Forces such as electromagnetism and gravity reach across the Universe; they are the long-ranged forces in current physics. And yet, in many applications---theoretical and otherwise---we only have access to finite domains of the world. For instance, in computations of entanglement entropy, e.g. for black holes or cosmic horizons, we raise boundaries to separate the known from the unknown. In this talk, I will argue we do not understand gauge theory as well as we think we do, when boundaries are present.

For example: It is agreed by all that we should aim to construct variables that have a one to one relationship to the theory's physical content within bounded regions. But puzzles arise if we try to combine definitions of strictly physical variables in different parts of the world. This is most clearly gleaned by first employing the simplest tool for unique physical representation---gauge fixings---and then proceeding to stumble on its shortcomings. Whereas fixing the gauge can often shave off unwanted redundancies, the coupling of different bounded regions requires the use of gauge-variant elements. Therefore, the coupling of regional observables is inimical to gauge-fixing, as usually understood. This resistance to gauge-fixing has led some to declare the coupling of subsystems to be the raison d'être of gauge [Rov14].

Here I will explicate the problems mentioned above and illustrate a possible resolution. The resolution was introduced in a recent series of papers [Gomes & Riello JHEP '17,Gomes & Riello PRD '18,Gomes, Hopfumuller, Riello NPB '19]. It requires the notion of a connection-form in the field-space of gauge theories. Using this tool, a modified version of symplectic geometry---here called ‘horizontal’---is possible. Independently of boundary conditions, this formalism bestows to each region a physically salient, relational notion of charge: the horizontal Noether charge. It is relational in the sense that it only uses the different fields already at play and relationships between them; no new “edge-mode” degrees of freedom are required.

The guiding requirement for the construction of the relational connection-form is simply a harmonious melding of regional and global observables. I show that the ensuing notions of regional relationalism are different from other attempts at resolving the problem posed by gauge symmetries for bounded regions. The distinguishing criterion is what I consider to be the ‘acid test’ of local gauge theories in bounded regions: does the theory license only those regional charges which depend solely on the original field content? In a satisfactory theory, the answer should be “yes". Lastly, I will introduce explicit examples of relational connection-forms, and show that the ensuing horizontal symplectic geometry passes this ‘acid test’."

This seminar will be livestreamed here.

**Philosophy of Physics Seminar Convenors for TT19: James Read and Simon Saunders**