Edward Witten Dirac Conversation

It's a pleasure to have Professor Edward Witten with us today, a very distinguished physicist from the Institute for Advanced Study in Princeton. Thank you. He, in fact, he has a long connection with ICTP. He was the first Dirac medalist in 1985. his important work on anomalies at the time when the salam was around which he shared with a great soviet physicist yakov zeldovich and then of course his list of his awards very long I will not go through that, but in 1990 he received also the Fields Medal. So he is a distinguished physicist as well as mathematician and he is here at ICTP this week for the huddle we have started at ICTP on quantum entanglement, space time and black holes, which explores these fascinating new connections between quantum entanglement, space time, and black holes. geometry, black holes. So before going into the history maybe let's start with the present and the future. I mean the past decade or so or even two decades one could say has revealed this amazing connection between Theories without gravity, theories with gravity, quantum mechanics, quantum field theory on one hand, and general relativity on the other hand. And this seems to be a very rich exploration which is, I think it's fair to say that the full... The implications of it are far from clear. That's very good. So yeah, maybe do you want to start with your thinking, what you're currently thinking about, what you think are the important points in this beautiful connection? Well, 20-30 years ago, people like Susskind and Ruft, trying to unravel problems about quantum black holes, had the idea of holography, that somehow... the physics in the interior and the presence of gravity is described holographically by some kind of ordinary quantum system on the boundary it sounded like a very wild idea but a couple of years later in nineteen ninety seven one martin by combinations in string theory, actually made what was then interpreted as a very concrete proposal for a holographic duality between ordinary theory on the boundary and quantum gravity in the interior. So we've known since 1997 that in some sense this kind of duality between ordinary physics on the boundary and physics with gravity inside is real, but understanding it has been another matter. and has come in different waves and as you said I think we're far from really getting to the bottom of things. So we can go back perhaps to the first string super string revolutions so you have been leading many of these developments which have been the leading light and We know that quantum field theory in some sense has been reigning supreme that whatever is the experimental observations are very well described by. We don't really need any further modification to our theory just to explain the data, but evidently this is not satisfactory for many theorists. There's really a problem with gravity. So quantum field theory works beautifully. to describe all the phenomena we know about that don't involve gravity. But it turns out that if you try to do gravity in the framework of quantum field theory, you run into a lot of trouble, basically because Einstein's highly nonlinear mathematics doesn't work well with quantum theory. So this was a puzzle for a long time, of course. It's a rather esoteric puzzle because it's... It's not clear how it can be put to the test experimentally, but conceptually we've got both quantum mechanics and gravity in the same world, so they have to work together somehow. And I don't think anyone really ever had any good ideas about it. But in the early 70s, people who actually were trying to solve a different problem stumbled in string theory upon a way to generalize ordinary quantum field theory in a way that makes gravity unavoidable rather than impossible. That was string theory. And the main reason, in my opinion, for all the interest in string theory ever since has been precisely that the pre-string framework in which we so successfully described the rest of the world of physics is really not compatible with gravity while string theory forces gravity upon us. Yeah, so that's, yeah, because sometimes we take your colloquium, I think, tomorrow will be on this topic, that one somehow takes some of the early successes of string theory, especially for the practitioners in the field for granted, you almost get kind of jaded by that success, but it's actually extraordinary. My title tomorrow is what every physicist should know about string theory. I know that younger colleagues tend to find it disheartening that under that title I lecture about things that are so old-fashioned. Yes, but even though they are old-fashioned, I mean by our standards, given the... Old-fashioned here means they go back to the 70s and 80s. That's right, yeah. But it's, we have not really fully understood. That's completely correct, yes. We don't, we're grappling with things that we stumbled upon bits and pieces of. There's been this incredible trail ever since, but we definitely haven't gotten to the bottom of any of it. That was true a moment ago when we were talking about holography and gravity, but it's also true when we discuss about what's really going on when quantum fields are generalized into strings. There's a fundamental mystery about that, what it really means. A dream is that all these things are tied together, maybe. Okay, we're hopefully we're missing some general lesson about The link between gravity and thermodynamics, likewise, we're presumably missing some general lesson about what's going on, what really is string theory even. Would it be a dream if all those general lessons would somehow fit together, but we'll have to see. Yeah, I had the good fortune of learning string theory from you, as a member. You had this wonderful... Of course I did. This was in the... when you were writing the Green Schwarzschild book. Well in the spring of 87 I told you. Yes, in the spring of 87. I was a student there. I didn't put two and two together, but yes that was roughly what you all said. And I remember it was attended by the people in the greater Princeton area, which is in the 50 kilometer radius. But, and it looked amazing, but within 10 years, somehow, despite all the successes of string theory, it was completely turned topsy-turvy. Was it? Well the... The subject was completely transformed by new discoveries within a decade of them. At that time, if you'd asked me for what was the best formulation I could give of what's the unsolved problem about string theory, I would have thought it was to be found... Einstein... string theory, at some sense, even at the classical level, is different from Einstein's theory, even before you quantize it. Einstein's classical theory, which was all he understood, he never did quantize it, was based on new ideas about the dynamics of space-time and the curvature of space-time. And string theory in some sense clearly generalizes those concepts in a way we don't understand very well. In 1987, I'm sure I would have said that the deep problem was to understand the generalization of the Einstein-Hilbert action and of Einstein's principle of general covariance. No, string theory. Well, string field theory is an attempt to answer that. I would do not believe I would have been committed to string field theory as the only approach, but some kind of classical theory whose quantization gave strings, I would have thought was the goal. But within a decade, there were non-perturbative dualities discovered that kind of made it clear in different ways that there's much more going on than that. So this was what is sometimes referred to as the second superstring. Yes. Although if I were doing the counting, it would be the third. The first superstring revolution should have been the period when Veneziano, Virasaro and others, Amati and others of that generation pioneered things. And then the first period I participated in should have been called the second one. And the phenomenon in the mid-90s I would have called the third superstring revolution. And holography was yet to come, so duality. Well, I'm not sure, I would give up after three, I wouldn't know how to put holography in the numbering. Yes, also, I mean, going back to duality, that also is another extremely surprising, and it's not only limited to string theory, also in quantum field theory, where people thought they had understood quantum field theory, now with the work of... You and Cyborg and we somehow think many other people, we think of quantum field theory in quite a different way. Well, of course in two dimensions it was known that there were all kinds of crazy phenomena and dualities. And I'm sure some people suspected it was also true in four dimensions, but we couldn't demonstrate it. Part of what's surprising is that all that, or to some of us it might be, to me since I tend to be conservative, it was a bit surprising that all that was true in four dimensions. But it was even more surprising that you could understand it. Yes. Because of the way that supersymmetry simplifies the study of the dynamics. Yeah, and perhaps it was even more surprising, I remember this You know, it's one of these events where I'm happy to say I was there. It was 1995, I think. Oh, the strings contrast? You see, it was really kind of extraordinary. I remember the buzz, and it was incredible. Because nobody thought that, okay, maybe four-dimensional gauge theories, but all the way to 10-dimensional string theory. It's a funny thing there because to a large extent in my presentation there I was putting together things that other people had really been saying. We just made little improvements here and there in some cases. No, it's very true that Sen and Townsend and Hull and many other people had. Yes, let's remember John Schwartz. In fact what got me started at the previous strings meeting which was in Berkeley in 1993, John Schwartz was much more excited than I'd seen him in a number of years. Yes. And what he was so excited about was the work he was doing with Sam on non-perturbative dualities. Yes. That, I didn't at the time know what to do with it, but that got me to start thinking in that direction. And it actually was only a few months after that that Natty, well Natty at Cyberg was working on dynamics of n equals one. Yes. And… He sort of recruited me to work with him on n equals 2. I think the reason he thought I'd be interested was that I had earlier discovered that n equals 2 Yang-Ngol theory is related to 4-manifold theory. And I think that he thought, and I know Michael Atiyah thought, that understanding the dynamics might have implications for 4-manifold theory. And that certainly proves to be correct. But I wasn't really thinking that way at the time. So I started working with Natty because his work on n equals 1 had been interesting. This was 1994. The discussion started in the fall of 93. But the papers emerged in the spring or summer of 94. So for a few months there wasn't much progress, but we knew that there were these duality conjectures going back to 100 and all of them, and wondered if they were relevant. And then during those months, A very famous paper by Aashok Sen came out, which for me was completely a turning point. This was his paper and the two months after that. Yeah, it's a beautiful paper. So I thought, see, even though what Sen and Schwartz had done and that Schwartz was so excited about was very pretty and very interesting, it was sort of similar in spirit, I thought. ...to things that it would have done before. Conceptually, it did involve... It was very nice that it worked, and it was striking, but Sen's calculation of the two monopole balance state... The diagram to find the SL to Z. Yes. Well, that was, I thought, a real conceptual advance in the methods. But also, it was... You see, if anyone had had the vision to try to do that calculation... It might have been done a few years before. So that taught me that we were probably missing other opportunities as well. So after that paper came out, we were, I think, more focused on the role of duality, the possible role of duality, than we'd been previously. But we didn't have a clear picture of what duality was supposed to mean for N equals 2. We'll eventually be interested in that. It's really. Well, at least for the simplest n equals 2 theory, it's a statement about the infrared, not about the fundamental microscopic description like in n equals 4. Anyway, but anyway, focus us more on the right direction of thinking about long poles and dylans, which ultimately were the right ingredients to use. Yeah, so and that the string duality is somehow all gelled within four or five months because I remember and you may not remember this, but I remember I met you in Jerusalem. There was a New Year's party in 1994. The beginning of 1994? End of 1994. And you said, oh, there is not much time left in 1994 for new ideas. But it was already the seeds of... This must have been there in your mind because I think in May we had this whole change of perspective and then which already other people like Helen Thompson were thinking about. Well I'd spent part of, well after the work with Cyberg, it then was clear that all this was relevant to foreign unfolds. So I spent part of the fall of 94 trying to work on that but my plan for what I wanted to do was too grandiose. Eventually, a few years later, with Greg Moore's help, we did what I originally wanted to do, that involved integration over the U-plane. Ah, U-plane, okay, and also it's related to Mach-Module. And eventually what you and I did just a year or two ago. But at some point, luckily, in the fall of 1994, I decided that was too complicated and I had to do easier things. But then the mathematicians told me anyway I should write up what I knew, even if I couldn't do it the way I wanted. No, but you also wrote this very interesting paper with Cameron, which was also in a way... I had the glimpses of type 2 heterot... I mean, in hindsight, the type 2 heterot... Yeah, yeah....deformed. Yeah, well, Kerman and I, we were counting in sometimes. But you had the right... Kerman......speculation there. Sort of, yeah. Well, anyway, we discovered very strange facts, and Kerman told me, well, we're going to understand these facts. I said, well, no, it's impossibly far away. But Kurman was right, because within a year or two... Yes, within a year it was more or less understood. But anyway, after the cyber-written papers, I spent a few months trying to do what was too hard on four manifolds, then had the sense to just write up a simple version and put the rest aside. Then I spent a few months... It bothered me that there were five string theories when only one was needed, so I thought maybe some of them were inconsistent. And I spent a few months at the beginning of 1995 trying to prove some of the string theories are inconsistent. Some global anomalies? Well, I wanted global anomalies but I couldn't find any. I remember I had two arguments. One, I can't remember at all now what it was. But the other, I thought that if you imagined the strong coupling behavior of type IIa, you'd find a contradiction. Because... How long? What was that? It looked like the BPS spectrum wasn't physically sensible, but at a certain point I realized that the BPS spectrum actually wasn't physically sensible. So after that I stopped thinking that these string theories were inconsistent, which kind of was a silly thing to even try for, because it was at the time almost known that they were dual to each other. It wasn't completely known, but there were... The two heterotic strings were dull, the type two strings were dull. Dahlia Ley and Polchinski had a story relating type one to type two. I guess there was the heterotic world and the type one type two world. Those hadn't yet been linked. Okay, you could have conceivably shown that one of the two worlds was inconsistent, but you couldn't have hoped to get it down to one string theme. But anyway... As you know, within a relatively short time it became clear that no one was going to get rid of any of the screen. They were all different limits of each other. That's right. Yeah, so that brings me to, I mean, I just want to ask you this more, it's a bit... philosophical question but I think it's important question because I think there seems to be some misunderstanding about I mean oftentimes I mean it's evident that these are some very deep concepts irrespective of how they will fit into a description of nature. It's not immediately clear. And often times in physics, even with... I mean this is a much more grand structure, but even with ideas like, let's say, Higgs mechanism, it took 50 years to find experimentally. So clearly, sometimes in popular imagination, I think somebody used this word, paparazzi, you know, going back to... Popper's principle of falsifiability applied in a rather naive way. That's okay if you can falsify it. But oftentimes, actual practitioners of physics are pursuing ideas with a long-term view, with not immediate falsifiable falsifiability. But that is nevertheless a kind of an internal criteria of what constitutes. important breakthroughs. I mean, there is no question in the minds of those who work in this field that big advances being made. Yeah, do you have... Well, one thing that's worth mentioning though is that apart from the dream of understanding physics at a deeper level, involving gravity. Work in string theory has been useful in shedding lights and more conventional problems in quantum field theory and even in condensed matter physics and as well with applications in mathematics. Apart from its intrinsic interest, those successes are one of the things that tend to give us confidence that we're on the right track. Yeah, yeah. Because, speaking personally, I find it implausible that a completely wrong new physics theory would give rise to useful insights about so many different areas. Yeah, actually I heard a very distinguished experimental physicist said this, that his approach to discovering new physics is to really improve the tools. I mean it's not an attitude that you might expect to hear because this is that if you have new tools, then new discoveries follow. It's not always that you go looking for a specific physical phenomena. And that I think is also true in theoretical explorations and there seems to be a really clearly well understood internal criteria for what constitutes important advance and what is just. Not such an important one. It's probably a difficult question to answer, but okay. You have been, I mean, string theory community has been clearly pursuing something quite deep and important. And it may very well be that experimental realization could be many years or decades. Unfortunately, that's quite possible. We have to hope for good fortune. It depends. There are things we can't control, how the nature of the answer, but how things work out in the real world. Yes. And the experiments we're able to do. So let me ask you a bit about your connection with the ICTP. I mean you had been, you came here, I think Salam was instituted this prize, the Dirac medal, and I think he was very proud to have chosen you, President Henry. I was actually a rather regular visitor at the ICTP for quite a number of years, starting around 1977. My wife is from southern Italy. We used to come to Italy often in the summers and very frequently spend some time here at the ICW. I must say that those are very exciting years to visit the ICDP, partly because of the environment that SOM created. Yes, and I'm very happy that you're here now for this week. Yes, I know that you come from an academic family. Your father was also a physicist, and your wife is also a theoretical physicist at Princeton. So, do you want to share something about what brought you to physics? Because I know that you didn't. You probably wanted to resist what your father was doing. I zigzagged a little bit, but then I discovered that my fate was physics. As you may know, my father, as you said, is a physicist, whose work has largely been in general relativity. But then you were in journalism or something? I zigzagged a little bit. You zigzagged a little bit, okay. Okay now, as you know, Salam created this organization, which is also something that is... dear to my heart is that apart from pursuing excellence and the frontiers of science, which is for example the Institute for Advanced Study is also a place like that, one of the impulses behind ICTP is to make the knowledge globally available, overcoming the barriers of geography, class and gender. The ICTP has done a great job in fostering physics and countries that have needed help in getting started. So even in a country like India, which as you know, the string theory community in India is now very strong, but I remember people are very important. If you go back to the 70s, I think that the OICTP deserves a lot of credit for how physics has advanced in India. Exactly. So and the third aspect is sort of promoting international cooperation through science. Yes. And you have been Perhaps a keen observer I think of this the SESAME project that you know in Jordan. I think ICTP played many at many junctures. I mean there were of course many contributors to this very important initiative but ICTP if I'm not wrong held an important conference in the HAD bringing together scientists from different countries who might not have been on politically friendly terms. But somehow this power of science as a common platform and a common language to bring them together and it looked like a kind of a pipe dream in fact at that time to have something in Jordan supported by Iran and Israel and... Yes, it seemed like science fiction and frankly whenever one hears what's happening in Sesame it sounds like news from an alternate universe. That's right. News from an alternate... universe where there's peace in the Middle East, but the countries are cooperating. Yes, I think, yeah, so that's one I wanted to ask you about that because, of course, difficult political, I mean, it's naive to expect that difficult political problems will get resolved through science diplomacy. But nevertheless, I think science can play at least a facilitator role. At least give the different countries an example of an alternative possibility. yeah to live together peacefully. Okay so now I want to ask you a bit about your other half of your distinguished career having to do with mathematics. So I mean you have made many very deep contributions to mathematics if I may read from the citations of your Fields Medal. Time and again he has surprised the mathematic community by a brilliant application of physical insight. Leading to new and deep mathematical theorems. And this, you know this famous quote of Eugene Wigner that, about the unreasonable effectiveness of mathematics and physics. But I want to ask you about this, the other way around, the unreasonable effectiveness of physics and mathematics. Evidently going back to Newton with calculus. Yes, certainly, I was about to say that. Calculus is very central. Mathematics became far more powerful after calculus was developed and it was developed because it was needed for physics. But then with Dirac, with quantum mechanics, it spurred a lot of... but I think you have brought in, and I think you can be singularly credited for this, really quantum field theory as a tool for addressing, you know, various methods in quantum field theory. So how do you, what is... I think the essential fact is that, first of all, quantum field theory is difficult to understand mathematically, because it's a little bit of a mystery why this is, but it's difficult to make it rigorous. So it's clearly an important tool in physics, and it has a lot of mathematical secrets, but mathematicians have a difficulty understanding it, so they have trouble learning the... mathematical lessons that can be deduced from quantum filter. So physicists, because they work with quantum filter all the time, either in particle physics or as well in condensed matter, have these potential tools at our disposal, which turn out to be mathematically powerful. But I think that's half of it. The other half of it is that string theory has, well, first of all, it's also very difficult for mathematicians to appreciate because prerequisite, basically, is to understand quantum flip theory as a physicist, and then you can understand what physicists are trying to do with string theory and proceed from there. But string theory seems to know a lot of mathematical secrets. Yes. And at the present stage of knowledge, physicists can work with that. Even though we don't understand string theory very well, we can still work with it more effectively than the mathematicians can. So there are opportunities there to extract some of these mathematical consequences. But you did try, I think, I mean, there was, you had this set of lectures, I remember, Delene at the Institute for Advanced Study. Yeah, how do you, how much do you think you succeeded in making? Well, there was certainly, we had a special program and then the lecture notes were published. In about 1996 and 7, roughly, trying to explain quantum fields and strings to mathematicians. I think there was only a moderate success. There was some success, but I've come to believe that it's actually really important to have some sense of how quantum field theory works in the real world. If you try to understand quantum field theory just as a mathematical subject, Without developing the intuition you get from its applications in the real world, it proves to be difficult. I guess I mean there's a clear problem of I mean maybe understanding QCD would be... Actually pure gauge theory rather than... Pure gauge theory, that's what I meant, yes. That would be a step towards... Well, if that problem were solved that would be... It would mean that... Well, it's a little hard to speculate what would come after because it would depend on how... difficult and technical the solution was, but potentially quantum field theory would be more accessible mathematically if that kind of thing were understood. But the clay problem as defined is extremely difficult because it asks not just for an existence proof but for proof of the mass gap, which I actually worry makes it too hard. Yeah, even the existence... I feel that the existence could be proved without proving the existence of the mass gap. The existence is basically a question about the ultraviolet. Limitless. You need some control of the infrared. I see, I see. But you see, you might be able to prove that the theory exists with parallel decay of correlations in the IR without being able to prove that they have exponential decay. Think of the two-dimensional non-linear signal model, say, of a two-sphere. which was suggested by Polyakov a long time ago, it was a practice problem. There's a theta angle for generic values of the theta angle, it's massive in the infrared, but at theta equals pi, it's actually believed to have mass less than the infrared. So, it's not necessary for the existence of the UV theory. Right. That's what I want to say. Yes. And you can well imagine, in this case, because there's one over n expansion, Well, in this case you might be able to use the one-over-n expansion to prove results that would be true for a sufficiently large m. But in a case where you didn't have a one-over-n expansion, you could well imagine being able to prove existence and not being able to decide the question of a mascot. Yes, okay, maybe they should have two million dollars. But you mentioned Michael Atiyah, I mean, so, I mean, your work on supersymmetry anomalies had this beautiful connections with index theory. And then you've done all this, all the work on Seiberg-Witten theory, Knott theory, Donaldson theory, I mean, the amount number of applications are also geometric Langlands program. It's quite astonishing how much power quantum field theory has given to make some progress in difficult mathematical problems. But Atiyah was an important influence on your career and your thinking. So you want to say something about your experience with or what? Well, one thing I'd like to say is that he's the one who told me. about Montaigne and Olive duality. Even though it's a physics paper? Yes, I think he had probably been talking to Peter Goddard. Or Olive as well. Olive, probably. Well, we have to remember that before Montaigne and Olive, there was Goddard, Noitz, and Olive. Yes. Well, I don't really know. I'm almost certain that he had been talking to Goddard as well. Anyway. Well, what really happened was that, at least from my point of view, what happened was that the Yang-Mills Instanton and the work of solving the U1 problem using instantons, first of all got physicists excited about differential geometry. We had modern questions in topology and differential geometry we hadn't been familiar with. And then I first heard the names of Tia and Singer from a paper by Albert Schwartz. Albert Schwartz was the one, it helped to solve the one problem using a Fermion zero mode in an instant non-field. Similar to work of Jakiev and Reggie involving monopoles. It helped solve the Dirac equation in an instant non-field and found a zero mode. Then Albert Schwartz explained that the zero mode was a consequence of the Ittiaz-Zenger index theorem. Yes. And certainly, I was shown that paper by Sidney Coleman, and it was one of several important papers Sidney showed me. This one I think would have... You were a postdoc at Howard? That's right. This paper I think I would have learned about anyway. But some of the important papers Sidney showed me never became well known. And I think I would not have known about them otherwise. An important example, in view of my later work on the Jones polynomial, was Albert Schwartz's paper on analytic torsion. And what nowadays is called BF theory. He didn't know to call it BF theory. He called it a degenerate quadratic functional. Well, in hindsight, that's more or less had to do with the abelian limit of Chern-Simons theory. So, later on... Okay, Sydney showed me that paper when it came out in about 77 or 78, and I didn't think about it again until a decade later when I remembered it when it was handy. Anyway, going back, the instanton and its connection to the index theorem, I know there are other things that maybe Atiyah and Singha would have told you about, but were less obvious from my point of view, caused physicists and mathematicians to start talking more around 1976 and 7. Yes. And Atiyah visited at Harvard where I was a postdoc. And as a result of our discussions there, he invited me to visit him in Oxford at the end of 77. And I went, and for a few weeks we talked a lot about gauge theories, and by this time they had discovered the ADHM construction of instantons. You have to imagine that at that time, I was a little younger than either of us is right now. We were all younger. With a little bit of patience. And Atiyah probably had the disconcerting experience of discovering that whatever they did, a physicist would tell them that they should do something else that would help even more with the quantum theory. Anyway, we talked a lot for a few weeks, and then… Well, what did you tell him to do? Well, something that would help with quantum manuals. But in hindsight, an incident on gas only has limited use for. The quantum theory. But roughly 20 years later we learned that the ADHM construction and the instanton gas are useful for supersymmetric versions of the theory. But it would have been a stretch to think of that in 1977. Anyway though, toward the end of my stay in Oxford... You went to visit him in Oxford? For about a month. Toward the end of my stay in Oxford, he showed me that these two papers, I think of God and Roots and Olive and... Montaigne and Olive and suggested I go to London to talk to Olive about the Montaigne and Olive paper. I couldn't, well, so I did. I realized possibly part of what was going on is that after talking to me so much for a few weeks, he wanted to find something else for me to do for a day or two. But anyway, I went to London. Enjoy London. Yeah. Talked to, I think I'd not been in London before. And I didn't get to see much because all I did on that visit was talk to David Olive. By the time I arrived I was skeptical because the Montaigne and Olive story was expressed in terms of a purely bosonic angles, coupled to a scalar. There were a lot of problems with it. For starters, they needed the Higgs potential to vanish and that isn't really a well defined concept quantum mechanically. And you can go in for there and raise all kinds of technical objections. So I was a skeptic. But, as we were talking, we realized that the supersymmetric version of the conjecture would make more sense. And by the end of the day, we realized that maybe the facts they were talking about could justify better in n equals 2 or n equals 4 superangles theory, which led to the paper we wrote. Yeah. I think that was a very key insight because discovering the central charges in supersymmetric algebra is because Because this whole work on BPS states and to get any control over the strong coupling was would not have been possible. Well, it is true that it was important later, but it is also true at that time I drew the wrong conclusion. Which was? Well, you see the conclusion, okay. As I told you in this case, I was very impatient, but I also was very conservative in my attitudes. So I was unlikely, very reluctant to adopt what seemed like an extreme hypothesis, if the facts had a more conservative explanation. explanation. And I felt we had been able to explain the facts that Montaner and Roloff had pointed out without assuming a non-perturbative duality. We had been able to explain those facts just from supersymmetry. And that caused me to remain skeptical about Montaner and Roloff duality. In fact, I remained skeptical. So, to what we said before, well, the interaction with Schwartz in 93 and then Senn's paper six months later or whatever, exactly whenever it was. After Senn's paper, I gave up on being skeptical. We simplified the story before because there were a few other clues that were important in wearing down my skepticism before Sam's work, including your paper with Jeff Harvey. Yes, okay. And a few others. But anyway, it was partly because of my skepticism that I... But all of an eye had shown that... n equals 2 or n equals 4 were more reasonable arenas for Montaigne and all of duality. But if you took Montaigne, but then there was, okay, okay, which of them? I think, I'm having trouble remembering now, but I think it was Osborne. Yes, Osborne, Osborne, yes, Osborne put it to n equal to 4. Yeah, Osborne showed that if you worry about the monopole spin, you find it should be n equals 4 because the monopoles... It was multiplied, yes. The monopoles for n equals 2 are in hypermultiplots, but for n equals 4 they're in vector multiplets. So if you were taking the Montaigne and Olive conjecture seriously, which I wasn't at this time because I thought we'd found a conservative explanation of the facts, then you would have possibly been inspired to do what Osborne did, which was certainly an important thing to do in hindsight. So I didn't really go back to it seriously. Until 1993 when some of these other things were happening that had worn down my skepticism. The experience with non-prohibitive dualities in string theory, on the two-dimensional ones, had been part of it. Mirror symmetry. Mirror symmetry, for example, and even abelian teeter. But also, another paper that... had an impact on me was by Cal and Harvey and Strominger. Yes. On small instantons. Oh. As having had five bins. Well, I was puzzled about the meaning of the phenomenon they were finding at the core of the small instanton. Yes. And later development understood. That's a good thing. Yeah, right, right. But at the time we couldn't. But I wanted to ask you, for example, your work on index theory, I mean, written index, and it's a kind of a beautiful generalization of index theory, which Atiyah played over. Or was that a separate path? Did discussions with Atiyah inspire you to think in that direction, or this was something else? Well, Atiyah helped me understand what the index meant and why it was a topological invariant. But if you understand the concept of the index, You can generalize it to a supersymmetric theory. Right, right. Oh. So that was something... That was a little bit different. Where I was more directly influenced by Tia and Bott, though, was in Morse theory. Yes. So, at this Karjev Summer School, I forget which year it was, maybe 1979, for some reason, Tia and Bott had taken it upon themselves to educate us about Morse theory. I had most definitely never heard of Morse theory, even of the name Morse. My guess is that none of the physicists in the audience had heard of Morse Theory before, but I can't say for sure. Anyway, they gave us a couple lectures on Morse Theory and it was very cute. Interesting. But I didn't have anything to do with it. But later I was trying to understand why it was so difficult to spontaneously break supersymmetry. And since it was difficult, I kept looking at simpler and simpler models. And finally I got to what seemed like the simplest model of all, which was in quantum mechanics, in other words, zero plus one dimensions. with scalar superfields and superpotential, it was still hard to break supersymmetry. And finally, I realized, well, at a certain point, I think I was in a swimming pool in Aspen, Colorado, I remembered Bott's lecture about what he called baby Morse theory, and realized that the superpotential had been interpreted as a Morse function. So, that was directly influenced in more detail I would say by a TN bot. Just defining the index of a supersymmetric quantum field theory, I would say is not that hard if you have been exposed to the index in quantum. Yeah, once yeah. And I remember reading this paper of yours in India as an undergrad, which really inspired me to wanting to go to about supersymmetry breaking in 1983, 1982. Constraints and so on. Okay, so one last question I want to ask you. Going back to Eugene Wigner's original quote. I mean, it is true that to some extent the unreasonable effectiveness of mathematics could be explained by maybe the way our brain works. I mean, we somehow are able to abstract a triangle from a Wigley triangle and make some generalization and discover Euclidean geometry. But in some cases the connection is so distant, like complex numbers playing a role in quantum mechanics, something which is completely invented by human beings. being so important in the description of the real world. I don't know, I mean, you probably have nothing more to say. I have nothing to say except a slightly tongue-in-cheek remark that it's as if the universe had been created by a mathematician. By a mathematician. Okay, so it was a pleasure, Edward. Oh, thank you. Thank you.

He was the first Dirac medalist in 1985. his important work on anomalies at the time when the salam was around which he shared with a great soviet physicist yakov zeldovich and then of course his list of his awards very long I will not go through that, but in 1990 he received also the Fields Medal. So he is a distinguished physicist as well as mathematician and he is here at ICTP this week for the huddle we have started at ICTP on quantum entanglement, space time and black holes, which explores these fascinating new connections between quantum entanglement, space time, and black holes. geometry, black holes. So before going into the history maybe let's start with the present and the future. I mean the past decade or so or even two decades one could say has revealed this amazing connection between Theories without gravity, theories with gravity, quantum mechanics, quantum field theory on one hand, and general relativity on the other hand.

And this seems to be a very rich exploration which is, I think it's fair to say that the full... The implications of it are far from clear. That's very good. So yeah, maybe do you want to start with your thinking, what you're currently thinking about, what you think are the important points in this beautiful connection? Well, 20-30 years ago, people like Susskind and Ruft, trying to unravel problems about quantum black holes, had the idea of holography, that somehow...

the physics in the interior and the presence of gravity is described holographically by some kind of ordinary quantum system on the boundary it sounded like a very wild idea but a couple of years later in nineteen ninety seven one martin by combinations in string theory, actually made what was then interpreted as a very concrete proposal for a holographic duality between ordinary theory on the boundary and quantum gravity in the interior. So we've known since 1997 that in some sense this kind of duality between ordinary physics on the boundary and physics with gravity inside is real, but understanding it has been another matter. and has come in different waves and as you said I think we're far from really getting to the bottom of things.

So we can go back perhaps to the first string super string revolutions so you have been leading many of these developments which have been the leading light and We know that quantum field theory in some sense has been reigning supreme that whatever is the experimental observations are very well described by. We don't really need any further modification to our theory just to explain the data, but evidently this is not satisfactory for many theorists. There's really a problem with gravity. So quantum field theory works beautifully.

to describe all the phenomena we know about that don't involve gravity. But it turns out that if you try to do gravity in the framework of quantum field theory, you run into a lot of trouble, basically because Einstein's highly nonlinear mathematics doesn't work well with quantum theory. So this was a puzzle for a long time, of course. It's a rather esoteric puzzle because it's... It's not clear how it can be put to the test experimentally, but conceptually we've got both quantum mechanics and gravity in the same world, so they have to work together somehow.

And I don't think anyone really ever had any good ideas about it. But in the early 70s, people who actually were trying to solve a different problem stumbled in string theory upon a way to generalize ordinary quantum field theory in a way that makes gravity unavoidable rather than impossible. That was string theory. And the main reason, in my opinion, for all the interest in string theory ever since has been precisely that the pre-string framework in which we so successfully described the rest of the world of physics is really not compatible with gravity while string theory forces gravity upon us. Yeah, so that's, yeah, because sometimes we take your colloquium, I think, tomorrow will be on this topic, that one somehow takes some of the early successes of string theory, especially for the practitioners in the field for granted, you almost get kind of jaded by that success, but it's actually extraordinary.

My title tomorrow is what every physicist should know about string theory. I know that younger colleagues tend to find it disheartening that under that title I lecture about things that are so old-fashioned. Yes, but even though they are old-fashioned, I mean by our standards, given the...

Old-fashioned here means they go back to the 70s and 80s. That's right, yeah. But it's, we have not really fully understood. That's completely correct, yes. We don't, we're grappling with things that we stumbled upon bits and pieces of.

There's been this incredible trail ever since, but we definitely haven't gotten to the bottom of any of it. That was true a moment ago when we were talking about holography and gravity, but it's also true when we discuss about what's really going on when quantum fields are generalized into strings. There's a fundamental mystery about that, what it really means.

A dream is that all these things are tied together, maybe. Okay, we're hopefully we're missing some general lesson about The link between gravity and thermodynamics, likewise, we're presumably missing some general lesson about what's going on, what really is string theory even. Would it be a dream if all those general lessons would somehow fit together, but we'll have to see.

Yeah, I had the good fortune of learning string theory from you, as a member. You had this wonderful... Of course I did. This was in the... when you were writing the Green Schwarzschild book.

Well in the spring of 87 I told you. Yes, in the spring of 87. I was a student there. I didn't put two and two together, but yes that was roughly what you all said. And I remember it was attended by the people in the greater Princeton area, which is in the 50 kilometer radius.

But, and it looked amazing, but within 10 years, somehow, despite all the successes of string theory, it was completely turned topsy-turvy. Was it? Well the...

The subject was completely transformed by new discoveries within a decade of them. At that time, if you'd asked me for what was the best formulation I could give of what's the unsolved problem about string theory, I would have thought it was to be found... Einstein... string theory, at some sense, even at the classical level, is different from Einstein's theory, even before you quantize it. Einstein's classical theory, which was all he understood, he never did quantize it, was based on new ideas about the dynamics of space-time and the curvature of space-time.

And string theory in some sense clearly generalizes those concepts in a way we don't understand very well. In 1987, I'm sure I would have said that the deep problem was to understand the generalization of the Einstein-Hilbert action and of Einstein's principle of general covariance. No, string theory. Well, string field theory is an attempt to answer that.

I would do not believe I would have been committed to string field theory as the only approach, but some kind of classical theory whose quantization gave strings, I would have thought was the goal. But within a decade, there were non-perturbative dualities discovered that kind of made it clear in different ways that there's much more going on than that. So this was what is sometimes referred to as the second superstring.

Yes. Although if I were doing the counting, it would be the third. The first superstring revolution should have been the period when Veneziano, Virasaro and others, Amati and others of that generation pioneered things.

And then the first period I participated in should have been called the second one. And the phenomenon in the mid-90s I would have called the third superstring revolution. And holography was yet to come, so duality.

Well, I'm not sure, I would give up after three, I wouldn't know how to put holography in the numbering. Yes, also, I mean, going back to duality, that also is another extremely surprising, and it's not only limited to string theory, also in quantum field theory, where people thought they had understood quantum field theory, now with the work of... You and Cyborg and we somehow think many other people, we think of quantum field theory in quite a different way. Well, of course in two dimensions it was known that there were all kinds of crazy phenomena and dualities. And I'm sure some people suspected it was also true in four dimensions, but we couldn't demonstrate it.

Part of what's surprising is that all that, or to some of us it might be, to me since I tend to be conservative, it was a bit surprising that all that was true in four dimensions. But it was even more surprising that you could understand it. Yes.

Because of the way that supersymmetry simplifies the study of the dynamics. Yeah, and perhaps it was even more surprising, I remember this You know, it's one of these events where I'm happy to say I was there. It was 1995, I think. Oh, the strings contrast?

You see, it was really kind of extraordinary. I remember the buzz, and it was incredible. Because nobody thought that, okay, maybe four-dimensional gauge theories, but all the way to 10-dimensional string theory.

It's a funny thing there because to a large extent in my presentation there I was putting together things that other people had really been saying. We just made little improvements here and there in some cases. No, it's very true that Sen and Townsend and Hull and many other people had.

Yes, let's remember John Schwartz. In fact what got me started at the previous strings meeting which was in Berkeley in 1993, John Schwartz was much more excited than I'd seen him in a number of years. Yes. And what he was so excited about was the work he was doing with Sam on non-perturbative dualities.

Yes. That, I didn't at the time know what to do with it, but that got me to start thinking in that direction. And it actually was only a few months after that that Natty, well Natty at Cyberg was working on dynamics of n equals one.

Yes. And… He sort of recruited me to work with him on n equals 2. I think the reason he thought I'd be interested was that I had earlier discovered that n equals 2 Yang-Ngol theory is related to 4-manifold theory. And I think that he thought, and I know Michael Atiyah thought, that understanding the dynamics might have implications for 4-manifold theory.

And that certainly proves to be correct. But I wasn't really thinking that way at the time. So I started working with Natty because his work on n equals 1 had been interesting.

This was 1994. The discussion started in the fall of 93. But the papers emerged in the spring or summer of 94. So for a few months there wasn't much progress, but we knew that there were these duality conjectures going back to 100 and all of them, and wondered if they were relevant. And then during those months, A very famous paper by Aashok Sen came out, which for me was completely a turning point. This was his paper and the two months after that. Yeah, it's a beautiful paper.

So I thought, see, even though what Sen and Schwartz had done and that Schwartz was so excited about was very pretty and very interesting, it was sort of similar in spirit, I thought. ...to things that it would have done before. Conceptually, it did involve...

It was very nice that it worked, and it was striking, but Sen's calculation of the two monopole balance state... The diagram to find the SL to Z. Yes. Well, that was, I thought, a real conceptual advance in the methods.

But also, it was... You see, if anyone had had the vision to try to do that calculation... It might have been done a few years before. So that taught me that we were probably missing other opportunities as well.

So after that paper came out, we were, I think, more focused on the role of duality, the possible role of duality, than we'd been previously. But we didn't have a clear picture of what duality was supposed to mean for N equals 2. We'll eventually be interested in that. It's really.

Well, at least for the simplest n equals 2 theory, it's a statement about the infrared, not about the fundamental microscopic description like in n equals 4. Anyway, but anyway, focus us more on the right direction of thinking about long poles and dylans, which ultimately were the right ingredients to use. Yeah, so and that the string duality is somehow all gelled within four or five months because I remember and you may not remember this, but I remember I met you in Jerusalem. There was a New Year's party in 1994. The beginning of 1994?

End of 1994. And you said, oh, there is not much time left in 1994 for new ideas. But it was already the seeds of... This must have been there in your mind because I think in May we had this whole change of perspective and then which already other people like Helen Thompson were thinking about. Well I'd spent part of, well after the work with Cyberg, it then was clear that all this was relevant to foreign unfolds.

So I spent part of the fall of 94 trying to work on that but my plan for what I wanted to do was too grandiose. Eventually, a few years later, with Greg Moore's help, we did what I originally wanted to do, that involved integration over the U-plane. Ah, U-plane, okay, and also it's related to Mach-Module. And eventually what you and I did just a year or two ago.

But at some point, luckily, in the fall of 1994, I decided that was too complicated and I had to do easier things. But then the mathematicians told me anyway I should write up what I knew, even if I couldn't do it the way I wanted. No, but you also wrote this very interesting paper with Cameron, which was also in a way... I had the glimpses of type 2 heterot...

I mean, in hindsight, the type 2 heterot... Yeah, yeah....deformed. Yeah, well, Kerman and I, we were counting in sometimes.

But you had the right... Kerman......speculation there. Sort of, yeah. Well, anyway, we discovered very strange facts, and Kerman told me, well, we're going to understand these facts.

I said, well, no, it's impossibly far away. But Kurman was right, because within a year or two... Yes, within a year it was more or less understood.

But anyway, after the cyber-written papers, I spent a few months trying to do what was too hard on four manifolds, then had the sense to just write up a simple version and put the rest aside. Then I spent a few months... It bothered me that there were five string theories when only one was needed, so I thought maybe some of them were inconsistent.

And I spent a few months at the beginning of 1995 trying to prove some of the string theories are inconsistent. Some global anomalies? Well, I wanted global anomalies but I couldn't find any.

I remember I had two arguments. One, I can't remember at all now what it was. But the other, I thought that if you imagined the strong coupling behavior of type IIa, you'd find a contradiction.

Because... How long? What was that? It looked like the BPS spectrum wasn't physically sensible, but at a certain point I realized that the BPS spectrum actually wasn't physically sensible. So after that I stopped thinking that these string theories were inconsistent, which kind of was a silly thing to even try for, because it was at the time almost known that they were dual to each other.

It wasn't completely known, but there were... The two heterotic strings were dull, the type two strings were dull. Dahlia Ley and Polchinski had a story relating type one to type two. I guess there was the heterotic world and the type one type two world.

Those hadn't yet been linked. Okay, you could have conceivably shown that one of the two worlds was inconsistent, but you couldn't have hoped to get it down to one string theme. But anyway...

As you know, within a relatively short time it became clear that no one was going to get rid of any of the screen. They were all different limits of each other. That's right.

Yeah, so that brings me to, I mean, I just want to ask you this more, it's a bit... philosophical question but I think it's important question because I think there seems to be some misunderstanding about I mean oftentimes I mean it's evident that these are some very deep concepts irrespective of how they will fit into a description of nature. It's not immediately clear.

And often times in physics, even with... I mean this is a much more grand structure, but even with ideas like, let's say, Higgs mechanism, it took 50 years to find experimentally. So clearly, sometimes in popular imagination, I think somebody used this word, paparazzi, you know, going back to...

Popper's principle of falsifiability applied in a rather naive way. That's okay if you can falsify it. But oftentimes, actual practitioners of physics are pursuing ideas with a long-term view, with not immediate falsifiable falsifiability. But that is nevertheless a kind of an internal criteria of what constitutes.

important breakthroughs. I mean, there is no question in the minds of those who work in this field that big advances being made. Yeah, do you have... Well, one thing that's worth mentioning though is that apart from the dream of understanding physics at a deeper level, involving gravity. Work in string theory has been useful in shedding lights and more conventional problems in quantum field theory and even in condensed matter physics and as well with applications in mathematics.

Apart from its intrinsic interest, those successes are one of the things that tend to give us confidence that we're on the right track. Yeah, yeah. Because, speaking personally, I find it implausible that a completely wrong new physics theory would give rise to useful insights about so many different areas.

Yeah, actually I heard a very distinguished experimental physicist said this, that his approach to discovering new physics is to really improve the tools. I mean it's not an attitude that you might expect to hear because this is that if you have new tools, then new discoveries follow. It's not always that you go looking for a specific physical phenomena.

And that I think is also true in theoretical explorations and there seems to be a really clearly well understood internal criteria for what constitutes important advance and what is just. Not such an important one. It's probably a difficult question to answer, but okay. You have been, I mean, string theory community has been clearly pursuing something quite deep and important.

And it may very well be that experimental realization could be many years or decades. Unfortunately, that's quite possible. We have to hope for good fortune. It depends. There are things we can't control, how the nature of the answer, but how things work out in the real world.

Yes. And the experiments we're able to do. So let me ask you a bit about your connection with the ICTP.

I mean you had been, you came here, I think Salam was instituted this prize, the Dirac medal, and I think he was very proud to have chosen you, President Henry. I was actually a rather regular visitor at the ICTP for quite a number of years, starting around 1977. My wife is from southern Italy. We used to come to Italy often in the summers and very frequently spend some time here at the ICW.

I must say that those are very exciting years to visit the ICDP, partly because of the environment that SOM created. Yes, and I'm very happy that you're here now for this week. Yes, I know that you come from an academic family. Your father was also a physicist, and your wife is also a theoretical physicist at Princeton.

So, do you want to share something about what brought you to physics? Because I know that you didn't. You probably wanted to resist what your father was doing. I zigzagged a little bit, but then I discovered that my fate was physics.

As you may know, my father, as you said, is a physicist, whose work has largely been in general relativity. But then you were in journalism or something? I zigzagged a little bit.

You zigzagged a little bit, okay. Okay now, as you know, Salam created this organization, which is also something that is... dear to my heart is that apart from pursuing excellence and the frontiers of science, which is for example the Institute for Advanced Study is also a place like that, one of the impulses behind ICTP is to make the knowledge globally available, overcoming the barriers of geography, class and gender.

The ICTP has done a great job in fostering physics and countries that have needed help in getting started. So even in a country like India, which as you know, the string theory community in India is now very strong, but I remember people are very important. If you go back to the 70s, I think that the OICTP deserves a lot of credit for how physics has advanced in India. Exactly. So and the third aspect is sort of promoting international cooperation through science.

Yes. And you have been Perhaps a keen observer I think of this the SESAME project that you know in Jordan. I think ICTP played many at many junctures.

I mean there were of course many contributors to this very important initiative but ICTP if I'm not wrong held an important conference in the HAD bringing together scientists from different countries who might not have been on politically friendly terms. But somehow this power of science as a common platform and a common language to bring them together and it looked like a kind of a pipe dream in fact at that time to have something in Jordan supported by Iran and Israel and... Yes, it seemed like science fiction and frankly whenever one hears what's happening in Sesame it sounds like news from an alternate universe.

That's right. News from an alternate... universe where there's peace in the Middle East, but the countries are cooperating. Yes, I think, yeah, so that's one I wanted to ask you about that because, of course, difficult political, I mean, it's naive to expect that difficult political problems will get resolved through science diplomacy. But nevertheless, I think science can play at least a facilitator role.

At least give the different countries an example of an alternative possibility. yeah to live together peacefully. Okay so now I want to ask you a bit about your other half of your distinguished career having to do with mathematics.

So I mean you have made many very deep contributions to mathematics if I may read from the citations of your Fields Medal. Time and again he has surprised the mathematic community by a brilliant application of physical insight. Leading to new and deep mathematical theorems.

And this, you know this famous quote of Eugene Wigner that, about the unreasonable effectiveness of mathematics and physics. But I want to ask you about this, the other way around, the unreasonable effectiveness of physics and mathematics. Evidently going back to Newton with calculus. Yes, certainly, I was about to say that. Calculus is very central.

Mathematics became far more powerful after calculus was developed and it was developed because it was needed for physics. But then with Dirac, with quantum mechanics, it spurred a lot of... but I think you have brought in, and I think you can be singularly credited for this, really quantum field theory as a tool for addressing, you know, various methods in quantum field theory. So how do you, what is... I think the essential fact is that, first of all, quantum field theory is difficult to understand mathematically, because it's a little bit of a mystery why this is, but it's difficult to make it rigorous.

So it's clearly an important tool in physics, and it has a lot of mathematical secrets, but mathematicians have a difficulty understanding it, so they have trouble learning the... mathematical lessons that can be deduced from quantum filter. So physicists, because they work with quantum filter all the time, either in particle physics or as well in condensed matter, have these potential tools at our disposal, which turn out to be mathematically powerful. But I think that's half of it. The other half of it is that string theory has, well, first of all, it's also very difficult for mathematicians to appreciate because prerequisite, basically, is to understand quantum flip theory as a physicist, and then you can understand what physicists are trying to do with string theory and proceed from there.

But string theory seems to know a lot of mathematical secrets. Yes. And at the present stage of knowledge, physicists can work with that.

Even though we don't understand string theory very well, we can still work with it more effectively than the mathematicians can. So there are opportunities there to extract some of these mathematical consequences. But you did try, I think, I mean, there was, you had this set of lectures, I remember, Delene at the Institute for Advanced Study.

Yeah, how do you, how much do you think you succeeded in making? Well, there was certainly, we had a special program and then the lecture notes were published. In about 1996 and 7, roughly, trying to explain quantum fields and strings to mathematicians.

I think there was only a moderate success. There was some success, but I've come to believe that it's actually really important to have some sense of how quantum field theory works in the real world. If you try to understand quantum field theory just as a mathematical subject, Without developing the intuition you get from its applications in the real world, it proves to be difficult.

I guess I mean there's a clear problem of I mean maybe understanding QCD would be... Actually pure gauge theory rather than... Pure gauge theory, that's what I meant, yes. That would be a step towards... Well, if that problem were solved that would be...

It would mean that... Well, it's a little hard to speculate what would come after because it would depend on how... difficult and technical the solution was, but potentially quantum field theory would be more accessible mathematically if that kind of thing were understood.

But the clay problem as defined is extremely difficult because it asks not just for an existence proof but for proof of the mass gap, which I actually worry makes it too hard. Yeah, even the existence... I feel that the existence could be proved without proving the existence of the mass gap.

The existence is basically a question about the ultraviolet. Limitless. You need some control of the infrared. I see, I see.

But you see, you might be able to prove that the theory exists with parallel decay of correlations in the IR without being able to prove that they have exponential decay. Think of the two-dimensional non-linear signal model, say, of a two-sphere. which was suggested by Polyakov a long time ago, it was a practice problem.

There's a theta angle for generic values of the theta angle, it's massive in the infrared, but at theta equals pi, it's actually believed to have mass less than the infrared. So, it's not necessary for the existence of the UV theory. Right. That's what I want to say.

Yes. And you can well imagine, in this case, because there's one over n expansion, Well, in this case you might be able to use the one-over-n expansion to prove results that would be true for a sufficiently large m. But in a case where you didn't have a one-over-n expansion, you could well imagine being able to prove existence and not being able to decide the question of a mascot.

Yes, okay, maybe they should have two million dollars. But you mentioned Michael Atiyah, I mean, so, I mean, your work on supersymmetry anomalies had this beautiful connections with index theory. And then you've done all this, all the work on Seiberg-Witten theory, Knott theory, Donaldson theory, I mean, the amount number of applications are also geometric Langlands program.

It's quite astonishing how much power quantum field theory has given to make some progress in difficult mathematical problems. But Atiyah was an important influence on your career and your thinking. So you want to say something about your experience with or what?

Well, one thing I'd like to say is that he's the one who told me. about Montaigne and Olive duality. Even though it's a physics paper? Yes, I think he had probably been talking to Peter Goddard. Or Olive as well.

Olive, probably. Well, we have to remember that before Montaigne and Olive, there was Goddard, Noitz, and Olive. Yes.

Well, I don't really know. I'm almost certain that he had been talking to Goddard as well. Anyway.

Well, what really happened was that, at least from my point of view, what happened was that the Yang-Mills Instanton and the work of solving the U1 problem using instantons, first of all got physicists excited about differential geometry. We had modern questions in topology and differential geometry we hadn't been familiar with. And then I first heard the names of Tia and Singer from a paper by Albert Schwartz. Albert Schwartz was the one, it helped to solve the one problem using a Fermion zero mode in an instant non-field.

Similar to work of Jakiev and Reggie involving monopoles. It helped solve the Dirac equation in an instant non-field and found a zero mode. Then Albert Schwartz explained that the zero mode was a consequence of the Ittiaz-Zenger index theorem.

Yes. And certainly, I was shown that paper by Sidney Coleman, and it was one of several important papers Sidney showed me. This one I think would have...

You were a postdoc at Howard? That's right. This paper I think I would have learned about anyway. But some of the important papers Sidney showed me never became well known.

And I think I would not have known about them otherwise. An important example, in view of my later work on the Jones polynomial, was Albert Schwartz's paper on analytic torsion. And what nowadays is called BF theory. He didn't know to call it BF theory. He called it a degenerate quadratic functional.

Well, in hindsight, that's more or less had to do with the abelian limit of Chern-Simons theory. So, later on... Okay, Sydney showed me that paper when it came out in about 77 or 78, and I didn't think about it again until a decade later when I remembered it when it was handy.

Anyway, going back, the instanton and its connection to the index theorem, I know there are other things that maybe Atiyah and Singha would have told you about, but were less obvious from my point of view, caused physicists and mathematicians to start talking more around 1976 and 7. Yes. And Atiyah visited at Harvard where I was a postdoc. And as a result of our discussions there, he invited me to visit him in Oxford at the end of 77. And I went, and for a few weeks we talked a lot about gauge theories, and by this time they had discovered the ADHM construction of instantons.

You have to imagine that at that time, I was a little younger than either of us is right now. We were all younger. With a little bit of patience.

And Atiyah probably had the disconcerting experience of discovering that whatever they did, a physicist would tell them that they should do something else that would help even more with the quantum theory. Anyway, we talked a lot for a few weeks, and then… Well, what did you tell him to do? Well, something that would help with quantum manuals. But in hindsight, an incident on gas only has limited use for. The quantum theory.

But roughly 20 years later we learned that the ADHM construction and the instanton gas are useful for supersymmetric versions of the theory. But it would have been a stretch to think of that in 1977. Anyway though, toward the end of my stay in Oxford... You went to visit him in Oxford?

For about a month. Toward the end of my stay in Oxford, he showed me that these two papers, I think of God and Roots and Olive and... Montaigne and Olive and suggested I go to London to talk to Olive about the Montaigne and Olive paper.

I couldn't, well, so I did. I realized possibly part of what was going on is that after talking to me so much for a few weeks, he wanted to find something else for me to do for a day or two. But anyway, I went to London.

Enjoy London. Yeah. Talked to, I think I'd not been in London before.

And I didn't get to see much because all I did on that visit was talk to David Olive. By the time I arrived I was skeptical because the Montaigne and Olive story was expressed in terms of a purely bosonic angles, coupled to a scalar. There were a lot of problems with it.

For starters, they needed the Higgs potential to vanish and that isn't really a well defined concept quantum mechanically. And you can go in for there and raise all kinds of technical objections. So I was a skeptic.

But, as we were talking, we realized that the supersymmetric version of the conjecture would make more sense. And by the end of the day, we realized that maybe the facts they were talking about could justify better in n equals 2 or n equals 4 superangles theory, which led to the paper we wrote. Yeah.

I think that was a very key insight because discovering the central charges in supersymmetric algebra is because Because this whole work on BPS states and to get any control over the strong coupling was would not have been possible. Well, it is true that it was important later, but it is also true at that time I drew the wrong conclusion. Which was?

Well, you see the conclusion, okay. As I told you in this case, I was very impatient, but I also was very conservative in my attitudes. So I was unlikely, very reluctant to adopt what seemed like an extreme hypothesis, if the facts had a more conservative explanation.

explanation. And I felt we had been able to explain the facts that Montaner and Roloff had pointed out without assuming a non-perturbative duality. We had been able to explain those facts just from supersymmetry. And that caused me to remain skeptical about Montaner and Roloff duality.

In fact, I remained skeptical. So, to what we said before, well, the interaction with Schwartz in 93 and then Senn's paper six months later or whatever, exactly whenever it was. After Senn's paper, I gave up on being skeptical. We simplified the story before because there were a few other clues that were important in wearing down my skepticism before Sam's work, including your paper with Jeff Harvey. Yes, okay.

And a few others. But anyway, it was partly because of my skepticism that I... But all of an eye had shown that... n equals 2 or n equals 4 were more reasonable arenas for Montaigne and all of duality. But if you took Montaigne, but then there was, okay, okay, which of them?

I think, I'm having trouble remembering now, but I think it was Osborne. Yes, Osborne, Osborne, yes, Osborne put it to n equal to 4. Yeah, Osborne showed that if you worry about the monopole spin, you find it should be n equals 4 because the monopoles... It was multiplied, yes. The monopoles for n equals 2 are in hypermultiplots, but for n equals 4 they're in vector multiplets.

So if you were taking the Montaigne and Olive conjecture seriously, which I wasn't at this time because I thought we'd found a conservative explanation of the facts, then you would have possibly been inspired to do what Osborne did, which was certainly an important thing to do in hindsight. So I didn't really go back to it seriously. Until 1993 when some of these other things were happening that had worn down my skepticism.

The experience with non-prohibitive dualities in string theory, on the two-dimensional ones, had been part of it. Mirror symmetry. Mirror symmetry, for example, and even abelian teeter.

But also, another paper that... had an impact on me was by Cal and Harvey and Strominger. Yes. On small instantons.

Oh. As having had five bins. Well, I was puzzled about the meaning of the phenomenon they were finding at the core of the small instanton. Yes.

And later development understood. That's a good thing. Yeah, right, right. But at the time we couldn't. But I wanted to ask you, for example, your work on index theory, I mean, written index, and it's a kind of a beautiful generalization of index theory, which Atiyah played over.

Or was that a separate path? Did discussions with Atiyah inspire you to think in that direction, or this was something else? Well, Atiyah helped me understand what the index meant and why it was a topological invariant.

But if you understand the concept of the index, You can generalize it to a supersymmetric theory. Right, right. Oh.

So that was something... That was a little bit different. Where I was more directly influenced by Tia and Bott, though, was in Morse theory. Yes.

So, at this Karjev Summer School, I forget which year it was, maybe 1979, for some reason, Tia and Bott had taken it upon themselves to educate us about Morse theory. I had most definitely never heard of Morse theory, even of the name Morse. My guess is that none of the physicists in the audience had heard of Morse Theory before, but I can't say for sure. Anyway, they gave us a couple lectures on Morse Theory and it was very cute.

Interesting. But I didn't have anything to do with it. But later I was trying to understand why it was so difficult to spontaneously break supersymmetry.

And since it was difficult, I kept looking at simpler and simpler models. And finally I got to what seemed like the simplest model of all, which was in quantum mechanics, in other words, zero plus one dimensions. with scalar superfields and superpotential, it was still hard to break supersymmetry. And finally, I realized, well, at a certain point, I think I was in a swimming pool in Aspen, Colorado, I remembered Bott's lecture about what he called baby Morse theory, and realized that the superpotential had been interpreted as a Morse function.

So, that was directly influenced in more detail I would say by a TN bot. Just defining the index of a supersymmetric quantum field theory, I would say is not that hard if you have been exposed to the index in quantum. Yeah, once yeah. And I remember reading this paper of yours in India as an undergrad, which really inspired me to wanting to go to about supersymmetry breaking in 1983, 1982. Constraints and so on. Okay, so one last question I want to ask you.

Going back to Eugene Wigner's original quote. I mean, it is true that to some extent the unreasonable effectiveness of mathematics could be explained by maybe the way our brain works. I mean, we somehow are able to abstract a triangle from a Wigley triangle and make some generalization and discover Euclidean geometry.

But in some cases the connection is so distant, like complex numbers playing a role in quantum mechanics, something which is completely invented by human beings. being so important in the description of the real world. I don't know, I mean, you probably have nothing more to say. I have nothing to say except a slightly tongue-in-cheek remark that it's as if the universe had been created by a mathematician. By a mathematician.

Okay, so it was a pleasure, Edward. Oh, thank you. Thank you.

Transcript for:Edward Witten Dirac Conversation

Transcript for:
Edward Witten Dirac Conversation