Bosonization of non-relativstic fermions in 2-dimensions and collective field theory

We revisit bosonization of non-relativistic fermions in one space dimension.Our motivation is the recent work on bubbling half-BPS geometries by Lin, Luninand Maldacena (hep-th/0409174). After reviewing earlier work on exactbosonization in terms of a noncommutative theory, we derive an action for thecollective field which lives on the droplet boundaries in the classical limit.Our action is manifestly invariant under time-dependent reparametrizations ofthe boundary. We show that, in an appropriate gauge, the classical collectivefield equations imply that each point on the boundary satisfies Hamilton'sequations for a classical particle in the appropriate potential. For theharmonic oscillator potential, a straightforward quantization of this actioncan be carried out exactly for any boundary profile. For a finite number offermions, the quantum collective field theory does not reproduce the results ofthe exact noncommutative bosonization, while the latter are in completeagreement with the results computed directly in the fermi theory.Comment: references added and typos corrected; 21 pages, 3 figures, eps