Topological Invariants and Anyonic Propagators

TL;DR

This paper explores the topological and fractal properties of particles with fractional spins, deriving invariants and propagator representations that connect topology, spin, and fractal dimensions.

Contribution

It introduces a topological invariant related to fractional spin particles and provides a path integral formulation for anyonic propagators, linking topology and fractal geometry.

Findings

Derived the Hausdorff dimension for fractional spin particles.

Established a topological invariant involving spin and genus.

Presented a path integral representation for anyonic propagators.

Abstract

We obtain the Hausdorff dimension, $h=2-2s$, for particles with fractional spins in the interval, $0\leq s \leq 0.5$, such that the manifold is characterized by a topological invariant given by, ${\cal W}=h+2s-2p$. This object is related to fractal properties of the path swept out by fractional spin particles, the spin of these particles, and the genus (number of anyons) of the manifold. We prove that the anyonic propagator can be put into a path integral representation which gives us a continuous family of Lagrangians in a convenient gauge. The formulas for, $h$ and ${\cal W}$, were obtained taking into account the anyon model as a particle-flux system and by a qualitative inference of the topology.