- \mu^2 x^ix^i\left(dx^+\right)^2+dx^idx^i\, ,$$ where $i=1,2,\dots,8$ and five-form field strength $$\label{fff}
    F_{(5)}=2\mu dx^{+}\left(dx^{1}dx^{2}dx^{3}dx^{4}+dx^{5}dx^{6}dx^{7}dx^{8}\right)\, .$$ We consider the following coordinate transformation $$\label{transfrot}
\begin{split}
x^- =z^- &+\frac{\mu}{2}\left(C_1 z^1z^2 + C_2z^3z^4 + C_3z^5z^6 + C_4z^7z^8\right)\, , \\[2mm]
\left( \begin{array}{c}
 x^{2k-1} \\[2mm]
 x^{2k}
\end{array} \right) &=
\left( \begin{array}{cc}
 \cos(\eta_k \mu z^+) & -\sin(\eta_k \mu z^+) \\[2mm]
 \sin(\eta_k \mu z^+) & \cos(\eta_k \mu z^+)
\end{array} \right)
\left( \begin{array}{c}
 z^{2k-1} \\[2mm]
 z^{2k}
\end{array} \right)\, ,
\end{split}$$ where $C_{k}$ and $\eta_{k}$, $k=1, 2, 3, 4$, are parameters.

Note that the transformations for the transverse coordinates are rotations whose angles depend on the $\eta_k$ parameters, hence the name “[*rotated pp-wave* ]{}”.

The metric then becomes $$\label{rotmetric}
    \begin{split}
        ds^2=&-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=1}^{4}     \left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
        &- 2\mu \sum_{k=1}^{4}\left[(C_k-\eta_k)z^{2k-1}dz^{2k}+(C_k+\eta_k)z^{2k}dz^{2k-1}\right]dz^+\, ,
    \end{split}$$ while the five-form field strength is invariant under the coordinate transformation . It is straightforward to check that the metric contains all the s obtained in Section \[sec:stringtheory\]. In fact, for various values of the $C_k$ and $\eta_k$ parameters, we have the following possibilities

  ------------------------------------------- --------------- ----------------------
  $C_1=C_2=C_3=C_4=0$                          $\Rightarrow$  no flat direction;
  $C_1=\eta_1=1$ and $C_2=C_3=C_4=0$           $\Rightarrow$  one flat direction;
  $C_1=\eta_1=C_2=\eta_2=1$ and $C_3=C_4=0$    $\Rightarrow$  two flat directions.
  ------------------------------------------- --------------- ----------------------

String theory can be quantized on the general 