Massless spectrum at the bosonic self-dual circle ($2(N+\tilde{N}-2)+n^2+w^2=0$)

The self-dual circle mass equation and closed-string level matching exhaust the possibilities at level sum zero, one or two. Besides the usual neutral $(N,\tilde{N})=(1,1)$ states, charges $(n,w)=(\pm 1,\pm 1)$ supply singly excited chiral states, with the sector fixed by level matching. Ground-state charges $(\pm 2,0)$ and $(0,\pm 2)$ also become massless. Their compact energy cancels the uncompactified tachyon mass. With $d$ physical transverse components, these families have $d^2+4d+4$ independent polarizations.