Local Fractional Supersymmetry for Alternative Statistics

TL;DR

This paper develops a theoretical framework for local fractional supersymmetry, leading to a fractional supergravity and an equation of motion for particles with braid group statistics, extending supersymmetry concepts.

Contribution

It introduces a group theory-based approach to one-dimensional fractional supersymmetry and constructs a local fractional supergravity model with reparametrization invariance.

Findings

Derived an equation of motion for particles with braid group statistics.

Established a curved fractional superline formulation.

Proposed a quantization method for fractional supergravity.

Abstract

A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in $1D$ supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry {\it i.e.} a fractional supergravity which is then quantized {\it \`a la Dirac} to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given, by means of a curved fractional superline.