Superconformal algebra

I had earlier also asked a question about super conformal theories and I am continuing with that, now with more specific examples. I am quite puzzled with it given that I see no book explaining even the basics of this. I am merely picking this up from stray notes and papers and mostly discussions.  So for ${\cal N} = 2$ superconformal algebra in $2+1$ dimensions the symmetry group was $SO(3,2)\times SO(2)$ and possibly the primary states of this algebra are labelled by the tuple $(\Delta, j,h)$ where $\Delta$ is the scaling dimension and $j$ is the spin and $h$ is its $R$ charge (or whatever it means to call it the $R$ charge highest weight)

I guess that in this context by “primary state” one would mean those which are annihilated by the special conformal operators (K) as well as the $S$ operators of the algebra. Though confusingly enough the terminology seems to be used for operators as well.

• I would like to know where can I find the complete superconformal algebra to be listed which can enable me to find the $(\Delta, j,h)$ of the states which are gotten by acting other operators n the primaries. I guess its only the momentum operators and the supersymmetry that is left to create the tower of the “descendants”

I guess that once one knows the above algebra I can understand these mysterious tables that I see listing the “states” corresponding to the supersymmetry operators. If for the identity operator the tuple is $(0,0,0)$ then I would like to understand how for the $Q$ the possible labels are apparently $(0.5,\pm 0.5,\pm 1)$ and for $Q^2$ the labels are $(1,0,\pm 2)$, $(1,0,0)$ and $(1,1,0)$. For $Q^3$ the labels are $(1.5,0.5,\pm 1)$ and for $Q^4$ the labels are $(2,0,0)$.

I would like to know how the above labels for the operators are derived. It gets harder for me to read this since for the operators the R symmetry as well as the spinorial index are both being suppressed.

• Now consider a primary labelled by $(\Delta, j,h)$ such that it is in the long representation and hence $\Delta >j+\vert h\vert +1$. Then I see people listing something called the “conformal content” of this representation labelled by the above state. For the above case the conformal content apparently consists of the following states, $(\Delta, j,h)$, $(\Delta+0.5, j\pm 0.5,h\pm 1)$, $(\Delta + 1 , j,h \pm 2)$, $(\Delta +1 , j+1,h)$, twice $(\Delta + 1, j,h)$, $(\Delta + 1, j-1,h)$, $(\Delta + 1.5, j\pm 0.5,h \pm 1)$ and $(\Delta + 2, j,h)$

I would like to know what exactly is the definition of “conformal content” and how are lists like the above computed.

Similar lists can be constructed for various kinds of short representations like those labelled by $(j+h+1,j,h)$ ($j, h \neq 0$), by $(j+1,j,0)$, by $(h,0,h)$, by $(0.5,0,\pm 0.5)$, by $(h+1,0,h)$ and $(1,0,0)$. Its not completely clear to me a priori as to why some of these states had to be taken out separately from the general case, but I guess if I am explained the above two queries I would be able to understand the complete construction.

There are usually two books that I recommend that provide some background knowledge with a bias to the more mathematically inclined reader:

• Martin Schottenloher: "A Mathematical Introduction to Conformal Field Theory" (no superconformal field theory, but explains the connection to the Wightman axioms)

• Blumenhagen, Plauschinn: "Introduction to Conformal Field Theory, With Applications to String Theory". (With a section on supersymmetric CFT and boundary CFT).

It is true that most papers on CFT don't explain and don't define the most basic concepts, so that there is a significant gap if you go from an introduction to QFT to the CFT literature, but the books above should bridge that gap.

Sidenote: The state-operator correspondence is sometimes called the Reeh-Schlieder property, which characterizes QFTs where the Reeh-Schlieder theorem is true. In these QFTs there is a unique vacuum state |0> that is separating and cyclic for all local operator algebras so that there is a 1:1 correspondence of operators A with the state that the vacuum gets mapped to, A |0>.

• Thanks for your reply and references. As it stands for the next few weeks, I won't be having access to any books. It would be great if you can give me some some online references which explain how this charge assignment is done in superconformal field theory. Feb 9 '11 at 16:29

http://arxiv.org/abs/hep-th/9905111

where many questions are just answered. Search for conformal primary, superconformal primary, algebra, etc. Most of your questions are basic textbook stuff and you shouldn't be asking 3 paragraphs - which require 5 paragraphs to be answered - about every single simple step.

There is nothing confusing about using the term "primary field" for the operators because conformal field theories imply the state-operator correspondence, a one-to-one map between the operators and the states of the Hilbert space. This map can be simply obtained in the radial quantization and it is one of the first things you should have learned about conformal field theories. The transformation of the operators under the generators of a group is given by the commutator - but the result is nothing else than the operator corresponding to the transformed state (by the simple rule).

Another question: You can't determine the dimensions and charges of all the primary fields "just from the algebra". The spectrum obviously depends on the theory - and there are many conformal or superconformal theories that share the same algebra. It's not true that all physics is encoded in the symmetry.

The $(\Delta,j,h)$ charges of various states - primary fields in this case - are simply the eigenvalues. For operators, as I said, they're also eigenvalues, but the symmetry generator $G$ has to act on the operator $O$ as the commutator, $[G,O]$. For example, the identity's commutators universally vanish so all the dimensions and charges of the identity (constant) operator are equal to zero. For supercharges, they're equal to something else and some of the signs differ for different components of the supercharge. However, individual supercharges usually square to zero, so only some products of the form $Q^2$ or $Q^3$ are nonzero which is why the charges and dimensions of the higher powers of $Q$ are less ambiguous than if all the products of components of $Q$ were nontrivial.

When you know the definitions and some maths, you must be able to answer all your questions above because they're simple. If you won't be able to jump into the rhythm where you can answer most of similar questions yourself, or by finding the right literature by yourself, you should give CFTs up because there's no point in learning a math-loaded set of questions by the method of listening/memorizing the answer to every individual question that you hear from someone else. The only conceivable meaningful purpose of learning similar things - such as portions of maths and physics - is to become able to answer subsequent questions without the help of others. If it can't work for you, please just give it up.

• I have seen that review and I didn't find in it anything that helps understand the calculations that I alluded to in the question. I agree that knowing the basic definitions and some maths I should be able to figure out the above calculation but then I am unable to find out the correct literature which defines this particular scheme of charge assignment in superconformal field theories. I would be grateful if you can provide more relevant references (and I am not giving up any soon!) Feb 9 '11 at 16:25
• Also I am not in the plan of memorizing any results! Had that been my idea then I wouldn't have been asking any questions on this forum! Feb 9 '11 at 16:32