String vertex operator

A string vertex operator inserts an asymptotic string state at a point of the worldsheet. Its conformal weights encode the physical-state and mass-shell conditions.

These insertions are described by the state-field correspondence of a vertex operator algebra; an individual insertion is not the whole algebra.

An integrated closed-string vertex is inserted as $\int d^{2}zV(z,\bar{z})$. In conformal gauge its matter operator must have conformal weights $(1,1)$ so that the insertion is invariant under worldsheet coordinate changes.

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.

A closed-string polarization tensor specifies the target-spacetime tensor state multiplying $\partial X^{\mu }\bar{\partial }{X}^{\nu }{e}^{ik\cdot X}$. Its decomposition into symmetric trace-free, antisymmetric and trace parts separates the graviton, Kalb–Ramond field and dilaton polarizations.

Changing a massless closed-string polarization by a momentum-longitudinal tensor changes the integrated vertex by a worldsheet total derivative, equivalently by a BRST-exact state. The corresponding target-space transformations are linearized diffeomorphisms and two-form gauge transformations.

A string mass-shell condition relates the target momentum of a vertex operator to its oscillator level. Together with polarization constraints, it makes the matter vertex have the conformal weights required for an integrated insertion.

The bosonic closed-string tachyon vertex is the normal-ordered exponential $:e^{ip\cdot X}:$ integrated over its worldsheet position. Pairwise contractions of such exponentials generate the Koba-Nielsen factor.