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In an Abelian Chern--Simons theory, the equation of motion ties a quasiparticle's gauge charge $q$ to gauge flux $\Phi =2\pi \hslash K^{-1}q$. A quasiparticle of charge $q^{\prime }$ encircling it therefore acquires the Aharonov--Bohm phase $2\pi q^{\prime T}{K}^{-1}q$.

Represent quasiparticle worldlines by conserved currents $j_I^{\alpha }$ and add the minimal-coupling termUp to the corresponding sign convention for charge, variation of the Abelian Chern--Simons theory givesFor a static quasiparticle of integer gauge-charge vector $q$, integration over a small surrounding disk yields the Chern--Simons flux attachment relationA quasiparticle of charge $q^{\prime }$ carried once around this flux acquires the Aharonov--Bohm phaseso

The constituent-particle current has charge vector $t=(1,\dots ,1{)}^T$. Its integral over the same disk isFor $K=I_n$ and $q=e_J$, this gives $N=1$. The self-exchange angle is ${\theta }_{qq}=\pi e_J^T{e}_J=\pi$, so the excitation is a fermion with one unit of microscopic particle number and may be identified with an electron.

In the charge-flux composite model, particle $1$ acquires the Aharonov-Bohm effect phase $q_1{\Phi }_2/\hslash$ when it circles the flux of particle $2$. The reciprocal Aharonov-Casher effect contributes $q_2{\Phi }_1/\hslash$. Thus the full braid isThe assumption for $a=(q_0,0)$ and $b=(0,{\Phi }_0)$ saysFor any allowed particle $(n{q}_0,m{\Phi }_0)$, the full braid with $a^2=(2q_0,0)$ is $e^{2im{q}_0{\Phi }_0/\hslash }=1$; similarly every particle braids trivially with $b^2=(0,2{\Phi }_0)$. Under the stated operational identification,For $c=a\times b=(q_0,{\Phi }_0)$, exchanging two identical composites is half their full braid and givesTherefore $a$ and $b$ reproduce the bosonic $e$ and $m$ particles, in either order, and $c$ reproduces their fermionic fusion product $f$.