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Chern–Simons line bundle on Teichmüller space

Colin Guillarmou and Sergiu Moroianu

Geometry & Topology 18 (2014) 327–377

Let X be a non-compact geometrically finite hyperbolic 3–manifold without cusps of rank 1. The deformation space of X can be identified with the Teichmüller space T of the conformal boundary of X as the graph of a section in TT. We construct a Hermitian holomorphic line bundle 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 π VolR(X) + 2πiCS(X)), where VolR(X) is the renormalized volume of X and CS(X) is the Chern–Simons invariant of X. This section is parallel on for the Hermitian connection modified by the (1,0) component of the Liouville form on TT. As applications, we deduce that is Lagrangian in TT, and that VolR(X) is a Kähler 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 1 and the sixth power of the determinant line bundle.

Chern–Simons invariants, hyperbolic manifolds, renormalized volume
Mathematical Subject Classification 2010
Primary: 32G15, 58J28
Received: 11 October 2011
Revised: 19 January 2013
Accepted: 8 September 2013
Published: 29 January 2014
Proposed: Jean-Pierre Otal
Seconded: Walter Neumann, Yasha Eliashberg
Colin Guillarmou
École Normale Supérieure
45 Rue d’Ulm
75230 Paris Cedex 05
Sergiu Moroianu
Institutul de Matematică al Academiei Române
PO Box 1-764
RO-014700 Bucharest