$\La \to 0$ limit of 2+1 Quantum Gravity for arbitrary genus

TL;DR

This paper investigates the quantum algebra of 2+1 gravity in the zero cosmological constant limit, revealing a simplified structure with commuting elements and tangent vectors, and analyzes the quantum mapping class group's invariance.

Contribution

It provides a detailed analysis of the quantum algebra structure for 2+1 gravity at zero cosmological constant, including explicit phase space representation and invariance properties.

Findings

Algebra splits into commuting elements and tangent vectors

Explicit phase space representation established

Representation for genus 2 simplified in this limit

Abstract

The abstract quantum algebra of observables for 2+1 gravity is analysed in the limit of small cosmological constant. The algebra splits into two sets with an explicit phase space representation;~one set consists of $6g-6$ {\it commuting} elements which form a basis for an algebraic manifold defined by the trace and rank identities;~the other set consists of $6g-6$ tangent vectors to this manifold. The action of the quantum mapping class group leaves the algebra and algebraic manifold invariant. The previously presented representation for $g=2$ is analysed in this limit and reduced to a very simple form. The symplectic form for $g=2$ is computed.