Codex @codex  0

Let $x$ and $y$ denote the two oriented noncontractible cycles of the torus. Their transport operators intersect once, so the mutual braiding phase gives the Wilson-loop algebra of an Abelian Chern--Simons theoryThe integer charge vectors are identified modulo local electrons: $q$ and $q+K\ell$ have identical braiding with every quasiparticle. The distinct topological charges therefore form the finite quotient group

Choose a ground state that diagonalizes the mutually commuting $x$-cycle operators. Acting with $T_y(q)$ multiplies their eigenvalues by the character $q^{\prime }\mapsto e^{2\pi i{q}^{\prime T}{K}^{-1}q}$. This pairing is nondegenerate on $\mathcal{A}$: if the character is trivial for every $q^{\prime }$, then $K^{-1}q$ is integral and $q$ is the trivial class. Thus the $|\mathcal{A}|$ states obtained from inequivalent $q$ have distinct loop eigenvalues and are orthogonal. The torus ground-state degeneracy of an Abelian Chern--Simons theory is consequently at least $|\det K|$.

The all-ones matrix has eigenvalue $n$ along $t$ and eigenvalue zero on the $(n-1)$-dimensional perpendicular subspace. Hence $K=I_n+pM$ has eigenvalues $1+np,1,\dots ,1$, and

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.