Loop Hori Formulae for T-duality and Twisted Bismut-Chern Character

The main purpose of this paper is to establish the loop space formulation of T-duality in the presence of background flux. In particular, we construct a loop space analogue of the Hori formula, termed $\mathbf{\text{the}}\mathbf{\text{loop}}\mathbf{\text{Hori}}\mathbf{\text{map}}$, and demonstrate that it induces a quasi-isomorphism between the exotic twisted equivariant cohomologies on the free loop spaces of the T-dual sides. Spacetime, when viewed as the constant loops, is a submanifold of loop space. The duality that we prove on loop space restricts to the T-duality with $H$-flux on spacetime. This significantly refines the earlier work of the authors in 2015 where T-duality was established after localisation to the base space. The construction of the loop Hori map is an application of our generalization of the Bismut–Chern character in 2015, originally introduced in the loop space interpretation of the Atiyah–Singer index theorem by Atiyah–Witten and Bismut.