Improved and perfect actions in discrete gravity

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they—although being defined on a triangulation—reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of general relativity. We construct the perfect action in three dimensions with a cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge calculus with curved simplices, which arises naturally in this context.

Figure 1

Coarse triangulation $\mathcal{T}$ consisting of edged $E$, triangles $T$ and tetrahedra $\Sigma$.

Figure 2

Fine triangulation $\tau$ consisting of edged $e$, triangles $t$ and tetrahedra $\sigma$.