= Solution
The local gauge symmetries of the <Polyakov action> are <worldsheet diffeomorphisms> and <Weyl transformations>. Under a reparametrization $\sigma^\alpha\mapsto\sigma'^\alpha$, the embedding coordinates $X^\mu$ transform as scalars and the <worldsheet metric> as a covariant tensor; the contracted integrand is a scalar density. Infinitesimally, with the active convention, $\delta X=-\xi^\alpha\partial_\alpha X$ and $\delta\gamma=-\mathcal L_\xi\gamma$. A <Weyl transformation> sends $\gamma_{\alpha\beta}\mapsto e^{2\omega}\gamma_{\alpha\beta}$ and leaves $X$ fixed. In two dimensions the factors from $\sqrt{-\gamma}$ and $\gamma^{\alpha\beta}$ cancel. Target-space Poincaré transformations are additional global symmetries, rather than <worldsheet> gauge symmetries.
These gauge freedoms put the metric locally in <conformal gauge>, $\gamma_{\alpha\beta}=\eta_{\alpha\beta}$. Its variation must still be imposed: it gives the <worldsheet stress tensor> constraints $T_{++}=T_{--}=0$. The field equation is the <wave equation>, and periodicity of a <closed string> gives independent left- and right-moving oscillator sets. In a normalization with $\alpha_0^\mu=\widetilde\alpha_0^\mu=\sqrt{\alpha'/2}\,k^\mu$, their commutators and normal-ordered generators are
$$
[\alpha_m^\mu,\alpha_n^\nu]=m\eta^{\mu\nu}\delta_{m+n,0},\qquad
L_n=\frac12\sum_{r\in\mathbb Z}:\alpha_{n-r}\cdot\alpha_r:,
\qquad L_0=\frac{\alpha'k^2}{4}+N,
$$
with the same formulas for the tilded sector and zero mixed commutators.
The <closed-string physical-state Virasoro conditions> in the critical flat-space theory are
$$
\boxed{L_{n>0}|\Phi\rangle=\widetilde L_{n>0}|\Phi\rangle=0,
\quad(L_0-1)|\Phi\rangle=(\widetilde L_0-1)|\Phi\rangle=0.}
$$
They act on the oscillator <Fock space>, and physical spurious (null) states are quotiented out. Equivalently physical states are the suitable <BRST cohomology> classes. Positive modes constrain kets; their adjoints constrain bras, so one does not impose every positive and negative matter mode on the same ket. Subtracting the two zero-mode equations gives <closed-string level matching>, $N=\widetilde N$, and adding their mass-shell content gives $M^2=4(N-1)/\alpha'$. Quantum consistency fixes the intercept to one and, for just these flat coordinate fields, $d=26$: the matter <central charge> $d$ cancels the reparametrization ghost <central charge> $-26$. Other dimensions require additional <worldsheet> degrees of freedom or a noncritical completion.
For the residual symmetry, take $\sigma^\pm=\tau\pm\sigma$. The conformal Killing equations leave independent transformations $\sigma^+\mapsto f(\sigma^+)$ and $\sigma^-\mapsto\widetilde f(\sigma^-)$, with a compensating Weyl rescaling restoring the flat metric. On the Euclidean plane the corresponding holomorphic vector fields are $\ell_n=-z^{n+1}\partial_z$ and their antiholomorphic partners. They satisfy the <Witt algebra> $[\ell_m,\ell_n]=(m-n)\ell_{m+n}$. The conserved contour generators are the Virasoro modes $L_n=(2\pi i)^{-1}\oint z^{n+1}T(z)\,dz$. Their quantum <Virasoro algebra> is
$$
[L_m,L_n]=(m-n)L_{m+n}+\frac d{12}(m^3-m)\delta_{m+n,0},
\qquad[L_m,\widetilde L_n]=0,
$$
with a second independent copy for the right movers. These are the generators of the residual conformal transformations, including their quantum central extension.
The <two-dimensional quantum-gravity interpretation of strings> follows from what is integrated, not from an analogy: at genus $h$, the Polyakov path integral integrates over both $X$ and the metric, divided by diffeomorphism and Weyl gauge volume. <String perturbation theory> sums these surface topologies, weighted by $g_s^{2h-2}$ before external-state coupling factors. Gauge fixing leaves <worldsheet> ghost determinants and integrals over <worldsheet moduli>. Thus it is <two-dimensional quantum gravity> coupled to the $d$ coordinate matter fields. A flat gauge is local; general higher-genus surfaces retain conformal-structure moduli and need not admit a globally flat representative.
Target-space gravity is a different conclusion. At $N=\widetilde N=1$, the state $\epsilon_{\mu\nu}\alpha_{-1}^\mu\widetilde\alpha_{-1}^\nu|k\rangle$ is massless, transverse in both indices, and has longitudinal gauge identifications. Its symmetric traceless transverse polarization is a <graviton>; the antisymmetric and scalar polarizations give the <Kalb–Ramond field> and <dilaton>. The <graviton> gauge identification is precisely the linearized metric diffeomorphism. Moreover the leading <string-frame massless effective action> contains $\int\sqrt{-G}e^{-2\Phi}R$, and <worldsheet> Weyl invariance with constant <dilaton> and zero two-form requires $R_{\mu\nu}=0$ at leading order in $\alpha'$. Thus both the propagating massless spin-two state and its low-energy interactions reproduce gravity. \b[<Closed strings> contain the <graviton> and yield Einstein gravity in their low-energy massless sector], together with other fields and higher-derivative corrections; the bosonic <tachyon> instability remains.
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