The String Coffee Table

Yesterday I was contacted by Bert Schroer.

He asked if in the context of my recent hep-th/0403260 I could see any way to get an

intrinsic understanding of ‘string’ or ‘worldsheet’ as a somehow localized object in target space and which concepts would make that visible .

He said that the context of this question is the recent discovery by himself and collaborators, reported in

Jens Mund, Bert Schroer, Jakob Yngvason, String-localized quantum fields from Wigner representations (2004)

of what is called string-localized fields. In a certain way these fields describe semi-infinite ‘strings’ and have the crucial property, that their commutator vanishes iff the respective ‘string rays’ are strictly spacelike seperated. This in a sense generalizes the phenomenon of commuting spacelike fields of point particle theories and apparently also provides representations of the Poincaé group for vanishing mass and infinite spin.

In the above paper it is pointed out that this notion of ‘string-localization’ is quite different from the properties of string fields as they appear in string field theory. There, instead, the commutator vanishes when the center of mass of two strings is spacelike seperated, irrespective of the extension of these strings.

I answered that I didn’t see how the first quantized theory of the Nambu-Goto/Polykaov string could shed any light on these issues, but that the context of the constructions in Bert Schroer’s paper reminded me of tensionless strings, where there are massless states of arbitrary spin in the spectrum.

Today I received an email where Bert Schroer disagrees with these assessments and points out several other aspects of the question. I find the above concept of string-localization and its disagreement with string field theory interesting, but will probably not be able to make many further sensible contributions to these questions. Therefore, with Bert Schroer’s kind permission, I will reproduce his latest email here in the hope that maybe others can make further comments.

From time to time interested laymen long to know how to ‘visualize’ the states of the (super)string which appear as elementary particles. This is certainly due to the fact that many popular accounts of string theory contain vague sentences like

Different oscillation patterns of the string correspond to different elementary particles.

without any further qualification of this statement.

Of course in order to really understand this one has to acquaint oneself with the required formalism. But I think it is a fun exercise in physics pedagogy to try to come up with semi-heuristic mental pictures which provide the layman with more information than the general statement above while avoiding a complete mathematical development of the theory.

Here I want to collect some previous attempts on my part to meet this challenge. I’d be interested in knowing how others would approach this question.

The easier part is to develop a visualization of the NS and NS-NS sectors. This I have first tried here.

When I was asked for a visualization of spin-1/2 particle states of string I had to come up with some explanation for what goes on in the R, NS-R and R-R sectors. The best I could do is this:

I am on my way to the spring conference of the German Physical Society, the

Frühjahrstagung der Deutschen Physikalischen Gesellschaft

(in Ulm) where I am going to give a little talk on the stuff that I have been working on lately. Since everybody can simply announce participant talks at this conference this is not a big deal and I regard it as a warmup for later. This is maybe also the reason why many groups in Germany, notably in string theory, seem to ignore this conference altogether.

On the other hand, H. Nicolai will be there and talk about the cosmological billards that he has been working on together with T. Damour, M. Henneaux and others. As far as I understand they have the mind-boggling claim that by symmetry reducing generic supergravity actions to cosmological models and identifying the symmetry of the resulting mini-superspace (which generically leads to chaotic billard dynamics) one can guess a vast extension of this symmetry group and hence the mini-superspace-like propagation on this group, which is not mini at all anymore but a 1d nonlinear sigma model on this monstrous group, and that this is equivalent to full supergravity with all modes included!

Since this is done for the bososnic part of the action only, I once asked H. Nicolai if we couldn’t simply get the same for full supergravity by simply SUSYing the resulting 1d sigma model. Susy 1d sigma models are extremely well understood. We know that the number of supersymmetries corresponds to the number of complex structures on the target space and the supercharges are essentially the Dolbeault exterior derivatives with respect to these complex structures. Nicolai told me that I am not fully appreciating the complxity of this task, which may be right :-) Still, this sounds promising to my simple mind.

Reducing quantum gravity to a 1 dimensiuonal QM theory of course smells like BFSS Matrix Theory. I think I also asked Nicolai if he sees a connection here, and if I recall correctly the answer was again that things are more difficult than my question seemed to imply. :-)

On the other hand, sometimes simple-minded insights lead to the right ideas. In retrospect I am delighted that I had come across and discussed the form-field potentials on mini-superspace which generically give rise to the billiard walls and the chaotic dynamics discussed by Nicolai and Damour in my diploma thesis on supersymmetric quantum cosmology (see section 5.2).

In fact, the way that I treat supergravity in that thesis is precisely how I am imagining Nicolai et al. could try to susy their OSOE (One dimensional Sigma Model of Everything ;-), namely first symmetry reduce the bosonic theory and then susy the result (instead of symmetry reducing the susy theory as usual). Maybe this is crazy, maybe not…

Ok, who else will bet there? There are many LQG people. A. Ashtekar will give a general talk on LQG for non-specialist. Bojowald of course will talk about what is called ‘Loop Quantum Cosmology’. With a little luck I find an LQGist willing to discuss the ‘LQG-string’ with me.

I would also like to talk to K.-H. Reheren, who has announced a talk on algebraic boundary CFT, about Pohlmeyer invariants, but I am not sure if he considers it worthwhile talking to me… :-/

There will probably (hopefully!) be many more intersting talks and people. If so, I’ll let you know…

P.S. Maybe I should mention that on occasion of the 125th birthday of Albert Einstein the entire conference is devoted to this guy. I am looking forward to hearing Clifford Will ask “Was Einstein right?”.

(Update 03/24/04)

Here are some pictures from Ulm and the conference:

Einstein was omnipresent on his 125th birthday in his native town:

Parts of Ulm University have a very interesting architecture:

Ashtekar talks about the limitations of string theory:

C. Fleischhack discusses the step in LQG the analogue of which for the ‘LQG string’ is considered problematic by some people.

My talk on deformations of superconformal field theories:

I have been asked by students if I would like to talk a little about introductory string theory. Since it is currently semester break, we decided to make an experiment (which is unusual for string theory) and try to do an informal and inofficial seminar.

The background of the people attending the semiar is very inhomogeneous and a basic knowledge of special relativity and quantum mechanics is maybe the greatest common divisor. Therefore we’ll start with elementary stuff and will try to acquaint ourselfs with the deeper mysteries of the universe (such as QFT, YM, GR, CFT, SUSY) as we go along.

If I were in my right mind I’d feel overwhelmed with the task of conducting such a seminar, but maybe at least I can be of help as a guide who has seen the inside of the labyrinth before. Hence I’d like to stress that

I can only show you the door. You’re the one that has to walk through it.

;-)

In this spirit, the very first thing I can and should do is prepare a commented list of introductory literature. Here it is:

Probably everybody has seen the Call for Votes for the creation of a new USENET newsgroup with the proposed name ‘sci.physics.strings’ which was announced here before and which is supposed to be a place for interesting online discussion of string theory for researchers, graduate students and everybody else who is interested.

Votes are simply submitted by email, following the instructions given here. Deadline is 16 Mar 2004.

While we are still discussing Ioannis Giannakis’ proposal for how to get worldsheet SCFTs for RR backgrounds and now that I have worked out how the deformations that I am considering reduce to the canonical deformations known in the literature when truncated at first order and how they relate to the vertex operators of the respective background, so that I am finally prepared to seriously begin to think about RR-backgrounds in NSR formalism - while all this is happening (at least in my life) a possibly interesting testing ground for any such ambitions is being discussed in a recent paper:

M. Billo, M. Frau, I. Pesando, $\mathcal{N} = 1/2$ gauge theory and its instanton moduli space from open strings in R-R background.

The paper studies the 4d physics of Type II compactified on $R^6/(Z_2 \times Z_2)$ with an R-R 5-form which is wrapped around a 3-cycle turned on, such that in 4d one sees a constant R-R 2-form $C_{\mu\nu}$ which (for some reason that escapes me) is known as a ‘graviphoton background’.

The point of the paper is to compute the amplitudes invoving the respective R-R vertex and find the respective effective field theory, which is a non-commutative one with respect to the fermionic degrees of freedom, because one gets something like

for the fermion anticommutator.

Anyway, what caught my attention is that in the introduction it says:

It is a common believe that the RNS formalism is not suited to deal with R-R background; while this is true in general, it is not exactly so when the R-R field strength is constant. In fact, in this case one can represent the background by a R-R vertex operator at zero momentum which in principle can be inserted inside disc correlation function among open string vertices without affecting their dynamics.

So maybe this R-R background, being very simple and still non-trivial, would be a good testing ground for Ioannis Giannakis’ conjecture that construction of SCFTs for R-R backgrounds is possible.

Essentially, what one would need to do to check this is (according to the theory of superconformal canonical deformations) to take the R-R vertex (2.14) of that paper, integrate it at constant worldsheet time over the string and add the result to the (anti-)holomorphic stress tensor:

and do something analogous to the supercurrent. (Here $S$ are spin fields.)

Then study the SCFT generated by $T^\prime$ and $T_F^\prime$.

Furthermore, I would like to check what one gets when adding also the second-order correction and possibly the full exact SCFT deformation associated with this background.

If for some reason this does not work, can maybe the superconformal canonical deformations be adapted to Berkovit’s covariant superstring formalism?

There is an old paper

A. Chamseddine, An Effective Superstring Spectral Action

which I have mentioned before and which keeps haunting me.

This paper is from an era where some people (Chamseddine, Fröhlich and others) have tried to identify the spectral aspect of Connes’ noncommutative geometry - namely that emphasizing the role of the Dirac operator - in string theory. According to Alejandro Rivero this was before the flood released by BFSS Matrix Models, strings in $B$-field backgrounds, etc., where the algebraic aspect of noncommutative geometry - namely the noncommutativity! - is emphasized instead.

I am not sure why the original spectral string activity didn’t survive the great flood. But from reading section 3 of the above paper by Chamseddine I get the impression that maybe a certain idea was missing. That’s what I want to talk about here. That, and how superstring theory in terms of spectral triples relates to DDF operators, classical invariants of string, deformations of SCFTs, discrete differential geometry and all that.

Let me briefly indicate what this section 3 is concerned with:

In section 2 the author had discussed how the superconformal constraints of the Type II string for gravitational and Kalb-Ramond background gives rise to Dirac operators on loop space. He then outlines the role that he imagines these Dirac operators should play. He writes:

Most of the considerations of the last section could be looked at from the non-linear sigma model study and one may ask for the relevance of noncommutative geometry. The point of view we like to advance is that once a spectral triple $(\mathcal{A},\mathcal{H},D)$ is specified it is possible to define a noncommutative space and use the tools of noncommutative geometry. […]

For the example studied in the last section we have $\mathcal{A} = C^\infty(\Omega(M))$, the algebra of continuous functions on the loop space over the manifold $M$. Elements of the algebra are functionals of the form $f[X^\mu(\sigma)]$ where $\sigma$ parameterizes the circle. […]

There is also an advantage in treating this model with the noncommutative geometric tools as this would allow us to consider, in the future, more complicated examples which could only be treated by noncommutative geometric methods. […]

To illustrate, consider the operator $$D = Q_+ + Q_-$$ [the sum of the supercharges of the string] […]. restricting to states which are reparameterization invariant […] it is possible to build the universal space of differential forms. A one-form is given by $$\pi(\rho) = \sum_i f^i [D,g^i] = \sum_i \int d\sigma \left[ f^i\left[X\right] \left(\psi_+^\mu + \psi_-^\mu\right) \frac{\delta g^i}{\delta X^\mu} \right]_{P=0} \,.$$

The idea here is clear: Take the supercharges of the string as Dirac operators, identify the algebra on which these act, construct the corresponding spectral triple and – do something sensible with this machinery that boosts our understanding of string theory!

I want to suggest an approach similar but different to what is sketched at the end of this section 3. I want to show how we can construct an abstract differential calculus with exterior differential $\mathbf{d}$ and forms $\omega$ such that the equations $$(\mathbf{d}\pm \mathbf{d}^\dagger)\omega = 0$$ are equivalent to the full set of worldsheet constraints of Type II strings. I don’t claim that this is a particularly difficult problem and in fact the solution is rather trivial, but I think that something interesting can be learned from it.

(For the start, I will consider the classical superstring only, i.e. work at the level of Poisson brackets.)

The crucial differences of the approach that I am going to discuss here to that of Chamseddine will be the choice of algebra as well as the definition of differential forms.

Namely I will follow the approach to spectral geometry which is sketched here and which starts by defining an abstract exterior calculus over a given algebra $\mathcal{A}$.

For this algebra I do not choose the algebra of reparameterization invariant functions, but the algebra $\mathcal{A}$ generated by integrals (along the string at fixed worlsheet time) of fields of reparameterization weight 1/2.

Now let $\mathbf{d}$ be the 0-mode of the $K$-deformed exterior derivate over loop space, as described here and consider the differential calculus $\Omega(\mathcal{A},\mathbf{d})$ generated from $\mathcal{A}$ and $\mathbf{d}$.

The trick is that this way all exact forms of this differential calculus are automatically reparameterization invariant objects, because they are integrals over unit weight fields. Using this reparameterization invariance it is now easy to see that the equations $$[\mathbf{d},\omega] = 0$$ $$[\mathbf{d}^\dagger,\omega] = 0$$ for $\omega \in \Omega(\mathcal{A},\mathbf{d})$ (where the bracket is the graded Poisson bracket) indeed imply the full superstring constraints $$[G_m,\omega] = 0$$ $$[\bar G_m,\omega] = 0 \,.$$ That they are even equivalent to them follows from noting that every solution to the latter set of equations is indeed $\mathbf{d}$-exact! Namely this is the result of the construction of the classical supersymmetric DDF operators which shows that all physical states of the superstring are obtained from acting with $\mathbf{d}$ repeatedly. Indeed, by looking at the construction principle of the DDF invariants of the classical superstring one sees that they have precisely the form of an exact abstract differential form $$\omega = [\mathbf{d},a_1][\mathbf{d},a_2]\cdots [\mathbf{d},a_p] \,.$$

Phew, I am running out of time. I have an appointment at the cinema tonight which I am about to miss. I’ll have to continue this here tomorrow. Let me just say that my evil plan is to use the above construction to modify the algebra $\mathcal{A}$ and get deformed worldsheet theories this way. I believe that this was originally Chamseddine’s et al.’s plan. Find the spectral triple of superstrings and then deform. Before the flood, at least.

Ok, gotta run.