Refined analytic torsion

Abstract

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-Müller 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.