  We provide evidence for conjectural dualities between nonrelativistic
Chern-Simons-matter theories and theories of (fractional, nonAbelian) quantum
Hall fluids in $2+1$ dimensions. At low temperatures, the dynamics of
nonrelativistic Chern-Simons-matter theories can be described in terms of a
nonrelativistic quantum mechanics of vortices. At critical coupling, this may
be solved by geometric quantisation of the vortex moduli space. Using
localisation techniques, we compute the Euler characteristic
${\chi}(\mathcal{L}^\lambda)$ of an arbitrary power $\lambda$ of a quantum line
bundle $\mathcal{L}$ on the moduli space of vortices in $U(N_c)$ gauge theory
with $N_f$ fundamental scalar flavours on an arbitrary closed Riemann surface.
We conjecture that this is equal to the dimension of the Hilbert space of
vortex states when the area of the metric on the spatial surface is
sufficiently large. We find that the vortices in theories with $N_c = N_f =
\lambda$ behave as fermions in the lowest nonAbelian Landau level, with
strikingly simple quantum degeneracy. More generally, we find evidence that the
quantum vortices may be regarded as composite objects, made of dual anyons. We
comment on potential links between the dualities and three-dimensional mirror
symmetry. We also compute the expected degeneracy of local Abelian vortices on
the $\Omega$-deformed sphere, finding it to be a $q$-analog of the undeformed
case.
