The String Coffee Table

Yesterday, we had Thomas Thiemann here at DAMTP for a seminar on his quantization of the bosonic string. Although I did not expect too much, having him explain his paper for nearly two hours on the blackboard (and a real coffee table discussion afterwards) I think I understand now what is going on and why he doesn’t get a central charge.

As a first step let me repeat what he is doing stripping off some of the not so essential parts of his story and using symbols that are closer to the ones more familiar for us. Curiously, in the end, you can get along without mentioning the Pohlmeyer charges at all, so this is not where the disagreement with the usual story is rooted.

For simplicity, let us consider a one-dimensional target space with coordinate $X$. Then, the good objects are the currents $j=\partial X$ and the current in the other chiral half. This is what Thomas calls $Y_{\pm }$, where $\pm$ indicates left- or right-movers. But as always, it is sufficient to restrict attention to a single chiral half.

The next step is to pick some complete set of functions $(f_n)$ on the circle such that any function on the circle such that any function can be expressed as linear combination of $f_n$’s. Now we can define $j_n=\int f_{n}j(x)\mathrm{dx}$. The usual choice is to take the indices $n$ to be integers and and $f_n=e^{inx}$. With this choice what I called $j_n$ are actually the $a_n$’s.

Thomas makes a different choice but that shouldn’t affect the physics. He takes the lables to be finite unions of intervals on the circle and the $f_n$ to be characteristic functions of the intervals (that is the functions that are 1 on the interval and 0 outside the interval).

Next we have to work out how the diffeomorphisms of the circle act on the $j_n$’s. We are used to ask this question on the infinitesimal level and use the generators $L_n=-z^{n+1}\partial$ that have the usual simple action on $e^{\mathrm{inx}}$ that is expressed in

Thomas prefers to work with finite diffeomorphisms (rather than infinitesimal ones) but again, this is not essential. Again the action is sufficiently simple, the diffeomorphism just moves the intervals around pointwise.

So far everything was classical. Now we promote the $j_n$ to operators. We can be careful and prefer to quantize $e^{i{j}_n}$ rather than $j_n$ directly because those will be bounded operators. This is similar to quantum mechanics where careful people use Weyl operators $U=e^{ix}$ and $V=e^{ip}$ with commutation relations $UV=e^{i\hslash }VU$ as those are unitary operators while $p$ and $x$ are unbounded. This gives us the “kinematical algebra”.

The next step is the crucial one: One has to chose a Hilbert space for these operators to act on. The standard choice is to take the highest weight representation or Fock space by defining the Hilbert space to be generated by the $a_n$’s with negative $n$ from the vacuum that is annihilated by the $a_n$ with positive $n$.

This step is where Thomas’ construction is different from the usual one. He takes a different Hilbert space. He defines it using the GNS construction (a well established procedure in mathematical physics). This works by selecting a “state” $\omega$ on the algebra (as a reminder, a state is a positive function that assigns to each operator a number, that should be thought of the expectation value of that operator in that state) and defines that as the vacuum on which the Hilbert space is build by acting with all the operators.

In fact, our Fock space can be constructed in that way as well. Just define $\omega (a_n)=p{\delta }_{n,0}$, just the vacuum expectation value of $a_n$. If you want Weyl operators you have to work a bit more (using the CBH formula) but it can be done. Now, it is essential that the state is invariant under diffeomorphism (or the Virasoro algebra) to obtain a nice, well defined theory. And this is where the central charge comes in. Our Fock space is not invariant, it transforms with a phase given by the exponential of $i$ times the central charge. This is the analogue of the calculation Urs did for us in the old thread. You have to be in the critical dimension and include the reparametrization ghosts to make the Fock vacuum invariant.

But Thomas picks a different vacuum. Remember, his $j_n$ were indexed by intervals and he defines $\omega (e^{i{j}_n})$ to be 1 if $n$ as an interval is empty and 0 otherwise. This is obviously invariant under diffeomorphisms that move the intervals around. But it is legitimate and you obtain a different Hilbert space (that is is not separable because the orthonormal system is labelled by intervals rather than integers but this is only a consequence but not essential).

So is he allowed to do this? Why doesn’t this happen in ordinary quantum mechanics? Well, it happens as well (this example is also due to Thomas): There you usually take the Hilbert space $L_2(R)$ on which $e^{iax}$ as a multiplication operator and $e^{ibp}$ by translations by $b$. This is the coordinate representation. There are other choices, for example the momentum representation or the oscillator representation, but they are related by a change of basis (a unitary equivalence in techspeak) so they are not really different.

In fact, there is the Stone-von-Neumann theorem, that tells you that up to unitary equivalence this is your only choice. So you usually don’t think about this. However, there is a technical assumption in this theorem, namely that the representation should be weakly continuous in $a$ and $b$ meaning that any matrix elements of $e^{iax}$ and $e^{ibp}$ are continuous in $a$ and $b$. If you drop that assumption there is another possibility: As your Hilbert space you take what is generated by all $T_s=e^{is}$ for real $s$ and $U$ and $V$ act the same way as before but you take a different scalar product: You define $⟨T_s,T_{s\prime }⟩=\delta (s-s\prime )$. Again, this Hilbert space is not separable, the ON system is labelled by continuous $s$ rather than by integers, so it cannot be unitary equivalent to the standard Hilbert space and in fact the representation is not continuous: The expectation value (a special matrix element) for the translation by $a$ is $\delta (a)$ which is not continuous in $a$.

Obviously, in this Hilbert space it’s also a bad idea to take the derivative with respect to $a$, so $p$, the infinitesimal operator of translations, is not well defined. So usually this pathological Hilbert space realization is ruled out by the assumption of weak continuity but Thomas has promised to give me a reference to a paper that discusses the relevance of this pathological Hilbert space in some condensed matter system.

Going back to the string case, we can see that Thomas’ vacuum (and thus Hilbert space) is similar in spirit to this pathological Hilbert space in this quantum mechanical example. Again, it is not weakly continuous and the scalar product that is induced by his state has the same delta function characteristic. I understand he agrees that if one requires continuity one would probably be left only with the usual Fock space representation that gives rise to the critical dimension.

He is not saying that the rest of the world is doing something wrong but only that there is another, pathological possibility to quantize the string if one uses weak enough rules for the game called quantization.

There are two straight forward calculations that one might do if interested: The first is to work out what $\omega$ really is in the Fock space state and check that the diffeomorphism group is really only represented projectively (that is with the phase from the central charge) thus establishing that as one would expect the usually construction can also be rewritten in this GNS formalism. This should be an easy calculation that is basically the exponentiated version of Urs’ calculation and I have no doubts that it will work out in the end.

The other calculation is in fact a bit more challenging: The algebra of the $a_n$’s is obviously an infinite copy of Heisenberg algebras (the algebra of $p$’s and $x$’s). One could try to use the pathological quantum mechanical Hilbert space for each oscillator and check what one would get. My bet would be that it yields something closely related to Thomas’ Hilbert space representation.

Finally note that the Pohlmeyer charges didn’t play a role: All really needed was the trivial observation that Thomas’ state was invariant under diffeomorphisms and from that on we could use the $j_n$ fields that by themselves are not diffeomorphism invariant.

Remains one question: Having now established that this construction is not really in conflict with what we usually do in string theory (in fact adding a slight additional technical assumption probably rules out the pathological state and brings us back to our beloved Fock space construction) what does it teach us about loop quantum gravity. There similar tricks are played: From what I know, first the kinematical algebra is constructed (basically the parallel transporters along edges and their momenta, the “electric” fields, please correct me if I’m wrong) and then a diffeomorphism invariant state is constructed on which the Hilbert space is build. This state is the Ashtekar-Lewandowsky measure (or Abhay-Jurek measure for friends) that is also very singular similar to Thomas’ string vacuum. I vaguely remember somebody making the statement that this is the only choice there.

So maybe in the end, there is nevertheless some fruitful interplay between strings and LQG: It seems that in the critical dimension and including ghosts the usually Fock vacuum is in fact another, much better behaved, diffeomorphism invariant state, at least in 1+1 dimensions!

A while ago I began to think about how deformed worldsheet CFTs could be related to deformed BRST operators obtained from classical solutions of OSFT. I knew that I was in the dark probably trying to re-invent the wheel, being ignorant of lots of results (see the discussion on s.p.s.) - but one has to start somewhere.

Now I am glad that I have finally found a recent paper where pretty much precisely the question which I was concerned with is studied. It is

[Update 21 May 2004: The general point has been made already in 1990 by Ashoke Sen in

Ashoke Sen: On the background independence of string field theory (1990).

There in the abstract it says:

Given a solution ${\Psi }_{\mathrm{cl}}$ of the classical equations of motion in either closed or open string field theory formulated around a given conformal field theory background, we can construct a new operator ${\hat{Q}}_B$ [$={\hat{Q}}_{B\mathrm{original}}+[{\Psi }_{\mathrm{cl}}\star ,\cdot ]$ (my remark)] in the corresponding two dimensional field theory such that $({\hat{Q}}_B{)}^2=0$. It is shown that in the limit when the background field ${\Psi }_{\mathrm{cl}}$ is weak, ${\hat{Q}}_B$ can be identified to the BRST charge of a new local conformal field theory

]

J. Klusoň: Exact Solutions in SFT and Marginal Deformation in BCFT (2003)

In the introduction it says (p. 2):

Our goal is to show that when we expand [the] string field around [a] classical solution and insert it into the original SFT action $S$ which is defined on [a given] BCFT, we obtain after suitable redefinition of the fluctuation modes the SFT action $S^{\prime }$ defined on ${\mathrm{BCFT}}^{\prime \prime }$ that is related to the original BCFT by inserting [a] marginal deformation on the boundary of the worldsheet. […] To say differently, we will show that two SFT action $S$, $S^{\prime }$ written using two different BCFT, ${\mathrm{BCFT}}^{\prime \prime }$ which are related by marginal deformation, are in fact two SFT actions expanded around different classical solutions.

That’s the type of result what I was thinking about. Apparently there are old related results in

A. Sen & B. Zwiebach: A proof of local background independence of classical closed string field theory (1993)

[Update 20 May 2004: The file on the arXiv does not seem to properly compile. Here is a working pdf version. Thanks to Yuji Tachikawa!]

The main point of my previous ponderings was the, maybe not very deep but in any case maybe interesting, speculation that to a given classical solution ${\Phi }_0$ of OSFT the deformed BRST operator

can in fact be identified as the BRST operator of a new worldsheet CFT, corresponding to the background described by ${\Phi }_0$.

From the responses that I received I got the impression that to some people this seems maybe obvious or even trivial. But on the other hand it is hard to find literature on any specific details on how this works in given examples.

The only work done in this direction which I knew of was the one by Ioannis Giannakis, especially

Ioannis Giannakis, Strings in Nontrivial Gravitino and Ramond-Ramond Backgrounds (2002)

which used such deformations of BRST operators for closed superstrings to guess the deformations which should follow, as I begin to understand now, from superstring SFT at classical solutions describing Ramond-sector backgrounds. (The subtleties related to worldsheet supersymmetry in Ramond backgrounds have been discussed here.)

It would be nice if things like that could really be derived from SFT, and in particular I would like to see if the deformations that I discuss in hep-th/0401175 can be derived from SSFT - but that’s still a long way to go…

For these reasons I am glad to have found the above paper by Klusoň, where at least some aspects of the relation (O)SFT $\leftrightarrow$ (B)CFT are discussed.

The main restriction of Klusoň’s approach, from the above point of view, seems to be that he restricts attention to classical SFT solutions of the form

i.e. to solutions which naively seem to be gauge equivalent to the trivial solution ${\Phi }_0=0$. But apparently there is a subtlety related to finite gauge transformations, which can make this a non-trivial classical solution to SFT.

Anyway, as discussed on p.8 of that paper, expanding the SFT action around such a classical ‘background’ ${\Phi }_0$ yields the mentioned deformations of the BRST operator

as expected. The essential point of Klusoň’s paper is that we can now write down the SFT action in terms of $Q^{\prime }$ (equation (3.1)), massage it appropriately and demonstrate this way that it is precisely of the form of the SFT action that one would write down with respect to a (B)CFT obtained from marginal deformations with the operator $e^W$. In other words, the SFT action using $Q^{\prime }$ which describes excitations about the classical solution ${\Phi }_0$ is exactly what one obtains alternatively when all the correlators $⟨\cdots ⟩$ in the CFT-language version of the OSFT action

are replaced by their respective marginal deformed correlators

That’s nice, because marginal deformations of BCFTs have been studied in quite some detail, the canonical reference being apparently

A. Recknagel & V. Schomerus: Boundary deformation theory and moduli spaces of D-branes (1999).

That’s all very nice. But here is one related riddle that I have pondered all day and which seems to be simple, but which I couldn’t get a handle on yet:

Apart from the backgrounds discussed above, which correspond to marginal deformations, there is at least one further background which should be a simple testing ground for these deformations, I’d say, namely a pure gauge field background, i.e. with

(e.g. equation (2.34) of the review hep-th/0102085).

We know that the related BCFT differs from the unperturbed one just by the charged endpoints of the string in the given gauge field. The question to me is: Is this also the result that we obtain from looking at

In order to decide this I tried to reexpress the operator $[{\Phi }_A\star ,\cdot ]$ in the usual first quantized formalism, i.e. reexpressing the graded star-commutator with this particular ${\Phi }_A$ by polynomials in the ${\alpha }_n,c_n$.

I have tried to find a closed expression for this using the machinery in

T. Kawano & K. Okuyama: Open String Fields as Matrices (2001),

which seemed to come in handy, since the SFT vertices simplify greatly in that formalism (e.g. formula (2.24) in that paper), but of course one has to deal instead with the Bogoliubov transformation (2.19), which obscures things again, at least to me at the moment. So either I am not seeing the obvious or this is harder than I expected. Hints are appreciated.

Readers of this weblog will recall that we had discussed here two drafts which I have meanwhile submitted to JHEP.

One is

On deformations of 2D SCFTs, hep-th/0401175

which I originally presented in the entries ‘Classical deformations of 2D SCFTs Part I and Part II.

The other is

DDF and Pohlmeyer invariants of (super) string, hep-th/0403260

which originates in the thread Pohlmeyer charges, DDF states and string-gauge duality.

Now the referee reports for these submissions have arrived. Both papers have fortunately been accepted (one with a slight modification, see below), but there are some comments in the reports which I would like to briefly discuss here, since they concern issues which have been addressed here at the String Coffee Table and hence might be of interest.

Concerning the SCFT deformation paper the referee writes:

This paper discusses deformations of 2d superconformal field theories. As 2d SCFTs represent classical solutions of the string equations of motion understanding how they deform as we turn on spacetime fields is very crucial in unraveling their vacuum structure.

The only part of the paper that is problematic is the claim that there is a deformation that can be interpreted as a RR background. This is obviously not correct since the RR excitations couple to spin fields (both matter and ghost) while the deformation is not written in terms of these fields.

Otherwise the results of the paper are intersting and the paper should be published after the author modifies the paper by omitting his claim about the RR backgrounds.

Finally in the future I would advise the author to attempt to derive equations of motion for the spacetime fields. In other words to go beyond the classical level and discuss the issue of normal ordering.

Of course precisely the issue with RR backgrounds has on the one hand side been a motivation for this entire investigation and on the other hand I don’t claim that my construction sheds any new light on this particular problem (not yet at least :-).

Actually I mention RR backgrounds in two different contexts in this paper:

One, which comes from the bulk of the text, is the observation that a certain type of SCFT deformation which I describe is apparently best interpreted as describing a D-string in an RR 2-form background - not an F string in such a background! The D-string couples to the RR 2-form pretty much like the F-string couples to the NSNS 2-form, so this explains why at this point RR backgrounds make an appearance even though string fields do not. The rekation and distinction between the D-string in RR 2-form background and the F-string in NSNS 2-form background is a little subtle, but I do try to discuss that in the paper. Probably I need to emphasize the reason why in this context no spin fields make an appearance.

On the other hand, I had included one additrional remark where it is indicated how RR backgrounds should fit into the framework of that paper after all, but then of course using spin fields. The idea is that there should be a deformation of the worldsheet BRST operator even for these backgrounds (though there are subtleties, of course), along the general lines discussed in a previous entry. But of course the inclusion of RR backgrounds for the F-string this way is more like an idea for a research program, maybe, than a result. So perhaps I should really just remove that paragraph.

Concerning the DDF/Pohlmeyer paper the referee writes

This paper relates the so-called Pohlmeyer charges of the bosonic string to the standard DDF oscillators. It’s not clear to me, even having read the paper that there is any point to the Pohlmeyer construction. When acting on physical states (ie, after quantization), it has been argued that the Pohlmeyer charges yield only triivial information (like the total momentum) about the state.

However, it is certainly of value to recast them in terms of the standard DDF operators, which do act nontrivially on physical states. On emight then have a hope of seeing whether ther is any nontrivial content in the Pohlmeyer construction.

I think this paper should, therefore, be published.

I am glad that the paper has been accepted, but I am also surprised that the idea that somehow the Pohlmeyer invariants all are just made up of center-of-mass momentum and Lorentz generators seems to have spread quite far.

This idea seems to have originated in a discussion between Luboš Motl and Edward Witten where it was rediscovered that, while it is obvious that the Pohlmeyer invariants at first and second order are trivial, even the Pohlmeyer invariants at third order are trivial. This is well known, see for instance

D. Bahns: The invariant charges of the Nambu-Goto string and Canonical Quantization (2004) ,

and , while maybe surprising, doesn’t continue to hold for higher orders. Indeed, a generally accepted proof says that the Pohlmeyer invariants are complete in the sense that from their knowledge the worldsheet can locally be reconstructed.

In the above paper I have included in the conclusion some speculations what the Pohlmeyer invariants, being nontrivial, could be good for. But I concede that maybe they are not good for anything, this remains to be shown. The burden of proof is on those who claim otherwise. Meanwhile, the Pohlmeyer invariants and their relation to DDF invariants has attracted some attention simply because this has been related to the general question on how theories of gravity can be or have to be quantized - as we have discussed in gory detail before.

Tomorrow I’ll travel to Hamburg, where on Wednesday I’ll give a talk at the theory seminar of University of Hamburg, on behalf of a kind invitation by Thorsten Pruestel. We had first met at the last DPG spring conference, where I learned of the approach by Pruestel’s group concerning gauge theories with nonunitary parallel transport, which is an attempt to describe (possibly discretized) gravity by means of a special non-unitary component of a gauge connection. Prof. Fredenhagen is also interested in noncommutative field theories, and hence my talk will be on the stuff that Eric Forgy and myself developed a while ago

Eric Forgy & Urs Schreiber: Discrete Differential Geometry on $n$-Diamond Complexes (2004)

as well as its applications to field theory and string theory.

Here I’d like to give a first sketch of what I am going to say in that talk, mostly in the hope that Eric Forgy will spot the major omissions. :-)

(I am having problems with my internet connection, that’s why the following is not fully properly formatted. I am hoping to improve this entire entry tomorrow.)