The Derived Duality

Electromagnetic duality — the exchange of electric and magnetic fields — is one of the oldest symmetries in physics. For Maxwell’s equations in vacuum, swapping E and B fields produces another valid solution. Generalized Maxwell theories extend this to p-form gauge fields in arbitrary dimensions: a p-form theory in d dimensions is dual to a (d-p-2)-form theory.

This paper shows the duality is not just a physical symmetry — it’s a precise equivalence in derived differential geometry.

The moduli space of a generalized Maxwell theory, formulated through the Batalin-Vilkovisky formalism and differential cohomology, carries a derived geometric structure. Dirac charge quantization — the requirement that electric and magnetic charges come in integer multiples — is encoded naturally in this framework as a condition on the differential cohomology classes. The charge-quantized moduli spaces of dual theories are equivalent as derived geometric objects.

The equivalence is stronger than a mere matching of observables. Derived geometry encodes not just the solutions but the symmetries, the gauge redundancies, and the higher coherences between them. The duality maps all of this structure faithfully, not just the physical content.

Compactification — reducing a theory on a higher-dimensional space to a lower-dimensional one — is computed as a pushforward of sheaves of cochain complexes. The duality survives compactification because it’s an equivalence of the full derived geometric data, not just of particular solutions.

Electromagnetic duality was discovered as a trick for solving equations. It turns out to be a statement about the geometry of the spaces of solutions — and that geometry is precise enough to be an equivalence.