Non-commutative holonomies in 2+1 LQG and Kauffman's brackets

TL;DR

This paper explores the quantization of 2+1 gravity with positive cosmological constant in Loop Quantum Gravity, focusing on non-commutative holonomies and their relation to Kauffman's brackets, revealing new algebraic structures.

Contribution

It introduces a novel quantum holonomy operator in 2+1 LQG linked to Kauffman's brackets, connecting crossing actions with q-deformed identities using standard SU(2) states.

Findings

Explicit construction of the quantum holonomy operator.

Demonstration of the relationship between holonomy crossings and Kauffman's brackets.

Holonomy actions are described entirely in terms of SU(2) spin network states.

Abstract

We investigate the canonical quantization of 2+1 gravity with {\Lambda} > 0 in the canonical framework of LQG. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A\pm = A \PM \surd{\Lambda}e, where the SU(2) connection A and the triad field e are the conjugated variables of the theory. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A_{\lambda} = A + {\lambda}e acting on spin network links of the kinematical Hilbert space of LQG. We provide an explicit construction of the quantum holonomy operator, exhibiting 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 the result is completely described in terms of standard SU(2) spin network states.