Charge Algebras, Central Terms, and Corners

Globally integrable surface charges need not represent the algebra of their parameters. Their bracket can differ from the charge assigned to the commutator by an extension term; that term is central only when it is constant on the declared reduced phase-space sector. Field-dependent parameters and unmatched corner representatives require adjusted tests instead.

This page starts with finite, globally integrable, reference-normalized charges on one cut. It derives the ordinary and covariant charge brackets, the extension and cocycle tests, and the effect of references and corner improvements. A bounded Maxwell calculation supplies the full physical application, while compact Yang–Mills supplies the non-Abelian sign check. Nonintegrable flux brackets, Virasoro and affine applications, and higher-codimension BFV coherence are handed to their dedicated pages.

Required background. Surface Charges, Integrability, and Ambiguities supplies the declared phase space, complete presymplectic form, globally integrable $H_\epsilon$, parameter prescription, corner representative, reference, and flux status. Quantum Implementations, Projective Actions, and Central Extensions supplies cocycles, coboundaries, sector qualifications, and the distinction between central extensions and anomalies.

Helpful background. Hamiltonian Group Actions and Moment Maps supplies the site’s Poisson-sign package and the finite-dimensional version of the representation test.

From integrable charges to a representation test

Let $\mathcal P_0$ be the connected component passed from the preceding page. Quotient its declared null directions and work on one connected symplectic sector $\mathcal P_{\mathrm{red},0}$ with two-form $\Omega_\Sigma$. If a global quotient is unavailable, the calculation may instead be made on a symplectic leaf, but every conclusion is then leafwise. On an unreduced presymplectic space, Hamiltonian vectors are nonunique up to null directions and an ordinary Poisson bracket need not be defined until the charges descend. Casimirs that vary between leaves are instead a general Poisson-space phenomenon.

The inputs to the algebra test are not only a list of parameters.

Data that must agree before two surface charges are bracketed

Input	Required declaration
Reduced sector	The connected symplectic sector or leaf, its null quotient, and the domain of every charge.
Parameter family	The admitted parameters, their bracket, their kernel as transformations, and whether they depend on the fields.
Charge prescription	The complete ΩΣ, globally integrable Hε, normalization, and boundary or corner representative.
Cut and flux status	The common oriented cut on which the bracket is evaluated, or the transport and lateral-flux rule relating different cuts.
Descent	Proof that proper directions act trivially and that the parameter bracket descends modulo them.

For a function $F$ on $\mathcal P_{\mathrm{red},0}$, define its Hamiltonian vector field and Poisson bracket by

$$\iota_{X_F}\Omega_\Sigma=\boldsymbol\delta F, \qquad \{F,G\}=\Omega_\Sigma(X_F,X_G).$$

This convention implies

$$X_F[U]=\{U,F\}, \qquad [X_F,X_G]=-X_{\{F,G\}}.$$

For each fixed, field-independent parameter $\epsilon$, the preceding page has established

$$\iota_{R_\epsilon}\Omega_\Sigma =\boldsymbol\delta H_\epsilon, \qquad X_{H_\epsilon}=R_\epsilon.$$

Assume throughout this fixed-label calculation that the admitted parameter space is linear and that the charge prescription, including its reference normalization, is linear:

$$H_{a\epsilon+b\eta}=aH_\epsilon+bH_\eta.$$

This hypothesis is needed both for antisymmetry of the defect below and for the cancellation of its Hamiltonian terms under the parameter Jacobi identity.

The ordinary Poisson bracket can therefore be evaluated directly as a surface pairing:

$$\begin{aligned} \{H_\epsilon,H_\eta\} &=\Omega_\Sigma(R_\epsilon,R_\eta)\\ &=R_\eta[H_\epsilon] =-R_\epsilon[H_\eta]\\ &\mathrel{\widehat{=}} \int_S \boldsymbol k_\epsilon[R_\eta\Phi;\Phi]. \end{aligned}$$

This motivates the fixed-label covariant charge bracket

$$\{H_\epsilon,H_\eta\}_{\mathrm{cov}} :=R_\eta[H_\epsilon].$$

It equals the ordinary reduced-phase-space Poisson bracket only because the charges are globally integrable, the same complete $\Omega_\Sigma$ is used, the labels are fixed, and the comparison is made on one cut. A surface pairing may still be writable when any of these hypotheses fails, but it is then not automatically a Poisson bracket of finite charges. Barnich and Compère derive the corresponding covariant surface bracket and its representation theorem in an asymptotic setting at Barnich and Compère 2008, § 4.3, pp. 16–17, Open PDF; the formula above has been translated as a complete sign package to the site’s convention.

Deriving the extension term

There are two bracket signs in common use, so define both before calculating. Let $[\epsilon,\eta]_0$ be the bracket for which the transformation vectors form a homomorphism,

$$[R_\epsilon,R_\eta] =R_{[\epsilon,\eta]_0}.$$

Define the charge-compatible bracket by

$$[\epsilon,\eta]_{\mathrm{ch}} :=-[\epsilon,\eta]_0, \qquad [R_\epsilon,R_\eta] =-R_{[\epsilon,\eta]_{\mathrm{ch}}}.$$

The moment-map prerequisite uses direct fundamental fields for a left action; there its printed Lie bracket already has the charge-compatible role. In the canonical Yang–Mills convention used later, the ordinary pointwise color bracket has the $[\ ,\ ]_0$ role. Writing the relation explicitly prevents the positive charge formula from being combined with the unreversed color bracket.

Now use the anti-homomorphism $F\mapsto X_F$:

$$\begin{aligned} X_{\{H_\epsilon,H_\eta\}} &=-[R_\epsilon,R_\eta]\\ &=R_{[\epsilon,\eta]_{\mathrm{ch}}}\\ &=X_{H_{[\epsilon,\eta]_{\mathrm{ch}}}}. \end{aligned}$$

The representation defect is therefore

$$\boxed{ K(\epsilon,\eta) := \{H_\epsilon,H_\eta\} -H_{[\epsilon,\eta]_{\mathrm{ch}}}, }$$

and it obeys $X_K=0$. On the connected symplectic sector this makes $K(\epsilon,\eta)$ a real constant. Equivalently,

$$\boxed{ \{H_\epsilon,H_\eta\} =H_{[\epsilon,\eta]_{\mathrm{ch}}} +K(\epsilon,\eta). }$$

The conclusion is componentwise: a different connected sector can carry a different constant. On a general Poisson space it is first a Casimir, and on an unreduced presymplectic space one must show that it descends before calling it central. The symplectic weak-moment-map argument is developed in Cannas da Silva 2006, §§ 26.1–26.4, pp. 164–167, Open PDF.

Jacobi gives the cocycle condition

Antisymmetry of the Poisson bracket and of $[\ ,\ ]_{\mathrm{ch}}$ makes $K$ antisymmetric. Insert the extended algebra into the Poisson Jacobi identity. The terms proportional to $H$ cancel by the parameter Jacobi identity; constants Poisson-commute with every function. What remains is

$$\begin{aligned} 0={}&K([\epsilon,\eta]_{\mathrm{ch}},\zeta) +K([\eta,\zeta]_{\mathrm{ch}},\epsilon)\\ &+K([\zeta,\epsilon]_{\mathrm{ch}},\eta). \end{aligned}$$

Thus $K$ is a Lie-algebra two-cocycle. This is a necessary algebraic test, not proof that the Lie-algebra extension exponentiates to the chosen global group; topology, regularity, and possible integrality conditions remain.

References change representatives, not the class

Let the linear reference prescription be shifted by constants on the sector,

$$H'_\epsilon=H_\epsilon+C_\epsilon, \qquad \boldsymbol\delta C_\epsilon=0.$$

The Poisson bracket is unchanged, whereas the charge assigned to the parameter bracket acquires $C_{[\epsilon,\eta]_{\mathrm{ch}}}$. Hence

$$\boxed{ K'(\epsilon,\eta) =K(\epsilon,\eta) -C_{[\epsilon,\eta]_{\mathrm{ch}}}. }$$

A defect of this form is a removable coboundary. If no allowed linear reference removes $K$, the extension is nonremovable relative to the declared physical normalization class. Calling $[K]$ nonzero in ordinary Lie-algebra cohomology requires the stronger test that no linear one-cochain removes it. An algebraic trivialization excluded by the boundary prescription makes the extension prescription-relative, not an ordinary nonzero cohomology class. Barnich and Compère give the same normalization test for asymptotic surface charges at Barnich and Compère 2008, § 4.3, p. 17, Open PDF.

What may be concluded after comparing the charge and parameter brackets

Outcome	Required test	Licensed conclusion
Exact representation	K = 0 with the declared references and representatives.	The charges reproduce the charge-compatible parameter bracket.
Removable extension	K is a coboundary from an allowed linear reference shift.	A different normalization gives an exact representation.
Central extension	K is constant on the connected reduced sector and satisfies the cocycle condition; allowed reference shifts determine whether it is removable within the declared prescription.	The charges represent a central extension. Its ordinary cohomology class is nonzero only if no linear one-cochain removes K; otherwise any remaining nonremovability is prescription-relative.
Field-dependent extension	The adjusted defect depends on the fields and passes its adjusted Jacobi test.	An algebroid or field-dependent extension may exist; it is not a central term.
No accepted algebra	Closure, descent, antisymmetry, integrability, or Jacobi fails.	The construction stops until the failed input or prescription is repaired.

A classical central extension is not automatically a local Schwinger term, a projective quantum action, or an anomaly. A Schwinger distribution must survive integration and the boundary prescription; quantization can add operator-domain or ordering issues; and an anomaly is a separate obstruction to background-gauge invariance or gauging. These distinctions are developed in Contact Terms, Equal-Time Commutators, and Schwinger Terms and What Is an Anomaly?.

Field-dependent parameters and the adjusted bracket

Suppose now that $\epsilon=\epsilon[\Phi]$ and $\eta=\eta[\Phi]$ are smooth parameter functionals and that the charge family is linear in its parameter. Continue the fixed-label variation from the preceding page,

$$\boldsymbol\delta_{\!\Phi}H_\epsilon := \boldsymbol\delta H_\epsilon -H_{\boldsymbol\delta\epsilon}, \qquad \iota_{R_\epsilon}\Omega_\Sigma =\boldsymbol\delta_{\!\Phi}H_\epsilon.$$

The total differential of $H_{\epsilon[\Phi]}$ is not the Hamiltonian differential of $R_\epsilon$. In particular, a field-dependent smearing of a constraint is not made into an ordinary generator merely by substituting the smearing into a fixed-parameter formula. The subtraction above defines an operation on the charge family with its label held fixed.

Write the parameter change induced by a transformation as

$$\Delta_\epsilon\eta:=R_\epsilon[\eta].$$

Direct calculation of the vector-field commutator gives the bracket that closes the anchor,

$$[\epsilon,\eta]_{0,*} = [\epsilon,\eta]_0 +\Delta_\epsilon\eta -\Delta_\eta\epsilon,$$

and therefore the adjusted charge-compatible bracket is

$$\boxed{ [\epsilon,\eta]_{\mathrm{ch},*} = -[\epsilon,\eta]_0 -\Delta_\epsilon\eta +\Delta_\eta\epsilon. }$$

These brackets are defined modulo reducibility parameters if the map $\epsilon\mapsto R_\epsilon$ has a kernel. They must preserve the admitted parameter class and satisfy Jacobi on that quotient.

The adjusted charge bracket is

$$\begin{aligned} \{H_\epsilon,H_\eta\}_* &:= \boldsymbol\delta_{\!\Phi}H_\epsilon(R_\eta)\\ &=R_\eta[H_\epsilon] -H_{\Delta_\eta\epsilon}\\ &=\Omega_\Sigma(R_\epsilon,R_\eta). \end{aligned}$$

It must also obey

$$\{H_\epsilon,H_\eta\}_* =- \boldsymbol\delta_{\!\Phi}H_\eta(R_\epsilon).$$

Failure of this antisymmetry check means that the label transport, integrability prescription, or symplectic representative has been applied inconsistently. The adjusted representation defect is

$$K_*(\epsilon,\eta)[\Phi] := \{H_\epsilon,H_\eta\}_* -H_{[\epsilon,\eta]_{\mathrm{ch},*}}.$$

Unlike the fixed-label derivation, this equation does not imply $X_{K_*}=0$: $H_{\epsilon[\Phi]}$ is not an ordinary Hamiltonian for the total variation, and the adjusted bracket is an operation on a field-dependent family. If $K_*$ varies on phase space, it is a field-dependent extension, not a central term.

To state the Jacobi test without pretending that $\{\ ,\ \}_*$ is an ordinary Poisson bracket on arbitrary functions, treat $K_*(\epsilon,\eta)$ as an extension function that generates no new parameter transformation. Declare its action with a charge by $\{K_*(\epsilon,\eta),H_\zeta\}_* :=R_\zeta[K_*(\epsilon,\eta)]$, where this is the total derivative under the chosen transport of $\epsilon$ and $\eta$, and declare the bracket of two extension functions to vanish. The adjusted charge family then satisfies Jacobi precisely when

$$\sum_{\mathrm{cyclic}} \left( R_\zeta[K_*(\epsilon,\eta)] +K_*([\epsilon,\eta]_{\mathrm{ch},*},\zeta) \right)=0,$$

and when $[\ ,\ ]_{\mathrm{ch},*}$ itself satisfies Jacobi on the parameter quotient. If labels are held fixed during $R_\zeta$ instead, their transport terms must be added explicitly. If no such declared extension closes, there is no accepted adjusted charge algebra. For constant $K$ and field-independent parameters, the equation reduces to the ordinary cocycle condition above. Barnich and Troessaert give the modified parameter bracket and the generalized cocycle and split-dependence tests at Barnich and Troessaert 2011, § 2.3, p. 7, eq. (2.16), and § 3.2, pp. 8–10, eqs. (3.4)–(3.9), Open PDF. Their application is nonintegrable asymptotic gravity; only its algebraic structure is used here, with the transformation signs translated to the site’s convention. Speziale displays the fixed-label subtraction and the role of field-dependent diffeomorphisms and charge cocycles at Speziale 2026, §§ 3.4–3.6, pp. 26–33, arXiv v3 Open PDF. The displayed adjusted formulas are the direct anchor calculation in the site’s sign convention; they should not be transplanted to a source using a different action or Poisson sign one term at a time.

Current improvements and boundary charges

Let $\mathcal J_\epsilon$ be a current $(d-1)$-form with $\mathrm d\mathcal J_\epsilon\mathrel{\widehat{=}}0$, and let $Y_\epsilon$ be a globally defined $(d-2)$-form. The local improvement

$$\mathcal J'_\epsilon =\mathcal J_\epsilon+\mathrm dY_\epsilon$$

preserves the on-shell divergence because $\mathrm d^2=0$. On a hypersurface $\Sigma$ with boundary $S$, however,

$$\mathcal Q'_\epsilon[\Sigma] -\mathcal Q_\epsilon[\Sigma] = \int_\Sigma\mathrm dY_\epsilon = \int_S Y_\epsilon.$$

Thus “identically conserved” is a local statement; it does not mean “zero boundary charge.” The following table concerns four operator-current representatives. A background-only local counterterm is deliberately outside its rows: it changes source contact terms rather than adding the dynamical operator $\mathrm dY_\epsilon$ to the current.

Canonical, improved, identically conserved, and boundary-shifted current representatives

Representative	Local statement	Integrated charge	Boundary or corner consequence
Canonical current 𝓙ε	d𝓙ε ≈ 0 in the declared theory.	𝓠ε[Σ] = ∫Σ𝓙ε.	Its value still depends on the declared boundary problem and flux conditions.
Improved representative 𝓙ε + dYε	It has the same local divergence when Yε is regular and globally defined.	It agrees with 𝓠ε only when ∫SYε vanishes under the stated conditions.	Local matrix elements can change even when the integrated charge does not.
Identically conserved contribution dYε	d(dYε) = 0 off shell.	Its entire hypersurface integral is the surface term ∫SYε.	It is physically trivial only after that surface term and any singular or global obstruction are checked.
Boundary-shifted representative	The bulk local equation is unchanged.	𝓠′ε − 𝓠ε = ∫SYε survives.	The surface charge and possibly its algebra must be recomputed; facewise choices can leave a corner mismatch.

Quantum Currents, Improvements, and Conservation develops the operator-side improvement and its falloff test. Spurions, Local Counterterms, and Symmetry Response explains why a dynamical current improvement is not merely a background contact-term scheme change.

The current integral $\mathcal Q_\epsilon[\Sigma]$ is not automatically the surface Hamiltonian $H_\epsilon$ used in the earlier algebra. They coincide only after the complete current construction, boundary terms, and presymplectic prescription establish $\boldsymbol\delta H_\epsilon =\iota_{R_\epsilon}\Omega_\Sigma$ with the same normalization. The table classifies possible current representatives; it does not supply that identification.

Once that identification has been proved, if the symplectic structure and parameter anchor are held fixed while a differentiable Hamiltonian-charge improvement $I_\epsilon$ is added, $H'_\epsilon=H_\epsilon+I_\epsilon$, direct expansion gives

$$\begin{aligned} K'_{12}={}&K_{12} +\{H_1,I_2\} +\{I_1,H_2\}\\ &+\{I_1,I_2\} -I_{[1,2]_{\mathrm{ch}}}. \end{aligned}$$

For $I_\epsilon=C_\epsilon$ constant, this reduces to the reference coboundary. A genuinely state-dependent $I_\epsilon$ changes the Hamiltonian vector field on a nondegenerate phase space; if it arose from a simultaneous change of boundary potential, $\Omega_\Sigma$ and the charge pairing must be changed together instead of using this fixed-$\Omega$ expansion.

Corners compose by oriented Stokes data

Consider an oriented spacetime slab $U$ with

$$\partial U =\Sigma_2\cup(-\Sigma_1)\cup B, \qquad \partial B=S_1\cup(-S_2),$$

where $S_i=\partial\Sigma_i$ and $B$ is the lateral face. Suppose the symplectic-potential currents on the three faces are improved by $\mathrm d\boldsymbol\beta_2$, $\mathrm d\boldsymbol\beta_1$, and $\mathrm d\boldsymbol\beta_B$, respectively. Each $\boldsymbol\beta$ is a spacetime $(d-2)$-form and a field-space one-form. Oriented Stokes reduction gives

$$\begin{aligned} \Delta\Theta_{\partial U} ={}& \int_{\Sigma_2}\mathrm d\boldsymbol\beta_2 -\int_{\Sigma_1}\mathrm d\boldsymbol\beta_1 +\int_B\mathrm d\boldsymbol\beta_B\\ ={}& \int_{S_2} (\boldsymbol\beta_2-\boldsymbol\beta_B) -\int_{S_1} (\boldsymbol\beta_1-\boldsymbol\beta_B). \end{aligned}$$

The codimension-two terms cancel when all face representatives are restrictions of one compatible global representative. Otherwise define

$$\boldsymbol c_i := \boldsymbol\beta_i|_{S_i} -\boldsymbol\beta_B|_{S_i}.$$

One must either impose $\boldsymbol c_i=0$ or include the minimal corner completion

$$\Theta_{\mathrm{corner}} =- \int_{S_2}\boldsymbol c_2 + \int_{S_1}\boldsymbol c_1.$$

This is explicit compatibility data; it does not by itself introduce a new dynamical edge field. If the corner completion changes $\Omega_\Sigma=-\boldsymbol\delta\Theta_\Sigma$, the charge existence and algebra tests must be repeated with the completed form. The need to include boundary and corner contributions in the bounded variational prescription is explained at Harlow and Wu 2020, §§ 2.1–2.4, pp. 7–23, Open PDF.

There is also a genuine ambiguity transformation in which the bulk potential representative and the bounded corner datum are shifted together. When that joint change leaves the complete $\Omega_\Sigma$ and Hamiltonian invariant, it is a change of description rather than a new charge. Shifting only the bulk term and discarding its paired corner change is not that ambiguity.

There is a second, lower-codimension Stokes test. If a charge density $q_\epsilon$ is integrated over an open cut $F$ with $C=\partial F$, then an allowed representative change $q'_\epsilon=q_\epsilon+\mathrm dU_\epsilon$ gives

$$H'_\epsilon[F]-H_\epsilon[F] =\int_F\mathrm dU_\epsilon =\int_C U_\epsilon.$$

For two faces glued along $C$, matching $U_\epsilon$ and opposite induced orientations cancel the shared contribution. A mismatch survives as corner data. Exact improvements therefore vanish on a genuinely closed cut under the required regularity assumptions, not on every face considered in isolation.

A nonzero central cocycle as a control calculation

Before the gauge-theory application, it is useful to see that the test can produce a nonzero answer. On $(\mathbb R^2,\omega=\mathrm dq\wedge\mathrm dp)$, let translations be labelled by $\xi=(u,v)$ with

$$R_\xi=u\,\partial_q+v\,\partial_p, \qquad H_\xi=up-vq.$$

For $\eta=(u',v')$,

$$\{H_\xi,H_\eta\} =uv'-vu'.$$

The translation algebra is Abelian, so its parameter bracket and associated charge vanish. The displayed constant is therefore $K(\xi,\eta)$. A reference shift cannot remove it because $C_{[\xi,\eta]_{\mathrm{ch}}}=0$. This finite-dimensional example is not a boundary charge, but it independently checks the constancy, cocycle, and nonremovability logic before the field-theory calculation.

Worked application: Maxwell algebra and the Yang–Mills sign check

Return to source-free Maxwell theory on $M=\mathbb R\times\Sigma$, with $\Sigma$ compact, connected, and oriented, and

$$S=\bigsqcup_{a=1}^{N}S_a.$$

Use the bounded phase space of the preceding page: the pullback of $A$ is fixed on the timelike boundary, and an admitted fixed gauge parameter has constant boundary values $\lambda|_{S_a}=c_a$. Define

$$\Phi_a:=\int_{S_a}E^n\,\mathrm dS, \qquad \sum_a\Phi_a=0.$$

Choose reference fluxes $\bar\Phi_a$ in the same source-free Gauss sector, so $\sum_a\bar\Phi_a=0$. The globally integrable charge is

$$H_c = \sum_{a=1}^{N} c_a(\Phi_a-\bar\Phi_a).$$

The diagonal parameter is null, so one may equivalently use relative parameters $a_A=c_A-c_N$ and independent fluxes $x_A=\Phi_A$, $A=1,\ldots,N-1$.

Maxwell gauge transformations leave the electric field and every flux invariant. For another allowed parameter $d$, this gives the first check

$$R_d[H_c]=0.$$

The canonical symplectic form gives an independent check: both $R_c$ and $R_d$ have zero electric-field component, so

$$\Omega_\Sigma(R_c,R_d)=0.$$

Therefore

$$\boxed{ \{H_c,H_d\}=0, \qquad K(c,d)=0. }$$

Because the Abelian parameter bracket vanishes, a reference shift cannot manufacture a central term. The result is also conserved under the no-flux boundary prescription already checked on the preceding page; conservation and algebra have nevertheless been tested independently.

Three descriptions of the same zero-extension Maxwell algebra

Description	Parameters and generators	Algebra result
Gauge-orbit description	Quotient the based null subgroup; relative boundary constants remain as charged transformations, while the diagonal constant is null.	Every surviving transformation leaves the relative electric fluxes invariant, so the generators commute.
Charge description	Hc is the reference-subtracted weighted relative flux.	{Hc, Hd} = 0 and K(c, d) = 0.
Coulomb-gauge description	The residual parameters are harmonic extensions of the fixed boundary constants; the common constant has zero gradient and zero charge.	The residual extensions commute and reproduce the same zero algebra; residual status alone was not used to infer physicality.

For the canonical Maxwell surface density $q_c=c\,\star F|_S$, the representative used here has no additional $\mathrm dU_c$ term. On an open rectangular face $F$, an alternative allowed representative would shift the charge by $\int_{\partial F}U_c$; matching adjacent face prescriptions cancels their shared edge, exactly as in the corner calculation above.

Maxwell does not exercise a non-Abelian parameter bracket, so add a compact Yang–Mills checkpoint. Use the canonical conventions

$$\begin{aligned} R_\epsilon A_i^a&=(D_i\epsilon)^a,\\ R_\epsilon E^{ia}&=g f^{abc}E^{ib}\epsilon^c,\\ [\epsilon,\eta]_0^a&=g f^{abc}\epsilon^b\eta^c. \end{aligned}$$

For fixed admitted parameters whose boundary values close under this bracket,

$$[R_\epsilon,R_\eta] =R_{[\epsilon,\eta]_0}, \qquad [\epsilon,\eta]_{\mathrm{ch}} =-[\epsilon,\eta]_0.$$

On the Gauss surface the complete generator is the boundary charge

$$Q_S[\epsilon] =\int_S\mathrm d^2S\, \epsilon^aE^{\perp a}.$$

Its bracket is

$$\begin{aligned} \{Q_S[\epsilon],Q_S[\eta]\} &=R_\eta[Q_S[\epsilon]]\\ &=\int_S\mathrm d^2S\, \epsilon^a g f^{abc}E^{\perp b}\eta^c\\ &=-Q_S[[\epsilon,\eta]_0]\\ &=Q_S[[\epsilon,\eta]_{\mathrm{ch}}]. \end{aligned}$$

Thus $K=0$ for this canonical representative and boundary problem. The Abelian limit $f^{abc}\to0$ reproduces the Maxwell result. The canonical bulk transformation laws and Gauss generator are developed at Tong 2018, §§ 2.1.2 and 2.2.1, pp. 31–33 and 40–42, official full-notes PDF; the bounded surface reduction above follows here by integration by parts on the Gauss surface in the site’s normalization. A different boundary action, corner completion, parameter class, or symplectic representative requires a new calculation; this negative result is not a universal theorem that Yang–Mills boundary charges have no extensions.

A full non-Abelian Coulomb-gauge treatment is deliberately not inferred from this checkpoint. Its residual parameters can be field dependent, projection onto the gauge slice can modify the bracket, and Gribov or stabilizer issues can obstruct a single global slice. Those cases must use the adjusted test rather than the fixed-parameter formula.

Diagnostics and stop rules

A reproducible charge-algebra calculation can be run in the following order.

1. Confirm existence. Use only finite, globally integrable charges with a declared reference and representative.
2. Descend. Verify that null transformations act trivially, form the required ideal, and do not change the bracket representative.
3. Close the parameters. Check admissibility and Jacobi for the fixed or adjusted parameter bracket, modulo reducibility parameters when needed.
4. Match the brackets. Evaluate the reduced Poisson bracket and the covariant surface pairing independently when both are available.
5. Classify the defect. Test constancy or field dependence, antisymmetry, the appropriate cocycle condition, and allowed coboundaries.
6. Match faces and cuts. Carry the corner representative, orientation, label transport, and lateral flux whenever faces or cuts differ.
7. Cross-check a limit. Use an Abelian limit, a canonical calculation, or another independently defined representative.

The stop rule is strict. Do not assert a representation if a global charge does not exist, the parameter class is not closed, the bracket has not descended through null directions, or antisymmetry or Jacobi fails. Do not call a field-dependent defect central. Do not compare charges on different cuts without the flux and parameter-transport prescription. An unmatched corner representative or a newly introduced boundary field changes the input and restarts the affected tests.

Common failure modes

Using the familiar positive formula with the wrong parameter bracket. With the canonical Yang–Mills color bracket, $[R_\epsilon,R_\eta]=R_{[\epsilon,\eta]_0}$, so the charges reproduce its negative. Either retain that minus sign or use the charge-compatible bracket consistently.

Calling every extra term central. A central term must Poisson-commute with all observables on the declared sector; on a connected symplectic sector it is constant. A field-dependent extension requires an adjusted Jacobi test.

Treating a reference as a corner improvement. A reference adds a componentwise constant and changes $K$ by a coboundary. A corner improvement can be state dependent and can change the symplectic pairing itself.

Dropping an exact term on an open face. Stokes’ theorem turns it into a corner integral. It vanishes only for a closed cut or after compatible facewise cancellation under the required regularity assumptions.

Equating a classical cocycle with a quantum anomaly. Quantization, operator domains, global exponentiation, local Schwinger terms, and gauging obstructions are separate tests.

Calling a residual transformation physical. Preserving a gauge condition only identifies a residual parameter. Its charge and null status must still be evaluated on the original phase space or a justified reduced slice.

Check your understanding

These checks carry no score or completion status.

1. The Yang–Mills sign

Starting from $R_\eta E^{\perp a}=g f^{abc}E^{\perp b}\eta^c$, show that $R_\eta Q_S[\epsilon]=-Q_S[[\epsilon,\eta]_0]$. Which bracket gives the positive charge-algebra formula?

Solution

Invariance of the Lie-algebra inner product, or the antisymmetry of $f^{abc}$, gives

$$\epsilon^a f^{abc}E^{\perp b}\eta^c =-E^{\perp a}f^{abc}\epsilon^b\eta^c.$$

After integration this is $-Q_S[[\epsilon,\eta]_0]$. Defining $[\epsilon,\eta]_{\mathrm{ch}}=-[\epsilon,\eta]_0$ gives

$$\{Q_S[\epsilon],Q_S[\eta]\} =Q_S[[\epsilon,\eta]_{\mathrm{ch}}].$$

2. Removing a coboundary

Suppose $K(\epsilon,\eta)=C_{[\epsilon,\eta]_{\mathrm{ch}}}$ for a linear constant functional $C$. Find a reference-normalized charge map with zero defect.

Solution

Choose $H'_\epsilon=H_\epsilon+C_\epsilon$. The shifted defect is

$$K'(\epsilon,\eta) =K(\epsilon,\eta) -C_{[\epsilon,\eta]_{\mathrm{ch}}} =0.$$

This works because $C_\epsilon$ is constant on the sector. A state-dependent corner functional cannot be removed by the same reference argument.

3. A shared corner

Two oriented faces $F_1$ and $F_2$ meet along $C$, with opposite induced orientations. Their charge densities are improved by $\mathrm dU_1$ and $\mathrm dU_2$. What remains on the glued face?

Solution

Stokes’ theorem gives $\int_CU_1$ from $F_1$ and $-\int_CU_2$ from $F_2$. The shared contribution is

$$\int_C(U_1-U_2).$$

It cancels when the representatives match on $C$. Otherwise it is explicit corner data and the charge bracket must be recomputed with that mismatch or with a justified corner completion.

Where the analysis continues

Edge Modes, Subregions, and Factorization asks when a specified gluing problem justifies new boundary variables rather than only compatible corner data. Asymptotic Symmetry, Soft Limits, and the Boundary Interface adds falloff, flux, and infrared data at infinity.

Corners, Stratification, and Higher-Codimension Data owns theorem-level BFV layers and higher-codimension composition. Concrete Virasoro, affine Kac–Moody, and gravitational central terms belong to The Virasoro Algebra and the Stress Tensor, Affine Current Algebras and WZW Models, and AdS3/CFT2 and the Brown–Henneaux Central Charge.

References

• Barnich, Glenn, and Geoffrey Compère. “Surface Charge Algebra in Gauge Theories and Thermodynamic Integrability.” Journal of Mathematical Physics 49 (2008): 042901. DOI. Open PDF
• Barnich, Glenn, and Cédric Troessaert. “BMS Charge Algebra.” Journal of High Energy Physics 2011, no. 12 (2011): 105. DOI. Open PDF
• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764. Berlin: Springer, 2001; author PDF revised January 2006. 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
• Speziale, Simone. “GGI Lectures on Boundary and Asymptotic Symmetries.” arXiv:2512.16810v3 [hep-th] (2026). DOI. Open PDF
• Tong, David. Gauge Theory. Cambridge Part III Mathematical Tripos lecture notes, 2018. Official course page. Official PDF