CaltechAUTHORS
  A Caltech Library Service

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

Kapustin, Anton and Orlov, Dmitri (2003) Vertex Algebras, Mirror Symmetry, and D-Branes: The Case of Complex Tori. Communications in Mathematical Physics, 233 (1). pp. 79-136. ISSN 0010-3616. https://resolver.caltech.edu/CaltechAUTHORS:20160506-073753322

[img] PDF - Submitted Version
See Usage Policy.

560Kb

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20160506-073753322

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 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00220-002-0755-7DOIArticle
http://link.springer.com/article/10.1007/s00220-002-0755-7PublisherArticle
https://arxiv.org/abs/hep-th/0010293arXivDiscussion Paper
ORCID:
AuthorORCID
Kapustin, Anton0000-0003-3903-5158
Orlov, Dmitri0000-0002-2230-457X
Additional Information:© 2003 Springer-Verlag. Received: 3 May 2001. Accepted: 17 August 2002. Published online: 8 January 2003. We are grateful to Maxim Kontsevich for valuable comments and to Markus Rosellen for pointing out a gap in the reasoning of Appendix B in the first version of the paper. We also wish to thank the Institute for Advanced Study, Princeton, NJ, for a very stimulating atmosphere. The first author was supported by DOE grant DE-FG02-90ER40542. The second author was supported in part by RFFI grant 99-01-01144 and a grant for support of leading scientific groups N 00-15-96085. The research described in this publication was made possible in part by Award No RM1-2089 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF). Communicated by R.H. Dijkgraaf.
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG02-90ER40542
Russian Foundation for Fundamental Investigations (RFFI)99-01-01144
Russian Foundation for Fundamental Investigations (RFFI)N 00-15-96085
Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF)RM1-2089
Issue or Number:1
Record Number:CaltechAUTHORS:20160506-073753322
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160506-073753322
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66708
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:06 May 2016 20:28
Last Modified:03 Oct 2019 09:59

Repository Staff Only: item control page