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

Abstract

We investigate the canonical quantization of 2+1 gravity with Λ > 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 \surdΛe, 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_λ = A + λe 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.

4 pages; Proceedings of Loops'11, Madrid, to appear in Journal of Physics: Conference Series (JPCS)

References in corpus (3)

• Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
• Chern-Simons theory, Stokes' Theorem, and the Duflo map
• 2+1 gravity with positive cosmological constant in LQG: a proposal for the physical state

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