Vertex Algebras and Mirror Symmetry
TLDR
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.Abstract:
Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties. This paper could also be of interest to physicists, because it contains explicit descriptions of A and B models of Calabi-Yau hypersurfaces and complete intersection in terms of free bosons and fermions.read more
Citations
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Journal ArticleDOI
Elliptic genera of toric varieties and applications to mirror symmetry
Lev A. Borisov,Anatoly Libgober +1 more
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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Moduli, Motives, Mirrors
TL;DR: In this article, the authors present a survey of the mathematical developments in mirror symmetry since M. Kontsevich's report at the Zurich ICM and provide an extensive, although not exhaustive bibliography.
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Deformations of chiral algebras and quantum cohomology of toric varieties
Fyodor Malikov,V. Schechtman +1 more
TL;DR: In this paper, the authors reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) with differential deformation and derive the chiral de Rham complex over the projective space.
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All toric l.c.i.-singularities admit projective crepant resolutions
TL;DR: In this paper, the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions.
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Free-field approach to D-branes in Gepner models
TL;DR: In this paper, free-field construction of boundary states in Gepner models based on the free field realization of N=2 superconformal minimal models is presented. But the model is not considered in this paper.
References
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Book
Introduction to Toric Varieties.
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.
Journal ArticleDOI
A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory
TL;DR: In this paper, the prepotentials and geometry of the moduli spaces for a Calabi-Yau manifold and its mirror were derived and all the sigma model corrections to the Yukawa couplings and moduli space metric were obtained.
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Vertex algebras for beginners
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.
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.
Journal ArticleDOI
Chiral rings in N = 2 superconformal theories
TL;DR: In this paper, the properties of chiral operators in N = 2 superconformal theories were investigated under a one-parameter family of twists generated by the U(1) current.