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:

  1. 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.

  2. 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.

  3. 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?

  • $\begingroup$ @David, thanks very much for editing it. I am still adding info, so you may change after I am done - there are stuff I added which you remove about the Pfaffian Moore-Read states $\endgroup$ – wonderich Feb 26 at 23:28
  • $\begingroup$ Follow your change, I just add one sentence: "and possibly more intriguing couplings between different CS actions/sectors" $\endgroup$ – wonderich Feb 26 at 23:31
  • $\begingroup$ OK, done. We can delete all our comments now :-) $\endgroup$ – David Roberts Feb 26 at 23:39

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