About 2+1 dimensional superconformal algebra I would like to get some help in interpreting the main equation of the superconformal algebra (in $2+1$ dimenions) as stated in equation 3.27 on page 18 of this paper. I am familiar with supersymmetry algebra but still this notation looks very obscure to me.

*

*In the above equation for a fixed $i$, $j$, $\alpha$, $\tilde{\beta}$ the last term,
$-i\delta_{\alpha , \tilde{\beta}} I_{ij}$ will be a $\cal{N} \times \cal{N}$ matrix for $\cal{N}$ extended supersymmetry in $2+1$ dimensions. Is this interpretation right?

(..where I guess $I_{ij}$ is the vector representation of $so(\cal{N})$ given as, $(I_{ij})_{ab} = - i(\delta _{ia}\delta _{jb} - \delta _{ib} \delta _{ja})$..)

*

*Now if the above is so then is there an implicit $\cal{N} \times \cal{N}$ identity matrix multiplied to the first term, $i \frac{\delta_{ij}}{2} [(M'_ {\mu \nu}\Gamma_\mu \Gamma_\nu C)_{\alpha \tilde{\beta}} + 2D' \delta _ {\alpha \tilde{\beta}}] $
?

So I guess that the equation is to be read as an equality between 2 $\cal{N} \times \cal{N}$ matrices. right?

*

*Is there is a typo in this equation that the first term should have $(M'_ {\mu \nu}\Gamma^\mu \Gamma ^ \nu C)$ instead of all the space-time indices $\mu, \nu$ to be down?


*I guess that in $M' _{\mu \nu}$ the indices $\mu$ and $\nu$ range over $0,1,2...,d-1$ for a $d-$dimensional space-time (...here $d=3$..) and for this range in the Euclideanized QFT (as is the case here) one can replace $M'_{\mu \nu} = \frac {i}{4}[\Gamma _ \mu , \Gamma _ \nu]$. Is that right?


*One is using the convention here where the signature is $\eta_{\mu \nu} = diag(-1,1,1) = \eta ^{\mu \nu}$ and the Gamma matrices are such that $\Gamma^0 = C = [[0,1],[-1,0]], \Gamma ^1 = [[0,1],[1,0]], \Gamma ^ 2 = [[1,0],[0,-1]]$ and then the charge conjugation matrix $C$ satisfies $C^{-1} \Gamma ^\mu C = - \Gamma ^{\mu T}$ and $[\Gamma ^\mu , \Gamma ^\nu]_+ = 2 \eta ^\mu \eta ^\nu$
Then $M^{\mu \nu}\Gamma _\mu \Gamma _ \nu = -3i [[1,0],[0,1]]$

Now for a specific case of this equation let me refer to the bottom of page $8$ and top of page $9$ of this paper.

*

*In physics literature what is the implicit equation/convention that defines the representation of $SO(N)$ with heighest weights $(h_1, h_2, ... , h_{[\frac{N}{2}]})$?

I could not find an equation anywhere which defines the $h_i$s

*

*How does choosing the weights of the $Q$ operator to be as stated in the bottom of page 8 determine the values of $i$ and $\alpha$ that goes in the RHS of the anti-commutation equation described in the first half?

And how does it determine the same for the $S$ operator which because of Euclideanization is related as , $S^{'}_{i \alpha} = (Q^{'i \alpha})^\dagger $ (...I guess that the raising and lowering of indices doesn't matter here...)

*

*Now given the choice as stated in the bottom of page 8 in the paper above and the S-Q Hermiticity relation and the anti-commutation relation in the first half of this question how does one prove the relation claimed on the top of page 9 which is effectively, $[Q^{'i\alpha},S^{'}_{i\alpha}]_+ = \epsilon_0 - (h_1 + j)$
I guess $\epsilon_0$ is the charge under the $D'$ of the first half defined for an operator $A$ (say) as $[D',A] = -\epsilon _0 A$ though I can't see the precise definition of $h_i$s and $j$ in terms of the RHS of the Q-S anti-commutation relation as described in the first half of the question.

*

*Does anything about the above $[Q^{'i\alpha},S^{'}_{i\alpha}]_+ = \epsilon_0 - (h_1 + j)$ depend on what is the value of $\cal{N}$? I guess it could be $2$ as in this paper or $3$ and it would still be the same expression.

It would be great if someone can help with this.
 A: The first bullet point: no. $I_{ij}$ (for a fixed $i,j$) is just a generator of $SO(2n)$, not  its explicit matrix representative.  The commutation relation in general is an equation inside the Lie algebra. 
The second bullet point: no. 
The third bullet point: Yes and no. People in the field don't usually care where to put the indices, because we usually use the extended Einstein convention where $A_\mu B_\mu$ means $A_\mu B^\mu$, i.e., repeated indices are interpreted as put on either superscript or subscript appropriately and summed over to give a Lorentz invariant result. 
The fourth point: no. Again, $M_{\mu\nu}$ is just a generator, not its matrix representative. $\Gamma_{\mu}$ is, on the other hand, is an explicit matrix.
The fifth point: this question doesn't make sense, due to the fourth point above.
The sixth point: there's no unified convention. In this case it's explained in the footnote 5.
The last three bullet points: I guess you should reread the papers based on the answers so far, and ask again at physics.SE as a separate question if you still have questions.
