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The combinatorial mirror duality for Calabi–Yau hypersurfaces in toric varieties has been generalized by Lev
Borisov in the case of Calabi–Yau complete intersections in Gorenstein toric
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one can consider the polar duality for reflexive polytopes as a special case of the duality for convex
Gorenstein cones and of the duality for Gorenstein polytopes.
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A. Varchenko (1990), "Multidimensional hypergeometric functions in conformal field theory, algebraic K-theory, algebraic geometry", Proc. ICM-90, 281–300.
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Batyrev, V.; van
Straten, D. (1995). "Generalized hypergeometric functions and rational curves on Calabi–Yau complete intersections in toric varieties".
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Candelas, P.; de la Ossa, X.; Green, P.; Parkes, L. (1991). "A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory".
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M. Kreuzer, H. Skarke (2002), "Complete classification of reflexive polyhedra in four dimensions", Advances Theor. Math. Phys., 4, 1209–1230
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M. Kreuzer, H. Skarke (1998) "Classification of reflexive polyhedra in three dimensions", Advances Theor. Math. Phys., 2, 847–864
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I. Gelfand, M. Kapranov, S. Zelevinski (1989), "Hypergeometric functions and toric varieties", Funct. Anal. Appl. 23, no. 2, 94–10.
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L. Borisov (1994), "Towards the Mirror
Symmetry for Calabi–Yau Complete intersections in Gorenstein Toric Fano Varieties",
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et al. for computing the number of rational curves on Calabi–Yau quintic 3-folds can be applied to arbitrary Calabi–Yau
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M. Kreuzer, H. Skarke (1997), "On the classification of reflexive polyhedra", Comm. Math. Phys., 185, 495–508
1168:. The complete list of 3-dimensional and 4-dimensional reflexive polytopes have been obtained by physicists
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Batyrev, V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties".
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Batyrev, V.; Borisov, L. (1997). "Dual cones and mirror symmetry for generalized Calabi–Yau manifolds".
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A mathematical explanation of the combinatorial mirror symmetry has been obtained by Lev
Borisov via
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471:. In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special
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that states a
Knowledge editor's personal feelings or presents an original argument about a topic.
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L. Borisov (2001), "Vertex algebras and mirror symmetry", Comm. Math. Phys., 215, no. 3, 517–557.
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277:-dimensional convex polyhedra. The most famous examples of the polar duality provide
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Batyrev, V.; Nill, B. (2008). "Combinatorial aspects of mirror symmetry".
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1116:. The classification of 4-dimensional reflexive polytopes up to a
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1109:{\displaystyle {\frac {1}{k_{0}}}+\cdots +{\frac {1}{k_{d}}}=1}
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personal reflection, personal essay, or argumentative essay
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1477:M. Kreuzer, H. Skarke, Calabi–Yau data,
534:-hypergeometric functions introduced by
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702:{\displaystyle GL(d,\mathbb {Z} )}
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897:{\displaystyle N(4)=473800776.}
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337:-dimensional convex polyhedron
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464:{\displaystyle (P^{*})^{*}=P}
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1202:Homological mirror symmetry
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709:-isomorphism. The number
395:-dimensional faces of the
862:{\displaystyle N(3)=4319}
1390:Contemporary Mathematics
1185:conformal field theories
1181:vertex operator algebras
595:dual cone and polar cone
493:convex lattice polytopes
317:-dimensional faces of a
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757:{\displaystyle d\leq 4}
593:. Using the notions of
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917:{\displaystyle d}
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285:is dual to
1500:Categories
1230:: 493–535.
1213:References
892:473800776.
287:octahedron
220:April 2017
211:; try the
198:link to it
155:April 2017
139:improve it
93:April 2017
36:improve it
1396:: 35–66.
1241:Nill, B.
1078:⋯
1020:∈
1001:…
749:≤
546:), where
451:∗
441:∗
411:∗
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371:−
201:. Please
143:verifying
42:talk page
1377:: 71–86.
1299:16401756
1191:See also
1174:Polymake
1420:6817890
1279:Bibcode
137:Please
79:Please
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289:, the
194:orphan
192:is an
1438:(PDF)
1416:S2CID
1358:arXiv
1295:S2CID
1269:arXiv
1246:(PDF)
1406:ISBN
857:4319
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813:2
810:(
807:N
787:1
784:=
781:)
778:1
775:(
772:N
752:4
746:d
726:)
723:d
720:(
717:N
697:)
693:Z
689:,
686:d
683:(
680:L
677:G
657:d
637:)
634:d
631:(
628:N
608:d
574:P
554:A
522:A
479:d
459:P
456:=
447:)
437:P
433:(
407:P
383:)
380:1
374:k
368:d
365:(
345:P
325:d
305:k
265:d
240:)
234:(
222:)
218:(
168:)
162:(
157:)
153:(
135:.
106:)
100:(
95:)
91:(
87:.
52:)
48:(
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