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The quantum McKay correspondence for polyhedral singularities

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Abstract

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 ℂ3/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 [ℂ3/G].

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Jim Bryan

  2. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Amin Gholampour

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  1. Jim Bryan
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  2. Amin Gholampour
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Correspondence to Jim Bryan.

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Bryan, J., Gholampour, A. The quantum McKay correspondence for polyhedral singularities. Invent. math. 178, 655–681 (2009). https://doi.org/10.1007/s00222-009-0212-8

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  • DOI: https://doi.org/10.1007/s00222-009-0212-8

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