Chern–Simons line bundle on Teichmüller space

Guillarmou, Colin; Moroianu, Sergiu

doi:10.2140/gt.2014.18.327

Download this article
	For screen
	For printing

Recent Issues

The Journal

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

DOI: 10.2140/gt.2014.18.327

Abstract

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 Teichmüller 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 $exp (\frac{1}{π} {Vol}_{R} (X) + 2 π i CS (X))$ , 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 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 $ℒ^{- 1}$ and the sixth power of the determinant line bundle.

Keywords

Chern–Simons invariants, hyperbolic manifolds, renormalized volume

Mathematical Subject Classification 2010

Primary: 32G15, 58J28

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

Volume 18, issue 1 (2014)