Not Even Wrong

The last few weeks have seen the appearance of two papers giving very different proofs of a quite important result in algebraic geometry, resolving a question that had been open for a very long time, and in the process helping to make progress in the classification of higher dimensional projective algebraic varieties. Readers should be warned that this doesn’t have anything to do with physics, and my knowledge of this kind of mathematics is highly shaky, so I’m relying largely on second-hand information from people much better informed than myself.

The theorem in question concerns the “canonical ring” of a smooth projective algebraic variety X, which is the graded ring R(X) defined by
$$R(X)=\oplus_{n=0}^\infty H^0(X, nK)$$
Here K is the canonical line bundle (top exterior power of the cotangent bundle) of X, nK is its n’th tensor power, and $H^0(X, nK)$ is the space of holomorphic sections of the bundle nK. This is also called the pluricanonical ring.

The new theorem says that this graded ring is finitely generated, and this implies quite a few facts about projective algebraic varieties of any dimension. In particular it implies the main goal of the “minimal model program” (also known as the Mori program) for classifying higher dimensional algebraic varieties.

A proof of this theorem was claimed back in 1999 by Hajime Tsuji, but it appears that there are problems with this proof. The arXiv preprint went through many revisions, but was never refereed and published. A couple weeks ago, a group of four algebraic geometers (Caucher Birkar, Paolo Cascini, Christopher Hacon and James McKernan) posted a preprint on the arXiv claiming a proof. Yesterday, Yum-Tong Siu, a well-known complex geometer from Harvard, posted another preprint, giving a very different, more analytical, proof of this theorem. Siu notes that he has been lecturing on this proof for over a year, first at last year’s Seattle conference on Algebraic Geometry.

The mathematicians involved in creating these two proofs are well-known experts, and it seems likely that both proofs are correct. Given that there are two of a quite different nature, it now seems extremely likely that this theorem has been proved.

For more detailed explanations of this result and its implications, I’m afraid that you’re likely to require someone who knows a lot more about algebraic geometry than I do. Perhaps some of my more expert readers here can help out.

Yesterday at the KITP Michael Dine gave a very good survey talk on Prospects for a String Theory Phenomenology. It’s pretty much hype free, has a much more realistic point of view than most talks on string phenomenology that I’ve seen, and gives a good idea of the current state of the subject.

Dine claims that almost all string theorists now accept the existence of flux vacua, although only some have adopted the anthropic landscape philosophy of Susskind et. al. He describes some string phenomenologists as closing their eyes to the problems represented by the large number of these vacua, and just working on some of the more tractable examples in the hope that something will turn up that will allow them to make some connection to the real world. He himself is convinced by the Denef-Douglas argument that even identifying a single vacuum state with sufficiently small cosmological constant is impossible, so that one has to make statistical arguments. He tries to have some optimism that perhaps this statistical study will allow one to make some kind of prediction, perhaps about whether the scale of supersymmetry breaking is low or high, although so far this has turned out to be impossible.

The discussion at the end of the talk is very interesting, with Dine acknowledging that there are lots of reasons he may be barking up the wrong tree and saying that he would be happier if this turned out to be the case. He quotes Witten as telling him that what he is doing can’t work, that there isn’t much point in trying to do the calculations he is trying to do because:

A.: “You are probably not going to succeed”, and

B: “If that is all you can do it would be a great disappointment. We have this beautiful theory and we are going to get everything out of it” [i.e. it would be unpredictive].

The Ottawa Citizen today has an Op-Ed by string cosmologist Jim Cline, headlined The Big Idea That Won’t Die, with a subtitle “The fact that string theory is suddenly under attack only underscores its success as a path to a unified description of nature.” There’s a lot that it outrageous about this piece, beginning with the subtitle. Normally scientists don’t start going on about the success of their theories until they have some experimental evidence for them.

Most outrageous are Cline’s claims that Smolin’s book and mine are written in a “defamatory style”, and are “slandering” string theory. Since he gives no evidence for either of these claims, there’s not much to say about them except that they’re defamatory and slanderous.

Cline makes the standard claim that string theory should be accepted since it has legitimately triumphed in the marketplace of ideas, while clearly being rather upset about the success that critics of string theory have recently been having in this same marketplace. Somehow, overhyping string theory is a legitimate marketplace activity, pointing out its problems is not.

He makes many of the by now standard bogus claims about supposed predictions and tests of string theory. At some point I suppose I should write a FAQ about these, since the string theory hype machine keeps promoting these things in a less than honest way to a public that is not well-equipped to see through the hype. Here’s a pretty complete list of the bogus “predictions”

String theory predicts supersymmetry and extra dimensions. The LHC will test these predictions.

The problem is that there is no prediction of either the scale of supersymmetry breaking or the size of the extra dimensions; in string theory these could be anything. All we know is that the energy scales involved are at least a TeV or so, since otherwise we’d have seen these phenomena already. There’s no reason at all to expect the extra dimension scale to be observable at the LHC, even most string theorists think this is highly unlikely. There is a standard argument that the hierarchy problem could be explained by a low supersymmetry breaking scale, but this is already starting to be in conflict with the lack of any observations of effects of supersymmetry in precision electroweak measurements, and now string theorists seem very willing to say that supersymmetry may be broken at an unobservably high scale.

String theory predicts observable effects in the CMB or gravitational waves.

If you look into this, this is based on very specific cosmological scenarios such as brane inflation, and again string theory doesn’t tell you even what the energy scale of the supposed predictions is. Undoubtedly you can get “predictions” from specific models, once one chooses various parameters, but not observing these “predicted” effects would not show that string theory is wrong but just that a specific scenario is wrong, with many other possible ones still viable. There’s a new review article by Henry Tye where he claims that “string theory is confronting data and making predictions”, which isn’t true. It is only certain specific scenarios that he has in mind, he admits that other, equally plausible, scenarios (such as using not branes but moduli fields as the inflaton) make no predictions at all. For more about this, one can watch recent talks by Tye and Polchinski at the KITP.

The anthropic landscape predicts the value of the cosmological constant and will make other predictions.

The latest contribution to the anthropic landscape hype is from Raphael Bousso and is entitled Precision Cosmology and the Landscape. I’ve written many times about the problems with the cosmological constant “prediction”. Bousso claims that “there is every reason to hope that a set of 10^500 vacua will yield to statistical reasoning, allowing us to extract predictions”. He doesn’t give any justification at all for this, neglecting to mention arguments about the inherent computational intractability of this question, and the failure of the program to try and predict the answer to the one question that seemed most likely to be approachable: is the supersymmetry breaking scale low or high?

String theory makes predictions testable at RHIC.

There are lots of problems with this, but the main one is that the “string theory” involved is a different one than the one that is supposed to unify particle physics and quantum gravity.

Update: For more promotional material about string theory, you can buy a set of lectures by Jim Gates entitled Superstring Theory: The DNA of Reality. I haven’t seen the videos, but Gates is probably not indulging in the kind of claims about “predictions” of string theory being made by many others.

Update: A couple people have pointed out that a new paper has appeared pointing out that the one “prediction” of the landscape claimed by Susskind, that of the sign of the spatial curvature, isn’t sustainable. This issue was discussed here with Steve Hsu, who was blogging from a conference where Susskind made that claim, and wrote about it in more detail here. Hsu is one of the co-authors of the new paper.

Update: There’s a rather critical review of Lee Smolin’s book in this week’s Science magazine by Aaron Pierce entitled Teach the Controversy! Somehow I suspect Pierce did not write the headline, since he doesn’t seem to think much of opposition to string theory or that it is a good idea to encourage any dissent about it. In his review, he pretty much completely ignores the fact that string theory is supposed to be a unified theory, explaining the standard model as well as quantum gravity, discussing just the question of string theory as a theory of quantum gravity. This is rather odd since quantum gravity isn’t even Pierce’s specialty. I’m somewhat curious what he might think of my book, which is pretty much all about string theory’s failure as an idea about particle theory.

This evening a very interesting paper appeared on the arXiv, entitled Instantons Beyond Topological Theory I by E. Frenkel, Losev and Nekrasov. The authors are studying theories with a topological sector (supersymmetric quantum mechanics and 2d sigma models on a Kahler manifold, N=2 supersymmetric YM in 4d), but are interested in sectors of the theories that are not purely topological. I’m looking forward to reading the paper over the next few days, but it is a bit daunting. This paper is nearly 100 pages long, and it is only part I of three parts, and actually just the simplest part, that involving quantum mechanics.

HEPAP is meeting today and tomorrow, here’s the agenda. From the slides of the talk about NASA, the budget situation there for fundamental science missions doesn’t look good, and there is discussion of the upcoming NRC committee charged with figuring out which of the “Beyond Einstein” missions to allow to go forward. At Dynamics of Cats, Stein Sigurosson has been writing about this in terms of the missions being sent to Thunderdome, only one to emerge alive.

Slides from the talks last month at the conference in honor of Nigel Hitchin’s 60th birthday are available.

Joe Lykken has a nice review article about the standard model, in which he notes:

There is only one diagonal Yukawa coupling that is of order one, and that is the top quark Yukawa. But even this case is mysterious. The top Yukawa is not really of order one: it is equal to one! For example, using the 2005 combined Tevatron value for the pole mass of the top quark, the corresponding Yukawa coupling is 0.99 +/- 0.01. The entire particle physics community has chosen (so far) to regard this fact as a 1 per cent coincidence. I should point out that similar percent level equalities, e.g. supersymmetric gauge coupling unification or the ratio of the total mass-energy density of the universe to the critical density, have spawned huge theoretical frameworks bolstered by thousands of papers.

Difference is that, as far as I know, nobody has an idea why this Yukawa coupling should be one. Maybe this is a big clue…

Over at Backreaction, there’s an excellent posting about Does String Theory Explain Heavy Ion Physics?, one of the very few places to find a non-overhyped discussion of this topic.

Davide Castelvecchi has a well-done review of my book at his sciencewriter.org web-site.

At this week’s physics colloquium at Penn, Andre Brown reports that Robert Cahn emphasized that “half the particles needed for supersymmetry have already been discovered.” He also recalled a quote from another colloquium about supersymmetry: “Supersymmetry has stood the test of time. There is no evidence for supersymmetry.”

Update: A couple people have pointed out the following rather accurate cartoon.

Since string theory first became popular in 1984-5, attempts to connect it to particle physics have suffered from various problems. One of the most severe of these goes under the name of “moduli stabilization”. Six dimensional Calabi-Yau manifolds come in families, parametrized by “moduli”. The dimensions of these moduli spaces can be of order 100 or so.

Naively it might appear that string theories are characterized by a choice of a topological class of Calabi-Yaus (no one knows if the number of these is finite or infinite), and then a choice of each of the 100 or so parameters that fix the size and shape of the Calabi-Yau. According to the standard string theory ideology, this is not the right way to think, instead there is really only one string theory, with different moduli values corresponding to different states. The moduli parameters are supposed to be dynamical elements of the theory, not something parametrizing different theories.

The problem with this is that if you promote the moduli to dynamical fields, they naively correspond to massless fields, and thus give new long-range forces. So you have to explain away why we don’t see 100 or so different kinds of long-range forces, and the experimental bounds on such forces are very good. Some kind of dynamics must be found that will “stabilize moduli”, giving them a non-trivial potential. The moduli fields will then be fluctuations about the minima of this potential. If the quadratic piece of the potential is large enough, their mass will be high enough to have escaped observation.

One needs a potential with non-trivial minima, and has to ensure that the dynamics is not such that the moduli will run off to infinity. In recent years, ways of achieving this have been found that typically involve “flux compactifications”, i.e. choosing non-trivial fluxes through the topologically non-trivial holes in the Calabi-Yau. On the one hand, this seems to provide a long-standing solution to the problem of how to stabilize the Calabi-Yau, on the other hand, it appears that there is an exponentially large number of possible minima. This is the origin of the “Landscape” and the associated claims of 10⁵⁰⁰ or more possible vacuum states for string theory.

The constructions involved are famously exceedingly complex and ugly, with Susskind referring to them as “Rube Goldberg machines”, and one of their creators, Shamit Kachru, the “Rube Goldberg architect”. Very recently a new Reviews of Modern Physics article by Kachru and Douglas called Flux Compactification has appeared. It can be thought of as a manual describing how to construct and count these Rube Goldberg machines.

Many string theorists had long hoped that whatever method was found to stabilize moduli would have only a small number of solutions. Then, in principle one would get only a small number of possible models of particle physics for each topological class of Calabi-Yaus. If the number of these was finite and not too large (the known number of constructions is something like 10⁵-10⁶), then to see if string theory could make contact with particle physics, one would just have to do a moderately large number of calculations, check them against the real world, and hope that one matched. If it did, it would then be highly predictive.

The existence of the flux compactifications with stabilized moduli described in the Douglas-Kachru article has convinced many string theorists that this old dream is dead. Some have tried to claim that this is a good thing, that the exponentially large number of states allows the existence of ones with anomalously small cosmological constants, and thus an anthropic explanation of its value. The problem then becomes one of how to ever extract any prediction of anything from string theory. Small CCs are achieved by very delicate cancellations, and it appears to be a thoroughly calculationally intractable problem to even identify a single state with small enough CC.

Many string theorists are now claiming that this is not really a big deal. So what if there are lots and lots of string theory vacua, it’s just like the fact that there are lots and lots of 4d QFTs! For arguments of this kind, see recent comment threads here and here. There’s something fishy about this argument, since discussion of flux compactifications has from the beginning focused on whether it is possible to use them to make predictions, whereas no one ever was worrying about whether (renormalizable) QFTs were predictive or not.

The source of the problem lies in the combination of the large numbers of string theory vacua with their Rube Goldberg nature. Consistent 4d QFTs are characterized by a limited set of data (gauge groups, fermion and scalar representations, coupling constants), and it has turned out that among the simplest possible choices of such data lies the Standard Model. String theorists commonly describe the Standard Model as “ugly”, but it is among the simplest possible 4d QFTs, and is extremely simple and beautiful compared to something like the flux compactification constructions. One could hope that while flux compactifications are inherently rather complicated, one of the simpler ones might correspond to the real world. As far as I know there’s no evidence at all for this, such a hope appears to have nothing behind it besides pure wishful thinking. Some string theorists like Douglas and Kachru don’t seem to think this is possible, focussing instead on statistical counts of more and more complicated flux compactifications, hoping to find not a simple one that will work, but a statistical enhancement of certain complicated ones that would pick them out.

4d QFT is a predictive framework not because the number of possible such QFTs is small, but because our universe is described extremely accurately by one of a small number of the simplest of such QFTs. A few experiments are sufficient to pick out the right QFT, and then an infinity of predictions follow.

Is the QFT framework falsifiable? One could imagine that things had worked out differently, that instead of the Standard Model predictions being confirmed, each time a new experimental result came in, one could only get agreement with experiment by adding new fields and interactions to the model. It might very well be that the QFT framework could not be falsified, since one could always evade falsification by adding complexity. This happens very often with wrong ideas: they start with a simple model, experimental results disagree with this, but can be matched by making the model more complicated. As new experiments are done, if the original idea is wrong, it doesn’t get simply falsified, but the increasing complexity of the models needed to match experiment sooner or later causes people to give up on the whole idea.

This is very much what has happened with string theory. The simple models that got people excited about the idea of string theory unification don’t agree with experiment, with the moduli stabilization problem just one example. It appears one can solve the problem, but it’s a Pyrrhic victory: one is forced into working with a class of models so vast and so complicated that one can get almost anything, and never can extract any real predictions.

Douglas and Kachru do address the question of whether one can ever hope to get predictions out of this class of models, but their answer is that they can’t think of any plausible way of doing so. They mention various things that people have tried, but none of these ideas seem to work. The best hope was that counting vacua with different supersymmetry breaking scales would lead to a statistical prediction of this scale, but this has not worked out for reasons that they describe. In the end they conclude:

For the near term, the main goal here is not really prediction, but rather to broaden the range of theories under discussion, as we will need to keep an open mind in confronting the data.

This acknowledges that no predictions from this framework seem to be possible, and that continuing work in this area just keeps producing yet wider and wider classes of these Rube Goldberg machines. They are suggesting basically giving up not on string theory, which would be the usual scientific conclusion in this circumstance, but instead to for now give up on the theorist’s traditional goal of making testable predictions. They advocate not giving up on string theory no matter how bad things look, instead just continuing as before, hoping against hope that an experimental miracle will occur. Maybe astronomers will find evidence for cosmic superstrings, maybe the LHC will see strings or something that matches up with characteristics of one of the Rube Goldberg models. There’s not the slightest reason to believe this will happen other than wishful thinking, which has now been promoted to a new program for how to do fundamental science.

Update: Via Lubos, for those who don’t know what a Rube Goldberg machine is, two examples are here and here.

Bert Schroer has a new version of his paper that was discussed here earlier this year, now with the amended title String theory and the crisis in particle physics (a Samizdat on particle physics). He claims that the version reflects a change in viewpoint due to his participation in this and other weblogs, and I believe he would like the opportunity to discuss this further here. There’s also a posting about this at the weblog of Risto Raitio.

Update: Schroer, agreeing with his critics that his paper had too many typos, has sent me a corrected version, which is available here, for use until the arXiv version gets updated. He also agrees that an “s” should be a “z” in Samizdat…

Update: Schroer has a new paper out, which contains a review of AQFT and a discussion of light-front holography, with further comments on the relation to the Maldacena conjecture.

Penny Smith, a mathematician at Lehigh University, has posted a paper on the arXiv that purports to solve one of the Clay Foundation Millenium problems, the one about the Navier-Stokes Equation. The paper is here, and Christina Sormani has set up a web-page giving some background and exposition of Smith’s work. I should emphasize that I know just about nothing about this kind of mathematics, but I’m reporting on this here for two reasons:

1. It looks plausible that this really is important.

2. Penny Smith tells me that she is a regular reader of this weblog.

Update: There’s an informative news article about this on the Nature web-site.