### Referee reports on SCFT deformations and Pohlmeyer invariants

#### Posted by urs

Readers of this weblog will recall that we had discussed here two drafts which I have meanwhile submitted to JHEP.

One is

On deformations of 2D SCFTs, hep-th/0401175

which I originally presented in the entries ‘Classical deformations of 2D SCFTs Part I and Part II.

The other is

DDF and Pohlmeyer invariants of (super) string, hep-th/0403260

which originates in the thread Pohlmeyer charges, DDF states and string-gauge duality.

Now the referee reports for these submissions have arrived. Both papers have fortunately been accepted (one with a slight modification, see below), but there are some comments in the reports which I would like to briefly discuss here, since they concern issues which have been addressed here at the String Coffee Table and hence might be of interest.

Concerning the SCFT deformation paper the referee writes:

This paper discusses deformations of 2d superconformal field theories. As 2d SCFTs represent classical solutions of the string equations of motion understanding how they deform as we turn on spacetime fields is very crucial in unraveling their vacuum structure.

The only part of the paper that is problematic is the claim that there is a deformation that can be interpreted as a RR background. This is obviously not correct since the RR excitations couple to spin fields (both matter and ghost) while the deformation is not written in terms of these fields.

Otherwise the results of the paper are intersting and the paper should be published after the author modifies the paper by omitting his claim about the RR backgrounds.

Finally in the future I would advise the author to attempt to derive equations of motion for the spacetime fields. In other words to go beyond the classical level and discuss the issue of normal ordering.

Of course precisely the issue with RR backgrounds has on the one hand side been a motivation for this entire investigation and on the other hand I don’t claim that my construction sheds any new light on this particular problem (not yet at least :-).

Actually I mention RR backgrounds in two different contexts in this paper:

One, which comes from the bulk of the text, is the observation that a certain type of SCFT deformation which I describe is apparently best interpreted as describing a *D-string* in an RR 2-form background - not an *F string* in such a background! The D-string couples to the RR 2-form pretty much like the F-string couples to the NSNS 2-form, so this explains why at this point RR backgrounds make an appearance even though string fields do not. The rekation and distinction between the D-string in RR 2-form background and the F-string in NSNS 2-form background is a little subtle, but I do try to discuss that in the paper. Probably I need to emphasize the reason why in this context no spin fields make an appearance.

On the other hand, I had included one additrional remark where it is indicated how RR backgrounds should fit into the framework of that paper after all, but then of course using spin fields. The idea is that there should be a deformation of the worldsheet BRST operator even for these backgrounds (though there are subtleties, of course), along the general lines discussed in a previous entry. But of course the inclusion of RR backgrounds for the F-string this way is more like an idea for a research program, maybe, than a result. So perhaps I should really just remove that paragraph.

Concerning the DDF/Pohlmeyer paper the referee writes

This paper relates the so-called Pohlmeyer charges of the bosonic string to the standard DDF oscillators. It’s not clear to me, even having read the paper that there is any point to the Pohlmeyer construction. When acting on physical states (ie, after quantization), it has been argued that the Pohlmeyer charges yield only triivial information (like the total momentum) about the state.

However, it is certainly of value to recast them in terms of the standard DDF operators, which

doact nontrivially on physical states. On emight then have a hope of seeing whether ther is any nontrivial content in the Pohlmeyer construction.I think this paper should, therefore, be published.

I am glad that the paper has been accepted, but I am also surprised that the idea that somehow the Pohlmeyer invariants all are just made up of center-of-mass momentum and Lorentz generators seems to have spread quite far.

This idea seems to have originated in a discussion between Luboš Motl and Edward Witten where it was rediscovered that, while it is obvious that the Pohlmeyer invariants at first and second order are trivial, even the Pohlmeyer invariants at third order are trivial. This is well known, see for instance

D. Bahns: The invariant charges of the Nambu-Goto string and Canonical Quantization (2004) ,

and , while maybe surprising, doesn’t continue to hold for higher orders. Indeed, a generally accepted proof says that the Pohlmeyer invariants are *complete* in the sense that from their knowledge the worldsheet can locally be reconstructed.

In the above paper I have included in the conclusion some speculations what the Pohlmeyer invariants, being nontrivial, could be good for. But I concede that maybe they are not good for anything, this remains to be shown. The burden of proof is on those who claim otherwise. Meanwhile, the Pohlmeyer invariants and their relation to DDF invariants has attracted some attention simply because this has been related to the general question on how theories of gravity can be or have to be quantized - as we have discussed in gory detail before.

Posted at May 13, 2004 12:38 PM UTC
## Pohlmeyer charges

What does that sentence

mean?At the classical level, a “complete” set of (commuting, modulo the contraints) conserved charges means that the system is completely integrable. In quantum mechanics, the corresponding statement is that the physical Hilbert space breaks up into 1-dimensional simultaneous eigenspaces of the charges.

What does “complete” mean here?

If possible, please don’t answer with a characterization specific to the string. Rather, I’d like a definition of “complete” in the Pohlmeyer sense, which would apply to an arbitrary classical mechanical system.