 Closed-bosonic-string mass shell from primary weights

A matter vertex with left and right oscillator levels $N,\tilde{N}$ and momentum $k$ has weights $N+{\alpha }^{\prime }{k}^2/4,\tilde{N}+{\alpha }^{\prime }{k}^2/4$. 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.