String-frame metric ($g_{ab}^{(s)}$)

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.

For $D>2$, rescale the string-frame metric by $g^{(E)}=e^{-4\phi /(D-2)}{g}^{(s)}$. The Weyl transformation of the Ricci scalar makes the coefficient of the Einstein-frame Ricci term independent of the dilaton. Indeed $g^{(s)}=e^{4\phi /(D-2)}{g}^{(E)}$ contributes factors $e^{2D\phi /(D-2)}$ from the volume, $e^{-2\phi }$ from the original coupling and $e^{-4\phi /(D-2)}$ from the Ricci term; their product is one. Derivative terms supply the transformed dilaton kinetic term. The two-dimensional case is excluded from this particular rescaling.