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The quantum McKay correspondence for polyhedral singularities

Bryan, Jim and Gholampour, Amin (2009) The quantum McKay correspondence for polyhedral singularities. Inventiones Mathematicae, 178 (3). pp. 655-681. ISSN 0020-9910.

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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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Additional Information:© Springer-Verlag 2009. Received: 6 May 2008. Accepted: 3 July 2009. Published online: 28 July 2009. The authors thank the referees for valuable comments on both technical and expository aspects of this paper. The authors also warmly thank Davesh Maulik, Miles Reid, Rahul Pandharipande, Sophie Terouanne, Michael Thaddeus, and Hsian-Hua Tseng for helpful conversations. The authors also acknowledge support from NSERC, MSRI, the Killam Trust, and the Miller Institute.
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Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Mathematical Sciences Research Institute (MSRI)UNSPECIFIED
Miller Institute for Basic Research in ScienceUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20091103-152128062
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16570
Deposited By: Tony Diaz
Deposited On:04 Nov 2009 23:17
Last Modified:03 Oct 2019 01:14

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