Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant

TL;DR

This paper explores the relationship between geometrical and Chern-Simons formulations of (2+1)-dimensional gravity, demonstrating how grafting and Dehn twists are represented as Wilson loop observables and how the cosmological constant influences the gauge algebra structure.

Contribution

It establishes a detailed connection between geometrical constructions and the Chern-Simons phase space, introducing a novel ring-valued Wilson loop framework incorporating the cosmological constant.

Findings

Grafting corresponds to Wilson loop observables generating phase space transformations.

Dehn twists are generated by a different Wilson loop observable acting as a Hamiltonian.

The cosmological constant deforms the gauge algebra into a ring extension with a formal parameter.

Abstract

We relate the geometrical and the Chern-Simons description of (2+1)-dimensional gravity for spacetimes of topology $R\times S_g$, where $S_g$ is an oriented two-surface of genus $g>1$, for Lorentzian signature and general cosmological constant and the Euclidean case with negative cosmological constant. We show how the variables parametrising the phase space in the Chern-Simons formalism are obtained from the geometrical description and how the geometrical construction of (2+1)-spacetimes via grafting along closed, simple geodesics gives rise to transformations on the phase space. We demonstrate that these transformations are generated via the Poisson bracket by one of the two canonical Wilson loop observables associated to the geodesic, while the other acts as the Hamiltonian for infinitesimal Dehn twists. For spacetimes with Lorentzian signature, we discuss the role of the cosmological constant as a deformation parameter in the geometrical and the Chern-Simons formulation of the theory. In particular, we show that the Lie algebras of the Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz algebra over a commutative ring, characterised by a formal parameter $\Theta_\Lambda$ whose square is minus the cosmological constant. In this framework, the Wilson loop observables that generate grafting and Dehn twists are obtained as the real and the $\Theta_\Lambda$-component of a Wilson loop observable with values in the ring, and the grafting transformations can be viewed as infinitesimal Dehn twists with the parameter $\Theta_\Lambda$.