Why we use majorana fermions the SYK model? Does anyone know why do we use majorana fermions in the SYK model. why we can't use Dirac spinors? Is there any specific reason why we use majorana fermions in this model?
 A: SYK model is 0+1d quantum field theory or more precisely, finite dimensional quantum mechanical models.
Main object of this theory is 0+1d fermions, i.e representation of Clifford algebra $Cl(0,1)$. 
It is 1d algebra with relation:
$$
\psi\psi + \psi\psi = 1
$$
And obviously have real (Hermititan) representation.
This fermions may be also representation of some group, and in SYK model they are spinor representation of $SO(2n)$ group. 
This mean that they have extra index and satisfy $Cl(2n)$ relations:
$$
\psi_i\psi_j + \psi_j\psi_i = \delta_{ij}
$$
This algebra have matrix representation, that is Hermititan representation $\psi_i^\dagger = \psi_i$ by construction.
See for example Gábor Sárosi: AdS2 holography and the SYK model, section 4 or if you interested in more general approach Antoine Van Proeyen: Tools for supersymmetry, section 3
. 
To conclude, we choose fermion as Majorana fermions of $SO(0,1)$ space-time symmetry group and as Hermititan matrices of $SO(2n)$ symmetry.
We can also choose fermions as Dirac fermions of $SO(0,1)$ space-time symmetry group, but it is trivial: Dirac spinor is redusible representation and consist of 2 Majorana spinors.
See also around equation 77 Fermions..
