Blumenhagen Lüst and Theisen use the following real representation for the two-dimensional gamma matrices: \begin{equation} \rho^0 = \begin{pmatrix} 0 && 1\\ -1 && 0 \end{pmatrix}, \hspace{1cm} \rho^1 = \begin{pmatrix} 0 && 1\\ 1 && 0 \end{pmatrix},\tag{7.74} \end{equation} and define Dirac conjugation as $\bar{\psi} = \psi^\dagger \rho^0$. Their form of the superconformal gauge RNS action in Minkowski space is \begin{equation} S = -\frac{1}{4 \pi} \int d^2 \sigma \Big( \frac{1}{\alpha^\prime} \partial^\alpha X^\mu \partial_\alpha X_\mu + i \bar{\psi}^\mu\rho^\alpha \partial_\alpha \psi_\mu \Big).\tag{7.24} \end{equation} The $\psi^\mu$ are Majorana spinors (which are real if the gamma matrices are real). If both the spinors and the gamma matrices are real, why is there a factor of $i$ in front of the fermion kinetic term? Is the book wrong?
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