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LG/CY correspondence: the state space isomorphism. (English) Zbl 1245.14038
The authors use the construction of mirror pairs proposed by P. Berglund and T. Hübsch [Nucl. Phys. B 393 (1993; Zbl 1245.14039)]. Let \(X_W\) be a hypersurface in a weighted projective space, defined by a quasihomogeneous polynomial \(W\). Let \(X_{W^T}\) be a hypersurface in other weighted projective space, where \(W^T\) is a quasihomogeneous polynomial obtained by transposing the coefficients matrix of \(W\). For \(G \subset \mathrm{Aut}(\{W=0\})\) and \(G^T \subset \mathrm{Aut}(\{W^T=0\})\) satisfying a certain correspondence, the Calabi-Yau \(X_W/G\) and \(X_{W^T}/G^T\) are expected to be a dual pair. This approach to mirror symmetry allows to describe a vast range of cases which are not covered by a more geometric approach due to Batyrev and Borisov.
The duality between \(G\) and \(G^T\) was precisely stated by Berglund and Hübsch only in some cases. However, recently M. Krawitz [“FJRW rings and Landau-Ginzburg mirror symmetry”, arXiv:0906.0796] found a general construction for the dual group \(G^T\) and proved a mirror symmetry theorem for all invertible polynomials \(W\) and all admissible groups \(G\) in the Landau-Ginzburg setting. In the article under review the authors prove that \(X_W/G\) and \(X_{W^T}/G^T\) are a mirror pair of Calabi-Yau orbifolds, i.e. there is a 90 degrees rotation of the Hodge diamond of the Chen-Ruan orbifold cohomology: \[ H^{p,q}_{CR}([X_W/\widetilde{G}]; \mathbb{C}) \simeq H^{N-2-p,q}_{CR}([X_{W^T}/\widetilde{G}^T]; \mathbb{C}). \]
This theorem is a direct consequence of the Krawitz’s Landau-Ginzburg mirror symmetry theorem and the main theorem of this article: the cohomological Landau-Ginzburg/Calabi-Yau correspondence. It is understood as an isomorphism of the Chen-Ruan orbifold cohomology on the Calabi-Yau side and the state space of Fan-Jarvis-Ruan-Witten theory on the Landau-Ginzburg side: \[ H^{p,q}_{CR}([X_W/\widetilde{G}]; \mathbb{C}) \simeq H^{p,q}_{FJRW}(W,G;\mathbb{C}). \] The proof is accomplished by building a common combinatorial model for both theories.

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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