T-duality and the exotic chiral de Rham complex

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

arXiv commentMinor corrections and expository improvements, to appear in Comm. Math. Phys

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Cited by in corpus (1)

• T-duality, vertical holonomy line bundles and loop Hori formulae