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Dual fans and mirror symmetry. (English) Zbl 1375.14133
Summary: We show that the mirror constructions of Greene-Plesser, Berglund-Hübsch, Batyrev, Batyrev-Borisov, Givental and Hori-Vafa can be expressed in terms of what we call dual fans. To do this, we associate to a pair of dual fans a pair of toric Landau-Ginzburg models, and we describe a process by which each of the mirror constructions listed also produces a pair of toric Landau-Ginzburg models. Replacing mirror pairs by toric Landau-Ginzburg models is reversible, and our main result is that the dual fan models and the mirror pairs models coincide.

MSC:
 14J33 Mirror symmetry (algebro-geometric aspects) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Keywords:
mirror symmetry; Landau-Ginzburg; toric
Full Text:
References:
 [1] Batyrev, V. V., Quantum cohomology rings of toric manifolds, Journées de Géométrie Algébrique d’Orsay, Orsay, 1992, Astérisque, 218, 9-34, (1993) · Zbl 0806.14041 [2] 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 [3] Batyrev, V. V.; Borisov, L. A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, (Mirror Symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, (1997), Amer. Math. Soc. Providence, RI), 71-86 · Zbl 0927.14019 [4] Batyrev, V.; Nill, B., Combinatorial aspects of mirror symmetry, (Integer Points in Polyhedra—Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, Contemp. Math., vol. 452, (2008), Amer. Math. Soc. Providence, RI), 35-66 · Zbl 1161.14037 [5] Berglund, P.; Hübsch, T., A generalized construction of mirror manifolds, Nuclear Phys. B, 393, 1-2, 377-391, (1993) · Zbl 1245.14039 [6] Borisov, L., Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties, (1993) [7] Candelas, P.; de la Ossa, X. C.; Green, P. S.; Parkes, L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B, 359, 1, 21-74, (1991) · Zbl 1098.32506 [8] Clarke, P., Duality for toric Landau-Ginzburg models, (2008) [9] Clarke, P., A proof of the birationality of certain BHK-mirrors, Complex Manifolds, 1, 45-51, (2014) · Zbl 1320.32032 [10] Cox, D. A.; Little, J. B.; Schenck, H. K., Toric varieties, Grad. Stud. Math., vol. 124, (2011), American Mathematical Society Providence, RI · Zbl 1223.14001 [11] Demazure, M., Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér. (4), 3, 507-588, (1970) · Zbl 0223.14009 [12] Favero, D.; Kelly, T. L., Proof of a conjecture of Batyrev and nill, (2014), Amer. J. Math., in press · Zbl 1390.14124 [13] Favero, D.; Kelly, T. L., Derived categories of BHK mirrors, (2016) [14] Fulton, W., Introduction to toric varieties, Ann. of Math. Stud., vol. 131, (1993), Princeton University Press Princeton, NJ, the William H. Roever lectures in geometry [15] Gel’fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants, (Mathematics: Theory & Applications, (1994), Birkhäuser Boston, Inc. Boston, MA) · Zbl 0827.14036 [16] Givental, A., A mirror theorem for toric complete intersections, (Topological Field Theory, Primitive Forms and Related Topics, Kyoto, 1996, Progr. Math., vol. 160, (1998), Birkhäuser Boston Boston, MA), 141-175 · Zbl 0936.14031 [17] Greene, B. R.; Plesser, M. R., Duality in Calabi-Yau moduli space, Nuclear Phys. B, 338, 1, 15-37, (1990) [18] Hori, K.; Vafa, C., Mirror symmetry, (2000) [19] Kelly, T. L., Berglund-Hübsch-krawitz mirrors via shioda maps, Adv. Theor. Math. Phys., 17, 6, 1425-1449, (2013) · Zbl 1316.14076 [20] Krawitz, M., FJRW rings and Landau-Ginzburg mirror symmetry, (2010), ProQuest LLC Ann Arbor, MI, Thesis (Ph.D.)-University of Michigan · Zbl 1250.81087 [21] Oda, T., Torus embeddings and applications, (Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay, (1978), Springer-Verlag Berlin, New York), based on joint work with Katsuya Miyake [22] Shoemaker, M., Birationality of berglund-Hübsch-krawitz mirrors, Comm. Math. Phys., 331, 2, 417-429, (2014) · Zbl 1395.14034 [23] Vafa, C.; Warner, N., Catastrophes and the classification of conformal theories, Phys. Lett. B, 218, 51-58, (February 1989) [24] Witten, E., Phases of $$N = 2$$ theories in two dimensions, (Mirror Symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, (1997), Amer. Math. Soc. Providence, RI), 143-211 · Zbl 0910.14019
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