Chern-Simons line bundle on Teichm\"uller space

TL;DR

This paper constructs a Hermitian line bundle on Teichmüller space linked to hyperbolic 3-manifolds, revealing geometric structures and invariants like volume and Chern-Simons, with applications to Weil-Petersson geometry.

Contribution

It introduces a new Hermitian holomorphic line bundle on Teichmüller space related to hyperbolic 3-manifolds and establishes its properties and applications.

Findings

The line bundle's curvature is proportional to the Weil-Petersson form.

The renormalized volume acts as a Kähler potential for the Weil-Petersson metric.

Explicit isomorphism between the line bundle and the determinant line bundle for Schottky uniformization.

Abstract

Let $X$ be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of $X$ can be identified with the Teichm\"uller space $\mc{T}$ of the conformal boundary of $X$ as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on $\mc{T}$, with curvature equal to a multiple of the Weil-Petersson symplectic form. This bundle has a canonical holomorphic section defined by $e^{\frac{1}{\pi}{\rm Vol}_R(X)+2\pi i\CS(X)}$ where ${\rm Vol}_R(X)$ is the renormalized volume of $X$ and $\CS(X)$ is the Chern-Simons invariant of $X$. This section is parallel on $\mc{H}$ for the Hermitian connection modified by the $(1,0)$ component of the Liouville form on $T^*\mc{T}$. As applications, we deduce that $\mc{H}$ is Lagrangian in $T^*\mc{T}$, and that ${\rm Vol}_R(X)$ is a K\"ahler potential for the Weil-Petersson metric on $\mc{T}$ and on its quotient by a certain subgroup of the mapping class group. For the Schottky uniformisation, we use a formula of Zograf to construct an explicit isomorphism of holomorphic Hermitian line bundles between $\mc{L}^{-1}$ and the sixth power of the determinant line bundle.