  We introduce the concept of a photonic Dirac monopole, appropriate for
photonic crystals, metamaterials and 2D materials, by utilizing the
Dirac-Maxwell correspondence. We start by exploring vacuum where the reciprocal
momentum space of both Maxwell's equations and the massless Dirac equation
(Weyl equation) possess a magnetic monopole. The critical distinction is the
nature of magnetic monopole charges, which are integer valued for photons but
half-integer for electrons. This inherent difference is directly tied to the
spin and ultimately connects to the bosonic or fermionic behavior. We also show
the presence of photonic Dirac strings, which are line singularities in the
underlying Berry gauge potential. While the results in vacuum are intuitively
expected, our central result is the application of this topological
Dirac-Maxwell correspondence to 2D photonic (bosonic) materials, as opposed to
conventional electronic (fermionic) materials. Intriguingly, within dispersive
matter, the presence of photonic Dirac monopoles is captured by nonlocal
quantum Hall conductivity - i.e. a spatiotemporally dispersive gyroelectric
constant. For both 2D photonic and electronic media, the nontrivial topological
phases emerge in the context of massive particles with broken time-reversal
symmetry. However, the bulk dynamics of these bosonic and fermionic Chern
insulators are characterized by spin-1 and spin-1/2 skyrmions in momentum
space, which have fundamentally different interpretations. This is exemplified
by their contrasting spin-1 and spin-1/2 helically quantized edge states. Our
work sheds light on the recently proposed quantum gyroelectric phase of matter
and the essential role of photon spin quantization in topological bosonic
phases.
