spacetime supersymmetric string action and kappa-transformations

I'm currently studying superstring theory from the book 'String Theory and M-Theory' by Becker, Schwarz etc.

In particular, I'm working my way through the spacetime supersymmetric string action in the GS-formalism.

To construct such a spacetime supersymmetric action, one adds a second part, $S_2$ to the action, so that the total $(S_1 + S_2)$ is invariant under kappa-transformations (see page 155).

On page 158 the full variation of the action is written as \begin{align*} \delta S = \frac{4}{\pi} \int d^2 \sigma \epsilon^{\alpha \beta} \left( \delta \overline{\Theta}^1 P_{+} \Gamma_{\mu} \partial_{\alpha} \Theta^{1} - \delta \overline{\Theta}^2 P_{-} \Gamma_{\mu} \partial_{\alpha} \Theta^2 \right) \Pi_{\beta}^{\mu}. \end{align*} where the orthogonal projection operators $P_{\pm}$ are defined by \begin{align*} P_{\pm} = \frac{1}{2} \left( 1 \pm \gamma \right), \qquad \text{where} \quad \gamma = - \frac{ \epsilon^{\alpha \beta} \Pi_{\alpha}^{\mu} \Pi_{\beta}^{\nu} \Gamma_{\mu \nu} }{2 \sqrt{-G}} \end{align*} and $\Gamma_{\mu \nu} = \Gamma_{[ \mu} \Gamma_{\nu ]}$

It is then claimed that this action is invariant under the transformations \begin{align*} & \delta \overline{\Theta}^1 = \overline{\kappa}^1 P_{-} \\ & \delta \overline{\Theta}^2 = \overline{\kappa}^2 P_{+} \end{align*} for abitrary Majorana-Weyl spinors $\kappa^1$ and $\kappa^2$ of appropriate chirality.

However, I don't understand how I can see this explicitly. I.e. if I plug in these transformation in $\delta S$ above, how do I see that $\delta S = 0$? I don't think I understand the action of the operators $P_{\pm}$. What do these orthogonal projection operators do exactly?

Any clarification is welcome.

• Hint: $P_\pm$ are projectors, and therefore $P_+P_-\equiv P_-P_+\equiv 0$. – AccidentalFourierTransform Jun 4 '18 at 21:09
• How do I see the action is spacetime supersymmetric? – Kamil Jun 5 '18 at 11:47