  It is known that the localization length scaling of noninteracting electrons
near the quantum Hall plateau transition can be described in a theory of the
bosonic density operators, with no reference to the underlying fermions. The
resulting ``Liouvillian'' theory has a $U(1|1)$ global supersymmetry as well as
a hierarchy of geometric conservation laws related to the noncommutative
geometry of the lowest Landau level (LLL). Approximations to the Liouvillian
theory contain quite different physics from standard approximations to the
underlying fermionic theory. Mean-field and large-N generalizations of the
Liouvillian are shown to describe problems of noninteracting bosons that
enlarge the $U(1|1)$ supersymmetry to $U(1|1) \times SO(N)$ or $U(1|1) \times
SU(N)$.
  These noninteracting bosonic problems are studied numerically for $2 \leq N
\leq 8$ by Monte Carlo simulation and compared to the original N=1 Liouvillian
theory. The $N>1$ generalizations preserve the first two of the hierarchy of
geometric conservation laws, leading to logarithmic corrections at order 1/N to
the diffusive large-N limit, but do not preserve the remaining conservation
laws. The emergence of nontrivial scaling at the plateau transition, in the
Liouvillian approach, is shown to depend sensitively on the unusual geometry of
Landau levels.
