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:
$$ Z=\int [DA][DA_i] \exp(i (S_{1,nab}+S_{2,nab}+ \dots + S_{1,ab} + S_{2,ab} + \dots) $$ 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/sectors.
My longer introduction with some background:
The paper
Anton Kapustin and Natalia Saulina, "Topological boundary conditions in abelian Chern-Simons theory" Nucl.Phys.B 845 issue 3 (2011) pp393-435, arXiv:1008.0654, DOI:10.1016/j.nuclphysb.2010.12.017
deals with the relations between Lagrangian subgroups/submanifolds, 2d topological boundary and 3d abelian Chern–Simons theory. 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:
Tian Lan, Juven C. Wang and Xiao-Gang Wen, "Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy" Phys. Rev. Lett. 114 issue 7 (2015) 076402, arXiv:1408.6514, DOI:10.1103/PhysRevLett.114.076402
which 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. 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 proposes a classification of bimodule categories between modular tensor categories. However, Lan–Wang–Wen does not use a continuum TQFT or QFT language, thus the result is not exactly easy to be phrase by Kapustin–Saulina result.
However, based on Lan–Wang–Wen 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, the paper
Biao Lian and Juven Wang, "Theory of the disordered $\nu=\frac52$ quantum thermal Hall state: Emergent symmetry and phase diagram" Phys. Rev. B 97 issue 16 (2018) 165124, DOI:10.1103/PhysRevB.97.165124, arXiv:1801.10149
uses the Lan–Wang–Wen formulas to "bootstrap" the 2d surface defects of 3d TQFTs for the experimentally relevant system, such as the $\nu=\frac52$ 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 rephrase the Lan–Wang–Wen results into a 3d non-abelian Chern–Simons theory analogous to Kapustin–Saulina?
