Functional integration on two dimensional Regge geometries

TL;DR

This paper derives exact discretized expressions for the Liouville action on Regge geometries of sphere and torus topology, ensuring invariance under conformal transformations and connecting to continuum results.

Contribution

It provides a novel exact formulation of the Liouville action and integration measure for Regge geometries with explicit invariance properties.

Findings

Exact expressions for Liouville action on Regge surfaces

Invariance under SL(2,C) and modular transformations

Continuum limit matches known results

Abstract

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.