Nonlinear Supersymmetry, Quantum Anomaly and Quasi-Exactly Solvable Systems

TL;DR

This paper explores nonlinear supersymmetry in one-dimensional quantum systems, addressing anomaly issues, classifying systems based on superpotential behavior, and identifying connections to quasi-exactly solvable models and two-dimensional generalizations.

Contribution

It provides a classification of nonlinear supersymmetric systems, analyzes polynomial supercharges, and introduces anomaly-free classes linked to quasi-exact solvability.

Findings

Identification of anomaly-free classes of supersymmetric systems

Connection between supersymmetry and quasi-exact solvability

Generalization of nonlinear supersymmetry to two dimensions

Abstract

The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is ``cured'' by the specific superpotential-dependent term of order $\hbar^2$. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems.