On a canonical quantization of 3D Anti de Sitter pure gravity

We perform a canonical quantization of pure gravity on AdS3 using as a technical tool its equivalence at the classical level with a Chern-Simons theory with gauge group SL2ℝ×SL2ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right)\times \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right) $$\end{document}. We first quantize the theory canonically on an asymptotically AdS space -which is topologically the real line times a Riemann surface with one connected boundary. Using the “constrain first” approach we reduce canonical quantization to quantization of orbits of the Virasoro group and Kähler quantization of Teichmüller space. After explicitly computing the Kähler form for the torus with one boundary component and after extending that result to higher genus, we recover known results, such as that wave functions of SL2ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right) $$\end{document} Chern-Simons theory are conformal blocks. We find new restrictions on the Hilbert space of pure gravity by imposing invariance under large diffeomorphisms and normalizability of the wave function. The Hilbert space of pure gravity is shown to be the target space of Conformal Field Theories with continuous spectrum and a lower bound on operator dimensions. A projection defined by topology changing amplitudes in Euclidean gravity is proposed. It defines an invariant subspace that allows for a dual interpretation in terms of a Liouville CFT. Problems and features of the CFT dual are assessed and a new definition of the Hilbert space, exempt from those problems, is proposed in the case of highly-curved AdS3.