Analytic torsion for twisted de Rham complexes

Abstract

We define analytic torsion τ(X,ε,H) ∈ det H •(X,ε,H) for the twisted de Rham complex, consisting of the spaces of differential forms on a compact oriented Riemannian manifold X valued in a flat vector bundle ε, with a differential given by ∇ε + H Λ · , where ∇ε is a flat connection on ε, H is an odd-degree closed differential form on X, and H•(X, ε, H) denotes the cohomology of this ℤ2-graded complex. The definition uses pseudodifferential operators and residue traces. We show that when dim X is odd, τ(X, ε, H) is independent of the choice of metrics on X and ε and of the representative H in the cohomology class [H]. We define twisted analytic torsion in the context of generalized geometry and show that when H is a 3-form, the deformation H → H-dB, where B is a 2-form on X, is equivalent to deforming a usual metric g to a generalized metric (g,B). We demonstrate some basic functorial properties. When H is a top-degree form, we compute the torsion, define its simplicial counterpart, and prove an analogue of the Cheeger-Müller Theorem. We also study the twisted analytic torsion for T -dual circle bundles with integral 3- form fluxes.