= Solution
Choose a Euclidean <string worldsheet> and the usual normalization of the <Polyakov action>. For a closed oriented worldsheet $\Sigma$, the <string nonlinear sigma model> is
$$
\boxed{\begin{aligned}
S={}&\frac1{4\pi\alpha'}\int_\Sigma d^2\sigma\sqrt g\,g^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu\\
&+\frac{i}{4\pi\alpha'}\int_\Sigma d^2\sigma\,\varepsilon^{ab}B_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu
+\frac1{4\pi}\int_\Sigma d^2\sigma\sqrt g\,\Phi(X)R^{(2)}.
\end{aligned}}
$$
Here $\varepsilon^{12}=1$ is the alternating density, not the tensor with components $1/\sqrt g$. The imaginary coefficient in the <Kalb–Ramond field> term is the Euclidean continuation; in Lorentzian signature this term is real. $G$ is the spacetime metric and $\Phi$ the <dilaton>.
All three terms have <worldsheet diffeomorphism> invariance, and are covariant under <spacetime> coordinate transformations when $X$, $G$, $B$ and $\Phi$ are transformed together. Under a <Weyl transformation> $g_{ab}\mapsto e^{2\omega}g_{ab}$, $\sqrt g\,g^{ab}$ is invariant in two dimensions, so the metric coupling has classical <Weyl invariance>. The antisymmetric coupling is independent of the <worldsheet metric>, so it too has classical <Weyl invariance>.
The curvature coupling requires care. An infinitesimal <Weyl transformation> gives $\delta_\omega(\sqrt gR^{(2)})=-2\sqrt g\,\Delta_g\omega$. Integration by parts on the closed worldsheet therefore gives the <Weyl variation of a nonconstant dilaton coupling>:
$$
\delta_\omega S_\Phi=-\frac1{2\pi}\int_\Sigma\sqrt g\,\Phi(X)\Delta_g\omega=-\frac1{2\pi}\int_\Sigma\sqrt g\,\omega\Delta_g\Phi(X).
$$
\b[A constant dilaton coupling is topological and Weyl invariant; a general varying dilaton coupling is not separately classically Weyl invariant.] Its classical variation participates in canceling the quantum <worldsheet Weyl anomaly>, producing the improved background equations below.
To relate these background couplings to the flat-space <bosonic string mass spectrum>, expand $G_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}(X)$ and express $h$ and $B$ as superpositions of plane waves. The perturbation of the action is a sum of <integrated string vertex operators> of the form
$$
\int d^2z\,\epsilon_{\mu\nu}:\partial X^\mu\bar\partial X^\nu e^{ip\cdot X}:.
$$
This is the vertex of $\alpha_{-1}^\mu\widetilde\alpha_{-1}^\nu|p\rangle$ in the <massless first closed-string level>. Its <conformal weights> are $(1+\alpha'p^2/4,1+\alpha'p^2/4)$, so marginality gives $p^2=0$. The <primary operator> conditions give transversality, and the <string-state gauge redundancy> removes longitudinal components. The symmetric trace-free <polarization tensor> is the <graviton> and perturbs $G$; its antisymmetric part is the <Kalb–Ramond field> and perturbs $B$. The scalar physical polarization is the <dilaton>. At the level of background fields its vertex involves the metric trace and the curvature improvement, so an arbitrary off-shell metric trace should not simply be identified with the physical dilaton without this distinction.
The spacetime <gauge invariance> of the antisymmetric coupling is
$$
\boxed{B\longmapsto B+d\Lambda,\qquad \delta B_{\mu\nu}=\partial_\mu\Lambda_\nu-\partial_\nu\Lambda_\mu,\qquad H=dB.}
$$
Indeed its action is $S_B=i(2\pi\alpha')^{-1}\int_\Sigma X^*B$. Its variation is
$$
\delta S_B=\frac{i}{2\pi\alpha'}\int_\Sigma d(X^*\Lambda)=\frac{i}{2\pi\alpha'}\int_{\partial\Sigma}X^*\Lambda=0
$$
for a closed worldsheet, by <Stokes theorem>. In components this is the total derivative $\partial_a(\varepsilon^{ab}\Lambda_\nu(X)\partial_bX^\nu)$: the term containing $\partial_a\partial_bX^\nu$ vanishes by antisymmetry. Thus only the <Kalb-Ramond field strength> $H$ can enter the local spacetime equations. With boundaries, a compensating boundary gauge-field transformation is needed.
Separate the <dilaton> into a constant $\Phi_0$ and a varying part. The <Gauss-Bonnet theorem> gives
$$
\frac1{4\pi}\int_\Sigma\sqrt g\,R^{(2)}=\chi(\Sigma)=2-2g
$$
for a connected closed orientable genus-$g$ worldsheet. Its Euclidean path-integral weight contains
$$
\boxed{e^{-S_{\Phi_0}}=e^{-(2-2g)\Phi_0}=g_s^{2g-2},\qquad g_s=e^{\Phi_0}.}
$$
Consequently the sum over worldsheet topologies is the <string genus expansion> $\sum_{g\ge0}g_s^{2g-2}\mathcal A_g$, with each coefficient integrating over that genus's <worldsheet moduli>. Each extra handle costs $g_s^2$, which is the loop-counting factor of <string perturbation theory>. With the usual normalization of $n$ external closed-string vertices, the connected amplitude carries $g_s^{2g-2+n}$. This <dilaton Euler-characteristic weighting> is distinct from the expansion in $\alpha'$ of a fixed-worldsheet <string nonlinear sigma model>.
Finally quantize that two-dimensional theory. The background fields are its couplings. The <background field expansion of a string sigma model>, $X=x+\xi$ in target-space <normal coordinates>, contains curvature vertices quadratic in the fluctuation $\xi$. Contracting their two fluctuation indices produces the <Ricci tensor> multiplying $\partial x^\mu\bar\partial x^\nu$. Logarithmic short-distance contractions renormalize $G$, $B$, and $\Phi$, giving functional <sigma-model beta functions>. Gauge independence requires vanishing of the <worldsheet Weyl anomaly>, including the classical improvement from the varying <dilaton>, not merely an arbitrary choice of renormalization scale.
For clarity, in conventional leading-order normalization the improved coefficients are
$$
\begin{aligned}
\overline\beta^G_{\mu\nu}&=\alpha'\left(R_{\mu\nu}-\frac14H_{\mu\rho\sigma}H_\nu{}^{\rho\sigma}+2\nabla_\mu\nabla_\nu\Phi\right)+O(\alpha'^2),\\
\overline\beta^B_{\mu\nu}&=\alpha'\left(-\frac12\nabla^\rho H_{\rho\mu\nu}+\nabla^\rho\Phi\,H_{\rho\mu\nu}\right)+O(\alpha'^2),\\
\overline\beta^\Phi&=\frac{D-26}{6}+\alpha'\left((\nabla\Phi)^2-\frac12\nabla^2\Phi-\frac1{24}H^2\right)+O(\alpha'^2).
\end{aligned}
$$
Here $H^2=H_{\mu\nu\rho}H^{\mu\nu\rho}$. The constant term in the <dilaton> coefficient is the matter-plus-ghost <central charge> deficit. At $D=26$, setting these coefficients to zero gives the leading spacetime field equations
$$
\boxed{\begin{aligned}
R_{\mu\nu}-\frac14H_{\mu\rho\sigma}H_\nu{}^{\rho\sigma}+2\nabla_\mu\nabla_\nu\Phi&=0,\\
\nabla^\rho(e^{-2\Phi}H_{\rho\mu\nu})&=0,\\
R+4\nabla^2\Phi-4(\nabla\Phi)^2-\frac1{12}H^2&=0.
\end{aligned}}
$$
For the last equation, take the trace of the first, $R-H^2/4+2\nabla^2\Phi=0$, and combine it with $\overline\beta^\Phi=0$; this verifies the coefficients and signs. These are equivalently the equations from the leading <string-frame massless effective action>
$$
S_{\rm eff}\propto\int d^{26}x\sqrt{-G}\,e^{-2\Phi}\left(R+4(\nabla\Phi)^2-\frac1{12}H^2\right).
$$
Higher worldsheet orders give higher-derivative $\alpha'$ corrections, whereas additional handles give string-loop corrections controlled by $g_s$. Thus quantum consistency of a two-dimensional <string nonlinear sigma model> determines the dynamics of its spacetime background fields. These leading coefficient conventions agree with https://arxiv.org/html/0908.0333v3[the primary string-theory lectures' background-field calculation].
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