We study three dimensional O(N ) k and U (N ) k Chern-Simons theories coupled to a scalar field in the fundamental representation, in the large N limit. For infinite k this is just the singlet sector of the O(N ) (U (N )) vector model, which is conjectured to be dual to Vasiliev's higher spin gravity theory on AdS 4 . For large k and N we obtain a parity-breaking deformation of this theory, controlled by the 't Hooft coupling λ = 4πN/k. For infinite N we argue (and show explicitly at two-loop order) that the theories with finite λ are conformally invariant, and also have an exactly marginal (φ 2 ) 3 deformation. For large but finite N and small 't Hooft coupling λ, we show that there is still a line of fixed points parameterized by the 't Hooft coupling λ. We show that, at infinite N , the interacting non-parity-invariant theory with finite λ has the same spectrum of primary operators as the free theory, consisting of an infinite tower of conserved higher-spin currents and a scalar operator with scaling dimension ∆ = 1; however, the correlation functions of these operators do depend on λ. Our results suggest that there should exist a family of higher spin gravity theories, parameterized by λ, and continuously connected to Vasiliev's theory. For finite N the higher spin currents are not conserved.

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 λ ≡ N/k. We compute some correlation functions in this theory exactly as a function of λ, 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.

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d N = 4 superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2-and 3-point functions of Higgs branch operators in the full 3d N = 4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d N = 2 subalgebra of the N = 4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S 3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

We study the conformal bootstrap for a 4-point function of fermions ψψψψ in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the ψ × ψ OPE, and also on the central charge C T . We observe features in our bounds that coincide with scaling dimensions in the GrossNeveu models at large N . We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.