  Recent numerical simulations of flat band models with interactions which show
clear evidence of fractionalized topological phases in the absence of a net
magnetic field have generated a great deal of interest. We provide an
explanation for these observations by showing that the physics of these systems
is the same as that of conventional fractional quantum Hall phases in the
lowest Landau level under certain ideal conditions which can be specified in
terms of the Berry curvature and the Fubini study metric of the topological
band. In particular, we show that when these ideal conditions hold, the density
operators projected to the topological band obey the celebrated $W_{\infty}$
algebra. Our approach provides a quantitative way of testing the suitability of
topological bands for hosting fractionalized phases.
