Topological boundary conditions in abelian Chern-Simons theory

TL;DR

The paper develops a 2-categorical framework for topological boundary conditions in abelian Chern-Simons theory, showing that boundaries correspond to Lagrangian subgroups of the discriminant group ${\sf D}$ determined by the coupling form $K$, with boundary line operators governed by ${\sf D}/{\mathsf L}$ and a canonical boundary associator arising from monopole effects. It computes the boundary semi-braiding between bulk and boundary Wilson lines and provides explicit formulas for the boundary associator, all expressed via ${\sf D}$, ${\mathsf q}$, and the subgroup ${\mathsf L}$; integrality artefacts in gauge fixing do not affect physical content. The work also analyzes surface operators via the folding trick, establishes a notion of duality walls that implement quantum equivalences between theories with the same signature, ${\sf D}$, and ${\mathsf q}$, and aligns the results with Belov-Moore's classification, while outlining a broader program for the full 2-category of boundary conditions and potential nonabelian extensions. Overall, the study provides a concrete, higher-categorical classification of 3d abelian CS theories and their topological boundaries and defects, with implications for dualities and 2d reductions.

Abstract

We study 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.