 Leading metric-dilaton Weyl condition

For the conventional Euclidean dilaton coupling $+(4\pi {)}^{-1}\int \sqrt{h}R\phi$ and no antisymmetric background, the metric worldsheet Weyl anomaly vanishes at first order in ${\alpha }^{\prime }$ when $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$. The one-loop Ricci term follows from the background field expansion of a string sigma model. Under a Weyl transformation, the dilaton coupling varies as $-(2\pi {)}^{-1}\int \sqrt{h}\omega {\square }_h\phi$; the embedding equations replace this by the target Hessian contribution. Full Weyl invariance also requires the dilaton curvature coefficient, including the central-charge deficit, to vanish. See the Lorentzian dilaton coupling sign convention when translating an action with a specified Lorentzian sign.

The conventional coefficient agrees with the background equations in Tong's string theory notes, Section 7.2.3.