Analytic torsion for twisted de Rham complexes

Abstract

We define analytic torsion 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 a flat connection on E plus an odd-degree closed differential form H on X. The difficulty lies in the fact that the twisted de Rham complex is only Z_2-graded, and so the definition of analytic torsion in this case uses pseudo-differential operators and residue traces. We show that when dim X is odd, then the twisted analytic torsion is independent of the choice of metrics on X and E and of the representative H in the cohomology class of 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 establish 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-Muller Theorem. We also study the relationship of the analytic torsion for T-dual circle bundles with integral 3-form fluxes.

24 pages LaTeX; corrections and additions; Keywords: Analytic torsion, twisted de Rham cohomology, generalized geometry, $T$-duality

References in corpus (7)

• Twisted K-theory and K-theory of bundle gerbes
• Generalized geometry and the Hodge decomposition
• Complex valued Ray-Singer torsion
• Spectral boundary conditions for generalizations of Laplace and Dirac operators
• Refined Analytic Torsion
• A Cheeger-Mueller theorem for symmetric bilinear torsions
• A Refinement of the Ray-Singer Torsion

Cited by in corpus (3)

• Twisted Morava K-theory and E-theory
• Analytic Torsion of Z_2-graded Elliptic Complexes
• Massey Product and Twisted Cohomology of A-infinity Algebras