= Closed-bosonic-string mass shell from primary weights
{title2=$M^2=4(N-1)/\alpha',\quad N=\widetilde N$}
A matter vertex with left and right oscillator levels $N,\widetilde N$ and momentum $k$ has weights $N+\alpha'k^2/4,\widetilde N+\alpha'k^2/4$. An <integrated string vertex operator> must have weights $(1,1)$, giving the displayed mass shell and level matching with $M^2=-k^2$. Physical states are further identified modulo <null string states>, equivalently through <BRST cohomology>; merely counting all weight-one primaries would include gauge redundancies.
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