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

TL;DR

The paper presents a canonical quantization of pure AdS3 gravity by exploiting its classical equivalence to SL(2,R)×SL(2,R) Chern-Simons theory. It develops a constrain-first approach that localizes the phase space on Teichmüller space and employs Kahler quantization, revealing a Hilbert space built from Virasoro×Virasoro representations and Liouville-like CFT data, with normalizability imposing a continuous spectrum and a lower bound on operator dimensions. Topology-changing amplitudes are proposed as a projection mechanism that reduces the large Hilbert space to a holographic subspace consistent with Liouville theory, while non-geometric projections and c<1 considerations are explored as routes to discrete spectra or minimal-model-like duals. The analysis connects Teichmüller/Kähler structure to Liouville/CFT interpretations, clarifying the role of the mapping class group and boundary data in shaping the physical state space. Appendices corroborate the torus-with-boundary computations and coordinate transitions, enriching the link between geometric quantization and holographic dual descriptions.

Abstract

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 SL(2,R)xSL(2,R). 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 Kaehler quantization of Teichmuller space. After explicitly computing the Kaehler 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 SL(2,R) 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.

Figures (7)

• Figure 1: Cutting a surface $\Sigma$ with a single boundary along homotopy cycles produces a disk with a hole (the grey disk).
• Figure 2: A cut surface $T_{(1,1)}^C$ giving a one-hole torus upon identification of the square boundary segments. We define the holonomy $W(z)$ along each edge of the square so that we have $e^{-B} e^{-A} e^B e^A$ when we complete a cycle.
• Figure 3: Gluing a collar to $\Sigma$.
• Figure 4: A global diffeomorphism changes the homology basis of $\Sigma$. Here the cycle represented with a thin line transforms into that represented by a thick line. The change of basis also induces a change in the pant decomposition of the surface.
• Figure 5: Pant decomposition of a $g=2$, one-boundary Riemann surface