Communications in Mathematical Physics

, Volume 339, Issue 2, pp 663–697 | Cite as

Twisted Chiral de Rham Complex, Generalized Geometry, and T-duality

  • Andrew Linshaw
  • Varghese Mathai


The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold Z, and contains the ordinary de Rham complex at weight zero. Given a closed 3-form H on Z, we construct the twisted chiral de Rham differential D H , which coincides with the ordinary twisted differential in weight zero. We show that its cohomology vanishes in positive weight and coincides with the ordinary twisted cohomology in weight zero. As a consequence, we propose that in a background flux, Ramond–Ramond fields can be interpreted as D H -closed elements of the chiral de Rham complex. Given a T-dual pair of principal circle bundles \({Z, \widehat{Z}}\) with fluxes \({H, \widehat{H}}\), we establish a degree-shifting linear isomorphism between a central quotient of the \({i\mathbb{R}[t]}\)-invariant chiral de Rham complexes of Z and \({\widehat{Z}}\). At weight zero, it restricts to the usual isomorphism of S 1-invariant differential forms, and induces the usual isomorphism in twisted cohomology. This is interpreted as T-duality in type II string theory from a loop space perspective. A key ingredient in defining this isomorphism is the language of Courant algebroids, which clarifies the notion of functoriality of the chiral de Rham complex.


Vertex Operator Topological Type Vertex Operator Algebra Conformal Weight Vertex Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DenverDenverUSA
  2. 2.Department of Pure MathematicsUniversity of AdelaideAdelaideAustralia

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