Combinatorial mirror symmetry

From HandWiki

A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for [math]\displaystyle{ 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]\displaystyle{ k }[/math]-dimensional faces of a [math]\displaystyle{ d }[/math]-dimensional convex polyhedron [math]\displaystyle{ P }[/math] and [math]\displaystyle{ (d-k-1) }[/math]-dimensional faces of the dual polyhedron [math]\displaystyle{ P^* }[/math] and one has [math]\displaystyle{ (P^*)^* = P }[/math]. In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special [math]\displaystyle{ 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]\displaystyle{ A }[/math]-hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and Andrei Zelevinsky[5] (see also the talk of Alexander Varchenko[6]), where [math]\displaystyle{ A }[/math] is the set of lattice points in a reflexive polytope [math]\displaystyle{ 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]\displaystyle{ d }[/math] there exists only a finite number [math]\displaystyle{ N(d) }[/math] of [math]\displaystyle{ d }[/math]-dimensional reflexive polytopes up to a [math]\displaystyle{ GL(d,\Z) }[/math]-isomorphism. The number [math]\displaystyle{ N(d) }[/math] is known only for [math]\displaystyle{ d \leq 4 }[/math]: [math]\displaystyle{ N(1) =1 }[/math], [math]\displaystyle{ N(2) =16 }[/math], [math]\displaystyle{ N(3) = 4319 }[/math], [math]\displaystyle{ N(4)= 473 800 776. }[/math] The combinatorial classification of [math]\displaystyle{ d }[/math]-dimensional reflexive simplices up to a [math]\displaystyle{ GL(d,\Z) }[/math]-isomorphism is closely related to the enumeration of all solutions [math]\displaystyle{ (k_0, k_1, \ldots, k_d) \in \N^{d+1} }[/math] of the diophantine equation [math]\displaystyle{ \frac{1}{k_0} + \cdots + \frac{1}{k_d} =1 }[/math]. The classification of 4-dimensional reflexive polytopes up to a [math]\displaystyle{ 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

  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". https://personales.unican.es/santosf/anogia05/slides/Nill-anogia05.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. doi:10.1007/BF02101841. Bibcode: 1995CMaPh.168..493B. 
  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. 
  10. ↑ Kreuzer, M. (2008). "Combinatorics and Mirror Symmetry: Results and Perspectives". http://misgam.sissa.it/topstrings/08sissa.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.



Retrieved from "https://handwiki.org/wiki/index.php?title=Combinatorial_mirror_symmetry&oldid=2234531"