Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality

TL;DR

This paper develops a new exotic twisted equivariant cohomology theory for loop spaces, introduces a refined Bismut-Chern character, and applies these concepts to demonstrate T-duality in string theory.

Contribution

It defines a novel exotic twisted $S^1$-equivariant cohomology for loop spaces, introduces the twisted Bismut-Chern character, and proves a localisation theorem with applications to T-duality.

Findings

Established a localisation theorem for the cohomology theory.

Constructed the twisted Bismut-Chern character form.

Applied the theory to demonstrate T-duality in string theory.

Abstract

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.