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 I had noticed that in their first massive (still unpublished) paper that started the subject, Beilinson and Drinfeld credit Witten with “the main idea” (I wrote about this in detail here).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what Beilinson and Drinfeld were saying, I did put them in touch with Hitchin’s work, and actually, in their very long, unpublished foundational paper on geometric Langlands that you can find on the web, Beilinson and Drinfeld acknowledged me very generously, far overestimating how much I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.
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