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We perform a canonical quantization of pure gravity on AdS3 using as atechnical tool its equivalence at the classical level with a Chern-Simonstheory with gauge group SL(2,R)xSL(2,R). We first quantize the theorycanonically on an asymptotically AdS space --which is topologically the realline times a Riemann surface with one connected boundary. Using the "constrainfirst" approach we reduce canonical quantization to quantization of orbits ofthe Virasoro group and Kaehler quantization of Teichmuller space. Afterexplicitly computing the Kaehler form for the torus with one boundary componentand after extending that result to higher genus, we recover known results, suchas that wave functions of SL(2,R) Chern-Simons theory are conformal blocks. Wefind new restrictions on the Hilbert space of pure gravity by imposinginvariance under large diffeomorphisms and normalizability of the wavefunction. The Hilbert space of pure gravity is shown to be the target space ofConformal Field Theories with continuous spectrum and a lower bound on operatordimensions. A projection defined by topology changing amplitudes in Euclideangravity is proposed. It defines an invariant subspace that allows for a dualinterpretation in terms of a Liouville CFT. Problems and features of the CFTdual are assessed and a new definition of the Hilbert space, exempt from thoseproblems, is proposed in the case of highly-curved AdS3.

On a Canonical Quantization of 3D Anti de Sitter Pure Gravity