= Transversality of derivative-exponential primary vertices
{title2=$p\cdot\zeta=0,\quad p^\mu\zeta_{\mu\nu}=p^\nu\zeta_{\mu\nu}=0$}
For free embedding bosons, the <operator product expansion> of the stress tensor with $\zeta_\mu:\partial X^\mu e^{ipX}:$ has a cubic pole $-i\alpha'(p\cdot\zeta):e^{ipX}:/2$. Its vanishing is precisely the primary condition, with weight $1+\alpha'p^2/4$. The two-derivative closed-string vertex has the two displayed transversality conditions from its holomorphic and antiholomorphic cubic poles. Being primary does not require $p^2=0$; that extra condition follows when a physical integrated vertex must have <conformal weights> $(1,1)$.
Back to article page
