Geometric Reid’s recipe for dimer models

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Abstract

Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution \(Y\), this derived equivalence generalises the Fourier-Mukai transform relating the \(G\)-Hilbert scheme and the skew group algebra \(\mathbb {C}[x,y,z]*G\) for a finite abelian subgroup of \(\mathrm{SL }(3,\mathbb {C})\). We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of Cautis–Logvinenko (J Reine Angew Math 636:193–236, 2009) and Cautis–Craw–Logvinenko  (J Reine Angew Math arXiv:1205.3110, 2014) to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko (J Algebra 324:2064–2087, 2010).

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Change history

  • 27 January 2021

    The main results of [1], especially Theorem 1.1, Corollary��1.2 and Theorem 1.4, are correct as written.

References

  1. 1.

    Bender, M., Mozgovoy, S.: Crepant resolutions and brane tilings II: Tilting bundles (Preprint). arXiv: 0909.2013

  2. 2.

    Bocklandt, R.: Consistency conditions for dimer models. Glasg. Math. J. 54(2), 429–447 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Bridgeland, T.: Flops and derived categories. Invent. Math. 147(3), 613–632 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (electronic) (2001)

  5. 5.

    Broomhead, N.: Dimer models and Calabi–Yau algebras. Mem. Am. Math. Soc. 215(1011):viii+86 (2012)

  6. 6.

    Cautis, S., Craw, A., Logvinenko, T.: Derived Reid’s recipe for abelian subgroups of \({SL}_3(\mathbb{C})\). J. Reine Angew. Math. (to appear, 2014). arXiv: 1205.3110

  7. 7.

    Cautis, S., Logvinenko, T.: A derived approach to geometric McKay correspondence in dimension three. J. Reine Angew. Math. 636, 193–236 (2009)

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Craw, A.: An explicit construction of the McKay correspondence for \(A\)-Hilb \(\mathbb{C}^3\). J. Algebra 285(2), 682–705 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Craw, A., Ishii, A.: Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient. Duke Math. J. 124(2), 259–307 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Craw, A., Quintero Vélez, A.: Cellular resolutions of noncommutative toric algebras from superpotentials. Adv. Math. 229(3), 1516–1554 (2012)

  11. 11.

    Craw, A., Quintero Vélez, A.: Cohomology of wheels on toric varieties. Hokkaido Math. J. (to appear, 2014). arXiv: 1206.5956

  12. 12.

    Crawley-Boevey, W.: On the exceptional fibres of Kleinian singularities. Am. J. Math. 122(5), 1027–1037 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Davison, B.: Consistency conditions for brane tilings. J. Algebra 338, 1–23 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Gonzalez-Sprinberg, G., Verdier, J.-L.: Construction géométrique de la correspondance de McKay. Ann. Sci. École Norm. Sup. (4) 16(3), 409–449 (1983)

  15. 15.

    Gulotta, D.: Properly ordered dimers, \(R\)-charges, and an efficient inverse algorithm. J. High Energy Phys. 10, 014, 31 (2008)

    Google Scholar 

  16. 16.

    Hartshorne, R.: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, vol. 20. Springer, Berlin (1966)

  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, pp. 127–141. Res. Inst. Math. Sci. (RIMS), Kyoto (2008)

  18. 18.

    Ishii, A., Ueda, K.: Dimer models and the special McKay correspondence, (2009). (Preprint) arXiv:0905.0059, v2

  19. 19.

    Ishii, A., Ueda, K.: A note on consistency conditions on dimer models. In: Higher Dimensional Algebraic Geometry, RIMS Kôkyûroku Bessatsu, B24, pp. 143–164. Res. Inst. Math. Sci. (RIMS), Kyoto (2011)

  20. 20.

    Ishii, A., Ueda, K.: Dimer models and crepant resolutions (2013) (Preprint). arXiv:1303.4028

  21. 21.

    Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology 39(6), 1155–1191 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Kapranov, M., Vasserot, E.: Kleinian singularities, derived categories and Hall algebras. Math. Ann. 316(3), 565–576 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Kennaway, K.: Brane tilings. Int. J. Mod. Phys. A 22:2977–3038 (2007) (Preprint). arXiv:0710.1660

  24. 24.

    King, A.: Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45(180), 515–530 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Logvinenko, T.: Reid’s recipe and derived categories. J. Algebra 324, 2064–2087 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Mozgovoy, S.: Crepant resolutions and brane tilings I: Toric realization (Preprint). arXiv: 0908.3475

  27. 27.

    Mozgovoy, S., Reineke, M.: On the noncommutative Donaldson–Thomas invariants arising from brane tilings. Adv. Math. 223(5), 1521–1544 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Reid, M.: McKay correspondence. http://lanl.arxiv.org/abs/alg-geom/9702016

  29. 29.

    Tapia Amador, J.: Combinatorial Reid’s recipe for dimer models. Work in progress

  30. 30.

    Wilson, P.: The Kähler cone on Calabi–Yau threefolds. Invent. Math. 107(3), 561–583 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

A. Craw thanks Akira Ishii and Kazushi Ueda for a preliminary draft of their preprint [20]. Thanks also to Alastair King for a useful remark, and to the anonymous referee for many helpful comments. A. Craw and A. Quintero Vélez were supported in part by EPSRC grant EP/G004048/1.

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Bocklandt, R., Craw, A. & Quintero Vélez, A. Geometric Reid’s recipe for dimer models. Math. Ann. 361, 689–723 (2015). https://doi.org/10.1007/s00208-014-1085-8

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