Coupling Dynamical Quantum Diffeomorphisms to Matter Degrees of Freedom

We introduce matter degrees of freedom into a recently proposed 2D quantum gravity model based on the Virasoro group. Quantum diffeomorphisms have dynamical content in this model, thus spoiling their classical gauge nature. The algebra of observables is enlarged now with the inclusion of an ensemble of new operators closing the affine Kac-Moody algebra of the (non-compact) semi-simple group SL(2, R), and constituting the modes of a set of scalar fields. The gravity effect on those new fields is accomplished by the natural semi-direct action of the Virasoro group on the new subalgebra. While the model is rather entangled at the severe quantum regime, at the semi-classical level we recover the action of the scalar fields modified with an added gravitational interaction term...