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

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