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$S$ and $T$ matrices described modular transformations. Fusion coefficient and particle spins $(N^{ij}_k,s_i)$ describe the mutual and self statistics of anyons. Their relation is well known in category theory (Our papers http://arxiv.org/abs/1506.05768 and http://arxiv.org/abs/1507.04673 used those relations).

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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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There is certainly a way to preserve the time-reversal symmetry with open boundary conditions: just put the two edge spins into a singlet state. You may say that this is cheating, since essentially we will have periodic boundary conditions by pairing the edge spins up. This depends on perspective, but indeed there are no other ways to have a completely ...

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