Dilaton ($\Phi$)

The dilaton is a scalar closed-string background field coupled to the worldsheet Ricci scalar. Its constant mode determines the string coupling, while spacetime variation modifies the background-field equations.

In two worldsheet dimensions, ${\delta }_{\omega }(\sqrt{g}R)=-2\sqrt{g}{\Delta }_g\omega$. Thus the standard dilaton curvature coupling does not have classical Weyl invariance for a general spacetime-varying dilaton. A constant dilaton gives the topological Euler characteristic instead. In the quantum sigma model, this classical variation combines with the loop anomaly; vanishing improved sigma-model beta functions is the full background conformal condition.

With Euclidean curvature coupling $(4\pi {)}^{-1}\int \sqrt{h}\phi R$, a constant dilaton contributes ${\phi }_0\chi$ by the Gauss-Bonnet theorem. The genus-$g$ closed-string weight is $e^{-{\phi }_0\chi }=g_s^{2g-2}$ with $g_s=e^{{\phi }_0}$. If the coefficient is instead written as $\int \sqrt{h}\Phi R$, then in the same convention $\phi =4\pi \Phi$. The field normalization must be fixed before reading off the coupling.

With worldsheet signature $(-,+)$, $e^{i{S}_L}=e^{-S_E}$, and the curvature convention $R[e^{2\omega }h]=e^{-2\omega }(R[h]-2{\square }_h\omega )$, the usual negative Lorentzian kinetic action becomes positive in Euclidean signature. A positive Lorentzian term $(4\pi {)}^{-1}\int \sqrt{-h}R\Phi$ becomes a negative Euclidean curvature coupling. Therefore the conventional Euclidean dilaton with positive curvature coupling is $\phi =-\Phi$. The leading metric-dilaton Weyl condition is $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$, or $R_{ab}-2{\nabla }_a{\nabla }_b\Phi =0$ with that literal positive Lorentzian coupling. Keeping the same named field and reversing the action's curvature-coupling sign reverses the Hessian term. The continuation follows directly from $d\tau =-id{\tau }_E$ and $S_E=-i{S}_L$; the geometric scalar curvature continues without an extra arbitrary sign.