A Gravitational Effective Action on a Finite Triangulation

TL;DR

This paper introduces a gravitational effective action defined on a finite triangulation of a surface, linking edge-length variations to defect angles and interpreting this as a trace anomaly in a discrete setting.

Contribution

It constructs a new function of edge-lengths that captures gravitational effects and relates local rescaling to geometric defect angles, providing a discrete analog of gravitational action.

Findings

Defines a gravitational effective action on triangulated surfaces

Shows the variation of the action relates to defect angles

Interprets the variation as a trace anomaly

Abstract

We construct a function of the edge-lengths of a triangulated surface whose variation under a rescaling of all the edges that meet at a vertex is the defect angle at that vertex. We interpret this function as a gravitational effective action on the triangulation, and the variation as a trace anomaly.