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A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for ${\displaystyle d}$-dimensional convex polyhedra. 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 ${\displaystyle k}$-dimensional faces of a ${\displaystyle d}$-dimensional convex polyhedron ${\displaystyle P}$ and ${\displaystyle (d-k-1)}$-dimensional faces of the dual polyhedron ${\displaystyle P^{*}}$ and one has ${\displaystyle (P^{*})^{*}=P}$. In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special ${\displaystyle d}$-dimensional convex lattice polytopes which are called reflexive polytopes.

It was observed by Victor Batyrev and Duco van Straten that the method of Philip Candelas et al. 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 ${\displaystyle A}$-hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei Zelevinsky (see also the talk of Alexander Varchenko), where ${\displaystyle A}$ is the set of lattice points in a reflexive polytope ${\displaystyle P}$.

The combinatorial mirror duality for Calabi–Yau hypersurfaces in toric varieties has been generalized by Lev Borisov 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 and of the duality for Gorenstein polytopes.

For any fixed natural number ${\displaystyle d}$ there exists only a finite number ${\displaystyle N(d)}$ of ${\displaystyle d}$-dimensional reflexive polytopes up to a ${\displaystyle GL(d,\mathbb {Z} )}$-isomorphism. The number ${\displaystyle N(d)}$ is known only for ${\displaystyle d\leq 4}$: ${\displaystyle N(1)=1}$, ${\displaystyle N(2)=16}$, ${\displaystyle N(3)=4319}$, ${\displaystyle N(4)=473800776.}$ The combinatorial classification of ${\displaystyle d}$-dimensional reflexive simplices up to a ${\displaystyle GL(d,\mathbb {Z} )}$-isomorphism is closely related to the enumeration of all solutions ${\displaystyle (k_{0},k_{1},\ldots ,k_{d})\in \mathbb {N} ^{d+1}}$ of the diophantine equation ${\displaystyle {\frac {1}{k_{0}}}+\cdots +{\frac {1}{k_{d}}}=1}$. The classification of 4-dimensional reflexive polytopes up to a ${\displaystyle GL(4,\mathbb {Z} )}$-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.

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.

See also

• Toric variety
• Homological mirror symmetry
• Mirror symmetry (string theory)

References

1. Batyrev, V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties". Journal of Algebraic Geometry: 493–535.
2. Nill, B. "Reflexive polytopes" (PDF).
3. 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.
4. 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.
5. I. Gelfand, M. Kapranov, S. Zelevinski (1989), "Hypergeometric functions and toric varieties", Funct. Anal. Appl. 23, no. 2, 94–10.
6. A. Varchenko (1990), "Multidimensional hypergeometric functions in conformal field theory, algebraic K-theory, algebraic geometry", Proc. ICM-90, 281–300.
7. L. Borisov (1994), "Towards the Mirror Symmetry for Calabi–Yau Complete intersections in Gorenstein Toric Fano Varieties", arXiv:alg-geom/9310001
8. Batyrev, V.; Borisov, L. (1997). "Dual cones and mirror symmetry for generalized Calabi–Yau manifolds". Mirror Symmetry, II: 71–86.
9. 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.
10. Kreuzer, M. (2008). "Combinatorics and Mirror Symmetry: Results and Perspectives" (PDF).
11. M. Kreuzer, H. Skarke (1997), "On the classification of reflexive polyhedra", Comm. Math. Phys., 185, 495–508
12. M. Kreuzer, H. Skarke (1998) "Classification of reflexive polyhedra in three dimensions", Advances Theor. Math. Phys., 2, 847–864
13. M. Kreuzer, H. Skarke (2002), "Complete classification of reflexive polyhedra in four dimensions", Advances Theor. Math. Phys., 4, 1209–1230
14. M. Kreuzer, H. Skarke, Calabi–Yau data, http://hep.itp.tuwien.ac.at/~kreuzer/CY/
15. L. Borisov (2001), "Vertex algebras and mirror symmetry", Comm. Math. Phys., 215, no. 3, 517–557.

• Algebraic geometry
• Mathematical physics
• Duality theories
• String theory