Polyakov path integral

The Polyakov path integral integrates over string embeddings and worldsheet metrics, divided by worldsheet diffeomorphisms and Weyl transformations. Gauge fixing introduces worldsheet ghosts and moduli integrals.

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.

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.