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Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

Aldo Riello

TL;DR

This work develops a canonical framework for the symplectic reduction of Yang–Mills theory in bounded regions by introducing covariant superselection sectors (CSSS) labeled by boundary electric flux [f]. Within a CSSS, the reduced phase space inherits a natural, gauge-invariant 2-form Ω^{red,[f]}, which becomes fully canonical after appending a boundary Kirillov–Kostant–Souriau (KKS) term on [f], yielding a symplectic structure independent of the horizontality choice. Flux rotations arise as Hamiltonian vector fields on the reduced space only when the horizontal connection ϖ is flat, signaling nontrivial, non-dynamical transformations that modify the Coulombic field and energy. To go beyond CSSS, the paper analyzes Donnelly–Freidel edge modes and shows that their phase-space extensions break boundary gauge invariance and introduce gauge-fixing ambiguities, contrasting with the canonical CSSS construction. The results highlight a principled, ambiguity-free reduction within CSSS and clarify the role and limitations of edge modes for quasilocal YM dof, with implications for gluing and the interpretation of boundary degrees of freedom.

Abstract

I develop a theory of symplectic reduction that applies to bounded regions in Yang-Mills theory and electromagnetism. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang-Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call "flux rotations," generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang-Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka "edge modes." However, unless the new edge modes model a material physical system located at the boundary of the region, I argue that the phase space extension by edge modes is inherently ambiguous and gauge-breaking.

Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

TL;DR

This work develops a canonical framework for the symplectic reduction of Yang–Mills theory in bounded regions by introducing covariant superselection sectors (CSSS) labeled by boundary electric flux [f]. Within a CSSS, the reduced phase space inherits a natural, gauge-invariant 2-form Ω^{red,[f]}, which becomes fully canonical after appending a boundary Kirillov–Kostant–Souriau (KKS) term on [f], yielding a symplectic structure independent of the horizontality choice. Flux rotations arise as Hamiltonian vector fields on the reduced space only when the horizontal connection ϖ is flat, signaling nontrivial, non-dynamical transformations that modify the Coulombic field and energy. To go beyond CSSS, the paper analyzes Donnelly–Freidel edge modes and shows that their phase-space extensions break boundary gauge invariance and introduce gauge-fixing ambiguities, contrasting with the canonical CSSS construction. The results highlight a principled, ambiguity-free reduction within CSSS and clarify the role and limitations of edge modes for quasilocal YM dof, with implications for gluing and the interpretation of boundary degrees of freedom.

Abstract

I develop a theory of symplectic reduction that applies to bounded regions in Yang-Mills theory and electromagnetism. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang-Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call "flux rotations," generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang-Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka "edge modes." However, unless the new edge modes model a material physical system located at the boundary of the region, I argue that the phase space extension by edge modes is inherently ambiguous and gauge-breaking.

Paper Structure

This paper contains 51 sections, 7 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context and motivations
  3. Sketch of the reduction procedure
  4. Flux superselection sectors
  5. Flux rotations
  6. Completion of the reduced symplectic structure
  7. Superselection sectors vs. edge modes
  8. Mathematical setup
  9. The YM configuration space as a foliated space
  10. An example: the SdW connection
  11. The phase space of YM theory with matter as a fibre bundle
  12. Flat functional connections, gauge fixings, and dressings
  13. Symplectic geometry, the Gauss constraint, and flux superselection
  14. Off-shell symplectic geometry
  15. Off-shell symplectic geometry
  16. ...and 36 more sections

Key Result

Lemma A.1

The kernel of the SdW boundary value problem at $A\in{\mathcal{A}}$ is given by the irreducibility parameters of $A$.

Figures (1)

  • Figure 1: The (fiducial) infinite dimensional field space bundle $\Phi_{\mathsf{G}}\to\Phi_{\mathsf{G}}/{\mathcal{G}}$. ( Left) The choice of a gauge fixing section $\sigma:\Phi_{\mathsf{G}}/{\mathcal{G}}\to\Phi_{\mathsf{G}}$. ( Center) The generation of the equivariant horizontal foliation associated to $\sigma$ by means of field- independent gauge transformations. ( Right) A change in gauge fixing through the action of a field-dependent gauge transformation. The DF boundaries symmetries correspond to changes of leaf in the gauge fixing foliation depicted in the central panel.

Theorems & Definitions (17)

  • Definition A.1: Reducibility parameters
  • Definition A.2: Irreducible configurations of $A$
  • Definition A.3: SdW boundary value problem---cf. section \ref{['sec:SdW']}
  • Lemma A.1: Kernel of the SdW boundary value problem
  • proof
  • Proposition A.1: Uniqueness of $E_{\text{Coul}}$
  • proof
  • Lemma A.2
  • proof
  • Definition A.4: Flux rotations
  • ...and 7 more