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In the path integral of the bosonic string, we fix the gauge by setting the metric $ h $ to a reference metric $ \hat{h} $. A common choice is the conformal gauge: \begin{equation} h_{\alpha \beta} \longrightarrow \hat{h}_{\alpha \beta} = e^{2\phi(\sigma)} \eta_{\alpha \beta}, \end{equation} or, when the entire worldsheet has topological obstructions that do not allow a flat metric: \begin{equation} h_{\alpha \beta} \longrightarrow \hat{h}_{\alpha \beta} = e^{2\phi(\sigma)} g_{\alpha \beta}. \end{equation} Here $ g $ is some fixed metric. This appears in the string theory books by BLT (see Eq.~(3.66)) and Polchinski (before Eq.~(3.3.24)).

Why don't we use a further Weyl transformation to remove the factor $ e^{2\phi(\sigma)} $ and directly fix the metric to $\eta_{\alpha \beta} $ or $ g_{\alpha \beta} $?

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You can’t just fix the metric to be flat. The Weyl symmetry is broken at the quantum level, which makes the conformal factor (the Liouville field) dynamical, and on top of that, surfaces with handles or holes have genuine geometric moduli that survive any gauge fixing. The correct gauge choice is $$ h_{\alpha\beta} = e^{2\phi} \hat{h}_{\alpha\beta}(m) $$

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