= Solution
Insert the given <closed-string mode expansion> into the stress tensor and compare $T(z)=\sum_nL_nz^{-n-2}$. <Normal ordering> gives
$$
\boxed{L_n=\frac12\sum_{m\in\mathbb Z}:\alpha_{n-m}\cdot\alpha_m:}.
$$
The inverse of the worldsheet Laplacian gives the free-field operator-product expansion
$$
\boxed{X^\mu(z,\bar z)X^\nu(w,\bar w)\sim-\frac{\alpha'}2\eta^{\mu\nu}\log|z-w|^2}.
$$
Hence
$$
\partial X^\mu(z)\partial X^\nu(w)\sim-\frac{\alpha'}2\frac{\eta^{\mu\nu}}{(z-w)^2}.
$$
Extracting the Laurent modes by contour integration yields the <string oscillator> algebra
$$
\boxed{[\alpha_m^\mu,\alpha_n^\nu]=m\eta^{\mu\nu}\delta_{m+n,0}}.
$$
The vanishing of the worldsheet stress tensor becomes the <Virasoro constraints>
$$
L_n|\text{phys}\rangle=\bar L_n|\text{phys}\rangle=0\quad(n>0),
\qquad (L_0-a)|\text{phys}\rangle=(\bar L_0-a)|\text{phys}\rangle=0.
$$
Since $L_0=\alpha'k^2/4+N$ and the bosonic-string normal-ordering constant is $a=1$, these zero-mode constraints are the target-space <string mass-shell condition>
$$
M^2=\frac4{\alpha'}(N-1)=\frac4{\alpha'}(\bar N-1),
\qquad N=\bar N.
$$
Therefore
$$
\boxed{M_T^2=-\frac4{\alpha'},\qquad M_g^2=0.}
$$
The positive-mode constraints further require
$$
\boxed{k^\mu\epsilon_{\mu\nu}=0,\qquad k^\nu\epsilon_{\mu\nu}=0},
$$
and <null string states> identify polarizations that differ by momentum-longitudinal terms. The symmetric trace-free, antisymmetric, and trace sectors of $\epsilon_{\mu\nu}$ describe the graviton, <Kalb–Ramond field>, and <dilaton>, respectively.
Finally, the <state–operator correspondence> maps $\alpha_{-1}^\mu\bar\alpha_{-1}^\nu|k\rangle$ to
$$
\partial X^\mu\bar\partial X^\nu e^{ik\cdot X}.
$$
Adding its integrated <massless closed-string vertex operator> to the <string nonlinear sigma model> changes the background coupling by
$$
\delta S\propto\int d^2z\,\delta G_{\mu\nu}(X)\partial X^\mu\bar\partial X^\nu.
$$
Thus a Fourier mode $\delta G_{\mu\nu}(X)=\epsilon_{(\mu\nu)}e^{ik\cdot X}$ is precisely a linearized deformation of the target-space metric. Its transversality, mass-shell condition, and polarization gauge redundancy are the worldsheet statements that the deformation is marginal and is defined up to a linearized spacetime diffeomorphism.
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