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Critical dimension: anything goes?

After Lisa's talk, we had an interesting discussion with Eva Silverstein of Stanford University, one of the most insightful and powerful young string theorists. Her statement that provoked the discussion was:

  • 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) + ...
where "#" are unimportant constants. The term "-10" comes from the contribution of conformal and superconformal Faddeev-Popov ghosts. This beta function must be zero for the theory to be conformal - which is necessary for our ability to gauge-fix the metric to the conformal gauge and obtain meaningful finite-dimensional integrals defining the loop amplitudes (and it's necessary for the unphysical modes of gauge bosons and graviton to decouple in spacetime, among many other things).

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 + ...
Now they argue that there should be other Phi-dependent terms in the action arising from fluxes, proportional to other powers of "exp(Phi)". There are at least two of them. For constant "Phi" the action reduces to "-V", i.e. minus the potential energy, and with these three competing terms, the potential energy can have another minimum at a non-zero value of "exp(-Phi)" - draw a graph of "V(Phi) vs. Phi" that first increases (1), then decreases (2), and then increases again (3) and you will find the minimum between the regions 2 and 3. Stationary points of the potential energy for scalar fields represent solutions in which "Phi" is constant.

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:

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.
The usual way to find new consistent vacua in string theory has been to start from some vacua that are known to be consistent, and continuously perturb them and "walk" along the "landscape". (As long as your walking is controllable, this approach does not prevent you from stabilizing the dilaton, which we ultimately want to do, of course.) In my opinion, one can't start from a vacuum whose existence is not well-established (or, more precisely, it clearly does not work at the zeroth order) and argue that there exist new vacua of string theory just because a conjectured effective field theory may give you a consistent picture of itself. The classical consistency of effective field theory is not enough for the existence of a background of the full string 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.

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reader Anonymous said...


Where do you think this "anything goes" mentality came from? I remember the era of string theory in the 1980's and early 1990's didn't have problems of this sort.

reader Luboš Motl said...

Good question but my answer will be a pure speculation.

I just think that we feel the need to find new and new things, and the new things whose validity could be more or less reliably established have been partially exhausted, so people logically started to work on new things whose full validity can't really be justified.

The attitude "anything goes" with respect to the work of other people is mostly a consequence of the collective thinking in which the string theorists - who are (too) nice people today - want to be nice to their colleagues, especially because the colleagues are smart and nice people, too. ;-)

Another factor is the cosmological constant / landscape effect. The landscape has been the only thing that offers, at this moment, a semiconvincing picture that a very small Lambda is possible. Many people started to like it and view the plethora of possibilities as a sexy thing.

This development is a counter-revolution against the duality revolution. A point of dualities - as well as mirror symmetry and topology change etc. - was to show the complete uniqueness of the theory (which was originally a set of methods that led to many disconnected versions of the theory). Witten originally looked for these dualities because he wanted to kill 4 of 5 superstring theories in 10 dimensions.

Some of the recent trends are going exactly in the opposite direction. Many physicists not only prefer to ignore Occam's razor - which commands not to add new structures unless they are proved to be necessary - but they really enjoy to break this rule. Many friends of ours enjoy if absolutely any no-go theorem (or any statement, for that matter) is relativized, even if the arguments against the well-defined statements are far-from-rigorous. Well, we're in a kind of postmodern era. Hopefully it's just temporary.

reader Anonymous said...

What's wrong with noncritical strings? Weyl and diffeomorphism anomalies, you say, but is there anything wrong with them? Why is strict worldsheet diffeomorphism necessary? An extra dimension appears for noncritical strings, but is that really a problem?

reader Anonymous said...

Cosmic strings are noncritial strings. QCD strings are noncritical strings. Be more open minded.

reader Luboš Motl said...

I did not say that noncritical strings are completely wrong. The Liouville-like new directions with linear dilaton are an intrinsic part of my explanations, if you read carefully.

What I said about them is the conjecture that there is no unique consistent non-perturbative completion of theories of non-critical fundamental strings, and in some more unusual theories, there is not even a perturbative definition of these theories.

Mixing it with some non-fundamental strings is a confusion. Please don't confuse the things so heavily. It is absolutely clear from the article that I talk about the fundamental stringy worldsheets whose dynamics is defined by the usual worldsheet methods, and not about some other objects that happen to look one-dimensional.

It can be that some other stringy objects - non-critical strings - are deformations of "fundamental strings", but as long as one defines the theory (and proves its existence) using the non-worldsheet description only, they are not a subject of this article. This article is about backgrounds of string theory defined in terms of non-critical fundamental strings.

reader Anonymous said...

The distinction between fundamental and nonfundamental isn't as clearcut as you seem to think. In M-theory, strings are supposed to be emergent excitations... In other words, not "fundamental".

reader Luboš Motl said...

As the readers who are able to read can testify, I am not saying that there is a clearcut invariant distinction between fundamental and non-fundamental strings.

I am talking about the question whether some backgrounds of string theory exist or not.

One possible method to show that a background exists is to describe it in terms of perturbative string theory with a worldsheet CFT.

There may exist other ways to describe backgrounds - in terms of the holographic dual, for example. But Eva et al. clearly does not claim to use different methods. She uses an argument based on a worldsheet CFT. This is her only method how she argues that the background exists, and I think that the argument is incorrect.

reader Anonymous said...

Hi Lubos,

May be you would enjoy taking a look at the radical reinterpretation of hadrons by Brian DuPraw.

Might this view dissolve some of the snags that string (and other) theoreticians are bumping in to?

The URL is


reader Anonymous said...

Stop twisting the words of others yet again. I doubt she or any other theorist for that matter actually used the phrase "anything goes". I doubt that is their philosophy either. You're misrepresenting their critical exploratory attitude as "anything goes".

reader Anonymous said...

Knowing Lubos, he would accuse DuPraw of being a crackpot. I would simply say he is misinformed.

reader Luboš Motl said...

Someone may have not used the *words* "anything goes", but the concrete *statements* were said, and I insist that my description is completely honest. Any dimension was argued to be possible, which means - sorry - anything goes.

And yes, Brian DuPraw from is an obvious crackpot. If someone uses the word "misinformed" for crackpots, it's her problem. Is not the physics equivalent?

Is it also politically incorrect to say that some statements about physics are wrong?

reader Quantoken said...

That looks like crackpot to me, too. Anything that does not make any quantitative prediction is a crackpot. Certainly by that standard super string theory, LQG, twistor, etc. etc., are all crackpots since none of them makes any none-ambiguous, quantitative, and verifiable predictions. None of them can tell us why the particle masses happen to be the values that they are. Why G is this particular small value but none-zero.

But I would try to refrain from using the word crackpot, only because this word has been abused too much and is now used by any body to call anybody else that has a different opinion. It's beginning to lose its original English meaning.

Isn't it true that the correct theory has not been found, so any of the existing M theories could be wrong? The best case scenary is one of the M is correct and all the other M-1 are wrong. The worse scenary is All M are wrong. And certain even worse is less than (M-1) of them are wrong, because that means some of them are not even wrong

It's even more ironic that John Baez composed a "crackpot index", knowing full well the particular theory he worships, being a none main stream one, may have a higher than (M-1)/M chance of being a crackpot theory, making himself a crackpotter.

It is still OK to call Brian Dupraw a crackpot since there is a zero percentage of chance that his stuff bears any truth.


reader Anonymous said...


One of your anonymous respondents
is correct: your claimed quote
of mine was inaccurate. I assume
you are interested in a useful discussion
rather than a polemic argument; if
so you should be extremely careful
to quote people accurately.
Although it may be tempting to
ascribe to someone else a
highly exaggerated
version of their view, and then
criticize it, this amounts to
an elementary logical fallacy
(a straw man argument).

It is especially surprising that
you reiterate here a claim that
the statement that any dimension
works was made. Though
there probably are other choices,
in order to discuss specific models in our discussion
I immediately specialized to 2+8k dimensions to have a simple known
consistent GSO.

Anyway, the statement is that
although the 10D-11D
backgrounds are distinguished by
low energy SUSY, and by having
an exactly flat solution in
the prescribed dimension, this
does not translate in any known
way into a prediction of string
theory. The question that arose
in Lisa's talk was whether AdS10
was a possible background
of string theory.
I suggested that we have enough
forces to fix the dilaton and
other additional dimensions starting from supercritical dimensionality. You
initially asserted that supercritical limits of string theory require a timelike linear dilaton. I corrected this by reminding us that other ingredients such as orientifolds and fluxes exist, and exert competing forces on the dilaton. Hence I expect that AdS10 is very much in the realm of possibility, in the same way that Andy, Alex and I argued dS solutions arise by balancing such a set of forces.

I did refer to a claim of a
hard prediction
of 10 or 11 as the maximal number
of dimensions as misleading,
and in the heat of discussion
with you even referred to it as dishonest.
This is because I think that
the standard for
claiming a prediction
should be rather high. More
we can refer to the
10d and 11d starting points as model-dependent possibilities that
may be motivated by other
assumptions like low energy
SUSY. This way of stating it
has the advantage of not being circular, and it admits that these backgrounds are not known to arise as top-down predictions. This is not just a technicality, since as I mentioned above we see
concretely many other possible choices with sufficient forces to for example fix the moduli. I agree with you that the
models are complicated, and deserve scrutiny.

Finally, the claim
that balancing different
orders in the string coupling
requires a breakdown of perturbation theory is also too
quick. Since the landscape of
possibilities of string theory
appears to elicit such strong
emotional responses, let us
discuss this in another context
first: Banks-Zaks fixed points
in quantum field theory. There,
one finds a
weakly coupled fixed point by balancing beta functions at different orders.
The system is still weakly coupled because the effective coupling (including species enhancements) is much smaller than 1 by tuning
the flavor and color numbers.
A similar thing is happening the flux compactifications (whether starting from the critical dimension or not) by tuning flux and other quantum numbers.

None of the arguments made
claim "anything goes", but do
in my opinion
provide strong evidence that
a lot of backgrounds go.
This may ultimately mean
string theory is more like
quantum field theory--further
choices being required to model
the part of the universe we
have access to. Even if
this is the ultimate situation,
it doesn't affect the fact
that the theory can play a useful
role in
addressing basic problems of gravitational physics (including
black holes, topology, dark
energy, and eternal inflation).

I suggest you include this
corrected description of my
views rather than
the statement you claimed
to quote, if you want your site
to be accurate.


reader Luboš Motl said...

Dear Eva,

thank you for your reply. As you can see, I was discussing exactly the example D=18 that we talked about, too. It *is* of the form 2+8k. Someone can view it as a great compromise ;-) if you admit that the dimensions different from 2+8k are hard to realize (so far?) because of the difficulties with the GSO projections (whose structure would differ from the rules in D=10), but as everyone can see, you are insisting that all dimensions of the form 2+8k are equally good predictions of string theory, and no limit on the dimension exists, even with a constant dilaton.

Everyone can read your viewpoint on black and white (or blue and yellow?). ;-) I can quote you exactly, if you wish:

"I did refer to a claim of a hard prediction of 10 or 11 as the maximal number of dimensions as misleading, and in the heat of discussion with you even referred to it as dishonest."

Incidentally, I think that you used the word "dishonest" already during Lisa's talk, immediately after I asked whether we know any AdS_{10} background in string/M-theory (what I was asking about was some variation of massive IIA, or something along these lines), and it was unrelated to any heat. You also told me that the word "dishonest" was not addressed to me, but you left me wondering whom did you refer to. Some friends of ours who communicate string theory to the public? I definitely disagree that anyone in the world is dishonest for saying that string theory implies that quantum gravitational vacua are, according to the current knowledge of string theory, obtained from bulk D=10 or D=11 underlying theories via certain kinds of compactification, deformation, and adding local defects.

Consequently, it seems that I am not exaggerating and your (and the anonymous poster's) accusations that I was twisting something can now easily be checked to be unjustified.

There is no convincing evidence for the extraordinarily strong statement about the vacua in D=2+8k for any k being (almost?) on equal footing with D=10. There is no evidence for the existence of a consistent, constant dilaton string theory background in D=18, and so far, there is no convincing evidence for the existence of a consistent AdS_{10} times something background in string theory. The usual arguments for (at least) the perturbative consistency do not work here because the perturbative expansion breaks down; the non-perturbative arguments based on spacetime supersymmetry do not work either, of course, because SUSY is absent.

Once again, it is definitely not enough to argue for the stabilization of a single scalar field to support the claim that a whole new UV-consistent background of string theory exists. In the vacua you mention, there is no reason to separate the dilaton from the rest of the excited string states. At g=0, the only problem could seem to be the dilaton tadpole, but as you turn on the string coupling, this problem spreads to all other excited states, and the infinite majority of the equations of motion are likely to be violated. Even if there were a non-perturbative consistent completion of linear dilaton theories in D=18, there exists no reason whatsoever to think that the conjectured point where the dilaton stabilizes will look like a (quasi)local theory in D=18 with reasonably large (i.e. meaningful) geometry.

My guess is that this hope is most likely incorrect because in that case, there would probably exist a way to obtain the background from D=18 flat background by some new perturbation theory.

We can perhaps say that D=18 and AdS_{10} with constant dilaton remain in the "realm of possibility", but a more influential fact is that no real calculation of something physical in these theories has been done (and I think no one can be done, at least not with the current tools, simply because these theories are constructed to be incalculable, with higher order terms guaranteed to beat the leading terms) and no consistency check for the existence of such a background has been shown in literature.

The body of evidence for the existence of string theory (consistent even non-perturbatively) and its realistic physics is only available for the backgrounds that are connected to (compactifications of) the D=10 and D=11 i.e. the spacetime-supersymmetric vacua. It is plausible, of course, that by omitting the requirement of spacetime supersymmetry, one may find many other backgrounds, and perhaps above D=11. But this is not an established result of string theory. The established result is that we have a theory that contains many backgrounds describable as compactifications of D=10 and D=11 supersymmetric backgrounds (six basic supersymmetric types of bulk dynamics). As far as the available evidence seems to suggest, this is a hard prediction of the current state of string theory, and loopholes remain in the realm of possibility or, more precisely, the realm of speculations.

M-theory revolutionized the field of string theory also because the vacua above D=10 became possible. Saying that D=18 is now equally OK is equivalent to saying that someone has done a discovery as solid as as important as M-theory. Sorry, I don't think it's the case.

By the way, there exist other speculations in the opposite direction that you say. Some friends of ours believe that *no* non-supersymmetric backgrounds of string theory can exist and be stable. Michael Dine has been trying to find various anomalies of any non-supersymmetric compactification, for example. Although it seems that these non-SUSY theories are OK perturbatively, it remains a completely open question whether they are OK nonperturbatively.

There also exist many analyses that indicate that it is very typical that non-SUSY backgrounds will always decay to SUSY backgrounds. This is what Allan would find pretty likely, and it also does not seem to agree with the existence of a large number of new vacua that have nothing to do with the SUSY vacua we know.

More generally, many friends of ours are convinced that some supersymmetry, or even low-energy supersymmetry, is a prediction of string theory. Tom Banks would probably not be the only one who would sign such a statement. And it is also necessary to say that supersymmetry is a key expectation of string theory for near future new discoveries, as long as there are any predictions whatsoever - at least uncertain ones.

I am open-minded about these questions how critical spacetime SUSY is for string theory simply because there is not enough evidence in either direction. Spacetime SUSY breaking remains an unknown of the current string theory, together with all questions that depend on it. You may believe that the existence of a local minimum of *one* function is enough for the existence of a whole new string theory, but please try live with the fact that this belief of yours does not have to be shared by others.

Non-perturbative consistency of supercritical string theory is not an established result either, and my bet would be that it is not a correct assumption. If you think that non-perturbative physics of supercritical string theory is a textbook material, as you mentioned, please give us the references for your textbook. ;-) It will be great if you prove me wrong and show that these strange new vacua - new calculable (at least in principle) frameworks - exist, but such a proof cannot be obtained by accusations from dishonesty and twisting.

Thanks for your reply once again.

All the best

reader Anonymous said...

Honestly, though, if we didn't know any better about particle physics, DuPraw will make an excellent phenomenologist. And its not really his fault if no one taught him particle physics convincingly and why should he trust physicists just based upon their authority?

reader Anonymous said...

It's the sheer height of arrogance to go calling others crackpots and to do so often. Calling others crackpots serves a cohesive function in the in group dynamics of insecure physicists. It makes them feel like geniuses and boosts their ego and marks the boundary between the in group and the out group of dummies. It also serves the function of brainwashing insecure physicists into the orthodox ideology of physics and intimidate them from independent thought out of fear of ridicule and ostracization. It also leads to close-mindedness, which suits the dominant powers in physics because their pet theories get widely accepted. Instead of having to justify their pet theories and opinions, they can accuse their opponents of being crackpots and that will do the rhetorical trick. Most innovations in science comes from people who have been accused of being crackpots because the orthodoxy is too set in its ways to recognize the improvement. This is true not only for physics, but for other fields as well, under the label of "quack", or something similar.

reader Anonymous said...

Oh, and another thing; calling others crackpots shows the utter contempt in which physicists hold for others who don't think like them.

reader Anonymous said...

Maybe two physical comments again:

[1] One should not ignore the attractiveness of non-critical string theories (in less than 10 dimensions): Starting from a critical string theory, one has to feed the theory with a lot of information on the compactified dimensions (assumptions on moduli, the geometry etc.). In contrast, in a lower-dimensional non-critical string theory, one only relaxes the assumption of a constant dilaton. One could forget about the whole compactification problem. Since much
less information is needed, Ockham would prefer non-critical strings. For this reason, I think non-critical string theories deserve some or more attention.

[2] A known example of a 6d non-critical string solution (AdS5 x S1) with a constant dilaton is the Klebanov-Maldacena background (0409133). The brane set-up, which this background is derived from, suggests the solution to be dual to a 4d N=1 QFT in the conformal window. Vice versa, a 6d string background dual to a N=1 4d SCFT can only have the structure AdS5xS1. The effective action approach at least predicts the correct structure, although the real (AdS/S1) radii could be different from the ones obtained from the effective action. The solution from the effective action might therefore be not too bad. One should actually have a look how much higher-derivative terms correct the result found from the effective action. I would be interested in seeing a similar analysis as Tseytlin and others (eg. 0108106) did for AdS5xX5 in 10d.

reader Luboš Motl said...

I find it a bit confusing to talk about Occam's razor in the context of Klebanov-Maldacena and subcritical strings. These backgrounds are not (semi)realistic models of theories in d=4.

They may be dual to an N=1 QFT in the conformal window, but as far as I understand, they're not what string theory is supposed to be - a quantum theory of 4D gravity.

You either have a 6D gravity with linear dilaton, or - via the gauge-gravity duality - a 4D quantum field theory. Neither of them is 4D theory with 4D gravity. It's irrelevant what would Occam said about these backgrounds because these backgrounds are ruled out.

AdS/CFT is more or less orthogonal to the problem of describing this real world. As far as I know, there does not exist any phenomenologically plausible (at least qualitatively, and probably) backgrounds of string theory that are (even morally) disconnected from the standard SUSY D=10 or D=11 vacua.

I think we should not forget the difference between physics and mathematics. AdS/CFT is a fascinating mathematical tool to relate two classes of superficially very different physical theories, but it is not really a tool to study unification of gravity with other forces, which is what string theory is supposed to be good for, and the only thing that makes the primary goals of string theory directly relevant for physics of this real world.

Even if these non-critical backgrounds were completely (non-perturbatively) consistent theories (which I doubt), we're very far from having candidates for theory of everything from these classes. This contrasts with the conventional models based on spacetime SUSY and D=10 and D=11.

reader Anonymous said...

"Honestly, though, if we didn't know any better about particle physics, DuPraw will make an excellent phenomenologist."

No. For an instance, what about spin and Clebs-Gordan coefficients of his theory?. What about energy spectrum?

reader Anonymous said...

Ouch! You really got schooled by a woman physicist Lubos! That's gotta be tough for you, given that you think they have smaller brains.

reader j_jall said...

There is a great discussion going on at:

[URL=]Yin_Yang of spacetime[/URL]
[URL=]Inverse Square Law [/URL]
which is trying to find what could be the structured topology of spacetime.
Go and look and leave your comment.

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