= Solution
The <Polyakov path integral> integrates over both the <string embedding map> and the <worldsheet metric>. After the usual analytic continuation, a formal closed-string vacuum functional has the schematic form
$$
Z=\sum_{g\ge0}g_s^{2g-2}\int\frac{\mathcal DX\,\mathcal Dh}{\operatorname{Vol}(\operatorname{Diff}\times\operatorname{Weyl})}e^{-S_E[X,h]},\qquad
S_E=\frac1{4\pi\alpha'}\int\sqrt h\,h^{ab}\partial_aX\cdot\partial_bX\,d^2\sigma.
$$
The genus weights organize <string perturbation theory>; the coupling $g_s$ is related to the constant <dilaton>. External states are inserted as <string vertex operators>, with the usual additional external coupling normalization. Division by the gauge group prevents counting different parameterizations and Weyl representatives as distinct configurations.
Gauge-fix $h=e^{2\phi}\widehat h(m)$, where $m$ labels inequivalent conformal structures. <Conformal gauge> fixes the local metric redundancies, not the <worldsheet moduli>. Unmarked compact surfaces have no complex moduli at genus zero, one at genus one and $3g-3$ at genus $g\ge2$. Integrating over the appropriate moduli quotient includes large diffeomorphisms; at genus one it requires a modular fundamental domain, as in <torus modular invariance>.
The infinitesimal traceless diffeomorphism operator is
$$
(P_1c)_{ab}=\nabla_ac_b+\nabla_bc_a-h_{ab}\nabla_dc^d.
$$
Its <Faddeev-Popov determinant> is represented by anticommuting <worldsheet ghost fields> $c^a$ and traceless symmetric $b^{ab}$, with action proportional to $\int\sqrt h\,b^{ab}(P_1c)_{ab}$. In conformal complex coordinates this is the pair of chiral <bc systems> of weights $(2,-1)$. Ghost zero modes must be handled rather than included in an ordinary nonzero determinant: conformal Killing vectors correspond to residual gauge transformations, while antighost zero modes pair with surviving moduli variations. At genus zero three insertion positions can be fixed by the <Möbius transformations>; corresponding $c\widetilde c$ factors supply the ghost zero modes. At higher genus the measure includes antighost insertions paired with the moduli.
In a flat target, integrating the free coordinate fields gives a Gaussian determinant $[\det'(-\Delta_{\widehat h})]^{-d/2}$, with the zero-mode integral supplying target volume or momentum conservation. Vertex insertions give the corresponding free-field correlators. For example, on a sphere, exponential operators have the factor
$$
\left\langle\prod_i:e^{ik_i\cdot X(z_i)}:\right\rangle
\propto\delta^{(d)}\left(\sum_i k_i\right)\prod_{i<j}|z_i-z_j|^{\alpha'k_i\cdot k_j}.
$$
The remaining unfixed insertion positions and moduli are integrated with the ghost measure. Physical closed-string operators have matter conformal weights $(1,1)$; an unintegrated insertion is $c\widetilde c V$, while an integrated insertion uses $\int d^2z\,V$. This explains how on-shell scattering amplitudes arise from the metric-and-embedding functional integral.
The quantum measure can break the classical Weyl symmetry. Each free coordinate contributes central charge one, while each chiral reparameterization ghost system contributes $-26$. Hence $c_{\rm total}=d-26$, and the <worldsheet Weyl anomaly> vanishes for the ordinary flat bosonic string exactly at \b[$d=26$]. Only then can the conformal factor be removed as a gauge degree of freedom without adding further dynamics. For a subcritical matter system it generally produces a Liouville-type theory, unless an appropriate additional conformal sector restores the central charge balance.
Equivalently the gauge-fixed construction uses a nilpotent string <BRST operator>. Physical states are its closed states modulo exact states, expressing the <BRST cohomology> of the gauge constraints. In the critical theory the ghost and matter contributions make this operator nilpotent and the measure gauge independent. Degenerations of moduli describe long-tube propagation and factorization onto intermediate string states. The bosonic tachyon can still cause infrared divergences; anomaly cancellation is not a claim that every vacuum integral is finite or that the bosonic vacuum is stable. \b[The path integral sums embeddings and conformal structures, with ghosts removing local gauge overcounting and moduli retaining the genuine geometry.]
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