### Pohlmeyer charges, DDF states and string-gauge duality

#### Posted by Urs Schreiber

It’s half past two in the morning here in central Europe and a can’t sleep because my brain is thinking about Pohlmeyer charges! Honest. I have to write this entry if only to get some sleep until my girlfriend will wake me up for breakfast in a few hours. :-)

So let’s consider the classical (super)string. Physical observables are those that are *gauge invariant* with respect to the worldsheet Virasoro algebra, i.e. those that Poisson-commute with all the (classical for the moment) Virasoro generators. We know that these just generate two copies of the diffeo group on the circle (for the bosonic string) so our classical physical observables are nothing but rep invariant functionals. Two very familar sorts of rep invariant functionals are

- Integrals over rep-weight 1 densities

- Wilson lines.

Pohlmeyer takes the second exit and shows that the set of all imaginable Wilson lines around the string (at fixed worldsheet time) is a full (overcomplete, actually) set of invariant classical observables. I like this somehow, because it smells of string-gauge duality, a little. Actually he considers not the Wilson lines themselves but their Taylor coefficients. Anyway, these Pohlmeyer charges are obviously classically rep invariant but have a very convoluted algebra. Thinking about it for about 15 minutes I could not figure out how to adapt the Pohlmeyer charges to the superstring (which need not mean much).

Ok. Now I take instead the first exit and construct a complete set of invariant classical observables by the first method, namely by considering integrals over fields that have classical conformal weight 1. (I’ll make all of this precise tomorrow morning, promised.) That’s very easy because this job has already been done for us by the people whose initials are DDF and whose real names I’ll look up tomorrow, if anyone insists. :-) I mean, to do so for the classical string we can straightforwardly adapt the CFT construction of DDF states. It is even easier for the classical string because we do not need to use a lightlike momentum for the classical DDF states. And even better - we get the construction for the superstring for free, because the DDF states for the superstring are just as well known. Since the real DDF states span the whole Hilbert space of the quantum string I bet that the classical DDF are complete, too.

I think that’s good, because the classical DDF states do tell us something about the string’s spectrum even in Thiemann’s approach. Furthermore, while there is no sign of critical dimension at the classical level I claim that the classical DDF states at least give us the level-matching condition! And all this for the superstring, too.

Seems to me that IF Thiemann’s approach is mathematically viable then it would profit a lot from using classical DDF states (which in his context would be the same as the quantum states, essentially), because these do tell us manifestly about the string’s spectrum and these do easily generalize to the superstring.

Or so I think. Remeber, I should really be sleeping right now ;-)

— Note added later on: —

I have taken the time to write this idea up in more detail:

Urs Schreiber, DDF-like classical invariants of (super)string.

## Re: Pohlmeyer charges, DDF states and string-gauge duality

This is the content of an email which I have just sent to K. Pohlmeyer and K. H. Rehren:

[begin forwarded email]Dear K. Pohlmeyer, Dear K. H. Rehren,

last year, in a lunch-break discussion with D. Giulini at a symposium, I first heard about your approach of algebraically quantizing the Nambu-Goto action using a quantum deformation of the Poisson algebra of classical invariants.

I learned more details of your approach from the nice summary which is given in the recent paper by Thomas Thiemann.

We have been discussing this paper in some detail on a weblog called the String Coffee Table. While the central idea of this paper is problematic, Thiemann nicely highlights the

independentconcepts involved in the construction ofPohlmeyer charges. In the course of this discussion I began to wonder about an issue which is quite unrelated to Thiemann’s approach but pertains to your algebraic quantization program:While the

Pohlmeyer chargeshave certain nice properties, they, as far as I understand, satisfy a rather convoluted algebra, have no obvious relation to the usual spectrum of the quantum string and have no obvious generalization to the superstring. (Is that correct?)Therefore I wondered why one couldn’t alternatively start the algebraic quantization procedure using a different set of invariants (which should be linear combinations of the

Pohlmeyer invariants), namely the classical observables associated with the DDF-operators.Unless I am mixed up, the classical analogues of the DDF-operators do exist, Poisson-commute with all the Virasoro constraints and are complete. They should hence, unless I am making a mistake, be an equivalent alternative starting point for algebraic quantization.

The advantage that I see in using classical DDF invariants is that they satisfy a nice Poisson algebra (isomorphic to that of worldsheet oscillators), hence manifestly know about the massive spectrum of the string (except for the ground state mass and degeneracy) and are easily generalized to the superstring.

I have tried to sketch more details of this idea in this draft. I would very much enjoy if you could find the time to have a brief look at this text and let me know your opinion on the viability of this idea. To me it seems as if it might put the algebraic quantization of the (super)string in contact with the standard quantization, but if you think that I am wrong about this please let me know.

I am posting a copy of this email to the String Coffee Table in the hope to reach a larger audience of interested people and maybe hear of further opinions.

Best regards,

Urs Schreiber

[end forwarded email]