  We revisit bosonization of non-relativistic fermions in one space dimension.
Our motivation is the recent work on bubbling half-BPS geometries by Lin, Lunin
and Maldacena (hep-th/0409174). After reviewing earlier work on exact
bosonization in terms of a noncommutative theory, we derive an action for the
collective field which lives on the droplet boundaries in the classical limit.
Our action is manifestly invariant under time-dependent reparametrizations of
the boundary. We show that, in an appropriate gauge, the classical collective
field equations imply that each point on the boundary satisfies Hamilton's
equations for a classical particle in the appropriate potential. For the
harmonic oscillator potential, a straightforward quantization of this action
can be carried out exactly for any boundary profile. For a finite number of
fermions, the quantum collective field theory does not reproduce the results of
the exact noncommutative bosonization, while the latter are in complete
agreement with the results computed directly in the fermi theory.
