= Torus ground-state degeneracy of an Abelian Chern--Simons theory
For an invertible <K-matrix>, an <Abelian Chern--Simons theory> has
$$
\dim\mathcal H(T^2)=|\det K|
$$
topological <ground states> on a spatial <torus>. On a closed orientable surface of <genus of a surface> $g$, the corresponding count is $|\det K|^g$. <Lectures on the Quantum Hall Effect by David Tong>[Sections 5.2.3–5.2.4, printed pp. 166–168, torus loop algebra and higher-genus count]{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> matrix
$$
K=\begin{pmatrix}0&2\\2&0\end{pmatrix},
$$
the 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.
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