The Quantum McKay Correspondence for polyhedral singularities

Details The Quantum McKay Correspondence for polyhedral singularities The Quantum McKay Correspondence for polyhedral singularities

2008-03-26

arxiv.org/abs/0803.3766v2

Mathematics

Algebraic Geometry

Introduction rewritten. Issue regarding non-uniqueness of conifold resolution clarified. Version to appear in Inventiones

Scientific paper

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the crepant resolution conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].

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