Braids as a representation space of SU(5)

TL;DR

This paper models elementary particles as braids within the SU(5) symmetry group, explicitly deriving the associated Lie algebra operators and identifying gauge bosons as braid transformations, linking topological structures to particle physics.

Contribution

It explicitly constructs the SU(5) Lie algebra representation using braids, connecting topological braid groups to gauge symmetries in grand unified theories.

Findings

Derived raising operators for SU(5) from braid representations

Identified gauge bosons as braid transformations

Proposed a topological framework for particle symmetries

Abstract

The Standard Model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary, is still unknown. Along the lines of previous preon models positing a substructure to explain these aspects, Bilson-Thompson showed how the first family of elementary particles is realized as the crossings of braids made of three strands, with charges resulting from twists of those strands with certain conditions; in this topological model, there are only two distinct neutrino states. Modeling the particles as braids implies these braids must be the representation space of a Lie algebra, giving the symmetries of the Standard Model. In this paper, this representation is made explicit, obtaining the raising operators associated with the Lie algebra of $SU(5)$, one of the earliest grand unified theories. Because the braids form a group, the action of these operators are braids themselves, leading to their identification as gauge bosons. Possible choices for the other two families are also given. Although this realization of particles as braids is lacking a dynamical framework, it is very suggestive, especially when considered as a natural method of adding matter to loop quantum gravity.