Edge modes without edge modes

TL;DR

The work analyzes how gauge theories defined on finite regions with boundaries possess nonlocal, flux-encoded degrees of freedom governed by the Gauss constraint, and how these DOF glue across subregions without introducing extra edge-like bulk DOFs. Using Maxwell theory as a tractable model, it develops a canonical symplectic reduction that respects boundary gauge invariance, reveals flux superselection sectors, and derives a nonfactorizable but fully reconstructible gluing via a nonlocal term that replaces naive edge modes. The paper then extends the analysis to Maxwell with matter and finally to non-Abelian Yang–Mills, introducing covariant superselection sectors, functional connections (including SdW-type), flux rotations, and a geometric BRST perspective. Together these results clarify the role of the Gauss constraint, link boundary data to global DOF, and provide edge-mode-free mechanisms for entanglement and gluing analyses in gauge theories with boundaries.

Abstract

We discuss gauge theories of the Yang-Mills kind in finite regions with boundaries, and in particular the definition of the corresponding quasi-local degrees of freedom and their gluing upon composition of the underlying regions. Although the most of the technical results presented here has appeared in previous works by Gomes, Hopfmüller and the author, we adopt here a new perspective. Focusing on Maxwell theory as our model theory, in most of the text we avoid technical complications and focus on the conceptual issues related to symplectic reduction in finite and bounded regions, and to gluing$-$e.g. superselection sectors, non-locality, Dirac's dressing of charged fields, and edge modes. In this regard, the title refers to a gluing formula for the reduced symplectic structures, where the "edge mode" contribution is explicitly computed in terms of gauge-invariant bulk variables. Despite capturing most interesting features, the Abelian theory misses some crucial technical and conceptual points which are present in the non-Abelian case. To fill this gap, we dedicate the last section to a brief overview of functional connection forms, flux rotations, and geometric BRST, among other topics.

Figures (9)

• Figure 1: A graphical representation of example \ref{['Ex:R3']}.
• Figure 2: A graphical representation of $\Phi$ and of the action of finite and infinitesimal gauge symmetries over it. The two dots represent two configurations $\varphi\in\Phi$ and $\varphi^g$ related by a finite gauge transformation $g\in{\mathcal{G}}$; the line represents the orbit of ${\mathcal{G}}$ through the configuration $\varphi$; finally, the double arrow represents the value at $\varphi$ of the vector field $\xi^\sharp$ associated to an infinitesimal gauge symmetry $\xi\in\mathrm{Lie}({\mathcal{G}})$---notice that it is tangent to the orbit through $\varphi$.
• Figure 3: A graphical representation of the on-shell space $\Phi_o\subset\Phi$ (the pink surface) and of the reduced phase space $\Phi//{\mathcal{G}}$ (the pink line). Gauge transformations are tangent to $\Phi_o$.
• Figure 4: A graphical representation of symplectic reduction. The question mark in the bottom line indicates that the "projection" of $\iota^*\Omega$ from the on-shell to the reduced spaces is not warranted unless conditions \ref{['eq:SymplRedSummary']} are met.
• Figure 5: A graphical representation of symplectic reduction in the presence of boundaries. On the left: The on-shell space $\Phi_o$ is foliated by subspaces $\Phi_o^f$ characterized by a fixed flux $f$ (the coloured vertical strips); upon reduction, one obtains a symplectic foliation of $\Phi//{\mathcal{G}}$ by the superselection sectors $\Phi//_{\space f\space}{\mathcal{G}}$ (the coloured segments). On the right: The reduced space $\Phi//{\mathcal{G}}$ is "blown up" and represented as the space $\mathbb R^3$ equipped with the concentric-spheres symplectic foliation discussed in Example \ref{['Ex:R3']} and Figure \ref{['fig:R3']}; in this representation, every coloured sphere embodies a flux superselection sector $\Phi//_{\space f\space}{\mathcal{G}}$.