= Background field expansion of a string sigma model
Expand the <string embedding map> along target geodesics as $X=\exp_{\bar X}Y$. The quadratic Euclidean fluctuation action has a <covariant derivative> term and a curvature term $-R_{acbd}Y^aY^b\partial\bar X^c\partial\bar X^d$. The logarithmically divergent coincident contraction is proportional to $\alpha'g^{ab}$; its curvature contraction is the <Ricci tensor>. A compensating metric <counterterm> gives $\beta^g_{ab}=\alpha'R_{ab}+O(\alpha'^2)$. This is why the target gravitational equation arises from quantum <worldsheet Weyl anomaly> cancellation, rather than from the classical worldsheet metric variation. Infrared regulation and removal of equation-of-motion terms by field redefinitions specify the usual one-loop scheme.
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