Boundary Symmetry, Surface Charges, and Edge Modes

Enter this chapter by declaring the boundary problem, not by inspecting the boundary value of a gauge parameter. To decide whether a transformation is a physical boundary symmetry, first ask whether it preserves the chosen field space and boundary conditions, whether the complete generator one-form exists, and whether that one-form is globally integrable and nontrivial. The answer can be an inadmissible transformation, a proper null direction, an integrable charged symmetry, or a nonintegrable direction that needs further input. Once a charge exists, flux and the charge algebra characterize its evolution and representation; they are not prerequisites for recognizing one charged direction.

The chapter develops that decision process for bounded or asymptotic regions in quantum field theory. It connects boundary Ward identities, Hamiltonian surface charges, charge algebras, subregion edge data, and the interface with soft limits. It does not develop gravitational phase spaces, scattering theorems, holographic dictionaries, or theorem-first BV–BFV constructions; those applications and formalisms have dedicated continuations below.

Helpful background. Presymplectic Systems and the Covariant Phase-Space Ambiguity Map supplies the field-space one-form and Hamiltonian language used to test a surface charge. Localized Transformations and Ward–Takahashi Identities supplies the localized-variation and boundary-term logic used for Ward identities. Neither is required merely to choose a route through this overview.

Parent volume. Symmetry and Gauge Structure

Jump to: choose a route · apply the decision sequence · review the chapter

Diagnose your preparation

Use these checks to identify the shortest useful entry point. An Unsure result is a routing signal, not a barrier to the overview.

Localized variations and Stokes’ theorem. Ready: You can localize a continuous transformation, integrate by parts, and distinguish a bulk equation-of-motion term from a boundary contribution; enter Boundaries, Flux, and Boundary Ward Identities. Unsure: You drop every total derivative before specifying the integration region. Repair: Use Localized Transformations and Ward–Takahashi Identities.

Constraints and generators. Ready: You can distinguish a gauge constraint in the bulk from the boundary term needed to make its generator differentiable; enter Proper and Improper Gauge Transformations. Unsure: You call every transformation generated by a first-class constraint physically null, even when the generator has a surface term. Repair: Use Gauge Orbits, Gauss Constraints, and Stabilizers and the presymplectic background above.

One-forms versus charges. Ready: You can test whether a field-space one-form is exact and distinguish local closure from global path-independence; enter Surface Charges, Integrability, and Ambiguities. Unsure: You write a symbol $H_\epsilon$ as soon as a surface variation $\boldsymbol{\delta}H_\epsilon$ appears. Repair: Use Symplectic Forms, Hamiltonian Flows, and Poisson Brackets followed by the presymplectic page.

Brackets and extensions. Ready: You can separate a Poisson bracket of charges from the Lie bracket of parameters and recognize that a central term must satisfy an appropriate cocycle condition; enter Charge Algebras, Central Terms, and Corners. Unsure: You identify every extra boundary term with a quantum anomaly. Repair: Use Hamiltonian Group Actions and Moment Maps and then the charge-algebra page.

Subregions and gluing. Ready: You can explain why gauge constraints obstruct a naive tensor product across an entangling surface; enter Edge Modes, Subregions, and Factorization. Unsure: You assume that adding boundary variables is always unique or always physically necessary. Repair: Begin with Gauge Fields, Redundancy, and Observable Content and then use the edge-mode page.

Asymptotic limits. Ready: You can state a falloff condition, the class of transformations that preserves it, and the surface on which a limiting charge is evaluated; enter Asymptotic Symmetry, Soft Limits, and the Boundary Interface. Unsure: You infer a soft theorem solely from the existence of a large parameter at infinity. Repair: Read the first three chapter pages before the asymptotic interface.

Choose a route

Build the full structural account. Read boundary Ward identities $\to$ proper and improper transformations $\to$ surface charges $\to$ charge algebras $\to$ edge modes $\to$ the asymptotic interface. At the end, you can move from a localized variation to a bounded charge claim without conflating integrability, conservation, algebra, or factorization.
Decide whether a gauge direction is physical. Read proper and improper transformations $\to$ surface charges, add Quantum Implementations, Projective Actions, and Central Extensions, and then read charge algebras. At the end, you can distinguish inadmissible, null, charged, and nonintegrable directions on a declared phase space.
Derive a boundary balance law. Read boundary Ward identities $\to$ proper and improper transformations $\to$ surface charges. Add charge algebras when corners or field-dependent parameters enter. At the end, you can state what crosses the boundary and whether a charge is cut independent.
Study a subregion or gluing problem. Read Gauge-Invariant and Dressed Observables $\to$ proper and improper transformations $\to$ edge modes, using the surface-charge page when a Hamiltonian generator is claimed. At the end, you can separate algebraic centers, boundary frames, extended phase spaces, and physical edge excitations.
Approach a soft or memory statement. Read boundary Ward identities $\to$ proper and improper transformations $\to$ surface charges $\to$ the asymptotic interface. At the end, you can identify the additional falloff, limiting, matching, and scattering inputs needed beyond the universal boundary-charge structure.

The arrows give a productive reading order, not a logical implication. A focused lookup can start later when its declared phase space, boundary prescription, and prerequisite charge construction are already secure.

From boundary data to a physical classification

Let $\Sigma$ be a hypersurface with codimension-two boundary $S=\partial\Sigma$ , let $\Phi$ denote the fields, and let $R_\epsilon$ be the field-space vector generated by a parameter $\epsilon$ . Before classifying $R_\epsilon$ , declare the action including boundary and corner terms, the field space and admitted variations, the boundary or falloff conditions, the allowed parameter class, and the orientation. Changing any of these data can change the answer.

The site convention is $\boldsymbol{\omega}=-\boldsymbol{\delta}\boldsymbol{\Theta}$ and $\iota_X\Omega_\Sigma=\boldsymbol{\delta}H_X$ . Accordingly, the relevant field-space one-form is

\eta_\epsilon(\delta) := \bigl(\iota_{R_\epsilon}\Omega_\Sigma\bigr)(\delta) = \Omega_\Sigma(R_\epsilon,\delta).

After the complete boundary and corner prescription has been included, the bulk constraints or equations of motion commonly reduce this pairing to a surface expression,

\eta_\epsilon(\delta) \ \widehat{=}\ \int_{S=\partial\Sigma} \boldsymbol{k}_\epsilon[\delta\Phi;\Phi].

The hat records that the surface representation is not an off-shell identity in general. The form $\boldsymbol{k}_\epsilon$ depends on choices of symplectic-potential, improvement, and corner representative; those choices must either be fixed or carried explicitly. Only after the one-form is globally integrable does an additional reference normalization fix the additive constant in $H_\epsilon$ . A well-posed variational principle and a complete Hamiltonian variation are therefore upstream of the charge test, not optional repairs applied after a preferred answer is chosen. This ordering and the role of boundary and corner terms are developed in Harlow and Wu 2020, § 1, pp. 3–5; §§ 2.2–2.4, pp. 11–23, Open PDF and Speziale 2026, §§ 2.3–2.6, pp. 10–14; §§ 3.1–3.3, pp. 21–26, Open PDF.

When $\epsilon=\epsilon[\Phi]$ depends on the fields, the prescription must separate the variation of the parameter from the variation on which the charge one-form is evaluated. Apply the resulting adjusted variation before testing any of the three outcomes below, and use the associated adjusted bracket in the algebra test. Field dependence is not itself a failure of integrability.

Read the figure from top to bottom. The solid path records tests on one fixed phase space; dashed boxes mark a stopped or conditional conclusion. The dashed return arrow is different: an edge extension replaces the phase space and presymplectic form, so the entire classification must be repeated. The semantic table below gives the complete linear equivalent.

A four-stage decision ladder first declares the boundary problem and tests admissibility and a finite generator variation, then classifies the boundary pairing as null, integrable and charged, or nonexact before separately testing flux, algebra, and any optional edge extension. — A gauge parameter at a boundary acquires physical meaning only through the declared phase space and its generator. With $\eta_\epsilon(\delta)=\iota_{R_\epsilon}\Omega_\Sigma(\delta)$ , $H_\epsilon$ is constant on each connected component in outcome A because $\eta_\epsilon$ vanishes for every admitted variation; it is a proper gauge direction only when an identity-connected direction is also declared redundant. Outcome B has a globally integrable, nontrivial Hamiltonian variation $\eta_\epsilon=\boldsymbol{\delta}H_\epsilon$ . Outcome C lacks a global charge until local nonclosure or a nonzero period is resolved. Field-dependent parameters modify the variation and bracket before this classification rather than defining a fourth outcome. For A and B—or for C only after a compatible charge/flux prescription resolves the obstruction—conservation and algebra are subsequent tests. The displayed equation defines the flux sign: positive $\mathcal F_\epsilon[B_{12}]$ lowers $H_\epsilon$ from $S_1$ to $S_2$ . The diagram is schematic and not to scale.

The same logic is available without the image:

Question	Required test	Valid conclusion	Shortcut to reject
Is the transformation part of this boundary problem?	$R_\epsilon$ preserves the declared fields, conditions, and parameter class	It is admissible	$\epsilon\vert_S\neq0$ therefore it is physical
Does a candidate generator one-form exist?	The complete bulk-plus-surface variation is finite and well defined for every admitted $\delta\Phi$	A well-defined candidate generator one-form exists; its integrability can be tested	A bulk constraint alone is the full generator
Which boundary and corner representative is used?	Fix the symplectic-potential, improvement, and corner representative or carry its dependence explicitly	The charge variation has a declared prescription	An additive reference for $H_\epsilon$ fixes this earlier ambiguity
Is the surface one-form trivial?	$\eta_\epsilon(\delta)=0$ for every admitted variation	$H_\epsilon$ is constant on each connected component; it may be normalized to zero	$H_\epsilon$ vanishes on one reference state, therefore the direction is proper
Is it charged and integrable?	$\eta_\epsilon=\boldsymbol{\delta}H_\epsilon$ globally and $\eta_\epsilon\not\equiv0$	An integrable charged boundary symmetry	A symbol $\boldsymbol{\delta}H_\epsilon$ guarantees path independence
Does the parameter depend on the fields?	Fix the adjusted variation before testing A, B, or C and use the adjusted bracket in the algebra test	The same three outcomes remain available under the stated prescription	Field dependence by itself means nonintegrability
Is integrability obstructed?	Test local field-space closure and periods around noncontractible loops	No global Hamiltonian charge exists without further input	Every nonintegrable direction is unphysical
Is an existing or repaired charge conserved?	For transported $\epsilon$ , evaluate the boundary flux in $H_\epsilon[S_2]-H_\epsilon[S_1]+\mathcal F_\epsilon[B_{12}]=0$ using a compatible charge/flux prescription	Zero flux gives cut independence; nonzero flux may describe a physical, nonconserved charge	Integrable implies conserved
What algebra is represented?	Use the appropriate bracket, parameter dependence, and corner prescription	The charges may represent the symmetry algebra with central or field-dependent terms	Any extension is automatically a quantum anomaly
Does a subregion need extra variables?	Specify the gluing or factorization problem and construct $(\mathcal P_{\mathrm{ext}},\Omega_{\mathrm{ext}})$	An edge extension may restore the desired action or composition law	Edge modes are a unique universal repair

The three pairing outcomes concern a one-form on the chosen phase space, not the value of a charge on one configuration. In particular, an integrable charge may vanish on a special state while remaining nontrivial on nearby states. Conversely, a constant nonzero value can be shifted by a reference choice; only its variation and the declared normalization determine whether it carries physical information.

Conservation is a second question once $H_\epsilon$ exists. Let $B_{12}$ be the portion of the physical or asymptotic boundary between cuts $S_1$ and $S_2$ . We define the sign of $\mathcal F_\epsilon$ by writing the balance law as

H_\epsilon[S_2] - H_\epsilon[S_1] + \mathcal F_\epsilon[B_{12}] =0.

Positive $\mathcal F_\epsilon[B_{12}]$ therefore lowers $H_\epsilon$ from $S_1$ to $S_2$ . When the flux vanishes, the same transported parameter gives a cut-independent charge. When it does not, a compatible charge/flux prescription may still describe a physical, nonconserved charge whose change measures what crossed the boundary. Wald and Zoupas provide a structurally important gravitational example of separating the Hamiltonian variation, flux, and conservation prescription; this chapter uses the distinction without importing their gravitational conclusions into every QFT boundary problem Wald and Zoupas 2000, §§ 3–4, pp. 9–19, Open PDF.

Finally, an extended phase space is a new proposal, not a relabeling of the old one. Boundary frames or edge variables can make regional symmetry actions or gluing transparent, but different questions can support different extensions or no explicit extension at all. In their stated classical Yang–Mills and general-relativistic settings with field-independent gauge parameters, the gauge-invariance and degeneracy distinction is analyzed in Assanioussi et al. 2024, §§ 3.1–3.4, pp. 13–18, Open PDF. Complementary extended-phase-space and relational treatments appear in Donnelly and Freidel 2016, §§ 2.4–2.5, pp. 14–16, Open PDF; Riello 2021, § 5, pp. 24–29, Open PDF focuses primarily on a finite-region Maxwell analysis.

Exact chapter guide

1. Boundaries, Flux, and Boundary Ward Identities

Start here to see what changes when a localized transformation is integrated over a region with boundary. The page derives the bulk Ward identity together with boundary inflow or leakage terms, distinguishes a boundary condition from a boundary equation of motion, and interprets flux as a balance law rather than a failed conservation slogan. After it, you can state which operator or current crosses the boundary and which assumptions remove the flux. It uses the localized Ward-identity background named above and Differential Forms, Integration, Orientation, and Stokes Theorem. Continue to proper and improper transformations when the localized variation is a gauge transformation, or to surface charges when a Hamiltonian generator is sought.

2. Proper and Improper Gauge Transformations

This page asks when an allowed gauge transformation remains a null redundancy and when it acts nontrivially at a boundary. It constructs the complete generator, separates bulk constraints from surface terms, and rejects a classification based only on whether $\epsilon$ vanishes at $S$ . After it, you can identify admissible and identity-connected null candidates on a declared phase space. It uses the boundary Ward page, Gauge Orbits, Gauss Constraints, and Stabilizers, and the presymplectic background. Continue to surface charges to test integrability and reference dependence.

3. Surface Charges, Integrability, and Ambiguities

This page asks when the surface one-form $\eta_\epsilon$ integrates to a function $H_\epsilon$ on phase space. It tests local closure and global periods, tracks improvements and corner representatives, treats reference normalization, and explains the adjusted variation required by field-dependent parameters. After it, you can say whether a proposed charge exists globally and exactly which prescription it uses. It uses the proper/improper page and the presymplectic background. Continue to charge algebras or to the asymptotic interface according to the application.

4. Charge Algebras, Central Terms, and Corners

This page asks how integrable surface charges represent the algebra of allowed transformations. It distinguishes ordinary, adjusted, and covariant brackets; derives the possible extension term; checks its cocycle and reference dependence; and keeps corner composition explicit. After it, you can test whether a central or field-dependent term is structural, removable, or prescription dependent without calling it an anomaly by default. It uses the surface-charge page and Quantum Implementations, Projective Actions, and Central Extensions. Continue to edge modes for gluing or to anomaly chapters only when a genuine quantum obstruction has been established.

5. Edge Modes, Subregions, and Factorization

This page asks why gauge constraints complicate subregion factorization and what an edge extension actually accomplishes. It compares algebraic centers, boundary frames, dressed regional observables, and extended Hilbert or phase spaces, while separating auxiliary gluing data from dynamical boundary excitations. After it, you can formulate a regional observable problem without assuming a unique tensor product or a unique edge-mode prescription. It uses the proper/improper classification and Gauge-Invariant and Dressed Observables; surface charges are required when the extension is claimed to carry a Hamiltonian symmetry.

6. Asymptotic Symmetry, Soft Limits, and the Boundary Interface

This page asks which parts of the bounded-region analysis survive when the boundary is taken to an asymptotic limit. It organizes falloffs, preserving transformations, limiting charges, flux, matching conditions, and the extra analytic input needed to relate Ward identities to soft or memory statements. After it, you can identify the precise point at which a universal boundary charge construction becomes a scattering, gravitational, or holographic problem. It uses the Ward, proper/improper, and surface-charge pages. Continue to the dedicated soft-theorem or BMS treatments for developed applications.

Conventions and recurring threads

The site conventions apply throughout. In particular, the metric signature is $(+---)$ , and the site presymplectic sign and contraction order are the ones stated above. Some references reverse the definition of $\boldsymbol{\omega}$ or write the two arguments of $\Omega_\Sigma$ in the opposite order. Translate their displayed signs before comparing formulas; the null, exact, nonexact, and flux distinctions are unchanged.

Symbol	Meaning in this chapter	Distinction to preserve
$\Sigma$	Hypersurface carrying the presymplectic form	It need not be a complete Cauchy surface for the whole spacetime
$S=\partial\Sigma$	Codimension-two cut or corner	It is not the boundary segment through which flux propagates
$B_{12}$	Boundary segment between $S_1$ and $S_2$	Its orientation fixes the sign in the balance law
$R_\epsilon$	Field-space vector generated by an allowed parameter	A nonzero $\epsilon\vert_S$ does not by itself make $R_\epsilon$ physical
$\boldsymbol{k}_\epsilon[\delta\Phi;\Phi]$	Surface charge-variation form	It is distinct from a boundary current such as $k_\partial^a$ in a Ward identity
$\eta_\epsilon$	Field-space one-form obtained by contracting $\Omega_\Sigma$	It is not a charge until global exactness is established
$H_\epsilon[S]$	Integrable charge on a cut with a stated reference	Integrability does not imply conservation
$\mathcal F_\epsilon[B_{12}]$	Flux through the intervening boundary	Nonzero flux need not make the charge unphysical

Three threads recur across all six pages. First, boundary conditions select a theory or sector; they are not merely late computational conveniences. Second, improvements, corner terms, and reference choices can change representatives without erasing the need to state the prescription. Third, field-dependent parameters change the variation and bracket, so formulas derived for fixed $\epsilon$ cannot be transferred mechanically. See Speziale 2026, §§ 3.4–3.6, pp. 26–33, Open PDF for a 2026 synthesis of these structural issues.

What a boundary value cannot decide

A nonzero parameter is not yet a charge. The parameter must preserve the declared boundary problem, and the complete generator variation must define a nontrivial exact one-form. Dirichlet, Neumann, mixed, radiative, and asymptotic conditions can classify the same formal parameter differently.

A vanishing charge on one state is not a proper gauge transformation. A proper/null direction has zero pairing with every admitted variation, subject to the theory’s declaration of redundancy. An integrable charge can vanish on the vacuum and be nonzero elsewhere.

An integrable charge is not automatically conserved. Integrability asks whether a field-space one-form is a differential. Conservation asks whether the flux between cuts vanishes for the transported parameter.

A central term is not automatically an anomaly. A classical charge algebra can carry a central extension, and its representative can depend on corner or reference choices. A quantum anomaly is a failure of a quantum symmetry statement and requires its own regulator and consistency analysis.

Edge variables are not a universal physical spectrum. They may encode a boundary frame, restore a regional symmetry action, or support gluing. Whether they are auxiliary, superselected, or dynamical depends on the observable algebra, boundary dynamics, and chosen completion.

An asymptotic charge is not by itself a soft theorem. The asymptotic construction first requires a stated spacetime dimension and theory, falloffs, preserving transformations, matching or regularity conditions, and a compatible charge/flux prescription Speziale 2026, §§ 5.2–5.4, pp. 45–56, Open PDF. Turning that structure into a soft theorem additionally requires a scattering setup and a declared perturbative and infrared regime; those inputs belong to the Soft Theorems continuation.

Boundaries and canonical exits

If the remaining question is…	Continue to…	Why the question leaves this chapter
How boundary charges become soft factors in a scattering amplitude	Soft Theorems	The derivation needs asymptotic states, infrared limits, and amplitude factorization
How BMS symmetry, memory, and soft sectors fit together	BMS, Memory, and Soft Sectors as Holographic Data	Null-infinity geometry and gravitational matching are application-specific inputs
How a Noether charge enters black-hole entropy	Noether-Charge Entropy and Higher-Curvature Terms	Horizon generators, gravitational constraints, and entropy normalization require a curved-spacetime treatment
How boundary phase spaces become a BFV complex	Boundary Phase Spaces, Constraints, and the BFV Charge	The theorem-first graded construction adds ghosts, cohomological data, and gluing axioms
How centers and edge choices affect entanglement	Centers, Edge Extensions, and Distillable Entanglement	Entropic quantities depend on the regional algebra, operational task, and superselection structure
How to quantize after a gauge condition is chosen	The Faddeev–Popov Construction	Determinants, ghosts, residual transformations, and gauge-parameter dependence are quantization questions
Whether a boundary failure is a genuine quantum anomaly	Anomalies, Inflow, and Matching	Regulated quantum Ward identities and consistency conditions go beyond a classical charge extension

Review the chapter

A useful response should expose each independent decision rather than replace the sequence with “large gauge transformation.”

1. Classify a boundary transformation without using its boundary value

A parameter $\epsilon$ is nonzero on $S$ . List the tests required before it can be called a charged boundary symmetry.

A complete response first declares the action and boundary/corner terms, field space, admitted variations, boundary conditions, parameter class, and orientation. It checks that $R_\epsilon$ preserves those data and that the complete bulk-plus-surface generator variation is finite and well defined. It then fixes or tracks the boundary and corner representatives and tests whether $\eta_\epsilon$ is globally exact and nontrivial. After obtaining $H_\epsilon$ , it fixes the additive reference normalization and then tests flux, conservation, the appropriate bracket, and corner terms. The fact that $\epsilon|_S\neq0$ supplies none of those tests.

2. Separate a null direction from a charge that vanishes on the vacuum

Two transformations have $H_\epsilon=0$ on a chosen reference solution. For the first, $\eta_\epsilon(\delta)=0$ for every admitted variation. For the second, $\eta_\epsilon=\boldsymbol{\delta}H_\epsilon$ is nonzero on nearby solutions. Classify them.

The first has a componentwise constant Hamiltonian and may be normalized to zero. If it is an infinitesimal identity-connected direction that the theory declares redundant, it is a proper/null gauge direction. The second is an integrable charged symmetry whose charge happens to vanish on the reference state. A value at one state cannot replace the tangent-space test.

3. Diagnose a global integrability obstruction

Suppose $\boldsymbol{\delta}\eta_\epsilon=0$ locally, but the integral of $\eta_\epsilon$ around a closed loop in phase space is nonzero. Does a global $H_\epsilon$ exist?

No single-valued global Hamiltonian exists on that phase-space domain because the one-form has a nonzero period. One can restrict to a simply connected patch, change the phase-space domain when physically justified, or describe the obstruction explicitly, but local closure alone does not establish global path independence. A field-dependent parameter adds a separate issue: its variation must be included in the adjusted charge variation and bracket.

4. Interpret a nonzero flux

An integrable charge satisfies $H_\epsilon[S_2]-H_\epsilon[S_1]+\mathcal F_\epsilon[B_{12}]=0$ with $\mathcal F_\epsilon\neq0$ . Is the symmetry absent?

Not necessarily. The charge is not cut independent under the stated transport of $\epsilon$ , but the balance law can give it physical meaning: its change is the negative of the flux through the intervening boundary in this orientation convention. One must still verify that the flux and charge use the same boundary prescription and parameter transport.

5. Test a proposed charge-algebra extension

A bracket calculation produces an additional boundary term. What must be checked before calling it a central charge or an anomaly?

First specify the ordinary or adjusted bracket and any field dependence of the parameters. Check closure, antisymmetry, the relevant Jacobi or cocycle condition, and dependence on charge normalization, boundary representatives, and corners. A field-independent nontrivial cocycle can define a classical central extension. Calling it a quantum anomaly additionally requires a regulated quantum symmetry statement and its consistency conditions.

6. Evaluate an edge extension

A subregion construction adds a boundary frame and changes $(\mathcal P,\Omega)$ to $(\mathcal P_{\mathrm{ext}},\Omega_{\mathrm{ext}})$ . Which earlier conclusions survive automatically?

None of the old classification results survive merely by notation. Recheck the action and boundary terms, admitted variations, allowed transformations, generator variation, null directions, integrability, flux, and algebra on the extended phase space. Then state whether the new variables are auxiliary gluing data, superselection labels, or dynamical boundary degrees of freedom for the problem at hand.

7. Choose the correct continuation

Route four requests: an amplitude soft factor, a BMS memory relation, a regional entanglement entropy, and a BFV boundary charge.

Use the soft-theorem treatment for the first, the BMS/memory treatment for the second, the centers-and-edge entanglement treatment for the third, and the BV–BFV boundary construction for the fourth. Each adds essential domain-specific structure that this chapter deliberately treats only as an interface.

Continue from here

For the complete sequence, begin with Boundaries, Flux, and Boundary Ward Identities. If the localized identity and the declared phase space are already secure, enter Proper and Improper Gauge Transformations or Surface Charges, Integrability, and Ambiguities. Use the edge-mode or asymptotic page only after identifying which charge, regional algebra, or limiting problem is actually being extended. Return to Symmetry and Gauge Structure to choose another chapter.

References

Assanioussi, Mehdi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, and Ludovic Varrin. “On the Covariant Formulation of Gauge Theories with Boundaries.” Classical and Quantum Gravity 41, no. 11 (2024): 115007. DOI. Open PDF.
Donnelly, William, and Laurent Freidel. “Local Subsystems in Gauge Theory and Gravity.” Journal of High Energy Physics 2016, no. 9 (2016): 102. DOI. Open PDF.
Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, no. 10 (2020): 146. DOI. Open PDF.
Riello, Aldo. “Edge Modes without Edge Modes.” arXiv:2104.10182 [hep-th] (2021). Stable record. Open PDF.
Speziale, Simone. “GGI Lectures on Boundary and Asymptotic Symmetries.” arXiv:2512.16810v3 [hep-th] (2026). Stable record. Open PDF.
Wald, Robert M., and Andreas Zoupas. “A General Definition of ‘Conserved Quantities’ in General Relativity and Other Theories of Gravity.” Physical Review D 61, no. 8 (2000): 084027. DOI. Open PDF.