Liouville Theory, Accessory Parameters and 2+1 Dimensional Gravity

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 Image 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.