 Weyl variation of a nonconstant dilaton coupling

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.