Chern–Simons line bundle on Teichmüller space

Abstract

Let $X$ be a non-compact geometrically finite hyperbolic $3$–manifold without cusps of rank $1$. The deformation space $\mathcal{\mathscr{H}}$ of $X$ can be identified with the Teichmüller space $\mathcal{T}$ of the conformal boundary of $X$ as the graph of a section in $T^{\ast }\mathcal{T}$. We construct a Hermitian holomorphic line bundle $\mathcal{ℒ}$ on $\mathcal{T}$, with curvature equal to a multiple of the Weil–Petersson symplectic form. This bundle has a canonical holomorphic section defined by $exp\left(\frac{1}{\pi }{Vol}_R\left(X\right)+2\pi iCS\left(X\right)\right)$, where ${Vol}_R\left(X\right)$ is the renormalized volume of $X$ and $CS\left(X\right)$ is the Chern–Simons invariant of $X$. This section is parallel on $\mathcal{\mathscr{H}}$ for the Hermitian connection modified by the $\left(1,0\right)$ component of the Liouville form on $T^{\ast }\mathcal{T}$. As applications, we deduce that $\mathcal{\mathscr{H}}$ is Lagrangian in $T^{\ast }\mathcal{T}$, and that ${Vol}_R\left(X\right)$ is a Kähler potential for the Weil–Petersson metric on $\mathcal{T}$ 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 ${\mathcal{ℒ}}^{-1}$ and the sixth power of the determinant line bundle.