Abstract
For
an acyclic representation of the fundamental group of a compact oriented odd-dimensional
manifold, which is close enough to a unitary representation, we define a
refinement of the Ray-Singer
torsion associated to this representation. This new invariant can be viewed as an analytic counterpart of the
refined combinatorial torsion
introduced by Turaev. The refined
analytic torsion is a holomorphic function of the representation of the fundamental group.
When the representation is
unitary, the absolute value of the refined analytic torsion is equal to the Ray-Singer torsion, while its phase
is determined by the
eta-invariant. The fact that the Ray-Singer torsion and the eta-invariant can be combined into one
holomorphic function allows to
use methods of complex analysis to study both invariants. I will present several applications of this method.
(Joint work with Thomas
Kappeler)
