We define analytic torsion (X, E,H) 2 detH•(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 rE + H ^ · ,
where rE 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 dimX is odd,
(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 7! 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-M¨uller Theorem. We also study the
twisted analytic torsion for T -dual circle bundles with integral 3-
form fluxes.