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Vertex Algebras and Mirror Symmetry
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.Comment: 45 pages, Late
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