= Massless spectrum at the bosonic self-dual circle
{title2=$2(N+\widetilde N-2)+n^2+w^2=0$}
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 $(N,\widetilde N)=(1,1)$ states, charges $(n,w)=(\pm1,\pm1)$ supply singly excited chiral states, with the sector fixed by level matching. Ground-state charges $(\pm2,0)$ and $(0,\pm2)$ 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.
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