By contrast, the Maxwell equations describing electromagnetism can be understood to be the Hodge equations of a principal U(1)-bundle or circle bundle π π : P → → M with fiber U(1). That is, the electromagnetic field F is a harmonic 2-form in the space Ω Ω 2 ( M ) of differentiable 2-forms on the manifold M . In the absence of charges and currents, the free-field Maxwell equations are