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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.

##### MSC:
 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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##### References:
 [1] Batyrev, V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. algebraic geom., 3, 3, 493-535, (1994) · Zbl 0829.14023 [2] Batyrev, V.V.; Borisov, L.A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, (), 71-86 · Zbl 0927.14019 [3] Berglund, P.; Hübsch, T., A generalized construction of mirror manifolds, Nuclear phys. B, 393, (1993) · Zbl 1245.14039 [4] Boissière, S.; Mann, É.; Perroni, F., A model for the orbifold Chow ring of weighted projective spaces, Comm. algebra, 37, 503-514, (2009) · Zbl 1178.14056 [5] A. Chiodo, H. Iritani, Y. Ruan, in preparation. [6] Chiodo, A.; Ruan, Y., Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Preprint · Zbl 1197.14043 [7] Corti, A.; Golyshev, V., Hypergeometric equations and weighted projective spaces, Preprint · Zbl 1237.14022 [8] Dimca, A., Singularities and topology of hypersurfaces, Universitext, (1992), Springer-Verlag New York, 263+xvi pp · Zbl 0753.57001 [9] Dolgachev, I., Weighted projective varieties, (), 34-71 [10] Fan, H.; Jarvis, T.; Ruan, Y., Geometry and analysis of spin equations, Comm. pure appl. math., 61, 6, 745-788, (2008) · Zbl 1141.58012 [11] Fan, H.; Jarvis, T.; Ruan, Y., The Witten equation, mirror symmetry and quantum singularity theory, Preprint · Zbl 1310.32032 [12] Fan, H.; Jarvis, T.; Ruan, Y., The Witten equation and its virtual fundamental cycle, Preprint [13] Hori, K.; Walcher, J., D-branes from matrix factorizations. strings 04. part I, C. R. phys., 5, 9-10, 1061-1070, (2004) [14] Iano-Fletcher, A.R., Working with weighted complete intersections, (), 101-173 · Zbl 0960.14027 [15] Intriligator, K.; Vafa, C., Landau-Ginzburg orbifolds, Nuclear phys. B, 339, 1, 95-120, (1990) [16] M. Kontsevich, unpublished. [17] Krawitz, M., FJRW rings and Landau-Ginzburg mirror symmetry, Preprint · Zbl 1250.81087 [18] Kreuzer, M.; Skarke, H., On the classification of quasihomogeneous functions, Comm. math. phys., 150, 1, 137-147, (1992) · Zbl 0767.57019 [19] Kreuzer, M.; Skarke, H., All abelian symmetries of Landau-Ginzburg potentials, Nuclear phys. B, 405, 2-3, 305-325, (1993), Preprint · Zbl 0990.81635 [20] Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities, Preprint · Zbl 1200.18007 [21] Romagny, M., Group actions on stacks and applications, Michigan math. J., 53, 1, 209-236, (2005) · Zbl 1100.14001 [22] Steenbrink, J., Intersection form for quasi-homogeneous singularities, Compos. math., 34, 2, 211-223, (1977) · Zbl 0347.14001 [23] Vafa, C.; Warner, N., Catastrophes and the classification of conformal field theories, Phys. lett. B, 218, 51, (1989) [24] Witten, E., Phases of $$N = 2$$ theories in two dimensions, Nuclear phys. B, 403, 159-222, (1993) · Zbl 0910.14020
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