Volume 18, issue 1 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 6, 2483–2999
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
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 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.

Keywords
Chern–Simons invariants, hyperbolic manifolds, renormalized volume
Mathematical Subject Classification 2010
Primary: 32G15, 58J28
References
Publication
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
Authors
Colin Guillarmou
DMA, UMR 8553 CNRS
École Normale Supérieure
45 Rue d’Ulm
75230 Paris Cedex 05
France
Sergiu Moroianu
Institutul de Matematică al Academiei Române
PO Box 1-764
RO-014700 Bucharest
Romania