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The Polyakov action contains the dilaton couplingFor 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 expansionFor four external states on the sphere, this is $g_s^2$.

In Euclidean worldsheet signature, the string nonlinear sigma model action can be writtenThe factor of $i$ in the B-field term is absent in Lorentzian signature. Here $G$ is the target metric, $B$ the Kalb–Ramond field, and $\Phi$ the dilaton.

The string coupling controls topology change and string splitting and joining. A constant background dilaton ${\Phi }_0$ sets $g_s=e^{{\Phi }_0}$.

The string nonlinear sigma model couples the worldsheet embedding fields to target-space backgrounds such as a metric, a Kalb–Ramond field and a dilaton. Quantum Weyl invariance requires its beta functions to vanish.