Next article  Contents of Volume 18 (2022)