  Boundary conformal field theory is brought to bear on the study of
topological insulating phases of non-abelian anyonic chains. These
topologically non-trivial phases display protected anyonic end modes. We
consider antiferromagnetically coupled spin-1/2 su(2)$_k$ chains at any level
$k$, focusing on the most prominent examples; the case $k = 2$ describes Ising
anyons (equivalent to Majorana fermions) and $k = 3$ corresponds to Fibonacci
anyons. We prove that the braiding of these emergent anyons exhibits the same
braiding behavior as the physical quasiparticles. These results suggest a
`solid-state' topological quantum computation scheme in which the emergent
anyons are braided by simply tuning couplings of non-Abelian quasiparticles in
a fixed network.
