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I'm doing some introductory reading on anyons, and I'm a bit confused by the way the basis are changed.

Suppose we have the Ising anyons, which obey $1\times1=1$, $1\times\Psi = \Psi$, $1\times\sigma = \sigma$, $\Psi\times\Psi = \Psi$, $\Psi\times\sigma = \sigma$ and $\sigma\times\sigma = 1 +\Psi$. Then consider the fusion space wherein 3 $\sigma$ anyons are constrained to fuse to a single $\sigma$, and we have $$F^\sigma_{\sigma\sigma\sigma} = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix}$$ for the transform from the basis $\{\left|(\sigma\sigma)\sigma;1\sigma;\sigma\right>,\left|(\sigma\sigma)\sigma;\Psi\sigma;\sigma\right>\} := \{A_1,A_2\}$, to the basis with fusion right to left, $\{B_1,B_2\}$, bieng analagous to the first basis but with the parenthesis around the second two of anyons rather than the first two. So as I understand it, this means that, for example,$$A_1 = \frac{1}{\sqrt{2}}(B_1+B_2).$$ I'm wondering if there is a some physical explanation for what this actually is, i.e. within a material what does this equivalence correspond to, or is it just more quantum weirdness?

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