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

Understand

In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant $\Lambda>0$ 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 $A+=A + \sqrt\Lambda e$.
• As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection $A_{\lambda}=A+\lambda e$ on the kinematical Hilbert space of loop quantum gravity.