Anyon monodromy ($M_c^{ab}=R_c^{ab}{R}_c^{ba}$)

The full double braiding of two anyons. The anyon balancing identity gives $M_c^{ab}={\theta }_c/({\theta }_a{\theta }_b)$ in a permitted anyon fusion channel. Unlike a single anyon R-matrix entry, this scalar is invariant under an anyon vertex gauge transformation.

An anyon whose full anyon monodromy with every other charge is the identity in every allowed anyon fusion channel. The vacuum anyon is transparent. Transparent charges and their anyon fusion summands form a symmetric subcategory of the braided monoidal category. They can have quantum dimension greater than one.

Apply the anyon balancing identity to the vacuum anyon channel in $a\times \bar{a}$. Transparency gives $1=({\theta }_a{\theta }_{\bar{a}}{)}^{-1}={\theta }_a^{-2}$. Thus a transparent anyon has bosonic or fermionic topological spin. This does not require its quantum dimension to equal one.

For transparent anyons $a,b$, the hexagon identities for braiding for braiding express double exchange of $a\otimes b$ with any charge $x$ as the successive double exchanges of $a$ and $b$ with $x$. Both are identities. Naturality restricts the resulting identity to every simple anyon fusion summand $c$ of $a\otimes b$, proving that $c$ is transparent.