Refined analytic torsion for twisted de Rham complexes

摘要

Let $E$ be a flat complex vector bundle over a closed oriented odddimensional manifold $M$ endowed with a flat connection $\nabla$. The refinedanalytic torsion for $(M,E)$ was defined and studied by Braverman and Kappeler.Recently Mathai and Wu defined and studied the analytic torsion for the twistedde Rham complex with an odd degree closed differential form $H$, other than oneform, as a flux and with coefficients in $E$. In this paper we generalize theconstruction of the refined analytic torsion to the twisted de Rham complex. Weshow that the refined analytic torsion of the twisted de Rham complex isindependent of the choice of the Riemannian metric on $M$ and the Hermitianmetric on $E$. We also show that the twisted refined analytic torsion isinvariant (under a natural identification) if $H$ is deformed within itscohomology class. We prove a duality theorem, establishing a relationshipbetween the twisted refined analytic torsion corresponding to a flat connectionand its dual. We also define the twisted analogue of the Ray-Singer metric andcalculate the twisted Ray-Singer metric of the twisted refined analytictorsion. In particular we show that in case that the Hermtitian connection isflat, the twisted refined analytic torsion is an element with the twistedRay-Singer norm one.