The Wu-Yang dictionary translates between two descriptions of gauge theories: the physicist's language of fields and particles, and the mathematician's language of principal bundles and connections. Bhalla-Ladd, March, and Weatherall identify a tension in this translation that has been hiding in plain sight.
Physicists describe gauge bosons — photons, gluons, W and Z bosons — as excitations of gauge fields. These fields are sections of vector bundles associated with the principal bundle of the theory. The gauge boson IS the section, quantized. This is the “particle-first” ontology that underlies standard treatments of Yang-Mills theory.
The fiber-bundle approach describes the same physics using connections on principal bundles. The connection — a mathematical object specifying how to parallel-transport between fibers — encodes the gauge field. But connections are not sections of vector bundles. They transform differently under gauge transformations. The Wu-Yang dictionary maps between the two descriptions, but the mapping isn't structure-preserving in the way physicists usually assume.
The consequence is a dilemma. Either the particle-first approach contains surplus structure — physical content that the fiber-bundle description doesn't capture, meaning the two formulations aren't physically equivalent — or gauge bosons are not sections of vector bundles, meaning the standard ontological picture of what a gluon IS must be revised.
The paper's title references Magritte: “Ceci n'est pas un gluon.” The image of the gluon in the particle-first formulation is not the gluon in the bundle formulation. The two pictures show the same physics but disagree about what kind of mathematical object the fundamental entities are. The disagreement is not about calculation — both give the same predictions — but about what the theory says exists.