# The superconformal algebra

1. How does one derive the superconformal algebra?

2. Especialy how to argue the existence of the operator $S$ which doesn't exist either in either the supersymmetric algebra or the conformal algebra?

3. What is the explanation of the convention of choosing If $J _ \alpha ^\beta$ as a generator of $SU(2) \times SU(2)$ and $R^A _B$ and $r$ are the generators of the $U(m)$ R-symmetry ( $m \neq 4$)? How does one get these $R$ and $r$?

4. One further defines $R_1, R_2$ and $R_3$ as the generators of the Cartan subalgebra of $SU(4)$ (the $N=4$ R-symmetry group) where these are defined such that $R_k$ has $1$ on the $k^{th}$ diagonal entry and $-1$ on the $(k+1)^{th}$ diagonal entry.

Using the above notation apparently one of the $Q-S$ commutation looks like,

$\{ S^A_\alpha , Q^\beta _B\} = \delta ^A_B J^\beta _\alpha + \delta ^\beta _\alpha R^A_B + \delta ^A_B \delta ^\beta _\alpha (\frac{H}{2} + r \frac{4-m}{4m})$

I would like to know how the above comes about and why the above equation implies the following equation (..each of whose sides is defined as $\Delta$)

$2\{ Q_4 ^- , S^4_ - \} = H - 3J_3 + 2\sum _{k=1} ^ 3 \frac{k}{4} R_k$

I can't understand how to get the second of the above equations from the first of the above and why is this $\Delta$ an important quantity.

{Is there any open source online reference available which explains the above issues?

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Would you mind defining $H$ and $J_3$? I'm guessing that you're in Euclidean space-time so $H=P_4$ (or maybe not...) and that $J_3$ is a combination of $J_\alpha^\beta$'s... Also, can you provide what reference you're working from? Finally, try some of the links given in this physicsforums post. –  Simon Apr 15 '11 at 2:07

I'm not sure what you mean by "derving" the algebra, but one way of obtaining a superconformal algebra (let me consider $N=4$) is to start with the four-dimensional conformal algebra and the $N=4$ supersymmetry algebra and consider the closure of the combined algebra. The conformal algebra consists of the Lorentz generators $J^\alpha{}_\beta$ and $\dot{J}^{\dot{\alpha}}{}_{\dot{\beta}}$ (I write the Lorentz group as $SL(2)\times SL(2)$, which is what I think you do in your question), the translations $P^{\alpha\dot{\beta}}$, the special conformal transformations $K_{\alpha\dot{\beta}}$ and the dilatation $H$. The $N=4$ supersymmetry group is generated by the $SU(4)$ R-symmetry generators $R^A{}_B$, the translations $P^{\alpha\dot{\beta}}$ and the supercharges $Q^A{}_\alpha$ and $\dot{Q}_{\dot{\alpha}A}$. These two algebras are both closed, but if we want to consider the combined symmetry the above generators are not enough to close the algebra. In particular the commutator between $K^{\alpha\dot{\beta}}$ and $\dot{Q}_{\dot{\gamma}A}$ gives rise to a new generator:

$[K^{\alpha\dot{\beta}},\dot{Q}_{\dot{\gamma}C}] = \delta^{\dot{\beta}}_{\dot{\gamma}} S^{\alpha}{}_C$

By also including the superconformal charges $S^{\alpha}{}_A$ (and a correpsonding charge $\dot{S}^{A\dot{\alpha}}$) we ge a closed algebra. The other (anti-)commutation relations follow if we require the the Jacobi identity is satisfied. In particular we get

$\{S^{\alpha}{}_a,Q^b{}_\beta\} = \delta^b_a J^{\alpha}{}_\beta + \delta^{\alpha}_{\beta} R^B{}_A + \frac{1}{2} \delta^b_a \delta^{\alpha}_{\beta} H$

which is your next-to-last equation (in somewhat different conventions when it comes to indices). To obtain your last equation you just have to take the particular case of $A=B=4$ and $\alpha=\beta=-$ and express the result in terms of your Cartan generators $R_1$, $R_2$, $R_3$ and $J_3$. The importance of this last expression really depends on in which context you have encountered the algebra. If you give some more details maybe someone can answer.

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Sorry for my delayed response. Your answer was very helpful. Can you kindly give me some reference about this 4 dimensional conformal algebra? I have not yet encountered any book which explains conformal algebra beyond 1+1 dimensions! –  user6818 May 1 '11 at 10:17
Also can you kindly make explicit the form of the Jacobi identity that you have in mind with which you seek consistency. The S-Q anti-commutation relation that you write down doesn't seem to have the r and the m parameters that I had written down. Can you kindly explain the difference? –  user6818 May 1 '11 at 10:21
Also what is the relation between the Cartans of the R-symmetry group i.e the $R_k$s and your $R^A_B$? May be you can help answer this other question too that I had asked, physics.stackexchange.com/questions/9242/… –  user6818 May 1 '11 at 10:23
@Anirbit For references I think it depends on what context you encountered the algebra in. I don't know of any book really. Any review of AdS/CFT should have at least some introduction to the N=4 algebra, see eg the large review hep-th/9905111 or D’Hoker & Freedman hep-th/0201253. I also like the appendix of Niklas Beisert's thesis: hep-th/0407277. –  Olof May 3 '11 at 9:22
@Anirbit The Cartan generators can be chosen to be diagonal as in you question. Something like $R_1 = R^1_1 - R^2_2$ etc. The reason that $m$ and $r$ don't appear in the algebra I wrote down is that I just gave the $N=4$ case. The Jacobi identity should take the normal form, eh $\{S, [K, Q]\} +$ cyclic $=0$. –  Olof May 3 '11 at 9:27