# Witten Index, letter partition function and superconformal representations.

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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.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit

edited Dec 10, 2014

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

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
answered Feb 10, 2011 by (130 points)
@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.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
@Yuji The above is the central idea that I do not understand and this is further emphasized in a statement which says, "Any irreducible representation of the superconformal algebra may be decomposed into a finite number of distinct irreducible representations of the conformal algebra...which are in turn labelled by their own primary states...hence the state content of an irreducible representation of the superconformal algebra is completely specified by the quantum numbers of its conformal primaries"

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
@Yuji I would like to know of references or explanations about this central idea expressed in the above quote.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
Uhm, Minwalla's paper uses the standard notations in string theory; yes it doesn't follow what mathematicians usually use but they are mostly equivalent. So, if you'd like to do string theory, you need to learn to use that notation. Yes that makes it harder for us to communicate to mathematicians, but that's life. We manage.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
Anyway, all these papers assume you know (physically relevant parts of) the representations of non-super Poincare algebra and non-super conformal algebras. Have you studied them? For the 4d conformal algebra, see inspirebeta.net/record/100788 .

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
You also need to first learn the large-N limit in general, before understanding the equation you wrote between $F$ and $I_{st}$. Read and understand Coleman's "Aspects of Symmetry" for example.
@Yuji Thanks for your reply. I also realize that I am trying to understand too many things at the same time. And I am finding it a bit unmanageable at times. But I am forced in this situation for a couple of months now. I am hoping that after these months I will be able to take a more systematic approach. Can you tell me some good references to understand the state-operator correspondence in this generality? I had some passing learning of it from the few paragraphs about it in the $1+1$ dimensional case from Polchinski's book.
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