N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets

TL;DR

This work extends ${\cal N}=4$ superconformal Chern-Simons theories by including twisted hyper-multiplets, producing linear and circular quivers with alternating hyper- and twisted hyper-multiplets and recovering the Bagger-Lambert $SO(4)$ model as a simple example. The authors derive the full Lagrangian, Yukawa couplings, and bosonic potentials, show that the theory can be organized into super Lie algebras, and classify non-abelian theories via quivers. They establish a precise equivalence between the BL model and an extended GW construction with PSU$(2|2)$ symmetry, including mass deformations that preserve ${\cal N}=4$ and $SO(4)$; they also connect the abelian limit to the M-crystal model describing M2-branes on $(\mathbb{C}^2/\mathbb{Z}_n)^2$ orbifolds. The results broaden the landscape of ${\cal N}=4$ three-dimensional SCFTs, offering new avenues for brane realizations and holographic duals, and suggest further generalizations beyond the present framework.

Abstract

We extend the N=4 superconformal Chern-Simons theories of Gaiotto and Witten to those with additional twisted hyper-multiplets. The new theories are generically linear quiver gauge theories with the two types of hyper-multiplets alternating between gauge groups. Our construction includes the Bagger-Lambert model of SO(4) gauge group. A family of abelian theories are identified with those proposed earlier in the context of the M-crystal model for M2-branes probing (C^2/Z_n)^2 orbifolds. Possible extension with non-abelian BF couplings and string/M-theory realization are briefly discussed.

Figures (3)

• Figure 1: Linear quiver structure of the original and extended GW theories. The gauge groups can be either $U(N_i)$ or alternation between $O(N_i)$ and $Sp(M_i)$. Dashed lines denote hyper-multiplets and dotted lines denote twisted hyper-multiplets.
• Figure 2: Toric diagram and crystal for $(\mathbb{C}^2/\mathbb{Z}_n)^2$ (reproduced from mc3).
• Figure 3: Brane configuration for the extended GW construction.