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Suppose that the field $B $ can be decomposed into $B=gB'+\phi \delta B$ with the coupling constant $g $ and $\delta \phi = I$ where $ \phi$ is indicator function on maximum set intersection such that $\delta \phi $ lies completely on the support of $\delta B $. Now the integration haar measure can be decomposed into $d [B]=d [B'] d [\delta B \phi]$. ...


I think I got the answer. The wavefunction you parallel transport around the closed loop has to be degenerate so that once around the loop, the wavefunction can point in some other direction different from the initial one you started with. Hence there can be a nonzero Berry phase.


The definition of $D$ requires sum over all the anyons, not just the "basic" ones -- after all, you can not really say which ones are "more basic" than others. In certain realizations, like in toric code, it is easy to write down the operators that create $e$ and $m$, but there is no sense that they are more basic than $\psi$, since you can as well take $m$ ...

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