# How do the vector and scalar potentials transform under electromagnetic duality trnasfotmation?

Maxwell equations are invariant under the duality transformation. The electric and magnetic fields are defined in terms of these potentials. How do these potentials transform under duality?

On the level of the field strength tensor $$F$$, duality is the map $$F\mapsto {\star}F$$, where $${\star}$$ is the Hodge dual. Maxwell's equations in vacuum are $$\mathrm{d}F = 0 \quad \text{and} \quad \mathrm{d}{\star}F = 0$$ and by the Poincaré lemma we can therefore locally find 1-forms $$A$$ and $$A'$$ such that $$\mathrm{d}A = F$$ and $$\mathrm{d}A' = {\star}F$$. The components of $$A$$ - the four-potential - are the usual scalar and vector components of electrodynamics. Since the duality exchanges $$F$$ and $${\star}F$$, it acts on the potential $$A$$ by exchanging it with $$A'$$.