Functional approach to (2 + 1)-dimensional gravity coupled to particles

The quantum gravity problem of N point particles interacting with the gravitational field in 2 + 1 dimensions is approached by working out the phase-space functional integral. The maximally slicing gauge is adopted for a non-compact open universe with the topology of the plane. The momenta conjugate to the gravitational field are related to a class of meromorphic quadratic differentials. The boundary term for the non-compact space is worked out in detail. In the extraction of the physical degrees of freedom functional determinants related to the puncture formulation of string theory occur and cancel out in the final reduction. Finally, the ordering problem in the definition of the functional integral is discussed.