Curvature function and coarse graining

TL;DR

This paper investigates how discrete sets of holonomy functions relate to the characterization of fiber bundles and connections, showing that in the abelian case, curvature functions can effectively replace holonomy functions for this purpose.

Contribution

The paper demonstrates that a discrete set of functions based on curvature can characterize bundle structure and constrain connections in the abelian case, extending the understanding of holonomy-based characterizations.

Findings

Discrete holonomy sets do not fully characterize bundles.

Curvature functions can characterize bundle structure in the abelian case.

Holonomy functions are covered by the exponential of curvature functions.

Abstract

A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to "local details" ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function $F_S (A)$ defined below. It covers the holonomy function in the sense that $\exp{F_S (A)} = {\rm Hol}(l= \partial S, A)$.