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 Zˆ be the T-dual circle bundle over M with flux Hˆ. Han and Mathai recently constructed the Z₂-graded space of exotic differential forms Ak¯(Zˆ). It has an additional Z-grading such that the degree zero component coincides with the space of invariant twisted differential forms Ωk¯(Zˆ,Hˆ)Tˆ, and it admits a differential that extends the twisted differential dHˆ=d+Hˆ. The T-duality isomorphism Ωk¯(Z,H)T→Ωk+1(Zˆ,Hˆ)Tˆ of Bouwknegt, Evslin and Mathai extends to an isomorphism Ωk¯(Z,H)→Ak+1 (Zˆ). In this paper, we introduce the exotic chiral de Rham complex Ach,Hˆ,k¯(Zˆ) which contains Ak¯(Zˆ) as the weight zero subcomplex. We give an isomorphism Ωch,H,k¯(Z)→Ach,Hˆ,k+1 (Zˆ) where Ωch,H,k¯(Z) denotes the twisted chiral de Rham complex of Z, which chiralizes the above T-duality map.