Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence

Alessandro Chiodo; Hiroshi Iritani; Yongbin Ruan

doi:10.1007/s10240-013-0056-z

Alessandro Chiodo ¹ ; Hiroshi Iritani ² ; Yongbin Ruan ³

¹ Institut de Mathématiques de Jussieu, UMR 7586 CNRS, Université Pierre et Marie Curie Case 247, 4 Place Jussieu 75252 Paris cedex 05 France
² Department of Mathematics, Graduate School of Science, Kyoto University Kitashirakawa-Oiwake-cho, Sakyo-ku 606-8502 Kyoto Japan
³ Department of Mathematics, University of Michigan 48109-1109 Ann Arbor MI USA

Publications Mathématiques de l'IHÉS, Volume 119 (2014), pp. 127-216

Abstract

We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following a proposal of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of $\hat{Γ}$ -integral structures applied to the derived category of the hypersurface {W=0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.

Received: 2012-01-12
Accepted: 2013-05-10
Online First: 2013-06-20
Published online: 2014-06-07

MR Zbl

DOI: 10.1007/s10240-013-0056-z

Keywords: Integral Structure, Chern Character, Quantum Cohomology, Twisted Theory, Frobenius Manifold

Author's affiliations:

Alessandro Chiodo ¹; Hiroshi Iritani ²; Yongbin Ruan ³

¹ Institut de Mathématiques de Jussieu, UMR 7586 CNRS, Université Pierre et Marie Curie Case 247, 4 Place Jussieu 75252 Paris cedex 05 France
² Department of Mathematics, Graduate School of Science, Kyoto University Kitashirakawa-Oiwake-cho, Sakyo-ku 606-8502 Kyoto Japan
³ Department of Mathematics, University of Michigan 48109-1109 Ann Arbor MI USA

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     author = {Alessandro Chiodo and Hiroshi Iritani and Yongbin Ruan},
     title = {Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and {Orlov} equivalence},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {127--216},
     year = {2014},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {119},
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     zbl = {1298.14042},
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}

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Alessandro Chiodo; Hiroshi Iritani; Yongbin Ruan. Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence. Publications Mathématiques de l'IHÉS, Volume 119 (2014), pp. 127-216. doi: 10.1007/s10240-013-0056-z

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