Combinatorial mirror symmetry

A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for ${\displaystyle d}$-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 ${\displaystyle k}$-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.^[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 ${\displaystyle A}$-hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei Zelevinsky^[5] (see also the talk of Alexander Varchenko^[6]), 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 ^[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 ${\displaystyle d}$ 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.^{[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]

See also

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