We consider the conformal field theory of N complex massless scalars in 2+1
dimensions, coupled to a U(N) Chern-Simons theory at level k. This theory has a
't Hooft large N limit, keeping fixed \lambda = N/k. We compute some
correlation functions in this theory exactly as a function of \lambda, in the
large N (planar) limit. We show that the results match with the general
predictions of Maldacena and Zhiboedov for the correlators of theories that
have high-spin symmetries in the large N limit. It has been suggested in the
past that this theory is dual (in the large N limit) to the Legendre transform
of the theory of fermions coupled to a Chern-Simons gauge field, and our
results allow us to find the precise mapping between the two theories. We find
that in the large N limit the theory of N scalars coupled to a U(N)_k
Chern-Simons theory is equivalent to the Legendre transform of the theory of k
fermions coupled to a U(k)_N Chern-Simons theory, thus providing a bosonization
of the latter theory. We conjecture that perhaps this duality is valid also for
finite values of N and k, where on the fermionic side we should now have (for
N_f flavors) a U(k)_{N-N_f/2} theory. Similar results hold for real scalars
(fermions) coupled to the O(N)_k Chern-Simons theory.Comment: 49 pages, 16 figures. v2: added reference