 Massless closed-string vertex operator

At the first massless closed-string level, the matter vertex has the form ${ϵ}_{\mu \nu }:\partial X^{\mu }\bar{\partial }{X}^{\nu }{e}^{ik\cdot X}:$, with null momentum and a polarization tensor transverse in both indices. Its symmetric trace-free, antisymmetric and trace sectors describe the graviton, Kalb–Ramond field and dilaton.