  In this work we propose an effective low-energy theory for a large class of
2+1 dimensional non-Abelian topological spin liquids whose edge states are
conformal degrees of freedom with central charges corresponding to the coset
structure $su(2)_k\oplus su(2)_{k'}/su(2)_{k+k'}$. For particular values of
$k'$ it furnishes the series for unitary minimal and superconformal models.
These gapped phases were recently suggested to be obtained from an array of
one-dimensional coupled quantum wires. In doing so we provide an explicit
relationship between two distinct approaches: quantum wires and Chern-Simons
bulk theory. We firstly make a direct connection between the interacting
quantum wires and the corresponding conformal field theory at the edges, which
turns out to be given in terms of chiral gauged WZW models. Relying on the
bulk-edge correspondence we are able to construct the underlying non-Abelian
Chern-Simons effective field theory.
