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Large N Chern-Simons-matter theories and three dimensional bosonization

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I will discuss the conformal theories of N complex
scalars or fermions in 2+1 dimensions, coupled to a U(N) Chern-Simons (CS)
theory at level k. In the large N limit these theories have a high-spin
symmetry, and, as I will review, they are dual to Vasiliev's high-spin gravity
theories on four dimensional anti-de Sitter space. Maldacena and Zhiboedov
showed that the high-spin symmetry determines the 2-point and 3-point functions
of these theories at large N, up to two parameters. The duality to Vasiliev's
gravity suggests that there is some mapping between the CS+scalar and CS+fermion theories
(when one adds a quartic coupling to the scalar theories, to flow to their
"critical fixed point"). We compute explicitly the large N limit of
some 2-point and 3-point functions in these theories. We show that the results
match with the general results of Maldacena and Zhiboedov, and they are
consistent with an equivalence between the critical theory of N scalars coupled
to a U(N) CS theory at level k, and the theory of k fermions coupled to a U(k)
CS theory at level (N-1/2). We conjecture that this large N equivalence may be
an exact duality between the scalar and fermion theories also at finite N, thus
providing a bosonization of the fermionic theory. Similar results hold for real
scalars (fermions) when U(N) is replaced by O(N).