Towards the Turaev-Viro amplitudes from a Hamiltonian constraint

TL;DR

This work quantizes a classical deformation of the 3D BF phase space that encodes homogeneous hyperbolic geometries, yielding a quantum group kinematics based on $\mathcal{U}_q(\mathfrak{su}(2))$ with intertwiners solving the quantum Gauss law. The Hamiltonian constraint on a 3-valent face is quantized to a spin-1 operator whose action generates a recursion whose solution is the $q$-6j symbol, providing a concrete link to Turaev-Viro transition amplitudes for real $q$ and suggesting a path to TV-type spin foam models for 3D gravity with negative cosmological constant. The results establish a consistent kinematic and dynamical framework that connects discrete hyperbolic geometry to topological invariants, offering a first step toward controlled coarse-graining and spinor-based formulations in the curved-LQG setting. Future work will address remaining Pachner moves, spinor formulations, and potential four-dimensional extensions to the $BF+\Lambda B^2$ sector.

Abstract

3D Loop Quantum Gravity with a vanishing cosmological constant can be related to the quantization of the $\textrm{SU}(2)$ BF theory discretized on a lattice. At the classical level, this discrete model characterizes discrete flat geometries and its phase space is built from $T^\ast \textrm{SU}(2)$. In a recent paper \cite{HyperbolicPhaseSpace}, this discrete model was deformed using the Poisson-Lie group formalism and was shown to characterize discrete hyperbolic geometries while being still topological. Hence, it is a good candidate to describe the discretization of $\textrm{SU}(2)$ BF theory with a (negative) cosmological constant. We proceed here to the quantization of this model. At the kinematical level, the Hilbert space is spanned by spin networks built on $\mathcal{U}_{q}(\mathfrak{su}(2))$ (with $q$ real). In particular, the quantization of the discretized Gauss constraint leads naturally to $\mathcal{U}_{q}(\mathfrak{su}(2))$ intertwiners. We also quantize the Hamiltonian constraint on a face of degree 3 and show that physical states are proportional to the quantum 6j-symbol. This suggests that the Turaev-Viro amplitude with $q$ real is a solution of the quantum Hamiltonian. This model is therefore a natural candidate to describe 3D loop quantum gravity with a (negative) cosmological constant.

Figures (6)

• Figure 1: The precise discretization scheme to describe the deformed model HyperbolicPhaseSpace is still to be determined. We proceed here to the quantization of the model HyperbolicPhaseSpace and recover the Turaev-Viro amplitude as a solution of the Hamiltonian constraint, as well as intertwiners based on ${{\mathcal{U}}_{q}(\mathfrak{su}(2))}$. This model is therefore a natural candidate to describe 3D Euclidian LQG with $\Lambda<0$ based on ${{\mathcal{U}}_{q}(\mathfrak{su}(2))}$.
• Figure 2: An edge of the cell decomposition carries the variables $(\ell, u)$, or equivalently $(\tilde{\ell}, \tilde{u})$. To represent the constraint $\ell u = \tilde{u} \tilde{\ell}$, it is convenient to thicken the edge and turn it to a ribbon edge on which the constraint is the "commutativity" of the box.
• Figure 3: This is an open portion of a ribbon graph. The edges 1, 2, 7 meet at a ribbon vertex with the constraint $\tilde{\ell}_7^{-1} \ell_2 \ell_1={\bf 1}$, and the edges 1, 5, 6 meet at another vertex with the constraint $\ell_5\tilde{\ell}_6^{-1}\tilde{\ell}_1^{-1}={\bf 1}$. The edges 1, 2, 3, 4, 5 close to form a face with the constraint $\tilde{u}_1^{-1} u_2 u_3 \tilde{u}_4^{-1}\tilde{u}_5^{-1}={\bf 1}$.
• Figure 4: This is a face of degree 3 which is the support of a flatness constraint.
• Figure 5: On the left: the face $f_{126}$ of the graph $G$ on which we solve the quantum flatness constraint. On the right: the same region in the graph $G/f_{126}$.

Theorems & Definitions (2)

• Lemma 1
• proof