#### Volume 18, issue 1 (2014)

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Chern–Simons line bundle on Teichmüller space

### Colin Guillarmou and Sergiu Moroianu

Geometry & Topology 18 (2014) 327–377
##### Abstract

Let $X$ be a non-compact geometrically finite hyperbolic $3$–manifold without cusps of rank $1$. The deformation space $\mathsc{ℋ}$ of $X$ can be identified with the Teichmüller space $\mathsc{T}$ of the conformal boundary of $X$ as the graph of a section in ${T}^{\ast }\mathsc{T}$. We construct a Hermitian holomorphic line bundle $\mathsc{ℒ}$ on $\mathsc{T}$, 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 $X\phantom{\rule{0.3em}{0ex}}$. This section is parallel on $\mathsc{ℋ}$ for the Hermitian connection modified by the $\left(1,0\right)$ component of the Liouville form on ${T}^{\ast }\mathsc{T}$. As applications, we deduce that $\mathsc{ℋ}$ is Lagrangian in ${T}^{\ast }\mathsc{T}$, and that ${Vol}_{R}\left(X\right)$ is a Kähler potential for the Weil–Petersson metric on $\mathsc{T}$ 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 ${\mathsc{ℒ}}^{-1}$ and the sixth power of the determinant line bundle.

##### Keywords
Chern–Simons invariants, hyperbolic manifolds, renormalized volume
##### Mathematical Subject Classification 2010
Primary: 32G15, 58J28