= Leading metric-dilaton Weyl condition
{title2=$\overline\beta^g_{ab}=\alpha'(R_{ab}+2\nabla_a\nabla_b\varphi)$}
For the conventional Euclidean <dilaton> coupling $+(4\pi)^{-1}\int\sqrt h\,R\varphi$ and no antisymmetric background, the metric <worldsheet Weyl anomaly> vanishes at first order in $\alpha'$ when $R_{ab}+2\nabla_a\nabla_b\varphi=0$. 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 $-(2\pi)^{-1}\int\sqrt h\,\omega\Box_h\varphi$; 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 https://davidtong.org/pdfs/teaching/string-theory/string7.pdf[Tong's string theory notes, Section 7.2.3].
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