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A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for -dimensional convex polyhedra.[1] The most famous examples of the polar duality provide Platonic solids: e.g., the cube is dual to octahedron, the dodecahedron is dual to icosahedron. There is a natural bijection between the -dimensional faces of a -dimensional convex polyhedron and -dimensional faces of the dual polyhedron and one has . In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special -dimensional convex lattice polytopes which are called reflexive polytopes.[2]
It was observed by Victor Batyrev and Duco van Straten[3] that the method of Philip Candelas et al.[4] for computing the number of rational curves on Calabi–Yau quintic 3-folds can be applied to arbitrary Calabi–Yau complete intersections using the generalized -hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei Zelevinsky[5] (see also the talk of Alexander Varchenko[6]), where is the set of lattice points in a reflexive polytope .
The combinatorial mirror duality for Calabi–Yau hypersurfaces in toric varieties has been generalized by Lev Borisov [7] in the case of Calabi–Yau complete intersections in Gorenstein toric Fano varieties. Using the notions of dual cone and polar cone one can consider the polar duality for reflexive polytopes as a special case of the duality for convex Gorenstein cones [8] and of the duality for Gorenstein polytopes.[9][10]
For any fixed natural number there exists only a finite number of -dimensional reflexive polytopes up to a -isomorphism. The number
is known only for : , , , The combinatorial classification of
-dimensional reflexive simplices up to a -isomorphism is closely related to the enumeration of all solutions of the diophantine equation . The classification of 4-dimensional reflexive polytopes up to a -isomorphism is important for constructing many topologically different 3-dimensional Calabi–Yau manifolds using hypersurfaces in 4-dimensional toric varieties which are Gorenstein Fano varieties. The complete list of 3-dimensional and 4-dimensional reflexive polytopes have been obtained by
physicists Maximilian Kreuzer and Harald Skarke using a special software in Polymake.[11][12][13][14]
A mathematical explanation of the combinatorial mirror symmetry has been obtained by Lev Borisov via vertex operator algebras which are algebraic counterparts of
conformal field theories.[15]
^Batyrev, V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties". Journal of Algebraic Geometry: 493–535.
^Nill, B. "Reflexive polytopes" (PDF).
^Batyrev, V.; van Straten, D. (1995). "Generalized hypergeometric functions and rational curves on Calabi–Yau complete intersections in toric varieties". Comm. Math. Phys. 168 (3): 493–533. arXiv:alg-geom/9307010. Bibcode:1995CMaPh.168..493B. doi:10.1007/BF02101841. S2CID 16401756.
^Candelas, P.; de la Ossa, X.; Green, P.; Parkes, L. (1991). "A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory". Nuclear Physics B. 359 (1): 21–74. doi:10.1016/0550-3213(91)90292-6.
^I. Gelfand, M. Kapranov, S. Zelevinski (1989), "Hypergeometric functions and toric varieties", Funct. Anal. Appl. 23, no. 2, 94–10.
^A. Varchenko (1990), "Multidimensional hypergeometric functions in conformal field theory, algebraic K-theory, algebraic geometry", Proc. ICM-90, 281–300.
^L. Borisov (1994), "Towards the Mirror Symmetry for Calabi–Yau Complete intersections in Gorenstein Toric Fano Varieties", arXiv:alg-geom/9310001
^Batyrev, V.; Borisov, L. (1997). "Dual cones and mirror symmetry for generalized Calabi–Yau manifolds". Mirror Symmetry, II: 71–86.
^Batyrev, V.; Nill, B. (2008). "Combinatorial aspects of mirror symmetry". Contemporary Mathematics. 452: 35–66. doi:10.1090/conm/452/08770. ISBN 9780821841730. S2CID 6817890.
^Kreuzer, M. (2008). "Combinatorics and Mirror Symmetry: Results and Perspectives" (PDF).
^M. Kreuzer, H. Skarke (1997), "On the classification of reflexive polyhedra", Comm. Math. Phys., 185, 495–508
^M. Kreuzer, H. Skarke (1998) "Classification
of reflexive polyhedra in three dimensions", Advances Theor. Math. Phys., 2, 847–864
^M. Kreuzer, H. Skarke (2002),
"Complete classification of reflexive polyhedra in four dimensions", Advances Theor. Math. Phys., 4, 1209–1230
^M. Kreuzer, H. Skarke, Calabi–Yau data, http://hep.itp.tuwien.ac.at/~kreuzer/CY/
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