The String Coffee Table

The recent discussion about attempts to quantize the (super-)string by studying its algebra of classical invariants (i.e. of phase space functions that Poisson-commute with the (super-)Virasoro constraints) has shown that perhaps one should not forget about the special role played by the so-called DDF operators in the worldsheet (S)CFT.

These operators, named after DelGuidice, DiVecchia and Fubini, have the special property that they commute with the entire (super-)Virasoro algebra. Applying them to any physical state hence yields another physical state. Indeed, they are even complete in that they generate the spectrum generating algebra, i.e. all physical states of the string can be obtained by acting with DDF operators on a massless (or tachyonic) state.

The DDF operators can furthermore be chosen so as to satisfy the usual (super-)oscillator algebra and hence they neatly encode all the information about the string except for that contained in the worldsheet ground states themselves. It is straightforward to go from the quantum DDF operators to the corresponding classical invariant observables on the string’s phase space, and these DDF invariants should hence be an alternative to the Pohlmeyer charges. (But I have to admit that so far all I know about the theory of Pohlmeyer charges is what is summarized in Thiemann’s paper.)

Before having a closer look at the relation between Pohlmeyer charges and DDF invariants I would like to review the construction of the most general DDF operators in (S)CFT here.

In the standard textbook literature one can find

- in Green, Schwarz & Witten (using non-CFT language) the construction of

- transversal bosonic (section 2.3.2)

- transversal supersymmetric (section 4.3.2)

- longitudinal bosonic (p. 11)

and in Polchinski (using CFT language) the construction of

- transversal bosonic (eq. (8.2.29))

DDF states.

Here I give a summary and derivation (in CFT language) of all

- transversal and longitudinal, bosonic and fermionic

DDF states (for a free supersymmetric worldsheet theory).

This is also summarized in the following notes

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

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.

Last year there was a symposium called Strings meet Loops at the AEI in Potsdam at which researchers in the fields of String Theory and Loop Quantum Gravity were supposed to learn about each other’s approaches. In his introductory remarks H. Nicolai (being a string theorist) urged the LQG theorists to try to better understand how their quantization approach compares to known results.

Since the worldsheet theory of the (super)string is nothing but (super)gravity in 1+1 dimensions coupled to other fields it would be an ideal laboratory to compare the results of LQG in this setting to the usual lore, which in particular features the central extension of the Virasoro algebra as well as consistency conditions on the number of target space dimensions.

How does this model fit into the framework of canonical and loop quantum gravity?

Nicolai asked.

A search on the arXive showed that so far only one paper had appeared which did address aspects of this simple and yet somewhat decisive question:

Artem Starodubtsev, String theory in a vertex operator representation: a simple model for testing loop quantum gravity.

Starodubtsev concluded:

The suggested [LGQ-like] version of the Hamiltonian constraint leaves us with a theory which is considerably different from ordinary string theory. There are several indications that string theory in its usual form can probably not be recovered from the model obtained. […] the first version of Hamiltonian constraint is anomaly-free and the same is true of the diffeomorphism constraint.

When, after the symposium, I mentioned this reference to A. Ashtekar, a leading figure in LQG, he told me that he meanwhile was aware of this result and planning to analyze the problem in more detail.

Apparently this has borne fruit by now, since yesterday a paper by Th. Thiemann appeared on the arXive

Th. Thiemann, The LQG-String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space

which gives a detailed analysis of an LQG inspired canonical quantization of the 1+1 dimensional Nambu-Goto action for flat target space. The approach is a little different from that by Starodubtsev, but the results are similar in their unorthodoxy: Thiemann finds

- no sign of a critical dimension

- no ghost states

- no anomaly, no central charge

- no tachyon (and, indeed, not the rest of the usual string spectrum).

The claim is that all this is possible due to a quantization ambiguity that has not been noticed or not been investigated before: Instead of using the usual Fock/CFT representation and imposing the constraints as operator equations, Thiemann uses families of abstract representations of the operator algebra obtained by the GNS construction and solves the quantum constraints by a method called group averaging, or its more sophisticated cousin, the so-called Direct Integral Method.

Since these are the same methods used in LQG for quantizing the gravitational field in 3+1 dimensions it is somewhat interesting to see how vastly different the results obtained this way are from the standard lore. One might hence take this as a sign that the LQG approach to quantization is odd. But in some circles this is interpreted in just the opposite way, dreaming of the possibility that the new quantization method might improve on the standard approach to quantization in string theory. Indeed Thiemann himself speculates in his conclusions that his quantization prescription might

- solve the cosmological constant problem

- clarify tachyon condensation [?]

- solve the vaccum degeneracy puzzle

- help finding a working phenomenological model

- help proving perturbative finiteness beyond two loops .

To my mind these are surprisingly bold speculations.

I would much rather like to understand conceptually the nature of the apparent quantization ambiguity (if it really is one) that is the basis for all this. Do we really have this much freedom in quantizing the NG action? Why then do several different quantization schemes (BRST, path integral, lightcone quantization) all yield the standard result which strongly disagrees with the one obtained by Thiemann? What is the crucial assumption in Thiemann’s quantization that makes it different from the ordinary one?

I believe that these questions are what originally motivated H. Nicolai to initiate this investigation and their answer should teach us something.

In the remainder of this entry I shall try to look at some of the technical details of Thiemann’s paper, trying to understand what exactly it is that is going on.

We all know from Edward Nelson that

First quantization is a mystery.

But it should be possible to understand how precisely it is mysterious and how it is not.

[Note added later on:]

After an intensive discussion and some false attempts to explain what is going on inThomas Thiemann’s paper, he finally chimed in himself and we could clarify the issue at the technical level. The crucial point is the following:

Thomas Thiemann does not perform a canonical quantization of the Virasoro constraints if we want to understand under canonical quantization that a theory with classical first-class constraints $C_I$ is quantized by demanding

What Thomas Thiemann instead does (by his own account) is the following:

1) Find a representation $\hat U_\varphi$ of the classical symmetry group elements $\varphi$ on some Hilbert space. (Here the $\hat U_\varphi$ need not have anything to do with the quantized $\hat C_I$, and in the case of the ‘LQG-string they don’t have anything to do with them.)

2) Demand that physical states are invariant under the action of the $\hat U_\varphi$.

It is clear that this method explicitly translates the classical symmetry group to the ‘quantum’ theory and hence cannot, by its very construction, ever find any anomalies and related quantum effects.

An interesting aspect of this is that exactly the same method is used with respect to the spatial diffeomorphism constraints in Loop Quantum Gravity (while the Hamiltonian constraint is quantized more in the usual way). It must therefore be emphasized that LQG is not canonical quantization in the sense that the classical first-class constraints are not promoted to hold as expectation value equations in the quantum theory.

For me, this is the crucial insight of this discussion, and it shows that Hermann Nicolai’s question did address precisely the right problem. In the toy example laboratory of the Nambu-Goto string it is much easier for non-experts (like me) to follow the details and implications of what is being done, than in full fledged LQG. And it turns out, to my surprise, that what is being done is a speculative proposal for an alternative to standard quantum theory. This is not only my interpretation, but Thomas Thiemann himself says that the procedure, sketched above, for dealing with the constraints, should be compared to experiment to see if nature favors it over standard Dirac/Gupta-Bleuler quantization.

I am open-minded and can accept this in principle, but this has not been obvious to me at all, before. It means that, in the strinct sense of the word ‘canonical’, LQG is not canonical at all but rather similar in spirit to other proposed modifications of quantum theory, like for instance those proposed to explain away the black hole information loss problem by modifying Schroedinger’s equation.

I have tried to discuss some of these insights here.

At first it may look like a trivial problem whose solution should have been known for ages, being used all over the place for applications in mathematics, theoretical physics and engineering. But surprisingly, it has apparently not fully been understood yet. I am talking about the adaption of the full machinery of differential geometry to discrete spaces. In particular, I am being told that the metric aspects of such a theory still puzzle many researchers, the Hodge star operator, for instance, being notoriously hard to come by on general discrete spaces.

A while ago Eric Forgy has convinced me that it may be worthwhile to think about these issues, and after a very intesive collaboration we came up with what looks like an interesting approach to discrete metric differential geometry to us. Now we are trying to communicate our results with other researchers in the field.

Since most of this exchange is currently going on by e-mail and since this puts severe restrictions on the amount of true interaction and maybe cross-fertilization as soon as more than two people are involved, I was wondering if maybe we’d need some sort of discussion forum. This blog entry is supposed to be the entry point to such a discussion. To get started, I list some relevant liks to the current literature below. The list won’t be comprehensive at all at the moment, but I am planning to update it as we go along.

Strings on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ are a toy model for the more interesting (and more difficult) $\mathrm{AdS}_5 \times \mathrm{S}^5$ scenario. Here I’ll review some aspects of the analysis of type II strings in this background. The goal is to discuss a calculation of the superstring’s spectrum by first going to the (pp-wave) Penrose limit and then making a perturbative calculation in curvature corrections. Since strings on $\mathrm{AdS}_3 \times \mathrm{S}^3 \times \mathrm{T}^4$ are exactly solvable this is nothing but a warmup for more interesting cases where such a perturbative calculation is inevitable.

In this first part I define the setup by writing down the metric, giving a set of invariant vector fields and defining the Penrose limit in terms of these vector fields, which involves magnifying the vicinity of a lightlike geodesic moving around the equator of $\mathrm{S}^3$.

Then I discuss the exact light-cone spectrum for superstrings with respect to this lightlike direction extending a result given in

A. Parnachev, D. Sahakyan Penrose limit and string quantization in $\mathrm{AdS}_3 \times \mathrm{S}^3$.

I want to talk a little about doing the exercise of canonically analyzing the action of the (super) D-string.

My motivation is to try to find the spacetime interpretation of the classical SCFT that has been discussed in section (2.8) of this entry.

I don’t have much time since the computer room at University of Barcelona that I am currently using will close soon. Therefore I’ll begin with just a few observations concerning the bosonic D-string and hopefully say more in a followup.