Electric and magnetic one-form symmetries

Take a compact U(1) connection with curvature periods equal to integer multiples of two pi and unit electric Wilson charge. Work in oriented Lorentzian D equal to four with hbar and light speed one, no theta term, and the standard Maxwell coefficient one over twice the squared coupling. Away from insertions, both the curvature and its Hodge dual multiplied by inverse coupling squared are closed two-forms. Their surface integrals define magnetic and electric charges with different normalization factors.

Call these closed two-forms dual-current forms so that no implicit Lorentzian double-Hodge sign is hidden in a current label. The orientation and Hodge convention are fixed so a unit electric source has electric flux one and a unit monopole has magnetic curvature flux two pi. The associated symmetry angles have period two pi. The normalization can be changed, but then charge labels, background periods, and the coupling coefficients must change together.

Exponentiating either flux integral over a closed oriented surface gives a symmetry defect. When the surface links a line carrying the matching integer charge, the correlator acquires the corresponding angle times charge phase. The electric symmetry acts on Wilson lines, while the magnetic symmetry acts on magnetic disorder lines. Deforming a surface through a region free of charged insertions changes no flux integral by Stokes' theorem.

With electric sources the Maxwell equation contains their current, so the electric flux form is no longer closed across their worldlines. A monopole similarly modifies the Bianchi identity and spoils magnetic closure. The presence of external probe insertions is consistent with a symmetry Ward identity: they are its charged operators. Allowing the same charges as dynamical endpoints in the theory can instead explicitly break the corresponding one-form symmetry. Those two roles of a line must be distinguished.

If the only dynamical electric charges are multiples of an integer N, a symmetry angle remains exact when it acts trivially on every allowed endpoint. This leaves the N discrete rotations inside the original electric U(1). If several charges are available, the residual subgroup is determined by their greatest common divisor. A unit dynamical charge removes the entire electric one-form symmetry, even when that particle is heavy; at energies below its mass a larger symmetry may only be emergent or approximate.

The same arithmetic applies independently to a specified magnetic charge lattice, provided electric–magnetic consistency and global data have been declared. This bookkeeping does not by itself establish confinement, a Higgs phase, or a massless photon. In particular, a compact gauge theory with nonperturbative monopoles cannot be called exactly source-free simply because its classical equations were written without a matter term. Exact ultraviolet symmetry, approximate low-energy conservation, and spontaneous realization of a remaining symmetry are separate questions.