 Physical-state Virasoro conditions for an open string

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.