I am studying the free fermion CFT using the book by Di Francesco et al. I am very confused with the notion of Majorana fermions, sparked by the following introduction to the chapter:
In two dimensions, the euclidian action of a free Majorana fermion is
$$\frac{1}{2}g\int d^2x \ \Psi^{\dagger}\gamma^0 \gamma^{\mu}\partial_{\mu}\Psi$$
The action looks like the generic fermion action to me, so according to my understanding it should contain charged fermions as well, that thus are not Majorana. Are the authors simply restricting their following analysis to real fermions or is there a deeper reason that the fields are always real?
Best Answer
They are restricting their analysis to real fermions. Their dagger $\dagger$ must mean just transpose. You can see later on they break up $\Psi$ into components $\Psi=(\psi,\bar{\psi})$ and $\Psi^\dagger$ is broken up into the same two components without any notion of complex conjugation.