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 Aharony, Seiberg, and Tachikawa 2013, §§1–2, arXiv v5 PDF.

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. The current-and-background framework is developed in Gaiotto, Kapustin, Seiberg, and Willett 2015, §§3–4, arXiv PDF; Appendix F gives the explicit electric–magnetic mixed anomaly.

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

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

Locally, compact two-form backgrounds have one-form gauge transformations

$$B_e\longmapsto B_e+d\Lambda_e, \qquad B_m\longmapsto B_m+d\Lambda_m.$$

The three-form curvatures have integral periods; globally these fields and transformations require differential-cohomology data. Under an electric-background transformation,

$$\delta_{\Lambda_e}S_5 =2\pi i\int_{M_4} \frac{\Lambda_e}{2\pi}\wedge d\!\left(\frac{B_m}{2\pi}\right),$$

which is the boundary anomaly. The $S$ map in the ordered background column $(B_m,B_e)^T$ is

$$(B_m,B_e)\longmapsto(B_e,-B_m).$$

It sends $B_e\wedge dB_m$ to $-B_m\wedge dB_e$. On a closed five-manifold, integration by parts gives

$$-\int B_m\wedge dB_e =\int B_e\wedge dB_m,$$

so the anomaly class is preserved. On a manifold with boundary the difference is precisely the descent term that must be carried by the interface. The coefficient and global refinement remain fixed by the charge lattice.

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

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

as anomaly classes after pullback. Allowed local counterterms can change cocycle representatives, descent terms, and contact-term conventions, but they cannot change the anomaly class. Equality as a de Rham polynomial is necessary and may still 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, arXiv PDF.

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

Half-gauging a finite symmetry first produces an interface

$$\mathcal D:\mathcal T\longrightarrow\mathcal T/G.$$

It becomes an internal noninvertible defect of $\mathcal T$ only after specifying an equivalence $\varphi:\mathcal T/G\simeq\mathcal T$ that includes the global data. For a finite zero-form, group-like schematic, reverse fusion may take the form

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

rather than the identity; on states, the normalized average $|G|^{-1}\sum_g\widehat{\mathcal U}_g$ is the projector. For a $3+1$-dimensional finite one-form gauging wall, the reverse fusion is instead topology-dependent. A representative schematic on the three-manifold $W$ is

$$\overline{\mathcal D}\circ\mathcal D =\frac{1}{|G|} \sum_{S\in H_2(W;G)}\eta(S),$$

where $\eta(S)$ inserts the one-form-symmetry surface Choi et al. 2022, §§2.1–2.3 and 4.2, arXiv PDF. These formulas live in different dimensions and symmetry degrees; they are not interchangeable. If one side assigns such a defect while the other assigns an invertible object, 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.

The figure below consolidates the chapter’s main global-data mechanisms while keeping two physically different scopes separate. In the top panel, inspect how the three adjoint-$\mathfrak{su}(2)$ center-charge choices form one $S$/$T$ orbit. Its $SO(3)_\pm$ labels use the spin-theory convention; on nonspin manifolds, line spins and the refined discrete-theta data must also be specified. In the middle panel, compare active charge transport with the inverse-transpose period map. Only then move to the bottom panel, where continuous source-free Maxwell backgrounds, their mixed anomaly, and the $S$ wall are displayed as a separate example.

After adjoint screening, the three $\mathfrak{su}(2)$ center-charge choices are maximal-isotropic subgroups of $\mathbb Z_2^2$. The $SO(3)_\pm$ names follow the spin-theory convention and suppress nonspin and line-spin refinements. Active $S$ and $T$ transport uses $M_\gamma=M_\Pi^{-T}$; the continuous source-free Maxwell background, anomaly, and wall panel is a separate example. The matrix entries and orbit relations are exact in the stated conventions; the layout is schematic and not to scale.

The structured charge, pairing, orbit, anomaly, and wall data state every convention, scope boundary, relation, and limitation independently of the drawing.

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. In the chapter’s typed SQCD evidence matrix, the global-background and extended-probe row is therefore marked not tested rather than being inferred from the continuous anomaly match.

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 actively on charge columns by $\gamma\mapsto M_\gamma\gamma$ and on the pair of one-form backgrounds compatibly. Show that preserving the Dirac pairing requires

$$M_\gamma^TJM_\gamma=J, \qquad J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Verify this for the $S$ matrix. Then apply $(B_m,B_e)\mapsto(B_e,-B_m)$ to the inflow integrand and show by integration by parts why the closed-five-manifold anomaly class is unchanged.

Solution

The pairing is $\gamma^TJ\gamma'$. After transformation it becomes $\gamma^TM_\gamma^TJM_\gamma\gamma'$, so equality for all charges is equivalent to $M_\gamma^TJM_\gamma=J$. For $S=J$, direct multiplication gives $S^TJS=J$. The background map sends $B_e\wedge dB_m$ to $-B_m\wedge dB_e$, and

$$d(B_m\wedge B_e)=dB_m\wedge B_e+B_m\wedge dB_e.$$

The integral of this exact term vanishes on a closed five-manifold, so $-\int B_m\wedge dB_e=\int B_e\wedge dB_m$. The compensating sign therefore preserves both the antisymmetric charge pairing and the mixed anomaly class.

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. DOI. Open PDF.
• Benini, Francesco, Clay Córdova, and Po-Shen Hsin. “On 2-Group Global Symmetries and Their Anomalies.” Journal of High Energy Physics 03 (2019): 118. DOI. Open PDF.
• Choi, Yichul, Clay Córdova, Po-Shen Hsin, Ho Tat Lam, and Shu-Heng Shao. “Non-Invertible Duality Defects in 3+1 Dimensions.” Physical Review D 105 (2022): 125016. DOI. Open PDF.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF.