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The Aharonov-Casher effect is the electromagnetic dual in which a magnetic moment or flux acquires a phase by encircling electric charge.

A charge-flux composite carries electric charge $q$ and localized magnetic flux $\Phi$. Mutual encircling of $(q_1,{\Phi }_1)$ and $(q_2,{\Phi }_2)$ produces the phase $\exp [i(q_1{\Phi }_2+q_2{\Phi }_1)/\hslash ]$ from the Aharonov-Bohm and Aharonov-Casher effects.

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 $f$.