The String Coffee Table

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 $\Phi_0$.

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Update 3rd May 2004

I have now some LaTeXified notes.

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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!

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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)

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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

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:

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

then, due to the associativity of the star product this can equivalently be rewritten as an operator equation

because

(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

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:

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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 $S(\Psi)$, 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 $\Psi_\mathrm{cl}$.

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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

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

This is manifestly a deformation of the equation of motion

of the string described by the state $\psi$ in the original background. Hence it is consistent to interpret

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

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

and that in operator language this reads equivalenty

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

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 $\mathbf{d} + \mathbf{\omega}$ 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

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

But precisely this is what does happen in open string field theory in the formal integral. There we have

All this suggests that one should think of the deformed BRST operator as morally a gauge covariant exterior derivative:

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.

(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 $E_{10}/K(E_{10})$. 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 $A = (a_{ij})$ and is generated from the $3r$ Chevalley-Serre generators

which have commutators

(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 $f_i$.

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 $h_i$:

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

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

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 $[T^a,T^b] = f^{ab}{}_c T_c$.
• If $A$ is semidefinite, then the KM algebra is an infinite affine Lie algebra, or equivalently a current algebra in 1+1 dimensions: $[j^a_m, j^b_n] = m \eta^{ab}\delta_{m+n,0} + f^{ab}{}_c j^c_{m+n}$.
• 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

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:

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

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

then it is very easy to check, using the defining relations of the KM algebra, that

Using the fact that the non-vanishing inner products of the algebra elements are

one finally finds the Lagrangian describing geodesic motion of the coset space:

(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.

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 $\exp(i \lambda)$ induces of course a modification of the constraint

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

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

If we think of the ‘test field’ $\psi$ as that representing a single string, then it seems that one has to think of

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.

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

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

What is the impact in this context of the fact that the Pohlmeyer holonomies

are Virasoro invariants? We have a vague understanding (hep-th/9705128) of what the map from $A$ to $|\psi\rangle$ has to do with string field theory. Can something similar be said about the Pohlmeyer map from $|\psi\rangle$ to $A$?

What is the meaning on the string theory side of a Gaussian ensemble in $u(N)$, 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

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?

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

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.