Witten Index, letter partition function and superconformal representations.

Question

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

Answer 1

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

Comments

@Yuji Where in the RHS the sum is over all operators in the theory. Its not very clear to me as to what this means. Also this vaguely reminds me of the so called "state-operator" map and also of the equality seen in superconformal algebra that dilatation operator is the sum of the Hamiltonian and the 0-component of the special conformal transformation. But I can't knit the whole story.

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

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In general, Anirbit, you're trying to understand too many things at the same time. Forget about multitrace/singletrace operators for the moment and first study the representations of the (super)conformal algebras, following the reviews, articles and textbooks.

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

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And, do understand the state-operator correspondence first, again forgetting about the particular case of N=4 SYM (or other gauge theory) for the moment. Note that people often call $\Delta$ the Hamiltonian in the CFT literature, and the "partition function" in the CFT literature is often with respect to $\Delta$, not $H$. Your last equation should rather be $Tr x^\Delta = \sum_{operators} x^\Delta$, which is almost trivially true, given the state-operator correspondence.

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

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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.

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

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@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.

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

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@Yuji Thanks for your reply. What puzzles me about these books is that they don't seem to use this terminology of "single/multi trace" operators and Witten Index From talking to people my operational understanding is that one computes say the single trace Witten Index one is summing over all possible guage invariant combinations of fields that can be created from the basic fields and taking derivatives (and very often one wants to drop from this sum those combinations which have an overall derivative and account for them later).

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

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@Yuji But what is puzzling is that the full Witten Index can be calculated also by just knowing the index over the "conformal towers" over the superconformal primaries. These "conformal towers" are basically created by taking arbitrary derivatives of the basic fields. (and the reason for this nomenclature is not obvious to me). If $x$ is the fugacity for the conserved quantity in question and $F(x)$ is the index calculated over the conformal towers and $I$ is the (full) Witten Index and $I_{st}$ is the single trace Witten Index then in the "large $N$" limit apparently the following holds,

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