# Some questions about chapter I.1 (by Minahan) of the “Review of AdS/CFT Integrability”

These questions are in reference to this beautiful review article by Minahan - http://arxiv.org/pdf/1012.3983v2

I gained a lot by reading some of its sections but not everything is clear to me. I would like to ask a few questions to clarify some of the things in it.

• On page 5 between equation 3.7 and 3.8 it says that the R-charge representation of the $S$ are "reversed" compared to that of the $Q$. What does it exactly mean ?

• There it does not explicitly specify the commutation relationship between $M_{\mu \nu}$ and $S^a_\alpha$ and $\bar{S}_{\dot{\alpha}a}$. Should I assume that its similar to that with $Q_{\alpha a}$ and $\bar{Q}^a_\dot{\alpha}$ ? Like if I may think -

$[M^{\mu \nu},S^a_\alpha] = i \gamma ^{\mu \nu}_{\alpha \beta} \epsilon ^{\beta \gamma} S^a_{\gamma }$

$[M^{\mu \nu},\bar{S}_{\dot{\alpha} a} ] = i \gamma ^{\mu \nu}_{\dot{\alpha} \dot{\beta}} \epsilon ^{\dot{\beta} \dot{\gamma}} \bar{S}_{\dot{\gamma} a}$

?

• Comparing equation 3.12 to 3.9 I see some possible discrepancies. Is there a factor of $\frac{1}{2}$ missing with the term containing $D$ on the RHS of equation 3.12 ? In the same RHS of equation 3.12 in the $M_{\mu \nu}$ term why has the $\gamma ^{\mu \nu}$ of equation 3.9 become $\sigma ^{\mu \nu}$ ?

• In the statement just below equation 3.14 it says "Hence a primary operator with R-charges $(J_1,0,0)$ is annihilated by $Q_{\alpha 1}$ and $Q_{\alpha 2}$ if $\Delta = J_1$"...Is this a consistency statement?

From this how does it follow that thesame operator is also annihilated by $\bar{Q}^3_{\dot{\alpha}}$ and $\bar{Q}^4_{\dot{\alpha}}$

In the same strain can one also say that $\Delta = -J_1$ is consistent with the primary operator being annihilated by $Q^3_\alpha$ and $Q^4 _ \alpha$ ?

• I guess the above conclusions follow from taking different values of $a$ and $b$ in the equation 3.13. But one would get an extra factor of $\frac{1}{2}$ on the RHS of 3.13 if one puts in the factor of $\frac{1}{2}$ with the term containing $D$ on the RHS of equation 3.12 (..as I think it should be ..)

• On page 8 one creates bispinors $F_{+\alpha \beta}$ and $F_{-\dot{\alpha} \dot{\beta}}$ out of $F_{\mu \nu}$. I would like to know what is the intuition/motivation/reason for doing this ?

Is the bispinor version of $F$ still have the meaning of a field strength ? If so then how does it relate to the bispinor version (?) $D_{\alpha \dot{\beta}}$ of the covariant derivative ?

• Yeah, this is a handful for one question. Your first bullet point about the R-charge representations of $S$ and $Q$ could be a standalone question, for instance. In fact, probably each of the individual questions here could be asked on its own - at least, you could definitely split this into 4 or 5 individual posts. – David Z Jun 27 '11 at 23:15