Solutions of the Regge Equations on some Triangulations of $CP^2$

Simplicial geometries are collections of simplices making up a manifold together with an assignment of lengths to the edges that define a metric on that manifold. The simplicial analogs of the Einstein equations are the Regge equations. Solutions to these equations define the semiclassical approximation to simplicial approximations to a sum-over-geometries in quantum gravity. In this paper, we consider solutions to the Regge equations with cosmological constant that give Euclidean metrics of high symmetry on a family of triangulations of $CP^2$ presented by Banchoff and K\"uhnel. This family is characterized by a parameter $p$. The number of vertices grows larger with increasing $p$. We exhibit a solution of the Regge equations for $p=2$ but find no solutions for $p=3$. This example shows that merely increasing the number of vertices does not ensure a steady approach to a continuum geometry in the Regge calculus.