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5

He inserts the identity $I = \sum_{S'}|S'\rangle\langle S'|$. This gives $\langle S|Q_\alpha Q_\alpha^\dagger|S\rangle = \sum_{S'} \langle S|Q_\alpha |S'\rangle \langle S'|Q_\alpha^\dagger|S\rangle =\sum_{S'} |\langle S|Q_\alpha |S'\rangle|^2$.

3

Yes. OP is right. There is a minus. Since by convention the complex conjugation obeys $$(z w)^{\ast} ~=~ w^{\ast}z^{\ast}~=~(-1)^{|z|~|w|} z^{\ast}w^{\ast} \tag{1}$$ for any two supernumbers $z$, $w$ (of definite Grassmann parities $|z|$,$|w|$), we should also have $$(A f)^{\ast} ~=~(-1)^{|A| ~|f|} A^{\ast}f^{\ast} \tag{2}$$ for an operator $A$ and ...

3

Eq. (4.4.11g) is for a Majorana spinor SUSY charges $Q_a$, $a=1,2,3,4$, while eq. (3.1.31) is for left Weyl spinor SUSY charges $Q_{\alpha}$, $\alpha=1,2$.

3

Normally the notation $(n_b|n_f)$ denotes the dimension of a super vector space of Grassmann-even dimension $n_b$ and Grassmann-odd dimension $n_f$. When writing a super vector as a column vector, it is standard to order the Grassmann-even sector before the Grassmann-odd sector. However, the authors introduce a non-standard ordering $(n_{b_1}|n_f|n_{b_2})$ ...

2

In the Standard Model coupled to GR as an effective theory, the cosmological constant is predicted to be $m_{Pl}^4$ i.e. $10^{123}$ times the correct value (you mentioned the correct value). SUSY improves this situation by cancellations between superpartners (fermions contribute the same to the C.C. as their bosonic partners but with the opposite sign if ...

1

Worldsheet supersymmetry is the fermionic symmetry of the worldsheet RNS action under the worldsheet supsresymmetry transformations that look like $$\sqrt{\frac{2}{\alpha'}}X \mapsto \sqrt{\frac{2}{\alpha'}}X + \mathrm{i}\bar{\epsilon}\psi^\mu$$ and which I'm too lazy to type out for all fields (and which also depend on whether or not we're looking at the ...

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