Closed-string physical-state Virasoro conditions ($(L_0-1)|\Phi ⟩=({\tilde{L}}_0-1)|\Phi ⟩=0$)

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 $L_0={\alpha }^{\prime }{k}^2/4+N$, the result is $N=\tilde{N}$ and $M^2=4(N-1)/{\alpha }^{\prime }$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.