Η-Invariant AND FLAT VECTOR BUNDLES II

Abstract

We first apply the method and results in the previous paper to give a new proof of a result (holds in C/Z) of Gilkey on the variation of ninvariants associated to non self-adjoint Dirac type operators. We then give an explicit local expression of certain 77-invariant appearing in recent papers of Braverman-Kappeler on what they call refined analytic torsion, and propose an alternate formulation of their definition of the refined analytic torsion. A refinement in C of the above variation formula is also proposed.

The Mathematical Work of V. K. Patodi

“…[92] and [12]), which reflects other deep implications of the η-invariant and the analytic torsion, may shed a light on this mystery (cf. [72] where a holomorphic function on the representation space of fundamental group, with absolute value being equal to the associated Ray-Singer analytic torsion, is constructed by using generalized versions of the above ηinvariant and analytic torsion). On the other hand, relations between the η-form and the Bismut-Lott torsion form have been exploited in [73] and [51].…”