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.
Categories: math.DG hep-th math.QA