Round Table Talk: Conversation with Edward Witten
Edward Witten Professor, the School of Natural Sciences, The Institute for Advanced Study Hirosi Ooguri Kavli IPMU Principal Investigator Yukinobu Toda Kavli IPMU Assistant Professor Masahito Yamazaki Kavli IPMU Assistant Professor
Kyoto Prize and Fourth Visit to Kyoto Ooguri: First I would like to congratulate Edward on your Kyoto Prize. In every four years, the Kyoto Prize goes in the eld of mathematical sciences, and this is the rst prize in this category awarded to a physicist. Right to left: Edward Witten, Hirosi Ooguri, Masahito Yamazaki, and Yukinobu Toda. Witten: Well, I can tell you I’m deeply honored to receive this prize. students, professional researchers, discuss the current state of the eld Ooguri: It is wonderful that your and general public interested in and opportunities in research. work in the area at the interface mathematics, a broad range of You have already given two of mathematics and physics has people. The magazine has been interviews for Sugaku Seminar. In been recognized as one of the most particularly inspirational for high 1990, at the International Congress important progress in mathematics school students. I enjoyed reading it of Mathematicians in Kyoto, you as well as in physics. For those of us when I was a high school student, received the Fields Medal. On working in this area, this is also very and I still subscribe to it. Yuji that occasion, Tohru Eguchi had gratifying. As Yuji Tachikawa said at Tachikawa said he read your interview an interview with you. You also the workshop yesterday, you’re like in 1994 in the magazine, which partly had a discussion with Vaughan sunshine for all of us in this area of inspired him to go into this area. Jones, another Field Medalist at research. Witten: I was very pleased to hear the Congress, and I remember, you Witten: Actually in my acceptance that from Yuji. It was very nice of him expressed interest in generalizing speech a couple of days ago, I to say that. your work in the Chern-Simons remarked that I regard it also as a Ooguri: It is my hope that the theory with a spectral parameter, recognition of the eld, not just of me. magazine article about the which is very natural from the point Ooguri: This conversation will conversation today would also of view of integrable models. appear as an article in this Japanese inspire the next generation of young Witten: Yes, I very much wanted to magazine, Sugaku Seminar, as well students to go into, not necessarily nd an explanation along these lines as in the Kavli IPMU News. Sugaku mathematics, but more broadly of the“ integrability” that makes it Seminar is widely read among high science and engineering. I thought possible to get exact solutions of school students, undergraduate this would be a good opportunity to two-dimensional lattice models such
10 Kavli IPMU News No. 28 December 2014 as the Ising model. I was completely I was unable to incorporate the a discussion session we had at RIMS, unsuccessful, but just in the last spectral parameter was that I was Kyoto University, with you and Hiraku couple of years, something in the working in the context of three- Nakajima, where Nakajima explained spirit of what I wanted to do was dimensional topological eld theory. his work on the action of an af ne done by Kevin Costello. In three-dimensional topological Lie algebra on the cohomology of Ooguri: We were just talking about eld theory, in addition to the the moduli space of instantons. In Costello’s work before you arrived moves where knots cross̶that the interview you had with Tohru here. Do you think that it achieved is, in addition to the Yang-Baxter Eguchi, you mentioned progress in what you wanted to do at that time? relation̶you have further relations mirror symmetry and S-duality at the Witten: Yes. Integrable models have involving creation and annihilation. time and expressed a hope of a more many facets, and there is no one There are Reidemeister moves that uni ed view on duality encompassing way to understand everything. But I are valid in topological eld theory, gauge theory and string theory. Some would say that speci cally the kind of but are not relevant for integrable of this hope has been achieved in the explanation I was looking for is what systems. I couldn’t nd the spectral last 20 years, I think. Costello found. parameter because I was trying Witten: De nitely some of it has Ooguri: I see. to use topological eld theory. been achieved. One thing that was Witten: What Costello did involves Costello made a very simple twist, achieved in the couple of years after a very simple but beautiful twist on replacing a real variable by a complex that second interview was simply the three-dimensional Chern-Simons variable and then everything worked that there emerged a picture of non- theory, in which he simply replaced beautifully. I de nitely regard that as perturbative dualities in string theory, one of the three real dimensions of the explanation I was trying to nd, generalizing what happens in eld space by a complex variable z. unsuccessfully, around 1990. theory. However, there are other Ooguri: Going to four dimensions. Ooguri: I see. So, after 23 years, aspects that are still mysterious and Witten: This is a four-dimensional nally your question was answered. not clearly understood. world with two real coordinates and Now, in 1994, you visited Japan for On the bright side, the fact that a complex coordinate z. Costello the second time and gave a public four-dimensional gauge dualities and de ned a 4-form which was the lecture here in Kyoto. a lot of dualities in lower dimensions wedge product of the Chern- Witten: In fact, I have had several comes from the existence of a six- Simons 3-form with the 1-form dz. opportunities to visit Kyoto, dimensional conformal eld theory He studied this as the action of a including a visit for the Strings 2003 is a major insight in understanding 4-dimensional theory. There is a conference that you organized and dualities better. We haven’t gotten crucial technical detail: for this theory also this current visit. to the bottom of things because to make any sense, the differential Ooguri: So, for every one of your we don’t really understand the six- operator that is obtained by four visits to Japan including this one, dimensional theory, but just knowing linearizing the equations of motion you have actually come to the Kyoto that the matter should be understood must be elliptic modulo the gauge International Conference Center. in terms of the properties of the six- group. I think that this is a little Witten: Yes, but I know that there is dimensional theory is an advance in surprising, but it is true. And given a trip to Okinawa lined up. understanding duality that certainly that, he then has a generalization of Ooguri: That would be the Strings wasn’t there at the time of this last the Chern-Simons theory that does 2018 Conference that we are interview. not have the full three-dimensional planning at the Okinawa Institute of symmetry, but it does have a complex Science and Technology. We surely I Was a Skeptic about Duality Round variable, namely z. hope that you will be back in Japan Table If you think carefully, you will see in 2018. Ooguri: I should introduce our that integrability, the Yang-Baxter At the time of your visit in 1994, discussants today. Yukinobu Toda is equation, involves two-dimensional you were just nishing your work on a mathematician and an Associate symmetry, but not really three- the Seiberg-Witten theory and also Professor at the Kavli IPMU, and he dimensional symmetry. The reason the Vafa-Witten theory. I remember received his Ph.D. in 2006. Masahito
11 Yamazaki is a physicist and a new in string theory. state of two monopoles whose Assistant Professor at the Kavli IPMU, By this time there were clues in existence was predicted by the and he entered graduate school the literature, and a number of new duality. That inspired me to believe in 2006. They represent the young ones had been discovered in the that one could do more. generation of mathematicians and early 1990’s. The clue that in uenced With this inspiration, and trying physicists working on the interface of me the most was the work of John to nd more evidence for the string theory and gauge theory. Schwarz and Ashoke Sen, who duality conjectures, Cumrun Vafa At your Commemorative Lecture showed that the low-energy effective and I started to study the Euler the other day that was one of the action of the heterotic string on a six- characteristics of instanton moduli Kyoto Prize events, you reviewed your torus had properties consistent with spaces. It was not too hard to see career in this area. You said that you the existence of a non-perturbative that electric-magnetic duality of entered graduate school in 1973, SL(2, Z) duality. They didn’t have supersymmetric Yang-Mills theory when the asymptotic freedom was what I regarded as really decisive implied that the generating function just theoretically being discovered. evidence for that conjecture, but their of those Euler characteristics should You came to Japan the second time ideas were very suggestive. be a modular function. Luckily for in 1994 and gave the interview we It still was not clear to me how one us, mathematicians in a number of were just talking about, and it was could nd decisive evidence for non- cases－and this includes the work roughly 20 years after that. Now it perturbative dualities in spacetime. of Nakajima that you mentioned is 20 years after that, so I thought At least to me, the rst such evidence before－had computed these Euler we should start up by trying to catch appeared in a short but brilliant characteristics, or had obtained up with the second 20 years of your paper by Ashoke Sen on a two closely related results from which career and see what your thoughts monopole bound state in N=4 super the Euler characteristics could be have been on some of the most Yang-Mills theory. To me, that was understood. We found that the important progress, and then we can fundamentally new evidence for the expected modularity held in all cases. start from there and discuss them. Montonen-Olive duality conjecture. (In one case－the four-manifold CP2 We have already started to talk It convinced me that the duality had －we ran into a“ mock modular about some of the developments to be right, and equally important, it form,” a concept that was new to since the interview in 1994, but convinced me that it was possible to us at the time but has made many maybe you can expand on that and understand it better. subsequent appearances in gauge tell us what you think have been the Ooguri: I thought that Sen’s paper theory and string theory.) highlights in this area in the last 20 gave a strong evidence for the Also during this period, Nathan years. S-duality, but it was your paper with Seiberg had been using holomorphy Witten: Certainly, one of Vafa that convinced us. as a tool to analyze the dynamics the highlights has been the Witten: Thank you. Sen’s paper of supersymmetric gauge theories. understanding on non-perturbative showed that you could actually go He wanted to understand what dualities in string theory, as a result of well beyond the suggestive but happened in N=2 theories. We which we have a much wider picture somewhat limited arguments about started talking about it, and the of what string theory is. In 1994, we electric-magnetic duality that had Sen paper inspired us to think that knew about mirror symmetry and been known, and learn something duality would play a role. That was other two-dimensional dualities that fundamentally new. Until Sen’s paper, one of the clues that actually led to arise in string perturbation theory, I had felt that what we understood our work on what became Seiberg- and we were really just beginning to about electric-magnetic duality, even Witten theory. think that there are similar dualities in the work of Sen and Schwarz, which Ooguri: It may be hard to believe space-time: four-dimensional gauge had de nitely in uenced me, was in for young people like Masahito theory dualities that are analogous to the framework of what Montonen Yamazaki or Yukinobu Toda, but the two-dimensional dualities. But in and Olive had understood 20 years before 1994, S-duality at least for me 1994, it was really just a guess that before. But Sen did a simple and was something very hard to believe. something analogous might happen elegant calculation, nding a bound It was like a beautiful dream. It would
12 Kavli IPMU News No. 28 December 2014 be nice to have, but you cannot was a skeptic. But since I was there already mentioned. I remember that, realistically hope that something to see him, I didn’t want to just say at the Strings ’93 meeting in Berkeley, like that could possibly happen. As that his idea was nonsense. We tried John Schwarz was more excited than I said, the rst evidence was Sen’s to make sense out it. So we discussed I had ever seen him since January paper and in some sense Edward’s it in the context of supersymmetry, 1984. In January 1984, telling me work with Vafa nailed it. After that, simply because with supersymmetry about his latest work with Michael everybody believed it. the mass renormalization (and even Green, he said“ We are getting Yamazaki: That’s surprising, because the full effective potential) of a close,” but I didn’t understand what I thought that the paper of Claus scalar can be zero. This was the only he thought he was getting close to. It Montonen and David Olive is quite context in which it seemed to me turned out, however, that that was a old. Were people skeptical about the that the brilliant idea of Montonen few months before they canceled the idea? and Olive could make sense. By the anomalies. When John was so excited Witten: This might make you laugh, end of the day, we found that their at the Strings meeting at Berkeley, I but I’ll tell you my early history with formulas are valid in the context of decided that I should better take him the Montonen and Olive paper. First N=2 supersymmetry. So we wrote seriously. of all, I hadn’t heard of it until I was a paper on that, and it was quite a If you had looked at it with the visiting Oxford at the end of 1977. satisfying paper to write, but I drew same skepticism I had had since Michael Atiyah showed this paper to the wrong conclusion from that the Montonen and Olive paper, me and said I should go to London paper. The conclusion I drew was you would have said that Sen and to discuss it with Olive. So, I looked that we had explained their formulas Schwarz were just discussing low at the paper and got in touch with without needing to assume non- energy physics and did not have David Olive and arranged to visit him. perturbative duality. solid evidence about strong coupling But by the time I got to London, I was Ooguri: Right. That was the message behavior. But John’s enthusiasm put pretty skeptical. Have you looked at I got by reading your paper with enough of a dent in my skepticism their original paper? Olive, too. The seemingly miracle that I started looking more closely to Yamazaki: Yes, I have. phenomenon was explained simply the papers of Duff and other authors Witten: In their original paper, they by supersymmetry. on solitons in string theory. At some considered a bosonic theory of a Witten: So at the time, and for point, I think in the fall of 1993, Duff gauge eld and a real scalar (valued many years after, I felt that there sent me an assortment of his papers in the adjoint representation). They was not really a lot of evidence for and I took them to heart. I don’t assumed that the potential energy for non-perturbative duality in four remember right now all of the papers the scalar eld is identically zero, and dimensions. on solitons in string theory that I they found remarkable formulas for Thus, to return to Masahito’s looked at during this period, but particle masses that are valid precisely question, I was a skeptic about certainly one important one was by in this case. Their proposal of electric- electric-magnetic duality during these Callan, Harvey, and Strominger. magnetic duality was based on the years, but actually I was a skeptic on There is another part of the fact that their mass formula was two levels. One, I was skeptical that background to this period that I symmetrical between electric and it was true, and two, I was skeptical should explain. magnetic charge. that you could really say anything Mike Duff, Paul Townsend, However, I knew quantum eld about it even if it were true. and other physicists working on theory well enough to know that To give a more complete picture, in supermembranes had spent a couple saying that the potential energy the early 1990s, there were various of years in the mid-1980s saying Round for the scalar eld is 0 is not a novel clues, some from the work that there should be a theory of Table meaningful statement quantum of people like Mike Duff, and also fundamental membranes analogous mechanically. If it were, we would not Curt Callan, Jeff Harvey, and Andy to the theory of fundamental strings. have a gauge hierarchy problem in Strominger, studying solitons in string That wasn’t convincing for a large particle physics. So, by the time I got theory, and then also there was the number of reasons. For one thing to London to see Olive, I de nitely work of Sen and Schwarz that I a three-manifold doesn’t have an
13 After that, you even went to string dualities.
String Duality Revolution
Witten: By the end of 1994, we had the experience of non-perturbative dualities in eld theory both in two dimensions and in four dimensions. In the two-dimensional case, for example, if you study a sigma- model with Calabi–Yau target space (such as is important in studying the compacti cation of string theory), one nds that the quantum theory can be extended far beyond the classical geometry of the Calabi-Yau manifold. One nds a web of phase transitions Euler characteristic, so there isn’t a approximation wasn’t good. But between different geometrical and topological expansion as there is the idea of membranes as non- non-geometrical descriptions of in string theory. Moreover, in three perturbative soliton-like objects in the sigma-model, which represent dimensions there is no conformal string theory made a lot of sense different semi-classical limits of the invariance to help us make sense even if the details in some papers theory. The Montonen-Olive duality of membrane theory; membrane were dubious. I was still a bit of a conjecture, as re ned by later authors, theory is non-renormalizable just like skeptic about what one can do with said that something similar happens General Relativity. this idea, but for reasons I’ve been in N=4 super Yang-Mills theory in There are all kinds of technical explaining, I was paying a lot more four dimensions, and Seiberg and objections, but at some point around attention. And that is actually why, I in 1994 had found something 1990 or 1991, instead of trying to when the Sen paper on the two- somewhat similar for N=2. think of membranes as fundamental monopole bound state came out, I Certainly there was a dream that objects, people working in this area was ready to completely change my something similar might happen in started thinking of membranes and outlook. string theory. Not only there was a other p-branes as non-perturbative Sen’s paper showed that one can dream, but there were a lot of papers objects that might exist in string do something new about strong in which people had pointed out theory. In general terms, this idea coupling and it was clear that if one pieces of such a story. I have already did make sense. In more detail, the had been inspired the way Sen was, mentioned some of these papers. situation was more complicated. If one could have done what he did 10 Another important paper was written you actually looked at the papers, or 15 years before. So, it showed that by Chris Hull and Paul Townsend in some of them made a lot of sense we had been missing opportunities. the spring of 1995. They wanted to because they had a classical soliton That de nitely changed the direction say that Type IIA superstring theory is solution with good properties. (Even in my work. It led to the paper the same as M-theory on a circle. The then, the solutions usually had with Vafa that you’ve been kindly only thing they really didn’t do was unusual properties that in some mentioning, and it helped put Seiberg to try to make it more quantitative. cases were clues to later discoveries.) and me on the right track for doing There’s a potential contradiction Other papers made a little bit less what we did in 1994 and then... which is that in Type IIA superstring sense, because the classical solution Ooguri: This is a great story that theory you don’t see 11 dimensions. involved a singularity that appeared shows that chance favors the But it turns out, as I realized a little in a region in which the classical prepared mind, as Pasteur said. later, that this question has a very
14 Kavli IPMU News No. 28 December 2014 simple answer. The 11-dimensional limit is a region of strong coupling from the point of view of Type IIA superstring theory, and the eleventh dimension isn’t visible for weak coupling. It soon became clear that the same thing was true in other cases. For example one might hope that Type I superstring theory and the SO(32) heterotic string will be the same. There is an obvious immediate contradiction: the theories have the same massless spectrum and low energy interactions, but beyond the low energy limit they look completely different. The answer is simply that if you match up the low energy In the moduli space of Calabi- strange. I personally didn’t focus eld theories, you will nd that Ya u manifolds, there are a variety on that question, but Strominger weak coupling in one is strong of singularities. Some questions explained that such a singularity coupling in the other. involving such singularities had been actually re ects a non-perturbative Once one starts thinking along important in my own work. quantum effect. The singularity those lines, it turns out that Ooguri: You are referring to the work arises when a charged black hole everything works. What were the involving linear sigma-models. becomes massless and it shows that implications? This way of thinking Witten: That is correct, and also quantizing a classical theory can’t do certainly led to a more uni ed picture my work (with Harvey, Vafa, and justice to string theory: there are non- of what string theory is. But very soon, Lance Dixon) on orbifolds. I had perturbative quantum effects even in there were further developments been interested in cases in which the what one might have wanted to call showing that the traditional ways classical geometry has a singularity the classical limit. of asking questions were probably but the quantum sigma-model Ooguri: You say there’s no analogous inadequate. In the 1980s, I was does not; these cases illustrate result in eld theory, this is a really convinced that in some sense the difference between ordinary genuinely string theory phenomenon. string theory should be based on a geometry and its generalization in Witten: I think so. Lagrangian that would generalize the classical limit of string theory. Ooguri: So, did you think this was an the Einstein-Hilbert Lagrangian for What I had not taken seriously is evidence that there is no Lagrangian gravity; it would have a symmetry that in general, as one deforms the description in string theory? group which would generalize the moduli of a Calabi-Yau manifold, Witten: It is evidence that you can’t diffeomorphism group. So there one can nd singularities of the fully do justice to string theory in would be a new classical theory of classical geometry that do also lead terms of quantizing a classical theory. geometry－with non-perturbative to singularities of the corresponding I don’t want to say that there isn’t a two-dimensional dualities built in as sigma-model. classical theory, because I believe that Round classical symmetries. One would then Such a singularity appears in string from some point of view there is. Table generate string theory by quantizing theory even in the classical limit, so Ooguri: Yes, as an approximate this classical theory. if you try to interpret string theory description, but you’re saying that But by the early 1990s, there was as a classical theory that then gets you cannot start from classical theory a troublesome detail that I personally quantized, it looks like the classical and apply quantization procedure... did not pay much attention to. theory has a singularity, which is Witten: We can’t fully understand
15 string theory by quantizing an was corrected by Paul Aspinwall in a a fact which we were convinced underlying classical theory. In some paper that he wrote in the summer of by this time－and on the other sense it is an intrinsically quantum of 1995. Aspinwall explained the hand N=4 super Yang-Mills theory mechanical theory. following: In M-theory at an ADE in four dimensions with gauge I don’t want to say you can’t derive singularity, you have only the hyper- group U(n) can arise from a system string theory by quantizing a classical Kahler moduli, but in string theory of n parallel D3-branes in Type IIB theory, but you can’t fully do justice at an ADE singularity, there also are superstring theory. Combining these to it that way, I think. B- eld moduli. The conformal eld two facts and taking the low energy But let us remember that even in theory becomes singular when the limit, Green was able to deduce the eld theory, Montonen-Olive duality B- eld moduli are zero; the orbifold Montonen-Olive duality of N=4 super means that the same theory has describes a non-singular situation in Yang-Mills theory with gauge group different classical limits, showing which the B- eld moduli are not zero. U(n). It is simply inherited from the that no one classical limit is really When the B- eld moduli vanish, Type IIB superstring theory specialized distinguished. there is a breakdown of the classical to the D3-branes. Ooguri: But in that case, you have a description that’s just analogous to That was an important early Lagrangian description. what Strominger had shown in his example of deducing a gauge theory Witten: Yes, in the Montonen-Olive paper on the Calabi-Yau singularity. duality from a string theory duality. case, one has a classical Lagrangian, It leads to enhanced gauge symmetry Even before all of this had in fact many of them. String theory that, from the standpoint of Type happened, Mike Duff and Ramzi is a little bit worse because even in IIA superstring theory, has a non- Khuri in 1993 had written a paper on what you want to call the classical perturbative origin. what they called string/string duality. limit, there are phenomena that you Strominger had considered a They had said there should be a self- really can’t make much sense of from charged black hole that arises from a dual string theory in six dimensions the classical point of view. wrapped three-brane, while here the which looked at in two different ways Ultimately, Strominger’s work relevant particle is a wrapped two- would give electric-magnetic duality illuminated something I’d missed. brane. But the idea is similar. of gauge theory in 4 dimensions. In the talk I gave at the Strings Ooguri: So this was the beginning It was actually a brilliant idea. The Conference in 1995 about non- of interaction between gauge theory only trouble was they didn’t have an perturbative dualities in string theory, idea and string theory idea where example in which it worked. and also in the corresponding paper non-Abelian, non-perturbative I realized in mid-1995 that if one “( String Theory Dynamics In Various dynamics of gauge theory can took the heterotic / Type II duality Dimensions”), there was one detail emerge from limits of string theory. on K3 and T 4 and compacti ed on that didn’t completely make sense. Witten: Right. Another important but another two-torus, one would get Type IIA superstring theory on a K3 extremely simple paper that helped an example rather similar to what manifold was supposed to be dual to show the implications of string Duff and Khuri had suggested. They the heterotic string on a four-torus, theory for non-perturbative duality had had in mind self-duality of a and in that context, I could see that in gauge theory was written by string theory, but the example that enhanced gauge symmetry resulted Michael Green in 1996. By this time, I considered was a duality between from the K3 surface developing Joe Polchinski and his collaborators two different string theories. Still an ADE singularity. But an ADE had basically shown that in modern the idea was similar. By the end singularity in classical geometry language a system of n parallel of 1995, Duff and I and some is just an orbifold singularity, and branes has U(n) gauge symmetry. I other authors had an example that perturbation theory remains valid had written a paper at the end of followed even more precisely what in string theory at an orbifold. The 1995 showing why that was useful, had been proposed two years before. orbifold does not generate a non- but Green wrote a very simple paper That involved the E8 x E8 heterotic perturbative gauge symmetry. For a with the following observation. Type string on a K3 surface with equal few months, I was puzzled. Actually, IIB superstring theory has a non- instanton numbers in the two E8’s. In I was making a simple mistake which perturbative duality symmetry－ all of these cases, one could deduce
16 Kavli IPMU News No. 28 December 2014 Montonen-Olive duality from a string duality. This is somewhat analogous tried to say in my Commemorative theory duality. to saying that some statement in Lecture for the Kyoto Prize. There is As we’re talking, I remember more number theory follows from the a difference between knowing what and more papers from the years Riemann conjecture. If somebody is true and knowing why it is true. In 1995-6 that were very dramatic at shows that something follows from this case, you have a mathematical that time, but that were also, honestly, the Riemann conjecture, one may or proof, but you’re still asking why, pretty easy to do in most cases. I was may not view this as an explanation and physicists ultimately don’t reminded of this yesterday by the of the statement in question, but know. All that we can do is to offer lecture by Hiraku Nakajima at the at least it puts the statement in a bigger conjectures, of which this is Kyoto Prize Workshop. Hiraku started bigger framework. Montonen-Olive a manifestation. But we don’t really by kindly remembering three lectures duality provided an analogous bigger understand the bigger conjectures. I gave at the Newton Institute in framework in my work with Vafa, Ooguri: In physicists’ perspective, 1996. The lectures concerned three and soon afterwards there was a still this duality has been geometrized as papers that I had written (respectively bigger framework, which was that symmetry in six dimensions. with co-authors Lev Rozansky, Ami the Montonen-Olive duality follows Witten: But the six-dimensional Hanany, and Nathan Seiberg). The from the existence of a certain six- theory is pretty mysterious. papers t together nicely. They were dimensional theory. It also follows in Toda: Is it not dif cult to understand fun to write and the lectures were various other ways from string theory the relationship between S-duality also fun to give. But what stands out dualities and I have mentioned a and six dimensional theory? in my memory is that at that time few of these constructions. But most Ooguri: The relation is very clear, but insights like that were more or less physicists would probably say the then you have to make sense of the out on the surface. It was quite a fun most complete framework that we six-dimensional theory itself. time to be working in this eld. I am have for Montonen-Olive duality is its Witten: We actually know quite a hoping that during my active career, relation to the six-dimension theory. lot about the behavior of the six- there will be another period like that. Ooguri: Yukinobu was asking for dimensional theory, though we do some mathematical explanation. not understand much about how it Difference between Knowing At that time, some hint of the should be constructed or understood What Is True and Why It Is True mathematical explanation was microscopically. Toda: I am an algebraic geometer. Nakajima’s work, about the symmetry One of the deepest discoveries I was originally working on some of the moduli spaces of instantons. about the behavior of the six- classical aspect of algebraic geometry, From the mathematical point of dimensional theory was made by but I became interested in some view, what the Vafa-Witten theory Juan Maldacena in 1997. He showed relationships between algebraic was computing were generating that it could be solved for large N in geometry and string theory inspired functions of the Euler characteristics terms of supergravity. Unfortunately, by your work. You have been of instanton moduli spaces. the regime in which the theory is discussing S-duality and modular Witten: Nakajima’s discovery of solved by supergravity isn’t the same forms. This was surprising from a the af ne Lie algebras was a kind regime in which we usually have to mathematical point of view - I cannot of proof, and actually a miraculous study it to understand the questions see why modularity appears. Do you discovery. But it still leaves one you’re asking. have an insight about that from a wondering where the af ne Lie Ooguri: I understand that the large N mathematical point of view? algebra symmetry came from. limit is not S-duality invariant. Witten: Vafa and I, of course, had a Toda: Right. After computations of Witten: Yes, that is correct. Round reason, which was the Montonen- the Euler characteristics, we know Maldacena’s solution of the Table Olive duality conjecture. What we that it’s a modular form but we don’t theory for large N works and makes did was to show that modularity know why it is modular, even for the complete sense. It does not directly of a certain generating function simplest example. help us in understanding Montonen- of Euler characteristics is a kind Witten: I completely agree. What Olive duality, because it involves of corollary of Montonen-Olive you’re saying is actually something I studying the theory in a different
17 region of parameters that is not are related to the classical problem in Quantum Entanglement invariant under duality. Or to put algebraic geometry. it differently, if one tries to apply Ooguri: You also attend string theory Ooguri: We are still discussing what Maldacena’s solution to understand seminars. What do you gain by was happening in the ’90s. Now, Montonen-Olive duality, one has to attending them and interacting with we should move on to the new work in a region of parameters in physicists? millennium. What do you think have which the description that Maldacena Toda: I think there are many kinds been highlights in the past 14 years? gave is not useful. of people in string theory. Some Witten: Part of the answer is that But the existence and success people’s works are close to me, the gauge-gravity duality that was of Maldacena’s solution de nitely like Donaldson-Thomas invariants introduced by Maldacena is very deep. increases the con dence of physicists and derived category of coherent Even today people are still discovering that the six-dimensional theory exists sheaves. In their seminars, I can interesting new facets of it. An and that all the canonical statements learn something, but that is almost a important example was the work of about it are true, even though seminar of mathematics. Shinsei Ryu and Tadashi Takayanagi we don’t understand everything. Ooguri: A mathematician told me on entanglement entropy in gauge/ It’s a little bit like mathematicians that physicists are like generating gravity duality. They discovered a discovering that some new functions of conjectures. Some really interesting generalization of consequences of the Riemann physicists are more useful for the Bekenstein-Hawking entropy of conjectures are true. This gives one mathematicians than others. For a black hole. Although I have not more con dence in the Riemann example, Hiraku Nakajima was telling personally worked on this subject, conjecture, but it doesn’t mean one me that he particularly likes Edward’s the developments have been pretty understands the Riemann conjecture. lectures, because even though he interesting and may contain deeper Ooguri: Yukinobu, what is your view doesn’t understand the motivations clues about quantum gravity. If I on current activities in physics? For and where ideas come from, some could see the right way to do this, example, yesterday Nakajima was of the statements Edward makes then I would probably work in this saying that it took him 18 years have sharp mathematical meanings area myself, but at least so far I to understand what Edward was to them, just like the equation that don't. But it’s one of the things I’d doing in his lectures in Cambridge, Tachikawa was quoting yesterday recommend watching most closely. and Kenji Fukaya was saying that at the workshop and these are Apart from Ryu and Takayanagi, sometimes he doesn’t understand something that mathematicians can I also would de nitely recommend even the statement because you don’t work on. the papers of Horacio Casini, in understand the right-hand side and Yamazaki: But then sometimes some cases co-authored with left-hand side of equations physicists people want to know the logic Marina Huerta. One of these papers write, for example. You have been behind it. I can make a statement addressed the following question. A at the Kavli IPMU for several years, that makes sense mathematically, black hole has a Bekenstein-Hawking interacting with physicists, so do you and mathematicians can try to prove entropy. Roughly 20 years ago, Jacob have any perspective to offer...? it. But they de nitely want to know Bekenstein considered the following Toda: Of course, I don’t know what’s happening. question. Suppose an object falls anything about string theory, but Witten: In any given case, I can’t into a black hole. The object has sometimes I look at papers and some guarantee that there isn’t a an entropy. When it falls into a calculations, and try to translate simpler answer. But the view of black hole, its entropy disappears physics words into mathematics, say most physicists about many of into the hole. The black hole gains D-branes to sheaves or BPS states the problems that we have been mass when it absorbs the object, to stable objects. Then I have lots discussing is going to be that the so its entropy goes up. The second of things to learn from the physics best setting for these questions is in law of thermodynamics says the side, and lots of problems to solve, the quantum eld theories that are total entropy should increase in this although I don’t understand their important in physics. process, so in other words the black physics origin. I also found that they hole entropy increases by at least
18 Kavli IPMU News No. 28 December 2014 the entropy that the infalling object there is no box keeping it there. (I say entropy, one suspects it’s probably had before approaching the black “almost certainly” because relativistic an important clue, but it might take hole. This tells you basically that if an quantum mechanics does not permit a younger person than me with object has given energy and is small us to say that a particle is de nitely fresh thinking to see what it is an enough to t inside a black hole of present in a given region.) Here for important clue to. given mass, then there’s an upper a single particle, we can de ne its I do want to mention one more bound to its entropy. Bekenstein energy, and we can identify (within contribution in that direction, which proposed such a bound, and people general limits of relativistic quantum is by Casini and Maldacena with called it the Bekenstein bound, mechanics) the region that it is Rafael Bousso and Zachary Fisher but for a long time no one could con ned in, but it is hard to make (BCFM). Years ago, Bousso had formulate precisely what this bound sense of the entropy of a single formulated a covariant version of the was supposed to say. particle. For a long time, there were Bekenstein bound; it is well adapted I am reminded here of what many papers discussing this, many to problems in cosmology. Everything Fukaya said about the relationship dozens and probably hundreds of I have said about the Bekenstein between physics and mathematics, papers, for the most part with limited bound has an analog for the Bousso where he remarked that it can be insight. Then, there was a simple bound. When you understood what hard to formulate precisely the terms and quite brilliant paper by Horacio it meant, it wasn’t very interesting, that enter some of the statements Casini, who showed that the right and when it was interesting, you made by physicists. In the case of concept is entanglement entropy couldn’t understand what it meant. the Bekenstein bound, the situation and that it can always be de ned In the recent work of BCFM, a precise was as follows. In a situation in in a natural way and does enter in formulation and proof of the Bousso which the concepts (the size, energy, a universal Bekenstein-like bound. bound is given, at least for quantum and entropy of the infalling object) This paper was well ahead of the eld theory in at spacetime. have clear meaning, the Bekenstein prevailing thinking in the eld and it Ooguri: I see that this new joint bound was trivially true and not very was a number of years, I think, before activity between quantum gravity interesting. For example, consider people widely appreciated it. and quantum information theory a gas consisting of many particles Ooguri: For example, Casini’s paper has become very exciting. Clearly bouncing around in a box. Here the solved the species problem that I had entanglement must have something size of the system and its energy and been puzzled about for some time to say about the emergence of entropy all have a clear meaning. and gave a convincing explanation spacetime in his context. The Bekenstein bound was true but that it’s not an issue. Witten: I hope so. I’m afraid it’s hard not very interesting because it was Witten: There were what people to work on, so in fact, I’ve worked satis ed by a very wide margin. You thought were counter-examples to with more familiar kinds of questions. could ask, could you nd a situation the Bekenstein bound, and so some I have spent a lot of the last decade where the Bekenstein bound is people, and I was one, thought or maybe even little bit more than close to being saturated? You can that if the bound was true, it was a decade maybe by now working accomplish this by considering not a statement about quantum eld on a succession of problems that a whole gas of particles but a single theories that can be coupled to probably were a little bit more out particle in a box. More exactly, gravity, not about all quantum eld of the mainstream than most of my this gets close to saturating the theories. But Casini showed that this previous work. Also, I simply worked Bekenstein bound if we can ignore was completely wrong. He gave a on these problems much longer the mass of the box, but that is an precise meaning to all the terms that than I usually worked on any one Round unrealistic assumption. To get close entered the Bekenstein bounds, and problem in the past. I guess the three Table to saturating the Bekenstein bound, he showed that it was a universal problems that best t what I have just we really should consider a single statement of quantum eld theory said have been gauge theory and the particle that at a given time is almost that follows from general principles. geometric Langlands program, gauge certainly contained in a certain That was extremely illuminating and theory and Khovanov homology, and region in spacetime even though like other work on entanglement superstring perturbation theory.
19 Superstring perturbation theory is already have in mind application to constructing Khovanov homology best understood in terms of super Khovanov homology? using a space of repeated Hecke Riemann surfaces, a fascinating Witten: The answer is“ no”: in modi cations. I did not initially know mathematical subject that I hope those years, I knew about Khovanov what to make of those clues but they mathematicians will get interested homology and I was frustrated to were a sort of red ag hanging out in. Super Riemann surfaces are not understand it, but I had no there. a generalization of ordinary idea it was related to geometric Hecke transformations are one Riemann surfaces to include odd or Langlands. I was frustrated at not of the most important ingredients anticommuting variables. There is a understanding Khovanov homology, in geometric Langlands. What fascinating algebro-geometric theory because I felt that my work on the they mean in terms of physics had that people partially developed in Jones polynomial ought to be a good bothered me for a long time, and the 1980s and then abandoned. It starting point for understanding eventually had been the last major would be great if it gets revived. By Khovanov homology, but I just stumbling block in interpreting the way, we are having a workshop could not see how to proceed. geometric Langlands in terms of next May at the Simons Center for (From a mathematical point of view, physics and gauge theory. Finally, Geometry and Physics in Stony Brook Khovanov homology is a re nement while on an airplane ying home and algebraic geometers might be or“ categori cation” of the Jones from Seattle, it struck me that a interested in it. polynomial of a knot.) Actually, Sergei Hecke transformation in the context Ooguri: Do you think there will be a Gukov, Albert Schwarz and Vafa had of geometric Langlands is simply an genuine new mathematics coming already given (in 2004) a physics- algebraic geometer’s way to describe out in these fermionic dimensions? based interpretation of Khovanov the effects of an“ ’t Hooft operator” Witten: I am sure that the algebraic homology, drawing in part on earlier of quantum gauge theory. I had geometry of super Riemann surfaces work of Ooguri and Vafa. But I found never worked with ’t Hooft operators, is exciting, and unfortunately, a it perplexing and a little frustrating but they were familiar to me as they lot of what was understood in the that the relation of this to gauge had been introduced in the late 1980s existed only in the form of theory was so indirect and remote. I 1970s as a tool in understanding unpublished notes or letters. I hope wanted to nd a more direct route quantum gauge theory. The basics of that our workshop will help change but for several years I found this how to work with ’t Hooft operators this situation. dif cult. and what happens to them under Eventually, however, some electric-magnetic duality were well- developments in the mathematical known, so once I could reinterpret Khovanov Homology literature helped me understand Hecke transformations in terms of ’t Yamazaki: I was attending your that Khovanov homology should Hooft operators, many things were lecture yesterday and you were be understood using the same clearer to me. explaining how you came to the ingredients that are used to Cautis and Kamnitzer had idea that Khovanov homology can understand geometric Langlands. I interpreted Khovanov homology in be written as N=4 super Yang- didn’t understand all of these clues, terms of the B-model of a space of Mills integrated over an unusual but I learned from two of them. One repeated Hecke transformations. integration cycle. One thing that was the work of Dennis Gaitsgory on Kamnitzer also conjectured in impressed me there was that your what mathematicians call quantum another paper that there would be previous papers were the crucial geometric Langlands (I am not sure an alternative description in terms input, namely your work with Anton this is the name a physicist would of an A-model of the same space. Kapustin in which you formulated the use) showing that the q parameter Technically, it was hard to nd the Kapustin-Witten equation, and also of quantum geometric Langlands right A-model. I really wanted to subsequent work you did with Davide is related to the q parameter of understand the A-model, because Gaiotto on boundary conditions in quantum groups and the Jones that was the approach in which one N=4 super Yang-Mills theory. When polynomial. The other was the work could expect to achieve manifest you worked on these papers, did you of Sabin Cautis and Joel Kamnitzer three- or four-dimensional symmetry.
20 Kavli IPMU News No. 28 December 2014 My main goal in studying Khovanov was the B-model. Since it doesn’t something to do with this gauge homology was to nd a description have manifest three-dimensional theory dynamics? with manifest symmetry and a clear symmetry, I decided to concentrate Witten: Well, I didn’t forget about relationship to the gauge theory on the A-model, but if I ever have a it, but since－as I already told you－I description of the Jones polynomial. couple of months to spare, I would was skeptical about Montonen-Olive I eventually succeeded in doing this. try to explain as a physicist the duality, I didn’t seriously try to relate One of the trickiest elements was Cautis and Kamnitzer B-model. I’m it to Langlands duality and I didn’t try that the gauge elds have to obey a reasonably optimistic I could do that to learn what Langlands duality was. subtle boundary condition that I call and I think it would be illuminating. I did not learn anything more about the Nahm pole boundary condition. The only problem is that there are these matters until the late 1980s. (The basic idea that leads to the a lot of things like that－interesting Then I learned just super cially about Nahm pole boundary condition was loose ends that I think I could clarify the Langlands correspondence. If introduced by Werner Nahm more if I spend a few months on them. one knows even a little bit about than 30 years ago, in his work on the Langlands correspondence and a magnetic monopoles.) Luckily for me, Langlands Correspondence and little bit about conformal eld theory I was familiar with the Nahm pole Gauge Theory Dualities on a Riemann surface, one can see boundary condition and its role in Ooguri: That the Langlands an analogy between them. I wrote a electric-magnetic duality because of correspondence has something to paper that was motivated by that but work that I had done with Davide do with S-duality was there even in then I realized that my understanding Gaiotto a few years earlier. the late ’70s. When was it that you was too super cial to lead to I suspect that the mathematics actually realized the signi cance of anything deep, so I abandoned the world could appreciate my work on it? matter for a number of years. Khovanov homology in the short to Witten: I didn’t give the complete Ooguri: I remember when I was a medium term and that the obstacle to explanation of my interaction with postdoc at the Institute for Advanced this is largely a lack of familiarity with Michael Atiyah in 1977. He told me Study in 1988 and 1989, Robert the Nahm pole boundary condition. about two things that were new Langlands himself was actually quite With this in mind, I have been to me. One was the Montonen- interested in conformal eld theory. I working with Rafe Mazzeo trying to Olive paper and the other was the am not sure exactly which aspect he give a detailed mathematical theory Langlands correspondence, which was interested in, however. of that boundary condition. We plays a central role in number theory Witten: I don’t think he was have written one paper formulating but which I had never heard of. He motivated by the Langlands rigorously the Nahm pole boundary had noticed that the dual group correspondence. But I think his condition in the absence of knots, of Langlands and the dual group work was in uential. Even though and we are trying to generalize this that enters the Montonen-Olive in a sense he didn’t precisely make to include the knots. The necessary conjecture (and which had been any major breakthrough himself, he inequalities are available but some introduced earlier by Peter Goddard, helped to nd the questions that details are not yet in place. Jean Nuyts, and Olive) were the same. stimulated the later development Yamazaki: I see. That’s a very nice On this basis, Atiyah suspected that of Stochastic Loewner Evolution, story of the physics-mathematics the Langlands correspondence has which has had a major impact interaction. You were partly something to do with the Montonen- on mathematics and has even motivated by the important papers in Olive conjecture. enlightened physicists about new mathematics and interpreted them as Ooguri: So that was in the late ’70s? ways to think about some questions Round a physicist. Then you have your own Witten: It was December of 1977 in conformal eld theory. I think Table physics story and you are now trying or January of 1978. That was when I Langlands was an in uence behind to bring it back to mathematics. visited Oxford for the rst time. this work, but I do not believe his Witten: As I have mentioned, the Ooguri: Did you take that seriously interest in conformal eld theory version that Cautis and Kamnitzer already at that time that the was motivated by the the Langlands were actually able to understand Langlands correspondence had correspondence or by gauge theory
21 dualities. This is my impression from the Jones polynomial in 1988－ can nd on the web, Beilinson and interacting with him over the years. there was a crucial difference. They Drinfeld acknowledged me very As I have already remarked, in seemed to have complex critical generously, far overestimating how the late 1980s, after spending some points that made exponentially large much I had understood. All that had time trying to develop the analogy contributions, and this normally is really happened was that based on between conformal eld theory not possible in physics. I am not sure a guess, I told them about Hitchin’s and the Langlands correspondence, if this point bothered anyone else, work, and then I think that made I concluded reluctantly that the but it bothered me. It turned out that all kinds of things obvious to them. analogy in the form I was developing this was a good question to think Maybe they felt I knew some of those was way too super cial, so I stopped. about, since I eventually found a nice things, but I didn't. But anyway, there But then around 1990, I heard about explanation, and this was a turning were ample reasons in those years to new work of Alexander Beilinson and point in enabling me to understand think that geometric Langlands had Vladimir Drinfeld on the geometric Khovanov homology via gauge something to do with physics, but as Langlands correspondence. This had theory. you can see I still couldn’t make any a few consequences. First of all, it The work of Beilinson and Drinfeld sense out of it. con rmed that my understanding on geometric Langlands bothered me Ooguri: So, what inspired you to of what the duality would mean in in much the same way. They were return to this? physics was way too super cial. What using familiar ingredients of physics Witten: A decade later there was they had was much more incisive but they were using them in ways a workshop at the Institute for and much more detailed than my that did not seem to t. It looked Advanced Study on geometric rather primitive analogy between like somebody had taken a bunch Langlands for physicists. Were you the Langlands correspondence and of chess pieces, or perhaps here in there? conformal eld theory. Their work Japan I should say a bunch of shogi Ooguri: I was invited, but there was a con rmed that physics that I knew pieces, and placed them on the board con ict of schedule, so I couldn’t go. I was relevant. But I was troubled, at random. The way that the pieces missed it. because they were using conformal were arranged did not make any Witten: There were two long series eld theory in a way that didn’t sense to me. That bothered me but I of lectures and then there were a make any sense to me. They studied could not do anything about it. couple of outliers. The long series conformal eld theory at negative Actually, the very little bit of were very well done, but they integer levels－in physics positive what Beilinson and Drinfeld were did not help me very much. Mark integers are more natural here－ saying that I could understand made Goresky gave a long series of lectures and used it in ways that looked quite me wonder if the work of Nigel aiming to tell physicists what is the strange. Hitchin would be relevant to them, Langlands correspondence. The As I explained yesterday in my so I pointed out to them Hitchin's only trouble for me was that to the lecture at the Kyoto Prize Symposium, paper in which he had constructed extent that one can explain this topic for a number of years, the“ volume commuting differe ntial operators in a couple of lectures, assuming conjecture” concerning the Jones on the moduli space of bundles essentially no knowledge of algebra polynomial (formulated and on a curve. Differently put, Hitchin beyond the de nition of a eld (in developed starting around 2000 by had in a certain sense quantized the algebraic sense), I was familiar Rinat Kashaev, Hitoshi Murakami, the classical integrable system that with the Langlands correspondence and Jun Murakami, among others, he had constructed a few years already. Namely I didn’t really know and explained to me in large part before. Although I understood anything about it, but I knew as by Sergei Gukov) bothered me. scarcely anything of what Beilinson much as one could explain in a few Although their statements bore a and Drinfeld were saying, I did put hours starting from zero. So I couldn’t super cial resemblance to physically them in touch with Hitchin’s work, really get much out of those lectures. well-motivated statements－in and actually, in their very long, In addition, Ed Frenkel (who had fact to statements that I myself unpublished foundational paper been the prime mover behind the had made in my original paper on on geometric Langlands that you occurrence of this workshop) gave
22 Kavli IPMU News No. 28 December 2014 a series of lectures that, as far as I moduli space, viewed differently, and couldn’t explain it to anybody was concerned, were basically about would give a mirror symmetry else. And in a sense, I still feel the shogi board on which the pieces between a B-model in a certain that way for the following reason. have been arranged at random. I complex structure and an A-model Physicists with a background in string really couldn’t get much out of those in a certain symplectic structure. This theory or gauge theory dualities lectures either because I already mirror symmetry was supposed to can understand my paper with knew that people working on the be the true geometric Langlands Kapustin on geometric Langlands geometric Langlands were taking duality, not the approximation. but for most physicists this topic is familiar pieces from the shogi set Actually, the reason I started working too detailed to be really exciting. On and arranging them on the board at on geometric Langlands with Anton the other hand, it is an exciting topic random as far as I was concerned. Kapustin was that he had studied for mathematicians, but dif cult to There were a couple of additional generalized complex geometry in understand because too much of lectures that weren’t part of any two-dimensional dualities. In that the quantum eld theory and string series. One of them was by David world, a family of dualities can theory background is unfamiliar (and Ben-Zvi. He told us about what was degenerate, and a mirror symmetry dif cult to formulate rigorously). supposed to be an approximation can degenerate to a holomorphic That paper with Kapustin may to the geometric Langlands duality. unfortunately remain mysterious to correspondence. I think he was When one starts thinking mathematicians for quite some time. talking largely about the work along these lines, it soon makes Yamazaki: Maybe that means that of another mathematician, Dima a lot of sense that the geometric we have to wait an extra 10 or 15 Arinkin. What was supposed to be Langlands duality is really a mirror years before... the approximation to the geometric symmetry, which can degenerate to Witten: We indeed may have to. I Langlands correspondence was a holomorphic duality, and that this think it’s actually very dif cult to see T-duality on the bers of the Hitchin is the approximation Ben-Zvi taught what advance in the near term could bration. This was described by Ben- us about. I became convinced that make the gauge theory interpretation Zvi in a complex structure in which that had to be right. There were of geometric Langlands accessible the bers of the Hitchin bration still a few hurdles to overcome. The for mathematicians. That’s actually are holomorphic, so the T-duality is a most dif cult one I already described one reason why I’m excited about holomorphic duality. It was already earlier. One does not get to rst base Khovanov homology. My approaches known to physicists that the T-duality with the Langlands correspondence to Khovanov homology and to on the bers of the Hitchin bration without Hecke operators, so it geometric Langlands use many of the comes from Montonen-Olive duality was necessary to have a physical same ingredients, but in the case of in four dimensions, and of course ever interpretation of Hecke operators in Khovanov homology, I think it is quite since Atiyah’s observations of 1977- terms of ’t Hooft operators of gauge feasible that mathematicians could 8, I had been aware of the possibility theory. It was also necessary to know understand this approach in the near that some version of the Langlands how to interpret the A-model of future if they get excited about it. I correspondence might be associated the cotangent bundle of a complex believe it will be more accessible. If to Montonen-Olive duality. But what manifold M in terms of differential I had to bet, I think I have a decent about the fact that Ben-Zvi was only operators on M. This actually was chance to live to see gauge theory claiming to deduce from T-duality fairly close to things that Kapustin and Khovanov homology recognized an approximation to geometric had done earlier. Once these points and appreciated by mathematicians, Langlands duality, rather than the were understood, it was pretty clear and I think I’d have to be lucky Round real thing? At a certain point, I to me as a physicist what is geometric to see that in the case of gauge Table started to suspect that the reason Langlands duality. theory and the geometric Langlands for this was simply that Ben-Zvi was But it was very hard to write a correspondence－just a personal describing the situation in the wrong paper about it. It took about a year. guess. complex structure. The idea was For that year, I felt like someone who Toda: Do you think your that the same T-duality of Hitchin’s had discovered the meaning of life idea of the S-duality and the
23 geometric Langlands can be somehow Savdeep Sethi and Michael Green theory are much more accessible as applied to the honest Langlands and then continued by Green with part of physics. For example, I did not program? many collaborators. In the original understand what Hiraku Nakajima Witten: I see that as being far away. work, Sethi and Green were trying explained yesterday at the Kyoto For me personally－it’s a dream that to understand certain low energy R 4 Prize Workshop, but I think that an eventually number theory would interactions in Type IIB superstring understanding might involve some make contact with physics some time, theory in ten dimensions (here R is of the things that were clear after but I doubt it will be soon. the Riemann tensor). They made an working with geometric Langlands. I There are all kinds of areas where amazing discovery, I would say: the can’t promise but it is worth a try. speci c number theory formulas answer is given by a certain non- Just one obvious thing is that appear in physics, and these may be holomorphic Eisenstein series of although Nakajima did not have clues that the dream will come true weight 3/2. Although my knowledge time to explain the whole picture, at one day. But to really get me excited, of number theory is very super cial, the end of his lecture he was telling somehow the number theory would I think that this sort of thing is much us about the af ne Grassmannian. have to enter the physics in a more closer to the interests of modern Isomorphism classes of t’ Hooft structural way. I’m not that interested number theorists than the kind of operators are associated to cycles in a speci c formula that comes out classical modular forms that usually in the af ne Grassmannian, so if a of a physics calculation in a more or appear in two-dimensional conformal mathematician tells you about the less ad hoc fashion. Number theory eld theory. af ne Grassmannian, you probably would have to be more integrated Ooguri: Those objects which are not want to think about at least part of with the physics to get me excited, totally modular have also appeared in what you are hearing in terms of and I don’t see that happening soon. number theory. ’t Hooft operators. I can make no In my work, I concentrated on the Witten: That is correct. A lot of promises, but I feel it would de nitely geometric form of the Langlands things that number theorists like be worth a try to understand correspondence because I could have appeared in physics, and some what Nakajima was saying from a see that there was hope to really have even appeared in my own work. physicist’s viewpoint. understand it in the context of the Plenty has been found to show that I am sure, at any rate, that there physics-based tools that were at the physics theories that we work on is much more that can and should hand. There might be something as string theorists are interesting in be done to understand much more like that one day for the Langlands number theory. They know something of geometric representation theory correspondence of number theory, about number theory but personally from a physical viewpoint. In fact, but probably a lot is missing and we I don’t see an opportunity to really part of the original work of Beilinson do not know what has to happen make contact in a structural way with and Drinfeld on geometric Langlands rst. I feel that the reason I was number theory in the foreseeable has still not been understood to my able to make progress was that my future. I can’t even formulate what it satisfaction. Here I have in mind the focus was much more narrow than would mean to make such contact, use of conformal eld theory at what trying to understand the Langlands so I can’t even properly tell you what they call the critical level (level -h, correspondence of number theory. we can’t do but I think the time is not where h is the dual Coxeter number) Toda: The relationship between right to do it. to construct the A-model dual of S-duality and geometric Langlands Anyway, that’s why I personally certain B-branes (the ones that are was surprising to me, as the number concentrated on geometric associated to opers, in the language theory seems to be a research area Langlands rather than on number of Beilinson and Drinfeld). Davide far from physics. theory, and geometric Langlands was Gaiotto and I obtained a few years Witten: Nevertheless there have hard enough. It was a lot of work ago a reasonable understanding of been many developments, which to understand it, but I think that what electric-magnetic duality does one day may be seen as important having understood it, many things to the variety of opers, but I still do clues. One of the deepest was that mathematicians do involving not really feel I understand its relation started roughly fteen years ago by geometric aspects of representation to conformal eld theory. However, in
24 Kavli IPMU News No. 28 December 2014 the last few years physicists working question, although there are many It has seemed unclear that a lot on supersymmetric gauge theories areas of current interest on which I of physicists would really get excited in four dimensions and their cousins probably cannot give useful advice, about the sort of details that I was in six dimensions have made several there is one bit of advice that I trying to ll in. So one of my hopes discoveries involving the role of actually would offer to algebraic has been that mathematicians will conformal eld theory at the critical geometers. I do recommend super get excited about developing super level, so the time may well be right to Riemann surfaces. I’m sure there’s Riemann surface theory. I can’t resolve this point. a deep theory there. I can’t promise promise but I think they should. Ooguri: I think we have about 10 anything about how quickly it will Ooguri: Do you expect also new more minutes. Is there any nal emerge. A deep theory will probably physics insight coming out from question? only be developed in the near term if the more precise understanding of enough people get excited about it. perturbative string theory? How to Work with Maybe the workshop we are having Witten: The answer may depend on Mathematicians next spring at the Simons Center will what you mean by physics insight. Toda: I have some general help make that happen but I can’t I think that one understands better question. What kind of problems promise anything. what superstring perturbation theory mathematicians should work on? Ooguri: It’s certainly true that, means if one formulates it in terms Witten: Well, there are lots of when people were working on of integration on the moduli space problems that algebraic geometers the niteness and vanishing of the of super Riemann surfaces. That is study that involve dualities studied by cosmological constant in perturbative insight of a sort. However, I do not physicists. In many of those cases, I string theory 25 - 30 years ago, it was see any evidence at the moment that will not be able to give much advice not totally satisfactory. The complete incorporating super Riemann surfaces as I am not an expert on recent understanding only came after in the way that we understand developments. In some cases, I am still your proper description in terms of perturbation theory will help us struggling to understand things that geometry of super Riemann surfaces. with non-perturbative questions, physicists did quite some time ago Witten: Thank you, Hirosi, and I’m for example, or with understanding that are very relevant. Just to give one glad you think so. Not all physicists better the symmetry structure of example, the Gopakumar-Vafa and agree, because it is possible to express string theory, or whatever may be the Ooguri-Vafa formulas have been very everything in terms of picture- correct concept. in uential for algebraic geometers, changing operators and so on, hiding Yamazaki: Let me ask my last but as a physicist, I was never satis ed the super Riemann surfaces. I think question. You’re working partly in that I understood them. So I actually personally that when one does that, the area of mathematical physics. spent a lot of time in the last year one doesn’t understand properly You have a lot of discussion with with a student (Mykola Dedushenko) what the formulas mean. But not mathematicians, and also write math trying to understand these formulas everybody agrees. papers. better. In this work, I was doing some I think one reason that the theory Witten: Well, I write math papers of the homework that I’d have to of super Riemann surfaces stopped in very special cases where I think understand before even trying to developing in the 1980s was that something I could actually do would answer your question. physicists became satis ed with their be illuminating. Recent examples have We more or less have the paper partial understanding in which the been my work with Ron Donagi on nished on the Gopakumar-Vafa and super Riemann surfaces were hidden. some foundational questions about Ooguri-Vafa formulas, I am glad to There is a tremendous beauty to this the moduli space of super Riemann Round say. subject that I think is simply missed surfaces, and the work with Rafe Table Ooguri: You will talk about it if you try to understand things Mazzeo that I mentioned before on next week at the Kavli IPMU (later that way. I care about it enough to the Nahm pole boundary condition. published as a paper, http://arxiv.org/ have spent several years by now on Yamazaki: I see. So, my question abs/1411.7108). spelling out details of the description is – what’s the advice if a physicist Witten: Going back to Yukinobu’s in terms of super Riemann surfaces. wants to work with a mathematician
25 effectively? right place led you to the question, we do not really understand string Witten: It’s really dif cult to give which you eventually solved and also theory properly as physics－we do advice. Usually producing rigorous it led to new developments. not understand the core ideas behind proofs requires very detailed methods. Witten: Yes. Another case was when it. At an even more basic level, the That makes it hard for a physicist I felt that Beilinson and Drinfeld had mathematicians are still not able to and so I myself have only done that the shogi pieces jumbled on the fully come to grips with quantum in very special cases where I thought chessboard. eld theory and therefore things something was really missing that coming from it are surprises. So for was actually simple enough that I Message to Young Students both of those reasons, I think that the could help do it, if I had the right physics and math ideas generated collaborator. Some physicists would Ooguri: Pieces are placed in a wrong are going to be surprising for a long want to go into more detail and learn way, but, if you look at it in different time. the techniques for rigorous proofs in dimensions, perhaps they are totally I think there are de nitely exciting a particular area, but most physicists aligned. opportunities for young people to I think will only be happy and I have a nal question. In the come and help explain what it all successful doing that in very special interview with Tohru Eguchi, 20 years means. We don’t understand this cases like the ones I’ve picked. ago in Sugaku Seminar, he asked you properly. We got a wider perspective Yamazaki: I see. Is it also true that in about the prospects at the interface in the 1990s when it became clear many of your works, the conversation of mathematics and physics, and that the different string theories are with some mathematicians has been you replied saying that the area had uni ed by non-perturbative dualities an inspiration for you? been certainly growing very strongly and that string theory in some sense Witten: This usually happens when and you predicted that the progress is inherently quantum mechanical. something a mathematician has done would continue in the foreseeable But we’re still studying many involves an aspect of the physics future. Certainly your prediction different aspects of a subject whose that hasn’t been understood and has been amply veri ed in the last core underlying prin ciples are not which doesn’t make sense to me. I 20 years. Given this, my question is, clear. As long as that is true, there mentioned earlier one case involving again, what is your prospect for the are opportunities for even bigger the volume conjecture. For years I next 20 years? Can you also give discoveries by today’s young people. could not understand the results in advice on the future of the eld for But if I could tell you exactly what this area because complex critical young students who will be reading direction you had to go in, I would be points were making exponentially this article? there. large contributions. I kept putting it Witten: First of all, in the last 20 Ooguri: Thank you very much for aside, not able to make progress. years, not only has this interaction taking time to talk to us. It has been Finally, in the summer of 2009, of math and physics continued to be fun. Congratulations again for your I attended a conference at the very rich but it has developed in such Kyoto Prize. Hausdorff Institute in Bonn on the diversity that very frequently exciting Witten: Thank you so much for your 20th anniversary of the Chern- things are done which I myself am kind words on the Kyoto Prize, and Simons theory. I heard more lectures able to understand embarrassingly also thank you for the discussion, on the volume conjecture. To me, little about, because the eld is because the discussion has helped it was just embarrassing to not expanding in so many directions. me remember how much we have understand why exponentially large I am sure that this is going to advanced in the last 20 years. contributions were coming in. I feel continue and I believe the reason it Ooguri: Let’s meet again 20 years vindicated in hindsight for worrying will continue is that quantum eld from now to assess our progress in so much about this question because theory and string theory, I believe, the next 20 years. the answer turned out to be really somehow have rich mathematical Witten: Let’s try. For that we will useful. secrets. When some of these secrets have to all exercise and keep t. Yamazaki: I see. In that case the come to the surface, they often come feeling that the pieces are not in the as surprises to physicists because
26 Kavli IPMU News No. 28 December 2014