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We define exotic twisted \({\mathbb{T}}\) - 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 \({\mathbb{T}}\) -equivariant cohomology of LZ .We establish a localisation theorem for the completed periodic exotic twisted \({\mathbb{T}}\) -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. Communicated by N. A. Nekrasov
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