The String Coffee Table

Okay, I’m a bit embarassed but I’ve finally gotten around to contributing something to the String Coffee Table. I’m embarassed because Jacques first contacted me in mid-June and here I am making my first post in mid-September.

I just got back from Columbus, where I spent a few days at a great workshop titled Quantum Theory of Black Holes. I decided at the last minute to trek down from Ann Arbor, making the three and a half hour drive with a couple of intrepid grad students. This was spurred, in no small part, by a combination of excellent speakers and no registration fees. When we showed up we were surprised to find that, besides the people who were actually speaking, there were very few participants. I can’t begin to explain how happy this made me. Having such a small group made every talk very interactive. Many of the talks went well over the allotted time because of this, but getting to hear everyone’s opinion on each topic was worth it. Furthermore, having a small group meant that there was no waiting in line to talk and ask questions during the break.

Once I’ve had a chance to look over my notes from the talk I’ll try and put together a more detailed post on some of the physics. One of the main themes that came out of the talks had to do with whether or not two-dimensional black holes exist in String Theory. The short version of this is that, in situations where we have a complete Matrix-Model description of what is going on, people seem to be finding evidence that there aren’t black holes (at least in the singlet sector). Clearly, this involves quite a few caveats, but it was interesting to see several people approaching the same conclusion. Unfortunately, I missed Herman Verlinde’s talk this morning where I think he was going to announce similar conclusions. I’ll try to refine this statement and add some comments on the physics later.

One of the reasons that there was such a small group at the workshop was that there were quite a few remote participants. The talks were all held at Ohio State’s Access Grid site. This means, in theory, that any other group with an Access Grid site can participate fully. There was a giant projection screen in the room with the speaker’s slides, as well as live video feeds of all the remote locations. One of the talks on the first day was Sumit Das, broadcasting from Kentucky. Unfortunately, there seemed to be quite a few bugs and we missed large portions of the talk. There was one awkward moment when the “live” feed of Sumit began to loop over and over. One by one, we all came to grips with the horrifying conclusion that Sumit was, in fact, an android that had crashed, and was now stuck in an interminable loop repeating the same point from his slides. Luckily, it was just a problem with the connection. In all seriousness, it’s a great concept with a few bugs that need ironing out. I’m told that the host site should have a microphone at each seat so that the remote sites can hear all of the questions and comments from the audience. The OSU site did not have all of its microphones, so I’m sure that the remote audiences missed out on some of the great back-and-forth between the speakers and the audience.

Readers of this blog will recall that I was involved in some thinking about Pohlmeyer invariants.

It all started with Th. Thiemann’s paper ‘The LQG-String’ in which the Pohlmeyer invariants were used in an attempt to find an alternative quantization of the relativistic string (which had some relevance to approaches of higher dimensional quantum gravity). This requires to find a consistent quantization of their algebra.

I thought that such a consistent quantization should be obtainable by expressing them in terms of the standard DDF invariants, which have a well known standard quantization and form a closed algebra.

When asking (I, II) K. Pohlmeyer, K.-H. Rehren and D. Bahns if this has been considered I received negative answers of various kinds, most of them saying that it cannot work in principle. At the same time Luboš Motl reported a conversation with Edward Witten about the suspicion that maybe the Pohlmeyer invariants contain no real worldsheet information at all. Related doubts were voiced by others.

Due to this I finally decided to write up how I thought the Pohlmeyer invariants could be re-expressed in terms of DDF invariants, and I had intensive discussion about these ideas with K.-H. Rehren at the DPG meeting in Ulm. The insight finally published as hep-th/0403260 was among the reasons for H. Nicolai to invite me to the AEI and for R. Flume to invite me to speak in the theory seminar in Bonn.

This may explain my surprise when a couple of days ago I received an email by A. P. Isaev, telling me that he had published precisely this result already in the early 1980s in

A. P. Isaev: Integrals of the motion of a closed relativistic string, Theor. Math. Phys. 54 (1983) 134 (original Russian version published in 1981)

and

V. I. Borodulin & A. P. Isaev: On the infinite set of integrals of motion for a closed supersymmetric string, Phys. Lett. B 117 (1982) 69.

Of course I apologize to A. P. Isaev for not having been aware of this old work - I was apparently not the only one. I will include a respective reference in my paper on the arXiv.

I have been a little silent lately, not because there were no interesting things to talk about, but, on the contrary, because John Baez and I decided to write all these interesting things about non-abelian surface holonomy and its description in terms of loop space formalism and 2-group theory up in a paper, the process of which is now taking up most of my time.

But all these considerations are rather general mathematical in the sense of not being directly concerned with any specific physical interpretation. I must not loose myself in general abstract nonsense, much fun as it may be, but try to understand how it relates to physics, and to strings in non-abelian 2-form backgrounds in particular.

Concerning this issue I had already long, detailed and very helpful disucssion mostly with Jens Fjelstand and with Amitabha Lahiri, but some basic issues remain a little mysterious, such as

What precisely are the states of string for non-abelian 2-form backgrounds?

and

How can the formal requirement for preferred reference points (i.e. the requirement to work in based loop space and the presence of source vertices in 2-group theory) be understood in terms of the worldsheet theory of strings?

I had made some suggestions concerning these points in hep-th/0407122, but some things became clear to me only upon working out the relation between loop space formalism and 2-group theory in the above mentioned notes. Now a referee (rightly I must admit) complains that I do not explain well in hep-th/0407122 why I restrict the discussion to based loops. This precisely addresses the above two points.

While a complete understanding of the worldsheet theory of strings in non-abelian 2-form backgrounds might still be elusive, I believe that a lot of very interesting and consistency-checked things can be said and should be taken very seriously. In order to do so and to comply with the referee’s suggestion, I therefore want to add a section to hep-th/0407122 which more thoroughly discusses the conceptual isues that one has to face when dealing with worldsheets in non-abelian 2-form backgrounds, so as to make the necessity of the construction in that paper more transparant.

A first version of such a new section is now included as section 2, pp. 5 of hep-th/0407122-replacement, which is reproduced below. I would very much enjoy discussing the issues addressed there in more detail. In particular, if we can come up with more details I would enjoy collaborating on some publication.