This post is meant to ask for proper References to fill a gap in the literature.

> **My short question** is that are there known and precise ways to formulate 2d topological boundary conditions" for certain but generic 3d non-abelian Chern-Simons (CS) theory? If the answer is yes, this can be a bridge between the abelian case done in Kapustin-Saulina, and the non-abelian case (for modular tensor category as 3d [=2+1d] topological order) done in Lan-Wang-Wen, References given below.

Given the form of CS theory on a 3-manifold $M^3$ as:

$$
Z=\int [DA] \exp(i (S_{1,nab}+S_{2,nab}+ \dots + S_{1,ab} + S_{2,ab} + \dots)
$$
with an action of non-abelian CS:
$$S_{j,nab}=\frac{k}{4\pi}\int_{M^3} \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)$$
for a gauge group $G_j$
and an action of abelian CS
$$
S_{j,ab}=\frac{K_{IJ}}{4 \pi}\int_{M^3} A_I dA_J.
$$
and possibly more intriguing couplings between different CS actions.

> **My longer introduction with some background**: 

1. [Kapustin-Saulina](https://linkinghub.elsevier.com/retrieve/pii/S0550321310006723) deals with the relations between Lagrangian subgroups/submanifolds, 2d topological boundary and 3d **abelian Chern-Simons theory**. 

In [Kapustin-Saulina, "Topological boundary conditions in abelian Chern-Simons theory" Nucl.Phys.B845:393-435,2011, arXiv:1008.0654, DOI: 10.1016/j.nuclphysb.2010.12.017](https://linkinghub.elsevier.com/retrieve/pii/S0550321310006723), it says: ... *topological boundary conditions in abelian Chern-Simons theory and line operators confined to such boundaries. From a mathematical point of view, their relationships are described by a certain 2-category associated to an even integer-valued symmetric bilinear form (the matrix of Chern-Simons couplings). We argue that **boundary conditions correspond to Lagrangian subgroups in the finite abelian group classifying bulk line operators (the discriminant group)**. We describe properties of boundary line operators; in particular we compute the boundary associator. We also study codimension one defects (surface operators) in abelian Chern-Simons theories. As an application, we obtain a classification of such theories up to isomorphism, in general agreement with the work of Belov and Moore.*

The only Reference that I know of which work out a certain generalization of Lagrangian subgroups/submanifolds or 2d topological boundary for 3d non-abelian Chern-Simons theory is this:



 2. [Lan-Wang-Wen](https://arxiv.org/abs/1408.6514) deals with the relations between Lagrangian subgroups/submanifolds, 2d topological boundary and 3d **non-abelian topological order described by modular tensor category, which includes  3d non-abelian Chern-Simons theory**. 
  


In [Lan-Wang-Wen, "Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
" Phys. Rev. Lett. 114, 076402 (2015), arXiv:1408.6514,
DOI: 10.1103/PhysRevLett.114.076402](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.076402), it states that:
*Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal{W}_{}$, whose entries are the fusion-space dimensions $\mathcal{W}_{ia}$, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.*

Mathematically, [Lan-Wang-Wen](https://arxiv.org/abs/1408.6514) proposes a classification of bimodule categories between modular tensor categories. However, [Lan-Wang-Wen](https://arxiv.org/abs/1408.6514)  does not use a continuum TQFT or QFT langauge, thus the result is not exactly easy to be phrase by Kapustin-Saulina result. 

However, based on [Lan-Wang-Wen](https://arxiv.org/abs/1408.6514) setup, if we know the modular data (modular S and modular T matrices) of TQFT, we can  "bootstrap" the 2d surface defects of 3d TQFTs. 

For example, a work of [Lian-Wang arXiv: 1801.10149](https://arxiv.org/abs/1801.10149) [Phys. Rev. B 97, 165124 (2018)
DOI: 10.1103/PhysRevB.97.165124](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.165124) 
use the Lan-Wang-Wen formulas to "**bootstrap**" the 2d surface defects of 3d TQFTs
for the experimentally relevant system, such as the  ν=5/2 Quantum Thermal Hall State, including:

- non-Abelian Pfaffian (Pf) Moore-Read state

- non-Abelian anti-Pfaffian state

- non-Abelian Particle-Hole Pfaffian state


> Rephrase my question: How to bridge the [Lan-Wang-Wen](https://arxiv.org/abs/1408.6514) results into a 3d non-abelian Chern-Simons theory analogous to [Kapustin-Saulina](https://linkinghub.elsevier.com/retrieve/pii/S0550321310006723)?