On the Statistical Origin of Topological Symmetries

TL;DR

This paper explores various advanced symmetries in quantum systems, revealing their interrelations and expressing them through bosonic and orthofermionic operators, thus uncovering a deep connection between different symmetry types.

Contribution

It provides explicit constructions of symmetry generators, relates orthosupersymmetry to fractional supersymmetry, and demonstrates that all orthosupersymmetric systems have topological symmetries.

Findings

Symmetry generators expressed via boson and orthofermion operators

Orthosupersymmetry of order p linked to fractional supersymmetry of order p+1

All orthosupersymmetric systems exhibit topological symmetries

Abstract

We investigate a quantum system possessing a parasupersymmetry of order 2, an orthosupersymmetry of order $p$, a fractional supersymmetry of order $p+1$, and topological symmetries of type $(1,p)$ and $(1,1,...,1)$. We obtain the corresponding symmetry generators, explore their relationship, and show that they may be expressed in terms of the creation and annihilation operators for an ordinary boson and orthofermions of order $p$. We give a realization of parafermions of order~2 using orthofermions of arbitrary order $p$, discuss a $p=2$ parasupersymmetry between $p=2$ parafermions and parabosons of arbitrary order, and show that every orthosupersymmetric system possesses topological symmetries. We also reveal a correspondence between the orthosupersymmetry of order $p$ and the fractional supersymmetry of order $p+1$.