Virasoro constraint

The worldsheet metric equation in conformal gauge sets the stress tensor to zero, $T_{++}=T_{--}=0$. In oscillator language these are the Virasoro constraints.

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.

The covariant physical-state conditions impose positive-mode Virasoro constraints on kets and the adjoint conditions on bras, with intercept $a$ in the zero mode. They are a string version of Gupta-Bleuler quantization. Imposing every positive and negative matter mode on the same state would conflict with the Virasoro central extension. In critical bosonic string theory, $a=1$ and the matter central charge is $26$, canceled by the reparameterization ghosts.

With the oscillator symplectic form of an open string, the quadratic generators $L_n={\sum }_k{\alpha }_k\cdot {\alpha }_{n-k}/2$ obey $\{{\alpha }_k,L_n\}=-ik{\alpha }_{k+n}$. Their Poisson brackets close without a central term, so they are first-class constraints. Quantization requires normal ordering and can produce the Virasoro central extension.