Dilaton Euler-characteristic weighting ($g_s^{-\chi }=g_s^{2g-2}$)

A constant dilaton contributes ${\Phi }_0\chi (\Sigma )$ to the Euclidean worldsheet action by the Gauss-Bonnet theorem. With string coupling $g_s=e^{{\Phi }_0}$ and Euler characteristic $\chi =2-2g$, the path-integral weight is $g_s^{2g-2}$. Canonically normalized external closed-string vertices add one power of $g_s$ each.