The 4D SUSY algebra can be written as

$$\{ Q_{\alpha}^{A} , Q_{\beta}^{B \dagger} \} = 2 m \delta^{AB} \delta_{\alpha \beta} + 2 i Z^{AB} \Gamma^0_{\alpha \beta}, \tag{B.2.37} $$

in a particular reference frame. One can find this formula in the Appendix B, page 448 of Polchinski's String Theory vol.II.

I am confused with the $'i'$ before the central charge. If we do a Hermitian conjugate on both side:

$$\{ Q_{\alpha}^{A \dagger} , Q_{\beta}^{B} \} = 2 m \delta^{AB} \delta_{\alpha \beta} - 2 i Z^{AB} (\Gamma^0_{\alpha \beta})^* $$

and then exchange $(A,\alpha)$ with $(B,\beta)$, the LHS is invariant. But the RHS is

$$2 m \delta^{AB} \delta_{\alpha \beta} - 2 i Z^{BA} (\Gamma^0_{ \beta \alpha})^* = 2 m \delta^{AB} \delta_{\alpha \beta} - 2 i Z^{BA} (\Gamma^0)^{\dagger}_{ \alpha \beta}.$$

Since $Z_{AB}$ is anti-symmetric and $(\Gamma^0)^{\dagger} = -\Gamma^0$, It seems that we have the wrong sign before the central charge term:

$$2 m \delta^{AB} \delta_{\alpha \beta} - 2 i Z^{AB} (\Gamma^0)_{ \alpha \beta}.$$

I think I made a mistake but I can not figure out where is it.

This post imported from StackExchange Physics at 2018-04-18 20:42 (UTC), posted by SE-user JQ Skywalker