New nonlocal symmetries and conservation laws are derived for Maxwell's
equations using a covariant system of joint vector potentials for the
electromagnetic tensor field and its dual. A key property of this system, as
well as of this class of new symmetries and conservation laws, is their
invariance under the duality transformation that exchanges the electromagnetic
field with its dual. The nonlocal symmetries of Maxwell's equations come from
an explicit classification of all symmetries of a certain geometric form
admitted by the joint potential system in Lorentz gauge. In addition to scaling
and duality-rotation symmetries, and the well-known Poincare and dilation
symmetries which involve homothetic Killing vectors, the classification yields
new geometric symmetries involving Killing-Yano tensors related to
rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's
equations are constructed from these geometric symmetries by applying a formula
that uses the joint potentials and directly generates conserved currents from
any (local or nonlocal) symmetries of Maxwell's equations. This formula is
shown to arise through a series of mappings that relate, respectively,
symmetries/adjoint-symmetries of the joint potential system and
adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived
as by-products of the study of cohomology of closed 1-forms and 2-forms locally
constructed from the electromagnetic field and its derivatives to some finite
order for all solutions of Maxwell's equations. The only nontrivial cohomology
is shown to consist of the electromagnetic field (2-form) itself as well as its
dual (2-form), and this 2-form cohomology is killed by the introduction of
corresponding potentials.