  Topological phases of matter arise in distinct fermionic and bosonic flavors.
The fundamental differences between them are encapsulated in their rotational
symmetries - the spin. Although spin quantization is routinely encountered in
fermionic topological edge states, analogous quantization for bosons has proven
elusive. To this end, we develop the complete electromagnetic continuum theory
characterizing 2+1D topological bosons, taking into account their intrinsic
spin and orbital angular momentum degrees of freedom. We demonstrate that
spatiotemporal dispersion (momentum and frequency dependence of linear
response) captures the matter-mediated interactions between bosons and is a
necessary ingredient for topological phases. We prove that the bulk topology of
these 2+1D phases is manifested in transverse spin-1 quantization of the
photon. From this insight, we predict two unique bosonic phases - one with even
parity $C=\pm 2$ and one with odd $C=\pm 1$. To understand the even parity
phase $C=\pm 2$, we introduce an exactly solvable model utilizing non-local
optical Hall conductivity and reveal a single gapless photon at the edge. This
unidirectional photon is spin-1 helically quantized, immune to backscattering,
defects, and exists at the boundary of the $C=\pm 2$ bosonic phase and any
interface - even vacuum. The contrasting phenomena of transverse quantization
in the bulk, but longitudinal (helical) quantization on the edge is addressed
as the quantum gyro-electric effect (QGEE). We also validate our bosonic
Maxwell theory by direct comparison with the supersymmetric Dirac theory of
fermions. To accelerate the discovery of such bosonic phases, we suggest two
new probes of topological matter with broken time-reversal symmetry:
momentum-resolved electron energy loss spectroscopy and cold atom near-field
measurement of non-local optical Hall conductivity.
