Dilaton Euler-characteristic weighting
= Dilaton Euler-characteristic weighting
{title2=$g_s^{-\chi}=g_s^{2g-2}$}
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.