Torus ground-state degeneracy of an Abelian Chern--Simons theory

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.

Quasiparticle transport around the two fundamental cycles of a torus gives loop operators obeyingThe induced pairing on ${\mathbb{Z}}^n/K{\mathbb{Z}}^n$ is nondegenerate, so this algebra has a representation of dimension at least $|\det K|$.