Hamiltonian Gauge theory with Corners: Constraint Reduction and Flux Superselection

We study the Hamiltonian formulation of gauge theory on spacetime manifolds endowed with a codimension-$1$ submanifold with boundary. The latter is thought of as a corner component for the spacetime manifold. We characterise the reduced phase space of the theory whenever it is described by a local momentum map for the action of the gauge group $\G$, by adapting Fr\'echet reduction by stages to the case of gauge subgroups. The local momentum map decomposes into a bulk term called constraint map defining a coisotropic constraint set, and a boundary term called flux map. The first stage, or \emph{constraint reduction} views the constraint set as the zero locus of a momentum map for a normal subgroup $\mathcal{G}_\circ\subset\mathcal{G}$, called \emph{constraint gauge group}. The second stage, or \emph{flux superselection}, interprets the flux map as the momentum map for the residual action of the \emph{flux gauge group} $\underline{\mathcal{G}}\doteq\mathcal{G}/\mathcal{G}_\circ$. Equivariance is controlled by cocycles of the flux gauge group $\underline{\mathcal{G}}$. Whereas the only physically admissible value of the constraint map is zero, the flux map is largely unconstrained. As a result, the reduced phase space of the theory, when smooth, is only a partial Poisson manifold $\underline{\underline{\mathcal{C}}}=\mathcal{C}/\mathcal{G} \simeq \underline{\mathcal{C}}/\underline{\mathcal{G}}$. Its symplectic leaves---defined through the flux map---are called \emph{flux superselection sectors}, for they provide a classical analogue of, and a road map to, the phenomenon of quantum superselection. To corners, we further assign a natural symplectic Lie algebroid over a Poisson manifold, $\mathsf{A}_{\partial} \to \mathcal{P}_{\partial}$, and show how the submanifold of on-shell configurations $\mathcal{C}_{\partial}\subset\mathcal{P}_{\partial}$ is also Poisson. We interpret $\C_\pp$ as defining the Noether charge algebra. Both $\mathcal{C}_{\partial}$ and $\underline{\underline{\mathcal{C}}}$ fibrate over a common \emph{space of superselections}, labeling the Casimirs of both Poisson structures. We showcase the formalism by explicitly working out the first and second stage reductions for a broad class of Yang--Mills theories, where $\underline{\underline{\mathcal{C}}}$ is found to be a Weinstein space, and discuss further applications to topological theories.