Maxwell's equations are part of a gauge theory, implying the existence of a certain group of symmetries. In this case the group is abelian, namely U(1). This is also known as the circle group, i.e. the group of complex numbers of magnitude 1 under multiplication. Maxwell's equations are described in terms of a vector potential or connection 1-form A (depending on who you ask - let's call it a 1-form). Then there is an associated curvature F = dA where "d" is the antisymmetrised derivative on differential forms. (You could also think of this as an antisymmetric (0,2) tensor field). Maxwell's equations are: dF = 0 (vacuous as dd = 0) and d*F = 4pi*j, where "j" is the electromagnetic current and "*" is the Hodge star. The six components of F may be thought of, viewed in terms of their spatial transformations once we pick a preferred time direction, as a vector E (the electric field) and a pseudovector B (the magnetic field). These are related by a duality transformation. We are free to change our particles by multiplying by a phase from U(1), which corresponds to transforming A by a gradient (i.e. A -> A + df). This gives us an abelian Yang-Mills theory. Nonabelian gauge groups, like SU(2), lead to nonabelian Yang-Mills theories which look roughly the same but with commutators.