Analytic torsion for twisted de rham complexes

Varghese Mathai University of Adelaide Siye Wu University of Hong Kong

Differential Geometry mathscidoc:1609.10226

Journal of Differential Geometry, 88, (2), 297-332, 2011
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.
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  author={Varghese Mathai, and Siye Wu},
  booktitle={Journal of Differential Geometry},
Varghese Mathai, and Siye Wu. ANALYTIC TORSION FOR TWISTED DE RHAM COMPLEXES. 2011. Vol. 88. In Journal of Differential Geometry. pp.297-332.
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