Exotic twisted equivariant cohomology of loop spaces and the twisted Bismut-Chern character

Fei Han and I define exotic twisted circle-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 circle-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted circle-equivariant cohomology for loop spaces.