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.
Highlights
In the covariant symplectic formalism for field theories, it is standard to require the symplectic flow of the symmetry to be generated by functionally differentiable charges
In view of these ambiguities, the presence of boundaries often seems to beg for the introduction of edge modes; they can be used both to cancel unwanted charges—which may arise due to overly weak boundary conditions on the gauge degrees of freedom,or to reinstate gauge symmetries—which may have been lost due to overly strong ones
Barring some obstructions—posed by the existence of field configurations with global symmetries—field-space can generically be understood as a principal fiber bundle, wherein we introduce a connection-form
Summary
In the covariant symplectic formalism for field theories, it is standard to require the symplectic flow of the symmetry to be generated by functionally differentiable charges. In the presence of boundaries, if no supplementary conditions are introduced a somewhat surprising (and well-documented) feature arises: the constructed symplectic charges may differ from the constraints by a boundary term, and will not vanish on the constraint surface In some discussions, this is taken to mean that these boundary charges carry information about physical, rather than ‘pure gauge’, symmetries; in other discussions more stringent boundary conditions are imposed, eliminating the ‘pure gauge’ charges. Connection-forms encode field-variations in terms of the fields themselves—they split variations into physical and gauge with respect to the field content itself They can be non-local but are always regional, meaning they can be consistently defined intrinsically in subregions of space. These relations—which do not always exist—provide a direct link between boundary charges, dressings, and gauge invariance, unified through the concept of field-space covariance
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