zbMATH — the first resource for mathematics

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.

14J33 Mirror symmetry (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI arXiv
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.