Functional integration on two-dimensional Regge geometries Original Research Article

By adopting the standard definition of diffeomorphisms for a Regge surface we give an exact expression of the Liouville action both for the sphere and the torus topology in the discretized case. The results are obtained in a general way by choosing the unique self-adjoint extension of the Lichnerowicz operator satisfying the Riemann-Roch relation. We also give the explicit form of the integration measure for the conformal factor. For the sphere topology the theory is exactly invariant under the SL(2, C) transformations, while for the torus topology we have exact translational and modular invariance. In the continuum limit the results flow into the well-known expressions.