Addendum to "Classical and Quantum Evolutions of the de Sitter and the anti-de Sitter Universes in 2+1 dimensions"

TL;DR

This paper revisits the phase space reduction of 2+1 dimensional Einstein gravity with a positive cosmological constant, providing a corrected construction that accounts for the geometry of SO(3,1) connections.

Contribution

It introduces a corrected physical phase space for positive cosmological constant cases, unifying different sectors through a new parametrization.

Findings

Constructed a correct phase space considering SO(3,1) geometry

Unified the two sectors of the physical phase space

Provided a new parametrization for the phase space

Abstract

The previous discussion \cite{ezawa} on reducing the phase space of the first order Einstein gravity in 2+1 dimensions is reconsidered. We construct a \lq\lq correct" physical phase space in the case of positive cosmological constant, taking into account the geometrical feature of SO(3,1) connections. A parametrization which unifies the two sectors of the physical phase space is also given.