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

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.