I spent yesterday afternoon down in Princeton, and attended a talk by Witten at the Institute on his work relating gauge theory and the geometric Langlands program. He says that his paper with Kapustin is done, it’s about 220 pages long, and will appear on the arXiv in Monday’s listings. So Sunday night, this link to hep-th/0604151 should start working. He’s also working on a book on the subject, where he would be the sole author.
At the start of talk, Witten noticed that many of the Institute’s mathematicians were there, and warned them that they had come to the wrong talk, since it was one aimed at physicists. Pierre Deligne got up and left, but others, including Sarnak and Langlands himself, did stay for the whole thing, although I’m not sure how much they got out of it.
Witten began by giving an outline of the talk, emphasizing six main ideas that were crucial to what he wanted to explain. He also listed as number zero the idea of geometric Langlands itself, saying he would talk about it at the end if he had time (he didn’t). The six main ideas were:
1. From a certain twisting of N=4 supersymmetric Yang-Mills one can construct a family of 4d TQFTs parametrized by a sphere. The twisting is the same sort that occurs in his original TQFT for Donaldson theory, in that case coming from N=2 supersymmetric Yang-Mills. The TQFTs he considers have an S-duality, part of a larger SL(2,Z) symmetry.
2. Compactifying the theory on a Riemann surface leads to topological sigma models, based on maps from the Riemann surface into the Hitchin moduli space MH of stable Higgs bundles. The four dimensional S-duality corresponds here to a mirror symmetry of these topological sigma models.
3. Wilson and ‘t Hooft operators of the 4-d gauge theory act on the branes of the topological sigma models. Branes mapped in some sense to a multiple of themselves by these operators are called electric or magnetic “eigenbranes” respectively.
4. Electric eigenbranes correspond to representations of the fundamental group, this is one side of the geometric Langlands correspondence.
5. The ‘t Hooft operators of the gauge theory correspond to the Hecke operators of the geometric Langlands theory although these are now defined on the space of Higgs bundles, not G-bundles.
6. Using a certaing co-isotropic brane on MH, magnetic eigenbranes give D-modules on the moduli space of bundles. The electric-magnetic duality coming from S-duality in the gauge theory relates electric and magnetic eigenbranes, giving the geometric Langlands duality between representations of the fundamental group of the Riemann surface in the Langlands dual group, and “Hecke eigensheaves” on the moduli space of G-bundles.
By the time he got to the 6th of these ideas, he was running out of time and things got very sketchy.
Witten made clear that this work doesn’t directly give dramatic new physics or mathematics, but rather just explains some tantalizing relations between gauge theory and Langlands duality, ones that were first noticed in work of Goddard, Nuyts and Olive in 1976, and pointed out to him by Atiyah way back then. The geometric Langlands program is famous among mathematicians for its difficulty (I still have trouble getting my brain around the concept of a Hecke eigensheaf..), and for its tantalizing nature, bringing together a range of different mathematical ideas (many involving conformal field theory, although these seem to be different than what Witten is doing). The new relations between this subject and supersymmetric gauge theory and TQFTs that Witten has unearthed may very well lead to some very interesting new mathematical developments in the future. Undoubtedly it will take people a while to make their way through the new 220 page paper and absorb all that he and Kapustin have worked out since last summer.
Wow.
Wow, wow.
Wow, wow, wow.
Seriously though: is there any way to give a rough intuitive feel for what a Hecke eigensheaf is? My vague memory of a talk I heard once is that it’s a way to define, given a particular curve on a given space, a sheaf on the moduli space of curves on the space? Is there a handwaving explanation of what it is and why we want it?
I have no idea, but I found this several-100-page definition by Beilinson and Drinfeld:
http://www.math.uchicago.edu/~arinkin/langlands/hitchin/
If we have to read these sorts of things as ‘background’ to the 200+ pages…well, I don’t know how old you guys are…
The first line of section 1.1.1 reads: let Y be a smooth equidimensional algebraic stack over C.
About the Witten/Kapustin paper at http://www.arxiv.org/abs/hep-th/0604151, Peter said:
“… The six main ideas were:
1. From a certain twisting of N=4 supersymmetric Yang-Mills one can construct a family of 4d TQFTs parametrized by a sphere. …”.
Does that mean that the physics of the paper is no good if conventional 1-1 fermion-boson supersymmetry is no good?
Does that mean that the paper is likely to be interesting math but vacuous as to physics?
Could that explain “… Witten noticed that many of the Institute’s mathematicians were there, and warned them that they had come to the wrong talk, since it was one aimed at physicists. Pierre Deligne got up and left …” ?
Tony Smith
http://www.valdostamuseum.org/hamsmith/
PS – A quote from Burton Richter about such supersymmetry:
“… I would say that supersymmetry is a pure “social construct” with no supporting evidence despite many years of e ort. It is okay to continue looking for supersymmetry as long as it doesn’t seriously interfere with real work (top, Higgs, neutrinos, etc.). …”.
Does that mean that the physics of the paper is no good if conventional 1-1 fermion-boson supersymmetry is no good?
Tony
Witten clearly knows perfectly well that 4d TFTs make much better physics than SUSY.
There is little truly “new” physics in the paper. What it does is show how a fair chunk of the structures that show up in Geometric Langlands appear in compactifications of this particular twisted N=4 SYM theory. He also works out the details of the twistings, the D-branes, the Wilson and ‘t Hooft operators, the various moduli spaces, etc. What one hopes when one finds new connections between mathematics and physics is that each can inform the other. This is certainly not like SW theory where Witten revolutionized the study of the invariants of 4-manifolds, but hopefully this is a good beginning. Maybe we’ll even learn a bit about gauge theory from all of this.
Some mathematicians got up and left, in all probability, because Witten was giving the talk for a physics audience. The language and backgrounds needed are often very different.
For Hecke operators, physics-wise they’re very easy. In this theory, you’ve compactified the 4D theory on a Riemann surface cross the upper half plane. There is some boundary condition (ie, D-brane) on the boundary of the upper half plane. One can put an ‘t Hooft loop in the 4D theory located at a point on the Riemann surface and parallel to the boundary of the upper half plane. From far away, this looks like a different boundary condition, giving an operation on D-branes. The data of a point on the Riemann surface and a representation of the Langlands dual of the gauge group (which is associated to the ‘t Hooft loop just as a representation of the gauge group is associated to a Wilson loop) give a Hecke operators. This Hecke operators acts on the derived category just like the ‘t Hooft loop acts on D-branes which are objects in the derived category.
On the math side, I can’t quite decipher my notes, but I do have something like this. I don’t vouch for any of it. Given a vector bundle (all associated to a given G-principal bundle, maybe?) on a Riemann surface, we can pick a point and a subspace of the fiber at that point. We can then look at the space of sections that are restricted to live in that subspace at the chosen space. This is the space of sections of a new vector bundle (read ‘coherent sheaf’, perhaps). It turns out that the space of inequivalent operators has to do with pairs of vector bundles that are isomorphic on the complement of a point (making these, in a sense, local operators). With a little work, this gets related to a loop grassmanian and eventually a double coset space which turns out to be exactly the coweight lattice mod the Weyl group, ie, a representation of the Langlands dual group.
That probably doesn’t help, does it.
Aaron, thanks for the physics definition of the Hecke operator; at least that formulation is easy to grasp. Unfortunately I don’t quite see how to relate it to the math talk I once heard, or how to make sense of your math paragraph, but I suppose that’s because of things like your “with a little work”…. 🙂
Are you sure you attended the talk for physicists?
Speaking seriously, why physicists not directly interested in mathematics of things like very-SUSY gauge theories should try to read that paper?
If the “family of 4d TQFTs parametrized by a sphere”
is constructed from “N=4 supersymmetric Yang-Mills”
then
how do those “4d TFTs make much better physics than SUSY” ?
(Which Kea says is something that “Witten clearly knows perfectly well”. )
Aren’t those 4d TFTs plagued by the SUSY in the N=4 supersymmetric Yang-Mills from which they are constructed ?
If not, then where did the SUSY go ?
Are Witten/Kapustin saying that their 4d TFTs break the SUSY,
and are they presenting this work as a physical mechanism of SUSY symmetry breaking ?
If the SUSY is physically wrong, and is not broken by their 4d TFTs, then wouldn’t their model be unrealistic as a physics model ?
Of course, as in index theory, Donaldson theory, etc, SUSY can be very useful in mathematics, so again it seems to me that the paper is likely to be very interesting math, but vacuous as physics.
Tony Smith
http://www.valdostamuseum.org/hamsmith/
Tony,
The theories being studied here are not the sort of physical supersymmetric theories that some people believe are physically realistic. They are TQFTs derived from supersymmetric gauge theories, much like the TQFT used in Donaldson theory. Witten is not claiming they are relevant to the real world, just that they lead to interesting mathematics.
Peter,
the danger of such a talk to a physicists audience by a famous personality with a rich pre-Atiyah physics past is of course that all the problems of placing sophisticated mathematics on top of conceptually underdeveloped physical ideas will agrevate the present crisis. The interesting sociological aspect is that top mathematicians seemed to have lost some of their unqualified admiration with which they looked at the physicist’s magic.
Peter said “… Witten is not claiming …[the TQFTs in hep-th/0604051]… are relevant to the real world, just that they lead to interesting mathematics. …”. I certainly agree with the claim of “interesting mathematics”.
However, as to “the real world”, look at the reaction of a very smart conventional superstring theorist, Aaron Bergman. He said:
“… There is little truly “new” physics in the paper. …
He [Witten] also works out the details of the twistings, the D-branes, the Wilson and ‘t Hooft operators, the various moduli spaces, etc. …
For Hecke operators, physics-wise they’re very easy. … This Hecke operators acts on the derived category just like the ‘t Hooft loop acts on D-branes which are objects in the derived category. …
What one hopes when one finds new connections between mathematics and physics is that each can inform the other. …
Some mathematicians got up and left, in all probability, because Witten was giving the talk for a physics audience. The language and backgrounds needed are often very different. …”.
It seems to me that Aaron Bergman, a physicist, is saying that hep-th/0604051 DOES contain what he considers to be physics (in fact, not even “new” physics, and so therefore stuff already well known to be physics),
and that he hopes that the math of hep-th/0604051 will be seen to be connected to physics (and therefore to the physical real world).
As Bert said, there is “… danger …[in]… such a talk to a physicists audience …”,
particularly if it is so physics-oriented that Witten felt called upon to have “… warned … mathematicians … that they had come to the wrong talk …”.
Again, please let me repeat that hep-th/0604051 (from what discussion I have seen about it) does seem to me to be likely to be very interesting mathematically, as was the use of SUSY in index theory, Donaldson theory, etc.
Why did the authors submit it to the physics archive hep-th instead of the math archive (and why did the Cornell arXiv moderators allow it to be placed in hep-th) ?
They could have submitted it to the math archive (since it is likely to be very good math) with a cross-listing to hep-th (since it uses some techniques and structures that are discussed in papers on hep-th).
However,
they submitted it to the physics archive hep-th which seems to me to indicate that the authors DO think of the paper as being physics, which implies describing physical reality – or has conventional superstring theory so distorted the definition of “physics” that description of physical reality is no longer part of it ?
Tony Smith
http://www.valdostamuseum.org/hamsmith/
Tony
thanks for your pertinent question. I do not see any scientific answer, again it seems to be sociological.
The moderation of hep-th/ which should be geared to particle theory is indeed highly biased.
Distler rejected a single cross listing in hep-th of my paper about the protagonist of QFT Pascual Jordan (I think it is important to know one’s own history and my article contains some little known facts which have profoundly influenced my present research on advanced QFT). It is clear that he did not approve of the political introduction (the only reason why I sent it to physics/, and not to hep-th). His note of cross rejection reads:
Dr. Schroer,
Your paper does not appear to deal even indirectly with theoretical high
energy physics, and therefore seems inappropriate for cross-listing in hep-th.
We appreciate your cooperation.
arXiv moderation
(this was his answer to my letter: Dear arXiv-moderation,
I kindly ask you to rescind your decision against cross-listing my paper physics/0603095 in hep-th. My reason is that about 80% of the essay deals with the protagonist of QFT Pascual Jordan (including some new fact which I found through the study of his old Zeitschrift fuer Physik papers) and having only one cross listing cannot be interpreted as any abuse of the cross-listing system.
with friendly regards
Bert Schroer)
I of course could have answered him that this is an updated small part of an extensive paper on Jordan which was not cross listed but rather posted some time ago on hep-th without any problem, but I got tired of this nonsense.
Speaking seriously, why physicists not directly interested in mathematics of things like very-SUSY gauge theories should try to read that paper?
You probably shouldn’t if you don’t like this sort of thing.
I should say, BTW, that much of the point of this paper is that many things in the geometric Langlands program that seem like hopelessly abstruse math from the physics prespective (of course mathematicians feel otherwise) turn out to be natural objects in the physics, in fact, objects that we’ve known and loved for years.
Tony writes:
“However, they submitted it to the physics archive hep-th which seems to me to indicate that the authors DO think of the paper as being physics, which implies describing physical reality”
I wouldn’t say that. Theories with N > 1 SUSY or topological field theories have been considered “physics” for some time, and rightly so — they have much in common with things that describe physical reality, but they are easier to solve. That they are unrealistic does not mean they are not physics.
Think of it this way: the study of Newton’s laws in the context of frictionless ramps and pulleys, which every student learns, is considered “physics”, even though it’s really just a dramatic oversimplification of reality. If you can’t understand the dramatically simplified, solvable examples, good luck trying to understand the mechanics of real-world systems! TQFTs and N=2 SUSY QFTs and such things have an analogous relationship to the Standard Model.
Yup, that’s why there is a “th” at the end of “hep-th”, not a “ph”.
Bert,
I don’t think what Witten is doing here will aggravate the present crisis, which I see as driven mainly by ideological attachment to the failed idea of string-based unification. There are problems with the way physicists approach the use of mathematics in their work, but I don’t think these are the main reason for the current sad state of particle theory.
Actually the fact that this new work of Witten has little to do with string theory, but instead relates 4d gauge theory and interesting mathematics, seems to me rather encouraging. Perhaps it will cause people to move on from failed string theory projects to something more promising. It will be very interesting to see what effect this work has. The mathematics involved is rather difficult, and the QFT techniques are pretty intricate, so I’m not so sure how many people will be willing or able to follow Witten into research in this area. But then, nothing much else is going on.
Tony,
The use of QFT to do new things in math is very much on the boundary of the two subjects, and I don’t see anything to be gained by saying it is “not physics”, or “not math”. It’s both, potentially leading to both new insights about math and about QFT, and there’s no reason for it not to be on hep-th. The problem with hep-th these days is not that it is too mathematical, it is that it’s full of worthless work about completely obscure and pointless results supposedly related to string theory. Some of this is mathematically sophisticated, most of it isn’t.
My reaction to Peter’s post was similar to that in the first five comments on this thread. Does anybody want to make an estimate of how many people in the world will have both the expertise and the sustained interest to carefully read this paper when it appears? Is there anyone reading this comment who is planning to undertake the task?
I’m not making this comment to denigrate Witten’s work on this subject. I’m genuinely curious about the number of people who are expert enough in both math and physics to work in this area.
Levi wrote:
Does anybody want to make an estimate of how many people in the world will have both the expertise and the sustained interest to carefully read this paper when it appears? Is there anyone reading this comment who is planning to undertake the task?
I think a lot of people will read it, though maybe not so many will read it carefully and in detail. Aaron’s comments make it pretty clear that the physics is nothing exceptionally weird; it should be comprehensible to many, many people. The technical details of current work in geometric Langlands seem much less approachable, at least from the perspective of a physicist, but Witten is coming at this from a physics perspective so I’m sure he will explain some of this in understandable language. Furthermore, he’s an exceptionally clear expositor and his papers tend to be surprisingly readable even when they’re rather technical.
Here’s a link to a paper dicussing the relationship between Langlands and CFT, written for physicists, evidently.
http://www.arxiv.org/abs/hep-th/0512172
dear anonymous,
the analogy with friction forgets the fact that (so far) only one strongly coupled gauge interaction is relevant for physics: QCD. From a fundamental point of view, QCD is a successfully closed issue, despite the fact that doing computations around 1 GeV is hard. Since the s and c quark masses are around 1 GeV, it seems unlikely that QCD has some hidden mathematical simplicity: more likely brute-force lattice will continue to be the only way of getting physically relevant results.
Actually the fact that this new work of Witten has little to do with string theory, but instead relates 4d gauge theory and interesting mathematics, seems to me rather encouraging.
The topological string is very important here, particularly with the identification of the compactified theory as an A- or B-model and the boundary condition technology that goes with that.
I don’t know how many people will end up reading the thing, but I can tell you that it includes me for what it’s worth. We had a semester long seminar on geometric Langlands and physics here at UT which definitely helps. That and having David Ben-Zvi around to answer questions on the math side.
Aaron,
Does the “topological string” really come into this? As far as I can tell (not having yet read the 220 page paper…), what is involved are TQFTs based on a fixed Riemann surface or Riemann surface with boundary. This is 2d QFT, not string theory if you’re not summing over genera.
This is a semantic argument probably not worth having, but in their full glory, the Hecke operators act on derived categories and the like which correspond to the possible branes in the theory. Call that whatever you want; that’s the physics.
Personally, I’d call 2d QFT on a fixed 2d background “2d QFT”, not string theory, even when there’s a boundary. But, that’s just me.
“Personally, I’d call 2d QFT on a fixed 2d background “2d QFT”, not string theory, even when there’s a boundary. But, that’s just me.
LOL! And then he goes on to say its a matter of semantics! What a ridiculous comment by Bergman. But, that’s just me.
It is evident that most people have no clue whatsoever about the Langlands Program.Also, it is absurd that people are arguing whether the paper should be in hep-th. Has anyone beside Aaron have anything concrete to say? I look forward to see the analysis of the 220 page paper (not just what Witten said in the Simon’s workshop)
Levi – thanks for the link, but you ask Does anybody want to make an estimate of how many people in the world will have both the expertise and the sustained interest to carefully read this paper when it appears?
Let me offer up the answer of Mathew 22:14 Many are called, but few are chosen
Saint John Chrysotom opined: “Among thousands of people there are not a hundred who will arrive at their salvation, and I am not even certain of that number, so much perversity is there among the young and so much negligence among the old.”
The theological evidence makes it pretty clear that only Aaron and 99 others are likely to be among the saved.
………….(the old and negligent)
most recent anonymous,
From the introduction to the paper that I linked to above:
“In recent lectures E. Witten outlined a possible scenario of how the two dualities – the Langlands duality and the S-duality – could be related to each other. It is based on a dimensional reduction of a four-dimensional gauge theory to two dimensions and the analysis of what this reduction does to “D-branes”. In particular, Witten argued that the t’Hooft operators of the four dimensional gauge theory recently introduced by A. Kapustin become, after the dimensional reduction, a Hecke “eigensheaf”, an object of interest in the geometric Langlands correspondence.”
So it really *is* a semantic argument.
Peter,
your view that the present crisis of particle physics is solely caused by a mental perversion called string theory does not hold water. It is simply not true that particle physics was healthy up to the day when sinister string theory appeared on the scene. Neither can one expect that the crisis in particle physics would evaporate suddenly if only the protagonists would see (and admit) that they are heading into the blue yonder where there will be no physical landing place.
Such paradigmatic changes in the delicate relation between the weight physical concepts in relation to sophisticated mathematical formalism do not happen out of a sudden, they rather announce their coming a long time before. The profound misunderstandings in the role of Euclideanization and that of the artistic role of the path integral (at one time in the not so distant path more than 90% of papers on gravity in the QFT setting were written in the euclidean Riemannian setting) were such warning signs.
Particle physics in particular in its local quantum physical setting has its own powerful autonomous conceptual structure and to make it jump over a ready made mathematical stick (or a more floppy string) like a dog does not have much of the chance. The first mathematician who saw that was of course von Neumann and his present day’s followers are asking local quantum physics what kind of mathematical concepts it leads to. Differential geometry you can impose on functional integrals but not on the principles underlying QFT.
The point which Tony Smith made is absolutely crucial and your struggle against the aberrations of string theory will be futile if you do not recognize the root of problems and keep looking only at the symptoms.
Bert and Peter – Conceptual roadblocks are not exactly new in physics or any other science. In physics, these are usually surmounted with the help of new experiments. The crisis in particle physics, such as it is, is due to the fact that we have neither new data nor important unexplained experimental facts. I suspect that new breakthoughs may have to wait for new data – or at least for a theory that makes an accessible prediction.
CIP (and Bert),
Sure, the fundamental problem is that physics is used to making progress by getting new clues from experiment and it hasn’t gotten any in a long time. But, even with a lack of new experimental clues, theorists still can learn new things and slowly make progress, although it’s a lot harder. To do this, I think they need to try many different things, since it is very unclear what is the right direction to go in. The kind of new algebraic thinking about the fundamentals of QFT that Bert advocates is one direction that should be pursued, I also think there are geometrical ideas motivated by path integrals that also deserve attention. The problem with the dominance of string ideology has been the way it has driven out the kind of other very different ideas that should be tried, of very different kinds.
The link works! Right. References first, hmmm: Gualtieri, Runkel et al on CFT, Hausel, plus all the obvious ones…dammit, I might have to try and read this…but I’m not expecting to get far in a hurry. 🙂
Peter previously mentioned the helpful article:
Mikhail Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, in Functional Analysis on the Eve of the 21st Century, Vol. 1, Birkhauser, Boston, pp. 119-151.
several comments here have discussed the general issue of crisis in physics, so I hope it will not be off topic to provide a link followup to Peter’s post of 25 January “Die Physik steckt in der Krise”
http://www.math.columbia.edu/~woit/wordpress/?p=334
To some extent that post and comments thereto concerned Reiner Hedrich, who was funded by the German Science Foundation to study the situation in theoretical physics and was interviewed for the article in Die Zeit.
Hedrich posted this article on arXiv today
http://arxiv.org/abs/physics/0604171
String Theory – From Physics to Metaphysics
In the earlier chain of comments Andre Bresges gave a bit of background on Hedrich and the author, Max Rauner, of the article in Die Zeit
http://www.math.columbia.edu/~woit/wordpress/?p=334#comment-8199
and Eli Rabett provided this translation
http://rabett.blogspot.com/2006/01/lost-this-is-something-of-new.html
since the earlier coverage of Hedrich and his study of (at least confusion and misdirection, if not) crisis in theoretical physics was interesting, I thought some might wish to have a look at Hedrich’s 30-page paper.
The line “Although multiverse scenarios and anthropic selection are not only motivated by string theory, but lead also to a possible explanation for the fine tuning of the universe, they are concepts which transcend the framework defined by the epistemological and methodological rules which conventionally form the basis of physics as an empirical science. ” is so exceedingly polite 🙂
Witten on page 5: A second major gap [in this paper] is that we do not shed light on the utility of two-dimensional CFT for the geometric Langlands’ program. Sigh.
Kea,
That’s the strange thing about Witten’s recent work: it seems to involve quite different uses of 2d QFT to get Langlands duality than the 2d CFT ideas that have motivated the mathematicians doing geometric Langlands (and that are discussed for instance in the Frenkel review article). Well, this means there is still a lot about this story which is not understood, which means there is still a lot to do…
Who,
Thanks for pointing that out. A quick read of it leaves me with the impression that he’s avoiding just drawing the simplest conclusion from all the material about the problems of string theory that he has gathered: it’s a wrong idea about unification and should be abandoned.
But they reference Runkel et al, and they seem to use Riemann (2D) surfaces like a skein theorist would, and take a look at Kapustin’s [41] for motivation, and and and…
Why don’t they even comment on the utility of category theory here? Is it because it’s all obvious to them?
Frankly, I’d rather be talking about geometric Langlands here…
Thank you. Perhaps you could summarise some of your views of this paper so far.
I’m waiting for the book :).
Seriously, it’s too long to completely absorb. Much of it is showing that things that intuitively ought to work really do so, like the particular twisting of N=4 SYM, the action of S-duality, and the details of the sigma model you get when compactifying. (On that point, it’s much more readable to me than Vafa’s work on the subject, but, then, I almost always find Vafa incomprehensible.) I like the sections on the action of the ‘t Hooft and Wilson operators. After various skims, it looks like the new math, if anywhere, is hiding in the book. I’m not sure what it leaves to say about the physics of this particular model, though. It’s an impressive exegesis.
What the paper does, I think, is bring Langlands to physicists. What I’m not sure it does is bring the physics to the mathematics. But that’s not a criticism; it’s probably too much to ask Witten to revolutionize every mathematical subject he thinks about, and we haven’t even seen the book yet. As usual with when physics meets mathematics, it amazes me how many of the things that snow up in the physics had already been anticipated by the mathematicians. I guess they’re smart, too.
The good news (for me at least), is that it doesn’t kill the project I’ve been thinking about with a few other people on the subject. You’d like it; it has categories in it.
🙂 Cheers
The discussion of the Witten paper reminds me of the bon mot about Schwinger.
“Other people publish to show how to do it. Julian Schwinger publishes to show that only he can do it.”
Of course, the similarity is superficial. Schwinger’s physics papers contained testable predictions – the Lamb shift, the gyromagnetic ratio, etc.
The discussion of the Witten paper reminds me of the bon mot about Schwinger.
“Other people publish to show how to do it. Julian Schwinger publishes to show that only he can do it.”
That’s remarkably inapposite. Witten is one of the best expositors in the field.
“That’s remarkably inapposite. Witten is one of the best expositors in the field.”
That’s rather like saying that Jesus Christ was one of the most important characters in the New Testament. EW completely revolutionized the way papers are written. Which makes his current tedious divagations all the more painful. We are all waiting for him to show us the way forward, and this is what we get? I mean, does anyone really believe that this is going anywhere? A sad contrast to http://arxiv.org/abs/hep-th/0106109, let alone the 1998 papers…..