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, 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.