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

## TL;DR

This paper develops an effective low-energy theory for 2+1D non-Abelian topological spin liquids, linking quantum wire models with Chern-Simons theory and conformal field theories at the edges.

## Contribution

It explicitly connects quantum wire constructions with Chern-Simons bulk theory for non-Abelian topological phases, including edge conformal field theories.

## Key findings

- Derived the edge conformal field theory as chiral gauged WZW models.
- Established the bulk-edge correspondence for these topological phases.
- Provided a unified framework linking quantum wires and Chern-Simons theory.

## Abstract

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.