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This question concerns the comparison of topological boundary conditions of TQFTs on a manifold with some boundary.

For example, we can consider defining the TQFT on a $D^3$ ball with a topological boundary condition on the boundary $S^2$. Another example, we can consider defining the TQFT on a $D^2 \times S^1$ solid torus with a topological boundary condition on the boundary $T^2$. In physics, this "topological boundary condition" may also refer as "gapped boundary condition."

Now Freed-Hopkins-Lurie-Teleman (FHLT) paper discusses the 3-2-1-0 extended TQFT. FHLT also discusses the topological boundary conditions, for example on page 35, a paragraph on "3-folds with boundary" and the page 38 "§9.4. Surfaces with boundary in another model for t-gerbe theory."

My question: If we take the TQFT as a Chern-Simons theory (such as an Abelian Chern-Simons theory with a symmetric bilinear form, e.g. based on this paper), I like to focus on the properties of the FHLT topological boundary conditions, and compare it with the topological boundary conditions from a Lagrangian subspace of discriminant group (for the symmetric bilinear matrix) of Abelian Chern-Simons theory given by this paper.

  1. Does FHLT topological boundary conditions contain the Lagrangian subspace of discriminant group given by this paper)?

  2. Does FHLT topological boundary conditions differ from the topological boundary conditions of Lagrangian subspace of discriminant group given by this paper)? Does FHLT topological boundary conditions introduce more than the Lagrangian subspace of discriminant group given by this paper)?

p.s. Readers can find the page 35 of FHLT paper discusses a Lagrangian subspace. Readers can find that the page 38 "§9.4. Surfaces with boundary in another model for t-gerbe theory" discusses the Morita equivalence. It is known from the Lagrangian subspace approach for topological boundary conditions result in, for example, "A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent." However, my concern is that what are new ingredients of FHLT topological boundary conditions that differ from the topological boundary conditions of the Lagrangian subspace of discriminant group?

Thank you for the patience reading this post.

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