# Witten Index, letter partition function and superconformal representations.

Except in a few papers I have seen so little written about this that I am not sure I can even frame this question properly.

• I would like to know of expository references and explanations on the concept of "single/multi trace letter partition function" and how it connects to Witten Index and superconformal field theory.

I haven't been able to find any reference which explains the concept of letter partition function and how techniques from representation theory get used to calculate them. (especially in the context of superconformal representations)

For example one can see between page 15 and 30 of this paper to see some usages of this.

As said above this technology comes up often in the context of superconformal group representations. I would be happy see references which give explanations about them.

In superconformal representations one often lists out "long" and "short" representations labelled by the "primaries" and then one calculates the Witten Index of them. (which apparently always vanishes for the long ones) To give an example of a case where Witten Index is calculated,

So for ${\cal N} = 2$ superconformal algebra in $2+1$ dimensions the symmetry group is $SO(3,2)\times SO(2)$ 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 would like to know what are the precise eigenvalue equations used to do the above labeling.

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. The Witten Index of the above is supposed to be $0$ and I guess it was supposed to be obvious without explicitly enumerating the labels.

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 queries I would be able to understand the complete construction.

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## 1 Answer

I would recommend you to read this paper from 2008. It contains more review materials in it than the one you quoted.

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@Yuji Thanks for your reply! I have seen that paper but haven't yet been able to go through it in all its glory. It would be helpful if you could tell me what is the eigenvalue equation that is implicitly being used to define that 3-tuple and what is the definition of the conformal content and how it is constructed. I guess it would be easier for me to get through the paper if you could write in a short explanation along the above. – Anirbit Feb 11 '11 at 6:46
@Yuji Also what confuses me with this terminology of short and long is that if I go by say the definitions of them as in the book by Weinberg then that is related to whether the mass of a massive supermultiplet saturates its lower BPS-like bound defined by the central charge. But in these theories there seems to be R-symmetry and hence central charge should be 0. Then what is the bound with respect to which short and long is being defined. I can't intuitively see what is the difference in the physics of short and long representations. – Anirbit Feb 11 '11 at 6:48
@Yuji Just a side question - Are you Yuji Tachikawa who recently wrote the paper on exactly marginal deformations with Seiberg, Wecht, Green and Komargodski? – Anirbit Feb 11 '11 at 6:50
@Anirbit: Yes I am. I just changed my profile on MO to use my full name. – Yuji Tachikawa Feb 11 '11 at 9:58
@Anirbit: as for the physics question: you just need to study the irreducible representations of superconformal algebra. This is a bit different from those of the super-Poincare algebra, but has a similar flavor. That's why they are also classified into long and short representations. Anyway, you really need to go through the review sections on that paper, or the review sections on arXiv.org/abs/hep-th/0510251 , or Minwalla's review arXiv.org/abs/hep-th/9712074 . – Yuji Tachikawa Feb 11 '11 at 10:09