[Paper Review] A cohomological description of connections and curvature over posets

John E. Roberts, Giuseppe Ruzzi|arXiv (Cornell University)|Apr 7, 2006

Homotopy and Cohomology in Algebraic Topology6 references4 citations

TL;DR

This paper develops a cohomological framework for gauge theories on partially ordered sets (posets), modeling principal bundles, connections, and curvature using non-abelian Čech-type cohomology. It establishes that connections are 1-cochains with curvature as their 2-coboundary, proves an analogue of the Ambrose-Singer theorem, and shows flat connections correspond to homomorphisms from the poset's fundamental group to the structure group G.

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.

Motivation & Objective

• To develop a geometric framework for gauge theories in algebraic quantum field theory where spacetime is modeled as a poset rather than a manifold.
• To generalize principal bundles, connections, and curvature to posets using cohomological methods.
• To establish a correspondence between flat connections and homomorphisms from the fundamental group of the poset to the structure group G.
• To define and analyze holonomy and gauge transformations in this cohomological setting.
• To provide a foundation for constructing algebraic quantum field theories based on bundle-theoretic structures over posets.

Proposed method

• Define the simplicial set of singular simplices on a poset K, using order-preserving maps from the standard simplex poset to K.
• Introduce 1-cohomology of a poset with values in a group G, identifying 1-cocycles with principal bundles.
• Define connections as 1-cochains satisfying a compatibility condition with the 1-cocycle (transition functions).
• Define curvature as the 2-coboundary of a connection, generalizing the standard differential geometric notion.
• Construct the group of gauge transformations as automorphisms of the 1-cocycle, acting via conjugation on connections.
• Prove an analogue of the Ambrose-Singer theorem using holonomy defined along paths in the poset.

Experimental results

Research questions

• RQ1 How can the notions of principal bundle, connection, and curvature be generalized from manifolds to posets?
• RQ2 What is the cohomological interpretation of connections and curvature in the context of a poset with values in a non-abelian group G?
• RQ3 How do flat connections relate to homomorphisms from the fundamental group of the poset to G?
• RQ4 Can holonomy and gauge transformations be defined in this cohomological framework, and do they satisfy analogues of standard theorems?
• RQ5 What is the structure of the gauge group acting on connections, and how does it depend on the topology of the poset?

Key findings

• Connections on a principal bundle over a poset are described as 1-cochains whose curvature is the 2-coboundary of the connection.
• Non-flat connections exist, demonstrating that curvature is non-trivial even in the poset setting.
• Flat connections are in one-to-one correspondence with group homomorphisms from the fundamental group of the poset to G.
• Holonomy along paths in the poset is well-defined and satisfies a version of the Ambrose-Singer theorem.
• The group of gauge transformations acts on connections via conjugation, and for abelian G or simply connected posets, this group is isomorphic to G.
• The action of the gauge group on connections is not free, as the trivial connection is fixed by all gauge transformations.