"gauge fixed world-sheet action"

Question

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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Answer 1

• 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.

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Comments

I am getting quite confused as to how they keep shifting between the "+-" notation and the "\alpha \beta" notation. I have seen those 4.6 and 4.7 exercises there in 4.6 they seem to be postulating a "new" definition of what is $T_{++}$ and what is $T_{--}$ and its not clear as to how or why they are related to $T_B(z)$. And what happened to our familar definition of the stress-tensor as $T_{\mu \nu} = (\partial_{\mu} \phi)\frac{\partial L}{\partial (\partial^{\nu} \phi)} - \eta_{\mu \nu}L$ ?

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What I typed above seems to have gotten completely messed up! Can you kindly be a little more explicit. In 4.6 excersise for example they claim $\delta_{+} (\psi_{-}\partial_{+} \psi_{-} + \psi_{+}\partial_{-} \psi_{+}) = \partial_+(\psi_{+}\partial_{-} \psi_{+}) - \partial_{-} (\psi_{+} \partial_{+} \psi_{+})$ This is not clear to me!

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Everything I am writing seems to get messed up! Can you kindly fill in some more details about those exercises - I have seen them but got lost midway.

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@user6818, +- refers to the light-cone coordinates: $x_+ = x_0 + x_1$, $x_- = x_0 - x_1$, which become $z$ and $\bar{z}$ in Euclidean signature. The definition of $T$ is simple: it is the Noether current associated to the $z\rightarrow z+\epsilon(z)$ symmetry. What you see in the book is just a trick for computing it. The formula you write for $T_{\mu\nu}$ is correct, but you should be careful with two things: $\psi$ anticommute and $\eta_{+-}=\eta{-+}=1/2$ (the diagonal entries are zero).

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good answer, +1. User: your first comment got invalidated because you wrote a quotation mark instead of a dollar in front of the first $T_{++}$ in it. So all maths and non-maths got exactly reverted afterwards. You still had time to edit it and fix it, a few minutes!

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I took the liberty of fixing the Tex, there's got to be a better way...

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