Spinning Particles, Braid Groups and Solitons

TL;DR

This paper develops techniques to compute the fundamental groups of configuration spaces of particles with internal structures on manifolds, generalizing braid groups, and explores their implications for spin, statistics, and solitons in quantum theories.

Contribution

It introduces a generalized braid group framework for particles with internal degrees of freedom and analyzes their representations to understand spin and statistics on various manifolds.

Findings

Half-integer spin quantizations occur on manifolds with spin_c structures.

Generalized braid groups are computed for particles with spin on arbitrary manifolds.

Existence of half-integer spin theories is linked to the manifold's spin_c structure.

Abstract

We develop general techniques for computing the fundamental group of the configuration space of $n$ identical particles, possessing a generic internal structure, moving on a manifold $M$. This group generalizes the $n$-string braid group of $M$ which is the relevant object for structureless particles. In particular, we compute these generalized braid groups for particles with an internal spin degree of freedom on an arbitrary $M$. A study of their unitary representations allows us to determine the available spectrum of spin and statistics on $M$ in a certain class of quantum theories. One interesting result is that half-integral spin quantizations are obtained on certain manifolds having an obstruction to an ordinary spin structure. We also compare our results to corresponding ones for topological solitons in $O(d+1)$-invariant nonlinear sigma models in $(d+1)$-dimensions, generalizing recent studies in two spatial dimensions. Finally, we prove that there exists a general scalar quantum theory yielding half-integral spin for particles (or $O(d+1)$ solitons) on a closed, orientable manifold $M$ if and only if $M$ possesses a ${\rm spin}_c$ structure.