Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory

TL;DR

This paper introduces a computational method to analyze the homology of causal sets, helping to identify when they resemble continuous manifolds by examining the stability of homology groups as a function of a thickening parameter.

Contribution

The paper develops a numerical tool to compute and study the homology of causal sets, providing a new way to test for manifoldlikeness based on homological stability.

Findings

Homology groups stabilize at a certain scale in simulated 2d and 3d causal sets.

Rapid fluctuations in homology occur below the stability scale.

Stability of homology groups indicates potential manifoldlikeness.

Abstract

We present a computational tool that can be used to obtain the "spatial" homology groups of a causal set. Localisation in the causal set is seeded by an inextendible antichain, which is the analog of a spacelike hypersurface, and a one parameter family of nerve simplicial complexes is constructed by "thickening" this antichain. The associated homology groups can then be calculated using existing homology software, and their behaviour studied as a function of the thickening parameter. Earlier analytical work showed that for an inextendible antichain in a causal set which can be approximated by a globally hyperbolic spacetime region, there is a one parameter sub-family of these simplicial complexes which are homological to the continuum, provided the antichain satisfies certain conditions. Using causal sets that are approximated by a set of 2d spacetimes our numerical analysis suggests that these conditions are generically satisfied by inextendible antichains. In both 2d and 3d simulations, as the thickening parameter is increased, the continuum homology groups tend to appear as the first region in which the homology is constant, or "stable" above the discreteness scale. Below this scale, the homology groups fluctuate rapidly as a function of the thickening parameter. This provides a necessary though not sufficient criterion to test for manifoldlikeness of a causal set.