Abelian Chern--Simons theory

An Abelian Chern--Simons theory with integer symmetric invertible matrix $K$ describes Abelian anyons whose braiding phases are determined by $K^{-1}$.

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 integral bilinear form multiplying the Abelian Chern--Simons gauge fields. Integer unimodular basis changes send $K$ to $WK{W}^T$ without changing the phase.

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 $2\pi q^{\prime T}{K}^{-1}q$.

A Lagrangian subgroup is a maximal set of mutually bosonic anyons that can consistently condense at a gapped boundary. An anyon can join the condensate only if it braids trivially with every already-condensed anyon.

An Abelian K-matrix Chern--Simons theory has $|\det K|$ topologically distinct ground states on a torus.

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 $|\det K|$.