I'm reading a book on AdS/CFT by Ammon and Erdmenger and chapter 3 covers supersymmetry. This isn't my first look at SUSY but it's my first in depth look to really try to understand it, and when they talk about constructing a Lagrangian for $\mathcal{N}=1$ chiral superfields they write the most general form,

$$ \mathcal{L} = \underbrace{K(\Phi^k\Phi^{k\dagger})_{|\theta^2\bar{\theta^2}}}_{\text{D-term}} + \underbrace{\left( W(\Phi^k)_{|\theta^2} + W^{\dagger}(\Phi^{k\dagger})_{|\bar{\theta^2}}\right)}_{\text{F-terms}} $$

Initially this rustled my Jimmies because they had just spent the preceding section having me jump through hoops to deal with all the other component fields then decided only to use these 3, but then they address this directly: "In the Lagrangian, only the D-term ... and the F-terms enter". Unfortunately, this is as detailed as the explanation gets and I was hoping someone could please explain why this is the case and why the remaining 6 component fields don't show up?

I posted this in PhysicsForums and failed to get an answer unfortunately, so hopefully this sees more attention

I'm reading a book on AdS/CFT by Ammon and Erdmenger and chapter 3 covers supersymmetry. This isn't my first look at SUSY but it's my first in depth look to really try to understand it, and when they talk about constructing a Lagrangian for $\mathcal{N}=1$ chiral superfields they write the most general form,

$$ \mathcal{L} = \underbrace{K(\Phi^k\Phi^{k\dagger})_{|\theta^2\bar{\theta^2}}}_{\text{D-term}} + \underbrace{\left( W(\Phi^k)_{|\theta^2} + W^{\dagger}(\Phi^{k\dagger})_{|\bar{\theta^2}}\right)}_{\text{F-terms}} $$

Initially this rustled my Jimmies because they had just spent the preceding section having me jump through hoops to deal with all the other component fields then decided only to use these 3, but then they address this directly:

"In the Lagrangian, only the $D$-term ... and the $F$-terms enter".

Unfortunately, this is as detailed as the explanation gets and I was hoping someone could please explain why this is the case and why the remaining 6 component fields don't show up?

I posted this in PhysicsForums and failed to get an answer unfortunately, so hopefully this sees more attention