path integral quantization

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We would like to perform the path integral

$Z = \int\!\mathcal{D}X\,\mathcal{D} g\, e^{-\frac{1}{4\pi\alpha'}\int\!d^2\sigma\,\sqrt{g}g^{ab}\partial_a X^\mu \partial_b X_\mu}\,$ ,

however, since the action is diffeomorphism and Weyl invariant, the integral over $g_{ab}\,$ overcounts equivalent configurations and is indeed divergent. We would be better off to fix the gauge to (say) the conformal gauge $g_{ab} = e^{\phi} \delta_{ab}\,$ or the unit gauge $g_{ab} = \delta_{ab}\,$ . Furthermore, to ensure that such a change of metric leaves the path integral unchanged, we need to be aware of any anomalies that may arise.