Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary actions in Ponzano-Regge discretization, Quantum groups and AdS(3)

Martin O'Loughlin

TL;DR

This work develops a boundary-focused analysis of 3D quantum gravity using the Ponzano–Regge discretization of BF theory, deriving boundary actions and exploring how different boundary conditions give rise to topological versus dynamical boundary degrees of freedom. It shows how the discretized path sum acquires diffeomorphism-invariant structure through Pachner moves and introduces quantum-group regularization via the Turaev–Viro construction, connecting the bulk theory to a boundary theory reminiscent of Liouville theory in AdS$_3$ contexts. In the Lorentzian case, the large-$j$ limit points to a boundary Liouville-like structure and a direct link to string theory on $AdS_3$, with a proposed correspondence between deformation parameters and central charges that bridges TV invariants, quantum groups, and Liouville CFT. The paper also sketches how these ideas could extend to $3+1$ dimensions, suggesting dynamical triangulations and horizon-boundary dynamics as a path toward a microscopic understanding of black-hole entropy in a non-supersymmetric setting.

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 AdS(3) are found.

Boundary actions in Ponzano-Regge discretization, Quantum groups and AdS(3)

TL;DR

This work develops a boundary-focused analysis of 3D quantum gravity using the Ponzano–Regge discretization of BF theory, deriving boundary actions and exploring how different boundary conditions give rise to topological versus dynamical boundary degrees of freedom. It shows how the discretized path sum acquires diffeomorphism-invariant structure through Pachner moves and introduces quantum-group regularization via the Turaev–Viro construction, connecting the bulk theory to a boundary theory reminiscent of Liouville theory in AdS contexts. In the Lorentzian case, the large- limit points to a boundary Liouville-like structure and a direct link to string theory on , with a proposed correspondence between deformation parameters and central charges that bridges TV invariants, quantum groups, and Liouville CFT. The paper also sketches how these ideas could extend to dimensions, suggesting dynamical triangulations and horizon-boundary dynamics as a path toward a microscopic understanding of black-hole entropy in a non-supersymmetric setting.

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 AdS(3) are found.

Paper Structure

This paper contains 11 sections, 48 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Ponzano Regge from BF-theory
  3. Symmetry and normalization
  4. Regularization
  5. Boundaries
  6. Quantum discrete boundaries
  7. Two-dimensional discrete path sums
  8. Lorentzian manifolds, Liouville theory and string theory on $AdS_3$
  9. $3+1$ dimensions
  10. Angular momentum identities
  11. Limits of $6j$ symbols

Figures (7)

  • Figure 1: A tetrahedron and the the part of the dual lattice that it intersects.
  • Figure 2: An edge of the lattice showing the corresponding dual face
  • Figure 3: A pair of boundary triangles and their tetrahedra. Dotted lines are the dual lattice. Bold dotted lines highlight a dual face cut by the boundary.
  • Figure 4: Boundary discretization arising from the boundary of bulk tetrahedra and the dual lattice (trivalent graph).
  • Figure 5: An example of part of a horizon configuration showing an isosceles "excitation".
  • ...and 2 more figures