Notes on a few quasilocal properties of Yang-Mills theory

TL;DR

This paper investigates the non-local features of Yang-Mills gauge theories with boundaries, using a geometric approach to characterize how fields are glued across regions, revealing that radiative modes and local charges suffice without introducing new boundary degrees of freedom.

Contribution

It provides a geometric framework for understanding the gluing of Yang-Mills fields across regions, distinguishing radiative and Coulombic components, and analyzing topologically non-trivial cases in 1+1 dimensions.

Findings

Radiative components and local charges are sufficient for gluing.

Global radiative modes are influenced by topological features and Aharonov-Bohm phases.

No new boundary degrees of freedom are needed in the formalism.

Abstract

Gauge theories possess non-local features that, in the presence of boundaries, inevitably lead to subtleties. In this article, we continue our study of a unified solution based on a geometric tool operating on field-space: a connection form. We specialize to the $D+1$ formulation of Yang-Mills theories on configuration space, and we precisely characterize the gluing of the Yang-Mills field across regions. In the $D+1$ formalism, the connection-form splits the electric degrees of freedom into their pure-radiative and Coulombic components, rendering the latter as conjugate to the pure-gauge part of the gauge potential. Regarding gluing, we obtain a characterization for topologically simple regions through closed formulas. These formulas exploit the properties of a generalized Dirichlet-to-Neumann operator defined at the gluing surface; through them, we find only the radiative components and the local charges are relevant for gluing. Finally, we study the gluing into topologically non-trivial regions in 1+1 dimensions. We find that in this case, the regional radiative modes do not fully determine the global radiative mode (Aharonov-Bohm phases). For the global mode takes a new contribution from the kernel of the gluing formula, a kernel which is associated to non-trivial cohomological cycles. In no circumstances do we find a need for postulating new local degrees of freedom at boundaries. $\text{The partial results of these notes have been completed and substantially clarified in a more recent, comprehensive article from October 2019. (titled: "The quasilocal degrees of freedom of Yang-Mills theory").}$