K-matrix ($K$)

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.