## April 14, 2004

### Power Supply

#### Posted by Urs Schreiber

I am currently visiting the Albert-Einstein institute in Potsdam (near Berlin). Hermann Nicolai had invited me for a couple of days in order to talk and think about Pohlmeyer invariants and related issues of string quantization.

It so happened that when I checked my e-mail while sitting in the train to Berlin I found a mail by Thomas Thiemann, Karl-Henning Rehren and Dorothea Bahns in my inbox, containing a pdf-draft of some new notes concerning what they flatteringly call “Schreiber’s DDF functionals” but what really refers to the insight that the Pohlmeyer invariants are a subset of all DDF invariants.

Glad that my journey should have such a productive beginning I read through the notes and began typing a couple of comments - when I realized that my notebook battery was almost empty.

Here is a little riddle: What are all the places in a german “Inter City Express” train where you can find a 230V power supply?

Right, there is one at every table. But when it’s the end of the Easter holidays all tables are occupied and when nobody is willing to let you sit on his (or her) lap then that’s it - or is it?

Not quite. For the urgent demands of carbon-based life forms there is fortunaly a special room - and it does have a socket, just in case anyone feels like shaving on a train. I spare you the details, but in any case this way when I arrived at the AEI the discussion had already begun. :-)

After further discussion of Thiemann’s and Rehren’s comments with Kasper Peeters and Hermann Nicolai we came to believe that there are in fact no problems with quantizing the Pohlmeyer invariants in terms of DDF invariants. I wrote up a little note concerning the question if there are any problems due to the fact that the construction of the DDF invariants requires specifying a fixed but arbitrary lightlike vector on target space. One might think that this does not harmonize with Lorentz invariance, but in fact it does. I am still waiting for Thiemann’s and Rehren’s reply, though. Hopefully we don’t have to fight that out on the arXive! ;-)

It turned out that I am currently apparently the only one genuinly interested in what the Pohlmeyer invariants could be good for in standard string theory. It seems that everbody else either regards them as a possibility to circumvent standard results - or as an irrelevant curiosity.

Here is a sketchy list of some questions concerning Pohlmeyer invariants that I would find interesting:

The existence of Pohlmeyer invariants gives us a map from the Hilbert space of the single string to states in totally dimensionally reduced (super) Yang-Mills theory. Namely, every state |psi> of the string (open, say) gives us a map from the space of u(N) matrices to the complex numbers, defined by

(1)$M:u\left(N\right)\to C$
(2)$A↦⟨\psi \mid \mathrm{TrP}\mathrm{exp}\left(\int {A}_{\mu }{𝒫}^{\mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}d\sigma \right)\mid \psi ⟩.$

Does conversely every state on $u\left(N\right)$ define a state of string? Apparently the answer is Yes. .

What is the impact in this context of the fact that the Pohlmeyer holonomies

(3)$\mathrm{TrP}\mathrm{exp}\left(\int {A}_{\mu }{𝒫}^{\mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}d\sigma \right)$

are Virasoro invariants? We have a vague understanding (hep-th/9705128) of what the map from $A$ to $\mid \psi ⟩$ has to do with string field theory. Can something similar be said about the Pohlmeyer map from $\mid \psi ⟩$ to $A$?

What is the meaning on the string theory side of a Gaussian ensemble in $u\left(N\right)$, as used in Random Matrix Theory?

I have a very speculative speculation concerning this last question: We know that the IKKT action is just BFSS at finite temperature. But the BFSS canonical ensemble

(4)$\mathrm{exp}\left(\mathrm{const}\mathrm{Tr}{P}^{2}+\mathrm{interaction}\right)$

is just the RMT Gaussian ensemble, up to the interaction terms. It might be interesting to discuss the limit in which the interaction terms become neglibile and see what this means in terms of the Pohlmeyer map from gauge theory to single strings.

Incidentally, without the interaction terms we are left with RMT theory which is known to describe chaotic systems. This seems to harmonize nicely with the fact that also in (11d super-) gravity, if the spacetime point interaction is turned off (near a spacelike singularity) the dynamics becomes that of a chaotic billiard.

Somehow it seems that the Pohlmeyer map relates all these matrix theory questions to single strings. How can that be? Can one interpret the KM algebra of 11d supergravity as a current algebra on a worldsheet?

Sorry, this is getting a little too speculative. :-) But it highlights another maybe intersting question:

What is the generalization of the Pohlmeyer invariant to non-trivial backgrounds?

I have mentioned somewhere that whenever we have a free field realization of the worldsheet theory (like on some pp-wave backgrounds) the DDF construction goes through essentially unmodified and hence the Pohlmeyer invariants should be quantizable in such a context, too.

But what if the background is such that the DDF invariants are no longer constructible, or rather, if their respective generalization ceases to have the correct properties needed to relate them to the Pohlmeyer invariants?

In summary: While it is not clear (to me at least) that the Pohlmeyer invariants can help to find (if it really exists) an alternative quantization of the single string, consistent but inequivalent to the standard one, can we still learn something about standard string theory from them?

Posted at April 14, 2004 11:09 PM UTC

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