are the Pauli matrices and $\omega_{a,b,c}$ are the spin connections. The non-vanishing components of the five-form field strength are $F_{+1234}=F_{+5678}=2\mu$.

We can write the action as $$\label{feract}
\begin{split}
    S_{lc}^{F}=& \frac{i}{2\pi  p^+ }\int d\tau d\sigma \Bigg\{{\left(S^1\right)^T}  \left[\partial_{+}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right]S^1\\
    +& {\left(S^2\right)^T}  \left[\partial_{-}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right]S^2 -2f {\left(S^1\right)^T} \Pi S^2\Bigg\}\, .
\end{split}$$ where $S^A$, $A=1,2$, is a eight component real spinor and we introduced the matrix $\Pi=\gamma^{1234}$, where $\gamma_i$ are $8\times 8$ Dirac matrices [^6]. Moreover, $\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$. The equations of motion are

\[eqmotferm\] $$\begin{aligned}
    &\left(\partial_{+}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right)S^{1}-f\Pi S^{2}=0\, ,\\
    &\left(\partial_{-}-\frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\right)S^{2}+f\Pi S^{1}=0\, .\end{aligned}$$

It is useful to observe that a field of the form $$S^{A}=e^{\displaystyle \frac{f}{2}\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}\tau}\Sigma^{A}$$ satisfies the above equations if the fields $\Sigma^{A}$ obey the equations of motion of the fermionic fields in the usual pp-wave background [@Metsaev:2001bj; @Metsaev:2002re]: $$\partial_{+}\Sigma^{1}-f\Pi \Sigma^{2}=0\, ,~~~~~~\partial_{-}\Sigma^{2}+f\Pi \Sigma^{1}=0\, ,$$ whose solutions are

$$\begin{aligned}
    &\Sigma^{1}=c_0\, e^{-i f \tau}S_0 - \sum_{n>0}c_n e^{-i \omega_{n}\tau}
    \left(S_n e^{i n \sigma}+\frac{\omega_{n}-n}{f} S_{-n}e^{-i n \sigma} \right)
    +\textrm{h.c. },\\
    &\Sigma^{2}=-c_0\, e^{-i f \tau}i\Pi S_0 - i \Pi\sum_{n>0}c_n e^{-i \omega_{n}\tau}
    \left(S_{-n} e^{-i n \sigma}-\frac{\omega_{n}-n}{f} S_{n}e^{i n \sigma} \right)+\textrm{h.c. },\end{aligned}$$

where, for all values of $n$, $\omega_{n}=\sqrt{n^2+f^2}$, while $c_n =
\frac{1}{\sqrt{2}}[1+(\frac{\omega_{n}-n}{f})^{2}]^{-1/2}$.

The fermionic conjugate momenta can be computed from the action $$\lambda^{A}=\frac{i}{2\pi}S^{A}\, ,$$ and the fermionic part of the Hamiltonian can be written in 