Given an acyclic representation $\alpha$ of the fundamental group of a compact
oriented odd-dimensional manifold, which is close enough to an acyclic unitary
representation, we define a refinement $T_\alpha$ of the Ray-Singer torsion
associated to $\alpha$, which can be viewed as the analytic counterpart of the
refined combinatorial torsion introduced by Turaev. $T_\alpha$ is equal to the
graded determinant of the odd signature operator up to a correction term, the
metric anomaly, needed to make it independent of the choice of the
Riemannian metric.
¶ $T_\alpha$ is a holomorphic function on the space of such representations of the
fundamental group. When $\alpha$ is a unitary representation, the absolute value
of $T_\alpha$ is equal to the Ray-Singer torsion and the phase of $T_\alpha$ is
proportional to the $\eta$-invariant of the odd signature operator. The fact
that the Ray-Singer torsion and the $\eta$-invariant can be combined into one
holomorphic function allows one to use methods of complex analysis to study both
invariants. In particular, using these methods we compute the quotient of the
refined analytic torsion and Turaev’s refinement of the combinatorial torsion
generalizing in this way the classical Cheeger-M¨uller theorem. As an
application, we extend and improve a result of Farber about the relationship
between the Farber-Turaev absolute torsion and the $\eta$-invariant.
¶ As part of our construction of $T_\alpha$ we prove several new results about
determinants and $\eta$-invariants of non self-adjoint elliptic operators.