3d Quantum Gravity: Coarse-Graining and q-Deformation

TL;DR

This work investigates coarse-graining in 3d quantum gravity within the Ponzano-Regge spinfoam by examining invariance under the 4\text{-}1 Pachner move. It develops extended BE-type identities with length and volume insertions via holonomy actions and triple graspings, revealing explicit propagation rules for edge lengths and tetrahedron volumes under coarse-graining and connecting these to Hamiltonian constraints. The study then links these results to the q-deformation of the 6j-symbol, showing that the leading-order q-derivative introduces a volume term that encodes a cosmological constant, while preserving Pachner move invariance order-by-order in \(q\); numerical tests support the interpretation and suggest avenues for all-orders understanding. Collectively, the paper clarifies how coarse-graining interacts with geometric observables and cosmological-constant deformations in 3d gravity, providing a geometrical and algebraic framework that could extend to higher dimensions and Lorentzian signatures through q-deformed topological models and their continuum limits.

Abstract

The Ponzano-Regge state-sum model provides a quantization of 3d gravity as a spin foam, providing a quantum amplitude to each 3d triangulation defined in terms of the 6j-symbol (from the spin-recoupling theory of SU(2) representations). In this context, the invariance of the 6j-symbol under 4-1 Pachner moves, mathematically defined by the Biedenharn-Elliot identity, can be understood as the invariance of the Ponzano-Regge model under coarse-graining or equivalently as the invariance of the amplitudes under the Hamiltonian constraints. Here we look at length and volume insertions in the Biedenharn-Elliot identity for the 6j-symbol, derived in some sense as higher derivatives of the original formula. This gives the behavior of these geometrical observables under coarse-graining. These new identities turn out to be related to the Biedenharn-Elliot identity for the q-deformed 6j-symbol and highlight that the q-deformation produces a cosmological constant term in the Hamiltonian constraints of 3d quantum gravity.

Figures (11)

• Figure 1: A tetrahedron with spins living on its six edges.
• Figure 2: 3-2 Pachner move
• Figure 3: 4-1 Pachner move
• Figure 4: From the tetrahedron to the dual spin network graph: each triangle represented by a dual vertex and each edge by a dual link, the dual graph also has the combinatorics of a tetrahedron.
• Figure 5: The holonomy operator around the dual cycle $(456)$ corresponding to a summit of the tetrahedron induces a tent move moving the summit and thus shifting the spins $j_{4},j_{5},j_{6}$, which can be interpreted as a 4-1 Pachner move creating a vertex within the tetrahedron.