Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity

TL;DR

The paper addresses canonical quantization of non-commutative holonomies in 2+1 gravity with a positive cosmological constant within loop quantum gravity, by studying the holonomy of $A_\lambda = A + \lambda e$ on the kinematical Hilbert space. It shows that the crossing of a quantum holonomy with a transversal edge yields a $q$-deformed crossing at the kinematical level, and that using the Duflo map to resolve operator ordering yields the exact crossing $A + A^{-1}$ with $A = e^{i o \hbar \lambda/4}$, matching a Kauffman-bracket-like structure; other orderings, including fully symmetric or flux-based approaches, fail to reproduce this. These results illuminate how quantum-group–like features can arise in a purely canonical setting and hint at how to relate canonical LQG to covariant spin-foam amplitudes such as Turaev–Viro, though a full connection requires implementing the curvature constraint and addressing potential anomalies. Overall, the work provides a concrete, SU(2) kinematical framework in which non-commutative holonomies exhibit q-deformed algebraic relations, offering a potential bridge between canonical LQG and topological quantum field theory approaches to 2+1 gravity with $\Lambda>0$.

Abstract

In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant $Λ>0$ in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection $A$ and the canonically conjugate triad field $e$. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of $A+=A + \sqrtΛe$. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection $A_λ=A+λe$ on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity. The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.

Figures (2)

• Figure 1: Cellular decomposition of the space manifold $\Sigma$ (a square lattice in this example), and the infinitesimal plaquette holonomy $W_p[A]$.
• Figure 2: Graphical representation of the action of two quantum holonomies $h_{\eta}(A_{\lambda})$ and $h_{\gamma}(A_{\lambda})$. The three dimensional structure depicted as over-crossing or under crossing encodes operator ordering. In this way the picture on the left denotes the operator action $h_{\eta}(A_{\lambda}) \triangleright h_{\gamma}(A_{\lambda})$ while the one on the right denotes $h_{\gamma}(A_{\lambda})\triangleright h_{\eta}(A_{\lambda})$.