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.
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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.
Best Answer
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>.