Boundary Symmetry, Surface Charges, and Edge Modes
At a boundary, an allowed gauge transformation can acquire a physical charge. This chapter explains how boundary conditions, flux, and the phase-space form determine that possibility, and why charges, edge modes, and subsystem factorization require separate choices.
Enter this chapter
Section titled “Enter this chapter”Begin with Boundaries, Flux, and Boundary Ward Identities to identify the boundary conservation law.
Helpful background. Presymplectic Systems and the Covariant Phase-Space Ambiguity Map and Localized Transformations and Ward–Takahashi Identities supply presymplectic degeneracy and localized symmetry identities.
Topics
Section titled “Topics”- Boundaries, Flux, and Boundary Ward Identities — Separates interior insertions from wall-supported terms and flux.
- Proper and Improper Gauge Transformations — Tests which admissible transformations remain null gauge directions.
- Surface Charges, Integrability, and Ambiguities — Distinguishes a finite charge variation from a globally integrable charge.
- Charge Algebras, Central Terms, and Corners — Explains extensions of charge algebras and corner contributions.
- Edge Modes, Subregions, and Factorization — Compares ways to extend subregion descriptions in constrained systems.
- Asymptotic Symmetry, Soft Limits, and the Boundary Interface — States the extra assumptions relating boundary balance to soft behavior.
How the ideas fit together
Section titled “How the ideas fit together”Contracting an admissible transformation with the phase-space two-form gives a candidate charge variation. A null contraction signals redundancy in that phase space; a nonzero contraction may yield a physical charge only if it is integrable. Conservation then additionally requires the relevant flux to vanish. See Harlow and Wu 2020, §§ 2–3.
The boundary-symmetry decision diagram keeps these tests in order. Edge-mode extensions can help describe a cut system, but they do not eliminate the need to specify the chosen observable algebra, gluing, and boundary conditions.
Review the chapter
Section titled “Review the chapter”Can a well-defined integrable charge vary between two boundary cuts?
Solution
Yes. Integrability asks whether the variation defines a function on phase space. Conservation compares that function at different cuts and is controlled by flux. An integrable charge with nonzero intervening flux obeys a balance law instead of remaining constant.
Where to go next
Section titled “Where to go next”Continue to Gauge Fixing, BRST, and BV for a local algebraic description of gauge redundancy.