Topological boundary conditions in abelian Chern-Simons theory

TL;DR

This paper explores the mathematical structure of topological boundary conditions and line operators in abelian Chern-Simons theory, providing a classification framework and detailed properties of boundary operators.

Contribution

It introduces a 2-category framework for boundary conditions, relates them to Lagrangian subgroups, and classifies abelian Chern-Simons theories up to isomorphism.

Findings

Boundary conditions correspond to Lagrangian subgroups in the discriminant group.

Computed the boundary associator for boundary line operators.

Classified abelian Chern-Simons theories up to isomorphism.

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.