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

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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 \(\widehat {\Gamma }\)-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.

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Abbreviations

W(x 1,…,x N ):

weighted homogeneous polynomial (Section 1.1)

X W :

Calabi-Yau hypersurface defined by W in \(\mathbf {P}(\underline {w})\) (Section 1.1)

μ d :

the group of d-th roots of unity

H :

state space (\(H(W,\boldsymbol {\mu }_{d}) \text{\ or\ } H_{{\operatorname {CR}}}(X_{W})\), Sections 2.1.1, 2.2.1)

H′:

narrow/ambient part (Section 2.3.1)

H″:

broad/primitive part (Section 2.3.1)

{t i}:

linear co-ordinates on the state space associated to a basis {T i } (Sections 2.3, 2.4, 3.5.2)

\(\widehat {\Gamma }\) :

Gamma class (Section 2.4.4)

\(\overline {H}\) :

state space of twisted theories (\(H_{{\operatorname {ext}}}\text{\ or\ } H_{{\operatorname {CR}}}( \mathbf {P}(\underline {w}))\), Section 3.5.2)

\(\mathcal {M}\) :

global Kähler moduli space (P(1,d)∖{0,v c}, Section 5.1)

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, UMR 7586 CNRS, Université Pierre et Marie Curie, Case 247, 4 Place Jussieu, 75252, Paris cedex 05, France

    Alessandro Chiodo

  2. Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

    Hiroshi Iritani

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109, USA

    Yongbin Ruan

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  1. Alessandro Chiodo
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  2. Hiroshi Iritani
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  3. Yongbin Ruan
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Correspondence to Hiroshi Iritani.

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Chiodo, A., Iritani, H. & Ruan, Y. Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence. Publ.math.IHES 119, 127–216 (2014). https://doi.org/10.1007/s10240-013-0056-z

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