Hamiltonian gauge theory with corners: constraint reduction and flux superselection
Aldo Riello, Michele Schiavina
TL;DR
The paper develops a general Hamiltonian framework for local gauge theories on manifolds with corners by decomposing the local momentum map into a bulk constraint part and a boundary flux part, enabling a two-stage (Fréchet) reduction. The first stage reduces by the constraint gauge group to a constraint-reduced space (Weinstein-type locally), while the second stage reduces by the residual flux gauge group to a partially Poisson space whose symplectic leaves are flux superselection sectors. Corners induce a boundary symplectic/Lie algebroid structure that encodes off-shell corner data and interfaces with BV-BFV ideas, connecting the classical reduction to quantum superselection and potential quantization. The running Yang–Mills example illustrates the framework: YM yields a constraint-reduced space that is Weinstein locally, and flux sectors arise as symplectic leaves labelled by boundary flux data, with Abelian and non-Abelian cases treated distinctly. The work provides a model-independent, geometrically transparent path from classical gauge theory with corners to potential quantum sector decomposition, with explicit constructions for YM, Chern–Simons, and BF theories and links to BV–BFV formalisms.
Abstract
We study gauge theories on spacetime manifolds with a codimension-$1$ submanifold with boundary. 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 $\mathcal{G}$, by means of Fréchet reduction by stages. The momentum map decomposes into a bulk term called constraint map, defining a coisotropic constraint set, and a boundary term called flux map. In the first stage, constraint reduction, the constraint set is the zero of a momentum map for a normal subgroup $\mathcal{G}_\circ\subset\mathcal{G}$, called constraint gauge group. In the second stage, flux superselection, the flux map is the momentum map for the residual action of the flux gauge group $\underline{\mathcal{G}}\doteq\mathcal{G}/\mathcal{G}_\circ$, which also controls equivariance. The reduced phase space of the theory, when smooth, is then only a partial Poisson manifold $\underline{\underline{\mathcal{C}}}\simeq \underline{\mathcal{C}}/\underline{\mathcal{G}}$. Its symplectic leaves 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 symplectic Lie algebroid over a Poisson manifold, $\mathsf{A}_{\partial} \to \mathcal{P}_{\partial}$, and show how on-shell configurations $\mathcal{C}_{\partial}\subset\mathcal{P}_{\partial}$ are also Poisson. Both $\mathcal{C}_{\partial}$ and $\underline{\underline{\mathcal{C}}}$ fibrate over a common 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.