T-duality and the exotic chiral de Rham complex

TL;DR

This paper extends the concept of T-duality in string theory to an exotic chiral de Rham complex framework, establishing isomorphisms between twisted and exotic differential forms on dual circle bundles.

Contribution

It introduces the exotic chiral de Rham complex and demonstrates its isomorphism with the twisted chiral de Rham complex, generalizing T-duality to a chiral setting.

Findings

Defined the exotic chiral de Rham complex on the T-dual bundle.

Established an isomorphism between twisted and exotic complexes.

Chiralizes the T-duality map in differential geometry.

Abstract

Let $Z$ be a principal circle bundle over a base manifold $M$ equipped with an integral closed $3$-form $H$ called the flux. Let $\widehat{Z}$ be the T-dual circle bundle over $M$ with flux $\widehat{H}$. Han and Mathai recently constructed the $\mathbb{Z}_2$-graded space of exotic differential forms $\mathcal{A}^{\bar{k}}(\widehat{Z})$. It has an additional $\mathbb{Z}$-grading such that the degree zero component coincides with the space of invariant twisted differential forms $\Omega^{\bar{k}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$, and it admits a differential that extends the twisted differential $d_{\widehat{H}} = d + \widehat{H}$. The T-duality isomorphism $\Omega^{\bar{k}}(Z,H)^{\mathbb{T}} \rightarrow \Omega^{\overline{k+1}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$ of Bouwknegt, Evslin and Mathai extends to an isomorphism $\Omega^{\bar{k}}(Z,H) \rightarrow \mathcal{A}^{\overline{k+1}}(\widehat{Z})$. In this paper, we introduce the exotic chiral de Rham complex $\mathcal{A}^{\text{ch},\widehat{H},\bar{k}}(\widehat{Z})$ which contains $\mathcal{A}^{\bar{k}}(\widehat{Z})$ as the weight zero subcomplex. We give an isomorphism $\Omega^{\text{ch},H,\bar{k}}(Z) \rightarrow \mathcal{A}^{\text{ch},\widehat{H},\overline{k+1}}(\widehat{Z})$ where $\Omega^{\text{ch},H,\bar{k}}(Z)$ denotes the twisted chiral de Rham complex of $Z$, which chiralizes the above T-duality map.