ABJM models in N=3 harmonic superspace
I. L. Buchbinder, E. A. Ivanov, O. Lechtenfeld, N. G. Pletnev, I. B. Samsonov, B. M. Zupnik
TL;DR
The paper presents a novel ${\cal N}{=}3$, $d{=}3$ harmonic superspace formulation of the ABJM Chern-Simons-matter theory, realizing three manifest off-shell supersymmetries while the remaining three mix superfields and close on shell; the action uses two bifundamental hypermultiplets and two CS gauge superfields with no explicit superpotential, and the sextic ABJM scalar potential emerges after auxiliary-field elimination. It then extends the construction to ${\rm U}(N)\times{\rm U}(M)$ and other gauge/matter configurations, showing ${\cal N}{=}6$ supersymmetry and ${\rm SO}(6)$ R-symmetry with on-shell closure, and demonstrates how the full ${\rm SO}(6)$ can enhance to ${\rm SO}(8)$ in the ${\rm SU}(2)\times{\rm SU}(2)$ (BLG) limit. The analysis provides a concrete, off-shell superfield route to study hidden symmetries and their algebraic structure, including the mechanism by which the ABJM sextic potential is generated without an explicit superpotential. The work suggests broader applications to M2-brane dynamics, AdS$_4$/CFT$_3$ correspondence, and potential extensions to ${\cal N}{=}8$ theories via ${\cal N}{=}3$ harmonic superspace techniques.
Abstract
We construct the classical action of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model in the N=3, d=3 harmonic superspace. In such a formulation three out of six supersymmetries are realized off shell while the other three mix the superfields and close on shell. The superfield action involves two hypermultiplet superfields in the bifundamental representation of the gauge group and two Chern-Simons gauge superfields corresponding to the left and right gauge groups. The N=3 superconformal invariance allows only for a minimal gauge interaction of the hypermultiplets. Amazingly, the correct sextic scalar potential of ABJM emerges after the elimination of auxiliary fields. Besides the original U(N)xU(N) ABJM model, we also construct N=3 superfield formulations of some generalizations. For the SU(2)xSU(2) case we give a simple superfield proof of its enhanced N=8 supersymmetry and SO(8) R-symmetry.