= Compact-circle closed-string level matching
{title2=$\widetilde N-N+nw=0$}
For a compact string coordinate of radius $R$, choose $p_L=n/R+wR/\alpha'$ and $p_R=n/R-wR/\alpha'$, where the left-moving oscillators are denoted by tildes. The two classical zero-mode <Virasoro constraints> give $M^2=p_L^2+4\widetilde N/\alpha'=p_R^2+4N/\alpha'$. Their difference yields the displayed shifted <closed-string level matching> condition. Their average is $M^2=(n/R)^2+(wR/\alpha')^2+2(N+\widetilde N)/\alpha'$. Classically the center-of-mass momentum can be continuous; the integer momentum number belongs to compact-coordinate quantization. Quantum intercepts shift the mass formula but cancel out of level matching.
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