We have some discussions in Phys.SE. about the braiding statistics of anyons from a Non-Abelian Chern-Simon theory, or non-Abelian anyons in general.

May I ask: **under what (physical or mathematical) conditions, when we exchange non-Abelian anyons** in 2+1D, or full winding a non-Abelian anyon to another sets of non-Abelian anyons of a system, **the full wave function of the system only obtain a complex phase, i.e. only $\exp[i\theta]$ gained** (instead of a braiding matrix)?

Your answer on the conditions can be freely formulated in either physical or mathematical statements. This may be a pretty silly question, but I wonder whether this conditions have any significant meaning... Could this have anyon-basis dependence or anyon-basis independence. Or is there a **subset** or **subgroup** or **sub-category** concept inside the full sets of anyons implied by the conditions.

This post imported from StackExchange Physics at 2014-06-04 11:39 (UCT), posted by SE-user Idear