Английская Википедия:Combinatorial mirror symmetry
Шаблон:Multiple issues A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for <math>d</math>-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 <math>k</math>-dimensional faces of a <math>d</math>-dimensional convex polyhedron <math>P</math> and <math>(d-k-1)</math>-dimensional faces of the dual polyhedron <math>P^*</math> and one has <math>(P^*)^* = P</math>. In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special <math>d</math>-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 <math>A</math>-hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei Zelevinsky[5] (see also the talk of Alexander Varchenko[6]), where <math>A</math> is the set of lattice points in a reflexive polytope <math>P</math>.
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 <math>d </math> there exists only a finite number <math>N(d)</math> of <math>d</math>-dimensional reflexive polytopes up to a <math>GL(d,\Z)</math>-isomorphism. The number <math>N(d)</math> is known only for <math>d \leq 4</math>: <math> N(1) =1</math>, <math>N(2) =16</math>, <math>N(3) = 4319</math>, <math>N(4)= 473 800 776.</math> The combinatorial classification of <math>d</math>-dimensional reflexive simplices up to a <math>GL(d,\Z)</math>-isomorphism is closely related to the enumeration of all solutions <math>(k_0, k_1, \ldots, k_d) \in \N^{d+1} </math> of the diophantine equation <math> \frac{1}{k_0} + \cdots + \frac{1}{k_d} =1 </math>. The classification of 4-dimensional reflexive polytopes up to a <math>GL(4, \Z) </math>-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
References
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ 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", Шаблон:Arxiv
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ 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/
- ↑ L. Borisov (2001), "Vertex algebras and mirror symmetry", Comm. Math. Phys., 215, no. 3, 517–557.
