Analytic Torsion for Twisted De Rham Complexes

We define analytic torsion τ (X,E ,H) ∈ detH(X, E ,H) for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X valued in a flat vector bundle E , with a differential given by ∇ +H ∧ · , where ∇ is a flat connection on E , H is an odd-degree closed differential form on X, and H(X, E ,H) denotes the cohomology of this Z2-graded complex. We show that when dimX is odd, τ (X, E ,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H ]. We also explain the relation to generalized geometry. We establish some basic functorial properties and, when H is a top-degree form, compute the torsion and relate it to a simplicial analogue. We also establish the relationship of an invariant form of the analytic torsion for T -dual circle bundles with 3-form fluxes.