Generalized Symmetries, Global Forms, and Anomalies under Duality

A duality must transport every symmetry that acts on genuine operators, including higher-form and higher-group symmetries, together with their background fields and anomalies. Matching continuous Lie algebras and local operators can miss different global forms, different line spectra, or a topological response that obstructs equivalence.

Required background. Electric–magnetic charge lattices and global form supplies the extended-operator data, while anomaly polynomials and inflow supplies the background-field test. Helpful background. Higher-group symmetry explains coupled background transformations.

Symmetry is a background-field functor

The most robust description of a global symmetry assigns a partition function to every allowed background:

$$(M,\mathcal B)\longmapsto Z_{\mathcal T}[M;\mathcal B].$$

The collection $\mathcal B$ may include an ordinary connection $A$, a two-form gauge field $B_2$ for a one-form symmetry, higher-degree fields, spin or spin-$c$ structure, and discrete cocycles. Gauge transformations can mix these fields when the symmetry is a higher group.

A duality with background map $f$ should satisfy

$$Z_A[M;\mathcal B_A] =e^{iC[M;\mathcal B_A]} Z_B[M;f(\mathcal B_A)]$$

for the backgrounds included in the claim. The phase $C$ is an allowed local counterterm. Its quantized or fractional part matters: an arbitrary background-dependent phase cannot be used to erase an anomaly.

This equation automatically tests more than flat-space Ward identities. Nontrivial bundles probe global form, torsion charges, contact terms, and symmetry-protected phases.

Genuine operators determine the global form

For gauge algebra $\mathfrak g$, the global group and matter representations determine which Wilson, ’t Hooft, and dyonic lines are genuine. Their charges form an allowed subset $L$ of an electric–magnetic lattice, constrained by mutual locality:

$$\langle\gamma,\gamma'\rangle\in\mathbb Z, \qquad \gamma,\gamma'\in L.$$

Screening identifies charges that differ by dynamical matter. Choosing a maximal mutually local set fixes a polarization and, in many cases, the global form and a discrete theta angle. A duality transformation must map

$$(\Gamma,\langle\ ,\ \rangle,L) \longmapsto (\Gamma',\langle\ ,\ \rangle',L')$$

integrally. Matching only $\mathfrak g$ leaves $L$ undetermined.

For example, $SU(N)$ admits fundamental Wilson lines, while $PSU(N)$ does not; the latter admits different magnetic bundles and discrete theta choices. Electric–magnetic duality can exchange these theories rather than act within one of them. The correct duality orbit is an orbit of complete global theories, as illustrated systematically in Aharony, Seiberg, and Tachikawa 2013, §§1–2.

Higher-form currents and backgrounds

In four-dimensional source-free Maxwell theory,

$$j^{(2)}_m=\frac{F}{2\pi}, \qquad j^{(2)}_e=\frac{G}{2\pi}$$

are conserved two-form currents, $dj_m^{(2)}=dj_e^{(2)}=0$, in the absence of magnetically or electrically charged matter. They generate magnetic and electric one-form symmetries acting on line operators. Dynamical charges break the corresponding continuous symmetry to a subgroup or remove it. This current-and-background formulation, including mixed anomalies, is developed in Gaiotto, Kapustin, Seiberg, and Willett 2015, §§3–4.

Couple backgrounds $B_e$ and $B_m$ to these currents. Their simultaneous gauging can have a mixed anomaly represented schematically by five-dimensional inflow

$$S_{5}=2\pi i\int_{Y_5} \frac{B_e}{2\pi}\wedge d\!\left(\frac{B_m}{2\pi}\right), \qquad \partial Y_5=M_4,$$

with the precise differential-cohomology refinement and coefficient fixed by the charge lattice. Under $S$ duality, electric and magnetic backgrounds exchange, with a sign required by the symplectic pairing. The inflow class must transform accordingly.

This example shows why a classical rotation of $(F,*F)$ is insufficient. The background fields, their periods, and their anomaly define the quantum symmetry being exchanged.

Transporting an anomaly polynomial

For continuous symmetries in even spacetime dimension $d$, an anomaly polynomial $I_{d+2}$ encodes perturbative anomalies. If the duality maps backgrounds by $f$, then

$$I^{A}_{d+2}(\mathcal F_A) =I^{B}_{d+2}(f(\mathcal F_A))$$

in the appropriate integral cohomology class, up to changes generated by allowed local counterterms. Equality as a de Rham polynomial is necessary but may miss torsion and global anomalies.

For a four-dimensional theory with an abelian symmetry, terms can include

$$I_6 =\frac{k_{ABC}}{6} \frac{F_A}{2\pi}\frac{F_B}{2\pi}\frac{F_C}{2\pi} -\frac{k_A}{24} \frac{F_A}{2\pi}p_1(T).$$

The coefficients are traces over chiral fermion charges in a UV Lagrangian, but the resulting ’t Hooft anomalies are RG invariant. Under a duality, first translate the symmetry basis—including accidental currents and quotient identifications—then compare coefficients. Matching numbers in mismatched bases is meaningless.

Higher groups mix the transformation laws

An ordinary symmetry $G_0$ and a one-form symmetry $G_1$ need not form a direct product. A Postnikov class can require a background constraint of the schematic form

$$dB_2=\beta(A),$$

or a mixed transformation

$$B_2\longmapsto B_2+d\Lambda_1+\omega_2(A,g).$$

Then $A$ and $B_2$ cannot be mapped independently. A duality that correctly matches ordinary currents but sends $B_2$ as if it were closed can violate the coupled gauge law and hence the symmetry itself. The constituent groups, Postnikov class, and coupled background transformations of a 2-group are stated explicitly in Benini, Córdova, and Hsin 2019, §2.2 and §§3–4.

Operationally, record the symmetry’s classifying space, background constraints, and action on every extended operator. Pull back the entire structure under the duality map, not only the individual group names.

Noninvertible images

Gauging can turn an invertible symmetry or duality wall into a noninvertible defect. Its fusion may be

$$\mathcal D\times\overline{\mathcal D} =\sum_{g\in G}\mathcal U_g$$

rather than the identity. The sum records a condensation or projection over symmetry defects. If one side of a proposed duality has this fusion rule while the other side assigns an invertible line, the extended-operator dictionary fails even if local correlators match.

Anomalies of noninvertible symmetries are encoded more generally by categorical and defect data, but the same principle survives: fusion, junctions, background couplings where defined, and obstruction classes must be transported.

A global-data comparison table

Object	Source theory	Target theory	Required relation
Faithful ordinary symmetry	$G_A/K_A$	$G_B/K_B$	Isomorphism after operator kernels are removed
Higher-form symmetry	group and charged defects	group and charged defects	Degree-preserving map or declared exchange under dimensional operations
Charge lattice	$\Gamma_A,L_A$	$\Gamma_B,L_B$	Integral pairing-preserving map of genuine subsets
Background fields	cocycles/connections and constraints	corresponding fields	Compatible map of gauge transformations and bundles
Local anomaly	$I_{d+2}^A$	$I_{d+2}^B$	Equality after pullback and allowed counterterms
Global/torsion anomaly	bordism or inflow class	corresponding class	Equality in the relevant generalized cohomology/bordism group
Defect fusion	products and junction spaces	images	Monoidal compatibility, including sums and projectors

The table prevents a local anomaly match from standing in for the global comparison.

Common pitfalls

Matching symmetry algebras but not faithful groups. Quotients by centers change bundles, representations, and anomalies.

Comparing only perturbative anomaly polynomials. Torsion and global anomalies can vanish in de Rham cohomology yet distinguish theories.

Mapping higher-form backgrounds independently in a higher group. Their gauge transformations are coupled; the Postnikov data must also match.

Exercises

Let an $SL(2,\mathbb Z)$ matrix act on charge columns by $\gamma\mapsto M\gamma$ and on the pair of one-form backgrounds compatibly. Show that preserving the Dirac pairing requires

$$M^TJM=J, \qquad J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Verify this for the $S$ matrix and explain the implication for the mixed electric–magnetic anomaly.

Solution

The pairing is $\gamma^TJ\gamma'$. After transformation it becomes $\gamma^TM^TJM\gamma'$, so equality for all charges is equivalent to $M^TJM=J$. For $S=J$, direct multiplication gives $S^TJS=J$. Therefore $S$ exchanges the electric and magnetic backgrounds with the compensating sign that preserves their antisymmetric pairing. The mixed inflow term is carried into the same anomaly class rather than into an unrelated response.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. arXiv:1305.0318.
• Benini, Francesco, Clay Córdova, and Po-Shen Hsin. “On 2-Group Global Symmetries and Their Anomalies.” Journal of High Energy Physics 03 (2019): 118. arXiv:1803.09336.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. arXiv:1412.5148.