Amount of SUSY preseved by branes Consider 10d superstring theory, a D5 brane extended in the 012789 directions, and an NS5 brane extended in the 012345 directions. I have read that this configuration preserves 1/4 of the supersymmetries. I am trying to figure out why, but I am struggling.
My understanding is that, given background supercharges $Q_L$ and $Q_R$, branes preserve SUSY's of the form $\epsilon_LQ_L+\epsilon_RQ_R$, where for the D5 brane above, $$\epsilon_L=\gamma_0\gamma_1\gamma_2\gamma_7\gamma_8\gamma_9\epsilon_R$$ while for NS5 branes (I believe this can be shown by S-dualising the above), $$\epsilon_{L/R}=\pm\gamma_0\gamma_1\gamma_2\gamma_3\gamma_4\gamma_5\epsilon_{L/R}.$$
So I combine these two to find $$\epsilon_L=-\gamma_5\gamma_4\gamma_3\gamma_7\gamma_8\gamma_9\epsilon_R\tag{1}$$ where I used $\gamma_i^2=-1$. I am not quite sure how to proceed -- in fact, apparently the correct relation would be $$\epsilon_L=\gamma_0\gamma_1\gamma_2\gamma_6\epsilon_R\tag{2}$$ but I am not quite sure how to show this.
 A: Turns out (as I imagined...) that it is nothing profound. Just multiply both sides of (1) by the chirality operator, $\gamma_{11}=\gamma_0\gamma_1\dots\gamma_9$. On the left-hand side, $\gamma_{11}\epsilon_L=\epsilon_L$. On the right hand side, the 3,4,5,7,8,9 gamma matrices within $\gamma_{11}$ annihilate those already present (since $\gamma^\mu\gamma^\mu=1\eta^{\mu\mu}$), leaving only the 0, 1, 2, and 6 gamma matrices and thus giving the result (2). I am leaving this here in case anyone else has the same issue I had.

Just to expand:
We have that $$\epsilon_L=\gamma_7\gamma_8\gamma_9\gamma_3\gamma_4\gamma_5\epsilon_L$$
Now $\gamma_7\gamma_8\gamma_9\gamma_3\gamma_4\gamma_5$ squares to $1$, so its eigenvalues are $\pm 1$; it is traceless, so half of its e-values are $1$ and half $-1$. Hence, the equation above preserves half of the components of $\epsilon_L$. $\epsilon_R$ is uniquely fixed in terms of $\epsilon_L$, so the total number of SUSY's preserved is $1/4$ of the original amount.
