Large N renormalization group flows in 3d mathcal{N} = 1 Chern-Simons-Matter theories

Abstract

We discuss 3d mathcal{N} = 1 supersymmetric SU(N) and U(N) Chern-Simons-matter theories, with Nf matter superfields in the fundamental representation of SU(N) or U(N). In the large N ’t Hooft limit with fixed ’t Hooft coupling λ these theories have one (for Nf = 1) or two (for Nf> 1) exactly marginal deformations in the superpotential. At finite N these couplings acquire a beta function. We compute the beta function exactly for λ = 0, at leading order in 1/N. For Nf = 1 we find four fixed points, one of which is triply-degenerate. We show that at large N there are at most six fixed points for any λ, and conjecture that there are exactly six, with three of them stable (including a point with enhanced mathcal{N} = 2 supersymmetry). The strong-weak coupling dualities of mathcal{N} = 1 Chern-Simons-matter theories map each of these fixed points to a dual one. We show that at large N the phase structure near each of the three stable fixed points is different. For Nf> 1 we analyze the fixed points at weak coupling, and we work out the action of the strong-weak coupling duality on the marginal and relevant superpotential couplings at large N (which was previously known only for Nf = 1). In addition, we compute in these theories the 2-point and 3-point functions of the lowest gauge-invariant singlet superfield at large N, for all values of λ and of the superpotential couplings, and use them to test the large N dualities. This computation is one of the ingredients needed for a computation of the beta function at order 1/N for all λ, which we leave for future work. We also discuss Chern-Simons-matter theories with extra Hubbard-Stratonovich type singlet fields, and suggest dualities between them.