A unified geometric framework for boundary charges and dressings: non-Abelian theory and matter

TL;DR

The paper develops a unified, geometry-based framework for boundary charges and dressings in non-Abelian gauge theories with matter by introducing a field-space connection $\varpi$ that defines horizontal (physical) directions. It shows how horizontality yields a gauge-covariant, boundary-compatible symplectic structure and explains how global (reducible) configurations generate nontrivial physical charges while generic gauge transformations are screened. Different realizations of $\varpi$ (e.g., Singer–DeWitt for YM, Higgs for matter, and Lorentz for vielbein gravity) illuminate how boundary data, edge modes, and dressings arise as geometric consequences of field-space structure rather than new degrees of freedom. The work clarifies the relation between dressings, gauge fixings, and boundary composition, and connects the constructed boundary charges to physically meaningful global quantities, thereby providing a robust, relational description that can address Gribov-type obstructions and gluing of regions. Its framework has broad implications for understanding gauge invariance, the role of boundaries in field theory, and the nonperturbative structure of the path integral.

Abstract

Boundaries in gauge theories are a delicate issue. Arbitrary boundary choices enter the calculation of charges via Noether's second theorem, obstructing the assignment of unambiguous physical charges to local gauge symmetries. Replacing the arbitrary boundary choice with new degrees of freedom suggests itself. But, concretely, such boundary degrees of freedom are spurious---i.e. they are not part of the original field content of the theory---and have to disappear upon gluing. How should we fit them into what we know about field-theory? We resolve these issues in a unified and geometric manner, by introducing a connection 1-form, $\varpi$, in the field-space of Yang-Mills theory. Using this geometric tool, a modified version of symplectic geometry---here called `horizontal'---is possible. Independently of boundary conditions, this formalism bestows to each region a physical notion of charge: the horizontal Noether charge. The horizontal gauge charges always vanish, while global charges still arise for reducible configurations characterized by global symmetries. The field-content itself is used as a reference frame to distinguish `gauge' and `physical'; no new degrees of freedom, such as group-valued edge modes, are required. Different choices of reference fields give different $\varpi$'s, which are cousins of gauge-fixing like the Higgs-unitary and Coulomb gauges. But the formalism extends well beyond gauge-fixings, for instance by avoiding the Gribov problem. For one choice of $\varpi$, would-be Goldstone modes arising from the condensation of matter degrees of freedom play precisely the role of the known group-valued edge modes, but here they arise as preferred coordinates in field space, rather than new fields. For another choice, in the Abelian case, $\varpi$ recovers the Dirac dressing of the electron.

Figures (9)

• Figure 1: A pictorial representation of the field-space ${\Phi}$ seen as a principal fiber bundle. We have highlighted a configuration $\varphi$, its (gauge-transformed) image under the action of $R_g:\varphi\mapsto\varphi^g$, and its orbit $\mathcal{O}_\varphi \cong {\mathcal{G}}$. We have also represented the quotient space of 'gauge-invariant configurations' ${\Phi}/{\mathcal{G}}$. On the left hand side of the picture, we have 'zoomed into' a representation of $\varphi$ and $\varphi^g$ as sections of a vector bundle over the spacetime region $M$ (here, we are assuming $\varphi$ to be the scalar field $\phi$ valued in $W$).
• Figure 2: In this representation $\Phi$ is the page's plane and the orbits are given by concentric circles around $\varphi_1$; $S$ is a section. The field $\varphi_2$ is a generic field in ${\Phi}$. The field $\varphi_1$ has a nontrivial stabilizer group: its orbit is reduced to a lower dimensional manifold (in this case just a point). The projection of $\varphi_1$ on ${\Phi}/{\mathcal{G}}$ sits at a qualitatively different point than that of $\varphi_2$: $\varphi_1$ is part of a lower dimensional stratum (in this case a 0-dimensional one).
• Figure 3: A pictorial representation of the split of ${\rm T}_\varphi {\Phi}$ into a vertical subspace $V_\varphi$ spanned by $\{\xi_\varphi^\sharp, \xi\in{\mathrm{Lie}({\mathcal{G}})}\}$ and its horizontal complement $H_\varphi$ defined as the kernel at $\varphi$ of a functional connection $\varpi$. With dotted lines, we represent a different choice of horizontal complement associated to a different choice of $\varpi$.
• Figure 4: The dressing path $\gamma$ from $\star$ to $\varphi$.
• Figure 5: The affine path from $A^\star=0$ to $A'=g^{-1}{\mathrm{d}} g$ is not vertical to the trivial path $\gamma$ from $A^\star$ to $A=0$: it passes through configurations with nonvanishing curvature.