Polyadic supersymmetry

TL;DR

The paper develops a polyadic extension of supersymmetry by applying a polyadization procedure to the 1D SQM toy model and constructing polyadic supercharges and Hamiltonians from polyadic sigma matrices $\bm{\Sigma}_{j}^{[\mathbf{n}]}$ that act on a $2(n-1)$-dimensional space. It introduces reduced arity brackets $\mathcal{R}_{(n)}^{[\mathbf{m}]}$ derived from the $n$-ary multiplication and builds $m$-ary polyadic superalgebras, yielding two main regimes: even reduced arity $m=2m'$ produces towers of higher-order even Hamiltonians, while odd reduced arity $m=2m'+1$ produces towers of higher-order odd supercharges with only the odd sector. The framework is instantiated with explicit constructions for $n=3$ and $n=4$, showing binary and higher-arity Hamiltonians and higher-order supercharges, including $3$rd- and $4$th-order operators, organized in block-matrix forms that generalize the standard Witten SQM. These results reveal a rich algebraic landscape where polyadic supersymmetry generates novel operator towers and potential new descriptions of multidegenerate quantum states, beyond conventional $N$-extended or multigraded SQM, with concrete formulas for the operators and their (anti)commutation relations. The work thus provides both a mathematical generalization of Lie superalgebras to $n$-ary and reduced-arity structures and a physically motivated set of toy-model realizations that highlight the distinctive features of polyadic supersymmetry.

Abstract

We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the $n$-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the $N$-extended and multigraded SQM. While constructing the corresponding supersymmetry as an $n$-ary Lie superalgebra ($n$ is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of $2\leq m<n$ and a related series of $m$-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity $m$ we obtain a tower of higher order (as differential operators) even Hamiltonians, while for $m$ odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.