"Gauge fixed world-sheet action" My question is in reference to the action in equation 4.130 of Becker, Becker and Schwartz.
It reads as,
$$S_{matter}= \frac{1}{2\pi}\int (2\partial X^\mu \bar{\partial}X_\mu + \frac{1}{2}\psi^\mu \bar{\partial} \psi_\mu + \frac{1}{2}\tilde{\psi}^\mu \partial \tilde{\psi}_\mu)d^2z.\tag{4.130}$$

*

*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,
$$T_B(z) = -2\partial X^\mu(z)\partial X_\mu (z) - \frac{1}{2}\psi^\mu(z)\partial \psi _\mu (z) = \sum _{n=-\infty} ^{\infty} \frac{L_n}{z^{n+2}}\tag{4.131}$$
and
$$T_F(z) = 2i\psi^{\mu} (z) \partial X_{\mu} (z) = \sum _{r=-\infty}^{\infty} \frac{G_r}{z^{r+\frac{3}{2}}}.\tag{4.133}$$

*

*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?
 A: *

*Consider Wick-rotating the action back from Euclidean signature into Lorentzian: you replace $\partial\rightarrow \partial_+$, $\bar{\partial}\rightarrow \partial_-$ and the fields are replaced by $\psi\rightarrow\psi_+$ and $\tilde{\psi}\rightarrow\psi_-$.

*The action you wrote has an $N=1$ superconformal symmetry (which, I believe, was one of the first SUSY examples) with generators $T_B(z)$ and $T_F(z)$ (sometimes called $T(z)$ and $G(z)$ respectively). Here $T_B$ is the usual energy-momentum tensor (generator of conformal symmetries), while $T_F$ is the supercurrent (supersymmetry generator). See exercises 4.6 and 4.7 for their derivation ($T_F$ is called $J$ there).

*The conservation of the energy-momentum tensor gives $\bar{\partial} T=0$ on the worldsheet (the punctured complex plane $z\neq 0$), i.e. $T$ is holomorphic. Recall, that $z=\exp(2(\tau+i\sigma))$, so $z=0$ corresponds to $\tau\rightarrow-\infty$. Similarly, conservation of the supercurrent implies that $T_F$ is holomorphic.

*To (partially) fix the worldsheet diffeomorphism invariance, one usually considers the conformal gauge: $g_{ab}=h_{ab} e^\phi$, where $g_{ab}$ is the worldsheet metric, $h_{ab}$ the flat metric and $\phi$ the dilaton, which decouples. In the action you wrote, the worldsheet metric is flat, which is the gauge-fixing in this case.

