T-duality and the exotic chiral de Rham complex

Published 3 Aug 2020 in math.DG, hep-th, and math.QA | (2008.00632v2)

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.