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The local gauge symmetries of the Polyakov action are worldsheet diffeomorphisms and Weyl transformations. Under a reparametrization ${\sigma }^{\alpha }\mapsto {\sigma }^{\prime \alpha }$, 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 ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\prime }/2}{k}^{\mu }$, their commutators and normal-ordered generators arewith the same formulas for the tilded sector and zero mixed commutators.

The closed-string physical-state Virasoro conditions in the critical flat-space theory areThey 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, $N=\tilde{N}$, and adding their mass-shell content gives $M^2=4(N-1)/{\alpha }^{\prime }$. 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 ${\sigma }^{-}\mapsto \tilde{f}({\sigma }^{-})$, 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 iswith 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 $N=\tilde{N}=1$, the state ${ϵ}_{\mu \nu }{\alpha }_{-1}^{\mu }{\tilde{\alpha }}_{-1}^{\nu }|k⟩$ 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 ${\alpha }^{\prime }$. Thus both the propagating massless spin-two state and its low-energy interactions reproduce gravity. 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.