Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 3 / a / Solution

In the Polyakov path integral, fix worldsheet diffeomorphism and Weyl symmetry to conformal gauge. The Faddeev-Popov determinant supplies the worldsheet ghosts, and anomaly cancellation selects $D=26$. Insert $m$ integrated closed-string tachyon vertex operators $e^{i{p}_i\cdot X}$ on the sphere. The zero mode of $X$ gives ${\delta }^{(26)}({\sum }_i{p}_i)$, while Gaussian contractions give the Koba-Nielsen factor ${\prod }_{j<l}|z_j-z_l{|}^{{\alpha }^{\prime }{p}_j\cdot p_l}$. Dividing by the conformal Killing group $SL(2,\mathbb{C})/{\mathbb{Z}}_2$ and using the sphere power of the string coupling gives the displayed amplitude with $g_s^{m-2}$.