Lorentzian dilaton coupling sign convention

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.