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

Abstract

We perform a canonical quantization of pure gravity on AdS 3 using as a technical tool its equivalence at the classical level with a Chern-Simons theory with gauge group SL(2, R) × SL(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 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 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 AdS 3 .

Monodromic vs geodesic computation of Virasoro classical conformal blocks

“…The above Einstein/Liouville relation conforms with the well-known correspondence between two-dimensional Virasoro conformal blocks and physical states in three-dimensional quantum SLp2, Cq Chern-Simons theory[25]. See, e.g., refs [26,27,28,29,30]. for more discussion of quantum 3d gravity and the Liouville theory.…”

Geometric actions for three-dimensional gravity

“…In the particular case of a ϕ independent b 0 , action (3.36) has been obtained in [8] after solving the constraints of Chern-Simons theory based on a semisimple group G on a spatial disk with a source. In the context of AdS 3 gravity it has been used in [48] and more recently in [49].…”

Symplectic realization of two interacting spin-two fields in three dimensions

“…The greatest puzzles in modern cosmology such as Dark Matter [2] and Dark Energy [3], which attempt to explain the primordial and late time accelerating expansion of the current universe, have been strong motivators for a plethora of alternative gravity theories beyond original Einstein's General Relativity (GR), both at the ultraviolet and infrared scale [4][5][6]. Besides that, the issue concerning the compatibility between renormalizability and the unitarity of gravity at the quantum regime [7,11], has also led many theorists to search for extensions of GR and their three-dimensional cousins; because they are, to our knowledge, another framework for addressing some conceptual features of the four-dimensional GR and some fundamental issues of quantum gravity [8][9][10]. Nonetheless, modifications and/or extensions to the theory of GR are highly restricted according to Lovelock's theorem [12,13].…”