Monopoles, Dirac Strings and Generalised Symmetries

Dirac's formulation of magnetic monopoles is shown to be equivalent to Maxwell theory coupled to 2-form gauge fields so that it has a local 1-form symmetry, with the 2-form gauge fields given in terms of the 2-form current densities associated with the Dirac strings. The field equations of Dirac's theory do not depend on the positions of the Dirac strings provided that they do not intersect the worldlines of any electrically charged particles; this constraint is called the Dirac veto. It is shown that Dirac's action is independent of the positions of the Dirac strings and that this corresponds a local 1-form symmetry. The electric and magnetic 1-form symmetries have a mixed anomaly, and the Dirac veto is shown to correspond to a restriction to gauge transformations for which the anomaly vanishes. The extension to $p$-form gauge fields in d-dimensions coupled to charged branes is discussed, together with the possibility of cancelling the anomaly by embedding in a higher-dimensional theory and so avoiding the veto.