= Solution
The first massless level of a <closed string> contains a spin-two field, an antisymmetric two-form and a scalar: the <graviton>, <Kalb–Ramond field> and <dilaton>. The polarization redundancies found in Question 2 become, in position space, the linearized transformations $\delta G_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$ and $\delta B_{\mu\nu}=\partial_\mu\Lambda_\nu-\partial_\nu\Lambda_\mu$. Thus the massless spin-two field already carries the gauge symmetry of linearized <general relativity>.
To describe slowly varying backgrounds, promote these polarizations to fields of the embedding coordinates. The Euclidean <string nonlinear sigma model> is
$$
S_\sigma=\frac1{4\pi\alpha'}\int d^2\sigma\left(\sqrt h\,h^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+i\epsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+\alpha'\sqrt h\,R^{(2)}\Phi(X)\right).
$$
Here $\epsilon^{ab}$ is the antisymmetric coordinate density. Quantum consistency requires <worldsheet Weyl anomaly> cancellation, not merely the classical background equations. At first order in $\alpha'$ the <bosonic sigma-model Weyl anomaly coefficients> give
$$
R_{\mu\nu}+2\nabla_\mu\nabla_\nu\Phi-\frac14H_{\mu\rho\sigma}H_\nu{}^{\rho\sigma}=0,\qquad\nabla_\rho(e^{-2\Phi}H^{\rho\mu\nu})=0,
$$
where $H=dB$ is the <Kalb-Ramond field strength>. The scalar anomaly condition, combined with the trace of the metric condition in the critical dimension, gives
$$
R+4\nabla^2\Phi-4(\nabla\Phi)^2-\frac1{12}H^2=0.
$$
These are the Euler–Lagrange equations of the leading <string-frame massless effective action>
$$
S_{\rm eff}^{(s)}=\frac1{2\kappa_0^2}\int d^{26}x\sqrt{-G}\,e^{-2\Phi}\left[R(G)+4(\nabla\Phi)^2-\frac1{12}H^2+O(\alpha')\right].
$$
For example, variation of $B$ gives the divergence equation above. Varying $\Phi$ and integrating its kinetic variation by parts gives $R+4\nabla^2\Phi-4(\nabla\Phi)^2-H^2/12=0$. Combining this with the metric variation yields the Ricci equation. Thus the string's quantum conformal consistency reproduces target-space gravity coupled to matter. Equivalently, expanding string scattering amplitudes at small <momenta> gives the same <action>: massless exchange poles are reproduced by these fields, while analytic terms from massive exchange become local higher-derivative interactions.
The apparent <dilaton>-dependent coefficient of $R$ is a frame choice. Put $\phi=\Phi-\Phi_0$ and, in dimension $D>2$, define the <Einstein-frame metric>
$$
g_{\mu\nu}^{(E)}=e^{-4\phi/(D-2)}G_{\mu\nu},\qquad\kappa^2=\kappa_0^2e^{2\Phi_0}.
$$
To see the cancellation explicitly, write $G=e^{2\omega}g^{(E)}$, $\omega=2\phi/(D-2)$. Then
$$
\sqrt{-G}\,e^{-2\Phi}R(G)=e^{-2\Phi_0}\sqrt{-g^{(E)}}\left[R_E-2(D-1)\nabla_E^2\omega-(D-1)(D-2)(\nabla_E\omega)^2\right].
$$
The Laplacian term is a boundary term. Combining the last term with the original positive <dilaton> kinetic term produces $-4(\nabla_E\phi)^2/(D-2)$. The three inverse metrics in $H^2$ give its remaining exponential factor. Hence
$$
S_{\rm eff}^{(E)}=\frac1{2\kappa^2}\int d^Dx\sqrt{-g^{(E)}}\left[R_E-\frac4{D-2}(\nabla_E\phi)^2-\frac1{12}e^{-8\phi/(D-2)}H_E^2+O(\alpha')\right].
$$
For the uncompactified bosonic theory $D=26$, these coefficients are $-1/6$ and $e^{-\phi/3}$. The Ricci term is now the <Einstein-Hilbert action>; its metric variation gives the <Einstein field equations> with <stress-energy tensor> from the scalar and two-form. \b[The leading low-energy theory is <general relativity> coupled to the massless string fields.]
The approximation requires characteristic <momenta> $E\sqrt{\alpha'}\ll1$ and small curvature in string units. Massive string modes have masses of order $1/\sqrt{\alpha'}$ and can be integrated out, generating corrections with extra powers of $\alpha' E^2$ or $\alpha' R$. The <string genus expansion> supplies independent loop corrections controlled by $g_s=e^{\Phi_0}$; at weak coupling tree-level gravity dominates. Dimensional analysis gives $\kappa_{26}^2\propto g_s^2(\alpha')^{12}$.
<Compactification> supplies lower-dimensional gravity and additional matter: metric and two-form components give <gauge fields>, internal geometry gives scalar <moduli>, and light modes of <Kaluza-Klein theory> must be kept if their masses lie below the chosen cutoff. <Open strings> add <gauge fields>, and supersymmetric string theories add <spacetime> <fermions>. Their particular matter spectrum depends on the <compactification> and projection; universal massless spin-two interactions retain the gravitational form. Finally, the bosonic massless-sector <action> is a formal truncation around a tachyonically unstable vacuum. It does not remove that instability. Tachyon-free superstring backgrounds provide stable applications of this low-energy gravitational interpretation.
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