A note on $SU(1,1|n)$ and $OSp(6|2)$ superconformal mechanics

TL;DR

The article develops one-dimensional superconformal mechanics with SU(1,1|n) and OSp(6|2) symmetries in the presence of non-Abelian bosonic currents. It shows that for $N>4$ these currents must obey algebraic constraints, and demonstrates a pragmatic diagonalization approach to obtain concrete current realizations; rank-1 (harmonic) constructions yield consistent supercharges and Hamiltonians expressible via Casimir operators. In the $OSp(6|2)$ case, currents are realized as $su(4)$ generators built from harmonics, enabling closure of the algebra and a Casimir-based Hamiltonian; a parallel but more intricate analysis is given for $SU(1,1|n)$, where eigenvalue patterns constrain possible current representations. The work emphasizes that the current solutions are not necessarily exhaustive and points to alternative representations (e.g., non-harmonic in specific instances) and open questions about general currents in osp(n|2) systems.

Abstract

In this article we consider the construction of the superconformal mechanics that realize $SU(1,1|n)$ and $OSp(6|2)$ symmetries and involve interactions with non-Abelian bosonic currents. If is shown that for $N>4$ supersymmetries the currents involved have to satisfy the algebraic equations. General considerations on methods of solving these equations are given. In the obtained particular solutions currents are expressed in terms of semi-dynamical variables (harmonics), and, on one instance, coordinates and momenta.