Not Even Wrong

A wonderful long-promised paper by Dan Freed, Mike Hopkins and Constantin Teleman entitled Loop Groups and Twisted K-theory II has just appeared. They have advertised it in the past under various names such as “K-theory, Loop Groups and Dirac Families”, but their latest way of organizing their work seems to be to relabel the two-year old Twisted K-theory and Loop Group Representations (which recently has been updated, improved and expanded with new material) as “Loop Groups and Twisted K-theory III”. Working backwards it seems, they now advertise a “Loop Groups and Twisted K-theory I” as still to appear, hopefully in less than two years.

I don’t mean to give them a hard time about this. They are doing wonderful work, continually refining and improving on their results, and the paper is worth the wait. At the moment I don’t have time to do them justice by explaining much about their results or the conjectural relations that I see to quantum field theory, but I wrote a little bit about this a while back in another context. In the future I’ll try and find time to write some more entries about this material.

Also related to this is a new paper of Michael Atiyah and Graeme Segal called Twisted K-theory and cohomology which discusses the relation of twisted K-theory to twisted and untwisted cohomology.

Teleman has also recently made available on his web-site a preliminary version of notes from his fascinating talk at the algebraic geometry conference in Seattle this past summer, entitled Loop Groups, G-bundles on curves. He starts off with some philosophy he claims comes from lessons learned in working with moduli of bundles:

(i) K-theory is better than cohomology
(ii) Stacks are better than spaces
(iii) Symmetry

The first and third points I’m well aware of, and he has convinced me to spend some more time learning about stacks by his next point, which I hope may clarify some issues that confused me when I was writing my notes on Quantum Field Theory and Representation Theory. According to Teleman, the fundamental K-homology class of a classifying stack BG gives a notion of “integration over BG” in K-theory that corresponds precisely to that of taking the G-invariants of a representation. This idea has been a fundamental motivation for me for quite a while. It seems to me that one fundamental question about the path integral formulation of the standard model is “why are we looking at the space of connections and trying to integrate over it?” The K-theory philosophy gives a potential answer to this: we’re looking at the space of connections because it is the classifying space of the gauge group, and we’re integrating over it because we want to be able to pick out the invariant piece of a gauge group representation. I’ll try and write up more about this later, especially if learning some more about stacks ends up really clarifying things for me as I hope.

On a somewhat different topic, Teleman recently gave a very interesting talk at Santa Barbara entitled The Structure of 2D Semi-simple Field Theories.

John Baez has a very interesting new paper on the arXiv this evening entitled Calabi-Yau Manifolds and the Standard Model. In it he points out that the standard model gauge group (which he carefully defines as SU(3)xSU(2)xU(1)/N, where N is a six-element subgroup that acts trivially on the standard model particles) is the subgroup of SU(5) that preserves a splitting of C⁵ into orthogonal 2 and 3 dimensional complex subspaces. Furthermore, if you think of SU(5) as a subgroup of SO(10), then the spinor representation of SO(10) on restriction to the standard model group has exactly the properties of a single generation of the standard model.

Baez would like to think of SO(10) as the frame rotations in the Riemannian geometry of a 10d manifold X. The SU(5) is then the holonomy subgroup picked out by a choice of Calabi-Yau complex structure on the manifold. One way to get such an X is as the product of R⁴ and a compact 6-manifold M⁶, picking Calabi-Yau structures on both manifolds in the product. What is happening here is related to an old idea I wrote a paper about a very long time ago (see Nuclear Physics B, vol. 303, pgs. 329-342, from 1988). By picking an orthogonal complex structure on R⁴, one picks out a U(2) in SO(4) (the Euclideanized Lorentz group), and it is tempting to identify this with the electroweak U(2). This is one part of what is happening in Baez’s construction. It’s very hard though to see what to do with this within the standard gauge theory framework; this is true both for my old idea and for Baez’s newer one. Maybe string theorists can come up with some way of implementing this idea of thinking of the standard model gauge group in terms of the Riemannian geometry of the target space of a string. If so I might even get interested in string theory…..

I don’t immediately see from Baez’s paper why the hypercharge assignments come out right. I need to sit down and work that out, but it’s getting late this evening. There are some other issues his paper raises that I’d like to think about, and maybe I’ll finally get around to doing some work to see whether what I’ve learned about spin geometry in recent years has any use in this context.

I also noticed today that Baez is advertising for students to come to UC Riverside to study Quantum Mathematics. I like the term, and for many students who really care about mathematics and fundamental physics, this would be worth thinking about.

Please, commenters who want to write about their favorite ideas about standard model geometry, try and stick to any aspects of this directly related to Baez’s paper.

Lawrence Krauss has an essay in today’s New York Times about science, religion and string theory, covering much the same material discussed here in a recent posting. There are postings about this from Mark Trodden at Cosmic Variance and Lubos Motl on his blog. In comments at Cosmic Variance, Lubos tries to make the rather bizarre claim that the status of the theory of evolution is much the same as that of string theory. I don’t notice any string theorists writing in there to tell him that he is full of it.

Meanwhile, in the real world, the Kansas Board of Education has voted to change the definition of science. Krauss has been very involved in this controversy in recent years, fighting the good fight against Intelligent Design and Creationism. I suspect he’s all too aware of the danger posed by string theorists like Lubos intent on muddying the waters about the question of what is solid, testable science, and what isn’t.

Update: Over at Cosmic Variance, see some of the reaction Krauss is getting to his criticisms of string theory.

As a commenter here noted last night, and other commenters have discussed in the last posting, Steven Weinberg has just put on the arXiv an article entitled Living in the Multiverse. In it, he correctly points out that theoretical physics was immensely successful during the twentieth century as it adopted a fundamental paradigm of exploiting symmetries and quantum mechanical consistency conditions, using these to develop extremely powerful and predictive theories. Initial hopes for superstring theory were that it would lead to further progress along similar lines, but these have not worked out at all.

Faced with the failure of superstring theory to provide any new predictions based on a useful new symmetry principle or consistency condition, instead of drawing the obvious conclusion that it’s just a wrong idea about how to get beyond the standard model, Weinberg instead proposes to dump the lessons of the success of twentieth century physics:

Now we may be at a new turning point, a radical change in what we accept as a legitimate foundation for a physical theory. The current excitement is of course a consequence of the discovery of a vast number of solutions of string theory, beginning in 2000 with the work of Bousso and Polchinski.

What Weinberg sees as “excitement” is what some others have characterized as “depression and desperation”. His “radical change in what we accept as a legitimate foundation for a physical theory” seems to be to give up on the idea of a fundamental theory that predicts things and instead adopt the “anthropic reasoning” paradigm of how to do physics. Weinberg goes through various examples of his own recent work of this kind, announcing that the probability of seeing a vacuum energy of the observed value is 15.6% (this seems to me to violate my high school physics teacher’s dictum about not quoting results to insignificant figures, but I’m not sure how you’d put error bars on that kind of number anyway). He also quotes approvingly recent anthropic work of Arkani-Hamed, Dimopoulos and Kachru, as well as that of his colleague Jacques Distler. All he has to say about the underlying string theory motivation for all this is that “it wouldn’t hurt in this work if we knew what string theory is.”

In his final comments he acknowledges that this new vision of fundamental physics is not as solidly based as the theory of evolution. Describing the strength of his belief in it, he says “I have just enough confidence about the multiverse to bet the lives of both Andrei Linde and Martin Rees’s dog.” One can’t be sure exactly what that means without knowing how he personally feels about Andrei Linde, or cruelty to innocent dogs.

Weinberg’s article is based on a talk given at a symposium in September at Cambridge on the topic “Expectations of a Final Theory”. I haven’t been able to find out anything else about this symposium, and would be interested to hear any other information about it that anyone else has. The article will be published in a Cambridge University Press volume Universe or Multiverse?, edited by Bernard Carr (the president of the Society for Psychical Research), about which I’ve posted earlier here.

I’m curious whether this Cambridge symposium was one of the infinite number of such things funded by the Templeton Foundation. Next week the Vatican will be sponsoring a Templeton-funded conference held in the Vatican City on the topic of Infinity in Science, Philosophy and Theology. It will feature a talk by Juan Maldacena on “Infinity as Simplification”, and is part of a larger Vatican/Templeton project called Science, Theology and the Ontological Quest. This project is designed to promote the vision of scientific research outlined by Pope John Paul II in two encyclical letters, including the rule that scientific research must be “grounded in the ‘fear of God’ whose transcendent sovereignty and provident love in the governance of the world reason must recognize.”

Update: Lubos Motl has some comments on the Weinberg article. This is one topic on which we seem to be in agreement.

Update (much, much later, May 2022): Rereading this posting many years later, I decided to check on the question of Templeton funding raised here. The Weinberg article was published in the volume Universe or Multiverse?, and the Acknowledgements section there has:

First and foremost, I must acknowledge the support of the John Templeton Foundation, which hosted the Stanford meeting in 2003 and helped to fund the two Cambridge meetings in 2001 and 2005.

There’s a new paper out tonight by Nikita Nekrasov entitled Lectures on curved beta-gamma system, pure spinors, and anomalies. Motivated by questions about the covariant superstring quantization method being studied in recent years by Berkovits, Nekrasov considers a sigma model with target space the space of “pure spinors”. For more about pure spinors I suggest consulting “Spin Geometry” by Lawson and Michelson, but in general they are a subspace of the full spinor space with remarkable properties. In R²ⁿ, a pure spinor determines a complex structure on R²ⁿ, one that doesn’t change when you multiply the spinor by a complex scalar. Furthermore, modding out by the action of the complex scalars, the space Q(2n) of projective pure spinors is a Kahler manifold, isomorphic to O(2n)/U(n). This is a projective algebraic variety, and geometric quantization of it gives back the space of spinors. There’s quite a lot of beautiful geometry in this story.

Unfortunately, in the Berkovits story the target space of the sigma model is not Q(2n), which is smooth and has every nice property one could ask for, but the space of pure spinors themselves which is a cone over Q(2n), and has a singularity at the origin. How to handle this singularity is the problem Nekrasov is addressing. This is a rather technical business, one about which I’m no expert (and I’m not sure there are many experts out there on this topic other than Berkovits and Nekrasov).

At the end of his paper Nekrasov makes what appear to be some remarkable comments. He describes two ways to deal with the singularity. The first is to just remove it and work with a non-compact target space. In his paper he shows that this removes certain potential anomalies, but he comments that doing this causes “some unclear issues with the definitions of string measure”. The second way to deal with the singularity is to blow it up, working with the total space of a complex line bundle over Q(2n). Nekrasov claims that if you do this the superstring “would cease to be consistent beyond tree and one-loop level, thereby killing at once the landscape [48] problem.” The reference is to Susskind’s anthropic landscape paper, although Nekrasov refers to Susskind as “Sussking”.

I’m assuming this is some sort of perverse joke, since if the superstring is inconsistent on flat ten-dimensional space, there’s every reason to believe it’s also going to be inconsistent on curved 10d spaces and what gets killed is not just the landscape, but the whole idea of unification based on the 10d superstring. Nekrasov goes on to end with the comment that “This is of course one of the unrealized, so far, hopes to solve some pressing predictive issues of string theory by capitalizing on its unusual, from the conventional quantum field theory point of view, perturbation theory”, referring to a 1987 paper of Greg Moore that I don’t have access to at the moment.

I’m curious to hear what people more expert in this subject think of all this. There are various relevant blog entries: Robert Helling and Urs Schreiber on Nekrasov’s talk a couple weeks ago about this in Hamburg, a recent posting by Jacques Distler, and a report on a talk by Berkovits at the KITP in August by Andrew Neitzke. For some relevant papers on the arxiv, see a paper by Berkovits and Nekrasov from earlier this year as well as quite a few papers by Berkovits and other collaborators written over the last few years.

Update: A commenter wrote in to point out that the Moore paper is available on-line as a scan of the preprint at KEK.

After my post appeared, there were later posts on this topic by Jacques Distler and Lubos Motl. Lubos seems to agree with me that Nekrasov’s comment about an inconsistency in the quantization of the superstring in flat 10d killing the landscape is rather bizarre, since such an inconsistency would probably then hold in all backgrounds.

Funny, but if you look at trackbacks for the Nekrasov paper, they’re there for Distler and Motl’s blog entries but not mine, even though mine appeared earlier. I guess whatever the moderation policy is for trackbacks these days, I’m in a separate category.

Update: After inquiring with the arXiv about what was going on about this trackback, I just heard that it has been posted. It’s still unclear to me what their moderation system is.

Last week the Perimeter Institute ran a Workshop on Cosmological Frontiers in Fundamental Physics and someone wrote in to point out to me that the talks are now available on-line. Much of the workshop was about mainstream cosmology, especially the more speculative ideas about inflation and what signals might be found in the CMB.

The particle physics component was heavily weighted towards Landscapeology (talks by Denef, Kachru, Kleban), with a workshop-ending talk entitled “50 Years since the LHC” by Nima Arkani-Hamed. Arkani-Hamed’s talk was supposed to be a prediction of what things would look like 50 years after the LHC, and he ended it with the prediction that “the LHC will put the last nail in the coffin of mono-vac theories” and that “the Landscape will be with us to stay”. Along the way he went over various possibilities for what the LHC might see and their implications for whether the cosmological constant and the weak scale are anthropically determined. He said that he believed the cosmological constant was anthropically determined, and half the time he thought the weak scale was too, the other half of the time he thought it wasn’t. He argued for giving up on coming up with new mechanisms for electroweak symmetry breaking since “working on the (N+2)nd variant on the (N+1)st model of EWSB is not worth it”, saying that instead people should participate in the LHC Olympics and work on the “inverse problem” of figuring out from LHC data the 105 parameters of the MSSM or some other model of beyond standard model physics.

Near the beginning of his talk he gave a graph representing (as a function of time) the average string theorist’s view of the probability that string theory could be used to calculate standard model parameters. This started out near 1 in 1985, dropping to a small number in 1995 after the duality revolution showed that strongly coupled strings didn’t get rid of the wide range of possible backgrounds, and further dropping to “a number close to the fine tuning of the cosmological constant” in 2000 after the advent of the Landscape and the non-zero cosmological constant. The same graph also included a plot for the views of “clueless popularizers and science journalists”, which had only recently started to head down slightly from 1. He claimed that such people are ten years behind the times and that it is “not going to be pretty” when they catch up with the current views of string theorists and realize the theory can’t predict anything.

While at Perimeter, (according to Lubos Motl) Arkani-Hamed was talking to LQGer Laurent Freidel about Doubly-Special-Relativity in three dimensions. It’s going to be a lot of fun to watch what Lubos has to say if and when some of his senior colleagues start working on LQG, something that seems entirely possible since string theory is now moribund, whereas LQG is in a much livelier state.

Landscapeology still seems to be making headway at taking over particle theory and ensuring that it becomes a pseudo-science. If you’ve got $250 and can get to an access grid facility, next week you can participate by video conferencing in a workshop at Ohio State on Strings and the Real World, which will have one day out of three devoted to the Landscape. I would have thought the title of the workshop would get into trouble with false advertising laws, since one thing that is clear is that absolutely none of the talks will be even slightly relevant to the real world.

For a horrific vision of where particle theory is headed, check out the website of the String Vacuum Project. The idea seems to be to get particle theorists spending their time developing software to do numerical computations searching amongst the infinite variety of the Landscape to find something or other. The section of the web-site on the connection of any of this to real particle phenomenology remains to be written.

The latest Cosmic Log column on msnbc.com concerns Lawrence Krauss’s new book Hiding in the Mirror and the author asked Krauss a question I’m expecting that physicists will be hearing more and more often as time goes on: “Why is string theory science but intelligent design isn’t?”

Krauss gives a response that isn’t completely convincing. He says that “the difference is that Ed Witten and the other good string theorists will, if an experiment comes along that demonstrates that supersymmetry isn’t discovered in a definitive way, be the first to say the theory is wrong.” This isn’t really true. Since the scale of supersymmetry breaking is unknown, one can’t hope to experimentally definitively show supersymmetry is not there. And the question at issue is string theory, not supersymmetry. Will string theorists abandon the theory when supersymmetry is not found at the LHC? We’ll see in a few years, but I already see them hedging their bets and many undoubtedly will not see the lack of supersymmetry at LHC energies as proving string theory wrong.

The behavior of string theorists that Krauss identifies as most like religion is the argument that “the theory is so beautiful it must be true.” I actually don’t hear many string theorists making this argument these days. If the theory actually were beautiful in the sense of providing some impressive new understanding of physics in terms of some simple, compelling mathematical or physical idea, that actually would be a good reason for believing in it, although not a completely conclusive one. All attempts so far to connect the theory to real physics lead to hideously complicated and ugly constructions. Some string theorists such as Susskind, argue that one should believe in string theory anyway, and it is this argument which seems to me to be more like religion than science. It’s my impression that Susskind and others are believing something for sociological and psychological reasons, something for which they have no rational, scientific argument. This behavior is not distinguishable from that of many of the intelligent designers, and if it becomes more widespread it ultimately threatens to do real damage to the public perception of science in general and theoretical physics in particular.

Krauss gets closer to the real difference between string theorists and intelligent designers when he says that string theorists “are trying to come up with predictions that actually do something”. More sensible string theorists are well aware that what they are doing isn’t going to be part of science until they figure out a way to use it to make real predictions that can be tested. In general, given a new speculative idea, it will not be obvious how to figure out all of its implications and see whether it can lead to real predictions. It can take years of work for this to become clear, and this sort of work is definitely science. On the other hand, if after a lot of work, there still is no indication that an idea can produce predictions, the continued pursuit of it at some point stops becoming science and starts becoming something more like religion. Susskind and other anthropic landscapeologists have already gone past this point: they have no plausible idea about how to ever get real predictions out of their framework. String theorists who argue that the theory is still too poorly understood, that more work is needed to understand whether there is some way around the radical non-predictivity implied by the landscape, are nominally still doing science. But at some point, as years pass without any progress in this direction, and evidence mounts that hopes for ways to get predictions aren’t working out, this activity stops being science and it too starts being a non-scientific activity pursued for sociological and psychological reasons. We’re close to that point, if not already past it.

Update: There’s a defense of string theory against the charge that it’s like intelligent design over at Kasper Olsen’s blog. I don’t find it very convincing, since it doesn’t address at all the question of how string theory is ever going to do what a real science is supposed to do: make falsifiable predictions. Much of Olsen’s list actually strikes me as a recitation of a catechism of supposed reasons why string theory is so wonderful, rather than a serious scientific argument. Some of these are also highly dubious (e.g. “the Standard Model can be reproduced in a very simple way”), they’re things that one has to be a true believer to say, since they really don’t accord with reality.

One commenter (Gavin), gave a very good reason for distinguishing string theory from intelligent design: “the former is trying to explain something that is already explained, while string theory is trying to solve a mystery” and he correctly notes that while string theory’s scientific credentials may be weak, the problem is that there aren’t really good alternatives (LQGers may argue with this…). John Baez’s comment about the relationship of math and physics was also quite nice.

Sir Michael Atiyah is here in the United States this month. Evidently he was at the Institute in Princeton last week during the Deligne conference, talking to Witten. Last Friday he gave a public talk at an AMS conference at the University of Nebraska on The Nature of Space, and will be giving another one tomorrow in Santa Barbara on the same topic.

Yesterday at the KITP he gave a talk on his own very speculative ideas about physics entitled Does the Universe Have a Memory?. He began by subtitling his talk “Crazy thoughts of an old man”, and noting that when he was at the Institute Witten had listened to him politely and then after a micro-second given him four reasons why his ideas wouldn’t work. One motivation he gave was that the current situation of string theory was somehow like Ptolemaic epicycles, with a fundamental idea that would drastically simplify everything still missing.

The speculative idea he was promoting was that perhaps quantum mechanics should be changed so that the future depends not just on the present, but on the history of the system during some short period before the present. So dynamics would be somewhat non-local in time. He hopes for some connection to the Connes version of the standard model, but this was all very vague. All in all, I fear that I wish Atiyah would go back to working on the relation between K-theory and physics….

Update: In a comment Doug provides a link to Atiyah’s public lecture.

There have been several recent interesting talks at the KITP in Santa Barbara as part of their program this semester on Mathematical Structures in String Theory. Last week Greg Moore gave a beautiful talk on Mathematical Aspects of Fluxes. It’s a sad commentary on the state of the field that he felt it necessary to begin his talk by apologizing that the work he was describing didn’t seem to have anything useful to say about the Landscape.

This week Graeme Segal spoke about On the Locality of the Statespace of Quantum Field Theory. He is trying to understand the right way to axiomatize the way in which the Hilbert space of a QFT depends on the boundary of space-time in a local fashion. In his talk he worked out things for the case of a free scalar on an arbitrary manifold. Also this week, Peter Teichner and Stephan Stolz spoke on their work on generalized cohomology and QFT, motivated by trying to understand the relation of elliptic cohomology and conformal field theory. Teichner gave the first part of the talk on Tuesday, Stolz gave the second part on Thursday. For more about their work, see some earlier comments of mine, their paper What is an elliptic object?, and an earlier survey talk by Teichner at the KITP.

The latest issue of Nature has an essay on Hermann Weyl by Frank Wilczek. The essay mainly advertises Weyl’s book Philosophy of Mathematics and Natural Science, originally published in 1926, but updated for the English translation in 1949. I’m embarassed to say I’ve never read this, despite my fascination with Weyl, so I guess I better go out and get ahold of a copy.