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T-duality and the exotic chiral de Rham complex

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 $Ω^{\bar{k}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$, and it admits a differential that extends the twisted differential $d_{\widehat{H}} = d + \widehat{H}$. The T-duality isomorphism $Ω^{\bar{k}}(Z,H)^{\mathbb{T}} \rightarrow Ω^{\overline{k+1}}(\widehat{Z}, \widehat{H})^{\widehat{\mathbb{T}}}$ of Bouwknegt, Evslin and Mathai extends to an isomorphism $Ω^{\bar{k}}(Z,H) \rightarrow \mathcal{A}^{\overline{k+1}}(\widehat{Z})$. 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 $Ω^{\text{ch},H,\bar{k}}(Z) \rightarrow \mathcal{A}^{\text{ch},\widehat{H},\overline{k+1}}(\widehat{Z})$ where $Ω^{\text{ch},H,\bar{k}}(Z)$ denotes the twisted chiral de Rham complex of $Z$, which chiralizes the above T-duality map.