Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators

TL;DR

This paper explores the structure of parasupersymmetric quantum mechanics, classifies Hamiltonians within this framework, and introduces topological invariants analogous to Witten indices, linked to Fredholm operators.

Contribution

It provides a classification scheme for parasupersymmetric Hamiltonians and introduces topological invariants related to Fredholm operators in this context.

Findings

Classification scheme for parasupersymmetric Hamiltonians

Introduction of topological invariants similar to Witten indices

Explicit algebraic expressions for invariants in certain systems

Abstract

The general features of the degeneracy structure of ($p=2$) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a forth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for the topological invariants of a class of parasupersymmetric systems is provided.