Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 4 / d / Solution

The zero-mode condition is automatic:Using $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$ and oscillator annihilation of the vacuum giveswhich vanishes when $t_{\mu \nu }{\alpha }_0^{\nu }=-v_{\mu }$. Similarly,which vanishes when $v\cdot {\alpha }_0=-t^{\mu }{}_{\mu }/2$. Higher positive Virasoro modes annihilate a level-two state. These are precisely the stated physical-state conditions.