mode expansions

\[bosmodeex\] $$\begin{aligned}
Y^k&=i \sum_{n=-\infty}^{+\infty} \frac{1}{\sqrt{\omega_n}}\left(a_{n}^{k}e^{-i (\omega_n \tau -n\sigma)}- \left(\tilde{a}_{n}^{k}\right)^\dagger e^{i (\omega_n \tau -n\sigma)}\right)\, , \\
\bar{Y}^k&=i   \sum_{n=-\infty}^{+\infty} \frac{1}{\sqrt{\omega_n}}\left(\tilde{a}_{n}^{k}e^{-i (\omega_n \tau -n\sigma)}- \left(a_{n}^{k}\right)^\dagger e^{i (\omega_n \tau -n\sigma)}\right)\, .\end{aligned}$$

The bosonic Hamiltonian now reads $$\label{Hcomplexfield}
     H_{lc}^{B}=  \frac{1}{4\pi p^+}\int_{0}^{2\pi}d\sigma \sum_{k=1}^{4}\left(\dot{\bar{X}}^k \dot{X}^k+ (\bar{X}^k)'(X^k)'
     +f^2 \left(1-\eta_{k}^{2}\right)\bar{X}^{k}X^{k}\right)\, .$$ Then we quantize the theory imposing the canonical equal time commutation relations $$\label{etcr}
    \left[a_{n}^{k},a_{m}^{k'}\right]=0\, , \qquad \left[a_{n}^{k},(a_{m}^{k'})^{\dagger}\right]=\left[\tilde{a}_{n}^{k},(\tilde{a}_{m}^{k'})^{\dagger}\right]=\delta^{kk'}\delta_{nm}\, .$$ We obtain the following bosonic spectrum in this background $$\label{rotbosH}
\begin{split}
    H_{lc}^{B}=&  \frac{1}{ p^+}\sum_{n=-\infty}^{+\infty}  \sum_{k=1}^2
    \left[\left(\omega_n + \eta_k f\right) M_{n}^{(k)}+\left(\omega_n - \eta_k f\right) \tilde{M}_{n}^{(k)}\right. \\
    +&\left.\left(\omega_n + \eta_{(k+2)} f\right) N_{n}^{(k)}+\left(\omega_n - \eta_{(k+2)}  f\right) \tilde{N}_{n}^{(k)}\right]\, ,
\end{split}$$ where $\omega_n = \sqrt{n^2 + f^2}$ for all $n\in \mathbb{Z}$ and the number operators are defined as $$M_{n}^{(k)}=a_{n}^{k\dagger}a_{n}^{k}\, , ~~ \tilde{M}_{n}^{(k)} =\tilde{a}_{n}^{k\dagger}\tilde{a}_{n}^{k}   \, ,~~N_{n}^{(k)}=a_{n}^{(k+2)\dagger}a_{n}^{(k+2)}\, , ~~\tilde{N}_{n}^{(k)} =\tilde{a}_{n}^{(k+2)\dagger}\tilde{a}_{n}^{(k+2)}$$ for $k=1,2$.

Fermionic sector
----------------

We now work out the fermionic part of the spectrum. The light-cone gauge and $\kappa$-symmetry gauge fixing condition are $$z^+ =  p^+ \tau, \qquad \Gamma^{+}\theta^A=0\,$$ where $\theta^A$, with $A=1,2$, is a Majorana-Weyl spinor with $32$ components. The Green-Schwarz fermionic light-cone action is then given by [@Metsaev:2002re] $$\label{GSaction}
    S_{lc}^{F}= \frac{i}{4\pi p^+}\int d\tau d\sigma \left[ \left(\eta^{\alpha\beta}\delta_{AB}-\epsilon^{\alpha\beta}\left(\sigma_{3}\right)_{AB}\right)\partial_{\alpha}z^+ \bar{\theta}^A \Gamma_+ \left(\mathcal{D}_{\beta}\theta\right)^B\right]\, ,$$ with covariant derivative $$\mathcal{D}_{\alpha}=\partial_{\alpha}+\frac{1}{4}\partial_{\alpha}z^+ \left(\omega_{+\rho\sigma}\Gamma^{\rho \sigma}-\frac{1}{2\cdot 5!}F_{\lambda\nu\rho\sigma\kappa}\Gamma^{\lambda\nu\rho\sigma\kappa}i\sigma_2 \Gamma_+ \right)\, ,$$ where $\sigma_{k}$’s 