- It's completely dishonest to say that 10 or 11 are preferred dimensions predicted by string theory because everything works in other dimensions, too. For example, I can construct AdS_{d} for any "d" with constant dilaton, and all such backgrounds exist in string theory. The dimensions "10" or "11" are not distinguished in any invariant way.
Those who know that I find it dangerous if a field of science starts to say that "anything goes", especially if there is not enough evidence for such a potentially postmodern approach, can predict that we were not exactly in a full agreement, exactly if you notice some strong words in Eva's assertion. ;-)
So let me say a couple of basic statements. In perturbative string theory, the condition "D=10" comes from the Weyl invariance on the worldsheet. The beta-function for the dilaton "Phi" (the classical values of the fields like "Phi(X)" play the role of "coupling constants" that define the two-dimensional theory on the worldsheet, and the beta function measures how much these couplings depend on the scale if you perform a renormalization group flow) contains terms like
- beta_{Phi} = #.(D-10) + #.(Nabla Phi)^2 + #.(Box Phi) + ...
So how do we guarantee it's zero? Well, the simplest solution is that we set "D=10", and the dilaton to constant. This is the canonical way to cancel the leading terms in the beta function for the dilaton. Are there other possibilities? Yes, you can set "D" to any other number, as long as your dilaton "Phi" is a linear function of spacetime coordinates in such a way that the "(Nabla Phi)^2" term cancels the "(D-10)" term. For linear dilaton, the "(Box Phi)" term is still zero. Of course, one can also add mild non-constant dilaton, e.g. one that satisfies the equation "Box Phi = 0 + #.(Nabla Phi)^2 + ...". I wrote the term "zero" for the main idea of the equation to remain transparent.
OK, Eva now claims that she can keep the dilaton "Phi" constant, and still allow "D" to be different from ten. These tricks are described in her papers
The second was written with Alex Maloney and Andy Strominger. How is it supposed to work? You re-interpret the requirement of the vanishing beta-function beta_{Phi} as an equation of motion in an effective field theory whose action contains terms like
- S_{eff} = ... + e^{-2 Phi} (Nabla Phi)^2 + ...
Do I believe that such backgrounds are part of string theory? Not really. My objection is that one can't use the low-energy spacetime effective action for the dilaton (and a finite number of other fields) unless one can show that it is a consistent truncation. This may be done in 10 or 26 dimensions but it's just not the case here. Normally, we use the spacetime effective action in the cases in which we know that there exists a flat solution of the equations of motion in which the low energy truncation simply picks the light fields - and the non-flat configurations may be obtained by continuous deformations from the flat background, at least locally (which may often be enough, assuming some degree of locality).
The situation here is more subtle. The term "(D-10)" in "beta_{Phi}" that we want to cancel is of order "1/alpha'", and it is very large for small "alpha'". Therefore, the effects (flux energy) that cancel it are also large, and drastically influence the spectrum and dynamics of the hypothetical string theory. One can't truncate the stringy spectrum into a small number of the low-lying modes because one does not really know the spectrum.
More generally, I find it hardly justifiable to start with a theory that has big problems at the zeroth order (the non-zero term "#.(D-10)" in "beta_{Phi}") and to rely on some higher-order terms to cure all these problems. This is just not how a consistent expansion can be done. A consistent expansion - in alpha' in this case - must start with a background/theory that works at the leading order, while the subleading corrections are cancelled order by order. It's OK to use the Fischler-Susskind mechanism and cancel higher-order problems by modifying the lower-orderactions etc. (which is not that different from the usual counterterms in quantum field theory that cancel the loop divergences), but it's not correct to do the opposite thing. If there is a problem at the leading order (which means a large problem - in this case a wrong central charge), we can't rely on subleading terms (which are supposed to be small) to cancel these problems because such an assumption is equivalent to admitting that the perturbation theory breaks down.
I would admit that it may be plausible that a new background like one of Eva exists "somewhere", but one can't prove its existence by starting from an inconsistent zeroth order theory. It may be nice to believe that some unusual conspiracy guarantees that the problems go away, but in my opinion, the default assumption should be that such theories that start from an anomalous starting point are inconsistent. More modestly, the people who write down the low-energy action for the metric, dilaton, B-field, and the Ramond-Ramond fields in any dimension should realize that their proposals will be very controversial. Otherwise we could also start from any inconsistent theory with any kind of tadpoles, and claim that some extra terms that undoubtedly exist will stabilize and cure the sick starting point and lead to a consistent theory.
Let me say the things differently. The only place in which one is fully allowed to use the spacetime effective action are those cases in which we know controllable backgrounds - most typically, flat space - in which the (very) low-energy degrees of freedom are (very) separated from the rest. This includes 11-dimensional M-theory (supergravity) or the five 10-dimensional string theories, but not an 18-dimensional theory that Eva took as an example.
Don't get me wrong. I don't claim that one can always exactly define the number of hidden dimensions. There are dualities and other effects - and the existence of the dimensions whose size is stringy or Planckian is a matter of convention because the Kaluza-Klein modes have comparable masses as the excited strings or black hole states and they can't really be separated. But 10 or 11 are the maximal numbers that can be decompactified and identified as "geometry" in any kind of superstring theory we know of (the word "super" here refers to worldsheet supersymmetry).
This discussion about "how many backgrounds are part of the real string theory" does not affect just these hypothetical 18-dimensional vacua. Similar questions also arise in various flux compactifications and non-critical string theory. In the context of flux compactifications, it seems to me that in many cases it will turn out that the language of effective field theory would be unjustified. In many examples, people use the same effective field theory that one uses for vanishing fluxes, even when it is known that the fluxes must change the physics considerably. Tom Banks has written various papers that warn against the unlimited belief that the universal low-energy effective theory is a correct description, and although some statements of him may be hard to swallow, he has certainly a point. See, for example:
- hep-th/0306174
- hep-th/0309170 with Dine and Gorbatov
It must be emphasized that the spectrum of low-energy states depends on the point on the "landscape" of string theory. Some states and fields may become light, and we must also satisfy their equations of motion. There is no universal low-energy effective theory for all of string theory. If one has a consistent theory with massive states, the classical solutions may simply set all the massive fields to zero, and neglect them in the analysis of the low energy effective action. But the word "consistent" is important. I don't think that one can start from an inconsistent starting point, assuming - without any justification - that a restricted set of fields form the low-energy spectrum, and only solve the equations of motion of this conjectured effective field theory - simply because we don't know whether this effective field theory is justified and whether it approximates any consistent theory. Let me write this important statement - which I think should be a weaker and less controversial observation than some of Tom Banks' comments - as a displayed "equation":
- If we want to prove the existence of a consistent, UV-complete string state or background "XY" in the landscape, it is not enough to demonstrate a self-consistent effective field theory picture of the "neighborhood" of the point "XY" in the landscape (and demonstrate the stabilization of the scalar fields), using an effective field theory whose degrees of freedom were chosen arbitrarily, assuming that the picture would work. Such a step is particularly insufficient if no point in the vicinity of "XY" can be shown to lead to a consistent and UV-complete theory.
These "jumps" require too much faith. And frankly speaking, it would be pretty discouraging if the string theorists were not only saying that they can't determine which of the 10^{300} compactifications is correct, but they could not even say anything about the total number of spacetime dimensions that these vacua naturally have.
The jump associated with a discrete change of the flux can be a "small jump" as long as the flux is dissolved over a large manifold, and this may be the right regime in which the effective description may become approximately trustworthy. But jumping from "D=10" to "D=18" with the hope that two new terms in the potential for one (!) particular field are enough to restore the consistency is just too a big jump of faith, and I have no good reasons to believe that there exist consistent 18-dimensional superstring theories constructed in this rather arbitrary way.
Eva might solve a non-linear equation for the dilaton but does she solve the equations for the infinite number of other fields in the string spectrum? She only "solves" them by believing that they can be set to zero, but if she neglects to study these other fields in her model of the landscape, there is no evidence showing that such a solution exists. In conformal field theory, we know that the massive spacetime fields should should be set to zero because otherwise their vev's contribution to the action is not marginal. But if we start from a non-conformal theory, this argument fails. One must consider the most general deformation by the excited stringy states - or equivalently, the "effective" string field theory for all these excited fields - when she tries to restore the conformal invariance. And then she's not guaranteed to find a solution for all these excited fields, I think.
Non-critical string theory
These considerations are also related to the question of non-perturbative completeness and uniqueness of non-critical string theory. Are the non-critical string theories as well established and non-perturbatively consistent as the 10-dimensional superstring theory, for example? I am skeptical about these claims. Many questions like that were discussed on Wednesday night - the discussion was led by Tadashi Takayanagi and it was a useful evening. (Incidentally, Tadashi said something about their recent work that identified the matrix model as a truncated boundary string field theory for the open strings living on the ZZ-branes.)
The conjectured duality between the old matrix models and the two-dimensional string Liouville theory is clearly a variation on the topic of the AdS/CFT correspondence. A gravitational theory (1+1-dimensional string theory) is described by a gauge theory in 0+1 dimensions - a typical example of holography. The worldsheet is the continuum limit of the Feynman diagrams. I knew this picture back in 1998, and the question was and is how exactly the details can be realized.
The relevant dual branes - replacing the D3-branes for "AdS_5 x S^5" - are supposed to be the "ZZ" D0-branes in 1+1 dimensions. (The only reason why it is believed to be the case is that the only other branes, the space-filling FZZT branes, do not have the tachyon field. In this sense, there is only "negative" evidence, not a positive one.) The only degrees of freedom living on these D0-branes are contained in a scalar tachyon in the adjoint of U(N), and its effective dynamics is the matrix model. There are several problems, however. One can't really write down the "near-horizon geometry" of the D0-branes embedded in flat space, so that one could argue that the decoupling of the near-horizon geometry in the gravitational picture is equivalent to the decoupling of the low-energy gauge fields in the open string picture. In two dimensions, these things don't work. One of the reasons is that the D0-branes are not small perturbations of the background. Moreover, we can't really say where the D0-branes are located in the Liouville spacetime. Is it at some finite point, or at infinite distance in the strongly coupled region?
Also, I don't see any good reason why the two-dimensional string-theoretical observables should be uniquely well-defined at the non-perturbative level. Unlike "d>3" where quantum gravity is hard, "d=2" quantum gravity is pretty easy. It has no local excitations. There are many detailed modifications which can perturb the action of the matrix model. There is no well-defined "strong coupling limit" of the Liouville theory. Well, the strongly coupled region is sick, and the tachyon condensate is good for elementary excitations to avoid this dangerous region.
But a well-defined strong-weak dual theory is the standard reason to argue that a theory is consistent and unique non-perturbatively in the string coupling - it works on both ends of "g" with a lot of details, and often also at some extra points in the middle, so where should the problems come from? If we avoid the strong coupling, it seems natural to say that we have no reason to think that the theory is unique and consistent non-perturbatively.
Two-dimensional string theory may be a nice toy-model that has some features of the full string theory, but we should be cautious in saying that it is a part of the "real" superstring theory which enjoys the same consistency and uniqueness standards. The evidence that the duality between the two-dimensional string theory and the old matrix models works well at the strong coupling is pretty poor - especially because of the sequences of potential contradictions found in the attempts to localize the black hole in the matrix model. This also means that if Eva constructed her conjectured 18-dimensional backgrounds by starting from the time-like linear dilaton which grows and eventually stabilizes around a finite stationary point, such an argument could be simply ruined by the possible fact that the linear dilaton non-critical backgrounds are only consistent perturbatively.
Bosonic M-theory as an analogy
In some sense, its status is pretty similar to the status of bosonic M-theory. Susskind and Horowitz conjectured (well, they were definitely not the first ones to consider this picture, but they were the seconds ones brave enough to publish it after Soo-Jong Rey) that bosonic string theory at strong coupling becomes 27-dimensional. A new dimension occurs, much like the 11-th dimension of M-theory. However, there is no Ramond-Ramond U(1) gauge field in bosonic string theory arising from the Kaluza-Klein U(1) gauge field in the 27-dimensional theory which seems as a problem. This is why they say that the extra dimension should be a line interval like in Horava-Witten theory, not a circle. The U(1) disappears. In Horava-Witten M-theory, there are anomalies cancelled by the E_8 gauge supermultiplet on the boundary. In this bosonic M-theory, there are no anomalies, and therefore the gauge group on the boundary can be anything.
(Susskind and Horowitz say, in my opinion incorrectly, that this absence of anomalies implies that there should be no gauge group on the domain walls.)
You see that things just don't seem to work and features are being fine-tuned to achieve some rough qualitative agreement. That's very far from imagining that there is a full quantitative duality. After all, there is no good reason to consider a strongly coupled limit of bosonic string theory. Bosonic M-theory has a bulk closed string tachyon, so something drastic happens with the spacetime as long as "g" starts to approach one. There is no good reason to believe that one can safely tune "g" to a very large value, while keeping the 26 dimensions. The conjectured dualities of the non-supersymmetric theories may be fun at the classical level and they may have many consistent features, but in the full physical theory, we don't know what they should mean.
If I summarize: I think that people have been recently thinking too much that "anything goes", and we should now try to spend some time to prove that most of the new classes of models and vacua that people have been proposing do not satisfy the full standards of string-theoretical consistency.
Matter-ghost separation
One of the philosophical generalizations that Eva used to suport hew viewpoint is that the only invariant information about perturbative string theory is that the total central charge is zero, and its "separation" to ghosts and matter (such as -26 and +26 in bosonic string theory or -15 and +15 for superstrings) is a matter of the background we choose, and all choices are O.K. It may sound as a proposal to unification, except that such a unification is not established. We know how to describe superstrings where the separation is -15 for ghosts and +15 for matter (or different separations in Berkovits' formalism which is equivalent) and all deformations we know how to define are those that preserve the separation of ghosts and matter. The deformations of superstrings may be added in the (0,0) picture, with a universal ghost prefactor. The fact that the Ramond-Ramond vertex operators in the integer pictures don't exist is one of the reasons why we can't honestly say that we can define perturbative string theory in the Ramond-Ramond backgrounds; this problem is probably avoided in Berkovits' formalism where all states may be written in the same "picture" i.e. with a universal dependence on ghosts.
Once again, it may be fun to say that all things that string theorists study - superstrings, bosonic strings, topological strings, supercritical and subcritical strings - belong to the same theory. But in my opinion, there is no convincing evidence for such an assertion. For example, bosonic string theory is a "different" theory from superstring theory and no plausible dynamical mechanism to connect them has been proposed as of today. Also, there are relations between topological string theory and the full string theory (topological strings compute special quantities in the full string theory), but once again, they're different theories whose Hilbert spaces should not be put together according to the present knowledge. So I would propose to avoid claims that the frameworks with different ghost structures on the worldsheet belong to the same theory because this conjecture is not supported by available evidence.
