I recently heard from Hirosi Ooguri that a transcript of a long conversation with Witten held at the time of his Kyoto Prize award has just appeared in the Kavli IPMU Newsletter. It’s a truly fascinating document, giving some great insights into Witten’s work at the boundary of math and physics and how he sees the state of ideas in this area. It’s wonderful that he was induced to give such a thoughtful and extensive explanation of both the history and significance of these various topics.

Just to pick out a couple examples, the discussion of geometric Langlands describes a lot of detailed history that I was unaware of. I had noticed that in their first massive (still unpublished) paper that started the subject, Beilinson and Drinfeld credit Witten with “the main idea” (I wrote about this in detail here). But there was nothing in what Witten has written that corresponds to what they did, and experts I talked to didn’t see how this came from Witten. Witten tells the true story this way:

Actually, the very little bit of what Beilinson and Drinfeld were saying that I could understand made me wonder if the work of Nigel Hitchin would be relevant to them, so I pointed out to them Hitchin’s paper in which he had constructed commuting differential operators on the moduli space of bundles on a curve. Differently put, Hitchin had in a certain sense quantized the classical integrable system that he had constructed a few years before. Although I understood scarcely anything of what Beilinson and Drinfeld were saying, I did put them in touch with Hitchin’s work, and actually, in their very long, unpublished foundational paper on geometric Langlands that you can find on the web, Beilinson and Drinfeld acknowledged me very generously, far overestimating how much I had understood. All that had really happened was that based on a guess, I told them about Hitchin’s work, and then I think that made all kinds of things obvious to them. Maybe they felt I knew some of those things, but I didn’t. But anyway, there were ample reasons in those years to think that geometric Langlands had something to do with physics, but as you can see I still couldn t make any sense out of it.

He also describes how he came to the idea of interpreting geometric Langlands as a form of mirror symmetry, inspired by things he learned from David Ben-Zvi at lectures about the Langlands program bringing together mathematicians and physicists at the IAS.

He contrasts his work in recent years relating Khovanov homology and gauge theory with the geometric Langlands work, saying that he thinks the Khovanov homology ideas are in a form such that mathematicians are more likely to be able to appreciate their roots in gauge theory:

I think it s actually very difficult to see what advance in the near term could make the gauge theory interpretation of geometric Langlands accessible for mathematicians. That’s actually one reason why I m excited about Khovanov homology. My approaches to Khovanov homology and to geometric Langlands use many of the same ingredients, but in the case of Khovanov homology, I think it is quite feasible that mathematicians could understand this approach in the near future if they get excited about it. I believe it will be more accessible. If I had to bet, I think I have a decent chance to live to see gauge theory and Khovanov homology recognized and appreciated by mathematicians, and I think I’d have to be lucky to see that in the case of gauge theory and the geometric Langlands correspondence – just a personal guess

About the geometric Langlands story, he thinks there is still much to be understood, including its connection to conformal field theory:

In fact, part of the original work of Beilinson and Drinfeld on geometric Langlands has still not been understood to my satisfaction. Here I have in mind the use of conformal field theory at what they call the critical level (level -h, where h is the dual Coxeter number) to construct the A-model dual of certain B-branes (the ones that are associated to opers, in the language of Beilinson and Drinfeld). Davide Gaiotto and I obtained a few years ago a reasonable understanding of what electric-magnetic duality does to the variety of opers, but I still do not really feel I understand its relation to conformal field theory. However, in the last few years physicists working on supersymmetric gauge theories in four dimensions and their cousins in six dimensions have made several discoveries involving the role of conformal field theory at the critical level, so the time may well be right to resolve this point.

Among the many other highly interesting comments, one was Witten’s take on the possible connection of quantum field theory to number theory. He has a long history with this, going back to conversations with Atiyah in 1977 in which Atiyah suggested some connection between Langlands and Montonen-Olive. Witten writes

I was skeptical about Montonen-Olive duality, I didn’t seriously try to relate it to Langlands duality and I didn’t try to learn what Langlands duality was. I did not learn anything more about these matters until the late 1980s. Then I learned just superficially about the Langlands correspondence. If one knows even a little bit about the Langlands correspondence and a little bit about conformal field theory on a Riemann surface, one can see an analogy between them. I wrote a paper that was motivated by that but then I realized that my understanding was too superficial to lead to anything deep, so I abandoned the matter for a number of years.

Later of course, he followed work on geometric Langlands and ultimately found the connection to gauge theory he worked out with Kapustin. As far as current prospects for connections to number theory, he has quite a few comments, but thinks the subject is still a dream that is not ripe:

For me personally−it’s a dream that eventually number theory would make contact with physics some time, but I doubt it will be soon. There are all kinds of areas where specific number theory formulas appear in physics, and these may be clues that the dream will come true one day. But to really get me excited, somehow the number theory would have to enter the physics in a more structural way. I m not that interested in a specific formula that comes out of a physics calculation in a more or less ad hoc fashion. Number theory would have to be more integrated with the physics to get me excited, and I don t see that happening soon. In my work, I concentrated on the geometric form of the Langlands correspondence because I could see that there was hope to really understand it in the context of the physics-based tools that were at hand. There might be something like that one day for the Langlands correspondence of number theory, but probably a lot is missing and we do not know what has to happen first.

This just gives a taste of the conversation, there is lots, lots more there, on a wide range of topics. Highly recommended reading for anyone with an interest in this area, I’ve never seen anything like it.

Thanks for the link.

Are there also interviews of the previous winners? I tried to find them but without success…

Peter, Your links are always useful but none more so than today’s link to the December 2014 issue of the IPMU News with its interview about physics and mathematics at IPMU of Edward Witten by Hirosi Ooguri. I would not have read this otherwise and could not put it down before reading all seventeen pages. Edward always speaks perspicuously and Hirosi cleverly steers the interview. The distinguished theoretician Hitoshi Murayama put mathematicians and physicists in adjacent offices at IPMU.

What can you say about this interview? Just amazing; it has it all and it’s of historical importance.

Maybe the only thing that is missing is a reference to the quest for the identification of the underlying symmetry of String theory responsible for its rigid structure that would further restrict the theory. Finding this symmetry is of immense importance since it will shed light to its underlying structure (potentially with phenomenological implications).

A glimpse of this symmetry can be seen in the Higher Spin algebra of Vasiliev’s theory in AdS which is expected to be a sector of String theory in its tensionless limit (where this symmetry would be unbroken).

Witten himself was one of the first people that motivated this in the context of Higher_Spin/ Free_CFT correspondence in his 2001 talk at the John Schwarz 60th birthday symposium.

http://theory.caltech.edu/jhs60/witten/1.html

Thanks for the link Peter and your positive reaction to the interview parallels mine. I’m sure you must have been asked this before, but I’m curious how you reconcile your appreciation of the fruitful ideas coming out of string theory with the otherwise polemical views you have of string theory as a science.

Tim Nguyen,

I really don’t want this to turn into yet another of the same tedious discussions about math and string theory that have often taken place here. One interesting thing about Witten’s commentary is that he makes very clear that no one really understands the origin of a lot of the fascinating structures he’s talking about or what the right way to think about them is. As he points out, these things make clear that there is much we don’t understand about QFT, much less string theory (which is becoming a very ill-defined term). To a large extent, the mindset of theorists for twenty years has been dominated by the conjecture that the big explanation is some mysterious 11d theory, and that if understood it would give a unified theory. I don’t see any progress on this, or actually any good reason to think that 11d is the way to a unified theory (quite the opposite…). Thinking about these mysteries with more of an open mind about what the new structures are that we’re looking for seems like a good idea, and I think Witten at his best (as is on display here) is often doing that.

Peter,

Apologies if these are repetitive comments (perhaps you could link me to relevant past discussions if you so wish), but it seems to me that string theory being on the one hand unsatisfactory as a science and on the other, being a rich source for mathematics, can be seen as independent issues. The same tree can produce both good fruit and bad fruit and we reap the benefits so long as we know which is which. Perhaps this was implicit in your remarks.

What applications have arisen from the mathematics invented by string theorists? If there aren’t any, how can it justifiably be called good mathematics?

Ian Welland,

Most mathematicians don’t think “good mathematics = applied mathematics”, and if you fired mathematics faculty whose research in recent years has not had significant applications, many of the best research universities in the world wouldn’t have much of a math department anymore. Even if you believe “good=applied”, pure math research often only finds applications very much later, even centuries later.

I’ll delete any more comments of this kind though. If you’re not interested in what Witten is talking about that’s fine, just ignore it.

Another thing that I would definitely expect to hear in a discussion like this and it was skipped for some reason is a reference to the Monstrous moonshine. This can also be linked to Witten’s pioneered work on AdS₃_QG/Monstrous_CFT₂ correspondence.

Three-Dimensional Gravity Revisited (http://arxiv.org/abs/0706.3359)

Woit, Excellent interview with Witten.

But, didn’t you find his outlook, while honest, a bit discouraging? e.g. gauge theory and Langlands won’t be appreciated by mathematicians anytime soon, number theory and physics won’t have any unification in any foreseeable future.

Justin,

Not at all, I found his comments realistic and actually encouraging. The relationship of mathematicians to gauge theory is a complicated and long story. It never has been the case that mathematicians are able to adopt physicist’s techniques for dealing with quantum Yang-Mills above 2d, for geometric Langlands or anything else. What mathematicians have gotten out of this is new ideas and inspiration about how to pursue their own methods (an old example was Seiberg-Witten, a nice new example is in Witten’s comments about Kevin Costello’s work). Ultimately I think new ideas will come here from each side doing what it does best, not one side adopting the other’s machinery.

About number theory and physics, I see lots of fascinating possibilities and connections, but agree with Witten that there is still no compelling picture of exactly what is going on here. I’d have been quite surprised if he claimed that major progress was imminent (few people work on this). What would have been discouraging would have been if he said he felt there was nothing there, that this couldn’t lead anywhere, but that’s quite different than what he was saying.

I believe that Beilinson and Drinfeld gave Witten so much credit for helping them because they recognize that true genius is often the ability to see connections where most people do not. Witten has demonstrated this genius on many occassions.

Pardon my ignorance, but I’m kind of curious at the hero worship of Witten in the physics community. Clearly a great mathematician, they don’t give you a Fields for nothing. My orals in grad school were presenting a JDG paper of his, brilliant bit of work. Took reading a whole book by John Roe to actually understand what he was doing. But physics? I understand that a big chunk of what people in physics work on, and have been working on for several decades, comes from Witten’s work, but what of it has actually been shown to be true? What has he worked on that has been experimentally verified? I’ve been to some neat talks on topological quantum field theory, fun stuff, but any known relation to the real world? Just curious…

Jeff M,

There really have been no major advances in theoretical particle physics that have been experimentally verified in now over 40 years. If you look at Nobel prizes, there have been no Nobel prizes for theoretical particle physics work done post-1973. So, yes Witten has not done anything at that level, but to be fair, neither has anyone else (he started grad school around 1973).

But he really has revolutionized how we think about quantum field theory. TQFT is mostly his doing. Anyone who looks at the history of this I think has to be in awe of how much he has done, with much of it way beyond what anyone else in the field was doing or capable of doing. The outcome of this so far has worked out much better for mathematics (his Fields medal is well-deserved, and that was awarded before Seiberg-Witten and much else) than for physics. But it’s entirely possible that when progress finally comes in the future, his ideas will be part of it. There really is a history of overlap between good physics and good mathematics, so the mathematical qft successes could very well be matched by success on the physics front.

He’s been I think overly enthusiastic about string theory unification, and slow to acknowledge that that idea hasn’t worked out, but at least you don’t hear him going on about the multiverse…

Peter,

“There really have been no major advances in theoretical particle physics that have been experimentally verified in now over 40 years.”Exact solutions in 2D statistical mechanics [e.g. Baxter’s hard hexagon model] and their critical limits, the 2D conformal field theories [e.g. the c=1/2 theory that describes the critical limit of the Ising model] are, in my opinion, major advances in theoretical physics, that are less than 40 years old and examples of these theories were verified experimentally in the early 1980’s in experimental studies of surface critical phenomena [my recollection is that these experiments were post Baxter’s work, but before Belavin et al.].

This is the only corner of string theory [in the broad sense of the word] that I’m aware of that has been experimentally verified. Non-canonical critical exponents were measured in the lab, and the theory was right on the money.

MathPhys,

My comment was intended to just be about theoretical HEP physics. Advances in QFT of course have many other applications. I don’t think Witten though has ever shown a lot of interest in condensed matter/stat mech physics, although some of his work has applications there and I’m sure he knows quite a bit about the subject. He doesn’t seem to be joining the current wholesale move of a lot of the string theory community into condensed matter.

Woit wrote,

“TQFT is mostly his doing.”

Witten did lots for the subject, and deeply deserved his fields medal. But, let’s not forget the importance of Atiyah who really suggested to Witten that knots might have a physical interpretation. Also, Floer’s results were critical as well. And Seiberg-Witten does have two names in the title.

Justin,

I wrote in detail about that history in my book, it’s chapter 10 there. I stand by the claim that “TQFT is mostly his doing”, although of course certainly other people were crucially involved.

TQFT gets a lot of attention from mathematicians, but it’s not what won Witten the respect of his fellow physicists. If I understand the history correctly, he was already famous by the time he started thinking about that stuff, for his work on 1/N expansions, on current algebra, on anomalies, on instantons, on chiral matter in supergravity, on positive energy in GR, and on SUSY breaking.

_What has he worked on that has been experimentally verified?_

He actually has worked on a few things which have been checked experimentally, but it’s not what made him famous. People listen to him because he’s done a lot to clarify the way people think about quantum field theories. It’s too bad that he hasn’t managed to predict any new resonances, but it’s a bit blinkered to think that’s the only worthwhile thing theoretical particle physicists can do. I’d say, for example, that Dirac’s synthesis of Heisenberg & Schrodinger’s work on quantum mechanics was far more important than his prediction of the positron. The former is critical for understanding quantum physics, whereas the latter was so dodgy that explaining it usually only confuses people who’re trying to learn the subject.

He doesn’t seem to be joining the current wholesale move of a lot of the string theory community into condensed matter.He’s a wise man. But my point is that conformal field theories, which are the building blocks of string theory, have verified experimentally, on a theory by theory basis. These theories with predictions based on Virasoro algebra, affine Lie algebras, expectation values of vertex operators, etc, make physical sense.

@MathPhys:

Can you elaborate or give references for your statement about experimental verification?

There are many experimentally realized system which behave like the Ising model or its cousins when they undergo second-order phase transitions. A monolayer of an inert gas on a graphite substrate is often quoted as a particularly clean system. Such systems were studied for many years before CFT – Ernst Ising got his PhD in 1925. It turned out, both empirically and by theoretical considerations, that similar statistical systems have identical and not just similar critical exponents, or completely different ones. This is called universality, and CFT yields an exhaustive classification of universality classes in 2D systems, and the RSOS models of Andrews-Baxter-Forrester gave concreate realizations of all classes as lattice models.

This is absolutely wonderful work – basically the only experimentally verified theoretical discovery during my entire non-career – and I have argued for 25 years that BPZ deserve the next Nobel prize for this. However, from the statphys point of view it is also clear that this is not a ToE, not even in statphys. CFT explains universality in 2D (the string world sheet is two-dimensional), but it has nothing to say about the experimentally more interesting 3D case. And we know that universality is also present in 3D, empirically and theoretically (epsilon expansion), so one could ponder if the explanation could be similar to 2D.

The experiments that I know of were performed at the U of Washington, Seattle, in the early 80’s. Theorists who were involved in the analysis of the results usually included M Schick. The critical exponents that were measured were non canonical, hence non trivial, and always agreed precisely with predictions based on 2D conformal field theories, which indeed came right after.

I only wish to say that the mathematics of string theory, that is the Virasoro algebra with a central extension, vertex operator algebras, etc, is more than recreational mathematics, and more than an abstraction in a platonic plane. It makes non trivial predictions that can be measured experimentally.

The fact that this mathematics is not sufficient to lead to a theory of everything is a different matter.