The String Coffee Table

String entropy and black hole correspondence

As with the previous entry, this one here is a reply to a question on sci.physics.strings which seems to have problems to propagate through USENET.

‘Mike2’ wrote in news:Mike2.17k9lf-100000@physicsforums.com

It seems clear that strings represent structure, and various quantities are calculated along the worldsheet. So can one calculate the entropy associated with the information contained in the quantities along the string?

Yes, there are many states of the string which have the same energy and taking the logarithm of this number gives you the entropy of the string at that energy.

There are lots of very deep questions associated to this entropy.

One is the so-called string-black hole correspondence. It is generally said that a black hole carries the highest amount of entropy per volume. But a simple calculation shows that this is true only up to a very small size of the black hole. As the black hole shrinks (due to evaporation by means of Hawking radiation) it will become very tiny and at some point the entropy of a highly excited single string of a given mass will be equal to that of the black hole of that mass. For even lower masses the string’s entropy will even be greater than that of the corresponding black hole. The point at which that happens is called the string/black hole correspondence point.

The interesting thing is that, despite the crudeness of the calculations used in this sort of correspondence, it gives an easy way to calculate the correct order-of-magnitude entropy of all kinds of black holes, Schwarzschild, rotating, various charged ones, etc. It also provides a nice heuristic picture of black hole entropy at the correspondence point. One can sort of imagine the different “bits of string” sitting on the horizon and the entropy comes from the different ways in which these bits are connected inside the whole by the string.

If you want to find out more about this effect see

Gary T. Horowitz & Joseph Polchinski: A Correspondence Principle for Black Holes and Strings (1996)

and

Thibault Damour & Gabriele Veneziano: Self-gravitating fundamental strings and black-holes (1999)

I have once written a little more detailed description of this correspondence principle on sci.physics.research:

In its more refined form this ‘principle’ amounts to noting that there is a critical excitation energy where a massive string collapses under its own gravity to the size of the order of the string scale (becoming a ‘string ball’) and that precisely at this point its rms radius coincides with its Schwarzschild radius and furthermore all its thermodynamical properties (temperature, entropy, radiation, decay rate) coincide, up to some unknown factors of order unity, with that of a BH of the same mass (e.g. hep-th/9907030).

(I am not sure how this relates to the D5/D1 brane models, which I don’t know well, but I seem to recall that these brane configurations are describable, and are described, by ‘effective long strings’, too.)

Anyway, the string/BH correspondence principle gives rise to a neat mental picture of the BH degrees of freedom which is actually rather similar to the LQG picture of a ‘pierced horizon’. It is roughly the following:

A highly excited string with the large mass $M\gg 1/l_s$ ($l_s=\sqrt{{\alpha }^{\prime }}$ is the string scale) is in strikingly good approximation a random walk of $n=\sqrt{N}$ steps of step size $l_s$, where $N$ is the level number of the string, i.e. $N=M^2{l}_s^2$. It follows that its entropy is to leading order

$$S_s\sim n\sim M{l}_s.$$

This is quite unlike the entropy dependence of a black hole, which goes as

$$S_{\mathrm{BH}}\sim M^q$$

with $q>1$. But it so happens that at the above mentioned correspondence point, which is reached when the mass of the string becomes the critical value

$$M_c=\frac{1}{g^2}\frac{1}{l_s}$$

($g$ is the string coupling) and where the string collapses under its self-gravity to a ball of diameter $\sim l_s$, the entropy of the string and that of a BH of the same size coincide. Still, the entropy of a random-walk-like string, even in the collapsed form, has a simple interpretation, it counts the number of decisions one can make while stepping along the random walk.

Now imagine how that collapsed ‘random walk’ looks like: A chain of n segments, each of length $l_s$ is restricted to lie within a ball whose diameter is also about $l_s$. A typical such state looks somewhat star-shaped with all the vertices of the random walk on the outside, forming a sphere. This sphere about coincides with the event horizon of a BH which has the same mass as our string. The edges of the random walk cross the interior of this sphere, pierce the horizon, deposit their vertex there, then return to a point near the corresponding antipode and so on, thereby covering the sphere with all n vertices, all about equally spaced (for a typical state). What is the mean area $A_v$ of the sphere occupied by one such vertex? It is the number of vertices divided by the area of the horizon, i.e.

$$A_v\sim n/l_s^{(d-1)}\sim n{g}^{(d-1)}/l_p^{(}d-1)\sim 1/l_p^{(d-1)}.$$

Here $d$ is the number of spatial dimensions and $l_p$ is the Planck length, given in string theory by

$$l_p^{(d-1)}=g^2{l}_s^{(d-1)}.$$

It follows that (for instance) in 1+3 dimensions each of the above vertices occupies an area of about a square Planck length of the event horizon.The entropy of this system is, due to the nature of a random walk and by the above formula, proportional to the number of vertices and hence to the area of the event horizon in Planck units.

This is the picture of black hole microstates at the string/BH correspondence point, i.e. for BH that are about to decay into a string state or for strings that are about to become black holes. (In fact the string ball configuration has been used to predict the signature of decaying black holes that may be detected in accelerators one day).

What happens to this crude picture when the mass is increased further? I am not sure how solid the knowledge obout the answer to this question is, but there is a lot of literature about “strings on the stretched horizon”. The basic idea is that once the BH description takes over the above mentioned vertices are somehow frozen on the event horizon. Since the temperature of the string and the Hawking temperature agree at the correspondence point and hence the rates of change of horizon area with mass do, one can show that further quanta of mass that one throws into a BH at correspondence point correctly translates into further string ‘vertices’ appearing on the horizon. But this only holds in the vicinity of the correspondence point. Farther away one has to take into account the fact that the energy of an object near a BH horizon is different when measured by an asymptotically far away observer. When this red-shifting effect is accounted for one can apparently consistently imagine the BH entropy being due to a (very) long string which is lying on the “stretched” event horizon in form of a random walk.

Of course, all this is nowhere near the technical sophistication of D5/D1 brane-system calculations. It is rather like a Bohr-atom model of quantum black holes.

Experiments probing quantum gravity

There still seems to be a problem with the newsserver at Harvard, from which the messages of sci.physics.strings emanate. For some reason whenever I take the time to write a longer message the server seems to decide that it has better things to do than propagating it to other newsservers. I hope we can solve this weird problem in the future (maybe it is just me not using the various programs correctly, somehow?) but for the time being I would like to make these lost messages available here at the Coffee Table, not the least because they are replies to questions that have been asked.

So here is the first one:

Alexander Blessing had asked, more or less, for experiments that could say something about quantum gravity.

There are indeed a couple of interesting ways, in principle at least, to look for new physics without using ever more expensive particle accelerators:

Here is a set of slides by Pisin Chen from SLAC discussing some implications of possible astrophysical observations for new physics, see also this link.

People are looking in particular at the following effects:

1) Ultra High Energy Cosmic Rays

2) Lorentz violation on small scales

3) Detection of dark matter

4) Search for additional spatial dimensions and violations of the equivalence principle

5) Spacetime granularity

I’ll give some links and comments for all these 5 points:

1) Ultra High Energy Cosmic Rays should on theoretical grounds be cut off at $5x10^{19}$ eV, the so-called Greisen-Zatzepin-Kuzmin (GZK) limit, due to the absorbtion of highly energetic rays by the cosmic microwave background.

But at least one experiment has claimed to have seen particles boyond the GZK limit, for a review see

Glennys R. Farrar, Tsvi Piran:
GZK Violation - a Tempest in a (Magnetic) Teapot?,
Phys.Rev.Lett. 84 (2000) 3527

I have heard that Smolin and some other people believe that they can explain the GZK violation with ‘doubly special relativity’, that this is furthermore somehow a prediction of LQG and that they are hoping that the GLAST experiment in 2006 will say something about this (but I am having
trouble finding more details on this speculation, in hep-th/0401087 Smolin vaguely talks about some such experiments).

Recently Edward Witten commented on this experiment where he said:

It is not clear at the moment that there is anything to explain. One of the two main experimental groups (AGASA) reported as of over a year ago that their high energy data are in accord with the expected GZK cutoff. The other group continues to report a discrepancy. We’ll see what happens as data improve.

(quote from the message at the above link).

2) People are apparently looking for violation of Lorentz invariance at small scales/large energies:

http://qom.physik.hu-berlin.de/prl_91_020401_2003.pdf

http://cfa-www.harvard.edu/Walsworth/Activities/posters/lliposter.pdf

From some theories of quantum gravity one might expect such effects, probably not from string theory.

3) Nature of dark matter.

Several people are trying to determine the presence and maybe the nature of dark matter.

For instance the DAMA collaboration and the UK Dark Matter collaboration as briefly summarized in this message by John Baez.

Of course the true nature of dark matter may be a very important clue for how the true theory of quantum gravity looks like, but current experiments are of course very far from saying anything about this.

4) Search for additional spatial dimenions and violations of the equivalence principle

There are potentially very interesting high-precision measurements of the equivalence principle, most notably by the EotWash group.

Variation of the usual $1/r^2$ force law of gravity on small scales could be related to extra dimensions and or torsion, for instance.

5) Spacetime granularity

From quantum gravity some people expect that on extremely small scales
spacetime will show some sort of foamy structure, maybe being topologically non-trivial. An old idea by Percival and collaborators is that atom interferometry, e.g. the 2-slit experiment with heavy stuff such as Buckminster Fullerenes (Nature Vol. 401, No. 6754, p. 680 (1999).) as done by Zeilinger’s group, may be sensitive to such a spacetime grabularity.

I had mentioned that in the past from time to time.
giving some references.

I am not sure what string theory really says about ‘quantum spacetime foam’ at small scales.

When we discussed Smolin’s paper it was argued by some people that string theory (on Minkowski space, say) predicts smooth spacetime
down to all scales. But this is not correct, as everybody knows and as I have tried to argue in more detail in this and this message once based on the discussion in Fedele Lizzi anmd Richard Szabo, Duality Symmetries and Noncommutative Geometry of String Spacetimes (1997) because as you try to probe the smooth background with highly energetic strings you will eventually see stringy effects and not be able to resolve the smoothness of the classical background at all, so that effectively it is not smooth on small scales.

So: Might stringy physics effect the phase of atoms/molecues used in matter interferometry ever so slightly?

Sigh.

Jacques Distler and Peter Woit have already commented on the latest attempt of Bogdanov & Bogdanov to convince laymen of their genius. One should really stop talking about this issue, since apparently the whole purpose of the exercise is to make everybody mention B&B, no matter in which sense. Apparently in lack of experts willing to support their ideas they rephrase comments of critics in such a way that it sounds like approvements.

A while ago I had written this summary of their mistakes (reproduced below). They thanked me personally for this ‘very accurate’ description of their work and said that I was among the very few who really understood the ‘big picture’ of their work. Well, that’s nice, because it allows me to use this certified authority to say in all clarity that the ideas summarized at the above link are indeed based on elementary misconceptions and invalid conclusions and hence make no sense.

Unfortunately it is precisely this little detail that surprisingly did not survive the adaption of my little text in their new book. Everybody who reads some french and knows what a topological field theory really is can compare my original summary of B&B’s mistakes with the respective excerpts I, II from their book to convince himself once and for all that their concern is not science.

I think I am allowed to say that. After all, I am among the few who understands them. ;-)

Unlooping the LQG string

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 $\left(f_n\right)$ 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 $j_n=\int f_{n}j\left(x\right)\mathrm{dx}$. 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 $$[L_n,a_m]=-m{a}_{n+m}.$$

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 $\omega \left(a_n\right)=p{\delta }_{n,0}$, 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 $\omega \left(e^{i{j}_n}\right)$ 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 $L_2\left(R\right)$ 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 $⟨T_s,T_{s\text{'}}⟩=\delta (s-s\text{'})$. 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 $\delta \left(a\right)$ 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!

CFTs from OSFT!

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

$$\tilde{Q}=Q+[{\Phi }_0\star ,\cdot ]$$

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

$${\Phi }_0=e^{-W}\star Q{e}^W,$$ 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

$$\tilde{Q}=Q^{\prime }=Q+[W,Q]+\cdots ,$$

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

$$S\sim ⟨I\circ \Psi \left(0\right)Q\Psi \left(0\right)⟩+\frac{2}{3}⟨f_1\circ \Psi \left(0\right)f_2\circ \Psi \left(0\right)f_3\circ \Psi \left(0\right)⟩$$

are replaced by their respective marginal deformed correlators

$$⟨{\Psi }_1\left(x_1\right){\Psi }_2\left(x_2\right)\cdots ⟩\to ⟨{\Psi }_1\left(x_1\right){\Psi }_2\left(x_2\right)\cdots {⟩}_W:=⟨e^W{\Psi }_1\left(x_1\right)e^W{\Psi }_2\left(x_2\right)\dots ⟩.$$

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

$${\Phi }_A=\int d^{D}k{A}_{\mu }\left(k\right){\alpha }_{-1}^{\mu }{c}_1\mid k⟩$$

(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

$$Q+[{\Phi }_A\star ,\cdot ]?$$

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.

Referee reports on SCFT deformations and Pohlmeyer invariants

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.

Talk: N=2 NCG, fields and strings

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.)

CFTs from OSFT?

Update 19 May 2004

I have finally found a paper which pretty much precisely discusses what I was looking for here, namely a relation between classical solutions of string field theory and deformations of the worldsheet (boundary-) conformal field theory. It’s

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

(see also this entry)

and it discusses how OSFT actions expanded about two different classical solutions correspond to two worldsheet BCFTs in the case where the latter are related by marginal deformations. In the words of the author of the above paper (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, $\mathrm{BCFT}\prime$ which are related by marginal deformation, are in fact two SFT actions expanded around different classical solutions.

In equation (2.31) the deformed BRST operator is given, which is what I discuss in the entry below, but then it is shown in (3.8) that this operator can indeed be related to a (B)CFT with marginal deformation.

One subtlety of this paper is that the classical SFT solutions which are considered are large but pure gauge and hence naively equivalent to the trivial solution ${\Phi }_0=0$, but apparently only naively so. To me it would be intreresting if similar results could be obtained for more general classical solutions ${\Phi }_0$.

---

Update 3rd May 2004

I have now some LaTeXified notes.

---

Here is a rather simple — indeed almost trivial — observation concerning open string field theory (OSFT) and deformations of CFTs, which I find interesting, but which I haven’t seen discussed anywhere in the literature. That might of course be just due to my insufficient knowledge of the literature, in which case somebody please give me some pointers!

---

Update 7th May 2004

I have by now found some literature where this (admittedly very simple but interesting) observation actually appears, e.g.

• equation (2.48) of

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

  which originates in

  Ashoke Sen: Equations of motion in non-polynomial closed string field theory and conformal invariance of two dimensional field theories (1990)

• section 5 of

  Isao Kishimoto & Kazuki Ohmori: CFT Description of Identity String Field: Toward Derivation of the VSFT Action (2001)

• and p. 8 of

  I. Aref’eva, D. Belov, A. Giryavets, A. Koshelev and P. Medvedev: Noncommutative Field Theories and (Super) String Field Theories (2002)

---

Here goes:

There have been some studies (few, though) of worldsheet CFTs for various backgrounds in terms of deformed BRST operators. I.e., starting from the BRST operator $Q_B$ for a given background, like for instance flat Minkowski space, one may consider the operator $${\tilde{Q}}_B:=Q_B+\hat{\Phi },$$ where $\hat{\Phi }$ is some operator such that nilpotency ${\tilde{Q}}_B^2=0$ is preserved.

By appropriately commuting ${\tilde{Q}}_B$ with the ghost modes the conformal generators ${\tilde{L}}_m^{\mathrm{tot}}$ of a new CFT in a new background are obtained (the new background might of course be gauge eqivalent to the original one).

See for instance

Mitsuhiro Kato: Physical Spectra in String Theories — BRST Operators and Similarity Transformations (1995)

and

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

One problem is to understand the operators $\hat{\Phi }$, how they have to be chosen and how they encode the information of the new background.

Here I want to show, in the context of open bosonic strings, that the consistent operators $\hat{\Phi }$ are precisely the operators of left plus right star-multiplication by the string field $\Phi$ which describes the new background in the context of open string field theory.

In order to motivate this consider the (classical) equation of motion of cubic open bosonic string field theory for a string field $\Phi$ of ghost number one: $$Q_B\mid \Phi ⟩+\mid \Phi \star \Phi ⟩=0,$$ where for simplicity of notation the string field has been rescaled by a constant factor.

(I am using the notation as for instance in section 2 of

Kazuki Ohmori: A Review on Tachyon Condensation in Open String Field Theories (2001).)

If we now introduce $\hat{\Phi }$, the operator of star-multiplication by $\Phi$ defined by $$\hat{\Phi }\mid \Psi ⟩:=\mid \Phi \star \Psi ⟩$$ then, due to the associativity of the star product this can equivalently be rewritten as an operator equation $${(Q_B+\hat{\Phi })}^2=0$$ because $$(Q_B+\hat{\Phi })\circ (Q_B+\hat{\Phi })\circ ={\underbrace{Q_B\circ Q_B\circ }}_{=0}+{\underbrace{Q_B\circ \Phi \star +\Phi \star Q_B\circ }}_{=\left(Q_B\Phi \right)\star }+{\underbrace{\Phi \star \Phi \star }}_{=(\Phi \star \Phi )\star }.$$ (Here it has been used that $Q_B$ is an odd graded (with respect to ghost number) derivation on the star-product algebra of string fields, that $\Phi$ is of ghost number 1 and that the star-product is associative.)

It hence follows that the equations of motion of the string field $\Phi$ are precisely the necessary and sufficient condition for the operator $\hat{\Phi }$ to yield a nilpotent, unit ghost number deformation $${\tilde{Q}}_B=Q_B+\hat{\Phi }$$ of the original BRST operator.

But there remains the question why ${\tilde{Q}}_B$, while nilpotent, can really be interpreted as a BRST operator of some sensible CFT. (Surely not every nilpotent operator on the string Hilbert space can be identified as a BRST operator!) The reason seems to be the following:

---

Update 21 May 2004

I have found out by now that what I was trying to argue here has already been found long ago in papers on background independence of string field theory. For instance on p.2 of

Ashoke Sen: Equations of motion in non-polynomial closed string field theory and conformal invariance of two dimensional field theories (1990)

it says:

In this paper we show that if ${\Psi }_{\mathrm{cl}}$ is a solution of the classical equations of motion derived from the action $S\left(\Psi \right)$, then it is possible to construct an operator ${\hat{Q}}_B$ in terms of ${\Psi }_{\mathrm{cl}}$, acting on a subspace of the Hilbert space of combined matter-ghost CFT, such that $({\hat{Q}}_B{)}^2=0$. ${\hat{Q}}_B$ may be interpreted as the BRST charge of the two dimensional field theory describing the propagation of the string in the presence of the background field ${\Psi }_{\mathrm{cl}}$.

---

We may consider, in the context of open bosonic string field theory, the motion of a single ‘test string’ in the background described by the excitatoins $\Phi$ by adding a tiny correction field $\psi$ to $\Phi$, which we want to interpret as the string field due to the single test string.

The question then is: What is the condition on $\psi$ so that the total field $\Phi +\psi$ is still a solution to the equations of motion of string field theory. That is, given $\Phi$, one needs to solve $$Q_B(\Phi +\psi )+(\Phi +\psi )\star (\Phi +\psi )=0$$ for $\psi$. But since $\psi$ is supposed to be just a tiny perturbation of the filed $\Phi$ it must be sufficient to work to first order in $\psi$. This is equivalent to neglecting any self-interaction of the string described by $\psi$ and only considering its interaction with the ‘background’ field $\Phi$ - just as in the first quantized theory of single strings.

But to first order and using the fact that $\Phi$ is supposed to be a solution all by itself the above equation says that $$Q_B\mid \psi ⟩+\mid \Phi \star \psi ⟩+\mid \psi \star \Phi ⟩=0.$$

This is manifestly a deformation of the equation of motion $$Q_B\mid \psi ⟩=0$$ of the string described by the state $\psi$ in the original background. Hence it is consistent to interpret $${\tilde{Q}}_B=Q_B+\{\hat{\Phi },\cdot \}$$ as the new worldsheet BRST operator which corresponds to the new background described by $\Phi$.

If we again switch to operator notation the above can equivalently be rewritten as $$\left\{\right(Q_B+\hat{\Phi }),\hat{\psi }\}=0,$$ where the braces denote the anticommutator, as usual.

Recalling that a gauge transformation $\Phi \to \Phi +\delta \Phi$ in string field theory is (for $\Lambda$ a string field of ghost number 0) of the form $$\delta \Phi =Q_B\Lambda +\Phi \star \Lambda -\Lambda \star \Phi$$ and that in operator language this reads equivalenty $$\widehat{\delta \Phi }=\left[\right(Q_B+\hat{\Phi }),\hat{\Lambda }]=0$$ one sees a close connection of the deformed BRST operator to covariant exterior derivatives.

As is very well known (for instance summarized in the table on p. 16 of the above review paper) there is a close analogy between string field theory formalism and exterior differential geometry.

The BRST operator $Q_B$ plays the role of the exterior derivative, the $c$ ghost correspond to differential form creators, the $b$-ghosts to form annihilators and the $\star$ product to the ordinary wedge ($\wedge$) product - or does it?

As noted on p.16 of the above review, the formal correspondence seems to cease to be valid with respect to the graded commutativity of the wedge product. Namely in string field theory $$\Phi \star \psi \ne \pm \Psi \star \Phi$$ in general.

But the above considerations suggest an interpretation of this apparent failed correspondence, which might show that indeed the correspondence is better than maybe expected:

The formal similarity of the deformed BRST operator ${\tilde{Q}}_B=Q_B+\hat{\Phi }$ to a gauge covariant exterior derivative $d+\omega$ suggests that we need to interpret $\Phi$ not simply as a 1-form, but as a - connection!

That is, $\Phi$ would correspond to a Lie-algebra valued 1-form and the $\star$-product would really be exterior wedge multiplication together with the Lie product, as very familiar from ordinary gauge field theory. For instance we would have expression like $$(d+\omega {)}^2=(d\omega ){\delta }^a{}_b+{\omega }^a{}_c\wedge {\omega }^c{}_b.$$

In such a case it is clear that the graded commutativity of the wedge product is broken by the Lie algebra products.

Is it consistent to interpret the star product of string field theory this way? Seems to be, due to the following clue:

Under the trace graded commutativity should be restored. The trace should appear together with the integral as in $$\int \mathrm{tr}{\omega }^a{}_c\wedge {\gamma }^c{}_b=\pm \int \mathrm{tr}{\gamma }^a{}_c\wedge {\omega }^c{}_b.$$

But precisely this is what does happen in open string field theory in the formal integral. There we have $$\int \Phi \star \Psi =\pm \int \Psi \star \Phi .$$

All this suggests that one should think of the deformed BRST operator as morally a gauge covariant exterior derivative: $${\tilde{Q}}_B=Q_B+\hat{\Phi }\sim d+\omega .$$

That looks kind of interesting to me. Perhaps it is not new (references, anyone?), but I have never seen it stated this way before. This way the theory of (super)conformal deformations of (super)conformal field theories might nicely be connected to string field theory.

In particular, it would be intersting to check the above considerations by picking some known solution $\Phi$ to string field theory and computing the explicit realization of ${\tilde{Q}}_B$ for this background field, maybe checking if it looks the way one would expect from, say, worldsheet Lagrangian formalism in the given background.

Billiards at half-past E10

(title?)

Last week I gave a seminar talk on cosmological billiards and their relation to proposals that M-theory might be described by a 1+0 dimensional sigma-model on the group of $E_{10}/K\left(E_{10}\right)$. I had mentioned that already several times here at the Coffee table and we had some interesting discussion over at sci.physics.strings. But while preparing the talk it occured to me that the basic technical observation behind this conjecture is so simple and beautiful that it deserves a seperate entry. I’ll summarize pp. 65 of

T. Damour, M. Henneaux & H. Nicolai: Cosmological Billiards (2002).

So how would the equations of motion of geodesic motion on a Kac-Moody algebra group manifold look like, in general?

A Kac-Moody (KM) algebra is a generalization of an ordinary Lie algebra. It is determined by its rank $r$, an $r\times r$ Cartan matrix $A=\left(a_{\mathrm{ij}}\right)$ and is generated from the $3r$ Chevalley-Serre generators $$\{h_i,e_i{f}_i{\}}_{i=1,\cdots ,r}$$ which have commutators $$[h_i,h_j]=0$$ $$[e_i,f_j]={\delta }_{\mathrm{ij}}{h}_j$$ $$[h_i,e_j]=a_{\mathrm{ij}}{e}_j$$ $$[h_i,f_j]=-a_{\mathrm{ij}}{f}_j.$$ (One should think of the SU(2) example where $h=J^3$, $e=J^{+}$ and $f=J^{-}$.)

The elements of the algebra are obtained by forming multiple commutators of the $e_i$ and the $f_i$. $$E_{\alpha ,s}=[e_{i_1},[e_{i_2},[\cdots ,[e_{i_{p-1},e_{i_p}}]\cdots ]$$ $$E_{-\alpha ,s}=[f_{i_1},[f_{i_2},[\cdots ,[f_{i_{p-1},f_{i_p}}]\cdots ].$$ Here, as always in Lie algebra theory, the $\alpha$ are the so called roots, i.e. the ‘quantum numbers’ with respect to the Cartan subalgebra generators $h_i$: $$[h_i,E_{\alpha ,s}]={\alpha }_i{E}_{\alpha ,s}$$ and $s=1,\cdots \mathrm{mult}\alpha$ is an additional index due to possible degeneracies of the $\alpha$s. Not all of these elements are to be considered different, but instead one has to mod out by the Serre relations $$\mathrm{ad}\left(e_i{)}^{1-a_{\mathrm{ij}}}\right(e_j)=0$$ $$\mathrm{ad}\left(f_i{)}^{1-a_{\mathrm{ij}}}\right(f_j)=0$$ which should be thought of as saying that the $E_{\alpha }$ are nilpotent ‘matrices’ with entries above the diagonal, while the $E_{-\alpha }$ are nilpotent with entries below the diagonal.

(I’d be grateful if anyone could tell me why these Serre relations need to look the way they do…)

A set of simple roots ${\alpha }^i$ generates, by linear combination, all of the roots and the Cartan matrix is equal to the normalized inner products of the simple roots $$a_{\mathrm{ij}}=2\frac{⟨{\alpha }^i\mid {\alpha }^j⟩}{⟨{\alpha }^i\mid {\alpha }^i⟩},$$ where the product is taken with respect to the unique invariant metric (just as for ordinary Lie algebras).

The nature of the KM algebra depends crucially on the signature of the Cartan matrix $A$. We have three cases:

- If $A$ is positive definite, then the KM algebra is just an ordinary finte Lie algebra $[T^a,T^b]=f^{\mathrm{ab}}{}_c{T}_c$.

- If $A$ is semidefinite, then the KM algebra is an infinite affine Lie algebra, or equivalently a current algebra in 1+1 dimensions: $[j_m^a,j_n^b]=m{\eta }^{\mathrm{ab}}{\delta }_{m+n,0}+f^{\mathrm{ab}}{}_c{j}_{m+n}^c$.

- If, however, $A$ is indefinite, we obtain an infinite KM algebra with exponential growth, which is, I am being told, relatively poorly understood in general. But this is the case of interest here!

A general element of the group obtained from such an algebra by formal exponentiation is of the form $$𝒱={\underbrace{\exp \left({\beta }^i{h}_i\right)}}_{=𝒜}{\underbrace{\exp \left(\sum _{\alpha ,s}{\nu }_{\alpha ,s}{E}_{\alpha ,s}\right)}}_{=𝒩}.$$ If we make the coefficients functions of a single parameter $\tau$ then we get the tangent vectors $P$ to the trajectory in the group ‘manifold’ traced out by varying $\tau$ by writing: $$P:=\frac{1}{2}(\dot{𝒱}{𝒱}^{-1}+{\left(\dot{𝒱}{𝒱}^{-1}\right)}^T).$$ This is exactly as for any old Lie group, the only difference being that we have here projected onto that part which is ‘symmetric’ with respect to the Chevalley involution which sends $$h_i^T=h_i$$ $$e_i^T=f_i$$ $$f_i^T=e_i.$$ The algebra factored out this way is the maximal compact subalgebra of our KM algebra, so that we are really dealing with the remaining coset space (which, unless I am confused, should hence be a symmetric space).

Now the fun thing which I wanted to get at is this: If we define generalized momenta $j_{\alpha ,s}$ such that $$\dot{𝒩}{𝒩}^{-1}=\sum _{\alpha ,s}{j}_{\alpha ,s}{E}_{\alpha ,s},$$ then it is very easy to check, using the defining relations of the KM algebra, that $$P={\dot{\beta }}^i{h}_i+\frac{1}{2}\sum _{\alpha ,s}{j}_{\alpha ,s}\exp \left({\alpha }_i{\beta }^i\right)(E_{\alpha ,s}+E_{-\alpha ,s}).$$ Using the fact that the non-vanishing inner products of the algebra elements are $$⟨h_i\mid h_j⟩=g_{\mathrm{ij}}$$ $$⟨E_{\alpha ,s}\mid E_{\beta ,s}⟩={\delta }_{s,t}{\delta }_{\alpha +\beta ,0}$$ one finally finds the Lagrangian describing geodesic motion of the coset space: $$ℒ\propto ⟨P\mid P⟩=g_{\mathrm{ij}}{\dot{\beta }}^i{\dot{\beta }}^j+\frac{1}{2}\sum _{\alpha ,s}\exp \left(2{\alpha }_i{\beta }^i\right)j_{\alpha ,s}^2.$$ (Here $g$ is the invariant metric of the algebra.)

The point is that there is a free kinetic term in the Cartan subalgebra plus all the off-diagonal kinetic terms which all couple exponentially to the Cartan subalgebra coordinates.

It is obvious that for very large values of $\beta$ the off-diagonal terms ‘freeze’ and leave behind effective potential walls which constrain the motion of the $\beta$s to lie within the Weyl chamber of the algebra, namely that poly wedge associated with the simple roots (all other roots generate potential walls which lie behind those of the simple roots.)

Anyone familiar with classical cosmology immediately recognizes the above Lagrangian as being precisely of the form as those mini/midi superspace Lagrangians that govern the dynamics of homogeneous modes of general relativity. There the ${\beta }^i$ are the logarithms of the spatial scale factors of the universe.

Indeed, it can be checked to low order that the Lagrangian of $E_{10}$ in the above sense reproduces that of 11d SUGRA when the latter is accordingly suitably expanded about homogeneous modes. That’s the content of

T. Damour, M. Henneaux & H. Nicolai: $E_{10}$ and the ‘small tension’ expansion of M Theory (2002).

But the crucial point is that there are many more degrees of freedom in the $E_{10}$ sigma model than can correspond to supergravity. There are indications that these can indeed be associated with brane degrees of freedom of M-theory:

Jeffrey Brown, Ori Ganor & C. Helfgott: M-theory and $E_{10}$: Billiards, Branes, and Imaginary Roots,

which, unfortunately, I still have not read completely.

New York, New York

I have visited Ioannis Giannakis at Rockefeller University, New York, last week, and by now I have recovered from my jet lag and caught up with the work that has piled up here at home enough so that I find the time to write a brief note to the Coffee Table.

Ioannis Giannakis has worked on infinitesimal superconformal deformations and I became aware of his work while I happened to write something on finite deformations of superconformal algebras myself. In New York we had some interesting discussion in particular with regard to generalizations of the formalism to open strings and to deformations that describe D-brane backgrounds.

The theory of superconformal deformations was originally motivated from considerations concerning the effect of symmetry transformations of the background fields on the worldsheet theory. It so happened that while I was still in New York a heated debate concerning the nature of such generalized background gauge symmetries and their relation to the worldsheet theory took place on sci.physics.strings.

People interested in these questions should have a look at some of the literature, like

Jonathan Bagger & Ioannis Giannakis: Spacetime Supersymmetry in a nontrivial NS-NS superstring background (2001)

and

Mark Evans & Ioannis Giannakis: T-duality in arbitrary string backgrounds (1995) ,

but the basic idea is nicely exemplified in the theory of a single charged point-particle in a gauge field $A$ with Hamiltonian constraint $H=(p-A{)}^2$. A conjugation of the constraint algebra and the physical states with $\exp \left(i\lambda \right)$ induces of course a modification of the constraint $$(p-A{)}^2\to (p-A+d\lambda {)}^2$$ which corresponds to a symmetry tranformation in the action of the background field $A$. In string theory, with its large background gauge symmetry (corresponding to all the null states in the string’s spectrum) one can find direct generalizations of this simple mechanism. (Due to an additional subtlety related to normal ordering, these are however fully under control only for infinitesimal shifts or for finite shifts in the classical theory.)

More importantly, as in the particle theory, where the trivial gauge shift $p\to p+d\lambda$ tells us that we should really introduce gauge connections $A$ that are not pure gauge, one can try to guess deformations of the worldsheet constraints that correspond to physically distinct backgrounds. This is the content of the theory of (super)conformal deformations. My idea was that there is a systematic way to find finite superconformal deformations by generalizing the technique used by Witten in the study of the relation of supersymmetry to Morse theory. The open question is how to deal consistently with the notion of normal ordering as one deforms away from the original background.

In order to understand this question better I tried to make a connection with string field theory:

Consider cubic bosonic open string field theory with the string field $\phi$ the BRST operator $Q$ for flat Minkowski background and a star product $\star$, where the (classical) equations of motion for $\phi$ are $$Q\phi +c\phi \star \phi =0,$$ for some constant $c$.

In an attempt to understand if this tells me anything about the propagation of single strings in the background described by $\phi$ I considered adding an infinitesimel ‘test field’ $\psi$ to $\phi$ and checking what equations of motion $\psi$ has to satisfy in order that $\phi +\psi$ is still a solution of string field theory. To first order in $\psi$ one finds $$(Q+c\{\phi \star ,\cdot \})\mid \psi ⟩=0.$$ If we think of the ‘test field’ $\psi$ as that representing a single string, then it seems that one has to think of $$Q^{\prime }:=Q+c\{\phi \star ,\cdot \}$$ as the deformed BRST operator which corresponds to the background described by the background string field $\phi$.

It is due to the fact that $a\star b$ and $b\star a$ in string field theory have no obvious relation that I find it hard to see whether $Q^{\prime }$ is still a nilpotent operator, as I would suspect it should be.

But assuming it is and that its interpretation as the BRST operator corresponding to the background described by $\phi$ is correct, then it would seem we learn something about the normal ordering issue referred to above: Namely as all of the above string field expressions are computed using the normal ordering of the free theory it would seem that the same should be done when computing the superconformal deformations. But that’s not clear to me, yet.

The campus of Rockefeller University.

Power Supply

I am currently visiting the Albert-Einstein institute in Potsdam (near Berlin). Hermann Nicolai had invited me for a couple of days in order to talk and think about Pohlmeyer invariants and related issues of string quantization.

It so happened that when I checked my e-mail while sitting in the train to Berlin I found a mail by Thomas Thiemann, Karl-Henning Rehren and Dorothea Bahns in my inbox, containing a pdf-draft of some new notes concerning what they flatteringly call “Schreiber’s DDF functionals” but what really refers to the insight that the Pohlmeyer invariants are a subset of all DDF invariants.

Glad that my journey should have such a productive beginning I read through the notes and began typing a couple of comments - when I realized that my notebook battery was almost empty.

Here is a little riddle: What are all the places in a german “Inter City Express” train where you can find a 230V power supply?

Right, there is one at every table. But when it’s the end of the Easter holidays all tables are occupied and when nobody is willing to let you sit on his (or her) lap then that’s it - or is it?

Not quite. For the urgent demands of carbon-based life forms there is fortunaly a special room - and it does have a socket, just in case anyone feels like shaving on a train. I spare you the details, but in any case this way when I arrived at the AEI the discussion had already begun. :-)

After further discussion of Thiemann’s and Rehren’s comments with Kasper Peeters and Hermann Nicolai we came to believe that there are in fact no problems with quantizing the Pohlmeyer invariants in terms of DDF invariants. I wrote up a little note concerning the question if there are any problems due to the fact that the construction of the DDF invariants requires specifying a fixed but arbitrary lightlike vector on target space. One might think that this does not harmonize with Lorentz invariance, but in fact it does. I am still waiting for Thiemann’s and Rehren’s reply, though. Hopefully we don’t have to fight that out on the arXive! ;-)

It turned out that I am currently apparently the only one genuinly interested in what the Pohlmeyer invariants could be good for in standard string theory. It seems that everbody else either regards them as a possibility to circumvent standard results - or as an irrelevant curiosity.

Here is a sketchy list of some questions concerning Pohlmeyer invariants that I would find interesting:

The existence of Pohlmeyer invariants gives us a map from the Hilbert space of the single string to states in totally dimensionally reduced (super) Yang-Mills theory. Namely, every state |psi> of the string (open, say) gives us a map from the space of u(N) matrices to the complex numbers, defined by

$$M:u\left(N\right)\to C$$

$$A\mapsto ⟨\psi \mid \mathrm{Tr}P\exp (\int A_{\mu }{𝒫}^{\mu }(\sigma \left)d\sigma \right)\mid \psi ⟩.$$

Does conversely every state on $u\left(N\right)$ define a state of string? Apparently the answer is Yes. .

What is the impact in this context of the fact that the Pohlmeyer holonomies $$\mathrm{Tr}P\exp (\int A_{\mu }{𝒫}^{\mu }(\sigma \left)d\sigma \right)$$ are Virasoro invariants? We have a vague understanding (hep-th/9705128) of what the map from $A$ to $\mid \psi ⟩$ has to do with string field theory. Can something similar be said about the Pohlmeyer map from $\mid \psi ⟩$ to $A$?

What is the meaning on the string theory side of a Gaussian ensemble in $u\left(N\right)$, as used in Random Matrix Theory?

I have a very speculative speculation concerning this last question: We know that the IKKT action is just BFSS at finite temperature. But the BFSS canonical ensemble $$\exp (\mathrm{const}\mathrm{Tr}{P}^2+\mathrm{interaction})$$ is just the RMT Gaussian ensemble, up to the interaction terms. It might be interesting to discuss the limit in which the interaction terms become neglibile and see what this means in terms of the Pohlmeyer map from gauge theory to single strings.

Incidentally, without the interaction terms we are left with RMT theory which is known to describe chaotic systems. This seems to harmonize nicely with the fact that also in (11d super-) gravity, if the spacetime point interaction is turned off (near a spacelike singularity) the dynamics becomes that of a chaotic billiard.

Somehow it seems that the Pohlmeyer map relates all these matrix theory questions to single strings. How can that be? Can one interpret the KM algebra of 11d supergravity as a current algebra on a worldsheet?

Sorry, this is getting a little too speculative. :-) But it highlights another maybe intersting question:

What is the generalization of the Pohlmeyer invariant to non-trivial backgrounds?

I have mentioned somewhere that whenever we have a free field realization of the worldsheet theory (like on some pp-wave backgrounds) the DDF construction goes through essentially unmodified and hence the Pohlmeyer invariants should be quantizable in such a context, too.

But what if the background is such that the DDF invariants are no longer constructible, or rather, if their respective generalization ceases to have the correct properties needed to relate them to the Pohlmeyer invariants?

In summary: While it is not clear (to me at least) that the Pohlmeyer invariants can help to find (if it really exists) an alternative quantization of the single string, consistent but inequivalent to the standard one, can we still learn something about standard string theory from them?

Billiards, random matrices, M-theory and all that

I am currently at a seminar on quantum chaos and related stuff. You cannot enjoy meetings like these without knowing and appreciating the Gutzwiller trace formula which tells you how to calculate semiclassical approximations to properties of the spectrum of chaotic quantum systems (like Billiards and particles on spaces of constant negative curvature) by summing over periodic classical paths.

One big puzzle was, and still is to a large extent, why random matrix theory reproduces the predictions obtained by using the Gutzwiller trace formula.

In random matrix theory you pick a Gaussian-like ensemble of matrices (orthogonal, symplectic or unitary ones) and regard each single such matrix as the Hamiltonian operator of some system. It is sort of obvious why this is what one needs for systems which are subject to certain kinds disorder. But apparently nobody has yet understood from a conceptual point of view why it works for single particle systems which are calculated using Gutzwiller’s formula. But there is quite some excitement here that one is at least getting very close to the proof that Gutzwiller does in fact agree with RMT, see

Stefan Heusler, Sebastian Müller, Petr Braun, Fritz Haake, Universal spectral form factor for chaotic dynamics (2004) .

One hasn’t yet understood why this agrees, only that it does so. My hunch is that it has to do with the fact that by a little coarse graining we can describe the classical chaotic paths as random jumps and that the random matrix Hamiltonians are just the amplitude matrices which describe these jumps.

But anyway. ‘Why all this at a string coffe table?’, you might ask.

Well, while hearing the talks I couldn’t help but notice the fact that I actually do know one apparently unrelated but very interesting example of a system which, too, is described both by chaotic billiards as well as by random matrices. This system is - 11 dimensional supergravity.

I had mentioned before the remarkable paper

T. Damour, M. Henneaux , H. Nicolai Cosmological Billiards (2002)

where it is discussed and reviewed how theories of gravity (and in particular of supergravity) close to a spacelike cosmological singularity decouple in the sense that nearby spacetime points become causally disconnected and how that leads to a mini-superspace like dynamics in the presence of effective ‘potential’ walls’ which is essentially nothing but a (chaotic) billiard on a hyperbolic space.

(This paper is actually a nice thing to read while attending a conference where everybody talks about billiards, chaos, coset spaces, symmetric spaces, Weyl chambers and that kind of stuff. )

So 11d supergravity in the limit where interactions become negligible is described by a chaotic billiard just like those people in quantum chaos are very fond of.

But here is the crux: 11d supergravity is also known to be approximated by the BFSS matrix model. Just for reference, this is a system with an ordinary quantum mechanical Hamiltonian

$$H=\mathrm{Tr}(\frac{1}{2}{\dot{X}}^i{\dot{X}}_i-\frac{1}{4}[X^i,X^j][X_i,X_j])+\mathrm{fermionic}\mathrm{terms},$$

where the $X^i$ are large $N\times N$ matrices that describe D0-branes and their interconnection by strings or, from another point of view, blobs of supermembrane.

Hm, but now let’s again forget about the interaction terms. Then the canonical ensemble of this system is formally that used in random matrix theory!

Am I hallucinating ot does this look suggestive?

I think what I am getting at is the following: Take Damour&Henneaux&Nicolai’s billiard which describes 11d supergravity. Now look at its semiclassic behaviour. It is known that this is governed by random matrix theory (But we have to account for some details, like the fact that the mini-superspace billiard is relativistic. Maybe we have to go to its nonrelativistic limit.) We realize that the weight of the random matrix ensemble is the free kinetic term of the BFSS model. Therefore we might be tempted to speculate that the true ensemble of randowm matrices which is associated with 11d supergravity away from the cosmological singularity is obtained by including the $[X,X{]}^2$ interaction term of the BFSS Hamiltonian in the weight. With this RMT description in hand, try to find the corresponding billiard motion. Will it coincide with the speculation made by DHN about the higher-order corrections to their mini-superspace dynamics?

In any case, I see that apparently random matrix theory (‘like every good idea in physics’ ;-) has its place in string theory. I should try to learn more about it.

Notions of string-localization

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.

Visualization of Superstring States

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:

DPG Symposium 2004

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: