Communications in Mathematical Physics

, Volume 301, Issue 3, pp 723–747 | Cite as

A Note on Dimer Models and McKay Quivers

  • Kazushi UedaEmail author
  • Masahito Yamazaki


We give one formulation of a procedure of Hanany and Vegh (J High Energy Phys 0710(029):35, 2007) which takes a lattice polygon as an input and produces a set of isoradial dimer models. We study the case of lattice triangles in detail and discuss the relation with coamoebas following Feng et al. (Adv Theor Math Phys 12(3):489–545, 2008).


High Energy Phys Dime Model Path Algebra Asymptotic Boundary Toric Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benvenuti S., Pando Zayas L.A., Tachikawa Y.: Triangle anomalies from Einstein manifolds. Adv. Theor. Math. Phys. 10(3), 395–432 (2006)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Broomhead, N.: Dimer models and Calabi-Yau algebras.[math.AG], 2009
  3. 3.
    Butti, A., Zaffaroni, A.: R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization. J. High Energy Phys. 0511, 019, 42 pp. (electronic) (2005)Google Scholar
  4. 4.
    Butti A., Zaffaroni A.: From toric geometry to quiver gauge theory: the equivalence of a-maximization and Z-minimization. Fortschr. Phys. 54(5–6), 309–316 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Davison, B.: Consistency conditions for brane tilings.[math.AG], 2009
  6. 6.
    Duffin R.J.: Potential theory on a rhombic lattice. J. Comb. Th. 5, 258–272 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feng B., He Y.-H., Kennaway K.D., Vafa C.: Dimer models from mirror symmetry and quivering amoebae. Adv. Theor. Math. Phys. 12(3), 489–545 (2008)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Fowler R.H., Rushbrooke G.S.: An attempt to extend the statistical theory of perfect solutions. Trans. Faraday Soc. 33, 1272–1294 (1937)CrossRefGoogle Scholar
  9. 9.
    Franco, S., Hanany, A., Martelli, D., Sparks, J., Vegh, D., Wecht, B.: Gauge theories from toric geometry and brane tilings. J. High Energy Phys. 0601, 128, 40 pp. (electronic) (2006)Google Scholar
  10. 10.
    Franco, S., Hanany, A., Vegh, D., Wecht, B., Kennaway, K.D.: Brane dimers and quiver gauge theories. J. High Energy Phys. 0601, 096, 48 pp. (electronic) (2006)Google Scholar
  11. 11.
    Franco, S., Vegh, D.: Moduli spaces of gauge theories from dimer models: proof of the correspondence. J. High Energy Phys. 0611, 054, 26 pp. (electronic) (2006)Google Scholar
  12. 12.
    Ginzburg, V.: Calabi-Yau algebras.[math.AG], 2007
  13. 13.
    Gulotta, D.R.: Properly ordered dimers, R-charges, and an efficient inverse algorithm. J. High Energy Phys. 0810, 014, 31 pp (2008)Google Scholar
  14. 14.
    Hanany, A., Kennaway, K.D.: Dimer models and toric diagrams., 2005
  15. 15.
    Hanany, A., Vegh, D.: Quivers, tilings, branes and rhombi. J. High Energy Phys. 0710, 029, 35 (2007)Google Scholar
  16. 16.
    Ishii, A., Ueda, K.: Dimer models and the special McKay correspondence.[math.AG], 2009
  17. 17.
    Ishii, A., Ueda, K.: On moduli spaces of quiver representations associated with dimer models. In: Higher dimensional algebraic varieties and vector bundles, RIMS Kôkyûroku Bessatsu, B9. Kyoto: Res. Inst. Math. Sci. (RIMS), 2008, pp. 127–141Google Scholar
  18. 18.
    Kasteleyn P.W.: Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Kato, A.: Zonotopes and four-dimensional superconformal field theories. J. High Energy Phys. 0706, 037, 30 pp. (electronic) (2007)Google Scholar
  20. 20.
    Kenyon, R.: An introduction to the dimer model. In: School and Conference on Probability Theory, ICTP Lect. Notes, XVII (electronic). Trieste: Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 267–304Google Scholar
  21. 21.
    Kenyon R., Okounkov A., Sheffield S.: Dimers and amoebae. Ann. of Math. (2) 163(3), 1019–1056 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kenyon R., Schlenker J.-M.: Rhombic embeddings of planar quad-graphs. Trans. Amer. Math. Soc. 357(9), 3443–3458 (2005) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lee, S., Rey, S.-J.: Comments on anomalies and charges of toric-quiver duals. J. High Energy Phys. 0603, 068, 21 pp. (electronic) (2006)Google Scholar
  24. 24.
    Martelli D., Sparks J., Yau S.-T.: The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds. Commun. Math. Phys. 268(1), 39–65 (2006)zbMATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Martelli D., Sparks J., Yau S.-T.: Sasaki-Einstein manifolds and volume minimisation. Commun. Math. Phys. 280(3), 611–673 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Mercat C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218(1), 177–216 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Mozgovoy, S., Reineke, M.: On the noncommutative Donaldson-Thomas invariants arising from brane tilings.[math.AG], 2008
  28. 28.
    Nakamura I.: Hilbert schemes of abelian group orbits. J. Alg. Geom. 10(4), 757–779 (2001)zbMATHGoogle Scholar
  29. 29.
    Okounkov, A., Reshetikhin, N., Vafa, C.: Quantum Calabi-Yau and classical crystals. In The unity of mathematics, Volume 244 of Progr. Math., Boston, MA: Birkhäuser Boston, 2006, pp. 597–618Google Scholar
  30. 30.
    Ooguri, H., Yamazaki, H.: Emergent Calabi-Yau geometry. Phys. Rev. Lett. 102(16), 161601, 4 (2009)Google Scholar
  31. 31.
    Reid, M.: Mckay correspondence., 1997
  32. 32.
    Stienstra, J.: Computation of principal A-determinants through dimer dynamics.[math.AG], 2009
  33. 33.
    Stienstra, J.: Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants. In Modular forms and string duality, Volume 54 of Fields Inst. Commun., Providence, RI: Amer. Math. Soc., 2008, pp. 125–161Google Scholar
  34. 34.
    Ueda, K., Yamazaki, M.: Homological mirror symmetry for toric orbifolds of toric del Pezzo surfaces.[math.AG], 2010

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan
  2. 2.Department of Physics, Graduate School of ScienceUniversity of TokyoBunkyo-ku, TokyoJapan

Personalised recommendations