Pauli-Lubanski scalar in the Polygon Approach to 2+1-Dimensional Gravity

TL;DR

This paper derives a conserved Pauli-Lubanski scalar in 2+1D gravity using 't Hooft's polygon approach, revealing an additional spatial shift and expressing the invariant in phase-space variables, with classical limit verification.

Contribution

It introduces a novel expression for the Pauli-Lubanski scalar in the polygon approach, incorporating an extra spatial shift and connecting it to phase-space variables.

Findings

The invariant includes an extra spatial shift Δ.

The scalar is expressed in terms of phase-space variables.

Classical limit of the invariant is verified.

Abstract

In this paper we derive an expression for the conserved Pauli-Lubanski scalar in 't Hooft's polygon approach to 2+1-dimensional gravity coupled to point particles. We find that it is represented by an extra spatial shift $\Delta$ in addition to the usual identification rule (being a rotation over the cut). For two particles this invariant is expressed in terms of 't Hooft's phase-space variables and we check its classical limit.