the form $$H_{lc}^{F}= \frac{i}{2\pi p^+ }\int^{2\pi}_{0}d\sigma
    \left({\left(S^1\right)^T}\dot{S^1}+{\left(S^2\right)^T}\dot{S^2}\right)\,$$ where we used the equations of motion . Now we quantize the theory imposing the canonical equal time anticommutation relations $$\left\{S_{n}^{a},\left(S_{m}^{b}\right)^{\dagger}\right\}=\delta^{ab}\delta_{nm}\,$$ and the fermionic Hamiltonian reads $$H_{lc}^{F}=\frac{1}{ p^+ }{\sum_{n=-\infty}^{+\infty}}S_{n}^{\dagger}
    \left(\omega_{n}+i\frac{f}{2}{\sum_{k=1}^{4}\eta_{k}\gamma^{2k-1,2k}}\right)S_{n}\, .$$ The matrices $i\,\gamma^{2k-1,2k}$ are commuting matrices and have eigenvalues $\pm 1$, each with multiplicity four. Since they commute we can find a set of common eigenvectors. Choosing this set as basis we can write the fermionic spectrum as $$\label{rotferH}
    H_{lc}^{F}= \frac{1}{ p^+}{\sum_{n=-\infty}^{+\infty}} \sum_{b=1}^{8}
    \left(\omega_n + \frac{f}{2} d_b \right)F_{n}^{(b)}\, ,$$ where $F_{n}^{(b)}$ are the fermionic number operators defined by the relation $$F_{n}^{(b)}=\left(S_{n}^{b}\right)^{\dagger}S_{n}^{b}\,$$ and where we have defined the coefficients $d_b$ as the following combinations of the $\eta_k$ parameters $$\begin{array}{lll}
        d_1 = -\eta_{1}-\eta_{2}+\eta_{3}+\eta_{4} \, , \phantom{qquad} & d_5 = -\eta_{1}+\eta_{2}+\eta_{3}-\eta_{4} \, ,\\[1mm]
        d_2 = -\eta_{1}-\eta_{2}-\eta_{3}-\eta_{4} \, , & d_6 = \eta_{1}-\eta_{2}+\eta_{3}-\eta_{4} \, ,\\[1mm]
        d_3 = \eta_{1}+\eta_{2}+\eta_{3}+\eta_{4} \, ,  & d_7 = \eta_{1}-\eta_{2}-\eta_{3}+\eta_{4} \, ,\\[1mm]
        d_4 = \eta_{1}+\eta_{2}-\eta_{3}-\eta_{4} \, ,  & d_8 = -\eta_{1}+\eta_{2}-\eta_{3}+\eta_{4} \, .
    \end{array}$$

At this point we can write the total light-cone Hamiltonian, $H_{lc}$, of type IIB string theory on the [*rotated pp-wave s*]{} $$\label{eq:rotH}
\begin{split}
    H_{lc}=&H_{lc}^{B} +H_{lc}^{F}= \frac{1}{ p^+}\sum_{n=-\infty}^{+\infty}  \left\{\sum_{k=1}^2
    \left[\left(\omega_n + \eta_k f\right) M_{n}^{(k)}+\left(\omega_n - \eta_k f\right) \tilde{M}_{n}^{(k)}\right]\right. \\
    