The <Polyakov action> contains the dilaton coupling
$$
S_\Phi=\frac1{4\pi}\int_\Sigma\sqrt h\,\Phi(X)R^{(2)}.
$$
For a constant <dilaton> $\Phi_0$, the <Gauss-Bonnet theorem> gives $S_\Phi=\Phi_0\chi(\Sigma)$, so the path-integral weight contributes $e^{-\Phi_0\chi}=g_s^{-\chi}$ with <string coupling> $g_s=e^{\Phi_0}$. A connected closed oriented genus-$g$ worldsheet has $\chi=2-2g$. Including one conventional factor of $g_s$ for each of $n$ external closed-string vertices gives the <string genus expansion>
$$
\boxed{\mathcal A_{g,n}\propto g_s^{,2g-2+n}}.
$$
For four external states on the sphere, this is $g_s^2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 1 / i / Solution

For a connected oriented closed-string worldsheet of genus $g$ with $n$ external closed-string vertices, the perturbative amplitude carries $g_s^{2g-2+n}$. The exponent combines the dilaton factor $g_s^{-\chi}$ for Euler characteristic $\chi=2-2g$ with one $g_s$ from each normalized vertex.


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