# Gauging the Boundary in Field-space

## TL;DR

This paper addresses the conceptual and technical issues of gauge theories at boundaries, proposing a connection-form approach in field-space to define physically meaningful charges and facilitate region composition.

## Contribution

It introduces a connection-form formalism in gauge theories that resolves boundary gauge ambiguities and defines relational charges independent of boundary conditions.

## Key findings

- The connection-form formalism yields a gauge-invariant notion of charge.
- It allows consistent composition of bounded regions in gauge theories.
- The approach clarifies the physical interpretation of boundary charges.

## Abstract

Local gauge theories are in a complicated relationship with boundaries. Whereas fixing the gauge can often shave off unwanted redundancies, the coupling of different bounded regions requires the use of gauge-variant elements. Therefore, coupling is inimical to gauge-fixing, as usually understood. This sort of reasoning has led some to declare the coupling of subsystems to be the raison d'\^etre of gauge. Indeed, while gauge-fixing is entirely unproblematic for a single region without boundary, it introduces arbitrary boundary conditions on the gauge degrees of freedom themselves -- these conditions lack a physical interpretation when they are not functionals of the original fields. Such arbitrary boundary choices creep into the calculation of charges through Noether's second theorem, muddling the assignment of physical charges to local gauge symmetries. The confusion brewn by gauge at boundaries is well-known, and must be contended with both conceptually and technically. Here I explicate the problems mentioned above and illustrate a possible resolution: in a recent series of papers the notion of a connection-form was put forward and implemented in the field-space of gauge theories. Using this tool, a modified version of symplectic geometry -- 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. Meanwhile, as required, the connection-form mediates a peaceful composition of regions, one compatible with the attribution of horizontal Noether charges to each region.