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Revisiting the Classical McKay Correspondence, Derived Equivalences and the Spectrum of Kleinian Surface Singularities: A Look Through the Mirror

Abstract

In this article, we revisit the classical McKay correspondence via homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. We further illustrate how this equivalence allows for the interpretation of the spectrum of a Kleinian surface singularity solely in terms of the representation-theoretic data of the associated binary polyhedral group.

2020 Mathematics Subject Classification:

14E16, 14-02, 14F08, 14J33, 14B05

Keywords

homological mirror symmetry, McKay correspondence, spectrum

Full text

Author Details

Enrique Becerra

Department of Mathematics
University of Miami
PO Box 249085, Coral Gables
FL 33124-4250, USA
e-mail: exb1015@miami.edu

Ludmil Katzarkov

Department of Mathematics
University of Miami
PO Box 249085, Coral Gables
FL 33124-4250, USA
e-mail: l.katzarkov@miami.edu

Ernesto Lupercio

Cinvestav
Departamento de Matematicas
Av. Instituto Politecnico Nacional 2508
Col. San Pedro Zacatenco, Alcaldia Gustavo A. Madero
Ciudad de Mexico, C.P. 07360, México
e-mail: lupercio@math.cinvestav.mx


References

  1. Vl. Arnold. On some problems in singularity theory. Proc. Indian Acad. Sci. Math. Sci. 90, 1 (1981), 1–9.
  2. V. Baranovsky. Orbifold cohomology as periodic cyclic homology. Internat. J. Math. 14, 8(2003), 791–812.
  3. V. V. Batyrev, D. I. Dais. Strong McKay correspondence, string-theoretic hodge numbers and mirror symmetry. Topology 35, 4 (1996), 901–929.
  4. E. Becerra, E. Lupercio, L. Katzarkov. The Stringy Spectrum of Orbifolds. IMSA Conference Higher Invariants in Equivariant and Geometric Topology, 2024.
  5. R. Bocklandt. A gentle introduction to homological mirror symmetry. London Math. Soc. Stud. Texts, vol. 99. Cambridge, Cambridge University Press, 2021.
  6. T. Bridgeland, A. King, M. Reid. The McKay correspondence as an equivalence of derived categories. J. Amer. Math. Soc. 14, 3 (2001), 535–554.
  7. E. Brieskorn. Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta Math. 2 (1970), 103–161.
  8. E. Brieskorn. Singular elements of semi-simple algebraic groups. In: Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 2, 279–284, 1970.
  9. P. Candelas, X. C. de la Ossa, P. S. Green, L. Parkes. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Phys. B 359, 1 (1991), 21–74.
  10. W. Chen, Y. Ruan. A new cohomology theory of orbifold. Comm. Math. Phys. 248, 1 (2004), 1–31.
  11. T. de Fernex, E. Lupercio, T. Nevins, and B. Uribe. A localization principle for orbifold theories. https://doi.org/10.48550/arXiv.hep-th/0411037, 2004.
  12. P. Du Val. On isolated singularities of surfaces which do not affect the conditions of adjunction (part I). In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 30, 453–459. Cambridge University Press, 1934, https://doi.org/10.1017/S030500410001269X.
  13. A. Durfee. Fifteen characterizations of rational double points and simple critical points. Enseign. Math (2) 25, 1–2 (1979), 131–163.
  14. W. Ebeling. Homological mirror symmetry for singularities. https://doi.org/10.48550/arXiv.1601.06027, 2016.
  15. P. I. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina. Introduction to representation theory. Stud. Math. Libr., vol. 59. Providence, RI, American Mathematical Society, 2011.
  16. H. Fan, T. Jarvis, Y. Ruan. The Witten equation, mirror symmetry, and quantum singularity theory. Ann. of Math. (2) 178, 1 (2013), 1–106.
  17. P. Gabriel. Unzerlegbare Darstellungen. I. Manuscripta Math. 6 (1972), 71–103; correction: ibid. 6 (1972), 309.
  18. G. Gonzalez-Sprinberg, J.-L. Verdier. Construction géométrique de la correspondance de McKay (Geometric construction of the McKay correspondence). Ann. Sci. Ecole Norm. Sup. (4) 16, 3 (1983), 409–449.
  19. M. Hazewinkel, W. Hesselink, D. Siersma, F. D. Veldkamp. The ubiquity of Coxeter-Dynkin diagrams (an introduction to the A − D – E problem). Nieuw Arch. Wisk. (3) 25, 3 (1977), 257–307.
  20. H. Kajiura, K. Saito, A. Takahashi. Matrix factorizations and representations of quivers. II: type ADE case. Adv. Math. 211, 1 (2007), 327–362.
  21. F. Klein. Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Leipzig, Teubner, 1884, https://eudml.org/doc/203220.
  22. M. Kontsevich. Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians: August 3–11, 1994, Zürich, Switzerland, 120–139. Basel, Birkhauser Verlag, 1995.
  23. A. I. Kostrikin. Introduction to algebra. Izdat. “Nauka”, Moscow, 1977 (in Russian); English translation by Neal Koblitz, Universitext, New York-Berlin, Springer-Verlag, 1982.
  24. J. McKay. Graphs, singularities, and finite groups. Uspekhi Mat. Nauk 38 3(231) (1983), 159–162.
  25. J. Milnor. Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, 2016.
  26. D. Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. https://doi.org/10.48550/arXiv.math/0302304, 2003.
  27. J. Steenbrink. Limits of hodge structures. Invent. Math. 31, 3 (1975/76), 229–257.
  28. J. Steenbrink. Mixed Hodge structures applied to singularities. In: Handbook of Geometry and Topology of Singularities III, 645–678. Cham, Springer, 2022.
  29. J. H. M. Steenbrink. Mixed Hodge structure on the vanishing cohomology. Nordic Summer School/NAVF, Symposium in Mathematics, Oslo, August 5–25, 1976, https://www.maths.ed.ac.uk/~v1ranick/papers/steenbrink2.pdf.
  30. B. Totaro. The resolution property for schemes and stacks. J. Reine Angew. Math. 577 (2004), 1–22.
  31. D. van Straten. The spectrum of hypersurface singularities. https://doi.org/10.48550/arXiv.2003.00519, 2020.
  32. A. N. Varchenko. Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface. Dokl. Akad. Nauk SSSR 270, 6 (1983), 1294–1297 (in Russian); English translation in Soviet Math. Dokl. 27, 3 (1983), 735–739.