Vertex Algebras, Mirror Symmetry, And D-Branes: The Case Of Complex Tori

TL;DR

This paper introduces a new, inclusive definition of vertex algebras and constructs N=2 superconformal vertex algebras from complex tori, linking algebraic, geometric, and mirror symmetry concepts with implications for D-branes.

Contribution

It provides a novel definition of vertex algebras that encompasses non-meromorphic fields and establishes criteria for isomorphisms and mirror relations of N=2 SCVA's associated with complex tori.

Findings

Criteria for isomorphic N=2 SCVA's from different tori.

Connection between N=2 SCVA isomorphisms and derived category equivalences.

Modification of Kontsevich's Homological Mirror Symmetry conjecture.

Abstract

A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat Kahler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted'' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds.