Edge Modes, Subregions, and Factorization

Gauge constraints obstruct a canonical tensor-product split because cutting a system and reducing its gauge redundancy are different operations. Gauss law matches the normal fluxes on the two sides of a cut, and a Wilson line crossing the cut cannot be assigned to two independent gauge-invariant regional algebras without extra endpoint data. One may instead choose a regional algebra with a center, reduce at fixed flux, introduce a boundary frame, or embed the physical states into an extended tensor product. Each construction solves a specified regional problem; none is a unique factorization of the original theory.

An edge-mode extension is especially useful when arbitrary gauge transformations at the cut should remain null directions of each regional phase space. It adds conjugate boundary data, cancels the cut contribution to the presymplectic pairing, and makes gluing expressible as flux matching followed by a diagonal reduction. The added variable is auxiliary unless a boundary condition, action, and Hamiltonian give it independent dynamics.

Required background. Proper and Improper Gauge Transformations supplies the declared phase space, the null-versus-charged test, and the rule that extending the phase space requires repeating that test. Gauge-Invariant and Dressed Observables supplies endpoint dressings and their nonuniqueness.

Helpful background. Presymplectic Systems and the Covariant Phase-Space Ambiguity Map supplies reduction by null directions and the boundary-pairing criterion.

Gauss law couples the two sides of a cut

Let a spatial slice without a physical boundary be divided into two regions,

$$\Sigma=R\cup_S\bar R, \qquad S=\partial R=-\partial\bar R.$$

The equality on the right records opposite boundary orientations. Take smooth Maxwell fields, or a regulated compact gauge theory, in one bundle sector and include charged matter when nonzero integrated regional flux is needed. With outward normals on both sides, define

$$e_R=E_R^\perp, \qquad e_{\bar R}=E_{\bar R}^\perp.$$

Smooth gluing requires

$$\boxed{e_R+e_{\bar R}=0.}$$

This is already a correlation between regional data. The same fact appears in three languages:

• on the constraint surface, the interface flux is the boundary value of the Gauss moment map;
• in the observable algebra, admitted flux data can lie in a center and label superselection sectors; and
• in a regulated Hilbert space, the two sides must carry matching dual boundary charges.

For compact regulated $U(1)$, let $e$ denote the complete set of quantized flux labels on the cut—not merely the total flux through a single closed surface. A typical physical decomposition is

$$\mathcal H_{\mathrm{phys}}(\Sigma) \simeq \bigoplus_e \mathcal H_R^{(e)} \otimes \mathcal H_{\bar R}^{(-e)}.$$

It is not the unrestricted product

$$\left(\bigoplus_e\mathcal H_R^{(e)}\right) \otimes \left(\bigoplus_{e'}\mathcal H_{\bar R}^{(e')}\right),$$

because terms with $e+e'\neq0$ fail the interface constraint. In a continuum description the sum may become a direct integral and its measure and operator domains must be specified. Donnelly gives the regulated Abelian and non-Abelian gluing picture in Donnelly 2014, pp. 2–6 and 9, Open PDF, while Donnelly and Freidel formulate the global space as a diagonal-invariant subspace of extended regional spaces in Donnelly and Freidel 2016, introduction and § 2.5, pp. 2–4 and 15–16, Open PDF.

Gauss law is not the only obstruction. Even after gauge matching, a separate continuum issue can remain: under the standard relativistic and regularity hypotheses reviewed by Yngvason, bounded-region local algebras are generically type III rather than type I, while the elementary Hilbert-space subsystem factorization is a type-I construction Yngvason 2005, pp. 135–140, Open PDF. The exact algebraic hypotheses, split inclusions, and cutoff limits are handed to Type-III Local Algebras, Entropy, and Cutoff Limits; this page isolates the additional gauge-specific obstruction.

The cut above is fiducial: after gluing, it must disappear from physical predictions. A material wall or an external timelike boundary is different. There the boundary condition may permit charge exchange, support a boundary action, or turn a surface transformation into a physical symmetry. Riello develops this distinction and a fully reduced alternative to independent edge coordinates in Riello 2021, introduction and §§ 4–5, pp. 2–6 and 18–30, Open PDF.

Four regional constructions

The words “the subsystem” do not determine a unique algebra or phase space. One must declare which operators may approach or cross $S$, which boundary transformations are quotiented, and which polarization is retained.

Different controlled responses to the subregion constraint

Construction	Retained or added data	Gluing rule	What does not follow
Regional observable algebra	Gauge-invariant operators assigned to the region; a chosen electric, magnetic, or trivial center	Match the shared central data and specify how cross-cut operators are recovered	The region alone does not select a unique center
Fixed-flux reduction	A reduced regional phase space at one prescribed interface-flux sector	Pair the sector with the opposite oriented sector on the other side	A conjugate edge coordinate is not required within that one sector
Boundary dressing or frame	A reference at the cut that completes open endpoints and compares gauge frames	Identify compatible dressed tangential data or transition functions	The frame and its dressing are not unique
Extended phase or Hilbert space	Regional boundary variables or representation factors before imposing interface invariance	Impose the flux moment map and quotient or project by the diagonal interface group	The extended product is kinematic, not the physical tensor product

For an electric-center choice in a regulated Abelian theory, normal flux commutes with the strictly interior gauge-invariant algebra and labels its blocks. A magnetic-center or a maximal-tree construction retains different boundary data, and some regulated choices have a trivial center. The center here is the center of the selected regional operator algebra, not the center of the gauge group. Casini, Huerta, and Rosabal compare these choices and the corresponding lattice constructions in Casini, Huerta, and Rosabal 2014, § 3.1 and §§ 4.1–4.3, pp. 5–16, Open PDF.

These constructions need not be equivalent. Fixing $e$ and reducing can describe gluing sector by sector without an independent frame coordinate. A frame becomes useful when one wants one regional space spanning several flux sectors, covariance under transformations with arbitrary boundary value, open endpoints, or a kinematic tensor product before projection. This is the central nonuniversality point: edge variables can be useful without being logically compulsory.

A Maxwell edge frame cancels the cut pairing

Work first on $R$ and allow the normal electric field at $S$ to vary. Include the standard matter contribution to the symplectic form when charged matter is present, but display its gauge-field part:

$$\Omega_R = \int_R d^3x\, \boldsymbol\delta A_i\wedge\boldsymbol\delta E^i.$$

With $\mathcal C=\partial_iE^i-\rho$, the differentiable generator in the chapter’s convention is

$$\begin{aligned} G_R[\lambda] &= -\int_R d^3x\,\lambda\mathcal C +\int_S d^2S\,\lambda e_R, \\ Q_R[\lambda] &= \int_S d^2S\,\lambda e_R. \end{aligned}$$

For a field-independent parameter and a tangent variation to the Gauss surface,

$$\left. \iota_{R_\lambda}\Omega_R \right|_{\mathcal C=0} = \boldsymbol\delta Q_R[\lambda] = \int_S d^2S\,\lambda\,\boldsymbol\delta e_R.$$

Thus a transformation with nonzero boundary value is not a null direction of this unextended regional form. Introduce instead a compact boundary frame $\phi_R$ with the same periodic identification as the $U(1)$ parameter and let

$$A_R\longmapsto A_R+d\lambda_R, \qquad \phi_R\longmapsto\phi_R+\lambda_R|_S.$$

Then $A_{R\parallel}-d_S\phi_R$ is invariant. Choose the extended form

$$\boxed{ \Omega_R^{\mathrm{ext}} = \Omega_R -\int_S d^2S\, \boldsymbol\delta\phi_R\wedge \boldsymbol\delta e_R. }$$

The sign is not decorative. Since $R_\lambda\phi_R=\lambda|_S$ and $R_\lambda e_R=0$ in Maxwell theory,

$$\begin{aligned} \left. \iota_{R_\lambda^{\mathrm{ext}}} \Omega_R^{\mathrm{ext}} \right|_{\mathcal C=0} &= \int_S d^2S\,\lambda\,\boldsymbol\delta e_R -\int_S d^2S\, (R_\lambda\phi_R)\boldsymbol\delta e_R \\ &=0. \end{aligned}$$

The enlarged gauge transformation is therefore null on the extended constraint surface. An open dressed matter insertion ending at $b\in S$ becomes

$$\widehat\Psi_\gamma(b,x) = e^{-ig\phi_R(b)} U_\gamma(b,x)\psi(x),$$

which is invariant because the original endpoint factor transforms by $e^{ig\lambda(b)}$. This makes the frame’s operational role concrete: it completes a boundary endpoint and cancels the regional presymplectic pairing.

The null gauge action is not the only action on the new variable. An independent frame shift

$$\Delta_\alpha\phi_R=-\alpha, \qquad \Delta_\alpha A_R=\Delta_\alpha e_R=0$$

has

$$\iota_{\Delta_\alpha}\Omega_R^{\mathrm{ext}} = \boldsymbol\delta \int_S d^2S\,\alpha e_R.$$

The same flux has moved from the generator of the enlarged gauge redundancy to the moment map of a surface-frame symmetry. The extension has not erased the charge; it has changed the phase space and the symmetry action. Admissibility, nullity, integrability, ambiguity, flux, and algebra must therefore all be recomputed. Donnelly and Freidel give the Yang–Mills boundary frame and its surface symmetry in Donnelly and Freidel 2016, §§ 2.1–2.5, pp. 8–16, Open PDF. Assanioussi and collaborators emphasize that gauge invariance of a presymplectic form is weaker than degeneracy along gauge directions and analyze the edge extension in Assanioussi et al. 2024, §§ 3.1–3.3, pp. 13–16, Open PDF.

Gluing is matching followed by reduction

Repeat the construction on $\bar R$. Under compatible regularity, polarization, bundle, and boundary-condition assumptions, gluing has the schematic symplectic-reduction form

$$\boxed{ \mathcal P(\Sigma) \simeq \left( \mathcal P_R^{\mathrm{ext}} \times_{\mathrm{matching}} \mathcal P_{\bar R}^{\mathrm{ext}} \right)\!/\!/\mathcal G_S. }$$

Here “matching” includes compatible dressed tangential data or a transition function. The moment-map equation includes $e_R+e_{\bar R}=0$, and the quotient removes the diagonal interface gauge group. If the two frames are identified on the matching locus, their symplectic terms cancel:

$$\begin{aligned} \Omega_{S,R}^{\mathrm{ext}} +\Omega_{S,\bar R}^{\mathrm{ext}} &= -\int_S d^2S\, \boldsymbol\delta\phi\wedge \boldsymbol\delta(e_R+e_{\bar R}) \\ &=0. \end{aligned}$$

At regulated quantum level the corresponding statement is

$$\mathcal H_{\mathrm{phys}}(\Sigma) \simeq \operatorname{Inv}_{\mathcal G_S} \left( \mathcal H_R^{\mathrm{ext}} \otimes \mathcal H_{\bar R}^{\mathrm{ext}} \right),$$

or the matched-sector sum displayed earlier. The tensor product belongs to the extension; matching and the singlet projector recover the physical space. Classical reduction does not by itself prove that reduction commutes with quantization.

Four checks catch most gluing mistakes:

1. Orientation: a common parameter gives $Q_R[\lambda]+Q_{\bar R}[\lambda]=0$ on the matching locus.
2. No-cut limit: after matching and reduction, no observable may depend on an arbitrary placement of $S$.
3. Sector count: every allowed flux label on one side is paired with its opposite or dual label on the other.
4. Cross-cut reconstruction: a Wilson line crossing $S$ is recovered by contracting its two dressed endpoints, not by counting two independent operators.

Center, frame, symmetry, and excitation are different

Four notions often called an edge mode must be separated.

1. A central label such as an Abelian electric-flux sector records a block of a chosen regional algebra.
2. A frame coordinate such as $\phi_R$ is conjugate to flux in an extended description and completes boundary dressings.
3. A surface symmetry acts on the frame while leaving the bulk gauge field fixed; its moment map can be the dressed flux.
4. A dynamical boundary excitation has evolution and an energy spectrum supplied by a physical boundary action, boundary condition, and Hamiltonian.

The first three do not imply the fourth. Adding a canonical pair to complete a symplectic construction supplies kinematics, not a boundary kinetic term. Conversely, a real physical boundary can select conditions under which an edge sector is dynamical. Ball and Ciambelli exhibit such a construction for Yang–Mills theory while also finding that the non-Abelian Hamiltonian need not split into independent bulk and edge pieces in Ball and Ciambelli 2026, §§ 3.1 and 4.1–4.2, pp. 4–10, official PDF. This is a result for their declared boundary problem, not evidence that every entangling cut carries new particles.

Boundary frames themselves remain construction-dependent. Intrinsic and extrinsic choices can lead to different gauge-invariant representatives, and several gauge-fixed descriptions can encode the same invariant data. A recent Maxwell analysis makes this many-to-one relation explicit in Araujo-Regado et al. 2025, §§ 4.2–4.3, pp. 31–36, Open PDF.

Compact Yang–Mills requires a group-valued frame

The Abelian canonical pair cannot be copied component by component. To see why, absorb the coupling into an anti-Hermitian compact-group connection and choose finite transformations

$$\begin{aligned} A^h&=h^{-1}Ah+h^{-1}dh, & E^{\perp h}&=h^{-1}E^\perp h, \\ u^h&=h^{-1}u. \end{aligned}$$

The group-valued frame $u:S\to G$ makes

$$\mathcal A_\parallel^u = u^{-1}A_\parallel u+u^{-1}d_Su, \qquad \mathcal E_\perp^u = u^{-1}E^\perp u$$

gauge invariant. A compatible edge symplectic potential is

$$\Theta_{\mathrm{edge}} = \int_S \left\langle E^\perp,\boldsymbol\delta u\,u^{-1} \right\rangle, \qquad \Omega_{\mathrm{edge}} = -\boldsymbol\delta\Theta_{\mathrm{edge}}.$$

For $U(1)$ with $u=e^{-\phi}$, this reduces to the sign of the Maxwell edge term above. For non-Abelian $G$, the Maurer–Cartan identity contributes a commutator term to $\Omega_{\mathrm{edge}}$: the boundary phase space is the pointwise cotangent bundle $T^*G$, not a list of independent Abelian pairs. The left action implements the enlarged gauge redundancy, while the commuting right-frame action carries the dressed flux charge. The group-valued frame, its two actions, and the dressed normal flux are developed in Donnelly and Freidel 2016, §§ 2.2–2.5, pp. 11–16, Open PDF.

Individual components $E^{\perp a}$ are not central in Yang–Mills theory. After a regional algebra and reduction have been chosen, gauge-invariant Casimir, coadjoint-orbit, or representation data can label sectors. Quantum gluing pairs dual boundary representations and contracts them through singlets or intertwiners Donnelly 2014, pp. 5–9, Open PDF. Riello explains the non-Abelian flux-sector qualifications in Riello 2021, § 7, pp. 32–33, Open PDF. This local construction is not a theorem about stabilizers, nontrivial bundles, Gribov regions, confinement, or anomaly cancellation; edge variables leave those separate problems unresolved.

Orbit, charge, and Coulomb-gauge descriptions

The Maxwell construction should give the same physical gluing condition in three descriptions. What changes is where the boundary information is stored.

Three descriptions of the same bounded Maxwell construction

Description	Regional statement	Interface datum	Glued conclusion
Gauge orbit	Cutting permits independent transformations on the two regions; adding the frame makes their boundary values null in each extended space	Relative frame or dressed transition data	Matching and the diagonal quotient recover the uncut orbit
Charge	The normal flux is the boundary moment map and labels fixed-flux sectors	eR + eR̄ = 0	Opposite charges pair and the common interface charge cancels
Coulomb gauge	Solving ∂iAi = 0 uses an inverse Laplacian with declared boundary data	Harmonic modes, normal flux, and the boundary Green function	Setting the frame representative to zero hides its role; it does not remove matching

In the orbit description, reduction before cutting uses one global gauge group, whereas separate regional reductions use two groups with independent boundary values. The extension records enough frame data to compare them and then removes the diagonal action during gluing.

In the charge description, $e_R$ is the interface moment map. A fixed-flux reduction is a legitimate alternative: reduce each region within a specified sector and pair it only with the sector $-e_R$. It avoids adding a conjugate coordinate inside that sector, but does not provide one regional phase space that moves among sectors.

In Coulomb gauge, the equation for the gauge parameter involves a Green operator for the spatial Laplacian. Its boundary condition and harmonic kernel are part of the gauge choice. A representative with $\phi_R=0$ can therefore move the frame information into a nonlocal dressing or the boundary Green data. Gauge fixing selects representatives of the same orbit problem; it does not turn the matched physical subspace into an unrestricted tensor product.

Abelian Chern–Simons zero modes across a cut

A finite-dimensional topological example isolates the same gluing issue. Take non-spin compact $U(1)$ Chern–Simons theory at even integer level $k$ on

$$M=M_1\cup_S M_2, \qquad \partial M_1=S, \qquad \partial M_2=-S,$$

and use the action normalization

$$I[A]=\frac{k}{4\pi}\int_M A\wedge dA.$$

The evenness assumption is part of the cited non-spin formulation. Odd level requires a spin or otherwise refined theory, which is outside this example Manoliu 1998, introduction, p. 3, Open PDF.

Classical bulk solutions are flat connections. At the tangent level, the boundary zero-mode space is

$$V=H^1(S;\mathbb R), \qquad \omega_S([a],[b]) = \frac{k}{2\pi}\int_S a\wedge b,$$

up to the displayed boundary-orientation convention. Large gauge transformations make the actual compact phase space a torus obtained by quotienting $V$ by the integral lattice, with the lattice’s $2\pi$ normalization tied to the convention for $A$.

For a handlebody $M_i$, restriction of flat bulk modes gives the linear subspace

$$L_i = \operatorname{im} \left[ H^1(M_i;\mathbb R) \longrightarrow H^1(S;\mathbb R) \right] \subset V.$$

The gluing diffeomorphism is understood to have transported both restriction images into the same copy of $V$.

Stokes’ theorem makes $L_i$ isotropic, and the long exact sequence together with Poincaré duality gives half the dimension of $V$; hence $L_i$ is Lagrangian. The compact images are Lagrangian subtori. Manoliu proves this statement for Abelian Chern–Simons flat-connection moduli in Manoliu 1998, §§ 2.2–2.3, pp. 9–12, Open PDF.

Matching the two restrictions is controlled, at this linearized zero-mode level, by the two-term complex

$$\mathsf C^\bullet: \qquad L_1\oplus L_2 \xrightarrow{\ d\ } V, \qquad d(\ell_1,\ell_2)=\ell_1-\ell_2.$$

Its cohomology is

$$H^0(\mathsf C^\bullet)=\ker d, \qquad H^1(\mathsf C^\bullet)=\operatorname{coker}d.$$

The kernel is the space of matched infinitesimal bulk zero modes. The cokernel measures failure of transversality and is precisely the information lost if one replaces a nontransverse gluing problem by an ordinary transverse intersection.

For a genus-one Heegaard splitting, each solid torus contributes a one-dimensional Lagrangian line in $V\simeq\mathbb R^2$.

• In the meridian-to-meridian gluing giving $S^1\times S^2$, the two lines coincide: $L_1=L_2=L$. Then $\ker d\simeq L$ is the diagonal continuous flat mode and $\operatorname{coker}d\simeq V/L$ records the excess direction.
• In the meridian-to-longitude gluing giving $S^3$, the two lines are complementary. The map $d$ is an isomorphism, so both kernel and cokernel vanish.

The second conclusion is only a real-linear test: other lens-space gluings can also be transverse over $\mathbb R$. Their distinction from $S^3$ lives in the integral lattice and compact-torus intersection, not in real cohomology alone. This calculation supplies a bounded physical bridge. The derived-intersection interpretation, its grading, compact torsion sectors, and full BV–BFV gluing conditions belong to Derived Intersections, Boundary Conditions, and Correspondences. This zero-mode complex is a finite-dimensional model of flat-moduli gluing; it neither constructs a Hilbert-space factorization nor defines an entropy prescription.

What the cited evidence licenses

The literature bearing on the interpretation and dynamical status of edge constructions was checked through August 3, 2026. The conclusions below are limited to the cited formulations and their stated hypotheses.

The sources support a deliberately limited conclusion. Donnelly and Freidel give an explicit extended-space and diagonal-gluing construction. Riello shows that fully reduced, fixed-flux gluing can instead reconstruct regional couplings without treating an independent edge coordinate as fundamental. Assanioussi and collaborators sharpen the motivation: invariance of the presymplectic form alone does not require an extension, while restoring degeneracy for the chosen gauge action can. Araujo-Regado and collaborators make the dependence on the selected reference frame explicit. Ball and Ciambelli show that a particular physical Yang–Mills boundary condition can support a dynamical edge sector.

These results are compatible once their questions are separated. They do not establish a unique regional algebra, a universal equivalence of center and extended-Hilbert-space prescriptions, a factorization theorem for continuum QFT, or a universal population of physical boundary particles. The final 2026 Yang–Mills article and the 2025 Maxwell reference-frame article were also checked for correction or withdrawal notices through the evidence cutoff; none was found.

Common pitfalls

Promoting a fiducial cut to a physical wall. Independent regional gauge transformations are bookkeeping before gluing. A physical boundary requires its own variational principle and boundary conditions.

Writing the extended tensor product as the physical space. The product is formed before the interface constraint. Flux matching and a diagonal quotient or singlet projection are still required.

Calling every edge variable an observable excitation. A frame may be an auxiliary coordinate or a dressing reference. Dynamics requires additional boundary input.

Treating the electric center as unique. The center depends on the selected regional algebra, regulator, boundary operator set, and polarization.

Using gauge fixing as a factorization proof. Coulomb or axial gauge can hide a dressing in nonlocal boundary data. Residual modes and flux matching remain.

Calling all non-Abelian flux components central. They transform in the adjoint representation and have a noncommutative moment-map algebra. Gauge-invariant representation or Casimir data can become central only after the regional algebra and reduction have been specified.

Confusing gauge and continuum obstructions. Removing the Gauss-law matching problem does not remove type-III nonfactorization for sharp continuum regions.

Check your understanding

These checks are for practice only; they are not registered assessments.

1. Opposite normals

Let one smooth electric field be restricted to both sides of $S$. Show that the two normal fluxes and the corresponding charges cancel.

Solution

The outward normals obey $n_{\bar R}=-n_R$. Hence $e_{\bar R}=n_{\bar R\,i}E^i=-n_{R\,i}E^i=-e_R$. For a common interface parameter,

$$Q_R[\lambda]+Q_{\bar R}[\lambda] = \int_S d^2S\,\lambda(e_R+e_{\bar R}) =0.$$

Using equality rather than a minus sign would correspond to expressing both fluxes with one common normal, not with two outward normals.

2. The edge-frame sign

Contract $\Omega_R^{\mathrm{ext}}$ with the enlarged Maxwell gauge direction and verify the invariance of $\widehat\Psi_\gamma$.

Solution

On Gauss tangents the bulk contraction is $\int_S\lambda\,\boldsymbol\delta e_R$. The edge contraction is $-\int_S(R_\lambda\phi_R)\boldsymbol\delta e_R =-\int_S\lambda\,\boldsymbol\delta e_R$ because $R_\lambda e_R=0$. They cancel. Meanwhile

$$e^{-ig\phi_R(b)} \longmapsto e^{-ig\lambda(b)}e^{-ig\phi_R(b)},$$

which cancels the factor $e^{ig\lambda(b)}$ carried by $U_\gamma(b,x)\psi(x)$.

3. Projecting the sector product

Start from $\mathcal H_R^{\mathrm{ext}}=\bigoplus_e\mathcal H_R^{(e)}$ and the analogous space for $\bar R$. Impose the Abelian interface constraint.

Solution

The unrestricted product contains all $\mathcal H_R^{(e)}\otimes\mathcal H_{\bar R}^{(e')}$. The diagonal gauge projector retains only pairs with $e+e'=0$, giving

$$\operatorname{Inv}_{\mathcal G_S} \left( \mathcal H_R^{\mathrm{ext}} \otimes\mathcal H_{\bar R}^{\mathrm{ext}} \right) \simeq \bigoplus_e \mathcal H_R^{(e)} \otimes \mathcal H_{\bar R}^{(-e)}.$$

Thus factorization holds for the extended kinematic space, while the physical space is the matched subspace.

4. When flux ceases to be central

Suppose $e(s)$ commutes with the original strictly interior Abelian algebra. What changes after its conjugate frame $\phi(s)$ is admitted as an operator?

Solution

The edge symplectic term gives a nonzero canonical bracket between $\phi(s)$ and $e(s')$, with its sign fixed by the displayed $\Omega_R^{\mathrm{ext}}$. Therefore $e$ no longer commutes with the full extended algebra. It can remain a central label only for a smaller algebra that omits the conjugate frame operator. This is why center choice and edge extension are related prescriptions, not identical facts.

Where the construction hands off

The present page stops before assigning an entropy to any center or extended space. Gauge Constraints, Centers, and Edge Data develops the regional-algebra dictionary, and Centers, Edge Extensions, and Distillable Entanglement separates center uncertainty, edge contributions, and operationally distillable resources.

The theorem-level compatibility of boundary fields, BFV charges, polarizations, residual modes, and gluing belongs to Edge Modes and Extended Observables at Gauge Boundaries. Gravitational and horizon applications require a new diffeomorphism phase space and are not obtained by replacing the Maxwell gauge parameter with a vector field; curved-spacetime gauge-edge and contact-term issues begin on Species, Gauge Edges, and Contact Terms.

The next page changes the physical problem from a finite cut to infinity: Asymptotic Symmetry, Soft Limits, and the Boundary Interface adds falloff conditions, infrared sectors, soft theorems, and memory.

References

• Araujo-Regado, Goncalo, Philipp A. Hoehn, Francesco Sartini, and Bilyana Tomova. “Soft Edges: The Many Links between Soft and Edge Modes.” Journal of High Energy Physics 07 (2025), 180. Open PDF.
• Assanioussi, Mehdi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, and Ludovic Varrin. “On the Covariant Formulation of Gauge Theories with Boundaries.” Classical and Quantum Gravity 41 (2024), 115007. Open PDF.
• Ball, Adam, and Luca Ciambelli. “Dynamical Edge Modes in Yang–Mills Theory.” SciPost Physics 20 (2026), 013. Official open PDF.
• Casini, Horacio, Marina Huerta, and José Alejandro Rosabal. “Remarks on Entanglement Entropy for Gauge Fields.” Physical Review D 89 (2014), 085012. Open PDF.
• Donnelly, William. “Entanglement Entropy and Nonabelian Gauge Symmetry.” Classical and Quantum Gravity 31 (2014), 214003. Open PDF.
• Donnelly, William, and Laurent Freidel. “Local Subsystems in Gauge Theory and Gravity.” Journal of High Energy Physics 09 (2016), 102. Open PDF.
• Manoliu, Mihaela. “Abelian Chern–Simons Theory.” Journal of Mathematical Physics 39 (1998), 170–206. Open PDF.
• Riello, Aldo. “Edge Modes without Edge Modes.” arXiv:2104.10182 [hep-th] (2021). Open PDF.
• Yngvason, Jakob. “The Role of Type III Factors in Quantum Field Theory.” Reports on Mathematical Physics 55 (2005), 135–147. Open PDF.