Wess Zumino model in two dimensions

[alialice]

Homework Statement

Hi!
I need some help to describe a Wess Zumino model in two dimensions:
spinors are real (because of the Majorana condition \theta=\theta^{\ast}) and have two components;
the superfield is:
\phi \left( x,\theta\right)=A\left( x\right) + i\bar{\theta}\psi\left(x\right) +\frac{i}{2} \bar{\theta}\theta F\left(x\right)
where:
A and F are scalars
ψ is a spinorial field.
The susy generator is:
Q_{\alpha}=\frac{\partial}{\partial \bar{\theta}^{\alpha}} -i \left(\gamma^{\mu} \right) _{\alpha} \partial_{\mu}

1) What are the supersymmetry transformations of the fields?

2) Which is the invariant action of the model?

Thank you very much if you could give me some help!

 

[member 553083]

Homework Equations N/AThe Attempt at a Solution 1) The supersymmetry transformations of the fields are:A: A'\left(x\right)= A\left(x\right)+ i\bar{\epsilon}\psi\left(x\right) ψ: ψ'\left(x\right)= ψ\left(x\right)+ i\epsilon F\left(x\right) F: F'\left(x\right)= F\left(x\right)+ i\bar{\epsilon}\gamma^{\mu}\partial_{\mu}\psi\left(x\right) 2) The invariant action of the model is:S= \int d^{2}x \left[ \frac{1}{2} \partial^{\mu}A \partial_{\mu}A + \frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi + \frac{1}{2}F^{2} \right]