Why does a SUSY Lagrangian only contain F and D terms?

[Maurice7510]

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?

Thanks in advance

 

[AndyW]

for any help or insight!

Hello! I am also familiar with AdS/CFT and supersymmetry, and I can offer some insight into why only the D-term and F-terms enter in the Lagrangian for $\mathcal{N}=1$ chiral superfields.

First, let's define what we mean by D-term and F-terms. The D-term is related to the gauge symmetry of the theory and is characterized by a term that involves the gauge field and its corresponding scalar field. On the other hand, the F-terms are related to the matter fields in the theory and are characterized by terms that involve the superpotential (W) and its conjugate (W^{\dagger}).

Now, in supersymmetric theories, we have a symmetry called supersymmetry that relates the fermionic and bosonic fields. This symmetry is broken in our current world, but it is believed to be a fundamental symmetry in nature. In order for a theory to be supersymmetric, it must have an equal number of fermionic and bosonic degrees of freedom. This is where the remaining 6 component fields come into play.

The chiral superfield, which is the building block for supersymmetric theories, contains 8 component fields - 4 fermions and 4 bosons. However, in order for the theory to be supersymmetric, we need to have an equal number of fermionic and bosonic degrees of freedom. This is achieved by imposing constraints on the chiral superfield, which reduce the number of component fields to 3 bosons and 3 fermions. These are the fields that appear in the D-term and F-terms of the Lagrangian.

In short, the remaining 6 component fields do not show up in the Lagrangian because they are not independent degrees of freedom. They are related to the 3 bosons and 3 fermions that do appear in the Lagrangian through the constraints imposed on the chiral superfield. I hope this helps to clarify things for you. Keep reading and exploring, and don't hesitate to ask for further clarification if needed!