Topological spin (${\theta }_a$)

The topological spin ${\theta }_a$ is the phase produced by a $2\pi$ rotation of anyon $a$, equivalently its self-exchange phase in an Abelian anyon theory. The two-dimensional spin-statistics relation is ${\theta }_a=e^{2\pi i{s}_a}$.

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.