Twisted chiral de Rham complex, generalized geometry, and T-duality

TL;DR

This paper extends the chiral de Rham complex to include a twisted differential influenced by background fluxes, establishing a T-duality isomorphism that connects geometric and cohomological structures in string theory.

Contribution

It constructs a twisted chiral de Rham differential and demonstrates a T-duality isomorphism using Courant algebroids, linking string theory backgrounds with vertex algebra structures.

Findings

The twisted chiral de Rham cohomology vanishes in positive weight.

In weight zero, it matches ordinary twisted cohomology.

A T-duality isomorphism is established between principal circle bundles with fluxes.

Abstract

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.