Anyon balancing identity ($R_c^{ab}{R}_c^{ba}={\theta }_c/({\theta }_a{\theta }_b)$)

In a ribbon anyon theory, twisting a composite charge $c\in a\times b$ is equivalent to twisting each input and applying their full anyon monodromy. The topological spin obeys ${\theta }_1=1$ and ${\theta }_{\bar{a}}={\theta }_a$; the latter is the ribbon duality convention, with no complex conjugation of the scalar.