The quantum McKay correspondence for polyhedral singularities

• 成果类型: Article
• 署名作者: Bryan, Jim; Gholampour, Amin
• 署名单位: University of British Columbia; California Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0212-8
• 发表日期: 2009
• 页码: 655-681
• 关键词: gromov-witten theory; conjecture; INVARIANTS; COHOMOLOGY; CURVES

摘要

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 a,(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 [a,(3)/G].