a_{α}^{I}

gives

\frac{1}{2 π ℏ} K_{I J} ϵ^{α β γ} \partial_{β} a_{γ}^{J} = j_{I}^{α} .

For a stationary quasiparticle of charge vector

q

, the time component implies that its enclosed gauge flux is

\oint a_{i}^{I} d x^{i} = 2 π ℏ (K^{- 1} q)^{I}

up to the orientation convention. Coupling a particle of charge

p

to this flux produces the full-braid phase

exp (2 πi p^{T} K^{- 1} q) = exp (2 i θ_{pq}), θ_{pq} = π p^{T} K^{- 1} q .

An exchange is half of the corresponding full braid. Reversing the orientation conjugates the phase.

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