Vertex Algebras, Mirror Symmetry, and DBranes: The Case of Complex Tori
Abstract
A vertex algebra is an algebraic counterpart of a twodimensional 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 antimeromorphic. To any complex torus equipped with a flat Kähler metric and a closed 2form 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 the 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 2form 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 2form. We also describe the implications of our results for BPS Dbranes on CalabiYau manifolds.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2003
 DOI:
 10.1007/s0022000207557
 arXiv:
 arXiv:hepth/0010293
 Bibcode:
 2003CMaPh.233...79K
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Quantum Algebra
 EPrint:
 70 pages, AMS Latex. v2: a gap in the reasoning of Appendix B has been filled, and a proof of the "Borcherds formulas" has been added