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, 14B05Keywords
homological mirror symmetry, McKay correspondence, spectrum
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
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