Topological boundary conditions in abelian Chern–Simons theory

Abstract

We study topological boundary conditions in abelian Chern-Simons theory and line operators confined to such boundaries. From the 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

Ground-state degeneracy for Abelian anyons in the presence of gapped boundaries

“…This section is a review of Abelian Chern-Simons theory, including topological boundary conditions, following [5,7].…”

“…Elementary topological boundary conditions correspond to special subgroups of D called Lagrangian subgroups [5,7]. By definition, a Lagrangian subgroup L ⊂ D consists of charges of quasiparticles which satisfy the following two properties:…”

“…Boundary domain walls may be characterized by the electric charges they carry, or equivalently by the holonomy of the U (1) N gauge field along a small semi-circle enclosing the domain wall. If we take the former viewpoint, then elementary boundary domain walls correspond to integer vectors Q i modulo elements of the form K ij j + L i + L j , where i is an arbitrary integer vector, as before, and L i and L i are arbitrary charge vectors which lie in the subgroups L and L , respectively [5]. Indeed, at the boundary domain wall both condensates with charges in L and L are present, and boundary domain walls which differ only by the emission or absorption of the condensate particles are indistinguishable.…”

“…Gapped boundary conditions correspond to topological boundary conditions in TQFT; for Abelian ChernSimons theory they have been studied in [5][6][7] while a more general theory based on fusion categories was developed in [8,9]. The problem of computing the ground-state degeneracy was previously addressed in [10], but only in the case when there are no boundary domain walls.…”

N $$ \mathcal{N} $$ =2 supersymmetric field theories on 3-manifolds with A-type boundaries

“…For example, given an isotropic vector v m such that v m κ mn v n = 0, we may consider boundary conditions δA = v m δA m µ dx µ and V = v m V m µ dx µ with arbitrary components δA µ and V µ on M 2 . For a general treatment of such boundary conditions in CS theory we refer the reader to [56].…”