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

TL;DR

This paper explores the McKay correspondence through homological mirror symmetry, establishing a derived equivalence that links vanishing cycles of Kleinian singularities with perfect complexes on orbifolds, and interprets the spectrum via representation theory.

Contribution

It provides a new perspective by connecting classical singularity theory with modern homological mirror symmetry, revealing a derived equivalence and a spectrum interpretation.

Findings

Derived equivalence between vanishing cycles and perfect complexes

Spectrum of Kleinian singularities expressed through representation theory

Enhanced understanding of McKay correspondence via mirror symmetry

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.