 Okay, good afternoon everybody. So it's very nice to have this, we have a large school going on on string theory and related topics. It's great to see scientists from all over the world here at ICTP. And today is a special occasion. ICTP has awarded the 2023 Dirac Medal to four physicists who have made wide-ranging contributions to string theory which is a mathematical framework in fundamental physics which aims to describe the entirety of the whole physical world. Very ambitious program. And the 2023 Dirac Medalists are Professor Jeffrey Harvey who is an Enrico Fermi Distinguished Service Professor of Physics at the University of Chicago. Professor Igor Klebanov who is a Eugen Higgins Professor of Physics and the Director of the Princeton Center for Theoretical Science at Princeton University. Professor Steven Schenker who is a Richard Herschel Weiland Professor of Physics at Stanford University. Leonard Saskin who is a Felix Block Professor of Theoretical Physics at Stanford University. And I have to say that it's particularly a pleasurable occasion for me because it's in a field in which I work in string theory and all these four people are some, you know, leaders in this field. Someone that I always admired both for their science and also as individuals. And I'm very happy that their pioneering contributions have organized the award sites, their pioneering contributions to perturbative and non-perturbative string theory and quantum gravity in particular to the aspects related to anomalies and quantum gravity. So anomalies, duality, black holes and holography. This year's medalists have made many original contributions to developing the theoretical framework of string theory with the goal of unifying all physical interactions including gravity. Their work has led to deep new insights about the physics of black holes and to major conceptual breakthroughs through the realization of new principles of duality and holography. I should just say one word about the Dirac Medal. ICTP's Dirac Medal is given in the honor of Paul Dirac, arguably one of the all-time greats in theoretical physics. Certainly one of the great physicists of 20th century whose contributions could rank with those of Newton or Einstein. Dirac was a very good friend of the center and he visited ICTP on a number of occasions. And his style of, I think the Dirac Medal was given for discoveries which may not yet have had experimental verification because Dirac was one of these people who really had this faith in the power of the human mind just to discover new principles just by pure thought. And of course many of the Dirac Medalists have gone on to, I think eight or nine of them have gone on to win the Nobel Prize later on. So it's a very prestigious medal and I'm very happy that we have this very distinguished set of people getting this medal this year. So with these words I will say more about their scientific contributions when we give the medal to each medalist. But it is also my pleasure to welcome her Excellency Ambassador Jane Minnes, permanent representative of the United States of America to UNESCO. I met her recently in UNESCO when ICTP celebrates its 60th anniversary and there was a very nice ceremony celebrating ICTP at the headquarters in Paris and Ambassador Minnes was there and thank you Ambassador for taking the time and she was thrilled to learn that in fact four Americans have gotten the Dirac Medal this year and she was very keen to visit us here. As you know United States rejoined UNESCO after a break sometime and ICTP is governed within a tripartite agreement it's a UN organization and we are governed in a tripartite arrangement together with the government of Italy the International Atomic Energy Agency within which ICTP was born you know at the time of Openheimer was the first chair of our scientific council and UNESCO and we have been a category one center of UNESCO since 1996 this tripartite agreement is in place. So therefore it's particularly important and I'm very happy that Ambassador has kindly agreed to be present for this ceremony because when the US joined UNESCO one of the priorities that were signaled by President Biden was STEM in Africa and contributions to science in developing countries and this is one of the very important missions of ICTP is to make advanced science globally available and to promote international cooperation through science and so I'm very happy that Ambassador is here so I will give the floor to her for the keynote speech. Thank you. Thank you Director Atish Dabokar. It's great to be here and it's great to see you again we've had the privilege to see each other a couple of occasions and the students and the professors should know that you have a terrific ambassador of the center and he's constantly talking about the great work that each of you do. I also want to recognize my colleague the Italian ambassador to UNESCO Ambassador Laborio Stilino who also is a big promoter of your work and was also a key feature of the 60th anniversary that Atish was just highlighting that was held at the UNESCO headquarters here in Paris just a couple weeks ago. So it's really a privilege to be with you today and for this important occasion recognizing the groundbreaking work of four distinguished professors. So Professor Jeffrey Harvey of the University of Chicago Professor Igor Klebinov of Princeton University Professor Steven Schenker of Stanford University and Professor Leonard Seskin of Stanford University. As each of you know, ICTP is a special place one that embodies the real intersection of the values and the mission of UNESCO with the best of the scientific community focused on physical and mathematical sciences. As I mentioned just a few weeks ago your director was here in Paris at the UNESCO headquarters celebrating the 60th anniversary and showcasing the global scope of the work that each of you do every day. It brings together top talent from around the world and it's a unique example of the cross disciplinary international cooperation. In the complex environment that we find ourselves in today we need more spaces where international collaboration is at the cornerstone of what we do. ICTP is globally recognized and at the forefront of inquiries into some of the most pressing fundamental questions of our times. And it's also these qualities that we try to embody at UNESCO which is why it was such a privilege for the United States to rejoin UNESCO this past summer and subsequently be elected to the board in November. ICTP is not just a UNESCO category one institute but the flagship center for scientific cooperation and discovery and it's a very powerful symbol of UNESCO's values and mandate. Now I could attempt to explain the individual research of the four distinguished physicists that are on the screen and two of them present with you today but I won't because they're each going to give a presentation and also I do not want my college science professor to be embarrassed by my interpretation of your work. So I'm going to leave that to the experts on the screen but what I will mention is the power of discovery. At the 60th anniversary celebration a few weeks ago the 2012 Nobel Prize winning physicist Sergey Karov addressed the group. It was a tough audience at UNESCO, a group of mostly ambassadors from different countries and very few scientific peers for the speaker to address but what he did was impactful. He discussed the power of discovery and the value of discovery, not research for a specific purpose or application but the power of discovery. He walked us through some of the greatest scientific discoveries of the last 200 years and the groundbreaking nature of each of them but that groundbreaking nature in the actual application of the discovery came 10, 15, 30, sometimes 50 years later. It wasn't immediately clear the transformational impact of the discovery. It's the scientific discovery of today that lays the foundation for tomorrow and that's why it's such an honor to recognize the four physicists who are the recipients of the prestigious Dirac Medal and I'm also very proud that they're Americans and leading the way in their research and representing Team USA so well. The Dirac Medal, as was mentioned, was first awarded 39 years ago in honor of Paul, Adrienne, Maurice Dirac, one of the greatest physicists of the 20th century and a steadfast supporter of ICTP. It recognizes not just the fundamental achievements and theoretical physics but breakthroughs with astounding resonance across the field and beyond. So for these accomplishments, Professor Harvey, Professor Klebenoff, Professor Schenker, Professor Susskind, on behalf of the United States of America, congratulations as recipients of the 2023 Dirac Medal. All four of you embody the essence of what it means to discover. The pursuit of discovery and your commitment to use your research to improve lives around the world. To all of you in the audience who are studying at ICTP, congratulations on your academic pursuits and commitment to collaboration across countries with scientists around the world. Thank you. So thank you Ambassador for the kind words and I once again extend a warm invitation to you to visit ICTP campus whenever the next opportunity arises. It's a beautiful place. You will enjoy visiting us and you can also experience the spirit of ICTP here in person. So I think now in terms of the program, we will hand over the medals in your presence first and then we will switch to their presentations. So and I will try to summarize their work. Since I work in the field, I will, I think I will, I can do some justice to that. Okay. So should we start, we'll start alphabetically. So, Professor Jeffrey Harvey, you know, I should say that he was actually my thesis advisor at Princeton. So he's like my physics guru in some ways. So it's a special pleasure for me. And we also had two years ago, the Dirac Medal was given to Pierre Hamon, another pioneer in our field. Who was the physics guru of Jeff? So it's really a particularly nice occasion. So Jeff Harvey together with Gross, Martinic and Rome discovered what is known as the heterotic string theory, which naturally leads to grand unified theories that can incorporate the standard model of particle physics. His work on orbit folds with Dixon, Wafa and Witten provided exact worship constructions of such models. He pioneered the study of solitons and brains, which has played a crucial role in the discovery and understanding of duality symmetries. His work on anomaly inflow with Callan has found important applications in condensed matter physics. So Professor Jeffrey Harvey. So now Igor, Professor Igor Klebenoff, I mean Igor was a senior student at Princeton when I had just started my PhD. So he was also somebody I looked up to, one of these big boys. And he is getting his dirac medal for foundational paper on the subject, Igor Klebenoff together with Gubsov and Polyakov developed a precise dictionary for the holographic anti-disseter conformal field theory correspondence. And he constructed examples of holographic gravitational duals of confining gaze theories together with Straszler, which have had many applications to model building. And just to give a one-line summary, it's really an extraordinary equivalence between a theory of gravity and a theory without gravity, which physicists are still grappling to understand fully. So congratulations Igor. So Steve Schenker, in some ways he was like my postdoc advisor. And I assure you it was not planless. It really happened. I mean he actually at that time was working on some very interesting problems and non-pertributive aspects of string theory, which is what convinced me to join him as my first postdoc. So Steve, and of course Lenny Suskind, he was the senior statesman in the field who has pioneered many ideas including the idea of holography, which I just mentioned too, even before people knew how to even formulate these thoughts. So Steve Schenker and Leonhard Suskind together with Banks and Fischler developed the first non-pertributive formulation of M-theory and string theory by providing a limiting procedure that describes the S-matrix. I will tell you more about the work of Suskind later, but the other influential contributions of Schenker include analysis of phase structure of lattice gauge theories together with Fratkin, classification of unitary two-dimensional conformal field theories with Friedan and Chu, covariant formulation of super string theory with Friedan and Martinek, non-pertributive formulations of string theory in low dimensions, which I mentioned with Douglas, and novel connections between chaos on black holes with Sadd and Stanford, something that he will talk about today. So congratulations Steve, it's great to see you. So congratulations Steve. Now we come to Professor Leonhard Suskind. As I mentioned his earlier work with Steve Schenker, but he was also very influential. I remember discussing with him about black holes and holography on a number of occasions, and he was even at the time when those ideas were not well understood. And he was among the first to recognize that dual models, which are models of strong interaction, the nuclear force could be interpreted in terms of strings, which has led to the whole new field of string theory. His other influential contributions include his work on Hamiltonian lattice gauge theories with Bogut, bariogenesis and technicular with DiMopoulos, and holography, which I mentioned, and connections between complexity theory and black holes, and also this work with Schenker, Banks and Fischler, which I mentioned. So congratulations Lenny. Thank you, Atish. Sorry Lenny, you are not in the frame, so if you can just go back. Yeah. No, you are not in the photo frame. You go back a little bit. Oh, I see. Can you see me now? Atish, is that good? And I'm also delighted to announce today that as part of the celebrations of the 60th anniversary, we are launching a special series of Dirac conversations. You might have seen some of them already on YouTube. Actually, it was just released a few hours ago, and for example, the conversation with Witten has already gone viral with 10,000 views, I'm told. So please go and look at it. But I think we did this interview with a number of almost 10 Dirac medalists. Today I recorded a very nice conversation with Jeff and Igor, and I think I will have to come to Bay Area, Lenny and Steve to record it over there, unless you come to ICTP sometime soon. So actually, I will be there in a couple of weeks, and maybe we will try to do an interview if we can. So it says, idea was to feature interviews with renowned scientists who have made major contributions to theoretical physics, and some of the names you will recognize, which we have already published. Luciano Mayani, Subhir Sachdev, Giorgio Parisi, Dan Tanson, Alessandra Bonanno, Thibaud Amour, and Pretorius, Francis Frans Pretorius, Peter Solar, Mikkel Parinello, Ngova Chau, who is a Fields Medalist, and Edward Witten, who is also a Dirac medalist and a Fields Medalist. So thank you very much. I think we can move to the next part of the talk, part of the ceremony, which will be the presentations by the Dirac medalist about giving a summary in 20 minutes of their work of their lifetime. It's a hard task. So I give the floor first to Jeff. Thank you very much, Atish, and the ICTP, and other medalists. It's a great honor to receive this prize. And I find it particularly meaningful for several reasons. One is, of course, Dirac's influence on our field. He was the originator of many profound ideas that still have great influence today. He was also one of the heroes of my PhD advisor, Pierre Ramon, who, as Atish pointed out, also was a winner of the Dirac model. And it's also very meaningful because of my connections to the ICTP and to Atish. I assure you that these prizes are not just given to friends of Atish. It may seem that way this time, but I know there were other people involved. But I've been to the ICTP many times, lectured at the summer school, organized at least one of the summer schools, and I've always found it to be an incredible place and an incredible experience to be here. So when asked to give, I'm not going to try to summarize my life in 20 minutes. That seems impossible. I was asked for a title, and I gave a kind of strange title, which was simple questions with complicated answers. And we usually think that the goal of understanding is the opposite of that. That is to find complicated situations that have simple answers. An example of that from our own field, I say, would be the strong interactions. So in the 60s, many strongly interacting mesons and baryons were found. It was a very complicated, interacting theory that was hard to describe. Many new tools were developed to try to understand it. But eventually we understood there's a very simple Lagrangian based on SU2 gauge theory. You can write it down in one line. And it provides a very simple explanation for an incredibly complicated phenomenon. So why should we be interested in the opposite of finding a, starting with a simple question and getting a complicated answer? Well, what I had in mind here was two things. First of all, I think many people in theoretical physics and also in mathematics feel like the work that they've done or the most important work they've done is often not constructing a new theory but uncovering or discovering a new theory. That is, there's a feeling of discovering something that exists already and your job is not to create it but to excavate it and to expose it to the rest of the world. And I've certainly had that feeling. And so if we think of trying to understand whether it's string theory or something else that is the underlying theory of the world and physical phenomenon, then every time that we can uncover some new part of this incredibly beautiful and intricate machine that we're digging out from the earth, it's always good to discover another part and we can figure out what that other part does. So sometimes you start digging in some place with a very simple question and you expose a whole new structure, something complicated. So that was one thing I had in mind when I gave this title of simple questions with complicated answers. And I guess the other was my own experience in my career because I feel like there have been a few times when I wrote papers that were basically answering a quite simple question. And sometimes I wasn't even sure that it was all that important but it seemed kind of interesting and sometimes other people were not all that impressed either. But eventually by following a track it led to very interesting and much more complicated and richer things than I certainly first imagined. But it's another example of starting with a simple answer and getting a simple question and getting a complicated answer. Now when getting an award like this I guess it's natural to reflect a little bit on your career and how you've gotten here. I must say the first two things that came to my mind when I reflected a little bit on that was first of all the importance of many work, of the work of many people in any accomplishment in science. There may be one person who eventually makes a creative leap and really understands something that seems very new but it's always built on the incremental work of others, partial results, hints and clues. And so in the work that I've done actually these weren't minor clues but I had major help from all of my collaborators and Atish mentioned many of them but there were a few that he didn't mention and I think I'm not sure if he mentioned Andy Strominger but he's been an important collaborator of mine and also Atish, Gary Gibbons and Fernando Ruiz were important collaborators of mine. So instead of trying to summarize everything I've done I want to just maybe illustrate a little bit this idea of getting to a complicated answer from a simple question with a particular aspect of a problem that I've looked at. And it actually starts in 1988 or 1989 when Atish was a graduate student and unlike most graduate students who come to you with a kind of please tell me what to work on and give me a problem, Atish approached me with a problem that he wanted to work on and believe it or not it was generalizing some work of Dirac. So that's one of the reasons I've chosen to talk about this. So Dirac had written a paper where he studied the so an accelerating electron will radiate and produce an electromagnetic field even if it's not radiating and that electromagnetic field will act back on the electron because the electron interacts with the electromagnetic field. Dirac analyzed this problem classically and showed that there were two effects one was a kind of infinite classical renormalization of the mass and the other was an effect that would damp the motion of the electron when it radiated. Now his treatment was not totally rigorous it was classical it's really a quantum problem and there's some issues with it but it has some phenomenological use and Atish wanted to generalize this to the back reaction on axion strings of axion radiation. There's a problem of interest back then it's still a problem of interest now because there's a possibility that axions play a role in the dark matter and that axion strings might have existed in the early universe or even exist now but I was not really interested in that problem I was, you know, wrapped up in super string theory and so I suggested that we turn this problem into a problem in super string theory because I knew that heterotic strings had some of the same properties as axion strings so essentially what I suggested to Atish is that we should look at the heterotic string we should assume a space time which was r21 times some circle of radius r times some six-dimensional space and this could be a Kalabiow space, it could be a six-torus or if we were doing strings in flat space or in ten dimensions it could be r6 and the idea was to take a string wrapped here and then take r to infinity and so that would just look like a macroscopic string extending along some direction in our space time and then the tension of the string is just the energy or the mass of that state over 2 pi r and we would evaluate this in the limit that r goes to infinity or rather we would evaluate the corrections to this coming from string theory so once you look at this setup and know a little bit about string theory it turns out that you can compute the corrections to this they come from diagrams in string theory which involve taking two vertex operators the operators that create these string states looking at their correlation function on a two-torus it's the one loop correction and then of course there would be higher loop corrections and while we focused on this and then later made a general argument about the higher order terms but I expected because string theory is ultraviolet finite that we would get a finite and maybe interesting answer from this but when we did the calculation or I think actually a t-first did the calculation the answer was zero and that seemed a little disappointing but then I thought well we should really understand why we got zero and so this led us into a long story including this work that I mentioned with Gary Gibbons and Fernando Ruiz and what we found out this was a consequence of the fact that these were so called BPS states that their mass was proportional to their charge and therefore not corrected in perturbation theory and in fact that in string theory there was an infinite stringy or string spectrum tower of these BPS states and these BPS states have the special property as I think many of you know that you can often track their existence and their properties from weak coupling to strong coupling and so in the development of string duality later on this played an important role because it gave you an infinite spectrum of states that had to also arise in any strong coupling limit of the original theory so in our paper or in the work that we did in essence we answered a very simple question the simple question was what are the space time fields outside a macroscopic string and this is just slightly more complicated than the question of what are the Coulomb fields outside an electron I mean this is something we learned in freshman physics and how to compute it in string theory it's a little more complicated because there's a metric, there's a B field, there's a dilaton but still that's what it's analogous to finding the Coulomb field but the fact that we found BPS states, that there was a tower of BPS states and that in supersymmetric gauge theory it was known that there were magnetic monopoles another invention of Drax and that these were BPS suggested to us that maybe string theory had some kind of duality and that this BPS property was a hint and so in our paper we actually said one might speculate that super strings themselves might arise as some sort of soliton sector of an underlying theory this was really rampant speculation but it turned out to be true now as was pointed out by, I guess I can, if I wander over to the other side is that a problem? Okay, so as was pointed out by Andy Strominger and then followed up in some work that I did with him and Kurt Callan the real analogy that we should have been thinking about was that in electromagnetism we compute the electric field by computing the flux of the electric field through a two sphere surrounding the point charge I'll write this in terms of differential forms because that generalizes most nicely to higher dimensions so here I'm putting a little number on it that indicates what kind of a form it is and F is the electromagnetic field strength and A is the vector potential so this has a kind of dual object that you can define by considering not F2 dual but F2 itself and this would correspond to a magnetic charge and of course we know that magnetic monopoles can be constructed using ideas of Dirac and then various improvements on those ideas and you should really ask yourself if this is the result and a quantization condition of course with the product of these two pi times an integer if this is true for a particle in four dimensions what should be the analog for a string in ten dimensions well for a string the string has to couple to something that's a two form rather than a one form because the world sheet is two dimensional and so the analogy would be that there is a fundamental string charge which you would compute by computing the integral of the star of H3 and in ten dimensions a seven sphere is the thing that encircles a string and this would be related to a charge which is the integral of H3 and this would be over a three sphere in ten dimensions that's the sphere that surrounds a five brain something with five spatial dimensions and one time dimension and of course there's again a quantization condition between these two charges so that led to a few questions what is a five brain? is there string five brain duality this was something that was considered by other people as well particularly Mike Duff and part of the answer to this first question was addressed in work with Kalin and Strominger but part of it remains mysterious in the sense that you can construct solutions that have this charge but when you have many of them on top of each other which works by supersymmetry you're led to a theory that we still don't really understand called the two zero theory or the zero two theory depending on which hand you like better left or right and part of the answer to number two really only emerged after the duality revolution of 1995 so I'm going to talk more about these things in tomorrow's lecture but I don't know quite when I started and how much time I have is anybody keeping track? okay I'm okay though but I want to just make a few comments about the duality revolution and the importance of kind of looking at the big picture and figuring out what calculation to do and following up on ideas so before 1994 there had been suggestions groups of British physicists including Noyce, Olive, Goddard, Montanon, I might be forgetting some of some kind of electromagnetic duality that would relate electric and magnetic charges and it would also relate weak coupling and strong coupling and it became true eventually that if this was going to be true it could only be true in Super Yang-Mills theory we now know that's not exactly true but this is still the simplest example of such a duality and it occurs, it can occur here because the monopole and let's say the W plus minus states are BPS, they sit in the same super-multiple and the formula for their mass is invariant if you change E to 4 pi over E where E is the coupling constant and change electric and magnetic charges but you could say that this was all just a consequence of N equals 4 supersymmetry and that there didn't have to be any kind of actual duality relating weak and strong coupling but Ashok Sen took a larger point of view he said what we should really do is not just look at exchanging electric and magnetic charges but rather we should borrow an idea of which I think he took from string theory which was the idea that in Yang-Mills theory in addition to the coupling E squared there's a parameter, the theta parameter, combine these into a complex variable and then there's a generalization of duality by a group of transformations called SL2Z and this generalizes the duality symmetry between electric and magnetic which corresponds to tau goes to minus 1 over tau, shifts of theta by 2 pi and combines them into a bigger group and what he then showed is that if duality was going to be true then there had to be states that had electric and magnetic charge which were any odd integer and 2 for the magnetic charge and that you could look for these states in a certain approximation where you only had to work at low energy and so he then found these states and this was the first example of a prediction that resulted from duality and didn't follow from supersymmetry and this completely changed people's perception of whether there was duality or not I think everybody in this audience or most almost everybody in this audience who's doing physics is so young that you grew up with the idea that there were dual theories that related weak and strong coupling but before 1994 essentially nobody believed it was possible in four dimensions there was a complete change in people's point of view and after this discovery it was clear that we should try to extend duality to string theory this idea of SL2Z really came from string theory you could embed this Yang-Mills theory into string theory so how is this going to work in string theory and well you might go over to the string 5 brain duality and try to implement that in your favorite string theory which for me happens to be the heterotic string but you don't really know what to do because we don't know how to quantize a 5 brain and what are we supposed to do so again and I feel like I've failed at this many times in my career I've failed to take the big point of view or to embed things in the largest possible structure but luckily I've contributed in small ways to people that have taken the bigger point of view so Holland Townsend and then Ed Whitten said well let's just look at the very big picture at the very in a very big picture we can just talk about what duality could possibly do and so we could just look at string theories that have n equals 8 supersymmetry that would be the 2a or 2b theories on a torus or we could look at theories that have n equals 4 that would be type 1 the heterotic I guess these on tori 2a or 2b k3 times some other torus and just ask ourselves like what are the possibilities what could possibly be the strong coupling description and in the analysis that Holland Townsend and then Whitten and much more generality did whenever there was a little problem like maybe you didn't have the right states they would just assume that it would work out basically and the details were then filled in in the upcoming years and so they sometimes had to assume that there were certain black hole states but very soon after Joel Polchinski explained that these were actually D-brained states and there were other ingredients missing so for example there wasn't any strong coupling limit of the e8 times e8 heterotic string but Horjave and Whitten explained that there should be m theories some 11-dimensional theory on an orbital fold and when it got to n equals 4 supersymmetry it turned out that to make the duality work you had to relate strings to 5-brains that were wrapped on 4-dimensional manifolds so this finally kind of realized the duality between strings and 5-brains but it wasn't sort of immediately obvious what to do you only could see what to do when you stood back tried to think as generally as possible put it into the broadest general context so I guess I kind of want to end here I'm going to talk a little bit more about this in the lecture that I give at the school tomorrow but I just want to emphasize to all the students here that it's very easy and natural to get caught up in the details of your computation that you're doing to focus on the particular questions that you're most immersed in and that's what we spend most of our time doing but it's really important to try to take some time to just relax and think about what's the largest possible picture that might be relevant to what you're trying to accomplish and I think it's only then when occasionally people have been able to make big breakthroughs by finding some picture that has an overall coherence and structure that's so beautiful and so encompassing that it must be true and then you can try to fill in the details later so I think that's a valuable lesson that one can take from some of these things that I played some role in and so I'll end here and thank you once again Ateesh and I see you next Thanks Thanks Jeff for this beautiful kind of 20 minute summary of the duality revolution that happened I think I need to add a kind of clarification just to dispel any misconceptions that the Dirac Medal is selected by a very eminent committee which consists of Alessandra Bonanno, Michael Green, David Gross, Juan Maldesena, Giorgio Parisi, Subir Sachdev and myself as the chair but as you can see there are two Nobel laureates there so Okay, so I pass the floor now to Igor Thank you very much Ateesh for this introduction and for this magnificent ceremony I'm very overwhelmed and grateful to ACTP and yeah it's a tremendous honor to be one of the Dirac Medalists and I should say that I'm very happy to be sharing the prize this year with some people who played a big role in my scientific life I came to Princeton University in the fall of 1982 as a very young graduate student and at the time Jeff was a very young brilliant assistant professor I believe and then halfway through my graduate career an amazing thing happened the first super string revolution where Jeff was one of the big players and I at the time actually was working on a large NQCD not on string theory on a kind of more pragmatic approach but the problem was the same, a large N limit of quantum chromodynamics so then soon after the string revolution broke out I like all other graduate students had to learn some string theory and I switched gears and then in 1986 I went on to be a postdoc at Stanford and Lenny Saskin became a kind of postdoctoral mentor to me I learned tremendous amount of stuff from him and of course over the years I also had a lot of great interactions with Steve Schenker so on to what I was planning to say you see the title, maybe the subtitles should be long titles followed by many complicated slides but anyways in the subtitle the talk is gauge string duality confinement refers to colored gluons and quarks confined into colorless hadrons and dimensional transmutation which I'll try to explain what this is but before then I want to talk, since this falls during the ICTP spring school 2024 just want to talk a little bit about my various visits to ICTP actually the first one was during a spring school in April 1991 and you can see this name so Jeff Harvey was one of the directors of that school and it turned out to be a tremendously stimulating school for me because I was asked to give lectures on large n matrix quantum mechanics and string theory in two dimensions and then back then I had more time so I actually spent a good part of the summer writing up these lectures and then it became one of the very first papers submitted to the famous archive you see the 019, the very first paper was 9108001 so I always joke that maybe the most remarkable thing about these lecture notes is that this was the 19th paper to ever appear on the archive but it was also kind of a bit of a beginning of journey on gauge string duality for me because here the gauge theory is very simple, it's just quantum mechanics that lives in zero plus one dimensions but string theory is two dimensional the title of the lecture is string theory in two dimensions and so this you can see these are two of the pictures that appear in this review it's a remarkable connection between Feynman graphs for this large n matrix quantum mechanics which generate random surfaces embedded in just zero plus one dimensions but that model turns out to be exactly solvable as was figured out by Brzezniks on Parisian Zuber and you get three fermions moving in some potential these are just fermions loaded to the Fermi Dirac C and you see like so this is actually something we later called C hat equal one string theory for a while people called it C equal one string theory but then a hat was added because it turned out that it was possible to even get string theory with world sheet fermions okay so here you see that there is an extra dimension kind of appearing in the process of this gauge string duality and then from 93 to 96 I was actually co-director of ICTP spring schools together with Robert Degraff and also Narain and Ranj Bardaimi and this is now from the first school I was co-director actually Toft was one of the speakers at that school so some of these schools actually included Dirac medal lectures so it's a great honor to be presenting one this year so now I will move on to more concretely what it appears I got this medal for so Jeff already alluded to this so there was this great period of discovery in the 50s and 60s different hadrons were discovered that went beyond neutrons and protons people found so-called strange particles they were mesons and baryons and behind all this we now basically know that there is a very simple Lagrangian which is can be written like on a t-shirt right it's just an on a billion gauge theory which encodes the gluons and their interactions and a bunch of quark fields psi these are Dirac fields we should say you're very very appropriate they're just Dirac fields with various labels flavor color but actually I've been for some reason just looking at Dirac's original paper it's absolutely amazing and how he actually originally now we're used to writing the Lagrangian but Dirac really loved the idea of the Hamiltonian and he wrote i h bar d by d t psi is h psi and h is just 4 by 4 matrix of course so good to keep this in mind I'll come back to this in my talk tomorrow but then so then what solidified last year we were busy celebrating 50 years of QCD and in particular the asymptotic freedom of strong interactions for which Gross and Wiltschek and Pollitz are won a Nobel Prize in 2004 we know that this part of standard model is particularly beautiful because it can stand alone theory being asymptotically free makes it well defined at short distances and the Tiven admits a lattice definition that matches onto the asymptotic freedom but going back to the original Young Mills paper they already posed the question why we can't really add the gauge invariant mass term for gluons so why don't we see these massless gluons roaming and propagating at speed of light so what we now know happens basically is that the colored quarks and gluons become confined and the mechanism for this confinement is still not fully well understood but we know it's of great practical importance had they not be confined we would not be like the way we are we're made of these confined hadrons but we can see on very short time scales actually propagation of colored objects for example and they eventually form jets of these confined hadrons so it's a very deep physical problem to which we don't yet have a full solution and I think in the work which I'll soon describe I at least was very motivated by building some models which are not just anti-decider conformal field theory but not anti-decider not conformal field theory duality and it's possible to do it in a way that in one asymptotic region the space is nearly ADS and it still has logarithmic running so that's a bit difficult to accomplish okay so just to talk about why we think the theory is confining well there is this Wilson formulation of non-abelian gauge theory which consists of link variables which are just the SUN matrices and the action is just made of plaquette terms which is trace of product of link variables around each plaquette the Wilson criterion for confinement is the area law for large Wilson loop which is equivalent to saying that when you separate the quark-anti-quark to large distance the potential instead of coulombic suddenly becomes linear and this is not hard to prove in the strong coupling expansion just expanding in these plaquette terms and then you tile this large Wilson loop by small plaquettes and you get the area law but what's hard is to show that this really happens in the opposite limit of we coupling on the lattice and there is numerical evidence but still no exact solution there so to obtain this continuum limit one needs to interpolate to the weak coupling on the lattice scale and then the following miracle has to happen so we know this QCD beta function from the Nobel Prize winning work this is the one loop coefficient and from it you can infer so this negative sign is crucial right which happens when the number of flavors is sufficiently small and then you watch this running and you see that this coupling will formally diverge even though it's very weak at short distances it will formally diverge at some scale and we call this scale lambda QCD so this mass scale is actually immensely suppressed relative to the lattice scale 1 over A0 so you want to take a kind of double limit where A0 the lattice spacing goes to 0 and the suppression factor becomes infinitely small this expansion and then we get this lambda QCD so there is a hint already at one loop but it's what appears to happen is that all the bound state masses bound states of gluons for example we call glubals their ratios are dimensionless and their scale as well as the string tension are governed by this lambda QCD so this was already suggested in the early Monte Carlo studies as well as the Hamiltonian formulation of QCD that Gogat and Souskind pioneered around the same time and this is the result that you see from the numerical work you see at short distances a Coulomb potential and then it pretty quickly turns over to the linear potential so the big question is can this asymptotic linearity be proven at least in some models so here is a picture from lattice of this confining flux tube its thickness is on the order of femtometer and it doesn't change much as you change the distance between the quark and anti quark now the importance of large n is that this flux tube cannot really the probability of it snapping but into two flux tube by creation of quark-anti-quark pair is suppressed by one over n that's a big idea of Toft and right around 50 years ago and it's simplified the problem a bit but has not rendered it exactly solvable so this large n Young Mills series one defines this Toft coupling lambda keeping it fixed as n goes to infinity and then the decay of the QQ bar pair into QQ two QQ bar pairs is like a decay of a meson into two mesons that probability is one over n so this defines for us a weakly coupled strength theory so now another thing that was already mentioned by Jeff is a crucial component of the second super string revolution that happened about 11 years after the first one besides the string dualities was the realisation that d-brains or d-brains are important they're the objects that are special kind of like defects on which open strings can end so even in a theory where in the bulk there are only closed strings once a closed string touches this d-brain that can open up and live on it as an open string giving these d-brains dynamics and actually Polchinsky in his very his breakthrough paper already said wrote that perhaps these d-brains are the fundamental Ramon Ramon charged elementary objects behind these brain solutions and supergravity but it was just seemed like a bit of a throwaway remark and the following several years people were trying to flesh it out there were many papers on this so one specific example that's really caught on is the case of three brains which realise you can stack them on top of each other and that gives SUN maximally supersymmetric and equal force supersymmetric gauge theory in four dimensions and certainly we are in four dimensions of three plus one dimensions but when you look at this macroscopic object it also creates a curved space around itself and one can study the correspondence between something that this Young Mills, Super Young Mills point of view and this curved space point of view so around the early 96 and on in collaboration with Steve Gobser who is unfortunately some of you know died in a tragic accident in 2019 he was at the time a graduate student and AWP who was a postdoc who set out comparing the entropy of this near extremal brain which is a slightly more complicated metric and it looked like it had a lot of features compared to just a black body entropy in three plus one dimensions just it had n squared degrees of freedom characteristic of n squared gluons and so on and then later on I looked at for example absorption cross-section where you absorb say gravitons from the asymptotic region into this throat region and there was some magical correspondence again with closed strings coupled to D brains and eventually it all crystallized after landmark paper by Juan Maldesena who pointed out that it's important it's one can at least take the low energy limit directly in the geometry and this throat region is five-dimensional anti-dissiduous space times five-dimensional sphere that is basically dual to the Seneca for Superan Mills theory and moreover when it's strongly coupled this space is weakly curved so one can just study supergravity limit and small corrections so this is a schematic slide on this this is from a semi-popular paper that Juan Maldesena and I wrote we tiled it with cows just to remind people that you know there is a spherical cow joke something idealized so this is a hyperbolic cow or something like that but we're still learning lessons from this hyperbolic cow so we got 26 years of rigorous research and tens of thousands of papers I wouldn't have thought back then that this would be like this but it's definitely cut on and one of the reasons is that one can develop two expansions one is the expansion in Feynman graphs just a week we coupled expansions in powers of the Toft coupling but then there is also expansion in one over square root of lambda that's due to the higher curvature corrections like curvature to the fourth power corrections to supergravity and one can have a lot of fun trying to match them there are some exact functions of Toft coupling now and now and now just to make a transition to confinement another thing that one can compute is the quark-anti-quark potential just by studying a string ending at infinity of ADS and then the pure ADS space because it's scale invariant the potential is exactly columnic so there is no sign of confinement but this how do you change the space to make it confining and the root of the idea appeared in some earlier papers by Sasha Polyakov who basically said that you need to put a kind of end to the space and then the string can go to you had the idea of warped space but more complicated than ADS that's basically what you need so now the slides will get a little more complicated I'll probably skip over a lot of technical details but essentially one thing that we need to do is first it's impossible to have a confining theory with unequal force supersymmetry because unequal force supersymmetry is very restrictive but if you first break supersymmetry to unequal one that's just supersymmetric load dynamics that can be confining so first one needs to figure out a way to make the theory unequal one supersymmetric one can do it through the so-called cone brain trick that I think started partly in my paper with Adwitan in 98 put the brains not at some generic point but at the apex of a cone which is a Kalabiyao space then the base of the cone changes from the five-sphere to some more complicated space which is called the Sasaki Einstein space so the simplest space like this is the conifold it's a very generic space in string theory it appears in many contexts it's a pretty generic conical singularity described by the equation sum of squares of four complex variables is equal to zero and one can describe the base of this cone for historical reasons called T11 in terms of angles completely explicitly you have two two-dimensional spheres and then an extra angle that's kind of mixed with these two two spheres but importantly the space has topology which is a two-dimensional sphere times three-dimensional sphere and with Witten we constructed the gauge theory the super conformal gauge theory that lives on the D3 brains on the conifold it's still like QCD like it has two SUN groups with some bifundamental super fields just matrices and we had a pretty explicit super potential and that was dual to ADS 5xT11 but now it's an equal one example of the duality but then it turns out that starting with this space you can reach a confining theory pretty easily one just needs to add a kind of Ramon-Ramon flux through the three cycle which is accomplished by wrapping five brains which Jeff mentioned around the two cycle and then you get another beautiful and well-known space called the deformed conifold all you do is you just add the constant on the right-hand side and magically the space describes confinement the metric of this warp deformed conifold first with Tatlian I wrote a paper but we didn't have the deformation and the space was singular but then with Matt Strassler we obtained this completely explicit looking metric with a warp factor and that turned out to give the confinement and here is the warp factor in ADS space is just blows up at r equals 0 and this is the ADS region but before we realized that the conifold gets deformed there was a naked singularity that we discovered with Tatlian and then once you deformed basically that's a way for the geometry to tell you I'm not acceptable but then once the deformation happens that's a way of generating this lambda QCD in this theory the solution becomes smooth and weakly curved everywhere and that describes confinement in this so in this particular model there is a kind of physicist proof of confinement the naked singularity is forbidden and the smooth space gives you a confining solution and a lot of effort went into studying the spectrum of bound states in this model this model exhibits dimensional transmutation because in the infrared there are logarithms so through the argument of these logarithms so it's very similar to what happens in QCD you generate this deformation this term dimensional transmutation is widely used as generating an infrared scale out of UV logarithmic running this theory is somewhat akin to n equal 1 supersymmetric gluodynamics but it has some other so it's best to be thought about as SU2M cross SUM gauge theory which also has some baryonic order parameters and it turns out that these baryonic order parameters can get vacuum expectation value as a result of which there is a goldstone boson and it's massless scalar superpartner in some ways it makes the theory even more QCD like because there is no mass gap just like in QCD there are pions, pions for massless quarks but the theory is still confining here there are also massless fields but the theory is still confining and the confining string this flux tube is nothing but this string that went to the bottom of the deformed conifold and one can I hope this is visible this is actually from an undergraduate thesis written years ago you can just compute this interpolation between collomic behavior and linear behavior very very simply by studying the shape of the classical string and this you see qualitatively is very similar to what you get on a lattice but this is essentially an analytic calculation or a calculation with very smaller there is no Monte Carlo here you just solve the equations for the non-Bugato action in this geometry so I would say that for this model which is akin to supersymmetric QCD the dual gravity does provide a hyperbolic cow approximation to QCD and of course a dream is to do something similar for large NQCD I wouldn't say this dream is at hand but we have to think long term so let me just finish by saying thank you ICTP and Grazie mille thanks Igor for this wonderful summary of the second or third I should say two and a half revolution of holography which many of you must be studying now in the school and this fascinating connection between the theory of gravity and the theory of strong interactions that he mentioned and now we move to Steve Schenker so thanks Steve for joining us at this ungodly hour at 5 o'clock in the morning and he will tell us about the sort of the ongoing excitement that continues in string theory with the work on this one of relation between quantum chaos and quantum gravity that he will tell us about so Steve okay well thanks very much let me let me try to share my screen and see if we can get this to work can people see that is it is can you see the screen yes I'm having a little trouble hearing you yes yeah everything is okay yes well I am glad to be here even if only remotely and I'm glad to teach you brought up the fact that this is I guess a little after six in the morning I'm a late riser so bear with me if things are not quite as clear as they might be let me echo Jeff and you who are saying that I'm really honored to receive this direct medal from the ICTP the ICTP is a vital institution in our field and I just want to acknowledge you and teach for doing the great work you're doing in nourishing this institution and well I really hope it can continue as long as physics is important in our world I'm particularly happy to share this award with Jeff, Igor and Lenny these are three physicists that I greatly admire known forever and I'm happy to count them all as friends as well as colleagues as a scientist I've always been fascinated about connections between apparently different areas of physics and so I thought today what I would talk about are some connections that have been occupying my attention during the last decade or so between the ideas in quantum gravity and quantum chaos these take place in the context mostly of studying objects called black holes which certainly are mainly thought about and certainly in the early days were thought about as mainly the stunningly simple astrophysical objects featureless you know elegant symmetrical things but this point of view changed actually around 50 years ago where basically largely through the work of Beckenstein and Hawking it was realized that these objects are not so simple they actually have the richness and complicated structure of thermal systems particular these authors pointed out that black holes should be thought of as having an entropy a degree of freedom count and in Hawking's famous work they were shown to actually have a temperature now these connections take on a new life in the context of this wonderful ADS CFT duality that Igor just described to us where you have a gravitational theory with a black hole in it and then you have a dual you sometimes call a holographic boundary description in terms of a gauge theory and this thermal object the black hole just corresponds to a thermal state of the gauge theory you heat the gauge theory up and you have these hot gluons running around you can see this red yellow and green gluons whistling around the boundary gauge theory and because of this remarkable duality the temperature of the gauge theory corresponds to the temperature of the black hole and the entropy of the gauge theory this n squared count corresponds to the entropy of the black hole well there are other common sort of features that you can study and make correspond thermal systems we know of are dissipative you know you drop some cream into your coffee and it sort of spreads out into a uniform state and black holes are too you see this by pinging a black hole and looking at the response now you know actually this has been observed experimentally remarkably or you ping a black hole in a violent way you imagine you take two black holes and that are near each other and these wonderful LIGO experiments on gravitational wave you can actually see the gravitational waves that are emitted from this process and they look kind of like this in the theoretical sketch the late stages where the black holes are almost coalesced there's this ringing down phenomena where things oscillate and decay and these are described by what are called quasi normal modes in the black hole this ring down and this quasi normal mode to take corresponds to thermal relaxation in the gauge theory so here's another correspondence between thermal behavior and black holes well there's another characteristic of thermal systems that I want to focus on today and that is thermal systems are typically chaotic when we learn about classical statistical mechanics we justify using statistical mechanics because we uniformly cover phase space and this we attribute to this very intricate pattern of the motion of gas of classical particles and the crucial ingredient in making this work is what we often call strong chaos sensitive dependence on initial conditions very colloquially known as the butterfly effect whereas if you change the initial point in phase space by a little bit it yields a dramatic change in the position at future times now you can quantitatively describe this by the rate of these divergence of these trajectories we say these nearby trajectories diverge exponentially with a Lyapunov exponent this is a number Lambis of L and this is a characteristic of how strong the chaos is and so we naturally are led to ask the question what is the gravitational analog of this kind of Lyapunov behavior and the first insights into this came from two important papers one by Haydn and Presco and the second by Sekito and Lenny Suskind and these insights were crucial but to make more progress to understand this connection more deeply we need a sharp diagnostic of this quantum Lyapunov behavior and this in the current wave of developments in string theory and black holes was actually developed by these authors affectionately known as ampsus and it's now called an out-of-time order quarrel it turns out is when you're thinking about connections in two different fields the kernel of this idea was actually discovered in condensed matter physics by Larkinov-Chinikov in the 1960s but the current mini body version of it was actually pointed to by these authors so this leads to the question what is the gravitational analog of this particular diagnostic well this is where I came into the subject I was brought into the subject by my collaborator Douglas Stanford with whom I've done a lot of the work I'm describing and I just want to acknowledge it's been a really fruitful and joyous collaboration during this past decade so after thinking about what we realized as Kutayev did also that Lyapunov behavior in black holes is described by a high-energy gravitational shockwave collision near the black hole horizon now such scattering was not new to us it was discussed previously and worked by Drayna Tuft and Kim and the Eric and Herman Verlinde and by interpreting their work in this new light we were able to show that this exponential, that this Lyapunov exponent is given by precisely this value the fact this has dimensions of an energy so having a KT here is not a surprise you turn it into an inverse time with an H bar and the 2 pi is the thing that you actually have to figure out motivated by this and other ideas that were floating around the field so we were able to argue using general ideas in quantum mechanics and quantum statistical mechanics the general quantum systems well there's some fine print here that I put at the bottom of the slide general quantum systems obey reasonable physical conditions obey the chaos path that this Lyapunov exponent in any reasonable physical system is actually less than or equal to this black hole value now this actually establishes a numerically precise refinement of a conjecture previously made by Sekino and Sustak they call the fast scrambling conjecture the black holes are the fastest scramblers in nature and in fact we see now that this is true in a tight quantitative sense that the Lyapunov behavior in any reasonable physical system is less than this black hole value well this is all very good and it's taught us a lot actually about strongly interacting thermal systems like these hot gauge theories but the gravitational correlates of quantum chaos we've discussed so far these relaxation, these quasi-normal modes or shockwave collisions these are classical things in general relativity things that general relativists could and did study many many years ago the question that we want to turn to now and that Douglas and I really wanted to turn to are there aspects of quantum chaos that have implications for the quantum gravitational behavior of black holes things that really involve quantum gravity in an intrinsic way well we think there are some and they go back to another characterization of quantum chaos that has a long history probably going back to Wigner in the 1950s and there's a long standing conjecture with an enormous amount of evidence that the energy levels of quantum chaotic systems that is the eigenvalues of the Hamiltonian are widely believed to have the same statistical properties as the eigenvalues of a random matrix you don't have to actually look at a particularly highly structured Hamiltonian you just take a matrix and stick random numbers in each of the entries and diagonalize it and that gives you the same statistical pattern of energy levels as a quantum chaotic system and I've always been drawn to universality and this has struck me as the more I think about it as a remarkable example of universality it's independent of the dimension of the quantum system of it being local or not and it's only weakly dependent on symmetry it's really an incredibly robust phenomenon and such a universal phenomenon in quantum chaotic systems should reflect a universal phenomenon in the quantum gravitational behavior of black holes so the hunt was on to find out what this phenomenon could be now unfortunately this is a really hard thing to study even given this wonderful ADS-CFT duality if you actually want to understand the individual energy levels let's say of the gauge theory that's just too difficult to study with our current level of technology fortunately around this time a simple tractable model of a black hole was developed it's a black hole in one space in one time dimension and it's called the Sachs-Devier-Katayev model developed by these authors in various guises and it's a remarkably simple system it's just a system of N interacting fermions an ordinary quantum mechanical system and one of the indications that it's got something to do with gravity is that it saturates the chaos down and in fact we now understand that in certain respects it really accurately reflects some gravitational dynamics of a low-dimensional black hole well this system is simple enough that we can actually study the energy levels of this toy black hole and their statistical properties numerically here's a piece of the energy spectrum of one of these SYK Sachs-Devier-Katayev black holes well we want to figure out if it looks like random matrix statistics so we need to diagnostic a random matrix statistics a good one is to take the Fourier transform of energy level differences this is sometimes called the spectral form factor and we can then just study the SYK model and we actually first did computer experiments when we really didn't know what was going on and it's a great virtue of this model that it's simple enough that we can actually do experiments when we don't know theoretically what's going on I would say that I've always been drawn to these kinds of systems in my work these simple models where you can really get in and do calculations even if the system you're interested in is beyond reach and so since we're looking for statistical information random matrix statistics we're going to do some averaging you might take many instances of this Sachs-Devier-Katayev model slightly different black holes and average over and this is what you see this is numerical data and you see as a function of this Fourier transform time you see something we call a slope, a ramp and a plateau and it turns out that this ramp plateau structure is the universal signal of random matrix statistics this particular shape the slope region is not universal it depends on which system you're looking at and so now we sharpen the question we can ask what is the signature or what is the origin of the ramp and plateau in gravity so using a bunch of clues that were supplied by the SYK level we realize that the ramp at least can be explained by a new geometry in addition to the ordinary black hole geometry that you add in the quantum mechanics in the path integral that Dirac pioneered describing gravity you add the black hole configuration and you also have to add this completely new configuration we call it the double cone and it's kind of a wormhole in space and time this is time and this is space in this picture and because it's a really new configuration it's a non-perturbed effect in quantum gravity kind of like tunneling and further such configurations such double cone things occur generically whenever black holes do and so this is consistent with the universal character of random matrix systems well we have to explain more than the ramp we have to explain this plateau as well this is turned out to be much more subtle it seems at least in simple models that you can explain the plateau by summing over even more complicated topologies if you will double cones with extra handles put on them and this is the state of the art is in this very nice work with Phil Sott my former student Douglas David Yang and Shengyu Yao my former student as well my current student actually well these wormholes turn out to play a role in all kinds of subtle quantum phenomenon black holes at least in simple models where we can understand them they describe the long time behavior of correlation functions they describe another wonderful aspect of chaos involving the eigenfunctions as well as the eigenvalues the eigenstate thermalization hypothesis the statistics of operator product coefficients in the gauge theory perhaps most well known closely related configurations seem to be the origin of this explanation of the page curve for the radiation from evaporating black holes here they're called what are called replica wormholes and then a very interesting work that I'm very excited about they seem to provide a mechanism for the formation of these mysterious objects called firewalls that's motivated a lot of research in this field so it seems like they're ubiquitous, they're useful but their existence raises a number of puzzling questions and one of them that I'll leave you with is that if you actually study one particular black hole let's say just the gauge theory that Igor described and looked at its energy levels and did their Fourier transform random matrix universality tells you you wouldn't get this blue curve you would get this red curve you would get a noisy curve whose envelope whose average is given by this rampant plateau so the double cone just describes the average behavior of the theory, not the detailed noisy wiggly spectrum and the detailed shape of this curve is basically depends on the actual numerical values the detailed numerical values of these energy levels so we're not done yet if we really want to understand the system we have to understand what new ingredients must be included in the bulk gravitational description to account for this noise now these new ingredients understanding this jagged red curve is not the only part we don't really understand what quantum gravity is until we understand the degrees of freedom that are necessary in order to compute this detailed spectrum so my hope is that by thinking about the degrees of freedom you need to produce these kinds of pictures we'll understand a little bit more about what really quantum gravity is and I hope that many of you in the audience will be able to join with us in trying to answer this central question that's motivated I think all of us in the field so I think with that I'll leave it and let me just thank the ICTP and all of you for your attention thank you okay thanks Steve for this wonderful talk about explaining some of the universal features understanding them from the black hole side now we will move to Lenny as last speaker and his title is as provocative as I would have expected from him on black holes and foundations of mathematics and he has often presaged developments including the holography and chaos so I look forward to what he has to say thank you can you hear me? hello? can you hear me? can't hear you yep okay you hear me okay I want to talk about physics not myself I'll mention only one thing about my long career I haven't been up this early since I was born at 4.15 in the morning 84 years ago so thank you Artish for the opportunity to relive that event okay thanks Lenny I would just I would request you to just step back a little bit so that we can see you okay but then I can't reach the computer screen okay okay don't bother don't bother continue I want to do something called sharing my content okay continue continue screen screen broadcast I think do you see my screen? do you see that? no not yet you don't see that well I didn't do it right then you have the green share screen button on the bottom no it's on the top oh it's on the top okay yeah screen now maybe I have to write do start broadcast I always forget let's try that one okay can you see that? hold on no you froze for some reason one more try yes it's not my fault this is just too early in the morning okay one more shot screen there's two things here one says screen broadcast the other says start broadcast I don't know which one I'm supposed to poke I think start broadcast you think so that's what I just did oh you did well let's try screen this time okay okay try the other one screen broadcast good good good it's better no I have to find the thing I looked at how about that yeah it's perfect good good perfect okay it's not it's not quite perfect because I don't like the cross hatching yeah but I'm gonna leave it anyway okay so as I said I want to talk about physics and not myself that's what I like best physics and that's what we're going to deal with the biggest problem with physics to my mind the biggest puzzle is that it exists at all that a creature just a handful of genes different than the monkeys could not only ask the questions about what the world is like what the physical world is like but answer them avoid all the nasty pitfalls and eventually come up with quantum mechanics general relativity the standard model and even today the beginnings of quantum gravity I say this mainly to say something good about the human race it's getting harder and harder to do sometimes okay what are the tools that we have used to get there the first actually it was the last one in a certain sense historically was the idea of experiment an idea of experiment controlled experiment doesn't go back that far thought experiments go far back further they go back way way way back to the Greeks and of course mathematics which also goes back to the Greeks or maybe even before the Greeks I don't know mathematics calculus geometry group theory topology complex functions differential equations all that stuff I'm going to talk today about the entry into physics of a different kind of mathematics the mathematics which is mathematics of the foundations of mathematics which grew up together with computer science the names the classic famous names and names like Turing, Church, Shannon, Goethe, von Neumann, Kumagorov that's on the classical side and on the quantum side it began with Feynman and a bunch of dot dot dots including names who are still active so I won't mention them right now there are just too many of them we're going to talk about quantum computational complexity why is it taking a common place in quantum theory of gravity in black holes so what is computational complexity and what does it have to do with the foundations of mathematics how difficult is a theorem there's an interesting question when I was a kid learning beginnings of Euclidean geometry I learned about theorems and I noticed that there were hard theorems and easy theorems in fact my teacher even told me that there's a very hard theorem called the four-color map theorem and I wondered sort of formulated this somewhere later is this idea of hard versus simple theorem, easy theorem, is it subjective idea or is it really an objective meaning to the hardness or the difficulty or the complexity of the theorem alright I think there is such an objective meaning the purple circle there is the circle that contains all possible sentences that you can formulate within some mathematical context within some axiomatic context all possible sentences you start with the axioms the axioms are the simple sentences and you can apply to those axioms the rules of logical inference and the rules of logical inference simply move you around in that space they move you from one sentence to another sentence and the complexity of a theorem t, t is the theorem the complexity of a theorem is simply the minimum number of steps that it takes you to go from the axioms to the theorem using the rules of inference a very simple idea that idea is the idea of computational complexity the idea, oh incidentally this is similar to asking what is the length of the absolute shortest path from one point to another in some kind of space that's the general idea a simple starting point a in this case the axioms but it could be more general a set of simple steps in our case it will be gates gates associated with a computer or quantum computer a target t and the complexity of t is simply the absolute minimum number of steps that it takes to go from a to t for example if we just happen to have a computer register with a bunch of bits they can be up or down the simple state that we might start with is just everybody up and then we can ask how many steps how many simple steps does it take to go to a general state to an arbitrary state now the answer in classical physics is very simple and small and number is small if the basic process is just a flip a bit it never takes more than n where n is the number of bits it never takes more than n steps to get from a to t and so complexity, what's called state complexity never gets very big in classical physics Feynman was the one who pointed out that quantum states can be vastly more complex in classical states the number of parameters for a classical state is just n, the number of bits for a quantum state a quantum state is a linear superposition of classical states two to the n of them with a set of amplitudes, psi i and so there are two to the n complex amplitudes describing a quantum state vastly more degrees of freedom in a quantum state than in a classical state that's what we need quantum computers to do quantum mechanics or to do numerically quantum mechanics in an inefficient way Feynman pointed this out in the context of computation he was asking the question why the hell is quantum mechanics so hard to do numerically and this is the reason if you have a quantum computer now a quantum computer we can think of is just a bunch of qubits on the left side that's the input some simple input you go where you're going by applying a bunch of gates the gates are just these two qubit interactions could be three qubit interactions and you let the system run until you get or until you hope you get where you want to go to some complex output and again the complexity of a quantum state is the minimum number of gates needed to prepare the target state from some simple state A now the maximum complexity I told you the maximum complexity classical is just n the number of bits quantum mechanically the maximum complexity that you can ever have is e to the n and that's much much greater than n furthermore almost all states have maximal complexity fundamentally this is a different way of describing chaotic systems as systems which generate complexity now given two states A and B one of modest complexity let's say polynomial complexity and one of exponential complexity how computationally difficult complex is it to distinguish A from B this is a different question and how complex A or B is it's a question of how complex it is computationally to try to distinguish the two of them one of modest complexity and one of exponential complexity surely the answer must be that it's not too hard to tell so Charlie Brown is going to be happy because he's going to be able to distinguish these two states but when he does the calculation he discovers that the complexity of complexity by that I mean how complex is it to distinguish these two states of modest versus large complexity he finds out that the distinguished modest complexity from high complexity itself is a very very difficult problem exponentially difficult what is the lesson the lesson is that quantum complexity even though it might not be very large is an extremely subtle ethereal fine spun elusive tenuous hyper-refined ghostly quantity a ghostly property of a state I got all these words from the thesaurus when I put in subtle but I think the best word from my mind comes from a wonderful paper by computer scientists Boulin Pfeffermann and Bazarani who are dealing with exactly the problem that I'm going to be describing they called quantum complexity unfeelable that it's very difficult to both measure and possibly difficult or exponentially difficult to measure or even to compute so unfeelable that it can hardly be an interesting property of a physical system like energy, entropy, or density and yet it is now expected and thought that the properties of the interior the physical properties of the interiors of black holes are governed by the property of complexity so let me tell you the conjecture of 2014 which is now largely accepted not the area of the horizon but the volume of the interior of the black hole is the complexity of its quantum state remember the area of the boundary of the black hole of the horizon is the entropy of the black hole the volume of the interior appears to be the complexity of its quantum state and it grows with time long after the black hole is settled down to thermal equilibrium well when I put this idea forward the response of most physicists was great great great volume equals complexity and they started calculating the volumes of lots of black holes checking it against some properties expected of complexity pretty much agreed when I explained these things to computer scientists who really do understand complexity the reaction was not nearly as happy the response of the computer scientists was full it's unfeelable how could it possibly correspond to an interesting measurable physical property of the interior of black holes like its volume my response was yes complexity is inaccessible very difficult very fine spun very unfeelable but so is the interior of a black hole I suspect the two inaccessibilities are dual to one another and that has largely turned out to be correct okay so here's the penrose diagram for a black hole I'm not going to describe it in any great detail I'm not going to describe it at all you don't know it look along the horizon along the horizon as the horizon forms it grows and it grows until some point the red dot where the black hole has come to thermal equilibrium that's where the area of the horizon and the entropy has finally reached its maximum and thereafter it just stays constant that time scale is very very short for a solar mass black hole I think it was about a millisecond very short time for black hole to come to thermal equilibrium and after that it's perfectly static and nothing whatever seems to go on at least as seen from the outside we were to plot the entropy growth as a function of time we would see ha ha a ramp and a plateau but it's a different ramp and a plateau the area versus the entropy and in about 10 to the minus 3 seconds for a solar mass black hole that it would max out at thermal equilibrium it's an example all of this of a general relativity thing area of the symbol two D's in it two upside down B's it's a symbol I've used for is dual to is dual to a quantum thing in this case entropy but something behind the horizon continues to grow long after thermal equilibrium here's the picture that goes with it you can see that those green dotted lines which represent in some sense the volume of the interior of the black hole continue to grow and grow and grow they grow of course the Penrose diagram is hard to read quantitatively but the answer is they grow linearly the volume and it's expected on general grounds of a combination of quantum mechanics chaos and other things that it continues to grow for a long time quantum fluctuations build up and do something non-classical inside the black hole how long does that take? it takes an exponential long time an exponentially long time exponentially in the entropy of the black hole for the black hole to stop growing in this way eventually it will stop growing so if we plot the volume of the black hole not the area as a function of time we'll see the classic ramp in the classic plateau same picture as for the entropy but on an incredibly different time scale it would take about 10 to the 78th sorry e to 10 to the 78th it doesn't matter whether it's years, seconds or anything else e to the 10 to the 78th years for the volume to max out what is this thing which grows linearly for an exponential time? but which is invisible or unfeelable from the outside? well complexity grows behaves in exactly this way again the same classic curve complexity starts out growing linearly grows for a very long exponential time exponential in the number of qubits which effectively means exponential in the entropy of a quantum computer until again it maxes out and it maxes out at a time which is exponential in the number of degrees of freedom of the system the conjecture the volume of the black hole interior is due to the complexity of the quantum state of the black hole let's see how much time do I have? stop me exactly when I've gone 20 minutes so maybe I've already gone 20 minutes okay so let me talk a little bit about complexity and firewalls when complexity increases the horizon is smooth let me not try to prove that that's a statement that's a conjecture it's a conjecture that as complexity increases the horizon behaves in a smooth classical way that you can fall through it and nothing special happens when complexity stops increasing that's the place in the time that we think shock waves form just behind the horizon the kind of shock waves that Steve was talking about before those shock waves are what are called firewalls just behind the horizon when complexity stops increasing now we can talk about a quantity which is the complexity of the time derivative of complexity what that means is how hard is it to compute given an initial state how hard is it to compute the time derivative of the complexity after a period and that is exponentially difficult even if the complexity itself is not so big it is very, as I said the complexity is so unfeelable that even when it's not big it's very difficult to compute that's the answer this is interesting a complexity the complexity of complexity in other words the complexity of knowing whether the complexity is increasing or decreasing is itself extraordinarily hard to predict now let's suppose Bob or somebody else wants to jump into a black hole he wants to know that that black hole doesn't have a firewall behind it so knowing how the black hole is formed in detail he tries to do a calculation of whether the complexity is increasing or decreasing at the moment that it's going to jump in and he finds it's just much, much too hard to do that it would take forever exponentially long e to the 10 to the 70, 80 years or worse than that extremely difficult calculation it is a calculation now before I talk a little more about this I want to talk about something somewhat philosophical idea called the church touring thesis this is something I think it's something in the foundations of mathematics and in the foundations of physics and in the foundations of computer science the church touring thesis said that no physical process can do a computation faster than a touring machine or faster than an ordinary digital calculator sorry it wrong statement no physical process can do a calculation that the computer cannot do to get the word faster that seems to be true as far as we know it seems to be true that any physical way of doing a computation using any physical system to do a computation you cannot do something that a computer or a touring machine cannot do that was interesting it's a principle of physics in a sense it tells you what physical systems are possible there's something called the extended church touring thesis it does say that no physical system can do a computation faster than a touring machine faster means qualitatively faster that appears to be a wrong statement we believe that quantum computers can do computations significantly faster than touring machines so something turned up that was called the quantum extended church touring thesis that says that no system can do a computation or learn the answer to a difficult question faster than a quantum computer as far as we know that's a correct statement well now let's ask the following question can bob jumping into the black hole find out whether there is or is not a firewall just behind the horizon in a short time remember now this would take a calculation to find out which would be exponentially difficult even with a quantum computer incidentally all bob has to do is jump in he instantly passes the horizon he instantly knows whether there was a firewall there and therefore whether the complexity was increasing or decreasing he's beaten the quantum extended church during thesis he's found out in almost no time at all well we think that the problem here is the statement of the problem what do we really mean by learning something in this context I think we should really mean what can be determined by the experimentalist outside the black hole the person doing the experiment outside the black hole and not the one jumping into the black hole the church, the extended church or the quantum extended church during thesis can be rescued in a simple way by just saying that outside the black hole doing this experiment from the outside perhaps in a repeatable way with many black holes can't not himself learn the answer to the question that a time faster than a quantum computer can do it now this leads to a conjecture the conjecture is a conjecture about how hard it is and how difficult it would be for bob who has jumped into the black hole found the answer almost instantly communicated outside the black hole ordinarily you might think the answer is you cannot communicate it at all but we do know that information can be transferred from inside the black hole the outside the black hole for example when the black hole evaporates the evaporation products carry out this information and so question is how long does it take to reconstruct from the Hawking radiation whether the interior did or didn't contain a firewall or happened to bob when he fell in and the answer must be in order to rescue the church touring the extended church touring pieces it must be at least as long as the computation would take to determine whether the complexity was increasing or not that seems to be very likely from work of Harlow and Hayton it does seem that the time that it would take to reconstruct anything about the interior of the black hole from assembling the Hawking radiation and doing a process on it itself would be exponentially long so it seems likely that the quantum extended church touring pieces is rescued is correct is a correct philosophical or perhaps a principle of physics but the interesting thing is that horizons the property of horizons the nature of horizons are what protect the quantum extended church touring pieces okay I think I'll leave you with that it's an interest I think it's an extremely interesting subject and it's telling us things about the meaning of the interior of black holes which I think is extremely subtle and perhaps at the root of what the interior of a black hole really means good I'm finished thank you and thank you and of course my colleagues in this class thank you so much for the honor of being recognized it's nothing better than being recognized by your colleagues and your peers thank you Ateesh okay thank you Lene thank you Steve and Jeff and Igor for these wonderful talks I'm sure there are many questions so I will be now we have some time to take questions and my colleague Professor Anise B.C. she will take the questions so questions so to the speakers can be shy so I have a question for Igor so what are your thoughts on the prospects of understanding the real world QCD with holographic tools thank you that was the last question on my slide and I think it's clearly not going to be in the supergravity limit because we know that due to asymptotic freedom that half coupling becomes small in the UV so we probably need some bigger set of tools that was applied to just find supergravity solutions like this deformed conifold solution but more generally I mean the idea of mapping doing some kind of large end mapping from gauge theory to some dual variables dual string variables I think people should keep exploring other tools that we know because in particular so not super symmetric say pure glue large end QCD we know defines a weekly coupled string theory but we don't know how to write down the world sheet formulation of it so we I think doing more detailed numerical studies of the theory of extrapolating to the large end limit has already produced new insights for example new insights about new degrees of freedom on the string world sheet and then another I think another thing in the spirit of Dirac I think it's useful to explore the Hamiltonian formulation of gauge theory because that gives us access to things that we can see using Monte Carlo and then we really see strings evolving splitting joining and that's also amenable to quantum computers and simulators so I expect a lot of research on that for years to come like the new connections with quantum simulators thank you very much question for professors asking I was amazed by this quantity 10 to the 37 it's much much bigger than the age of the universe so how do you get to this quantity maybe I'm getting to trouble by asking this but it's simple generally in quantum complexity the maximum complexity is exponential in the number of degrees of freedom which means exponential in the entropy the complexity is a measure of how long it takes to to get where you want to go how many steps you have to make minimum number of steps so the number of steps that you would have to make to get to the final most complex state of black hole would be exponential in its entropy the entropy the entropy itself is an exponentially big quantity it's been actually big in the sense that 10 to the 78 is the entropy of a solar dot mass black hole so it's e to the 10 to the 78 a rather large number that's kind of interesting I've discovered in my career that people are more afraid people meaning physicists in general much more afraid of very very large numbers than they are of infinite numbers if something comes out infinite you say oh okay it's infinite if something comes out e to the 10 to the 78th wow you get real scared that's the craziest thing I ever heard of can't mean anything so yeah big numbers are and this is far from the biggest number that occurs in this business but it is big you know of course if somebody if you say e to the 10 to the 78th that's the time that takes and somebody asks you what units are you using yeah doesn't matter more questions thank you for nice presentation I'm not I don't have so much experience in black holes but can you tell us about something like experimental aspects of black holes do we have some experimental news is that about black holes who are you asking about experimental no who are you who are you asking it doesn't matter like who knows well Steve gave a very good answer his answer from Malico the ring down the ring down of quasi normal modes it's not really a quantum process well it is a quantum process but it's also a very classical process but that's a very good experimental measurement of some of the things we're talking about that's any body else Jeff hi Lenny it's Jeff I wanted to ask a question so you were drawing an analogy between complexity black holes and foundations of mathematics this purple diagram and the idea of a minimum length of a minimum length of a proof of a statement in mathematics but in the foundations of mathematics we know that there are points outside of that purple purple area that are true but not provable in any finite number of steps so is there some analog of that in black hole physics I think it means there are configurations within a given setup of maximums and so forth where the complexity of a true statement can be infinite you'll just never get to it the girdle sentence this sentence can't be proved that's the girdle sentence you'll never get to it no matter how you'll never get to it or it's inverse or it's opposite no matter how long you run the thing but yet you think it's pretty darn true that the sentence cannot be proved I think mathematicians have found more and more interesting statements that are in this class I think if you believe the complexity ideas it's probably true that the vast majority of true theorems can't be proved and in black hole physics well I think that's just a statement that almost all states of a black hole are maximally complex and therefore almost all states have this property that the complexity can't be increasing and therefore likely have firewalls but those black holes have been around for each of the 10 to the 78 years not for the page time as Joe and colleagues claimed the claim now is that black holes are on firewalls which is a symptom of having reached maximum complexity after a time of order e to the s but the universe isn't that old that's true, neither am I I'll leave it there this is the question for Cezking thanks for your nice talk but I didn't know about this the complexity and the volume of the interior of black holes I was wondering if perhaps you can apply the same idea for the interior of a wormhole in the case of a wormhole or perhaps it has not any sense I don't know I'm not sure there's much difference between the interior of a black hole and the interior of a wormhole in some sense the interior of a black hole is a wormhole you can apply it to the idea of the interior of a wormhole but it's pretty much exactly the same thing this is the idea that what is a wormhole if you mean by a wormhole in Einstein-Rosenbridge it really is just the interior of a black hole so yeah I would say yes you can apply it to that I have a question to Professor Leonard you were talking about foundations of mathematics so I just want to ask and of course we are experiencing Dirac spirits let's say and I just want to ask do you think mathematical beauty is equally important to experiments yeah not in the way Dirac did I knew Dirac a little bit he was really an idealist in the sense of Plato he really did believe that mathematical beauty was more important than anything else I didn't know Einstein he died when I was 15 years old but I think Einstein did not have that same feeling he tended to operate by thinking about the Duncan experiments thought experiments and consistency of principles was like they were clashing paradoxes things which you believe two things but they were inconsistent with each other you have to sort them out and my approach to physics has always been through those paradoxes and inconsistent apparent inconsistencies first apparent inconsistency I was ever involved in was quark confinement how would there be fields describing quarks at the same time no quarks now to you these days that's not a paradox to us in 1969-70 that was a real paradox how can there be quantum fields and not at the same time have the particles associated with those quantum fields the grand paradox that's occupied all of us for so long and so forth and I've always been very very drawn the problems in which you have let's say two fundamental principles both of which you have very strong reason to believe but which clash with each other and I always found that more to my taste than asking whether a particular mathematical structure is beautiful in in some aesthetic sense so but each to his own there acted quite well with these ideas I have two questions for Steve Schenker the first question is about the features he described in his very nice talk to what extent do you expect they will be extendable to higher dimensional theories of gravity for example if one considers including the double con geometries in the path integral one might be bothered by the presence of closed time like curves that's just an example and the second question is about even in lower dimensions about theories which show some very nice features without averaging for example the tensor models which Egor and collaborators have explored where they fit in the scheme of things just your comments about them thanks for the questions those are both good questions the double cone geometry to make sense of it in particular this funny place where the tip of the cone seems to shrink to a point where you might worry about closed time like curves that needs to be regulated and there's a complicated story about how you regulate it but it seems to make sense and when you do that the closed time like curve issue is not not to be problematic you can construct such regulated solutions essentially whenever you have a black hole geometry by doing some cutting and pasting and identifications you can make one these configurations almost certainly occur in higher dimensions of theories of gravity so we have strong reasons to believe that any quantum chaotic system has this behavior and since we know higher dimensional theories of gravity by some of these wonderful gauge gravity dualities seem to be dual to quantum chaotic systems we expect these things to be there the question about averaging is a really naughty and difficult one and I think it represents the boundary of what we understand it seems we know if you believe in a random matrix universality any kind of averaging averaging over time will produce this feature that is quantitatively described by this double cone the question is what else is there that produces the noise it means that just this gravitational optics don't seem to be sufficient and so there needs to be something else going on and this presumably our degrees of freedom beyond simple gravity there are other possible ideas involving baby universes and alpha states but I I guess maybe let me take a more lower hour approach there's very interesting work of I guess my colleagues and I are thinking about to Chi Ming Chang and Ying Lin trying to understand this question in the BPS spectrum and there there are hints that some of the objects that started some of these gauge gravity dualities the D-brains might be important for understanding the detailed structure of what are called microstates so I guess my own view rather agnostic about whether or not we should think about the gravity that describes the double cone as some kind of effective description of a theory that does not have such wormholes or we should think about a theory with these space-time wormholes and something else and I suspect that both descriptions are true that there are multiple descriptions of quantum gravity one where you include these wormholes explicitly and one where if you will integrate them out they are just merely effective descriptions but I think understanding that and I should say in the SYK model one could see that happen but I think understanding that in these higher dimensional versions dual to super Yang-Mills theory is the frontier of what we're doing and I think it gets to this question about really what is the precise gravitational dual of let's say N equals 4 super Yang-Mills theory we don't know the answer to that we don't have a complete set of rules and that that I think is well, it's sobering and it's challenging and well I hope you'll be able to help us figure out the answer to that Thank you very much Thanks a lot I have a question for Sir Saskan you you proposed the complex decoil volume and recently people proposed a complex decoil general volume what's your what's your opinion I missed I didn't understand yes people proposed the word general volume equal complexity that reduction to your proposed volume equal complex you said some other kind of volume I didn't understand what is the general volume equal complexity what is the general volume the volume means the volume means the volume of a spatial slice of maximum volume slice but as other people have pointed out in particular yes, a lot of people have pointed out in particular Rob Meyers and others there are a whole family a large number of things which all behave the same way one of them was the action of the of the interior of the black hole that one I pointed out my colleagues but he pointed out there's a whole enormous family of different things which are all very similar to each other which all behave the same way which all grow the same way and they use the term complexity equals anything something like that so there are a large number of very similar things which all behave the same way there are also a large number of different exact definitions of what you mean by complexity and so maybe it's not surprising that the large number of different things you call complexity could be due to the large number of different geometric things that you can define behind the horizon which all seem to behave the same way there was one interesting fact that I didn't emphasize very much that the properties of complexity to growth, maximum and so forth you might think they might depend on details such as which geeks do you use to describe a quantum computer what is the particular starting point that you start with and the answer is that complexity numerically seems to be very robust that you can change details all over the place and not change significantly the way complexity grows its maximum, how big it is and so forth and that also seems to be true at the gravitational level that there are many many different things you can define in detail about behind the horizon, all of which seem to behave very much the same way now I'm not sure if that was the question you were asking or not but ok so thank you thank you very much I think it was very exciting session I would say and now we go to drink prosecco Steve and Lenny wish you were here to drink prosecco with us and go for a nice Italian dinner but for that you will have to come here again I wish I could have been here I wish I could have been there I echo that as well Atish but I think right now I'm going to go have a drink ok ok thank you thank you everybody wait wait wait just a second I think we would like to take the photo of all the four derachmentalists together is that right is that what you are going to try ok we want to try the photo of all the four derachmentalists together two avatars two avatars and two real it's complex but we are going to take a photo of all four of you ok it's fine do you see me? yeah yeah we'll see you and I think we'll just take a photo I don't see you but I don't need to see you for that is you don't need me to see you ok thank you thank you are we finished at it? yes yes we have a nice picture of you we'll send you ok thank you thanks Steve thanks to all of you