Analytic Torsion for Twisted De Rham Complexes

Abstract

Abstract. We define analytic torsion τ (X, E , H) ∈ det H• (X, E , 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 E , with a differential given by ∇ E +H ∧ · , where ∇ E 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. The definition uses pseudodifferential operators and residue traces. We show that when dim X 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 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.

Conformal invariants of twisted Dirac operators and positive scalar curvature

“…It is only in this case that one gets a generalization of the usual Dirac operator on X, in contrast to the case of the twisted de Rham complex, cf . [12,22,5,23,24,26]. When the compact spin manifold X has non empty boundary and assuming that the Riemannian metric is of product type near the boundary and that H satisfies the absolute boundary condition, we explicitly identify the twisted Dirac operator near the boundary to be CẼH " σ`B Br`C Proof.…”

Index type invariants for twisted signature complexes and homotopy invariance

“…As argued in [35], we can conclude that deformation of H by a B-field is equivalent to deforming the usual metric to a generalized metric. In this way, deformations of the usual metric and that of the flux by a B-field are unified to a deformation of generalized metric.…”

“…Using twisted Hodge theory [35], it is easy to see that exactly as in the untwisted case, the operator A H induces an isomorphism between the twisted Hilbert module de Rham cohomologies. Following [31] (see also [27]), we introduce a path (J t,H ) 0≤t≤2 of adjointable operators on the direct sum Hilbert module…”

“…Homotopy invariance, products. Here we omit the proofs of all the statements as they are straightforward generalizations of the corresponding statements in the case of manifolds without boundary [35,36] and the absolute and relative cohomology for manifold with boundary, cf. [25,41].…”

“…Using again the twisted Hodge decomposition theorem [35], this allows to deduce that T t,H induces an isomorphism between the twisted homologies. Indeed, one first works with smooth forms and then the exponentials of the twisted laplacians induce the identity operators on homologies.…”