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

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, ϖ, 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 ϖ's, which are cousins of gauge-fixings 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 ϖ, 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, ϖ recovers the Dirac dressing of the electron