Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra

TL;DR

This paper reveals the algebraic foundation of nonlinear holomorphic supersymmetry through the Onsager algebra, introduces new commuting charges, and generalizes the structure to pseudo-supersymmetry, impacting integrable models and quantum mechanics.

Contribution

It establishes the Onsager algebra as the core structure of nonlinear holomorphic supersymmetry and explicitly constructs the superalgebra for various orders, including new quadratic terms.

Findings

Identified the Onsager algebra as underlying n-HSUSY.

Constructed a new set of commuting charges including quadratic terms.

Generalized the algebraic structure to pseudo-supersymmetry and applications.

Abstract

Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order $n\in\N, n>1$, ($n$-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies $n$-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan-Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of $n$-HSUSY and fix it explicitly for the cases of $n=2,3,4,5,6$. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan-Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with $n$-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the $PT$-symmetric quantum mechanics.