Boundary actions in Ponzano–Regge discretization, quantum groups and $AdS_3$

Abstract

Boundary actions for three-dimensional quantum gravity in the discretized formalism of Ponzano-Regge are studied with a view towards understanding the boundary degrees of freedom. These degrees of freedom postulated in the holography hypothesis are supposed to be characteristic of quantum gravity theories. In particular it is expected that some of these degrees of freedom reside on black hole horizons. This paper is a study of these ideas in the context of a theory of quantum gravity that requires no additional structure such as supersymmetry or special gravitational backgrounds. Lorentzian as well as Euclidean regimes are examined. Some surprising relationships to Liouville theory and string theory in AdSs are found. e-print archive: http://lanl.arXiv.org/abs/gr-qc/0002092 796 Boundary actions in Ponzano-Regge discretization

Three-dimensional gravity, point particles and Liouville theory

“…Because the consistency of both this model and LFT would be based on using the same series of representations, it is very likely that this model will be related to quantum LFT, the later giving a holographic description of the former. A similar discussion of the relation between state sum models and boundary Liouville theory can be found in [30].…”

“…Thus, the particle interpretation of LFT vertex operators passes two tests: (i) CFT relation (30) between mass and conformal dimension becomes for these vertex operators the usual gravity relation between mass and the angle deficit (for small deficit angles); (ii) partition functions agree, in the semi-classical approximation b ≪ 1, l ≫ l p . As we mentioned in the text, this agreement of the partition functions holds in a much more general situation than that of a single point particle.…”

BTZ Black Hole Entropy and the Turaev–Viro Model

“…In [28] it was explained (in the classical group limit q → 1) that the Wigner 3jm symbols assigned to the boundary triangles arise naturally when one considers the construction of the Ponzano-Regge model from the first order action on a manifold with boundary, and the alternative choices corresponding to fixing either the boundary metric or the boundary connection were discussed. It would be interesting to investigate further the type of discrete two-dimensional theory that is induced on the boundary in the case of quantum groups, and the possible relationship with conformal field theory.…”

Holography for the Lorentz group Racah coefficients

“…It would be interesting to compare the "holography" that played role in this paper with the more conventional 3D holography, for example the one that has been looked at in [11,12]. The boundary projections that appear in that context seem vaguely similar.…”