Edge Currents and Vertex Operators for Chern-Simons Gravity

TL;DR

This paper explores the quantization of 2+1 dimensional Chern-Simons gravity, revealing how boundary conditions and sources influence edge currents and the algebraic structure, with implications for understanding quantum gravity in lower dimensions.

Contribution

It introduces a gauge-independent canonical quantization method for 2+1 gravity, classifies boundary conditions, and extends the formalism to include sources and vertex operators.

Findings

Edge currents form an $ISO(2,1)$ Kac-Moody algebra.

Boundary conditions determine the reduction to 1+1 gravity theories.

Vertex operators create sources and modify boundary algebra.

Abstract

We apply elementary canonical methods for the quantization of 2+1 dimensional gravity, where the dynamics is given by E. Witten's $ISO(2,1)$ Chern-Simons action. As in a previous work, our approach does not involve choice of gauge or clever manipulations of functional integrals. Instead, we just require the Gauss law constraint for gravity to be first class and also to be everywhere differentiable. When the spatial slice is a disc, the gravitational fields can either be unconstrained or constrained at the boundary of the disc. The unconstrained fields correspond to edge currents which carry a representation of the $ISO(2,1)$ Kac-Moody algebra. Unitary representations for such an algebra have been found using the method of induced representations. In the case of constrained fields, we can classify all possible boundary conditions. For several different boundary conditions, the field content of the theory reduces precisely to that of 1+1 dimensional gravity theories. We extend the above formalism to include sources. The sources take into account self- interactions. This is done by punching holes in the disc, and erecting an $ISO(2,1)$ Kac-Moody algebra on the boundary of each hole. If the hole is originally sourceless, a source can be created via the action of a vertex operator $V$. We give an explicit expression for $V$. We shall show that when acting