Refined analytic torsion for twisted de Rham complexes

Let $E$ be a flat complex vector bundle over a closed oriented odd dimensional manifold $M$ endowed with a flatconnection $\nabla$. The refined analytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler.Recently, Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex with an odd-degreeclosed differential form $H$, other than one form, as a flux and with coefficients in $E$. In this paper, wegeneralize the construction of the refined analytic torsion to the twisted de Rham complex. We show that the refinedanalytic torsion of the twisted de Rham complex is independent of the choice of the Riemannian metric on $M$ and theHermitian metric on $E$. We also show that the twisted refined analytic torsion is invariant (under a naturalidentification) if $H$ is deformed within its cohomology class. We prove a duality theorem, establishing arelationship between the twisted refined analytic torsion corresponding to a flat connection and its dual. We alsodefine the twisted analogue of the Ray–Singer metric and calculate the twisted Ray–Singer metric of the twistedrefined analytic torsion. In particular, we show that in case that the Hermitian connection is flat, the twistedrefined analytic torsion is an element with the twisted Ray–Singer norm one.

Keywords

determinant, analytic torsion, twisted de Rham complex

2010 Mathematics Subject Classification

58J52

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Published 2 September 2011