Electric and Magnetic One-Form Symmetries

In four-dimensional compact Maxwell theory, the equation of motion and the Bianchi identity close two different two-forms. They generate electric and magnetic one-form symmetries whose charged probes are Wilson and ’t Hooft lines. Electric matter sources the first conservation law and screens Wilson charge; monopoles source the second and screen magnetic charge. In the compact semisimple Yang–Mills examples below, the surviving finite symmetry also depends on the global gauge group, matter representations, discrete theta data, and the chosen genuine-line spectrum—not on the Lie algebra alone.

The controlled Maxwell model below has no physical boundary, uses $\theta=0$, and distinguishes external probe lines from dynamical particles. The symmetry statements are exact in the declared theory. They do not by themselves decide confinement, spontaneous breaking, electromagnetic duality, or the infrared phase.

Required background. Higher-Form Currents, Charges, Backgrounds, and Ward Identities supplies the closed-dual-current, charge-surface, background, and contact-term conventions. The Free Maxwell Field and Gauge Redundancy supplies the Maxwell action, equation of motion, Bianchi identity, Gauss law, and gauge redundancy.

Helpful background. Genuine Lines, Screening, and Charge Lattices supplies the genuine-versus-screened distinction, the Dirac pairing, and the need to choose an allowed line lattice. Global Form, Matter Representations, and the Faithful Gauge Group supplies central quotients, honest representations, and the reason that local Lie-algebra data are insufficient.

Computational companion. No runnable charge-lattice explorer is currently available. The quotient, character, and junction calculations below are analytic and self-contained.

Source-free compact Maxwell theory has two one-form symmetries

Work on an oriented spin Lorentzian four-manifold $M$, with the spin structure held fixed and no physical boundary in the region under study. Let $\mathfrak a$ be a faithfully normalized compact $U(1)$ connection,

$$\mathfrak a\longmapsto\mathfrak a+\mathrm d\lambda, \qquad \lambda\sim\lambda+2\pi,$$

with curvature $f=\mathrm d\mathfrak a$ locally and flux quantization

$$\frac{1}{2\pi}\int_{\Sigma_2}f\in\mathbb Z$$

on every closed oriented two-cycle $\Sigma_2$ on which the bundle is restricted. Begin with no dynamical electric charges and no dynamical magnetic monopoles. The action is

$$S_{\mathrm M}[\mathfrak a] =-\frac{1}{2e^2}\int_M f\wedge\star f.$$

The two closed charge forms are

$$\widetilde j_{e,2}=\frac{\star f}{e^2}, \qquad \widetilde j_{m,2}=\frac{f}{2\pi}.$$

They have the same degree but different origins:

$$\mathrm d\widetilde j_{e,2}=0 \quad\text{is the Maxwell equation}, \qquad \mathrm d\widetilde j_{m,2}=0 \quad\text{is the Bianchi identity}.$$

With the site’s Lorentzian Hodge convention, $\star^2=-1$ on two-forms. If the undualized currents are also displayed, consistency with $\widetilde j=\star j$ therefore gives

$$j_{e,2}=\frac{f}{e^2}, \qquad j_{m,2}=-\frac{\star f}{2\pi}.$$

The minus sign in $j_m$ is a Hodge-sign consequence, not a negative magnetic charge convention. The positively normalized magnetic charge is still $\int_{\Sigma_2}f/(2\pi)$.

In this source-free theory both closed forms generate exact continuous $U(1)^{(1)}$ symmetries. Electric charges and magnetic monopoles reduce the two factors by different mechanisms below.

For a closed oriented surface $\Sigma$, define

$$\begin{aligned} Q_e(\Sigma)&=\int_\Sigma\frac{\star f}{e^2}, & U_e(\alpha,\Sigma)&=\exp\!\bigl(i\alpha Q_e(\Sigma)\bigr), \\ Q_m(\Sigma)&=\frac{1}{2\pi}\int_\Sigma f, & U_m(\beta,\Sigma)&=\exp\!\bigl(i\beta Q_m(\Sigma)\bigr), \end{aligned}$$

where $\alpha,\beta\in\mathbb R/2\pi\mathbb Z$. Both generators live on codimension-two supports, both charged objects are lines, and both backgrounds are two-form higher connections. The surface is topological only while its deformation avoids charged worldlines, boundaries, and other declared obstructions. The two symmetries and their respective equation-of-motion and Bianchi origins are given in Gaiotto et al. 2015, § 4.1, arXiv v2, pp. 14–15, eqs. (4.1)–(4.2), Open PDF and reviewed in Bhardwaj et al. 2024, Example 2.2, arXiv v2, pp. 14–16, eqs. (2.49)–(2.55), Open PDF. Their field and coupling conventions translate to the faithfully normalized compact connection and charges declared here; the displayed coefficients are the page-local conventions.

The comparison is compact enough to keep the two conservation laws visibly separate.

Electric and magnetic data in the scoped Maxwell model

Test	Electric sector	Magnetic sector	What changes it
Closed charge form	⋆f/e2	f/(2π)	Charged contacts or a boundary flux term
Local origin	Equation of motion	Bianchi identity	Electric and magnetic sources enter different equations
Charged probe	Wilson line	’t Hooft line	The genuine-line spectrum is additional global data
Dynamical source	Electrically charged worldline	Monopole worldline	Endpoints screen the corresponding line charge
Finite residual	Electric charges generating Nℤ leave ℤN	Magnetic charges generating Mℤ leave ℤM	A dyonic spectrum requires the combined lattice quotient

The table does not assert that both residual factors can be chosen independently in an arbitrary theory. Mutual locality, the global form, spin or other tangential data, and discrete theta information must first select a consistent genuine-line spectrum.

Wilson and ’t Hooft lines carry different charges

Choose for the moment the standard absolute compact-$U(1)$ spectrum in which every integral pair $(n,m)\in\mathbb Z^2$ labels a genuine Wilson–’t Hooft line $L_{n,m}(C)$. The Wilson representative is

$$W_n(C)=\exp\!\left(i n\oint_C\mathfrak a\right), \qquad n\in\mathbb Z,$$

whereas the ’t Hooft label is imposed by the disorder condition

$$\frac{1}{2\pi}\int_{S^2_{\mathrm{link}}}f=m, \qquad m\in\mathbb Z.$$

The small sphere links $C$ once with the declared positive orientation. A singular boundary condition defines a candidate disorder line; its genuineness still depends on the full global operator data.

Let $C$ and $\Sigma$ be disjoint closed oriented supports in a linking ball, with every other insertion $\mathcal X$ outside the sweep and the normalized surface removable afterward. Fix the positive unit-link convention so that $U_e(\alpha)$ acting on $W_1$ gives $e^{i\alpha}$ and $U_m(\beta)$ acting on $L_{0,1}$ gives $e^{i\beta}$. Then

$$\chi_{n,m}(\alpha,\beta) =\exp\!\bigl(i\alpha n+i\beta m\bigr),$$

and repeated linking gives

$$\begin{aligned} &\left\langle U_e(\alpha,\Sigma)U_m(\beta,\Sigma) L_{n,m}(C)\mathcal X \right\rangle \\ &\qquad= \chi_{n,m}(\alpha,\beta)^{\operatorname{Lk}(\Sigma,C)} \left\langle L_{n,m}(C)\mathcal X\right\rangle . \end{aligned}$$

Reversing either support negates the linking number and inverts the unitary character. The Wilson and ’t Hooft charge/contact statements are worked out in Bhardwaj et al. 2024, Example 2.3, arXiv v2, pp. 21–22, eqs. (2.79)–(2.82), Open PDF. The line itself need not be topological: the topological object in this calculation is the symmetry surface moved through its allowed deformation domain.

Two backgrounds reveal a mixed anomaly

The two continuous symmetries can each be probed by a two-form background, but coupling both exposes an obstruction. Put

$$X=f-B_e$$

and use the local representative

$$S[\mathfrak a;B_e,B_m] =-\frac{1}{2e^2}\int_M X\wedge\star X +\frac{1}{2\pi}\int_M B_m\wedge X.$$

The electric one-form gauge transformation

$$\mathfrak a\longmapsto\mathfrak a+\Lambda_e, \qquad B_e\longmapsto B_e+\mathrm d\Lambda_e$$

leaves $X$ invariant. Under $B_m\mapsto B_m+\mathrm d\Lambda_m$, however, integration by parts on closed $M$ gives

$$\delta_m S =-\frac{1}{2\pi}\int_M \Lambda_m\wedge\mathrm dB_e.$$

Thus this counterterm scheme preserves electric covariance and displays the mixed electric–magnetic anomaly in the magnetic transformation. A local background counterterm can move the variation between the two symmetries but cannot remove it from both. This means that the two symmetries cannot both be gauged without additional inflow; it does not mean that either symmetry is explicitly broken, and it does not by itself define a higher group.

The finite-background response forms are

$$\widetilde j_e(B_e,B_m) =\frac{\star X}{e^2}-\frac{B_m}{2\pi}, \qquad \widetilde j_m(B_e)=\frac{X}{2\pi}.$$

At zero background they reduce to the two charge forms above. In this counterterm scheme the background equations read

$$\mathrm d\widetilde j_e=0, \qquad \mathrm d\widetilde j_m=-\frac{\mathrm dB_e}{2\pi}.$$

The response form $\widetilde j_e(B_e,B_m)$ is scheme-dependent. Under a magnetic background gauge transformation it obeys

$$\widetilde j_e\longmapsto \widetilde j_e-\frac{\mathrm d\Lambda_m}{2\pi}.$$

Its formal closure in this representative is therefore not simultaneous gauge invariance of both backgrounds.

The quadratic $B_e$ term is essential; the linear source coupling alone is not a complete finite-background action. Brennan and Hong write the counterterm-equivalent $B_m\wedge f$ scheme, in which the variation appears in the electric transformation; subtracting $B_m\wedge B_e/(2\pi)$ gives the representative above and moves it to the magnetic transformation. This anomaly and its counterterm dependence are developed in Brennan and Hong 2023, § 2.5.2, arXiv v2, p. 26, eqs. (2.109)–(2.113), Open PDF and Gaiotto et al. 2015, § 4.1, arXiv v2, p. 15, paragraph beginning “The two one-form symmetries,” Open PDF.

Matter and monopoles source different conservation laws

Let $\mathcal J_e^{(3)}$ and $\mathcal J_m^{(3)}$ denote the normalized distributional three-form currents of dynamical electric particles and monopoles. The sourced equations have the schematic normalization

$$\mathrm d\!\left(\frac{\star f}{e^2}\right) =\mathcal J_e^{(3)}, \qquad \mathrm d\!\left(\frac{f}{2\pi}\right) =\mathcal J_m^{(3)}.$$

An electric worldline obstructs deformation of $U_e$ across it but does not by itself source the Bianchi identity. A monopole worldline does the reverse. This separation is why “charged matter breaks the one-form symmetry” is incomplete unless the charge type is named.

For the chosen genuine lattice

$$\Lambda_{\mathrm{gen}}=\mathbb Z^2,$$

let $S_{\mathrm{dyn}}\subset\Lambda_{\mathrm{gen}}$ be the sublattice generated by dynamical endpoints. The protected line classes and their faithful one-form action are

$$\mathcal A_{\mathrm{line}} =\Lambda_{\mathrm{gen}}/S_{\mathrm{dyn}}, \qquad G^{(1)}_{\mathrm{faithful}} =\operatorname{Hom}(\mathcal A_{\mathrm{line}},U(1)).$$

If the nonzero dynamical electric charges generate exactly $N\mathbb Z$ and there are no monopoles, then

$$S_{\mathrm{dyn}}=N\mathbb Z\times\{0\}, \qquad \mathcal A_{\mathrm{line}} \simeq\mathbb Z_N\times\mathbb Z,$$

so the exact symmetry is $\mathbb Z_{N,e}^{(1)}\times U(1)_m^{(1)}$ in the declared absolute theory. A unit electric charge sets $N=1$ and removes the electric factor. Dually, magnetic charges generating $M\mathbb Z$ reduce the magnetic factor to $\mathbb Z_{M,m}^{(1)}$; a unit monopole removes it.

For dyons, separate electric and magnetic greatest-common-divisor rules are not enough. One must quotient the full charge lattice by the endpoint matrix, check mutual locality and the genuine-line choice, and then take the character group. Smith normal form is the appropriate bounded calculation. The electric quotient is derived in Bhardwaj et al. 2024, § 3.2.1, arXiv v2, pp. 29–31, eqs. (3.10)–(3.19), Open PDF; magnetic and dyonic lattices are treated in Bhardwaj et al. 2024, §§ 3.2.4–3.2.5, arXiv v2, pp. 36–39, eqs. (3.58)–(3.78), Open PDF.

First application: a finite surface network from charge-N matter

Keep the no-monopole assumption and let the dynamical electric charges generate exactly $N\mathbb Z$, with $N\geq2$. Hold magnetic backgrounds trivial and assume the residual electric $\mathbb Z_N^{(1)}$ symmetry is exact and has an anomaly-free, invertible group-like network in this restricted domain. Let

$$r=[n]_N\in\mathbb Z_N$$

be the unscreened electric charge of $W_n(C)$, and let $U_\alpha(\Sigma)$, $\alpha\in\mathbb Z_N$, be the closed oriented electric symmetry surface. In the normalization inherited from the continuous electric surface,

$$U_\alpha(\Sigma) =\exp\!\left(\frac{2\pi i\alpha}{N}Q_e(\Sigma)\right).$$

This exponential, rather than a local conserved current, is the exact finite symmetry operator. Its orientation laws are

$$U_\alpha(\overline\Sigma)=U_{-\alpha}(\Sigma), \qquad W_n(\overline C)=W_{-n}(C).$$

For the same controlled linking domain and positive unit-link convention used above,

$$\begin{aligned} &\left\langle U_\alpha(\Sigma)W_n(C)\mathcal X \right\rangle \\ &\qquad= \exp\!\left[ \frac{2\pi i}{N}\,\alpha r\, \operatorname{Lk}(\Sigma,C) \right] \left\langle W_n(C)\mathcal X\right\rangle . \end{aligned}$$

The surface fusion law is

$$U_\alpha\otimes U_\beta \simeq U_{\alpha+\beta\;\mathrm{mod}\;N}.$$

For two incoming sheets and one outgoing sheet meeting along an oriented line $K$, declared junction data have the type

$$\mathcal I_{\alpha,\beta}^{\ \gamma}(K): U_\alpha\otimes U_\beta\longrightarrow U_\gamma, \qquad \alpha+\beta-\gamma=0\pmod N.$$

The Wilson character is compatible with the incidence precisely when

$$\exp\!\left[ \frac{2\pi i}{N}\,r(\alpha+\beta-\gamma) \right]=1.$$

This congruence is necessary, but it neither constructs nor normalizes the junction and proves no associativity or coherence statement.

Equivalently, in a chosen cellular cochain model a flat finite background is a cocycle

$$b_2\in Z^2(M,\mathbb Z_N), \qquad \lambda_1\in C^1(M,\mathbb Z_N), \qquad b_2\longmapsto b_2+\delta\lambda_1,$$

so the gauge-equivalence class is $[b_2]\in H^2(M,\mathbb Z_N)$.

Its dual-cell representative is a network of the surfaces and junctions just described. Cocycle closure supplies the signed incidence condition, not the junction operator or its coherence data.

The screening endpoint is equally explicit. If $P:y\to x$ and a charge-$N$ field transforms as $\psi_N\mapsto e^{iN\lambda}\psi_N$, then

$$W_N(P)\longmapsto e^{iN[\lambda(x)-\lambda(y)]}W_N(P),$$

so

$$\overline\psi_N(x)W_N(P)\psi_N(y)$$

is gauge invariant. Therefore $W_N$ is screenable, $n\sim n+N$ as an unscreened class, and $W_1$ remains a faithful detector under the declared spectrum. Charge-one matter destroys this finite electric symmetry.

Charge-$N$ breaking of the continuous electric symmetry to $\mathbb Z_N^{(1)}$ appears at Gaiotto et al. 2015, § 4.1, arXiv v2, p. 17, paragraph beginning “Next, we add,” Open PDF. The group-like linking and junction network are bounded by Gaiotto et al. 2015, §§ 2–3, arXiv v2, pp. 6–13, especially eqs. (2.2), (2.6)–(2.8), and (3.1)–(3.4), Open PDF. This application exhibits an electric line, symmetry surfaces, and their line junction; it makes no claim about the magnetic background, the phase of the theory, or gauging.

Global form changes the Yang–Mills answer

For a connected compact semisimple gauge group $G$, the electric one-form symmetry is the subgroup of the center that acts trivially on every dynamical matter representation. In pure simply connected $SU(N)$ Yang–Mills it is $\mathbb Z_N^{(1)}$. If a Wilson line in representation $R$ has $N$-ality $r(R)$, then a surface labeled by $k\in\mathbb Z_N$ acts by

$$\exp\!\left[ \frac{2\pi i}{N}\,k\,r(R)\, \operatorname{Lk}(\Sigma,C) \right].$$

If the dynamical matter has $N$-alities $r_1,\ldots,r_s$, the surviving electric subgroup is

$$\mathbb Z_d^{(1)}, \qquad d=\gcd(N,r_1,\ldots,r_s).$$

This finite symmetry has topological surfaces but generally no local Noether two-form current.

Matter and global-form dependence of the bounded Yang–Mills comparison

Theory data	Electric one-form result	Reason or qualification
Pure SU(N)	ℤN	Every center element acts on Wilson N-ality
SU(N) with fundamental matter	Trivial	N-ality one screens every center class
SU(N) with adjoint matter only	ℤN	Adjoint matter has N-ality zero
PSU(N) or another quotient	Not fixed by the Lie algebra	Magnetic and dyonic genuine lines depend on the quotient and discrete theta data

The last row is the essential global-form warning. For example, pure $SU(2)$ has electric $\mathbb Z_2^{(1)}$ and no magnetic factor from $\pi_1$. In the standard spin $SO(3)_+$ theory the remaining $\mathbb Z_2^{(1)}$ symmetry is magnetic and its charged genuine line is pure magnetic. In $SO(3)_-$ the remaining $\mathbb Z_2^{(1)}$ symmetry and its charged line are dyonic. Thus even “$\mathbb Z_2$ one-form symmetry” is not a complete theory specification.

The center, matter-screening quotient, and $SU(N)$ examples are developed in Bhardwaj et al. 2024, §§ 3.3.1–3.3.4, arXiv v2, pp. 39–50, especially eqs. (3.137)–(3.153), Open PDF. The global form, genuine Wilson–’t Hooft spectrum, and discrete-theta distinction are developed in Aharony, Seiberg, and Tachikawa 2013, Introduction and § 2, arXiv v5, pp. 1–5 and 12–16, Open PDF. These are kinematic symmetry statements, not predictions of confinement or a vacuum condensate.

Flux operators expose a topology-sensitive algebraic gap

The same electric and magnetic fluxes give a bounded physical example of why Einstein causality does not imply naïve Haag duality. This is a separate free-field-strength-net calculation: its real Weyl parameters are not the integer compact-line lattice used above. Consider the vacuum Gaussian net generated by the free Maxwell field strengths in four-dimensional Minkowski space. On a compactified time slice $S^3$, take an unknotted solid torus $\mathcal R_0\simeq S^1\times D^2$ and its causal completion $\mathcal R$. The interior spatial complement is another solid torus with a once-linked core; its causal completion is $\mathcal R^\prime$. Let $\mathcal A_{\mathrm{add}}(\mathcal R)$ be the von Neumann algebra generated by bounded functions of electric and magnetic fields smeared in contractible subregions of $\mathcal R$.

Locality gives only

$$\mathcal A_{\mathrm{add}}(\mathcal R) \subseteq \mathcal A_{\mathrm{add}}(\mathcal R^\prime)^\prime.$$

The right-hand side is the dual or maximal algebra for this net and region:

$$\mathcal A_{\mathrm{max}}(\mathcal R) :=\mathcal A_{\mathrm{add}}(\mathcal R^\prime)^\prime.$$

Regularize spanning surfaces whose boundary loops wind a ring and form the bounded flux Weyl operators

$$V_B(q)=e^{iq\Phi_B}, \qquad V_E(\widetilde q)=e^{i\widetilde q\Phi_E}, \qquad q,\widetilde q\in\mathbb R.$$

Here $\Phi_B$ and $\Phi_E$ are smeared magnetic and electric fluxes. For linked core loops in complementary rings their normalization can be chosen so that

$$V_E^{\mathcal R}(\widetilde q)V_B^{\mathcal R^\prime}(q) =e^{iq\widetilde q} V_B^{\mathcal R^\prime}(q)V_E^{\mathcal R}(\widetilde q).$$

Each flux Weyl operator commutes with all additively generated observables in the causal complement, but the linked commutation relation prevents it from being generated by contractible smearings inside the ring. In this free net, the computation is

$$\begin{aligned} \mathcal A_{\mathrm{max}}(\mathcal R) ={}&\mathcal A_{\mathrm{add}}(\mathcal R) \vee \{V_B(q),V_E(\widetilde q)\}_{q,\widetilde q\in\mathbb R}, \\ \mathcal A_{\mathrm{add}}(\mathcal R) \subsetneq{}&\mathcal A_{\mathrm{max}}(\mathcal R), \end{aligned}$$

and likewise with $\mathcal R$ and $\mathcal R^\prime$ exchanged. The symbol $\vee$ denotes the von Neumann algebra generated by the listed subalgebras and operators. The causal complement has therefore been computed at the level relevant to the gap: it is ring-like, and its commutant contains the two topological flux classes missing from the additive ring algebra.

There is a further locality ceiling. The two full maximal algebras for the complementary rings do not commute with one another, as the linked Weyl relation shows. A local Haag-dual completion must choose mutually local flux charges. With

$$\Omega\!\left((q,\widetilde q),(q',\widetilde q')\right) =q\widetilde q'-q'\widetilde q,$$

locality requires $\Omega(L,L)\subset2\pi\mathbb Z$, while duality requires a maximal mutually local choice $L=L^\perp$. For a rank-two self-dual lattice, any lattice basis has symplectic pairing of magnitude $2\pi$. Thus the additive net is local but not dual; a Haag-dual completion restores duality at the expense of ring additivity and requires extra global charge data. One cannot independently add every real electric and magnetic flux family on both sides.

This is exact for the vacuum free-field Gaussian net, ring-like regions, and bounded smeared flux exponentials. It is not a sharp unsmeared-loop limit, not a theorem for arbitrary topology, and not a result for interacting compact QED or Yang–Mills. The calculation and equality above are given in Casini, Magán, and Martínez 2022, §§ 2–3.2, arXiv v2, pp. 3–11, especially eqs. (2.1)–(2.7), (3.1)–(3.4), and (3.14)–(3.23), Open PDF. The theorem-level distinctions among locality, additivity, dual nets, causal completion, and Haag duality belong to Isotony, Additivity, Duality, and Primitive Causality.

What one-form symmetry does not decide

These examples separate exact symmetry data from their dynamical realization.

• An exact electric or magnetic one-form symmetry does not tell whether it is spontaneously broken. Area, perimeter, and Coulombic laws require an infrared analysis.
• A finite surface group does not specify a genuine-line spectrum. Global form, discrete theta data, mutual locality, and endpoint content remain inputs.
• The mixed background anomaly obstructs simultaneous gauging without inflow. It neither erases the individual symmetry operators nor turns their coexistence into a higher group.
• Heavy charged matter can make a continuous symmetry look emergent below the mass threshold while the exact ultraviolet symmetry is only the finite subgroup. “Emergent” and “exact” must not be exchanged.
• A boundary adds flux and possible boundary-current terms. None of the closed-surface equations above is a boundary condition.
• At $\theta\neq0$, electric and magnetic labels mix through the Witten effect. The $\theta=0$ formulas cannot simply be reused.
• The topology-sensitive algebraic gap is a statement about which bounded operators are additively generated. It is not a failure of Einstein causality.

Common pitfalls

Calling the Bianchi identity an electric conservation law. In the normalization used here, $\mathrm d(\star f/e^2)=0$ is electric and follows from the equation of motion; $\mathrm d(f/2\pi)=0$ is magnetic and follows from the Bianchi identity.

Letting electric matter break the magnetic symmetry automatically. An electric endpoint screens Wilson charge. Magnetic breaking requires monopole or dyonic endpoint data that source $\mathrm df$.

Inferring a product of gcds for arbitrary dyons. Dyonic endpoints mix the two lattice directions. Compute the full quotient and check mutual locality before taking its character group.

Reading a finite symmetry from the Lie algebra. The same $\mathfrak{su}(2)$ algebra supports $SU(2)$ and $SO(3)_\pm$ theories with different genuine lines and one-form actions.

Replacing a junction by charge arithmetic. The congruence $\alpha+\beta-\gamma=0$ is an incidence condition. It does not supply the junction operator, its normalization, or coherence maps.

Deriving Haag duality from locality. Locality gives an inclusion. The free Maxwell ring fluxes make that inclusion strict.

Calling a Wilson line topological because a linked surface is topological. The surface action is deformation invariant in its complement. A Wilson line can remain metric- and renormalization-dependent.

Check your understanding

1. Which conservation law is sourced by an electric worldline, and which is sourced by a monopole worldline?

Checked answer

An electric worldline contributes to $\mathrm d(\star f/e^2)$, so it obstructs an electric surface sweep. A monopole contributes to $\mathrm d(f/2\pi)$, so it obstructs a magnetic surface sweep. Neither source should be silently inserted into the other identity.

2. Take $N=6$, $n=8$, $\alpha=4$, and a positive unit link. Compute the finite electric phase. Then test the junction $U_4\otimes U_5\to U_3$.

Checked answer

The unscreened class is $r=[8]_6=2$, so

$$\exp\!\left(\frac{2\pi i}{6}\,4\cdot2\right) =\exp\!\left(\frac{2\pi i}{3}\right).$$

The incidence is $4+5-3=6\equiv0\pmod6$, and therefore every character gives one around the junction. This passes the selection rule but does not construct or normalize $\mathcal I_{4,5}^{\ 3}$.

3. In $SU(4)$ Yang–Mills, compare fundamental matter, adjoint matter, and matter of $N$-ality two.

Checked answer

Fundamental matter has $r=1$, so $\gcd(4,1)=1$ and the electric one-form symmetry is trivial. Adjoint matter has $r=0$, so $\gcd(4,0)=4$ and the full $\mathbb Z_4^{(1)}$ survives. Matter of $N$-ality two leaves $\mathbb Z_2^{(1)}$. None of these kinematic results decides whether the surviving symmetry is spontaneously broken.

4. Vary the two-background Maxwell action under a magnetic transformation $B_m\mapsto B_m+\mathrm d\Lambda_m$.

Checked answer

Since $X=f-B_e$ and $\mathrm dX=-\mathrm dB_e$,

$$\frac{1}{2\pi}\int_M\mathrm d\Lambda_m\wedge X =-\frac{1}{2\pi}\int_M \Lambda_m\wedge\mathrm dB_e.$$

The Maxwell square is unchanged. This is the mixed-anomaly variation in the electric-preserving counterterm scheme, not explicit breaking of the magnetic symmetry at zero background.

5. Why does the Maxwell ring example obey locality but fail naïve Haag duality?

Checked answer

Locality gives $\mathcal A_{\mathrm{add}}(\mathcal R)\subseteq \mathcal A_{\mathrm{add}}(\mathcal R^\prime)^\prime$. The smeared electric and magnetic flux Weyl operators commute with the additive complement but cannot be assembled from contractible smearings in $\mathcal R$; their linked Weyl relation detects that omission. They enlarge the right-hand side, so the inclusion is strict without any failure of spacelike commutativity.

6. What must be supplied before the statement “the gauge algebra is $\mathfrak{su}(N)$” determines its one-form symmetry?

Checked answer

One must at least specify the global gauge group, allowed bundles, matter representations and their center charges, genuine Wilson–’t Hooft line spectrum, mutual-locality condition, discrete theta data, tangential structure, and any allowed endpoints or boundaries. The Lie algebra fixes only the local infinitesimal gauge data.

Continue to phases, dynamics, duality, and local nets

Breaking Higher-Form Symmetry and Diagnosing Phases is the immediate next chapter step; it owns area, perimeter, and Coulombic laws as symmetry realization tests. Line Operators, Screening, and Generalized-Symmetry Diagnostics develops confinement and string-breaking criteria, while Compact U(1) in 2+1 Dimensions and the Monopole Plasma is the dimension-specific controlled monopole-dynamics model.

Electric–Magnetic Charge Lattices and Global Form owns parent-action dualization, theta dependence, flux sums, and the global line-lattice map. Gauging a Higher-Form Symmetry promotes the background to dynamical data. Isotony, Additivity, Duality, and Primitive Causality supplies the theorem-first net-property distinctions that the Maxwell ring example only illustrates physically.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 2013, no. 8 (2013): 115. DOI. Open PDF, arXiv v5.
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv v2.
• Brennan, T. Daniel, and Sungwoo Hong. “Introduction to Generalized Global Symmetries in QFT and Particle Physics.” arXiv:2306.00912v2 [hep-ph], revised 2 June 2023. Stable record. Open PDF, arXiv v2.
• Casini, Horacio, Javier M. Magán, and Pedro J. Martínez. “Entropic Order Parameters in Weakly Coupled Gauge Theories.” Journal of High Energy Physics 2022, no. 1 (2022): 079. DOI. Open PDF, arXiv v2.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI. Open PDF, arXiv v2.