Polyadic supersymmetry

TL;DR

This paper develops a novel polyadic supersymmetry framework by extending traditional supersymmetric quantum mechanics with n-ary algebraic structures, leading to new forms of supercharges, Hamiltonians, and superalgebras.

Contribution

It introduces a polyadic analog of supersymmetry using n-ary sigma matrices and constructs new m-ary superalgebras with unique properties.

Findings

Polyadic supercharges and Hamiltonians are represented as cyclic shift block matrices.

New brackets with reduced arity m< n are discovered, forming m-ary superalgebras.

Higher order Hamiltonians and supercharges are obtained depending on the parity of m.

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.