= Lorentzian dilaton coupling sign convention
{c}
With worldsheet signature $(-,+)$, $e^{iS_L}=e^{-S_E}$, and the curvature convention $R[e^{2\omega}h]=e^{-2\omega}(R[h]-2\Box_h\omega)$, 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 $\varphi=-\Phi$. The <leading metric-dilaton Weyl condition> is $R_{ab}+2\nabla_a\nabla_b\varphi=0$, 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 $d\tau=-i\,d\tau_E$ and $S_E=-iS_L$; the geometric scalar curvature continues without an extra arbitrary sign.
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