In this talk I will discuss fractional Chern insulators with
emphasis on the analogy with more conventional continuum
Landau level physics — and on aspects that are qualitatively
new in the lattice setting such as Berry curvature
fluctuations, competing instabilities and novel collective
states of matter emerging in bands with higher Chern number.
I will also explain how the lattice setting naturally allows
for exotic extrinsic wormhole-like twist defects (aka
“genons”) that effectively increase the genus of space.