  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.
