Chern-Simons line bundle on Teichmuller space

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