  We formulate a theory of classical electrodynamics where the only admissible
electric charges are topological singularities in the electromagnetic field,
and charge quantization is accounted by the Chern theorem, such that Dirac
magnetic monopoles are not needed. The theory allows positive and negative
charges of equal magnitude, where the sign of the charge corresponds to the
chirality of the topological singularity. Given the trajectory ${\bf w}(t)$ of
the singularity, one can calculate electric and magnetic fields identical to
those produced by Maxwell's equations for a moving point charge, apart from a
multiplicative constant factor related to electron charge and vacuum
permittivity. The theory is based on the relativistic Weyl equation in
frequency-wavevector space, with eigenstates comprising the position, velocity,
and acceleration of the singularity, and eigenvalues defining the retarded
position of the charge. From the eigenstates we calculate the Berry connection
and the Berry curvatures, and identify the curvatures as electric and magnetic
fields.
