  We consider charge superselection sectors of two-dimensional quantum spin
models corresponding to cone localisable charges, and prove that the number of
equivalence classes of such charges is bounded by the Kosaki-Longo index of an
inclusion of certain observable algebras. To demonstrate the power of this
result we apply the theory to the toric code on a 2D infinite lattice. For this
model we can compute the index of this inclusion, and conclude that there are
four distinct irreducible charges in this model, in accordance with the
analysis of the toric code model on compact surfaces. We also give a sufficient
criterion for the non-degeneracy of the charge sectors, in the sense that
Verlinde's matrix S is invertible.
