Chern–Simons line bundle on Teichmüller space

Guillarmou, Colin¹ ; Moroianu, Sergiu²

¹ DMA, UMR 8553 CNRS, École Normale Supérieure, 45 Rue d’Ulm, 75230 Paris Cedex 05, France
² Institutul de Matematică al Academiei Române, PO Box 1-764, RO-014700 Bucharest, Romania

Geometry & topology, Tome 18 (2014) no. 1, pp. 327-377.

Résumé

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

Détail

DOI : 10.2140/gt.2014.18.327

Classification : 32G15, 58J28
Keywords: Chern–Simons invariants, hyperbolic manifolds, renormalized volume

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