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 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.

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