[Paper Review] Hamiltonian gauge theory with corners: constraint reduction and flux superselection

Aldo Riello, Michele Schiavina|arXiv (Cornell University)|Jul 1, 2022

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper develops a Hamiltonian framework for gauge theories on spacetimes with corners, introducing a two-stage reduction: first, constraint reduction via a momentum map for the constraint gauge group; second, flux superselection via the residual flux gauge group. The key result is that the reduced phase space is a partial Poisson manifold fibred over flux superselection sectors, providing a classical analogue of quantum superselection rules.

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, $\mathsf{A}_{\partial} o \mathcal{P}_{\partial}$, 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.

Motivation & Objective

• To formulate Hamiltonian gauge theory on spacetimes with codimension-1 boundaries (corners) using symplectic reduction techniques.
• To characterize the reduced phase space when the gauge group acts via a local momentum map, particularly in the presence of boundaries.
• To identify flux superselection sectors as symplectic leaves of the reduced phase space, arising from the momentum map of the residual gauge group.
• To construct a natural symplectic Lie algebroid on the corner manifold and relate it to the Noether charge algebra.
• To demonstrate the formalism in Yang–Mills theories and extend it to topological field theories like Chern–Simons and BF theory.

Proposed method

• Adapt Fréchet reduction by stages to infinite-dimensional gauge groups, decomposing the momentum map into bulk (constraint) and boundary (flux) terms.
• Define the constraint gauge group as the kernel of the constraint map, and the flux gauge group as the quotient of the full gauge group by the constraint subgroup.
• Use weak equivariance and cocycle conditions to control the action of the flux gauge group on the reduced phase space.
• Construct a Poisson structure on the off-shell corner data manifold, identifying its symplectic leaves as flux superselection sectors.
• Apply the formalism to Yang–Mills theory, showing that the second-stage reduction yields a Weinstein space in the abelian case.
• Establish a fibration of both the reduced phase space and the corner data over a common space of superselection labels, identifying Casimirs of both Poisson structures.

Experimental results

Research questions

• RQ1 How can Hamiltonian gauge theory be consistently formulated on spacetimes with corners using symplectic reduction?
• RQ2 What is the role of the constraint map and flux map in the momentum map decomposition for gauge theories with boundaries?
• RQ3 How do flux superselection sectors emerge from the residual gauge symmetry after constraint reduction?
• RQ4 What is the geometric structure of the corner data manifold, and how is it related to the Noether charge algebra?
• RQ5 How does the formalism apply to specific theories like Yang–Mills, Chern–Simons, and BF theory?

Key findings

• The reduced phase space of the theory is a partial Poisson manifold, denoted $\underline{\underline{\mathcal{C}}} = \mathcal{C}/\mathcal{G} \simeq \underline{\mathcal{C}}/\underline{\mathcal{G}}$, with symplectic leaves labeled by flux superselection sectors.
• The flux superselection sectors arise from the momentum map of the residual flux gauge group $\underline{\mathcal{G}} = \mathcal{G}/\mathcal{G}_{\circ}$, and are in one-to-one correspondence with the Casimirs of the corner Poisson structure.
• For Yang–Mills theory, the second-stage reduction yields a Weinstein space in the abelian case, indicating a well-behaved reduced phase space.
• The corner manifold $\mathcal{P}_{\partial}$ carries a natural partial Poisson structure, and its on-shell submanifold $\mathcal{C}_{\partial}$ is Poisson, interpreted as the Noether charge algebra.
• The formalism identifies a common space of superselection labels over which both $\underline{\underline{\mathcal{C}}}$ and $\mathcal{C}_{\partial}$ fiber, unifying the classical superselection structure.
• The Faddeev–Popov operator $\Delta_{A_0A}$ is shown to be invertible in a neighborhood of each connection $A_0$, enabling local Coulomb gauge fixing and Hodge-type decompositions in the presence of corners.