Publications mathématiques de l'IHÉS

, Volume 119, Issue 1, pp 127–216 | Cite as

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

Article

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.

Notation

W(x1,…,xN)

weighted homogeneous polynomial (Section 1.1)

XW

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

μd

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)

{ti}

linear co-ordinates on the state space associated to a basis {Ti} (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,vc}, Section 5.1)

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Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alessandro Chiodo
    • 1
  • Hiroshi Iritani
    • 2
  • Yongbin Ruan
    • 3
  1. 1.Institut de Mathématiques de Jussieu, UMR 7586 CNRSUniversité Pierre et Marie CurieParis cedex 05France
  2. 2.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  3. 3.Department of MathematicsUniversity of MichiganAnn ArborUSA

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