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.

The pure Abelian Chern--Simons theory has zero metric stress. For a source written as $-c\int \star J\wedge A$, holding the covariant components of the one-forms $J,A$ fixed makes the source equal to $-c\int \sqrt{-g}{g}^{\mu \nu }{J}_{\mu }{A}_{\nu }{d}^{3}x$. Variation using $\delta \sqrt{-g}=-\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }/2$ gives the displayed stress-energy tensor. If the independent source is instead the fixed two-form $\mathcal{J}=\star J$, the source action is metric-independent and its stress vanishes. The variational data must therefore be stated.

The K-matrix is the symmetric integer matrix multiplying the coupled Abelian gauge fields in an Abelian Chern--Simons theory. For invertible $K$, the anyon lattice of an Abelian Chern--Simons theory isAdding $Kn$ to a charge vector changes it by a local particle. Its full braiding with any integer charge $q^{\prime }$ changes by $2\pi n^T{q}^{\prime }$, leaving the phase unchanged.

For example, $K=(2)$ has two classes, represented by $q=0,1$. Encircling two copies of class $1$ gives $\exp (2\pi i/2)=-1$; fusing them gives class $0$. This illustrates how an integer coupling produces fractional anyon statistics.

An integer unimodular matrix $W$ changes the basis asBecause $W^{-1}$ is also integral, it maps charge classes bijectively and preserves $q^T{K}^{-1}\tilde{q}$. It changes the coordinates of the theory rather than its braiding data. If electromagnetic coupling is specified by a charge vector $t$, that vector must be transformed consistently as $t^{\prime }=Wt$. ^[ref]

For an invertible integer $K$-matrix, Abelian anyons are integer vectors $q$ modulo local particles $K{\mathbb{Z}}^N$. Their mutual full-braiding phase is $\exp (2\pi i{q}^T{K}^{-1}{q}^{\prime })$.

In an Abelian Chern--Simons theory, the equation of motion ties a quasiparticle's gauge charge $q$ to gauge flux $\Phi =2\pi \hslash K^{-1}q$. A quasiparticle of charge $q^{\prime }$ encircling it therefore acquires the Aharonov--Bohm phase $2\pi q^{\prime T}{K}^{-1}q$.

For a bosonic Abelian anyon theory, a Lagrangian subgroup of Abelian anyons is a subgroup of types of anyons closed under anyon fusion, with trivial topological spin for every member and trivial mutual braiding among members. Every anyon outside the subgroup must braid nontrivially with at least one member; pairwise compatibility alone is insufficient.

These conditions describe a maximal condensable collection for an appropriate fully gapped boundary. In the toric code, the vacuum together with the electric anyon gives one choice, and the vacuum together with the magnetic anyon gives another. Electric and magnetic anyons cannot both belong to the same collection because their mutual braiding is nontrivial. Additional physical context, including a nonchiral bulk and any imposed symmetry, matters when using this data to construct a boundary.^[ref]

For an invertible K-matrix, an Abelian Chern--Simons theory hastopological ground states on a spatial torus. On a closed orientable surface of genus of a surface $g$, the corresponding count is $|\det K{|}^g$. ^[ref]

The determinant counts the anyon classes in ${\mathbb{Z}}^N/K{\mathbb{Z}}^N$: the fundamental cell of the sublattice $K{\mathbb{Z}}^N$ has volume $|\det K|$ relative to the unit integer lattice. Transport around intersecting torus cycles gives the noncommuting Wilson-loop algebra of an Abelian Chern--Simons theory, which distinguishes these states.

For the toric code matrixthe quotient has four classes, represented by $(0,0),(1,0),(0,1),(1,1)$, so the torus count is four. These represent different global sectors rather than four locally distinguishable arrangements. In a finite microscopic realization, local perturbations can produce small finite-size splitting; the topological count describes the protected low-energy sector in the large-system limit.

Quasiparticle transport around the two fundamental cycles of a torus gives loop operators obeyingThe induced pairing on ${\mathbb{Z}}^n/K{\mathbb{Z}}^n$ is nondegenerate, so this algebra has a representation of dimension at least $|\det K|$.