Anomaly of the Electromagnetic Duality of Maxwell Theory

Every physicist knows that classical electromagnetism is described by Maxwell's equations, and that it is invariant under the electromagnetic duality S: (E, B) → (B, −E). The properties of the electromagnetic duality in the quantum theory, represented by an SL(2, Z) group acting on the lattice Z^2 of the electric and magnetic charges, might not be as well known to physicists in general. This is particularly true when going around a nontrivial path in the spacetime results in a duality transformation. In this work, we uncover a feature of the Maxwell theory and its duality symmetry in such a situation, namely that it has a quantum anomaly. We find that the anomaly of this system in a particular formulation is 56 times that of a Weyl fermion. The interpretation of this result is twofold: one is by using the bulk symmetry-protected topological phase in 4+1 dimensions characterizing the anomaly, and the other is by considering the properties of some (5+1)-dimensional superconformal field theory. Our result reproduces, as a special case, the known anomaly of the all-fermion electrodynamics—a version of the Maxwell theory where all particles of odd charge are fermions—discovered in the last few years.