# 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

asked Jan 30, 2011
edited Dec 10, 2014

## 1 Answer

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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 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.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
@Yuji Just a side question - Are you Yuji Tachikawa who recently wrote the paper on exactly marginal deformations with Seiberg, Wecht, Green and Komargodski?

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
@Anirbit: Yes I am. I just changed my profile on MO to use my full name.

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

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
And, as you're in TIFR, so ask any string theorists there about your question, before coming to MO. That would be much, much straightforward to do.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
@Yuji Thanks for your reply. I have in fact been looking at these very papers to understand stuff but somehow I find it very uncomfortable to read Minwalla's papers. Somehow I find his method of writing to be hard to relate to other standard literature (especially of mathematics). Hence I have been looking for other people's writings or text book references on the topic. (And about my TIFR affiliation: it is very likely that I will leave TIFR in a few months)

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
@Yuji One of the key arguments in the first of the Minwalla's papers that you linked to is that energy, $SO(3)$ highest weight and R-symmetry highest weights form a complete set of labels of the entire spectrum in the question.

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
Also, when string theorists say ".... are labeled by such and such", it doesn't usually mean "all ... are labeled by such and such" with mathematical rigor. It's better to interpret the sentence as "... which usually appear in physics are labeled by such and such". So, don't try to prove what are stated in string theory papers. Instead, work on a few examples and check that indeed, they have the properties as claimed by the author.

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
For physicists, the prototype is always the harmonic oscillator. So, we first combine the generators of (super)conformal algebras into creation and annihilation operators. Then, given a representation, take a state, and we apply the annihilation operator as much as possible. Eventually we get the states with lowest energy/lowest dimension in that representation. That states with lowest energy/dimension in the original group can still carry a representation of the little group.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
This representation of the little group is the label Minwalla is referring to.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
@Yuji Thanks a lot for your explanations. This labeling has been worrying me ever since. I was wondering if this was related to an idea that representations of a non-compact group are labelled by those of the maximal compact subgroup etc. But nothing like this seems to be proven or referred to in these papers. My understanding of CFT and supersymmetry is rooted in the chapter from the book by Polchinski and the third volume of Weinberg's QFT series respectively. Coming from this background the set-up in Shiraz's papers looks very unfamiliar.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
@Yuji I was wondering if there are other texts or approaches that if I am familiar with will help me understand these papers on index. (something apart from the Polchinki or Weinberg's books that I said above) I have rarely ever seen any exposition on CFT in 3+1 dimensions, thanks for the reference to that Mack's paper.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
You're already at the point textbooks won't help you, Anirbit. Start looking up references of a paper, references of references of the paper, (references of)^k the paper. Yes that will soon exponentially diverge, but you will find relevant materials that way. Mack's paper should be in k<4 of Minwalla's:) By the way, Terning's "modern supersymmetry" might have a section on 3+1(super) conformal algebra.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
@Yuji Thanks for your reply. I have had a look at Terning's book too. Terning infact has a section explaining the unitarity restriction by superconformal invariance that Shiraz had discovered. I find Terning's exposition more understandable!

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Anirbit
Yeah, but Terning's book is as advanced as you can get via textbooks. Note that Shiraz didn't really discover them; it's all in Dobrev&Petkova. Shiraz's exposition is far nicer than the original paper though.

This post imported from StackExchange MathOverflow at 2014-12-10 17:30 (UTC), posted by SE-user Yuji Tachikawa
@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
@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
@Yuji $I = \prod _ {n=1} ^ \infty \frac{1}{1-F(x^n)} = exp \left [ \sum _{m=1} ^ \infty \frac{I_{st} (x^m)}{m(1-x^m)} \right ]$ I would like to know of the derivation/explanation of the above equality which seems to hold in the above limit. Also in the same strain I would like to know of another equality which gets used in these papers which is as follows. If $\Delta$ is the dilatation operator and $H$ is the Hamiltonian then one seems to claim the following equality for the partition function, $Tr x^H = \sum _ {operators} x^ \Delta$

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