Studies of the ABJM Theory in a Formulation with Manifest SU(4) R-Symmetry

TL;DR

The paper reformulates the ABJM theory with manifest $SU(4)$ R-symmetry to explicitly verify both Poincaré and conformal supersymmetries, thereby establishing the full $OSp(6|4)$ superconformal symmetry. It starts from the abelian $U(1)\times U(1)$ case, clarifying the role of the $A_\pm$ decomposition, the constraint $F_- = dA_- = 0$, and the $Z_k$ projection that reduces supersymmetry to $N=6$ for $k>2$ (while $N=8$ is recovered for $k=1,2$). It then extends to the non-abelian $U(N)\times U(N)$ theory, deriving the quartic and sextic interactions ($L_4$ and $L_6$), fixing the cubic spinor variations, and showing the potential can be written as $V = \frac{1}{6}\mathrm{Tr}(N^{IA} N^I_A)$, with conserved superconformal currents confirming $OSp(6|4)$. The results provide a robust, manifestly symmetric framework for analyzing vacua, deformations, and the AdS$_4$/CFT$_3$ duality, aligning with and extending prior ABJM findings.

Abstract

We examine the three-dimensional N = 6 superconformal Chern--Simons theory with U(N) X U(N) gauge symmetry, which was recently constructed by Aharony, Bergman, Jafferis, and Maldacena (ABJM). Using a formulation with manifest SU(4) R-symmetry and no auxiliary fields, we verify in complete detail both the Poincare supersymmetry and the conformal supersymmetry of the action. Together, these imply the complete OSp(6|4) superconformal symmetry of the theory. The potential, which is sixth order in scalar fields, is recast as a sum of squares.