Communications in Mathematical Physics

, Volume 233, Issue 1, pp 79–136

Vertex Algebras, Mirror Symmetry, and D-Branes: The Case of Complex Tori

  • Anton Kapustin
  • Dmitri Orlov

DOI: 10.1007/s00220-002-0755-7

Cite this article as:
Kapustin, A. & Orlov, D. Commun. Math. Phys. (2003) 233: 79. doi:10.1007/s00220-002-0755-7


 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 Kähler 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 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 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Anton Kapustin
    • 1
  • Dmitri Orlov
    • 2
  1. 1.School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA. E-mail: kapustin@ias.eduUS
  2. 2.Algebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin str., GSP-1, Moscow 117966, Russia. E-mail: orlov@mi.ras.ruRU

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