Exotic Twisted Equivariant Cohomology of Loop Spaces, Twisted Bismut-Chern Character and T-Duality

We define exotic twisted T-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 in Bouwknegt et al. (Commun Math Phys 228: 17-49, 2002) and Mathai and Stevenson (Commun Math Phys 236: 161-186, 2003), which represents classes in the completed periodic exotic twisted T-equivariant cohomology of LZ. We establish a localisation theorem for the completed periodic exotic twisted T-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.