the others get other fourth roots of unity.

The ${\mathbb Z}_4$- and GSO-invariant states in this sector are of the form

  -----------------------------------------------------------------------------------------------------------------------------------------------
                         State                                                                   Count
  ---------------------------------------------------- ------------------------------------------------------------------------------------------
   $|m=0,4,8\rangle \otimes \left( \psi^{1-2}_{-1/2},   spacetime vectors, in the ${\bf 1}$, ${\bf 1}$, $\wedge^4 {\bf 8} = {\bf 70}$ of $su(8)$
         \overline{\psi}^{1-2}_{-1/2} \right)$         

    $|m=2,6\rangle \otimes \left( \psi^{3-4}_{-1/2},       1 hypermultiplet in $\wedge^2 {\bf 8} = {\bf 28}$, $\wedge^2 {\bf \overline{8}} =
         \overline{\psi}^{3-4}_{-1/2} \right)$                                      {\bf \overline{28}}$ of $su(8)$
  -----------------------------------------------------------------------------------------------------------------------------------------------

The (R,NS) sector in $k=2$ is closely related. Here, fields have the following boundary conditions: $$\begin{aligned}
X^{1-4}(\sigma + 2\pi) & = & + X^{1-4}(\sigma), \\
\psi^{1-4}(\sigma + 2 \pi) & = & - \psi^{1-4}(\sigma), \\
\lambda^{1-8}(\sigma + 2 \pi) & = & + \lambda^{1-8}(\sigma), \\
\lambda^{9-16}(\sigma + 2 \pi) & = & - \lambda^{9-16}(\sigma).\end{aligned}$$ Just as in the (NS,NS) sector, $E_{\rm left} = 0$ and $E_{\rm right} = -1/2$. Here, the left Fock vacua form a spinor of the low-energy $so(16)$.

The ${\mathbb Z}_4$-invariant states in this sector are of the form

  ------------------------------------------------------------------------------------------------------
                          State                                              Count
  ----------------------------------------------------- ------------------------------------------------
   $(\mbox{spinor}) \otimes  \left( \psi^{1-2}