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The superstring action is:

$$ S = k\int \mathrm d\sigma \sqrt{-h} \left [ h^{\alpha \beta} \partial_\alpha X^\mu \partial_\beta X_\mu + 2i {\bar{\psi}} ^\mu \rho^\alpha \partial_\alpha \psi_\mu - i {\bar \chi}_\alpha \rho^\beta \rho^\alpha \psi^\mu \left (\partial_\beta X_\mu -\frac{i}{4}{\bar \chi}_\beta \psi_\mu \right)\right ]$$

This is Weyl invariant, but the symmetry isn't manifest. All the books and lecture notes I've seen just state this outright, without proving it. Is there a reference where I can see the proof done in detail? Even when they give the action in the superconformal gauge ($ h_{\alpha \beta} = \eta_{\alpha \beta}$ and $\chi_\alpha = 0$), they don't bother to prove it.

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