${\cal N}=8$ supersymmetric mechanics with spin variables from indecomposable multiplets

TL;DR

The paper constructs two not-fully-reducible ${\cal N}=8$, $d=1$ indecomposable multiplets by deforming the ${\bf (1,8,7)}$ multiplet with couplings to ${\bf (8,8,0)}$ spin sectors. It provides manifest ${\cal N}=8$ superfield constraints, derives both off-shell and on-shell actions for each version, and proves that the two models are equivalent on-shell with spin variables forming the adjoint representation of ${\rm SO}(8)$. The results include multiple formulations (SU(4) covariant, ${\cal N}=4$ harmonic superspace, and octonionic covariance) and explicit on-shell ${\rm SO}(8)$-currents, highlighting a rich symmetry structure that extends to ${\rm OSp}(8|2)$ in the on-shell limit. These indecomposable systems extend the landscape of ${\cal N}=8$ supersymmetric mechanics and open avenues toward matrix-valued (spin Calogero-like) generalizations and fully manifest ${\cal N}=4$ superspace constructions.

Abstract

We define two new indecomposable (not fully reducible) ${\cal N}=8$, $d=1$ off-shell multiplets and consider the corresponding models of ${\cal N}=8$ supersymmetric mechanics with spin variables. Each multiplet is described off shell by a scalar superfield which is a nonlinear deformation of the standard scalar superfield $X$ carrying the $d=1$ multiplet ${\bf (1,8,7)}$. Deformed systems involve, as invariant subsets, two different off-shell versions of the irreducible multiplet ${\bf (8,8,0)}$. For both systems we present the manifestly ${\cal N}=8$ supersymmetric superfield constraints, as well as the component off- and on-shell invariant actions, which for one version exactly match those given in arXiv:2402.00539 [hep-th]. The two models differ off shell, but prove to be equivalent to each other on shell, with the spin variables sitting in the adjoint representation of the maximal $R$-symmetry group ${\rm SO}(8)$.