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

TL;DR

This paper investigates solutions to the Regge equations with cosmological constant on symmetric triangulations of CP^2, revealing that increasing vertices alone does not guarantee convergence to continuum geometry.

Contribution

It provides explicit solutions for p=2 and demonstrates the absence of solutions for p=3, highlighting limitations in the Regge calculus approach.

Findings

Solution found for p=2 triangulation.

No solutions found for p=3 triangulation.

Increasing vertices does not necessarily lead to continuum limit.

Abstract

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 Kuhnel. 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.