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In this work we propose an effective low-energy theory for a large class of2+1 dimensional non-Abelian topological spin liquids whose edge states areconformal degrees of freedom with central charges corresponding to the cosetstructure $su(2)_k\oplus su(2)_{k'}/su(2)_{k+k'}$. 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 ofone-dimensional coupled quantum wires. In doing so we provide an explicitrelationship between two distinct approaches: quantum wires and Chern-Simonsbulk theory. We firstly make a direct connection between the interactingquantum wires and the corresponding conformal field theory at the edges, whichturns out to be given in terms of chiral gauged WZW models. Relying on thebulk-edge correspondence we are able to construct the underlying non-AbelianChern-Simons effective field theory.

Effective Theories for 2+1 Dimensional Non-Abelian Topological Spin Liquids