  We define a Chern-Simons invariant for a certain class of infinite volume
hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau
function on a cover of the Hurwitz space, to the lifting of the function $F$
defined by Zograf on Teichm\"uller space, and another holomorphic function on
the cover of the Hurwitz space which we introduce. If the point in cover of the
Hurwitz space corresponds to a 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 relating determinants of Laplacians on $X$ in the
hyperbolic and singular flat metrics. Combining this with a result of Kokotov
and Korotkin, we obtain a similar expression for the isomonodromic tau function
of Dubrovin. 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 $X$.
