My question is in reference to the action in equation 4.130 of Becker, Becker and Schwartz.
It reads as,

$S_{matter}= \frac{1}{2\pi}\int (2\partial X^\mu \bar{\partial}X_\mu + \frac{1}{2}\psi^\mu \bar{\partial} \psi_\mu + \frac{1}{2}\tilde{\psi}^\mu \partial \tilde{\psi}_\mu)d^2z$

Its not clear to me as to why this should be the same as the Gervais-Sakita (GS) action as it seems to be claimed to be. Firstly what is the definition of $\tilde{\psi}$? (..no where before in that book do I see that to have been defined..) Their comment just below the action is that this is related to the $\psi_+$ and $\psi_-$ defined earlier but then it doesn't reduce to the GS action.

What is the definition of the "bosonic energy momentum tensor" ($T_B(z)$) and the "fermionic energy momentum tensor" ($T_F(z)$)? I don't see that defined earlier in that book either.

I am not able to derive from the above action the following claimed expressions for the tensors as in equation 4.131 and 4.133,

$T_B(z) = -2\partial X^\mu(z)\partial X_\mu (z) - \frac{1}{2}\psi^\mu(z)\partial \psi _\mu (z) = \sum _{n=-\infty} ^{\infty} \frac{L_n}{z^{n+2}}$

and

$T_F(z) = 2i\psi^{\mu} (z) \partial X_{\mu} (z) = \sum _{r=-\infty}^{\infty} \frac{G_r}{z^{r+\frac{3}{2}}}$

It would be helpful if someone can motivate the particular definition of $L_n$ and $G_r$ as above and especially as to why this $T_B(z)$ and $T_F(z)$ are said to be holomorphic when apparently in the summation expression it seems that arbitrarily large negative powers of $z$ will occur - though I guess unitarity would constraint that.

Why is this action called "gauge-fixed"? In what sense is it so?

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