Higher-Form Currents, Charges, Backgrounds, and Ward Identities

A continuous $p$-form symmetry relates four pieces of data once their normalizations and domains are fixed: a $(p+1)$-form current $j_{p+1}$, a closed dual current $\widetilde j_{d-p-1}=\star j_{p+1}$, charges obtained by integrating $\widetilde j$ on $(d-p-1)$-cycles, and a nondynamical $(p+1)$-form background $B_{p+1}$. The current and its dual contain the same local information after choosing the metric and convention; charges and background response are derived global and source data. Background covariance ties the four together. With a charged $p$-dimensional defect present, it becomes a distributional Ward identity whose contact term records the defect’s charge.

This page treats one ordinary continuous Abelian factor, with $0\leq p\leq d-1$, in Lorentzian signature with the site’s $(+---)$ convention. The local derivation uses compactly supported transformations, or a region with no physical boundary. Finite symmetries generally have no local Noether current; large background transformations, global anomalies, boundaries, explicit breaking, and nonlinear source terms are separate tests rather than consequences of the local divergence equation.

Required background. Higher-Form Symmetry from Operators and Linking supplies the operator-first definition, degree dictionary, linking action, and continuous-versus-finite distinction. Coupling to Background Gauge Fields and Bundles supplies the nondynamical-source convention, background covariance, global patching, and the distinction between probing and gauging.

Helpful background. de Rham Cohomology, Periods, Duality, and Intersection supplies the closed-versus-exact distinction, periods, dual cycles, and intersection signs. Parallel Transport and Holonomy supplies the compact-connection and Wilson-line conventions used in the application.

Conservation makes charge surfaces topological

Let $M$ be an oriented $d$-manifold, and let a continuous $U(1)$ $p$-form symmetry have a renormalized $(p+1)$-form current $j_{p+1}$. We keep the convention established on the preceding page,

$$\widetilde j_{d-p-1}=\star j_{p+1}.$$

Away from charged supports, explicit breaking, local or background-anomaly contributions, and physical boundaries, conservation is

$$\mathrm d\widetilde j_{d-p-1}=0.$$

Some sources call the closed form $\widetilde j$, rather than its Hodge dual $j$, the current. The two notations contain the same local information, but the Hodge star depends on the metric whereas the operator-first symmetry definition does not. Form degree is a distinct grading from engineering and scaling dimension and must not be confused with either one, even though symmetry can constrain scaling dimensions in special settings such as CFT.

For a closed oriented $(d-p-1)$-cycle $Y$, define

$$Q(Y)=\int_Y\widetilde j_{d-p-1}.$$

Suppose $Y_1-Y_0=\partial V$ for an oriented $(d-p)$-chain $V$. Stokes’ theorem gives the complete deformation test,

$$Q(Y_1)-Q(Y_0) =\int_V\mathrm d\widetilde j.$$

The charge is unchanged only when $V$ crosses no charged insertion and there is no breaking term, anomaly contribution, or boundary flux in the swept region. Closedness does not imply exactness: on a topologically nontrivial spacetime, periods of $\widetilde j$ can carry nonzero charge. For $p\leq d-2$, an improvement $\widetilde j\mapsto\widetilde j+\mathrm dk$ by a globally defined $(d-p-2)$-form $k$ leaves charges on closed cycles unchanged when no new boundary term appears. There is no such local-form improvement at $p=d-1$.

After fixing charge normalization, exponentiation produces the continuous topological operator

$$U_\varphi(Y)=\exp\!\left(i\varphi Q(Y)\right), \qquad \varphi\sim\varphi+2\pi.$$

For $p=0$, $j_1$ is the ordinary one-form current, $Y$ is a codimension-one charge surface, and a charged support is a point. For $p=1$ in four dimensions, $j_2$, $\widetilde j_2$, the background, and the charge surface all have degree or dimension two. The full cross-check is collected in the degree dictionary on the preceding page.

The current, charge, and topological-operator construction appears in the original operator formulation at Gaiotto et al. 2015, § 3, arXiv v2, pp. 11–13, especially eqs. (3.1)–(3.4), Open PDF and in the explicit differential-form treatment at Iqbal 2025, § 3.1, pp. 17–19, especially eqs. (3.1.1)–(3.1.7), Open PDF.

A higher-form background packages the Ward identity

A local background for the continuous symmetry is a $(p+1)$-form $B_{p+1}$. Its infinitesimal gauge parameter is a $p$-form $\Lambda_p$,

$$B_{p+1}\longmapsto B_{p+1}+\mathrm d\Lambda_p, \qquad H_{p+2}=\mathrm dB_{p+1}.$$

The background is fixed while the dynamical fields are integrated over. It is therefore a probe, not a field being gauged. Retaining the plus-source convention of the ordinary background-field chapter, its linear coupling at $B=0$ is

$$S_{\mathrm{src}}[B] =\int_M B_{p+1}\wedge\widetilde j_{d-p-1}.$$

This formula defines the first response. A finite background can require $B$-dependent currents, seagulls, or other local contact terms. The Maxwell example below displays that completion explicitly.

For compactly supported $\Lambda$, or on a closed $M$, graded Stokes gives

$$\begin{aligned} \delta_\Lambda S_{\mathrm{src}} &=\int_M\mathrm d\Lambda_p\wedge\widetilde j_{d-p-1} \\ &=(-1)^{p+1} \int_M\Lambda_p\wedge\mathrm d\widetilde j_{d-p-1}. \end{aligned}$$

In an anomaly-free theory without charged insertions, invariance for arbitrary $\Lambda_p$ recovers $\mathrm d\widetilde j=0$. This is the higher-form version of deriving an ordinary Ward identity by localizing a global transformation. Bhardwaj et al. use a background with unit-period holonomy and hence write an explicit factor $2\pi$; after rescaling to the site’s $2\pi$-periodic holonomy convention, their background and source degrees are the ones above Bhardwaj et al. 2024, § 4.1, arXiv v2, pp. 61–63, Definitions 4.1–4.2 and eqs. (4.7)–(4.17), Open PDF.

Charged defects produce distributional contacts

Let $C^p\subset M$ be a closed oriented support. Its Poincaré-dual distribution is the $(d-p)$-form $\Delta_C$ defined by

$$\int_M\omega_p\wedge\Delta_C=\int_C\omega_p$$

for every compactly supported smooth test $p$-form $\omega_p$. This equation fixes both the degree and the orientation sign. Reversing the orientation of $C$ sends $\Delta_C$ to $-\Delta_C$.

Suppose $\mathcal O_q(C)$ has charge $q$ in the convention

$$\mathcal O_q(C) \longmapsto \exp\!\left(iq\int_C\Lambda_p\right)\mathcal O_q(C).$$

For a normalized inserted functional, background covariance reads

$$\begin{aligned} &\left\langle \mathcal O_q(C)\,\mathcal X \right\rangle_{B+\mathrm d\Lambda} \\ &\qquad= \exp\!\left(iq\int_C\Lambda_p\right) \left\langle \mathcal O_q(C)\,\mathcal X \right\rangle_B, \end{aligned}$$

where any transformation of the other insertions in $\mathcal X$ must also be included. Differentiate at $B=0$, use the source variation above, and define the current insertion by differentiating the normalized functional. The smeared identity is

$$\begin{aligned} &\int_M\mathrm d\Lambda_p\wedge \left\langle \widetilde j\,\mathcal O_q(C)\,\mathcal X \right\rangle_{c} \\ &\qquad= q\int_C\Lambda_p \left\langle \mathcal O_q(C)\,\mathcal X \right\rangle . \end{aligned}$$

Here the subscript $c$ means connected between the current insertion and the rest; equivalently, retain the disconnected subtraction generated by differentiating a normalized correlator. Integrating by parts gives the local distributional Ward identity

$$\begin{aligned} \mathrm d \left\langle \widetilde j\,\mathcal O_q(C)\,\mathcal X \right\rangle_c ={}&(-1)^{p+1}q\,\Delta_C \\ &\times \left\langle \mathcal O_q(C)\,\mathcal X \right\rangle , \end{aligned}$$

plus the contacts of any charged operators in $\mathcal X$. The parity sign comes from moving $\mathrm d$ past a $p$-form test parameter. For $p=0$ it reproduces the negative contact in the site’s ordinary plus-source convention; for $p=1$ the contact is positive. Euclidean conventions often place an additional factor of $i$ in this formula. Iqbal gives that Euclidean line-defect form and defines the same Poincaré-dual distribution at Iqbal 2025, § 3.1, pp. 17–19, eqs. (3.1.1)–(3.1.7), Open PDF.

Integrating the contact equation over $V^{d-p}$ with $\partial V=Y_1-Y_0$ yields

$$Q(Y_1)-Q(Y_0) =(-1)^{p+1}q\,(V\mathbin{\boldsymbol{\cdot}}C).$$

Thus the local Ward contact and the global linking action are the same charge measurement in differential and integrated form. For a null-homologous $Y=\partial V$—equivalently, take $Y_0$ removable and $Y_1=Y$ in the sweep above—define $\operatorname{Lk}(Y,C)=V\mathbin{\boldsymbol{\cdot}}C$. With the convention $U_\varphi=\exp(i\varphi Q)$ used above, the resulting character is

$$\exp\!\left[ i(-1)^{p+1}\varphi q\,\operatorname{Lk}(Y,C) \right].$$

For $p=0$, the site’s ordinary-current convention instead names $\epsilon=-\varphi$ and writes $U(\epsilon)=\exp(-i\epsilon Q)$; for $p=1$ the phase above has the positive sign used in the Maxwell example. A crossing changes the charge; an allowed deformation through the complement does not. If $C$ is open, if $Y$ has a boundary, or if $M$ has a physical boundary, relative cycles and declared endpoint or boundary data replace this closed-cycle argument.

Small and large background transformations test different data

The formula $B\mapsto B+\mathrm d\Lambda$ is only a local description. A compact $U(1)$ higher-form background is globally a higher connection with patching data. In a de Rham description of its free part, a large transformation can shift $B$ by a closed form with $2\pi$-integral periods; torsion information is invisible to one global differential form. The partition function, rather than a chosen branch of its logarithm, must be covariant under such transformations.

This local-form/global-class distinction is explained at Freed 2002, Introduction, arXiv v2, pp. 1–2, especially p. 2, Open PDF. The theorem-level differential-cohomology construction remains outside this page.

Consequently, a local divergence equation cannot establish invariance under large transformations. A local anomaly adds a nonzero infinitesimal background variation; a global anomaly can leave every infinitesimal Ward identity intact while multiplying the partition function by a nontrivial phase. Neither is the same as explicit breaking, in which the fixed action or couplings fail to preserve the symmetry.

For a finite Abelian group—and, in particular, for an ordinary $p\geq1$ higher-form symmetry—there is usually no infinitesimal transformation to differentiate. An additive $G^{(p)}$ background is instead a $(p+1)$-cochain $b$ with

$$\delta b=0, \qquad b\longmapsto b+\delta\lambda_p,$$

away from declared background defects or higher-group mixing. The finite Ward statement is covariance of $Z[b]$ and of charged insertions. A dual-cell representative of $b$ is a network of codimension-$(p+1)$ symmetry defects; cocycle closure gives signed incidence at its junctions, but it does not construct or normalize the junction operators. A finite non-Abelian $p=0$ background instead uses flat principal-$G$ or Čech cocycle data with noncommutative transition and gauge laws; the additive cochain formula above does not describe it.

The distinction between small and large continuous transformations and the discrete cochain model is stated explicitly in Bhardwaj et al. 2024, § 4.1, arXiv v2, pp. 61–63, eqs. (4.7)–(4.17), Open PDF. The defect-network realization of finite backgrounds, including its anomaly and junction qualifications, is developed at Gaiotto et al. 2015, §§ 2–3, arXiv v2, pp. 7–13, especially eqs. (2.6)–(2.8) and (3.1)–(3.4), Open PDF.

Compact Maxwell theory makes the contact term explicit

Work on an oriented spin Lorentzian four-manifold, with the spin structure held fixed, $\theta=0$, and no physical boundary in the region used below. Let $\mathfrak a$ be a faithfully normalized compact connection,

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

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

$$\frac1{2\pi}\int_{\Sigma_2}f\in\mathbb Z$$

for every closed oriented two-cycle $\Sigma_2$ on which the bundle is restricted. Begin with no dynamical electric matter and keep magnetic backgrounds outside this calculation. The Maxwell action is

$$S_{\mathrm M}[\mathfrak a] =-\frac1{2e^2}\int_M f\wedge\star f.$$

The electric one-form current and its closed dual are

$$j_{e,2}=\frac{f}{e^2}, \qquad \widetilde j_{e,2}=\frac{\star f}{e^2}, \qquad \mathrm d\widetilde j_{e,2}=0.$$

The last equation is Maxwell’s equation in the source-free theory. The Bianchi identity $\mathrm df=0$ instead controls the magnetic current; this page does not mix the two sectors.

A local two-form electric background $B_e$ has the finite completion

$$S_{\mathrm M}[\mathfrak a;B_e] =-\frac1{2e^2} \int_M(f-B_e)\wedge\star(f-B_e).$$

It is invariant under the simultaneous local transformation

$$\mathfrak a\longmapsto\mathfrak a+\Lambda_1, \qquad B_e\longmapsto B_e+\mathrm d\Lambda_1.$$

At $B_e=0$, differentiating with respect to $B_e$ gives $\widetilde j_e=\star f/e^2$. At finite background the response is instead $\star(f-B_e)/e^2$. Expanding the square produces both the linear $B_e\wedge\star f/e^2$ coupling and the quadratic $B_e$ seagull required by finite covariance. Globally, $B_e$ and $\Lambda_1$ are compact higher-bundle data rather than arbitrary global forms.

For a closed oriented curve $C$, the Wilson line

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

transforms as

$$W_n(C)\longmapsto \exp\!\left(i n\int_C\Lambda_1\right)W_n(C).$$

The $p=1$ Ward identity therefore has no extra minus sign:

$$\begin{aligned} \mathrm d_x \left\langle \widetilde j_e(x)\,W_n(C)\,\mathcal X \right\rangle_c ={}&n\,\Delta_C(x) \\ &\times \left\langle W_n(C)\,\mathcal X\right\rangle , \end{aligned}$$

with additional contacts if $\mathcal X$ contains other charged insertions. Integrating over a three-chain that crosses $C$ once gives charge $n$; exponentiating that result gives the Wilson-line linking character.

Iqbal derives the electric current and Wilson contact at Iqbal 2025, § 4.1, pp. 25–27, especially eqs. (4.1.2)–(4.1.5), Open PDF. The invariant $(f-B_e)^2$ completion appears, with a magnetic background that we have set to zero, at Brennan and Hong 2023, § 2.2.1, arXiv v2, pp. 15–16, especially eq. (2.60) on p. 16, Open PDF; their Euclidean overall sign has been translated to the site’s Lorentzian convention. The original generalized-symmetry treatment gives the electric flux current and the compact $U(1)$ normalization at Gaiotto et al. 2015, § 4.1, arXiv v2, pp. 14–17, especially eqs. (4.1)–(4.4), Open PDF. Their current normalization differs by a conventional rescaling of the Maxwell coupling and symmetry angle; the invariant content is the closed electric flux and its Wilson charge.

Charge-N matter leaves a finite Ward identity

Now add dynamical electric matter whose nonzero charges generate exactly $N\mathbb Z$, with $N\geq2$. Assume there are no dynamical magnetic monopoles, keep magnetic backgrounds trivial, and restrict to an exact, non-anomalous, invertible group-like electric network with coherent topological junction data. The continuous electric flux form is no longer a conserved symmetry current: dynamical matter worldlines provide its sources. Every such source carries charge divisible by $N$, however, so crossing one changes $Q_e$ by a multiple of $N$.

Only the angles

$$\varphi=\frac{2\pi\alpha}{N}, \qquad \alpha\in\mathbb Z_N,$$

leave the exponentiated flux invariant across every dynamical worldline. The surviving symmetry is the exact finite group $\mathbb Z_N^{(1)}$, not a continuous symmetry with a smaller local current. Its closed surface operators can be written schematically as

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

The formula records the normalized exponentiated flux. The fundamental finite background is instead a cocycle

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

Let $r=[n]_N$ be the unscreened charge of $W_n(C)$. For closed oriented disjoint $C$ and $\Sigma$ in a linking ball, all other insertions outside the sweep, and the normalized surface removable after the sweep, fix a positive unit link of $U_1$ with $W_1$ to give $e^{2\pi i/N}$. Then

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

The same statement in a finite background is the covariance law

$$\begin{aligned} &Z[b_2+\delta\lambda_1;W_r(C)] \\ &\qquad= \exp\!\left[ \frac{2\pi i}{N}\,r\langle\lambda_1,C\rangle \right] Z[b_2;W_r(C)]. \end{aligned}$$

There is no infinitesimal derivative of this equation that produces a local Noether current. The finite Ward identity is the character itself.

Surface fusion and a declared two-in/one-out line junction are

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

The character passes the junction precisely because

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

This incidence check is necessary, but it neither constructs nor normalizes $\mathcal I$ and proves no associativity or coherence statement. A charge-$N$ field supplies an endpoint for $W_N$, so $n\sim n+N$ as screening classes; $W_1$ remains a faithful detector under the declared spectrum. Charge-one matter kills the residual electric one-form symmetry, while removing all electric matter strengthens it back to continuous $U(1)^{(1)}$.

Charge-$N$ matter and the surviving $\mathbb Z_N^{(1)}$ symmetry are described at Gaiotto et al. 2015, § 4.1, arXiv v2, p. 17, paragraph beginning “Next, we add matter fields of charge $n$,” Open PDF. The endpoint, screening quotient, and residual surface generators are derived at Bhardwaj et al. 2024, § 3.2.1, arXiv v2, pp. 29–31, especially eqs. (3.10)–(3.19), Open PDF.

Boundaries, breaking, and anomalies change the identity

The compact formula $\mathrm d\widetilde j=0$ has several distinct ways to fail or acquire extra terms.

• A charged contact is the symmetry action. The delta distribution at a declared charged support is not explicit breaking. It is how the current measures the charge.

• Explicit breaking is a bulk insertion. If fixed couplings do not transform covariantly, their variation adds a breaking operator to the Ward identity. A dynamical endpoint can similarly remove a charge sector from the faithful symmetry.

• A boundary carries flux or its own current. Stokes’ theorem adds a boundary integral. Bulk charge is conserved only after the prescribed boundary flux, boundary current, and endpoint data are included. Open charge surfaces are relative cycles, not the closed cycles used above.

• An anomaly is a background obstruction. A local anomaly adds a background-dependent variation; a global anomaly can appear only under a large transformation. Neither is repaired by declaring the symmetry explicitly broken.

• Finite covariance needs global data. A compact continuous background is a higher connection and a finite background is a cocycle. One globally written differential form misses patching and torsion sectors.

• Non-Abelian zero-form symmetry is covariant. At $p=0$ a non-Abelian background replaces $\mathrm d$ by the covariant derivative and acts on multiplets. The Abelian form derivation here should not be copied with matrices inserted ad hoc.

Detailed differential-cohomology models are outside this page. Hydrodynamic constitutive relations are also downstream: a conserved two-form current is an input to magnetohydrodynamics, not yet a constitutive theory.

Common pitfalls

Calling the dual current exact because it is closed. Conservation gives $\mathrm d\widetilde j=0$, not $\widetilde j=\mathrm dk$ globally. Nonzero periods are precisely what charge integrals detect.

Dropping the parity sign in the contact equation. With the site’s plus source $\int B\wedge\widetilde j$, graded integration by parts produces $(-1)^{p+1}$. Mixing that convention with a parity-adjusted source changes the $p=0$ contact sign.

Treating the linear source as the full background coupling. Finite covariance can require seagulls and a background-dependent response current. The Maxwell combination $f-B_e$ is the explicit check.

Using a local Ward identity to settle a global anomaly. Infinitesimal conservation does not test large transformations or torsion backgrounds.

Assigning a Noether current to a finite symmetry. A finite symmetry can have topological defects, cocycle backgrounds, characters, and anomalies without an infinitesimal current.

Confusing probing with gauging. $B$ is held fixed in $Z[B]$. Gauging requires a new sum or integral over allowed backgrounds and changes the operator spectrum.

Calling the Maxwell Bianchi identity electric conservation. In the normalization above, $\mathrm d\star f=0$ is the electric equation of motion; $\mathrm df=0$ is magnetic. Monopoles affect the latter at $\theta=0$.

Replacing a junction by a congruence. The signed label equation is only an incidence selection rule. Junction existence, normalization, local degrees of freedom, and coherence remain additional data.

Check your understanding

Use each answer outline to verify both the calculation and its domain.

1. For $p=0$ and for $p=1$ in $d=4$, list the degrees of $j$, $\widetilde j$, $B$, $\Lambda$, and $\Delta_C$.

   Answer outline

   For $p=0$, the degrees are $1$, $d-1$, $1$, $0$, and $d$. For $p=1$ in $d=4$, they are $2$, $2$, $2$, $1$, and $3$. In both cases $\mathrm d\Lambda\wedge\widetilde j$ and $\Lambda\wedge\mathrm d\widetilde j$ are top forms.

2. Derive the sign in the charged-defect contact equation from the plus-source convention.

   Answer outline

   On a closed region, $\int\mathrm d\Lambda_p\wedge\widetilde j =(-1)^{p+1}\int\Lambda_p\wedge\mathrm d\widetilde j$. Background covariance supplies $q\int_C\Lambda_p=q\int_M\Lambda_p\wedge\Delta_C$. Arbitrary $\Lambda_p$ therefore gives $\mathrm d\langle\widetilde j\mathcal O_q\rangle_c =(-1)^{p+1}q\Delta_C\langle\mathcal O_q\rangle$.

3. Let $Y_1-Y_0=\partial V$. When can $Q(Y_1)=Q(Y_0)$ fail even if the separated-point current is conserved?

   Answer outline

   It can fail when $V$ crosses a charged support, when an endpoint or explicit breaking insertion lies inside, when physical-boundary flux is omitted, or when a local/background anomaly contributes to the integrated Ward equation. A purely global anomaly can leave this local empty-sweep equality intact while obstructing covariance under a finite or large background transformation.

4. Expand the Maxwell background action and identify the response current at $B_e=0$ and at finite $B_e$.

   Answer outline

   Expanding $(f-B_e)\wedge\star(f-B_e)$ gives the Maxwell term, the linear coupling $+\int B_e\wedge\star f/e^2$, and the quadratic term $-\int B_e\wedge\star B_e/(2e^2)$. The response dual current is $\star f/e^2$ at zero background and $\star(f-B_e)/e^2$ at finite background. Omitting the quadratic term destroys finite covariance.

5. What survives when charge-$N$ matter is added to the pure Maxwell example?

   Answer outline

   The continuous electric flux is no longer a conserved symmetry current. Its exponentiated surfaces survive only at $\varphi=2\pi\alpha/N$, giving an exact $\mathbb Z_N^{(1)}$ action on $r=[n]_N$. The finite background is a $\mathbb Z_N$ two-cocycle, and the surface junction obeys the signed label condition. No local infinitesimal current or differential Ward identity survives for the finite group.

6. Explain why $Z[B+\mathrm d\Lambda]=Z[B]$ for every small transformation does not prove full background invariance.

   Answer outline

   Small transformations see only locally exact shifts and the local Ward identity. Large compact transformations can change holonomies by integral periods, and torsion backgrounds are not represented by global forms. A global anomaly can therefore multiply $Z$ by a phase while every infinitesimal divergence equation remains valid.

Continue to one-form models, gauging, and hydrodynamics

Electric and Magnetic One-Form Symmetries is the immediate next step. It develops both Maxwell sectors, matter and monopole breaking, global form, and the non-Abelian qualifications that were held outside this derivation.

Gauging a Higher-Form Symmetry promotes the background to summed-over higher-gauge data. Higher-Group Symmetry and Coupled Backgrounds explains how transformations mix rather than form a direct product. What Is an Anomaly? separates local and global obstructions from explicit breaking.

For real-time constitutive physics, Magnetohydrodynamics and Higher-Form Symmetries takes the conserved two-form current as hydrodynamic input. The theorem-level defect formulation belongs to Invertible p-Form Symmetries and Topological Defects.

References

• 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 (2023). arXiv. Open PDF, arXiv v2.

• Freed, Daniel S. “Dirac Charge Quantization and Generalized Differential Cohomology.” Surveys in Differential Geometry 7 (2002): 129–194. 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.

• Iqbal, Nabil. Lectures on Generalized Global Symmetries: Principles and Applications. SpringerBriefs in Physics. Cham: Springer Nature Switzerland, 2025. DOI. Open PDF.