{\beta}}\\
 -{(C^T)_{\dot{\alpha}}}^\beta & 0
\end{pmatrix}\,.$$ The matrices $\mathcal{C}\Gamma^\mu$ are symmetric, or equivalently $$(C\bar{\gamma}^\mu)^{\alpha\beta}\ =\  (C\bar{\gamma}^\mu)^{\beta\alpha}\,,\qquad
 (C^T\gamma^\mu)_{\dot{\alpha}\dot{\beta}}\ =\  (C^T\gamma^\mu)_{\dot{\beta}\dot{\alpha}}\,.$$

The world-sheet fermion $\psi^\mu(z)$ in the Ramond sector has zero-modes that satisfy the $SO(1,9)$ Clifford algebra $$\{\psi^\mu_0, \psi^\nu_0\}\ =\ 0\,.$$ The degenerate ground states therefore become the space-time spinor, on which $\psi^\mu_0$ act as space-time gamma matrices. We summarize here the related convention. We denote the ground state spinor as $\left(\begin{matrix}|{}^\alpha\rangle\\ |{}_{\dot{\alpha}}\rangle\end{matrix} \right)$, on which $\psi^\mu_0$ acts as $$\psi^\mu_0|{}^\alpha\rangle\ =\ 
|{}_{\dot{\alpha}}\rangle\frac{1}{\sqrt{2}}(\bar{\gamma}^\mu)^{\dot{\alpha}\alpha}\,,\qquad
\psi^\mu_0|{}_{\dot{\alpha}}\rangle\ =\
|{}^\alpha\rangle\frac{1}{\sqrt{2}}(\gamma^\mu)_{\alpha\dot{\alpha}}\,.$$ Then $\hat{\Gamma}_{11}$ defined by (\[gamma11\]) acts on the ground states as $$\hat{\Gamma}_{11}|{}^\alpha\rangle\ =\ |{}^\alpha\rangle\,,\qquad
 \hat{\Gamma}_{11}|{}_{\dot{\alpha}}\rangle\ =\ -|{}_{\dot{\alpha}}\rangle\,,$$ by which the definition of the GSO projection (\[GSO Ramond\]) is supplemented. Similarly, the BPZ conjugate of the ground state spinor $(\langle{}^\alpha|,\langle{}_{\dot{\alpha}}|)$ satisfies $$\langle{}^\alpha|\psi^\mu_0\ =\ 
\frac{i}{\sqrt{2}}(\bar{\gamma}^\mu)^{\dot{\alpha}\alpha}\langle{}_{\dot{\alpha}}|\,,\qquad
 \langle{}_{\dot{\alpha}}|\psi^\mu_0\ =\
- \frac{i}{\sqrt{2}}(\gamma^\mu)_{\alpha\dot{\alpha}}\langle{}^\alpha|\,,$$ with the normalization $$\langle{}^\alpha|{}_{\dot{\alpha}}\rangle\ =\ {C^\alpha}_{\dot{\alpha}}\,,\qquad
 \langle{}_{\dot{\alpha}}|{}^\alpha\rangle\ =\ 
-i{C^\alpha}_{\dot{\alpha}}\,.$$ The nontrivial matrix elements of $\psi^\mu_0$ are then given by $$\langle{}^\alpha|\psi^\mu_0|{}^\beta\rangle\ 
=\ \frac{1}{\sqrt{2}}(C\bar{\gamma}^\mu)^{\alpha