Combinatorial mirror symmetry

A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for $d$ -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 $k$ -dimensional faces of a $d$ -dimensional convex polyhedron $P$ and $(d-k-1)$ -dimensional faces of the dual polyhedron $P^{*}$ and one has $(P^{*})^{*}=P$ . In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special $d$ -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 $A$ -hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei Zelevinsky^[5] (see also the talk of Alexander Varchenko^[6]), where $A$ is the set of lattice points in a reflexive polytope $P$ .

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

References

^ 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/
^ L. Borisov (2001), "Vertex algebras and mirror symmetry", Comm. Math. Phys., 215, no. 3, 517–557.

[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.

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References