The String Coffee Table

It’s Sunday morning, I have had a little stroll through Ulm (‘in Ulm und um Ulm herum’ ;-). Einstein everywhere: The theater features a play about him, there are Einstein exhibitions, the bookstores have books about Einstein and relativity in their showcases, his face is there when you enter hotel lobbies.

C. N. Yang will give an ‘Einstein Lecture’ today, at Ulm University, but not before 19:30, so I have still some time to kill.

Of course I am working on my talk, which is due on Tuesday. Here is an outline of what I am going to say in that talk:

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SUPERSTRINGS FROM LOOP SPACE PERSPECTIVE

$\,\,\,\,\,\,\,\,$(DEFORMATION OF SCFTs AND COVARIANT HAMILTONIANS)

Investigations motivated by the loop space perspective on Type II superstring dynamics lead to insights concerning general deformations of two dimensional superconformal field theories as well as to a method for covariant perturbative calculations of superstring spectra on general backgrounds with applications to strings in RR backgrounds and test of AdS/CFT.

Content:

1) What is well known

2) What is not so well known

3) What is new

4) What is not fully understood yet

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1) WHAT IS WELL KNOWN

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- - - - consider an $N=2$, $D=1$ susy QM with supercharges $\mathbf{Q}_1$, $\mathbf{Q}_2$ and algebra$\{\mathbf{Q}_i,\mathbf{Q}_j\} = 2\delta_{ij} \mathbf{H}$

What deformation of the supercharges will preserve this algebra?

Witten (1982): Use polar form $\mathbf{d} = (-i Q_1 + Q_2)$.

$\Rightarrow$ algebra isomorphism must preserve nilpotency of $\mathbf{d}$ as well as adjointness relations:

- - - - scalar $W$ induces a potential (-> Morse theory)

- - - - 2-form W induces torsion (Froehlich et al.)

- - - - $\Rightarrow$ deformation operator $W$ induces background fields!

- - - - Generalization to global $N=1$, $D=2$ susy algebra: use a Killing vector $k$ and set

- - - - now $\{\mathbf{Q},\mathbf{Q}\} = \mathbf{H} \pm i\mathcal{L}_k$

- - - - $\mathcal{L}_k = \{\mathbf{d},k⇁\}$ is generator of translations along $k$!

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2) WHAT IS NOT SO WELL KNOWN

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Can this be generalized to local $N=1$, $D=2$ SUSY?

Yes! As above, but preserve nilpotency up to reparameterizations $\mathcal{L}_k$ now!

$\Rightarrow [\mathcal{L}_k, W] = 0$

- - - - Super Virasoro algebra of the superstring is of the above form!:

The Killing vector

is the reparameterization Killing vector on loop space.

$\Rightarrow$ Superstring is Dirac-Kähler on loop space

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3) WHAT IS NEW

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a) superstring backgrounds from deformations (hep-th/0401175)

- - - - deformations of the above form give indeed all superstring backgrounds!

- - - - hermitean part of $W$ is background vertex operator in (-1,-1) picture (pre-image under T_F, \bar T_F)

- - - - anti-hermitean part gives background gauge transformations

- - - - reproduces ‘canonical deformations’ (hep-th/9902194) when truncated at first order - - - - well known example for gauge transformation: T-duality, (hep-th/9511061)

- - - - same possible for: S-duality (hep-th/0401175)

b) covariant Hamiltonians (hep-th/0311064)

- - - - deformation technique allows concise algebraic expressions for SCFT objects in arbitrary backgrounds

- - - - covariant Schrödinger equation for superstring:

- - - - Generalizes to ALL of the above deformations by simply setting

with $[\mathcal{L}_v, W ] = 0$

- - - - this is so far just a rewriting of the super Virasoro constraints but yields

$\Rightarrow$ covariant Hamiltonian $H_v$ for ALL (stationary) backgrounds which computes string spectrum:

- - - - application to perturbative calculations of string spectra

e.g. $\mathrm{AdS}_3 \times S^3 \rightarrow$pp-wave (hep-th/0311064)

- - - - some computational aspects are simpler, some are more involved than in LCQ - but more generally applicable than LCQ

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4) WHAT IS NOT FULLY UNDRESTOOD YET

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- - - - how to deal with normal ordering beyond first order??

$\Rightarrow$ background equations of motion beyond first order??

- - - - RR backgrounds?? (cf. hep-th/0205219)

$\Rightarrow$ this would allow above perturbation technique in particular for AdS/CFT on $\mathrm{AdS}_5 \times \mathrm{S}^5$

- - - - finally: deformation technique makes conenction to NCG manifest, ‘spectral string’ (www-stud.uni-essen.de/~sb0264/p4a.pdf) (cf. Chamseddine, Froehlich)

This has been a very intensive day.

When I arrived (late) in A. Ahstekar’s talk this morning I had only five hours of sleep behind me, but I made it to the first coffee break without major casualties and resupplied myself with caffeine.

Ashtekar gave a general introductory lecture on LQG. Afterwards Peres talked about ‘Quantum Information and Relativity Theory’, being concerned with problems such as a quantum measurement in one frame may come before the measured event in another frame.

After the coffee break Clifford Will gave an extremely enjoyable talk on experimental tests of gravity. As I have said here, I learned that LISA will not be able to see primordial gravitational wave backgrounds due to the noise made by binary star systems in our galaxy.

After lunch the parallel sessions started. I had a problem, because I wanted to listen to the NCG stuff which was parallel to a session in which Folkert Müller-Hoissen gave a talk. Now, Eric and I have spent a lot of time with extending and generalizing the work by Dimakis and Müller-Hoissen and I had never met these authors before, so I decided to ditch the NCG session and sit in on ‘Symmetries, Integrability and Quantization’. It was sort of interesting, though I later learned that this way I missed a talk about NCG on Lorentzian spacetimes, which I really regret to have missed. I’ll need to talk to those people in private tomorrow.

Anyway, my hope was to get hold of Müller-Hoissen after the talks and get on his nerves by talking about discrete differential geometry. But unfortunately he was inolved in a discussion with somebody else about something else, which went on and on and on…

Finally I decided not to wait any longer and ran to the lecture auditorium to catch at least the second half of C. Fleischhack’s talk on ‘Progress and Pitfallls of LQG’. This was a very technical talk with lots of formulas with a huge amount of indices and symbol decorations. I have made a photograph of the point where he puts on the transparancy which says that now the spatial diffeomorphism constraints are solved. I believe that the field of LQG would maybe profit from deemphasizing technical details at this point and instead emphasizing the big crucial point: The diffeomorphism constraints are ‘solved’ without imposing Dirac constraint quantization.

I saw that Thomas Thiemann was in the audience, A. Ashtekar was, Hermann Nicolai was, and decided that it would be nice to continue our Coffee Table discussion about this point, which some people feel is a little problematic.

So as the talk was over I asked the lecturer about this point. He answered that this method is simpler than Dirac quantization and also has the advantge that also ‘large’ diffeomorphisms can be included, i.e. those that cannot continuously be connected to the identity (such as coordinate reflections).

When I began to argue that the method may be simpler but is not what Dirac tells us to do (for good reasons, like path integral and BRST formalism), A. Ashtekar approached me. He was very nice and helpful, as usual, and we went outside to further discuss things.

He checked if I am the one who had pointed him to the Coffee Table discussion and told me that he didn’t answer my email partly because he found some statements of the Coffee Table discussion overly offending. I think he is right about that and highly appreciate that he still very patiently talked and listened to me. Many thanks to Abhay Ashtekar, indeed.

First he said that the way LQG deals with the gauge constraints is not different from what one does in gauge theory. I replied that that has to do with the fact that the anomalies of the standard model happen to cancel, while it is not clear that those of canonical gravity would (without the ghost sector). I think he agreed.

I suggested that maybe LQG should then perhaps try to handle the ghost-extended Einstein-Hilbert action instead of the pure EH action. At least for 1+1 dimensional gravity this does indeed remove the anomaly, as is well known. I got the impression that A. Ashtekar found this idea is maybe worth considering (but I am not sure).

He told me that what he found problematic with the ‘LQG-string’ is that the method does not in any way seem to involve the fields on the background spacetime. I found this remark interesting, because it resonates with my own feelings concerning this point, which I have expressed here. A. Ashtekar hinted at some alternative approaches by himself and somebody else which are apparently under investigation, but I feel that I should not report on that here in public.

After this very illuminating discussion I was asked by somebody if I am working on LQG, because he had seen me pipe up in Fleischhack’s talk. After I had answered that to the negative we exchagned personal and scientific identities, and I was delighted to meet in Thorsten Prüstel somebody working on - guess what - discrete field theory on Lorentzian graphs.

We immediately had lots of things to talk about. I showed him Eric Forgy’s and mine pre-pre-print and he was very interested and invited me to give a talk about that at University of Hamburg. He himself is working on an interesting approach to get gravity from the nonunitary part of an extended gauge group on a graph field theory, roughly. I hope that when I am in Hamburg I will get the chance to learn more about that.

We already had to hurry to get to the ‘Welcome Party’. There we happened to sit next to Prof. Kastrup from Aachen. Kastrup was the thesis’ advisor of two of the leading figures in current LQG, namely Thomas Thiemann and Martin Bojowald. We learned how Thomas Thiemann as a student originally wanted to work on string theory and was later convinced to look into LQG.

I found it very interesting that Kastrup agreed with my assessment that LQG is not ‘canonical’ in the usual sense, because it does not represent the classical canonical coordinates and momenta as operators on a Hilbert space.

Indeed, in the afternoon I had heard the very interesting talk by Kastrup about quantization of integrable systems in angle/action variables. This is an interesting and subtle issue of canonical quantization, which can already be studied for the ordinary harmonic oscialltor.

The point is that by the naive correspondence rule one would think that, since the angle $\omega$ and the action $S$ are a pair of canonically conjugate coordinates and momenta (like the ordinary $q$ and $p$ are, too) there should be self-adjoint operators $\hat \omega$ and $\hat S$ which satisfy $[\hat \omega,\hat S] = i$.

But it is easy to convince oneself that this cannot work, which has to do with the fact that the angle is defined only modulo $2\pi$, or, equivalently, that $\omega$ and $S$ do not provide global coordinates on phase space, because the origin has the usual coordinate singularity of polar coordinates in the plane.

Kastrup showed how with taking much care one can instead construct two other operators $K_+$ and $K_-$ such that together with $K_0 = \hat S$ they give the Lie algebra of $\mathrm{SO}(1,2)$ and that from these the standard $\hat q$ and $\hat p$ can be reobtained. (This can be understood heuristically by thinking of the 2d phase space of the oscillator with the origin removed as a (‘light’-)cone.)

He concluded by saying that, while being equivalent to the quantization with $q$ and $p$, this could have experimental consequences in quantum optics. (I asked him about how this can be true, but unfortunately failed to understand his answer.)

Anyway, this is a nice example for how subtle ordinary canonical quantization itself can already be.

The talks today didn’t interest me much (things like ‘Einstein and art’) and I spent the time reading and answering my mail as well as preparing my own talk.

But I did have lots of very valuable conversations.

At lunch I met Hermann Nicolai and we had a long discussion about Pohlmeyer invariants, DDF invariants, LQG, the ‘LQG-string’, diffeomorphism anomalies in string theory, string field theory, Matrix Models and prejudices in quantum gravity.

To begin with, I was kind of surprised to learn that H. Nicolai, together with K. Peeters, is currently thinking about Pohlmeyer invariants himself. We discussed the known results regarding their quantization and I mentioned that I think that there is a solution to the apparent problems. H. Nicolai was interested and invited me to visit the AEI in April to talk about these ideas.

Then of course we discussed the ‘LQG-string’ and what can be learned from it about LQG itself. H. Nicolai said that he had hoped that the LQG people would find the anomaly for the string, which, as he said, he would have considered a breakthrough for the whole LQG field. But what has now actually been done, he said, reminded him more of certain artificial constructions in axiomatic field theory, which are also mathematically well defined but physically empty.

I asked him about what this now means for full LQG and he said that, similarly, he would expect that there should be an anomaly and that he finds the constructions done in LQG problematic.

I tried to understand how the same issue could be understood from within string theory, and he basically said that one would have to understand closed string field theory in order to tackle this question. But currently a satisfactory definition of closed string field theory is of course not known.

I said that I am wondering if we cannot learn anything in this regard from BFSS/IKKT Matrix Models. The authors of the IKKT/IIB model at least claimed that the permutation subgroup of the full $\mathrm{U}(N \to \infty)$ gauge group of the model becomes the diffeomorphism group in the limit. Heuristically, we can think of the matrices as describing discrete spacetime points and the permutation subgroup permutes these points, so is related to diffeos in some sense.

Hermann Nicolai didn’t know the details of this claim (me neither :-) and remarked that most permutations would give rather pathologic diffeos in the continuum limit. In any case, my uneducated guess is that the issue of diffeo anomalies and the like is hidden in the $N\to \infty$ limit of the matrix models, which is not well understood at all, as far as I know.

Since the time for the afternoon talk sessions was drawing near we stopped at this point. But as I was about to leave the cafeteria K.-H. Rehren approached me. I very much appreciated this, because from our previous online conversation I had gotten the, apparently wrong, impression that he wasn’t interested in my comments on Pohlmeyer invariants.

I was delighted that we immediately sat down, pulled out pens and paper and began doing algebraic calculations. I think that in a couple of minutes we could clarify issues that would have taken weeks by email, at the previous speed.

But then we really had to hurry, because I was the one supposed to give the next talk!

I am not sure if it is a good or a bad sign, but at this DPG spring conference of the faculties ‘Gravitation and Theory of Relativity’ and ‘Theoretical and Mathematical Foundations of Physics’ my talk is apparently the only one directly concerned with strings! But I couldn’t complain. With H. Nicolai, K.-H. Rehren and F. Müller-Hoissen in the audience I knew I was talking to people who I would have liked to ask about their opinion on my stuff at any rate.

Unfortunately, there wasn’t much time for questions and feedback. Let’s see what tomorrow brings… If nothing else, my talk will probably enter the annals of the DPG as the only one based on chalk and blackboard in the age of PowerPoint. ;-)

After the afternoon sessions I took care to catch one of the speakers on NCG whose talks I had missed the day before. I was lucky to get hold of a mathematical physicist of the name Paschke, who had given a talk on NCG on pseudo-Riemmanina manifolds. He had presented a technique where you slice a globally hyperbolic manifold in compact spatial leaves, perform ordinary NCG a la Connes on these spatial slices and then figure out how to glue the resulting spectral triples together to get a ‘spectral quadrupel’. At first it might seem that this way time is ‘commutative’ while only space is ‘noncommutative’, but the crux is apparently that it turns out that this is not the case and that in some sense also time becomes ‘noncommutative’. But I didn’t see this in any detail.

After having understood how Paschke is proposing to deal with NCG on pseudo-Riemannian spaces I tried to make him tell me what he thinks of Eric’s and mine approach which is supposed to deal, among other things, with pseudo-Riemannian discrete spaces.

Paschke emphasized that he finds it very problematic to break the compactness assumption of Connes’ approach, which technically means that one is (compact) or is not (non-compact) dealing with a $C^*$ algebra which has a unit, because then many of Connes’ theorems won’t hold if there is no unital $C^*$ algebra. That may be true, but I am under the strong impression that one can do interesting physics on non-compact noncommutative algebras nevertheless. But this will be a crucial point to be worked out if I want to communicate with the Connes school of NCG.

I have to run now to get to the debut performance (really, the official dress rehearsal) of Dirk D’ase’s ‘Einstein opera’.

Hello all,

my article in the popular German science magazine “bild der wissenschaft” (April 2004 issue) about a conference “Strings Meet Loops” last year and further aspects which Urs has mentioned is now available also as an extended version in English (but without pictures and illustrations).

In the printed issue there is another article about loop quantum cosmology (not yet translated).

Perhaps you are interested.

Best, Rudy

http://www.arxiv.org/abs/physics/0403112
Title: The Duel: Strings versus Loops
Authors: Ruediger Vaas
Comments: Extended version from: Ruediger Vaas: Das Duell: Strings gegen Schleifen.
Published in: bild der wissenschaft (2004), no. 4, pp. 44-49.
- Translated by Martin Bojowald and Amitabha Sen. - 10 pages, including 1 table and references
Subj-class: Popular Physics
Journal-ref: bild der wissenschaft (2004), no. 4, pp. 44-49

Abstract:
Physicists in search of the foundation of the world: how tiny objects
can create matter, energy and even space and time - and possibly
countless other universes. – This article is meant as an introduction
into quantum geometry (loop quantum gravity) and string theory, written
for the general audience. Partly, this article is also a conference
report and review, based on the “Strings Meet Loops” conference at the
Max-Planck-Institute for Gravitational Physics (Albert Einstein Institute),
Golm/Germany, in October 2003. – Keywords: quantum geometry, loop quantum
gravity, string theory, string cosmology, spin networks, spin foams,
anthropic principle, theory of everything, Abhay Ashtekar, Martin Bojowald,
Michael Douglas, Jerzy Lewandowski, Hermann Nicolai, Robert Oeckl, Fernando
Quevedo, Carlo Rovelli, Amitabha Sen, Lee Smolin, Leonard Susskind.

The above mentioned semi-popular article on loop quantum cosmology - i.e. Martin Bojowald et al.’s work about avoiding the big bang singularity - is now translated and available online.

Rüdiger Vaas: Der umgestülpte Urknall.
bild der wissenschaft, no. 4 (2004), pp. 50–55.
Translated by Amitabha Sen:
The Inverted Big-Bang
http://arxiv.org/abs/physics/0407071

Best wishes,
Rudy