Invertible p-Form Symmetries and Topological Defects

An invertible $p$-form symmetry in $d$ spacetime dimensions is represented by topological operators on codimension-$(p+1)$ submanifolds. Their fusion realizes a group, their inverse is implemented by orientation reversal or the inverse label, and their action on a $p$-dimensional charged operator is measured by linking. Topological deformability is not enough: if a defect has no inverse under fusion, it belongs to a noninvertible categorical symmetry rather than a $p$-form symmetry group.

Required background. Defects on Stratified Spacetimes supplies composition and isotopy data. Higher-Form Symmetry from Operators and Linking supplies the physical symmetry operators. Support, Codimension, and Operator Data fixes the support dimension of extended operators. Helpful background. Higher-Form Currents, Charges, Backgrounds, and Ward Identities connects the defect action to conserved currents and backgrounds.

Topological representatives and group law

Let $A$ be an abelian group. For $g\in A$, write $U_g(M^{d-p-1})$ for a symmetry defect supported on a closed oriented submanifold of codimension $p+1$. “Topological” means that correlation functions are unchanged under deformations of $M$ that do not cross charged insertions. Fusion on parallel supports satisfies

$$U_g(M)U_h(M)=U_{g+h}(M), \qquad U_0(M)=\mathbf1, \qquad U_g(M)^{-1}=U_{-g}(M).$$

For $p>0$, locality forces the group to be abelian: two defects may be exchanged in the transverse space without crossing a charged operator. Gaiotto, Kapustin, Seiberg, and Willett state the codimension, fusion, and linking formulation in Gaiotto et al. 2015, §3, printed pp. 7–11 (PDF).

A charged operator $W_q(C^p)$ carries a character $\chi_q\in\widehat A$. If $M$ links $C$ once, moving the topological defect around the operator gives

$$U_g(M)W_q(C)=\chi_q(g)\,W_q(C).$$

For several components, the exponent is multiplied by the oriented linking number. This is a categorical action by invertible endomorphisms: fusion of $U_g$ and $U_h$ multiplies their characters, and the inverse defect produces the inverse phase.

A background field for a finite $p$-form symmetry is a $(p+1)$-cocycle, modulo higher gauge transformations. The cocycle condition is the network version of defect conservation: symmetry sheets may meet only at junctions whose labels sum to zero. For continuous symmetries one uses a $(p+1)$-form connection, including its integral periods and global differential-cohomology data when those affect observables.

Four-dimensional one-form example

Consider a four-dimensional gauge theory with a $\mathbb Z_N$ electric one-form symmetry. Its topological operators are surfaces $U_m(\Sigma)$, $m\in\mathbb Z_N$, and a Wilson line $W_q(C)$ has center charge $q\in\mathbb Z_N$. With unit linking,

$$U_m(\Sigma)W_q(C) =\exp\!\left(\frac{2\pi i mq}{N}\right)W_q(C),$$

and for arbitrary disjoint supports the exponent is multiplied by $\operatorname{Link}(\Sigma,C)$. Fusion gives $U_mU_n=U_{m+n}$, while $U_{-m}$ is the inverse. The phase passes three independent checks: it is invariant under $m\mapsto m+N$, it is multiplicative under fusion, and it becomes one for center-neutral lines.

Equivalently, couple the theory to a background two-form $\mathbb Z_N$ gauge field $B$. A Wilson line of charge $q$ is not invariant under a one-form gauge transformation unless it is dressed by a surface or transforms by the corresponding character. This background-field formulation and the defect-linking formulation are equivalent only after charge normalization and global-form choices are matched; Gaiotto et al. 2015, §§3–4, printed pp. 7–18 (PDF) develops both and exhibits their anomaly and gauging consequences.

The exact first application is developed by Higher-Form Symmetry from Operators and Linking. The mathematical contribution here is the invertible defect action and its coherence. Whether the symmetry is broken, gauged, anomalous, or visible in a particular spectrum remains a physical question for that treatment.

Distinguishing nearby structures

An ordinary symmetry has $p=0$: its topological defects have codimension one and act on local operators. A one-form symmetry has codimension-two defects and acts on lines. A higher group is not merely a product of groups at several form degrees; its junction law includes a Postnikov class that mixes the corresponding backgrounds. Finally, a noninvertible topological defect may fuse to a direct sum rather than to one defect label.

The decisive adversarial test is to propose a deformable surface $D$ with

$$D\otimes D\cong\mathbf1\oplus X.$$

No object $D^{-1}$ can satisfy $D\otimes D^{-1}\cong\mathbf1$ if fusion dimensions are positive and $X\ne0$. Thus $D$ can still define a topological operator and a categorical action, but it cannot be an element of a one-form symmetry group. Calling it “group-like” would incorrectly erase its additional fusion channel.

Global geometry supplies another check. A symmetry surface that bounds a three-chain can be shrunk away only when the chain crosses no charged line and no background flux. On a manifold with torsion cycles, a flat background can have nontrivial holonomy even though its differential-form curvature vanishes. The correct charge is therefore a character of the global higher-form gauge field, not merely an integral of a locally defined form. This is why the gauge group’s global form and the allowed line lattice must accompany the linking phase.

Exercises

For $N=4$, determine which Wilson charges are invisible to the surface $U_2$.

Solution

The linking phase is $e^{2\pi i(2q)/4}=(-1)^q$. It is one precisely for $q=0,2$ modulo $4$. The surface therefore detects the quotient of the charge group by its even subgroup, not every $\mathbb Z_4$ charge.

References

• 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.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 2014, no. 4 (2014): 001. DOI; Open PDF.