  Here, we reveal our recent progress on a geometrical approach of quantum
physics and topological crystals linking with Dirac magnetic monopoles and
gauge fields through classical electrodynamics. The Bloch sphere of a quantum
spin-1/2 particle acquires an integer topological charge in the presence of a
radial magnetic field. We show that global topological properties are encoded
from the poles of the surface allowing a correspondence between smooth fields,
metric and quantum distance with the square of the topological number. The
information is transported from each pole to the equatorial plane on a thin
Dirac string. We develop the theory, "quantum topometry" in space and time, and
present applications on transport from a Newtonian approach, on a quantized
photo-electric effect from circular dichroism of light towards topological band
structures of crystals. Edge modes related to topological lattice models are
resolved analytically when deforming the sphere or ellipse onto a cylinder.
Topological properties of the quantum Hall effect, quantum anomalous Hall
effect and quantum spin Hall effect on the honeycomb lattice can be measured
locally in the Brillouin zone from light-matter coupling. The formalism allows
us to include interaction effects from the momentum space. Interactions may
also result in fractional entangled geometry within the curved space. We
develop a relation between entangled wavefunction in quantum mechanics,
coherent superposition of geometries, a way to one-half topological numbers and
Majorana fermions. We show realizations in topological matter. We present a
link between axion electrodynamics, topological insulators on a surface of a
cube and the two-spheres' model via merons.
