Article ID Journal Published Year Pages File Type
4665703 Advances in Mathematics 2014 58 Pages PDF
Abstract

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.

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Physical Sciences and Engineering Mathematics Mathematics (General)
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