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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]

Forthe vectors $q_e=(1,0{)}^T$ and $q_m=(0,1{)}^T$ haveThe K-matrix formula therefore gives bosonic self-exchange for $e,m$ and mutual full-braiding phaseFusion adds vectors. Becauseboth double fusions lie in $K{\mathbb{Z}}^2$ and braid trivially with every vector; hence they represent the vacuum in the anyon lattice of an Abelian Chern--Simons theory. Finally $q_f=q_e+q_m$ has $q_f^T{K}^{-1}{q}_f=1$, so its exchange phase is $e^{i\pi }=-1$. These are exactly the fusion and braiding data of the surface-code anyon model.

The block matrix isIf both $q\in \mathcal{L}$ and $q^{\prime }$ condense, choose their boundary excursions so that the two string operators cross once. Their commutator is their mutual full-braiding phase:Both operators act as the identity on every ground state, so consistency requiresWriting $q^{\prime }=(q_{\mathrm{top}}^{\prime },q_{\mathrm{bot}}^{\prime })$, this saysSet $l_{\mathrm{top}}=M^{-1}{q}_{\mathrm{bot}}^{\prime }$ and choose $l_{\mathrm{bot}}=0$. Then $Kl=(0,q_{\mathrm{bot}}^{\prime }{)}^T$, andTherefore every additionally condensable anyon has the formThe $Kl$ term is a local particle in the anyon lattice of an Abelian Chern--Simons theory, so the condensate is maximal modulo local excitations, as required for a Lagrangian subgroup of Abelian anyons.