Superconformal Chern-Simons Theories and the Squashed Seven Sphere

TL;DR

This work shows that a 3D $U(N)\times U(N)$ Chern-Simons-matter theory with bi-fundamental matter and a global $Sp(2)$ symmetry admits two supersymmetric completions: the known $\mathcal{N}=6$ theory and a new $\mathcal{N}=1$ theory. The $\mathcal{N}=1$ theory is constructed with an antisymmetric invariant tensor $\Omega_{AB}$, has a positive-definite bosonic potential and a moduli space $\mathcal{M}=(\mathbb{C}^4/\mathbb{Z}_k)^N$, and preserves no $\mathcal{N}=5$ extension; the level $k$ is not shifted. The authors propose a gravity dual for the $\mathcal{N}=1$ theory: M-theory on $AdS_4\times S^7/\mathbb{Z}_k$ with a squashed metric, where the isometry is $Sp(2)\times Sp(1)$ and reduces to $Sp(2)\times U(1)$ after the orbifold. This yields a new AdS$_4$/CFT$_3$ dual pair tied to a squashed seven-sphere and highlights a concrete instance where geometry controls supersymmetry in the dual field theory.

Abstract

We show that there are two supersymmetric completions of the three-dimensional Chern-Simons theory of level k with gauge group U(N)xU(N) coupled to four sets of massless scalars and spinors in the bi-fundamental representation, if we require Sp(2) global symmetry with the matter fields in the fundamental representation of SU(4). One is the N=6 superconformal theory recently studied in arXiv:0806.1218 [hep-th] and another is a new theory with N=1 superconformal symmetry. We conjecture that the N=1 theory is dual to M theory on AdS_4 x Squashed S^7/Z_k.