Tau function and Chern–Simons invariant

We define a Chern–Simons invariant for Schottky hyperbolic 3-manifolds of infinite volume. We then prove an expression relating the Bergman tau function on a fiber space over the Teichmüller space to the lifting of the function F defined by Zograf on Teichmüller space, and a holomorphic function on this space which we introduce. If the point in this space corresponds to a marked Riemann surface X, then this function is constructed from the renormalized volume and our Chern–Simons invariant for the bounding 3-manifold of X given by Schottky uniformization, together with a regularized Polyakov integral. We also obtain a relation between the Chern–Simons invariant and the eta invariant of the bounding 3-manifold, with defect given by the phase of the Bergman tau function of X.