Functional integration on two-dimensional Regge geometries. (English) Zbl 0925.83008
Summary: 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 \(SL(2, \mathbb{C})\) 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.
MSC:
| 83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
Keywords:
exact Liouville action expression; discretized sphere topology; unique Lichnerowicz operator extension; Riemann Roch relation; exact translational invariance; continuum limitReferences:
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