A first analysis regarding matter-dynamical diffeomorphism coupling

TL;DR

This paper explores the quantum and classical dynamics of matter coupled to two-dimensional gravity, revealing a complex algebraic structure involving Virasoro and SU(1,1)-Kac-Moody algebras and their implications for physical modes.

Contribution

It introduces a novel analysis of matter-gravity coupling in 2D models using extended algebraic structures, highlighting global features and fusion rules previously overlooked.

Findings

Quantum diffeomorphism operators have a dynamical content.

Coupling to gravity arises from the semi-direct algebraic structure.

Classical limit yields an explicit matter-gravity coupled action.

Abstract

A first attempt at adding matter degrees of freedom to the two-dimensional ``vacuum'' gravity model presented in gr-qc/9907071 is analyzed in this paper. Just as in the previous pure gravity case, quantum diffeomorphism operators (constructed from a Virasoro algebra) possess a dynamical content; their gauge nature is recovered only after the classical limit. Emphasis is placed on the new physical modes modelled on a SU(1,1)-Kac-Moody algebra. The non-trivial coupling to ``gravity'' is a consequence of the natural semi-direct structure of the entire extended algebra. A representation associated with the discrete series of the rigid SU(1,1) is revisited in the light of previously neglected crucial global features which imply the appearance of an SU(1,1)-Kac-Moody fusion rule, determining the rather entangled quantum structure of the physical system. In the classical limit, an action which explicitly couples gravity and matter modes governs the dynamics.