Chern-Simons line bundle on Teichmüller space
Résumé
Using Chern-Simons theory, we define a holomorphic Hermitian line bundle $\mathcal{L}_g$ over the deformation space $\mathcal{H}$ of a given convex co-compact hyperbolic $3$ manifolds $X$, identified to a quotient $\mathcal{T}_X$ of Teichmüller space $\mathcal{T}_g$ of the conformal boundary of $X$. The curvature of $\mathcal{L}_g$ is a constant times the Weil-Petersson form. We construct a holomorphic canonical section of $\mathcal{L}_g$ with norm given in terms of renormalized volume and argument given by Chern-Simons invariant of the conformally compactified metric. The manifold $\mathcal{H}$ can be seen as a submanifold of the tangent space $T\mathcal{T}_X$ of $\mathcal{T}_X$, projecting diffeomorphically to the base $\mathcal{T}_X$ and we use our canonical section to prove that $\mathcal{H}$ is Lagrangian in $T\mathcal{T}_X$ and that the renormalized volume is a Kähler potential for the Weil-Petersson metric on $\mathcal{T}_X$. Finally, we construct an explicit isometric isomorphism between the line bundle $\mathcal{L}_g$ and a power of the determinant line bundle on Teichmüller space.
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