# A puzzle from 'The origin of the hidden supersymmetry'

In the paper arXiv:1004.5489 The origin of the hidden supersymmetry, the author use {Qa,Qa}={Qb,Qb}=2H, {Qa,Qb}=0 for N=2 hidden SUSY, which is different from what I was taught: {Qa,Qa}={Qb,Qb}=0, {Qa,Qb}=2H. I think they are controversially different. Is any of them wrong? If not, could anyone help by explaining some more?

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The indices don't match for $\{Q_a,Q_b\} = 2H.$ What happens if you put $a=b$, for example? On the RHS, you should have some object that carries the right indices (such as $\delta_{ab}$, if $a,b$ live in some Euclidean space). – Vibert Mar 2 '13 at 12:12
Yes that was confusing. Now I fixed the notation. Thanks a lot. – Simon Mar 2 '13 at 20:06
Could anybody give a hand on this? Isn't it supposed not to be a hard question...? – Simon Mar 5 '13 at 18:02
You're thinking about this the wrong way. $a$ and $b$ are NOT fixed labels, i.e. you should never write down $\{Q_a,Q_a\} = \ldots$ or something like that. If you're really confused, you can write $\{Q_1,Q_1\} = \ldots,$ $\{Q_1,Q_2\} = 0$ etc. They are truly indices that can transform; you could rotate the supercharges, or write down $T_{\pm} = Q_1 \pm Q_2$, for example. The authors' $\{Q_a,Q_b\} = 2\delta_{ab} H$ is definitely right (up to a possible choice of conventions, of course). – Vibert Mar 5 '13 at 18:48
It's actually the same as with normal QM commutators you learned years ago. At the beginning, you write $[x,p_x] = \ldots$ etc., but at some point you grow up and just put $[x_i,p_j] = i\hbar \delta_{ij}$. The latter is manifestly $SO(3)$ invariant (if $i = 1,2,3$). – Vibert Mar 5 '13 at 18:52