1.1 Subchapter 1
may change by a Kähler transformation; see (13.57). However, we saw in Ex. 17.1 that Kähler transformations of N(X, \bar{X}) are forbidden. So N must be gauge invariant. As in global supersymmetry, the superpotential \mathcal{W} must be gauge invariant.
Exercise 17.2 Show that these statements lead to the following equations:
\begin{aligned}
&\partial_{\bar{j}} k_A{}^I = 0, \quad X^J \partial_J k_A{}^I = k_A{}^I, \\
&N_I k_A{}^I + N_{\bar{I}} \bar{k}_A{}^{\bar{I}} = 0, \\
&\mathcal{P}_A = \frac{1}{2}i \left( N_I k_A{}^I - N_{\bar{I}} \bar{k}_A{}^{\bar{I}} \right) = i N_I k_A{}^I = -i N_{\bar{I}} \bar{k}_A{}^{\bar{I}}, \quad \partial_{\bar{I}} \mathcal{P}_A = i N_{J \bar{I}} k_A{}^J, \\
&\mathcal{W}_I k_A{}^I = 0.
\end{aligned} \tag{17.18}
The real quantities \mathcal{P}_A are the moment maps associated with holomorphic symmetries of Kähler manifolds, as discussed in Sec. 13.4.1. Note that \mathcal{P}_A has (Weyl, chiral) weight (2, 0), and can thus not be changed by adding constants.
We now present the full expressions for the three superconformal Lagrangians of (17.15). The D-terms are obtained from (17.7) and the F-terms from (16.35), using components that are covariantizations of those found for global supersymmetry in Sec. 14.2. This leads to
\begin{aligned}
[N]_D e^{-1} &= N_{I\bar{J}} \left( -\mathcal{D}_\mu X^I \mathcal{D}^\mu \bar{X}^{\bar{J}} - \frac{1}{2} \bar{\Omega}^I \mathcal{D} \Omega^{\bar{J}} - \frac{1}{2} \bar{\Omega}^{\bar{J}} \mathcal{D} \Omega^I + F^I \bar{F}^{\bar{J}} \right) \\
&+ \frac{1}{2} \left[ N_{I\bar{J}\bar{K}} \left( -\bar{\Omega}^I \Omega^{\bar{J}} \bar{F}^{\bar{K}} + \bar{\Omega}^I (\mathcal{D} X^{\bar{J}}) \Omega^{\bar{K}} \right) + \text{h.c.} \right] \\
&+ \frac{1}{4} N_{I\bar{J}K\bar{L}} \bar{\Omega}^I \Omega^{\bar{J}} \bar{\Omega}^{\bar{K}} \Omega^{\bar{L}} \\
&- i N_I k_A{}^I D^A - \sqrt{2} \bar{\lambda}^A N_{I\bar{J}} \left( \Omega^I \bar{k}_A{}^{\bar{J}} + \Omega^{\bar{J}} k_A{}^I \right) \\
&+ \left[ \frac{1}{2\sqrt{2}} \bar{\psi} \cdot \gamma \left( N_{I\bar{J}} F^I \Omega^{\bar{J}} - N_{I\bar{J}} \mathcal{D} \bar{X}^{\bar{J}} \Omega^I - \frac{1}{2} N_{I\bar{J}\bar{K}} \Omega^{\bar{K}} \bar{\Omega}^I \Omega^{\bar{J}} + N_I P_R \lambda^A k_A{}^I \right) \right. \\
&+ \left. \frac{1}{8} i \varepsilon^{\mu\nu\rho\sigma} \bar{\psi}_\mu \gamma_\nu \psi_\rho \left( N_I \mathcal{D}_\sigma X^I + \frac{1}{2} N_{I\bar{J}} \bar{\Omega}^I \gamma_\sigma \Omega^{\bar{J}} + \frac{1}{\sqrt{2}} N_I \bar{\psi}_\sigma \Omega^I \right) + \text{h.c.} \right] \\
&+ \frac{1}{6} N \left( -R(\omega) + \frac{1}{2} \bar{\psi}_\mu \gamma^{\mu\nu\rho} R'_{\nu\rho}(Q) \right) - \frac{1}{6\sqrt{2}} \left( N_I \bar{\Omega}^I + N_{\bar{I}} \bar{\Omega}^{\bar{I}} \right) \gamma^{\mu\nu} R'_{\mu\nu}(Q),
\end{aligned}
[\mathcal{W}]_F e^{-1} = \mathcal{W}_I F^I - \frac{1}{2} \mathcal{W}_{IJ} \bar{\Omega}^I \Omega^J + \mathcal{W}_I \bar{\psi} \cdot \gamma \Omega^I + \frac{1}{2} \mathcal{W} \bar{\psi}_\mu P_R \gamma^{\mu\nu} \psi_\nu + \text{h.c.} \,,
\begin{aligned}
\left[ f_{AB} \bar{\lambda}^A P_L \lambda^B \right]_F e^{-1} &= -\frac{1}{4} f_{AB} \left[ 2 \bar{\lambda}^A P_L \mathcal{D} \lambda^B + \hat{F}_{\mu\nu}^{-A} \hat{F}^{\mu\nu -B} - D^A D^B \right. \\
&+ \left. \frac{1}{4} \bar{\psi} \cdot \gamma P_L \left( \frac{1}{2} \gamma^{\mu\nu} \hat{F}_{\mu\nu}^{-A} - i D^A \right) \lambda^B - \frac{1}{8} \bar{\psi}_\mu \gamma^{\mu\nu} P_R \psi_\nu \bar{\lambda}^A P_L \lambda^B \right] \\
&+ \frac{1}{2\sqrt{2}} f_{AB,I} \bar{\Omega}^I \left( -\frac{1}{2} \gamma^{\mu\nu} \hat{F}_{\mu\nu}^{-A} + i D^A \right) \lambda^B \\
&+ \left[ -\frac{1}{4} f_{AB,I} \left( F^I + \frac{1}{\sqrt{2}} \bar{\psi} \cdot \gamma \Omega^I \right) + \frac{1}{8} f_{AB,IJ} \bar{\Omega}^I \Omega^J \right] \bar{\lambda}^A P_L \lambda^B + \text{h.c.}
\end{aligned} \tag{17.19}
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