Exchangeability makes p-form symmetry groups Abelian and simple charges characters

Let $d\ge 2$, $1\le p<d$, and let $\mathcal{T}$ be a selected model with the following raw group-like defect data. Let $G$ be a locally compact Hausdorff topological group, with no Abelianity assumption. On every admitted closed codimension-$(p+1)$ support, supply invertible topological endodefects $U_g$ with total parallel fusion $U_g\star U_h=U_{gh}$, transparent identity and inverses. Supply crossing bijections ${\rho }_g$ on complete typed insertion data and background/boundary bijections ${\beta }_g$, preserving types, supports, algebraic relations and admissibility, with ${\rho }_{gh}={\rho }_g\circ {\rho }_h$, ${\beta }_{gh}={\beta }_g\circ {\beta }_h$ and identity actions at $e$. Require covariance $C_{\mathcal{T}}({\beta }_g(b);{\rho }_g(I))=C_{\mathcal{T}}(b;I)$ for every complete admitted insertion list $I$ and background $b$, including decorated networks, and invariance of each declared correlator under admissible isotopies transporting all defect data. Require effectiveness: equal crossing actions on the complete domain, equal background maps, and equal actions in all admitted correlators imply equal group labels. These are the model, topologicality, fusion, crossing, covariance and effectiveness data of the preceding symmetry definition, with its extra positive-form Abelianity restriction deliberately withheld. Suppose that for every $g,h\in G$ and every allowed configuration of remaining insertions and fixed data, the model admits two parallel copies of their codimension-$(p+1)$ symmetry supports inside a contractible neighborhood with a disk spanned by two independent normal directions, and an admissible isotopy exchanging their order without intersecting any other insertion or changing boundary or model data. Require that the model-supplied transport along this isotopy identifies the two composite crossing bijections on all complete typed insertion data and the two composite background maps; that is, it identifies ${\rho }_g\circ {\rho }_h$ with ${\rho }_h\circ {\rho }_g$ and ${\beta }_g\circ {\beta }_h$ with ${\beta }_h\circ {\beta }_g$. Then $G$ is Abelian, so these data satisfy also the positive-form group restriction in that definition.

For the separate character conclusion, assume additionally that the crossing action has a unitary Hilbert-space realization: there are a complex Hilbert space $\mathcal{H}$ and a group homomorphism $V:G\to U(\mathcal{H})$, with $V_e=I$ and $V_{gh}=V_g{V}_h$, such that crossing $U_g$ acts on bounded linear operators by $O\mapsto V_{g}O{V}_g^{-1}$. Let $O\in B(\mathcal{H})$ be a nonzero genuine $p$-dimensional operator, where $B(\mathcal{H})$ is the space of bounded linear operators with norm $\mathrm{\parallel }O\mathrm{\parallel }:={\sup }_{\mathrm{\parallel }\psi \mathrm{\parallel }=1}\mathrm{\parallel }O\psi \mathrm{\parallel }<\mathrm{\infty }$. Call $O$ simple for this result if crossing each $U_g$ sends it to a scalar multiple ${\chi }_O(g)O$, and assume the map $g\mapsto {\chi }_O(g)\in \mathbb{C}$ is continuous. Then ${\chi }_O:G\to U(1)$ is a continuous character, so the charge of $O$ belongs to the Pontryagin dual $\widehat{G}={\mathrm{Hom}}_{\mathrm{c}\mathrm{t}\mathrm{s}}(G,U(1))$. Neither conclusion is asserted for $p=0$; the character conclusion also makes no claim for nonscalar crossing actions.

Model, defect, crossing, effectiveness and character conventions; its positive-form Abelian restriction is the conclusion proved here

Gaiotto, Kapustin, Seiberg, and Willett, §3, PDF pp. 11–12

Bhardwaj et al., §§2.2.1–2.2.2, PDF pp. 12–20

The diagram records the exchange isotopy and the same-support fusion maps. Its top arrow is equality of correlators under [A1]; the proof above, not the diagram, uses effectiveness to infer equality of group labels.

The individual group conclusion is subject to the specified exchange isotopy. It does not assert that electric and magnetic flux operators commute jointly on every spatial topology. In Maxwell theory, torsion cycles can give a torsion-pairing commutator between the electric and magnetic flux operators, a mixed relation between distinct symmetry actions.

Gaiotto, Kapustin, Seiberg, and Willett, Appendix F, PDF pp. 70–71