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

Theory of classical electrodynamics with topologically quantized singularities as electric charges