Tau function and Chern-Simons invariant

TL;DR

This paper introduces a new Chern-Simons invariant for certain hyperbolic 3-manifolds and relates it to the Bergman tau function, the Schottky uniformization, and the isomonodromic tau function, revealing deep connections between geometry, analysis, and mathematical physics.

Contribution

It defines a novel Chern-Simons invariant for infinite volume hyperbolic 3-manifolds and links it to tau functions and geometric invariants on Riemann surfaces.

Findings

Established a relation between Chern-Simons invariant and Bergman tau function.

Connected the Chern-Simons invariant to the eta invariant and phase of tau functions.

Derived formulas linking hyperbolic geometry, tau functions, and determinants of Laplacians.

Abstract

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 $X$.