A cohomological description of connections and curvature over posets

TL;DR

This paper develops a cohomological framework for describing connections and curvature on posets, extending geometric notions like principal bundles to non-manifold settings relevant in algebraic quantum field theory.

Contribution

It introduces a cohomological approach to define and analyze connections and curvature on posets, generalizing classical differential geometry concepts to non-manifold structures.

Findings

Connections can be nonflat, with curvature as a 2-coboundary.

Flat connections correspond to homomorphisms of the poset's fundamental group into G.

An analogue of the Ambrose-Singer theorem is established.

Abstract

What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by finding a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group $G$. Interpreting a 1--cocycle as a principal bundle, a connection turns out to be a 1--cochain associated in a suitable way with this 1--cocycle; the curvature of a connection turns out to be its 2--coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into $G$. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem.