= Solution
Varying the <Abelian Chern--Simons theory> with respect to $a^I_\alpha$ gives
$$
\boxed{
\frac1{2\pi\hbar}K_{IJ}
\epsilon^{\alpha\beta\gamma}
\partial_\beta a^J_\gamma=j_I^\alpha}.
$$
For a stationary quasiparticle of charge vector $q$, the time component implies that its enclosed gauge flux is
$$
\oint a^I_i\,dx^i
=2\pi\hbar(K^{-1}q)^I
$$
up to the orientation convention. Coupling a particle of charge $p$ to this flux produces the full-braid phase
$$
\boxed{
\exp\!\left(2\pi i\,p^TK^{-1}q\right)
=\exp(2i\theta_{pq})},
\qquad
\boxed{\theta_{pq}=\pi p^TK^{-1}q}.
$$
An exchange is half of the corresponding full braid. Reversing the orientation conjugates the phase.
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