Last time I pointed to the paper by Green, Komargodski, Seiberg, Tachikawa and Wecht (GKSTW) I was asked to show what I would have done differently. Here is a sketch of some things that I would have put as a part of the main paper (this is not to be assumed to be comprehensive, or careful, nor will I add many references).

Part of the reason to do this is that I think some people out there might benefit from some aspects of this information. What follows is rather technical, so if you’re not well versed in basic SUSY, and on CFT’s, I apologize in advance: you should not read this then. What I describe bellow is an extended version of the appendix to GKSTW paper. In a good tradition of computer games, this is a third-party add-on. Or you can think of it as a cheap mash-up. Up to you.

Notice: I’ve had some problems with formatting and colors. Temporary fixes have been put in place. Expect updates to text as I find mistakes.

The main information that I would have added in the bulk of the paper is sketched in the appendix of GKSTW. This would be a more in depth discussion of the superconformal representations and the unitarity bounds. Also, how they relate to superspace. Many of these are somewhat well known to experts, but chasing references can be quite a mess.

The first thing is the superconformal group in 4-dimensions. I will just write the generators as a diamond, organized by dimension

The superindex indicates the dimension of the operator. I have suppressed various spinor indices, except on the rotations, where there are two different SU(2) subgroups of SO(4). In the spinor notation one of them rotates left spinor indices, and the other rotates the right spinor indices. The quantum numbers associated to these two will be called

There are also special conformal and superconformal generators (K,S), and there is an R-charge, which we have called R.

There is also the unitarity relation in the conformal group. These are that the elements of dimension zero are self-adjoint, while those of dimension k are adjoints of operators of dimension -k. In particular, we have that

and that

The usual notation is that Q has un-dotted indices and the one with the bar has dotted indices.

The most important commutation relations for us are that

and that

I’m going to skip all normalization constants in the algebra above (you have to look at references for that). For the , we change the sign of R in the equation, and we change the undotted indices by dotted indices.

(red equation)

Ok, so this sets up the algebra of the super-conformal group. Finally, some notation: A superprimary is a state annihilated by the S generators.

It is called a chiral superprimary if it also annihilated by the .

Chiral superprimaries have a dimension that is completely determined by their R-charge and spin. Let us call such a state

This follows from the commutation relation we had above in red. The idea is that because of unitarity we have that

but for a chiral primary this is zero. The adjoint of is and using the algebra one finds that

(green equation)

where the expression above is for eigenvalues of the corresponding operators. The full expression one gets is

(blueq equation)

if the original state is orthonormal. This is important. It tells us that if the operator remains chiral (is annihilated by the ), but it gets an anomalous dimension, then the green equality stops being satisfied. The blue equation then tells us that the dimension gets bigger (we assume for this that the R-charge does not change). This is the statement that the operator can become marginally irrelevant when it is a perturbation of the conformal group.

It also tells us that the field stops being a primary. This is part of the punchline of the paper: if an operator is not a primary, it is a descendant of another operator of lower dimension. We can do this twice, and we find that it descends from .

So the upshot is that one must have operators of lower dimension in the theory, with which the chiral operator combines to get an anomalous dimension. These other set of operators must have no descendant of the form , where is the precursor, because if such operators exist, the superconformal representation is complete and one can find another chiral primary from linear combinations of operators that is chiral and preserved.

So to get a chiral primary to be lifted it has to combine with operators of lower dimension that have a `null descendant’ after acting on it with two supercharges. These are usually the types of objects that lead to conserved current equations.

So far I have not mentioned superspace. If one adds superspace to the mix, then one needs to understand what’s the difference between covariant superspace derivatives D and the Q.

They are actually very similar, indeed , so up to conformal descendants D and Q operate on almost the same footing. This is important. A chiral superfield is one annihilated by . If the lowest component of the superfield is taken as an operator and it is a primary of the conformal group, then necessarily annihilates the operator.This equation is true if the superfield is chiral always.

But this does not mean it is annihilated by , so this is where the above applies. This relation to superspace tells us that there is an equation of the form

as an operator equation and the variable r is zero at the beginning. This makes a superfield that is semi-conserved when r=0.

When one puts the extra information of dimensions together, one finds that one needs a special type of pairing between superfields: the ones associated to current conservation equations and the chiral ones.

This splitting and joining of things to lift supersymmetries is reminiscent of index theory, so the right way to think about this is in terms of index theory. So if I were writing the paper, I would have also gone and dug up stuff into the superconformal index in four dimensions to see if there is a relation (http://arXiv.org/abs/hep-th/0510251).

The part above is the algebra of how things have to combine to lift chiral primaries.

The other half of the GKSTW paper is about the obstruction to deformation (when does a deformation cause the deformation itself to become irrelevant). This half was suggested originally in the work of Barak Kol done on 2002, called “On conformal deformations”, where a non-rigorous argument based on supergravity was suggested and exploited in some setups: there is a HIgss effect where a vector becomes massive.

Here, there is a bit of history to tell. Braka Kol and myself where at the Institute for Advanced Study in Princeton then. Barak wrote his paper then, but this happened in 2002 and that year was somewhat crazy for me. So although the result is very interesting I was very busy at the time and did not have time to pay true attention to it (which I would have under other less busy circumstances). Of course, Barak is a good friend and he contacted me asking that I ‘put history straight‘ on precedence of results after my first post on the GKSTW paper, so I agreed to that.

So history is as follows:

NSVZ first, then Leigh-Strassler. Wait a few years, Kol. Wait a few more years and then GKSTW. There are a lot of missing links in the middle of course , plus side trees and branches and a lot of other results from way back when. Getting all of this straight usually takes quite a while and is something that many times comes out wrong.

Finally, let me point out that I consider that most of the details of the sketch above are well known to experts and they would have reconstructed them on their heads while reading the GKSTW appendix and other parts of the text. Furthermore, anything much longer than what I wrote above, and I would have made a ‘review paper’ instead. Indeed, a proper review paper is due for the field, and to quote a famous paper “hopefully done by somebody else”

on May 26, 2010 at 10:19 amCyrilHi David,

Do you know any text to learn (4d) conformal perturbation theory from?

Thanks

Cyril

on May 26, 2010 at 1:25 pmdberensteinHi Cyril:

I don’t know any text where this is done carefully in 4 dimensions. You have a much better chance with 2d where the subject was originally developed to understand string backgrounds. For that, I think that the book of Joe Polchiniski would be a good place to start.

on May 26, 2010 at 12:07 pmLuboš MotlA crisp explanation, David!

The following sentence is a question, and really just a question at this moment.

When you say “operator remains chiral (is annihilated by Qbar)”, does the term “annihilate” mean that – for operators – the operator commutes with Qbar (which is what I would expect from the state-operator correspondence), or do you really mean a full-fledged annihilation by Qbar (from the left, from the right, or from both sides)? Or does the word “annihilate” for operators mean the vanishing commutator?

You may want to write a paper straightening this stuff a little bit. If you find it painful to send such a review to hep-th, you may always send it to pop-ph or something like that.

BK has kindly pointed out a mistake in my way of referring to the history, too, and I have equally kindly fixed the references.🙂

on May 26, 2010 at 1:22 pmdberensteinHi Lubos:

For operators it means vanishing commutator. For states it really means annihilation. Under the operator state correspondence one matches the way the conformal group naturally acts: it acts to the left on states lets say, and by commutators on operators.