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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.

A quantum Hall state is a two-dimensional electronic state exhibiting quantized transverse transport in a strong perpendicular magnetic field. In an ideal plateau regime, the longitudinal conductivity is suppressed while the Hall conductivity takes ${\sigma }_{xy}=\nu e^2/h$. Integer and fractional values of the filling-related label $\nu$ distinguish different regimes.

Landau levels organize the single-particle spectrum, while interactions are essential to fractional states. A measured plateau reflects a robust bulk response together with the role of edges and disorder; it is not obtained simply by replacing a classical carrier density with an integer in the ordinary Hall formula.