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A mirror theorem for toric stacks

Part of: Algebraic geometry: Foundations Special varieties Surfaces and higher-dimensional varieties Projective and enumerative geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  01 June 2015

Tom Coates ,
Alessio Corti ,
Hiroshi Iritani  and
Hsian-Hua Tseng
Tom Coates
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK email t.coates@imperial.ac.uk
Alessio Corti
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK email a.corti@imperial.ac.uk
Hiroshi Iritani
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan email iritani@math.kyoto-u.ac.jp
Hsian-Hua Tseng
Affiliation:
Department of Mathematics, Ohio State University, 100 Math Tower, 231 West 18th Ave., Columbus OH 43210, USA email hhtseng@math.ohio-state.edu
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Abstract

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We prove a Givental-style mirror theorem for toric Deligne–Mumford stacks ${\mathcal{X}}$. This determines the genus-zero Gromov–Witten invariants of ${\mathcal{X}}$ in terms of an explicit hypergeometric function, called the $I$-function, that takes values in the Chen–Ruan orbifold cohomology of ${\mathcal{X}}$.

Keywords

Gromov–Witten theorytoric Deligne–Mumford stacksorbifoldsquantum cohomologymirror symmetryGivental’s symplectic formalismhypergeometric functions

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14A20: Generalizations (algebraic spaces, stacks) 14M25: Toric varieties, Newton polyhedra 14J33: Mirror symmetry 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 10 , October 2015 , pp. 1878 - 1912
Creative Commons
Creative Common License - CCCreative Common License - BY
This article is distributed with Open Access under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
Copyright
© The Authors 2015

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