On recovering continuum topology from a causal set

Abstract

An important question that discrete approaches to quantum gravity must address is how continuum features of space-time can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the space-time continuum is a locally finite partial order. A new topology on causal sets using “thickened antichains” is constructed. This topology is then used to recover the homology of a globally hyperbolic space-time from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or “Hauptvermutung” of causal set theory.