 Abelian Chern--Simons theory

An Abelian Chern--Simons theory describes topological gauge fields in two spatial dimensions and time. In units $\hslash =1$, its bulk action can be writtenwhere $a^I$ are compact Abelian gauge field one-forms, $\wedge$ is the wedge product of differential forms, and the K-matrix is symmetric, integral and invertible. This supplies a long-distance description of gapped Abelian quantum Hall states. ^[ref]

A quasiparticle labeled by an integer vector $q$ couples to the corresponding combination of gauge fields. The full encircling phase of charges $q,q^{\prime }$ isThus global particle paths can carry measurable braiding information even though the pure bulk action has no metric-dependent local wave term. Boundaries require their own treatment; the bulk formula alone does not specify every allowed boundary condition.