The (unitary) "phase" factor for non-Abelian anyons satisfies the
(non-Abelian) Knizhnik-Zamolodchikov equation:

$$\big (\frac{\partial}{\partial z_{\alpha}} + \frac{1}{2\pi k} \sum_{\beta \neq \alpha} \frac{Q^a_{\alpha}Q^a_{\beta}}{z_{\alpha} - z_{\beta}}\big )U(z_1, ....,z_N) = 0 $$

Where $z_{\alpha}$ is the complex plane coordinate of the particle
$\alpha$ , and $Q^a_{\alpha}$ is the matrix representative of the $a-$th
gauge group generator of the particle $\alpha$ and $k$ is the level .

Please, see the following two articles by Lee and Oh (article-1,
article-2).

In the first article they explicitly write the solution in the case of
the two-body problem:

$$U(z_1, z_2) = exp( i\frac{Q^a_1Q^a_2}{2\pi k} ln(z_1-z_2))$$

The articles describe the method of solution:

The non-Abelian phase factor can be obtained from a quantum mechanical
model of $N$ particles on the plane each belonging possibly to a
different representation of the gauge group minimally coupled to a gauge field with a Chern-Simons term in the Lagrangian.

The classical field equations of the gauge potential can be exactly solved and
substituted in the Hamiltonian. The reduced Hamiltonian can also be exactly solved. Its solution is given by the action of a unitary phase factor on a symmetric wave function. This factor satisfies the Knizhnik-Zamolodchikov equation.

The unitary phase factor lives in the tensor product Hilbert space of
the individual particle representations. The wave function is a vector in this Hilbert space valued holomorphic function depending on the $N$ points in the plane.

This post imported from StackExchange Physics at 2014-04-05 03:32 (UCT), posted by SE-user David Bar Moshe