= Exceptional massless ground states of a bosonic circle
{title2=$R=|n|\sqrt{\alpha'}/2\ \text{or}\ R=2\sqrt{\alpha'}/|w|$}
In the critical <closed bosonic string>, compactification of one coordinate gives $M^2=n^2/R^2+w^2R^2/\alpha'^2+2(N+\widetilde N-2)/\alpha'$ and $\widetilde N-N+nw=0$. At oscillator level zero, level matching forces $nw=0$. Thus a pure <momentum> <ground state> becomes a massless <scalar field> at $R=|n|\sqrt{\alpha'}/2$, while a pure winding <ground state> does so at $R=2\sqrt{\alpha'}/|w|$. These are threshold states descended from the bosonic <tachyon>; they are additional to the neutral <graviton>, two-form, vectors, <dilaton> and radius modulus. At the <self-dual circle>, the four charges $(\pm2,0),(0,\pm2)$ contribute to the enhanced massless <scalar field> spectrum.
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