Ray–Singer type theorem for the refined analytic torsion

We show that the refined analytic torsion is a holomorphic section of the determinant line bundle over the space of complex representations of the fundamental group of a closed oriented odd-dimensional manifold. Further, we calculate the ratio of the refined analytic torsion and the Farber–Turaev combinatorial torsion. As an application, we establish a formula relating the eta-invariant and the phase of the Farber–Turaev torsion, which extends a theorem of Farber and earlier results of ours. This formula allows to study the spectral flow using methods of combinatorial topology.