= Weyl variation of a nonconstant dilaton coupling
{c}
{title2=$\delta_\omega S_\Phi=-(2\pi)^{-1}\int\sqrt g\,\omega\Delta_g\Phi(X)$}
In two worldsheet dimensions, $\delta_\omega(\sqrt gR)=-2\sqrt g\Delta_g\omega$. 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.
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