= Closed-string physical-state Virasoro conditions
{title2=$(L_0-1)|\Phi\rangle=(\widetilde L_0-1)|\Phi\rangle=0$}
Critical <closed bosonic strings> have independent left- and right-moving <Virasoro constraints>. Positive modes in both sectors annihilate physical kets, while both zero-mode conditions impose the mass shell and <closed-string level matching>. Quotient by physical <null string states>, equivalently use the appropriate <BRST cohomology>. With $L_0=\alpha'k^2/4+N$, the result is $N=\widetilde N$ and $M^2=4(N-1)/\alpha'$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.
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