Liouville theory, accessory parameters and 2+1 dimensional gravity

TL;DR

This paper establishes a link between the asymptotic behavior of the conformal factor, accessory parameters, and the Hamiltonian dynamics of particles in 2+1 dimensional gravity, confirming a conjecture by Polyakov.

Contribution

It proves a relation connecting the Liouville action, accessory parameters, and the Hamiltonian structure in 2+1 gravity, extending Polyakov's conjecture to more general singularities.

Findings

Proved the relation between conformal factor asymptotics and accessory parameters.

Established the Hamiltonian nature of particle dynamics in 2+1 gravity.

Extended the relation to include general elliptic and parabolic singularities.

Abstract

We prove a relation between the asymptotic behavior of the conformal factor and the accessory parameters of the SU(1,1) Riemann- Hilbert problem. Such a relation shows the hamiltonian nature of the dynamics of N particles coupled to 2+1 dimensional gravity. A generalization of such a result is used to prove a connection between the regularized Liouville action and the accessory parameters in presence of general elliptic singularities. This relation had been conjectured by Polyakov in connection with 2-dimensional quantum gravity. An alternative proof, which works also in presence of parabolic singularities, is given by rewriting the regularized Liouville action in term of a background field.