  We use the recently developed tools for an exact bosonization of a finite
number $N$ of non-relativistic fermions to discuss the classic Tomonaga
problem. In the case of noninteracting fermions, the bosonized hamiltonian
naturally splits into an O$(N)$ piece and an O$(1)$ piece. We show that in the
large-N and low-energy limit, the O$(N)$ piece in the hamiltonian describes a
massless relativistic boson, while the O$(1)$ piece gives rise to cubic
self-interactions of the boson. At finite $N$ and high energies, the low-energy
effective description breaks down and the exact bosonized hamiltonian must be
used. We also comment on the connection between the Tomonaga problem and pure
Yang-Mills theory on a cylinder. In the dual context of baby universes and
multiple black holes in string theory, we point out that the O$(N)$ piece in
our bosonized hamiltonian provides a simple understanding of the origin of two
different kinds of nonperturbative O$(e^{-N})$ corrections to the black hole
partition function.
