Phase space structure of Chern-Simons theory with a non-standard puncture

TL;DR

This paper explicitly constructs the symplectic structure of Chern-Simons theory on a punctured surface with a special puncture, facilitating models of open universes in 2+1 gravity.

Contribution

It introduces a novel treatment of a distinguished puncture in Chern-Simons theory, enabling curvature singularities with arbitrary Lie algebra coefficients.

Findings

Explicit symplectic structure derived for the phase space.

Inclusion of a distinguished puncture with cotangent bundle variables.

Framework for modeling open universes in 2+1-dimensional gravity.

Abstract

We explicitly determine the symplectic structure on the phase space of Chern-Simons theory with gauge group $G\ltimes g^*$ on a three-manifold of topology $R \times S$, where $S$ is a surface of genus $g$ with $n+1$ punctures. At each puncture additional variables are introduced and coupled minimally to the Chern-Simons gauge field. The first $n$ punctures are treated in the usual way and the additional variables lie on coadjoint orbits of $G\ltimes g^*$. The $(n+1)$st puncture plays a distinguished role and the associated variables lie in the cotangent bundle of $G\ltimes g^*$. This allows us to impose a curvature singularity for the Chern-Simons gauge field at the distinguished puncture with an arbitrary Lie algebra valued coefficient. The treatment of the distinguished puncture is motivated by the desire to construct a simple model for an open universe in the Chern-Simons formulation of $(2+1)$-dimensional gravity.