Elliptic genera of toric varieties and applications to mirror symmetry
Lev A. Borisov,Anatoly Libgober +1 more
161
TL;DR: In this article, it was shown that the elliptic genus of a Calabi-Yau manifold is a Jacobi form, and that the dimensions of the genus can be determined by the Hodge numbers.
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Abstract: The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities.
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Vertex Algebras and Algebraic Curves
Abstract: Introduction Definition of vertex algebras Vertex algebras associated to Lie algebras Associativity and operator product expansion Applications of the operator product expansion Modules over vertex algebras and more examples Vertex algebra bundles Action of internal symmetries Vertex algebra bundles: Examples Conformal blocks I Conformal blocks II Free field realization I Free field realization II The Knizhnik-Zamolodchikov equations Solving the KZ equations Quantum Drinfeld-Sokolov reduction and $\mathcal{W}$-algebras Vertex Lie algebras and classical limits Vertex algebras and moduli spaces I Vertex algebras and moduli spaces II Chiral algebras Factorization Appendix Bibliography Index List of frequently used notation.
763
Elliptic genera of singular varieties
Lev A. Borisov,Anatoly Libgober +1 more
TL;DR: In this article, the authors introduced the notions of the orbifold elliptic genus and the genus of singular varieties, and the relation between them was studied, and it was shown that the generating function for the genus π n, π ρ n for symmetric groups acting on $n$-fold products coincides with the one proposed by R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verklinde.
127
Vertex Algebras and Mirror Symmetry
TL;DR: In this article, the relation between these vertex algebras for mirror Calabi-Yau manifolds and complete intersections in toric varieties is established, which can be used to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algesbras.
86
•Posted Content
Elliptic Genera of Singular Varieties
Lev A. Borisov,Anatoly Libgober +1 more
TL;DR: In this article, the authors studied the relation between elliptic genus and singular genus of singular varieties and showed that the generating function for this genus coincides with the one proposed by Dijkgraaf, Moore, Verlinde and Verlin.
77
McKay correspondence for elliptic genera
Lev A. Borisov,Anatoly Libgober +1 more
TL;DR: In this article, a correspondence between singular elliptic genera of a global quotient was established, where the former is defined in terms of the fixed point set of the action, the latter is defined by the resolution of singularities.
References
•Book
Introduction to Toric Varieties.
William Fulton
- 12 Jul 1993
TL;DR: In this article, a mini-course is presented to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications, concluding with Stanley's theorem characterizing the number of simplicies in each dimension in a convex simplicial polytope.
3.7K
•Book
Vertex algebras for beginners
Victor G. Kac
- 01 Jan 1997
TL;DR: In this paper, a formal distribution a(z,w) = 2 QFT and chiral algebras is defined and the Virasoro algebra is defined, which is a generalization of the Wightman axioms.
1.7K
•Book
Topological methods in algebraic geometry
Friedrich Hirzebruch
- 01 Jan 1966
TL;DR: In this paper, the Riemann-Roch theorem for algebraic manifolds and complex analytic vector bundles is presented. But the authors do not consider the complexity of complex analytic line bundles.
1.5K
•Journal Article
Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties
TL;DR: In this article, it was shown that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families of algebraic compactifications of affine hypersurfaces.
1.4K