Hamiltonian gauge theory with corners: constraint reduction and flux superselection

Abstract

We study the Hamiltonian formulation of gauge field theories on manifolds with corners, and develop a method to characterize their symplectic reduction whenever the theory admits a local momentum map for the gauge group action. This is achieved by adapting reduction by stages to the case of Hamiltonian actions of gauge groups associated to gauge subgroups that emerge from the presence of corners.We start from a decomposition of the local momentum map into a bulk term called constraint map, and a boundary term called flux map. The first stage, or constraint reduction, views the constraint as the zero locus of a momentum map for a normal subgroup G• ⊂ G (called constraint gauge group). The second stage, or flux superselection, interprets the flux map as the momentum map for the residual action of the flux gauge group G/G•. Equivariance is controlled by cocycles of the flux gauge group.Whereas the only physically admissible value of the constraint map is zero, the flux map is (largely) unconstrained. Consequently, the reduction by stages generally outputs a Poisson space with symplectic leaves corresponding to level sets of the flux map, and thus to the coadjoint (or affine) orbits of the flux gauge group. We call such symplectic leaves flux superselection sectors, for they provide a classical analogue of (and a road map to) the phenomenon of quantum superselection. Finally, we construct a symplectic (action) Lie algebroid over a Poisson manifold of corner fields, A ∂ → P ∂ . This encodes the corner/flux gauge symmetries and provides a description of flux superselection intrinsic to the corner submanifold which in many cases of interest is local.We work out in detail the example of Yang-Mills theory and briefly discuss other applications to Chern-Simons theory and BF theory. ContentsA. RIELLO AND M. SCHIAVINA 3.4. Local momentum forms 19 3.5. Equivariance and corner cocycles 21 3.6. Central extension by a nontrivial cocycle 24 3.7. Yang-Mills theory: general setup 25 4. Constraint and Flux momentum maps 28 4.1. Constraint/flux decomposition: Main assumptions 28 4.2. Yang-Mills theory: constraint/flux decomposition 32 4.3. The flux map 33 4.4. Flux annihilators 34 4.5. Yang-Mills theory: flux annihilators 37 4.6. The constraint and flux gauge groups 40 4.7. Yang-Mills theory: flux gauge groups 45 5. Locally Hamiltonian symplectic reduction with corners 46 5.1. First stage symplectic reduction: the constraint 46 5.2. Yang-Mills theory: constraint-reduced phase space 50 5.3. Second stage symplectic reduction: flux superselections 53 5.4. Yang-Mills theory: flux analysis 58 6. Corner data 61 6.1. Off-shell corner data 62 6.2. The Poisson manifold of off-shell corner data and superselections 70 6.3. Equivalence with on-shell superselection sectors 76 6.4. An alternative set of pre-corner data 78 6.5. Yang-Mills theory: corner Poisson structure and superselections 79 7. Applications 84 7.1. Chern-Simons theory 84 7.2. Corner ambiguity and θ-Yang-Mills theory 87 7.3. BF theory 89 Appendix A. Remarks on infinite dimensional...

Null Hamiltonian Yang-Mills theory. Soft symmetries and memory as superselection

“…In Section 2 we outline the preliminaries of Hamiltonian gauge theories on manifolds with corners, thought of as boundaries of codimension-1 hypersurfaces over which the Hamiltonian theory is specified. This is mostly a review of [RS22].…”

“…Section 4 describes the superselection structure for null, Yang-Mills theory as a consequence of the general theorem [RS22,Theorem 1]. This is a short-hand version of the paper which gives direct access to Section 7.…”

“…In this section we review the theoretical framework for the symplectic reduction of a locally Hamiltonian gauge theory on a codimension-1 hypersurface with boundaries (Σ, ∂Σ). This framework was developed in a previous publication [RS22] to which we refer for details.…”

“…Throughout this section, we will use Maxwell theory on a spacelike surface as a running example. This theory, and its non-Abelian counterpart, has been treated in great detail in the dedicated sections of [RS22], as well as in [GHR19; GR21; Rie21a] (see also [Rie21b] for a more pedagogical overview).…”

“…Locally Hamiltonian gauge theory. In short, the notion of a locally Hamiltonian gauge theory is a special case of an infinite-dimensional Hamiltonian G-space for which there exist stronger, local, versions of all the defining quantities and relations-and in particular of the Hamiltonian flow equation-which therefore hold pointwise over Σ, at the level of densities (more on this in Remark 2.3): Definition 2.2 (Locally Hamiltonian gauge theory [RS22]). A locally Hamiltonian gauge theory over Σ is given by the specification of (i) a space of local fields φ ∈ P .…”