Integer-level charge identification in complex sine-Gordon theory ($Q\mathrm{mod}k,{\lambda }^2=4\pi /k$)

An additional integer-level completion identifies the charge labels modulo $k$ and has $k-1$ nonzero sectors. For even $k$, the two maximal labels describe one self-conjugate sector. This interpretation is specified in Dorey and Hollowood, section 2; it is additional to the local classical density.