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 convex polyhedron
-dimensional faces of the dual polyhedron
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
The combinatorial classification of
-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.