## May 13, 2004

### 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 do act 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

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

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### Pohlmeyer charges

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.

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.

Posted by: Jacques Distler on May 13, 2004 7:31 PM | Permalink | PGP Sig | Reply to this

### Re: Pohlmeyer charges

I think ‘complete’ as used by Pohlmeyer et al. is supposed to be meant in the usual sense, i.e. that there are as many mutually Poisson-commuting conserved (and gauge invariant) quantities as degrees of freedom, classically.

What is explicitly claimed to be proven in

K. Pohlmeyer & K.-H. Rehren:
THE INVARIANTS OF THE NAMBU-GOTO THEORY: THEIR GEOMETRIC ORIGIN AND THEIR COMPLETENESS (1987)

is that the knowledge of the Pohlmeyer invariants allows to reconstruct the string worldsheet up to spatial translations.

In fact a subset of the Pohlmeyer invariants is sufficient, but I don’t quite recall if this subset is demonstrated to consist of mutually Poisson-commuting Pohlmeyer-invariants (I must have the paper somewhere in this intimidating pile of printouts here somewhere…).

My main point was that this implies that the Pohlmeyer invariants contain more information than just the center of mass momentum and the angular momentum of the string.

This can also be seen explicitly, but it is quite tedious. The Pohlmeyer invariant at 4th order (4 indices) fills page 21 of hep-th/0403108 (in an operator ordering which destroys its quantum invariance, but anyway), which is the paper version of a masters thesis from a couple of years ago.

Posted by: Urs Schreiber on May 14, 2004 12:57 PM | Permalink | PGP Sig | Reply to this

### Integrability

I think ‘complete’ as used by Pohlmeyer et al. is supposed to be meant in the usual sense, i.e. that there are as many mutually Poisson-commuting conserved (and gauge invariant) quantities as degrees of freedom, classically.

Well, I’ve learned in discussions hereabouts never to assume that people use standard terms like “canonical quatization” or “complete set of charges” to mean what I think they mean.

Anyway, let’s assume Pohlmeyer et al are right: upon imposing the constraints, the classical bosonic string is integrable. What would the implications be if this persisted to the quantum theory, i.e. if the quantum theory were integrable?

Actually, I don’t even need that. I think it is enough to ask what would happen if the 4-index tensor charge that you mention above were conserved in the quantum theory.

Just as conservation of (spacetime) momentum imposes constraints on the (worldsheet) correlation functions, and hence on the string S-Matrix, so would conservation of this higher tensor charge.

Unfortunately, by fairly standard arguments, I think it implies that the string S-Matrix is trivial.

What am I missing?

Posted by: Jacques Distler on May 14, 2004 2:20 PM | Permalink | PGP Sig | Reply to this

### Re: Integrability

Let me try to understand your concern in terms of the DDF invariants, which are, I claim, a superset of the Pohlmeyer invariants, but better understood:

The DDF operators ${A}_{n}^{\mu }$ are an infinite set of mutually commuting operators that all commute with he constraints, and we can pick the subset of ‘number operators’

(1)${N}_{n}^{i}:={A}_{-n}^{i}{A}_{n}^{i}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\forall n\in N\phantom{\rule{thinmathspace}{0ex}},i\in \left\{2,3\cdots ,25\right\}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\mathrm{no}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sum}\right)$

plus the com momenta, which all mutually commute and whose expectation values/eigenvalues completely determine the physical state of a string up to null states.

What is it that makes this acceptable while a similar statement for the Pohlmeyer invariants seems dubious to you?

(Please note that I am by no means saying that what you say is wrong, I am just trying to understand your concern.)

Posted by: Ur s on May 14, 2004 2:40 PM | Permalink | PGP Sig | Reply to this

### Steve Parke

Whoops! I’m not sure whether the paper I wanted to refer you to was the one I linked to above, or this one.

Posted by: Jacques Distler on May 14, 2004 2:28 PM | Permalink | PGP Sig | Reply to this

### Re: Steve Parke

I have now had a look at

S. Parke: ‘How many conserved currents are necessary for the calculation of the S-matrix in the massive Thirring model?’

In the first paragraph on p. 2 it says:

These properties are known to be implied by the existence, in the quantized model, of an infinite number of conserved local currents […]

Could it be that the point is that the Pohlmeyer invariants as well as the DDF invariants are an infinite number of conserved objects, but they are not (integrals over) an infinite number of conserved local currents?

Posted by: Ur s on May 14, 2004 2:59 PM | Permalink | PGP Sig | Reply to this

### Re: Steve Parke

You’re right. Parke, in his argument, is certainly assuming a local (in spacetime) conservation law.

Let’s proceed differently. Assume that each physical single-string state is specified uniquely by an infinite set of commuting conserved charges. If you factorize the 2→2 S-matrix element, the charge of the intermediate string state exchanged is uniquely fixed to be the sum of the charges of the 2-string state that creates it.

But there’s one thing we know, from Veneziano: there’s an infinite number of poles from on-shell intermediate string states. Varying only the external momenta, I can “hit” all of these poles. Is that compatible with the notion that these intermediate string states (when they go on-shell) are uniquely characterized by their Pohlmeyer charges? It seems unlikely.

As to the DDF operators, they don’t, naively, yield an infinite set of commuting charges. They may commute with the constraints, but they don’t commute with each other.

Posted by: Jacques Distler on May 14, 2004 3:36 PM | Permalink | PGP Sig | Reply to this

### Re: Steve Parke

It seems unlikely.

As to the DDF operators, they don’t, naively, yield an infinite set of commuting charges. They may commute with the constraints, but they don’t commute with each other.

I think we can restrict to a subset of DDFs which does consist of mutually commuting operators, as I have indicated above and which is still complete in the sense that states are characterized by these mutually commuting charges up to null excitations. Not?

Posted by: Urs Schreiber on May 14, 2004 3:45 PM | Permalink | PGP Sig | Reply to this

### Re: Steve Parke

OK, I think I see the point.

These charges are not local (on the worldsheet). You need both locality and commuting with the Virasoro generators for the charges to be preserved by string interactions. We know that total oscillator number is not a conserved quantum number for the interacting string.

Leaving aside the question of their utility, haven’t you completely characterized these charges now? You’ve found a maximal commuting set (the ${N}_{n}^{i}$). The states of the free string do, indeed break up into 1-dimensional eigenspaces of these commuting charges.

And we know the (very simple) algebra satisfied by the full set of DDF operators.

Pretty much end-of-story, no?

Posted by: Jacques Distler on May 14, 2004 8:53 PM | Permalink | PGP Sig | Reply to this

### End of the story.

Pretty much end-of-story, no?

Yes. :-)

The reason why there is any interest in the Pohlmeyer invariants is that some people are hoping that using them the entire story can be rewritten, including that end.

I cannot really explain or defend this point of view, since it wasn’t my idea and I don’t fully see what the principle should be. But vaguely the idea is that there might be alternative ‘quantizations’ of the string and that these could be found by looking at algebras of classical invariants of string as well as their possible quantizations (whatever that may mean in detail).

Many years ago K. Pohlmeyer came up with the set of classical invariants that now carry his name, and since then some people have tried to understand what quantum extensions of their classical Poisson algebra might be. Taken as such, this is a horribly complicated task, because the algebra of Pohlmeyer invariants is very complicated already at the classical level. And the task had not been completed. But the hope by some people working on this task was and is (as one can see from reading the introductions of the relevant papers and as I have been told explicitly) that if it could ever be completed, that it might then yield a consistent quantum theory of the Nambu-Goto string without the usual effects such as critical dimension, etc.

This is a development originally quite independent of Thomas Thiemann’s approach, but of course the aims are similar.

The point that I want to make is that it has been overlooked that there is indeed at least one solution of Pohlmeyer’s program: It is obtained by simply realizing that the classical Pohlmeyer invariants are but a subset of all invariants in the enveloping algebra of the classical DDF oscillators. Using this insight, the Pohlmeyer program is solved by simply replacing the classical DDF invariants by the ordinary DDF operators. This leads to the above mentioned end of the story, namely simply to the ordinary quantization of the string, implying that the Pohlmeyer invariants are just a peculiar subset of the enveloping algebra of DDF invariants and that all the usual results, such as the critical dimension, apply.

(I suspect that one can find a subset of Pohlmeyer invariants which consists of mutually commuting invariants (e.g. $\left\{{Z}^{\mathrm{iijj}}{\right\}}_{i,j\in \left\{2,\cdots ,25\right\}}$ plus others) that are nothing put products of number operators ${N}_{n}^{i}$ times maybe center-of-mass momenta, but I haven’t fully checked this.)

I have received complaints that the use of DDF invariants does not solve the Pohlmeyer program, due to the fact that the construction of the DDF invariants involves an arbitrary but fixed lightlike target space vector $k$, a fact that has been claimed to somehow interfere with Lorentz invariance. I don’t think this is true. (Maybe here is some ambiguity concerning the rules of the game that people want to play.)

Anyway, the bottom line is: If you haven’t been interested in Pohlmeyer invariants before, then there is probably little reason to be interested in them now, since idications are that they are just a peculiar way for doing standard steps.

But one question might somehow still remain: Is it possible in principle to find ‘alternative’ quantizations of the string?

I doubt it, but some people are working hard trying to do just that.

Posted by: Urs Schreiber on May 17, 2004 12:12 PM | Permalink | PGP Sig | Reply to this

### Re: End of the story.

I wrote:

But one question might somehow still remain: Is it possible in principle to find ‘alternative’ quantizations of the string?

I doubt it, but some people are working hard trying to do just that.

Here is a recent publication arguing for these ‘alternative quantizations’ that even cites the discussion here at the String Coffee Table:

Bert Schroer: An anthology of non-local QFT and QFT on noncommutative spacetime (2004).

See in particular the first half of p. 19. There it says:

String-localization is a radically different property from the properties of quantized Nambu-Goto strings on which string theory is based. There exist two different quantizations of the N-G string. The more intrinsic approach is due to Pohlmeyer [46] and consists in extracting a complete set of invariants (called Pohlmeyer charges) which together with the generators of the Poincaré group form a closed Poisson algebra. This task has been almost completed and in this way the Nambu-Goto system was seen to be an integrable system in the classical sense. The quantization of this algebra i.e. the search for an algebra of operators which have commutation relations which mimic the Poisson structure as much as possible is still an open problem. The Pohlmeyer strings are not string-localized but their algebra of invariants share with the string-localized fields that there is no mysterious and unphysical distinction (resulting from the non-intrinsic canonical quantization) of d=10,26 rather they exist as Poincaré invariant theory in each dimension $\ge 3$.

The other approach is that which led to string theory proper and consists in quantizing before eliminating the constraints and afterwards taking care of the constraints in the spirit of the BPS cohomological approach. The localization aspect was recently investigated and was found that they are neither string localized nor do they fulfill the causal dependency property of a relativistic causal pointlike localization (for an on-line discussion of these aspects see [48].

Posted by: Urs Schreiber on May 18, 2004 11:07 AM | Permalink | PGP Sig | Reply to this

### Fairy Tales

I’ve already explained why such alternative quantizations are very-likely nonsense. But let me try again, this time by way of an analogous (failed) attempt at an alternative quantization.

Light-cone gauge is a manifestly-unitary quantization of the string, which is almost Lorentz-invariant. The only failure of Lorentz-invariance is in the $\left[{M}^{i-},{M}^{j-}\right]$ commutator. This ought to vanish, but instead, one gets a transverse antisymmetric tensor, ${B}^{\mathrm{ij}}=-{B}^{\mathrm{ji}}$, which vanishes only in 26 dimensions (and the correct choice of intercept).

That’s the usual story, but there’s another case where ${B}^{\mathrm{ij}}=0$, namely 3 dimensions! There being only one transverse coordinate, an anti-symmetric tensor must vanish.

So why don’t you read in all the textbooks that the bosonic string can be consistently quantized in 3 dimensions? After all, unlike quantizing the bosonic string in some random number of dimensions (which, even if you could succeed (you can’t), would be a rather pointless exercise), 3 dimensions is important. In 3 dimensions, the Ising model is dual to a theory of random surfaces, so finding a continuum string theory in 3-dimensions would be tantamount to finding a dual to the 3D Ising model.

If you could compute the critical exponents of the 3D Ising model from a string theory, you would be rich and famous (well, OK, maybe not rich …).

But finding a Lorentz-invariant, unitary quantization of the free string (which we just succeeded in doing — unlike Pohlmeyer, Thiemann, and the rest), is not sufficient.

We also need to find consistent, unitary, Lorentz-invariant string interactions. That’s a much harder problem, and one which fails for the 3D bosonic string in light-cone gauge.

If you have even a wacky idea for obtaining Lorentz-invariant interactions in this theory, let’s talk about it, write a paper, and become rich and famous.

As to Pohlmeyer, Thiemann and company, they are so far from even showing that they have a consistent quantization of the free string, that there’s no point in holding one’s breath waiting for them to construct consistent string interactions.

P.S.: Congratulations on the first citation to the Coffee Table. I don’t know that you can put it on your CV, but it is nice.

Posted by: Jacques Distler on May 18, 2004 2:30 PM | Permalink | PGP Sig | Reply to this

### The first!

I can’t deny that I am somewhat happy to see my very first peer-reviewed paper in (virtual) print. :-)

Thanks to everybody for the inspiring discussion here at the Coffee Table which was one major reason that this paper came into being in the first place.

Posted by: Ur s on May 14, 2004 3:03 PM | Permalink | Reply to this

### Re: The first!

Nice! :)

It’s Friday and you deserve at least a mini-celebration. Go and have some fun! :)

I know I plan to after this grueling week :)

Eric

Posted by: Eric on May 14, 2004 4:13 PM | Permalink | Reply to this

### Re: The first!

The fun that expects me this evening is hard labor in renovating our new flat. We are still not done, but with a little luck we can move this weekend! :-)

Posted by: Urs Schreiber on May 14, 2004 4:18 PM | Permalink | PGP Sig | Reply to this