Analytic Torsion for Twisted De Rham Complexes

Abstract

We define analytic torsion $\tau(X,\mathcal{E},H)\in \operatorname{det}H^{\bullet}(X,\mathcal{E},H)$ 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 $\mathcal{E}$, with a differential given by $\nabla^{\mathcal{E}}+H \land \cdot$, where $\nabla^{\mathcal{E}}$ is a flat connection on $\mathcal{E}$, $H$ is an odd-degree closed differential form on $X$, and $H^{\bullet}(X,\mathcal{E},H)$ denotes the cohomology of this $\mathbb{Z}_2$ graded complex. The definition uses pseudodifferential operators and residue traces. We show that when $\operatorname{dim} X$ is odd, $\tau(X, \mathcal{E},H)$ is independent of the choice of metrics on $X$ and $\mathcal{E}$ 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 $H \mapsto H-dB$, 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-Muller Theorem. We also study the twisted analytic torsion for $T$-dual circle bundles with integral 3-form fluxes.