Weyl variation of a nonconstant dilaton coupling (${\delta }_{\omega }{S}_{\Phi }=-(2\pi {)}^{-1}\int \sqrt{g}\omega {\Delta }_g\Phi (X)$)

In two worldsheet dimensions, ${\delta }_{\omega }(\sqrt{g}R)=-2\sqrt{g}{\Delta }_g\omega$. Thus the standard dilaton curvature coupling does not have classical Weyl invariance for a general spacetime-varying dilaton. A constant dilaton gives the topological Euler characteristic instead. In the quantum sigma model, this classical variation combines with the loop anomaly; vanishing improved sigma-model beta functions is the full background conformal condition.