Theories of Everything with Curt Jaimungal - Why Physicists Say We Don't Understand Quantum Field Theory

Episode Date: July 14, 2025

As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe In this episode, I speak with Professor Nikita Nekrasov, one of the most original minds in theoretical physics. We dive into why quantum field theory still isn’t fully understood, despite its experimental success and why a complete axiomatic foundation might not even be possible. Nikita walks us through his solution to the Cyberg-Witten puzzle, the birth of the Nekrasov partition function, and how exotic structures in four dimensions could underlie the chemistry of life. This conversation blends deep math, quantum weirdness, and personal stories from the front lines of discovery. If you’ve ever wondered what it really means to understand reality, this one’s for you. Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e Timestamps: 00:00 Introduction 01:13 Understanding Quantum Field Theory 04:38 The Journey to Gauge Origami 06:53 The Story of the Microsoft Partition Function 19:12 Connecting Quantum Mechanics and Field Theory 36:43 The Nature of Instantons 45:05 Exotic R4 and Its Implications 49:01 Dealing with Non-Compactness 52:25 The Emergence of Non-Commutative Geometry 1:04:29 Lessons from Mentors 1:20:11 Language as a Dynamic System 1:22:35 The Concept of Gauge Origami 1:30:48 Insights from Collaboration with Peers 1:37:15 Aspirations for Future Work 1:38:15 Advice for Aspiring Researchers Links Mentioned: •⁠ ⁠Nikita’s Papers: https://scholar.google.com/citations?user=bKV59LwAAAAJ&hl=en •⁠ ⁠Nikita’s Lecture: https://scgp.stonybrook.edu/archives/44062 •⁠ ⁠Two Dimensional Gauge Theories [Paper]: https://arxiv.org/pdf/hep-th/9204083 •⁠ ⁠Richard Borcherds [TOE]: https://youtu.be/U3pQWkE2KqM •⁠ ⁠Edward Frenkel [TOE]: https://youtu.be/n_oPMcvHbAc •⁠ ⁠Edward Frenkel’s Presentation [TOE]: https://youtu.be/RX1tZv_Nv4Y •⁠ ⁠Edward Frenkel’s Presentation [Part 2]: https://youtu.be/0AC-Ol1z5vI •⁠ ⁠String Theory Iceberg [TOE]: https://youtu.be/X4PdPnQuwjY •⁠ ⁠Roger Penrose [TOE]: https://youtu.be/sGm505TFMbU •⁠ ⁠Cumrun Vafa [TOE]: https://youtu.be/kUHOoMX4Bqw •⁠ ⁠Garrett Lisi [TOE]: https://youtu.be/z7ulJmfFvd8 •⁠ ⁠Chiara Marletto [TOE]: https://youtu.be/Uey_mUy1vN0 •⁠ ⁠Debunking “All Possible Paths” [TOE]: https://youtu.be/XcY3ZtgYis0 •⁠ ⁠Peter Woit [TOE]: https://youtu.be/TTSeqsCgxj8 •⁠ ⁠Leonard Susskind [TOE]: https://youtu.be/2p_Hlm6aCok •⁠ ⁠Seiberg-Witten Prepotential from Instanton Counting [Paper]: https://arxiv.org/pdf/hep-th/0206161 •⁠ ⁠Eva Miranda [TOE]: https://youtu.be/6XyMepn-AZo •⁠ ⁠Leptons and Quarks [Book]: https://www.amazon.com/LEPTONS-QUARKS-SPECIAL-COMMEMORATING-DISCOVERY/dp/9814603007 •⁠ ⁠Brian Greene [TOE]: https://youtu.be/O2EtTE9Czzo •⁠ ⁠David Wallace [TOE]: https://youtu.be/4MjNuJK5RzM •⁠ ⁠Jenann Ismael [TOE]: https://youtu.be/7kvXihDAOi0 •⁠ ⁠Claudia de Rham [TOE]: https://youtu.be/hNPMKy6RxCE •⁠ ⁠Yang-Mills [Paper]: https://arxiv.org/pdf/2504.19097 SUPPORT: - Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join - Support me on Patreon: https://patreon.com/curtjaimungal - Support me on Crypto: https://commerce.coinbase.com/checkout/de803625-87d3-4300-ab6d-85d4258834a9 - Support me on PayPal: https://www.paypal.com/donate?hosted_button_id=XUBHNMFXUX5S4 SOCIALS: - Twitter: https://twitter.com/TOEwithCurt - Discord Invite: https://discord.com/invite/kBcnfNVwqs #science Learn more about your ad choices. Visit megaphone.fm/adchoices

Some people will say they of course understand everything about quantum field theory, but we don't understand it as the complete structure. To really understand superposition as a quantum, you have to be a quantum entity. Professor Nikita Nekrasov unified previously disconnected fields, instanton moduli spaces, random partitions, integrable systems, and quantum strings. What was required was so novel that when he was presenting at conferences, other physicists couldn't tell if his results contradicted or confirmed theirs. His mentor David Gross advised him, keep your poker face.

That gamble paid off. It turns out, four dimensions aren't arbitrary. I'm Kurt Jaimungal, and I interview researchers regarding their theories of reality with technical depth. Today we explore how noncommutative geometry cures quantum singularities, why understanding superposition requires becoming quantum yourself, and Nikita's speculative connection between R4's exotic smooth structures and the chemistry undergirding life itself. From falling asleep during Witten's seminal lecture to late-night residue calculations

with Greg Moore, this is mathematical physics at its most human and revolutionary. Why don't we understand quantum field theory? So it probably depends on who you ask. So some people will say they of course understand everything about quantum field theory. Practitioners who use it, they will tell you that they understand because they know how to use it and they know how to do some calculations which will then can be maybe compared to experiment. And more often than not, the comparison is favorable. But we don't, maybe as theoretical physicists with some kind of mathematical ambitions, we don't understand

it as the complete structure built out of axioms, let's say. So you want an axiomatic quantum field theory that reproduces all the successes of the standard model? Something which, well, we would like to have some basic principles, like maybe not axioms, but some basic principles from which we can build structure from bottom to top, bottom up. We have many, many examples of what we think quantum field theory is, so many examples of quantum field theories. And there are many overlaps, but there are also missing kind of areas.

We don't know whether gravity is part of quantum field theory or it's something else which requires a different approach and so there are many questions like that. So there is of course a very practical approach which works with many theories which are involved in standard model which you can just simulate them on a computer, just do the largest field theory. There are various solities with that because not all structures which we know they are in smooth fields and fields defined on smooth manifolds. Manifolds may have different structures like spin structure.

All those things are kind of hard to represent faithfully on the lattice, but with some room for error, people do that. But that still looks like a bypass, not the whole thing. And we know examples of quantum field theories which probably don't have lattice description. And so the work which we do, some of us do, which we're actually trying to chip away at the unknown in connecting us to those theories by coming up with maybe mathematical conjectures which sort of follow from the fact that these theories exist and then checking them maybe by more standard means and then gaining more confidence that the reason for those conjectures to be there probably is valid.

Now we're going to build up to gauge origami, but before we do, I want to get to another one of your constructions. So Dirac said to Feynman, infamously, do you have an equation? And Feynman could have said to Dirac, well, do you have a diagram? And now I'm thinking you could have chimed in and said, well, do any of you have a partition function? Well, of course they had lots of partition functions. Yes, yes. But anyhow, you have one named after you.

So let's get to that story. I believe this was around 2002, where you cracked the cyber-witten instanton counting puzzle, which was in the mid-1990s. 1994, yeah. Right. And it was there where you devised the Nacrosoft partition function. So tell us about what that partition function is and what it was like coming up with that solution.

Was it a lightning bolt? Was it an incremental climb? What did that moment feel like? Yeah. So this is a story which is a kind of worth a book because it's a story worth of many sleepless nights, but also... There's a book called The Count of Monte Cristo, so yours would be The Count of Instant Hunts.

Right, right. Thank you for observing the analogy. You have to dig an underground tunnel from some predicament you are to actually get to the dry land. Some people say there is also vengeance involved, but in my story, no vengeance. It's all pure love. Are you sure? Well, at least that's my inner work. My inner work is to realize that all we do in science is joy, even though it feels like suffering along the way.

But the goal is to recognize beauty and maybe add a little bit of beauty and use it later just because beautiful things are good to have around and sometimes they're useful. So the story started, I don't know where it started actually, maybe started in 1992. Wheaton wrote a paper which was called Two-dimensional Gauge Theories Revisited. And in that paper, he proposed to use a technique from rather abstract mathematics at the time called equivalent localization to compute the path integral, so it's a Feynman integral over trajectories or field configurations, an example of a rather simple but nevertheless interesting gauge theory, two-dimensional Young-Mills theory. So our world is described to some approximation by four-dimensional Yang-Mills theory, quantum

Yang-Mills theory. But if you imagine a world in which there is only one space dimension and one time dimension, then you could study the simplified version of Yang-Mills theory. And that was a theory which was actually solved by several tools. But one tool which was kind of interesting was by Sasha Migdal, Alexander Migdal, who actually first defined it using lattice approximation. And then he found out that it was an interesting theory in which you can define it on any kind of lattice. You can make the lattice finer or economical, so use as little number of edges as possible,

given the topology of space-time, and the partition function will be the same. So it was an interesting theory which was almost topological. We would not call it now almost topological. It only depended on the area of space-time. So space-time, here I'm using a little bit of a jargon. The theory is interesting when space-time not physical spacetime when you have time and space, but what we sometimes call Euclidean spacetime. It's a manifold in which the notion of a distance between points is similar to the

notion of distance between points in space. There's ordinary mining metric. How it is connected to the space-time where the notion of distance involves events and so some distances can have negative square, for example. That's a story which we can discuss, but let's not get there. Okay. Let's not get there. Anyway, so Witten observed that this theory, the reason why Migdal was able to solve it and get a very interesting answer. So Witten observed that it had to do with certain topological structures present in

the final dimensional space, which you can associate to any two-dimensional surface, we call them Riemann surface. It's like a surface of a, well, your audience knows probably what Riemann surfaces are. So Riemann surfaces have, if they are intable, they have only one number which characterizes the topology and number of handles. Now Yang-Ning's theory deals with connections, with ways to transport things from one point to another in some bundle over the base space. The lowest energy configurations, lowest action configurations in Yanin's theory are the connections which have zero curvature, which means that if you transport things around small loops,

the result doesn't depend on which loop you choose as long as the loops are small. But if the loops are not small, if they wind around handles so they cannot be contracted to a point, then those transports can be non-trivial. This fact actually was observed in physics in the famous Argonne-of-Bomb experiment. It's a famous story which showed to the world and to physicists also that the connection, the vector potential actually has a meaning, has a physical meaning, not just the curvature. If you're familiar with Maxwell equations, you can write them only using the curvature, the field strength. For for a while people thought that the vector potential is not really meaningful because first of all it's not universally defined.

You can make the so-called gauge transformation and change it. But the fact that they transport, especially the transport observed by quantum particle depends on the path which can be non-contractable. It was a big deal. And so, on the two-dimensional human surface, because you have several handles, you can have those transports and they can have pretty much arbitrary values. And there is a space which parameterizes them, which is called the modular space of light

connections. These days, this space is, of course, very popular in the stories involving geometric landlands program. But anyway, so if your connection, if your gauge theory is defined with compact gauge group, then it's some compact space, although with singularities, because you have symmetries and you divide by symmetries and sometimes symmetries act not freely, so it's a singular space. But you can define its volume. It has a natural...

Well, it turns out it actually could be used as a phase space. It has a symplectic structure, which was actually found by, I guess, Atiyah and Bott in the early 80s. And so there is a natural number which you can associate to every human surface, which is the volume of that modular space computed in the Louisville measure. And so Witten computed this number by a very clever trick using localization applied to two-dimensional angles theory, got beautiful answers involving zeta functions. So some of us now call them Riemann-Witten zeta functions because for SU2 these are just Riemann zeta functions, but for other groups, they're kind of generalizations.

So these are interesting objects. Anyway, so in 1992, I was just beginning my studies. I was very much impressed by the paper. In fact, my friend and later collaborator, Andrei Losyev, suggested I study that paper and present it at a seminar. So you know I come from Moscow, so I was attending some string theory seminars there. The tradition there was that you study some paper and this was the beginning of archive. So it was the first, maybe second year of archive. So the papers from the West started coming in not by, you know, by, by sprint prints in some diplomatic posts.

Right. But as actual, you know, paper, you can print out if you have a printer, which of course nobody had, but it was a second order problem. And so you study the paper and you present it to your friends and colleagues and then they grill you for several hours. And then you study the paper and you present it to your friends and colleagues and then they grill you for several hours and then you understand that you have to study more and then you go back and you study more. And at that point you were an undergraduate? I was an undergraduate at the time.

So it was my probably third year after high school. Right, right. But I mean, I already knew what I wanted to do. So I mean, I was lucky to get interested in mathematical physics, in what I call mathematical physics early on. in what I call mathematical physics early on. And so somehow I got into this crowd of people, group of people discussing whatever was interesting. So I was very much impressed with this paper and I thought, okay, maybe one can use this localization to compute interesting things in other theories.

And I remember we were discussing with my other friend and also collaborator later, Sasha Gorsky, and we asked ourselves, why couldn't we apply this supersymmetric technique to quantum mechanics, especially quantum integrable systems. Later on, this spectrum was analyzed by Haldane, Duncan Haldane, who got Nobel Prize for many, many things. And so the way this spectrum was organized is now called Haldane statistics. So these particles, because of this particular interaction, they kind of behave as neither bosons nor fermions, but something that can meet between. And so this coupling constant can be an

interpolating parameter. Anyway, Gorski and I found that this model of particles moving on the circle and repelling each other with the potential which is proportional to inverse square of the distance, core distance, is actually part of the two-dimensional Agnus theory. So quantum mechanics is equivalent to two-dimensional theory. It's one of the interesting examples of deceiving dimension. This is part of the structure which I mentioned when I say that we don't quite understand quantum field theory because when we explain to engineers that quantum field theory is the continuous

limit of something you define in the lattice, of course an engineer will think that well the lattice is embedded into space-time of some dimensionality, so that's the dimensionality of the theory. But it turns out that you can define a theory in one number of dimensions, but then by looking at special observables, so by restricting the set of observables you're allowed to look at, you will not be able to distinguish it from the theory in the, let's say, lower dimensions and vice versa. So we think we live in three plus one dimensional space-time, but it might be just that we don't have access to observables which can probe high dimensional space time.

And eventually it's said that your work implies that particles live across multiple dimensions at once. Maybe particles is not the correct word here though. When I was talking about particles in colliderodior model, that was more of a mental picture like beads, like some small objects. So in particular in this case, the way this relationship between quantum mechanics and field theory was working was that the role of those particles was played by the eigenvalues of the holonomy, so the transport of the two-dimensional gauge field around the circle of space.

We found that those points in the circle behave as, so time continues to be the time, so those points in the circle behave as particles in the collogero-sazerlend model. The mathematicians call it collogero-moser-sazerlend model because Jürgen Moser, the Swiss mathematician, found a very neat way of explaining the integrability of that model, which we'll probably get to later. So how does this relate to your partition function? So we are like 10 years before that. Just a moment.

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Now you can enjoy The Economist and all it has to offer for less. Head over to their website www.economist.com slash toe to get started. Thanks for tuning in. And now let's get back to the exploration of the mysteries of our universe. Again, that's economist.com slash toe. So two years later, okay, so let's just keep it. So let's just remember that there is some relation between quantum many body systems and gauge theories with some supersymmetry. Okay. And that in gauge theory, certain things can be calculated exactly.

If you can recognize some symmetry and can use the mathematical tool of equivalent localization. Got it. Two years later, it was 1994, and I was traveling, one can say backpacking through Europe, and I found myself in Paris. When it just happened,

the International Congress of Mathematical Physics took place. So there was an interesting talk. There were two talks by Witten there who presented his work with Zyberg on Zyberg-Witten solution of n equals two gauge theory and on his work with Kumunwafa on N equals two gauge theory and on his work with Kumrunwafa on the strong coupling tests of s-duality. Yes. So that's s-duality of N equals four supremum theory. I spoke to Kumrun on this podcast. I'll place the link on screen and in the description as well.

Right. So I should admit that, I mean, okay, so it was 1994. I was just out of my, I just finished basically my undergraduate studies. I finished the military bootcamp and I just went to Europe and I was going to graduate school later. Yes. It was a very hot summer and at Wittenstock, on Zabrick-Witten theory, I fell asleep. So it was, I mean, I remember. I mean, I wasn't there, but I remember you telling the story.

Yes, I was asleep, but I must be ashamed. I am ashamed. I could not really comprehend the significance of what was going on, but I was impressed with the formula he wrote for certain quantity, which is called pre-potential of theory. Just to recall, the abroquitin solution was the ANSATS for what is called low energy effective action of some gauge theory. They said that the quantity which they wanted to compute, the one which determines this low energy physics, can be expanded in power series in instant dots. So what are the instant dots? So this gauge theory, the monobillion gauge

theory, there are configurations of gauge fields in the vacuum where they fluctuate in a way which changes the topology of the gauge bundle in which they are defined. So let me maybe unpack that. The thing which is not often appreciated is that the fact that our world is described by gauge fields means that maybe conceptually or maybe physically there are extra dimensions to the world we live in. Because what these gauge fields are, they describe the transport of things over our space-time, but which happened in a kind of additional space, which is fibered over our space-time. And this space could be a group. So typically, as a basic object, one takes the principal bundle. So the fiber of this vibration is just a group manifold, or it could be a space

in which the group acts. So for example, fermions of the standard model, they're all sections of some bundles which are associated with some principle bundle. Now one can take the totality of those fibers of a spacetime and look at the space, the resulting total space. So it will be a manifold of dimensionality, four plus the dimension of the group. So if the group is SU2, the total space will be seven dimensional. If the group is SU3, the total space would be 12 dimensional and so on. Some people like group E8, so the total space would be 252 dimensional.

At any rate, just given the fact that your base space is your four-dimensional space time and the fiber is the standard group, doesn't specify uniquely what would be the total space. So there are different ways of fibring this group over the base manifold. So the ways to parametrize, to enumerate the topology of those total spaces is the subject of the study in homotopic topology. And so there are special classes which are responsible for those classifications called characteristic classes found by Pontjagin and Churn. And so in doing gauge theory, we're actually doing a little bit of quantum gravity in the

sense that we are summing over different topologies of this total space. So when I say summing, I mean it in quantum mechanical sense. If you think about Feynman path integral, which is the kind of integral over paths, over possible trajectories of evolution of your system, in gauge theory, the evolution of the system involves the choice of the total bundle, total space of a principal G bundle. And so we're summing over the topologies of those bundles. Let me see if I can break this down for the audience so far. Yes.

Okay, and you tell me if I'm incorrect at any point. So space-time we ordinarily think of it like x, y, and z. And then, well, there's also a time. So you get x and y, the plane that most people know about, by doing x cross y. So that's something that people are familiar with. You take R, R1, and then you cross it to get R2. And then you're wondering, well, what is this fiber business that people are talking about?

Locally, meaning if you look closely, you can think of a fiber as another cross. So we're just crossing with a gauge group. So your example is already, so already in your example, we can give an example of a non-trivial bundle. So if your y variable is compact, it's actually not aligned by the circle. And now I'm adding x. Sorry, x was first, let's say y was first. Sure, sure, sure. So, I want to fiber the line over a circle. So one way I can just take a product, direct product, so I will get the space which I will

get will be a cylinder. I think I even have a scene drawn on my blackboard somewhere, but anyway. So it's a cylinder. It's a trivial bundle. So the base is a circle and the fiber is a cylinder. We would call it a trivial bundle. So the base is a circle and the fiber is a line. But now there's another option, which is to say that as I go around the circle, my line flips orientation.

And so it comes back as the same line but it's inverted. So I used the action of the group Z2, the two element group, which acts on the real line by multiplying X by minus one. So X goes to negative X. And so that gives you another space which is known as Möbius strip and that's a non-trivial bundle. But you can also fiber the circle over a sphere in a non-trivial way. So let's take our two donuts, which is probably a good place to start. Now imagine, so I would think of the... good place to start. Now imagine, so I would think of the, so donut is a product of a dimensional disk times a

circle. So now what I want to do. And these two donuts currently are disconnected. Yeah, so they're separate, two separate donuts. Unfortunately, I don't have donuts with me. Tim Hortons is what they would say here in Toronto. Right. Yes, yes.

Also coffee. Now I want to take the boundary of those donuts, of these two donuts. So those would be products of S1 times S1. So now I want to identify the first S1. So they will be the same S1, I glue them together, but the second S1, the fiber, I want to identify with a twist. In such a way that as I go around the first S1, the second one makes the full rotation. It's almost like this example with the Möbius strip, but now the rotation is 360 degrees, and it's a rotation of a circle, not the flip of a line. So if you make a little mental exercise, glue these things together, you will realize that

the resulting space is now a three-dimensional sphere. Aha. So one way to maybe imagine this is to actually embed one, so take one donut and then realize that the complement of this donut in the three-dimensional space is actually topologically also a donut if you add a point at infinity. Aha. And so the union of those two donuts is the complete three-dimensional space with the point at infinity, which is a three-dimensional sphere. And it turns out that if you're studying the parallel transport on this bundle with

non-trivial topological class, the minimal Yang-Mills energy configuration will be not flat so the curvature will be non-zero, but it will have a very interesting property. It will be self-dual or anti-self-dual. So this is something which is specific to four dimensions. So, if you list the coordinates of a four-dimensional spacetime, like x, y, z, and t, and then, and suppose we orient them, so I say x, y, z, and t, and now if you have a two-form, so a two-form is something which likes to take in two planes or two bi-vectors and produce numbers. So for example, I take x, y. But if I remember I had x, y, z, and t, so if I took x, y, then the remaining is z, t. And if I took x, z, the remaining would be y, t, and so on. So there is this

duality between two forms, which is given by the orientation of the four-dimensional space. And so the minimal energy configurations turned out to be such for which the curvature is self-dual. So if you apply duality to it in that sense, you get itself or maybe negative of that, depending on the sign of the topological class. So these are called instantons in the technical sense, but this is not the most general definition, of course, of what instanton is. And the significance of those solutions in quantum field theory is that these are configurations for which most of the action, in both literal and mathematical sense, happens in a kind of compact region

of spacetime. And here I should make a disclaimer that the space time I'm talking about is not the real space time. It's the Euclidean space time. Which incidentally is real. Which is incidentally real, but it's also imaginary. So in this real manifold, what we perceive as a time direction is actually imaginary. I don't like when people actually present it this way. I like to think of it as a part of the structure in which we study quantum fields on manyfolds

endowed with complex metrics in general. The metric can be real with different signatures. Square of the distance and time direction could be positive or negative, or it could be positive in all directions. So we call that Euclidean signature metric. But in general, we could and should also study the metrics which are just complex. So the metrics elements are complex, the values of the metric are complex. But that's maybe a slight digression.

I'm sorry, this is a bit technical. That's fine. Anyway, so Zybrouken-Witten wrote a formula, and they said that to really understand this formula from the first principles, one should derive it by doing honest path integral in supersymmetric gauge theory and evaluating those contributions to the effective action coming from those peculiar fluctuations of gauge fields, which happen in compact regions of Euclidean spacetime. And the moduli space of instantons, is that a connected space? So it is connected for each topological class.

So if you fix the topological charge, it is connected. But for different topological charges, they are disconnected distinct spaces. So a priori, mathematically you would have said, okay, I have just an infinite number of integrals to compute, why would they combine into something nice? So I would say that you never approach an interesting problem directly. It's very hard to study things face on, you need some kind of preparation work and maybe if you're lucky you will get to the point where you hit on the problem you're actually interested in or you dare to be interested in.

So at this point you were exploring, hoping that you were going to get to the solution, but allowing yourself to not get to the solution because it was interesting what you were exploring anyhow? Right. So the goal was so that at this point, I mean, I had to get familiar to familiarize myself with instantons, with modular spaces of instantons, what they are, what they look like, what do people know about them. So, one thing which became popular at the time was the study of quib varieties.

But they were not defined as a modular space of solutions of some partial differential equations. Baffel and Whitten mapped this property to the very non-trivial property of the four-dimensional gauge theory, which inverts the gauge coupling. So, somehow they related the coupling of gauge theory to the geometry of some, at the time, abstract two-dimensional torus elliptic curve. So that was part of one of the hints towards the BPS safety correspondence. Some structures found in the studies of four-dimensional gauge theory, structures involving supersymmetry and the cohomology of some supercharges are related to the structures found in the in two-dimensional

conformal field theory. Right, so just to linger on this point, so many people have heard of ADS-CFT and that involves differential geometry and harmonic analysis and string worldsheets and so on. But there's also something else called BPS-CFT which you helped popularize and articulate. Well, BPS-CFT was just maybe a joke term, which I coined at the time because it sounded like ADSCFT. But of course, it's just like, just as ADSCFT is a string duality,

it's an open-closed string duality in my mind. Even though some people claim it's just holography in its purest form, kind of open-close string duality in my mind. Even though some people claim it's just holography in its purest form and doesn't require string theory, I don't think it's the case. I think it's... Oh, you think it's string dependent, it's not just string inspired. I mean, it's kind of a... It's one of these things which makes string theory great.

It gives you a mechanism for holographic duality. And ADS-CFT in that sense is a projection of some ten-dimensional entity. There are versions of ADS-T which use the same theory. So it turns out that the integrals which one had to compute to get the Zybrick-Witten solution were independent of that parameter. So you can hope that if you send it to zero you get the same answer as if you don't. So this is one of the tricks which I learned essentially from Whitten. Whitten used in his studies of supersymmetry breaking, a quality which is now called Whitten index, which is also a partition function. It's a kind of a partition function of a quantum system

with supersymmetry, which differs from the conventional partition function where you average exponential negative Hamiltonian divide by temperature. You also insert the operator, which temperature, you also insert the operator which weights bosonic and fermionic states in your Hebel space with different signs. And so we can show that in many favorable circumstances, this partition function is independent of the temperature. And so you can compute it in different limits when temperature goes to zero or temperature goes to infinity. And when temperature goes to zero, this partition function receives contributions from ground

states only, from the vacuum. And when the temperature goes to infinity, it is usually the regime where things are computable. And so one of the consequences of this method is the physics proof of Iteya-Singer index theorem, which was used, for example, by Luis Alvarez-Gomez, my current director. So, if they can prove that there is a parameter in your Hamiltonian or in your action or in your system to which the observable you want to compute is insensitive, then you vary this parameter as much as you can and try to find the regime in which things become computable. And so in my story, one of these parameters was this parameter of non-commutativity.

Right. Unfortunately, so this, this, this fact, so this trick cured, you know cured one half of the problem, namely it dealt with this singularity of the modular space of instantons, which corresponds to small distances in spacetime. So things which happened at short distances. But since I'm interested in working over R4, so we want to understand physics on the four-dimensional flat space, I like mathematicians who use instant and modular spaces for things like invariance or for manifolds, which is the content of Donaldson theory. For them, the interesting spaces are complex spaces, even though sometimes

with very clever, clever tricks, they draw conclusions even for R4 and famously found that there are unlike all R dimensions, so as a smooth manifold, there is only one R1, there is only one R2, only one R3, only one R5, only one R6. There's only one Rn for any n except 4. And 4 dimensions, there are exotic Euclidean spaces, which is quite controversial. Just a moment. Do you think that fact is connected to the R4 of our space? The fact that our space-time is R4? Could be, yeah. Because it has to do with the fact that two-dimensional surfaces meet in four dimensions and they

don't meet in higher dimensions, generically. And so the fact that certain things meet often, like probabilistically, so if you throw random walkers, it's probably crucial for life. So let me just put it this way. Okay. Life needs interaction and propagation. So the exotic R4 is our world. The R4 is our world because there's an exotic R4 and if there wasn't then we wouldn't have enough interactions for us to be here? Right.

So I would say, I don't know what the causality is, but I would say that the reason Exotic R4 exists is probably... optimum points on your first five orders. Shop now at nofrills.ca. Important in things like having chemistry and things like that. So there is certain topology involved, you know, in having complex structures, in having biochemistry, depends on certain topological features of our space and space time. Some of these things are crucial in the construction of exotic R4.

So these things might not be unrelated. But since it's discussing things for which you have only one sample, it's hard to draw conclusions. Because I haven't heard anyone connect chemistry to R4's exotic property. Unless this is just a speculation of yours, you're not sure. No, maybe we could probably elaborate with that. I mean, I'm not I'm not strong in chemistry. So I probably if I if I venture in this direction, I'll probably say something silly.

But just I mean, I would say that the fact that four dimensional four dimensional. Well, okay. What do we know? We know that four dimensions are special in having interacting, propagating vector fields. So gauge theory as quantum theory is well defined in four dimensions and has very interesting properties. And so it is not well defined in five or higher dimensions and it's less interesting in fewer dimensions. People study it in three dimensions because it still has some qualitative features, but

there are certain things you get almost for free because you get mass scale for free and so on. In four dimensions, things happen just, you know, they're kind of on the boundary between things becoming trivial or ugly. So it's a four dimensions is interesting border case. And the fact that the possibility of exotic smooth structures in four dimensions uses the same kind of coincidences, which is all I'm trying to say about that. Okay, so getting back to the story.

So back to the story. So in physics, I need to, for physics applications, I need to compute meaningful integrals over the modular space of instantons on literally non-compact infinite R4. And this modular space is again non-compact because now those instantons, which are like pseudo particles, like events in spacetime, they can run away to infinity. So you can have a sequence of events which happen here and then on Saturn and then on in Andromeda and then go all the way to infinity. So there's no limit to that. So

again, I'm in trouble because I cannot integrate by parts. I cannot I mean, I'm not sure because I cannot integrate by parts. I'm not sure my integral is convergent. Life is difficult. So I need to invent another trick to somehow cure that possible divergence or non-compactness at least. Now, if we're doing something on R4, R4 has a beautiful symmetry. It's a symmetry of rotation.

Well, it has also symmetry of translations, but translations act without fixed points. So if we use translations, it will kill everything. So it's not the right symmetry in this case. The right symmetry to use is the symmetry of rotations. And so the group of rotations of Euclidian four-dimensional space is the group SO4, and its maximal torus, maximal Abelian sub-algebra subgroup is 2-dimensional, and it's generated by rotations in two orthogonal planes. Now, of course, if you just do some mathematical construction which is completely artificial. First of all, you might make mistakes, but also you lack

intuition. So it would be hard to know what is the right way to proceed. What are you doing? Meaning you lack physical intuition? Right. So this mathematical construction was nice, but I wanted to have some kind of physical understanding where these parameters come from and what would be the physical realization. Is there a physical meaning for such a deformation? Now, remember, I was doing this in the late 90s and at that time, even though the non-commutative field theories were already popular, and non-commutative deformation breaks Lorentz symmetry.

Because you see, if I tell you that x and y don't commute, the fact that x and y commute in one way and z and t commute in another way breaks the symmetry between x, y and z, which is part of the Lorentz symmetry. So people started discussing quantum field theories which were not Poincare invariant, but people were reluctant. So this partition function scale, like a partition function of a non-ideal gas, where the volume, the role of the volume of the space in which the gas was confined was played by the inverse product of these parameters epsilon 1, epsilon 2. And the leading term, the free energy, so it's the energy, free energy per unit volume was its I-recurrent pre-potential.

So first I discovered that experimentally just by expanding it term by term, because I felt like these partition functions were good for something because they were so beautiful. They fit nicely to each other, one instant on, two instant on, so on. And then when I saw that the one case people computed, the one instant on per term people computed and sometimes took it to two instant terms, people computed by very hard work, matched with what I got by relatively simple work, then I was convinced it must work. But of course I was not very confident in myself, so I had to go up to five instantons and get some help from numerics and got some errors along the way.

So I had to... Is that the one about the story where you were working for two weeks on a laptop, and then you were in a hotel room and it worked out? That's a different one. That's a different story. By that time I was much more confident. So that's when I published it. I made a conjecture in this paper that this partition function,

which was a deformation of, I mean, it gave more than the zebra-crucian pre-potential. It gave something else because it had two more parameters. And so the conjecture was that it gives the amplitudes of topological string. and the reason for that we can discuss. And also that it is given by some conformal field theory or some deformations of conformal field theory. So that the structure which I saw in this particular function was that of some two-dimensional CFT hiding in the supers-spectrum computation four dimensions. So that's that. It was the moment both of joy and frustration because

it felt like I should be able to prove it, that it's not just conjecture, but just to prove that indeed the curves that Zabrik and Whitten proposed and conjectured and people then extended their conjectures to other gauge groups and types of theories with various metric content should be provable, but I was too close and it was kind of, I couldn't do it. So it was, it took me another year and another happy, happy meeting. Just a fortunate meeting with the right person at the right time to be able to, to prove at least some of these conjectures. Mm-hmm. So that was the meeting with Andrei Okunkov, just on the train station in France, in Bure-sur-Rivette.

And so I knew Andrei from before, but we basically just played volleyball together. When I was in Princeton as a student, he was at Princeton as a postdoc. We just played volleyball, but we never really discussed physics or math. I knew that he was an expert in combinatorics. And maybe he knew something about partitions. And so I just stumbled upon him on the train station and he said, what are you doing? I said, what do I have this difficult problem when I have a sum over infinite set of partitions,

which I believe has this hidden structure there with curves emerging, but I don't know how to prove it. I mean, I don't have... It feels like it should have something to do with conformal field theory to dimensions, but I'm not sure exactly what. And he said, well, but you are in luck because I'm computing the partition sums from dusk till dawn and then from dawn till dusk. He also knew of a problem about random partitions for which the answer is given by some kind of a curve.

So, he knew about some emergent geometry in the problem involving random partitions. That was the famous limit shape of Vershik, Kerov, and Logachev, which mathematicians studied for completely different reasons, also in the late 70s. It was a different crowd of mathematicians interested in different problems. But fortunately, I met Andrei and he said, well, this is a similar problem, so maybe there is a similar solution. And again, it turned out that the problem which the mathematician solved was for the trivial case, but with some ingenuity.

And Andrei had a lot of ingenuity at the time with this, so he used a lot of interesting complex analysis ideas to transform the problem of computing my asymptotics to the problem of finding a limit shape. Limit shape is kind of the most probable geometric shape in the ensemble of random geometries you are given. We found that indeed it was the family of Zybr-Pruten curves that govern the asymptotics of my partition function. Wonderful. So you went volleyball.

Yes. Because you went to Paris for volleyball and you weren't doing much math there. But then I think in 98 or so you with Maxime Koncevich, I think you went there for volleyball, but then it ended up being a fortuitous lunch, something like that if I recall. You don't need to get into that story, but it's so... Well, volleyball is important for mathematicians for some reason. So I like, I mean, I'm real amateur at volleyball. Sometimes I manage to get a good serve, sometimes I don't.

But at IHS, which I visited in 1998 and where I worked later for many years, there was this tradition that in the summer, in the compound where visitors live, people would play volleyball. It used to be, the main driving force behind that used to be Professor Kirillov, Alexander Kirillov, the inventor of geometric quantization, who also was the advisor of Andrey Okunkov. So that's why we played volleyball because Kirillov invited us to play volleyball. And later, Maxim Konsevich became the volleyball guru. And so every year in the Buub-sur-Vette, people play volleyball because Maxim is the driving

force. Okay, so from Moscow to Princeton to Paris to New York, you've worked in these different cultures. Would you say that the culture of academia is different there, or at least the one that you interacted with? Or would you say there's more similarities than dissimilarities? Yeah, it's... Actually, I mean, I wouldn't say... I don't think they're so different.

I wouldn't... Right now, I mean, after having spent 30 years in doing physics and math in between. I think it's pretty much universal. I think that's what makes science beautiful. It's kind of universal. It's independent of the nationality. Yes. In Moscow, I had kind of a double, I led maybe a double life, maybe a triple life. I had two advisors. One was my advisor in particle physics, another was my advisor in, I guess, Singularity Theory. Both of them knew that I'm doing something else. I'm actually, I mean, I had some... My main interest

was in string theory, in kind of modern mathematical physics. But part of their kind of training, maybe part of the coming of age kind of thing, you have to work on traditional problems for a while. So I was computing radiative corrections to standard model parameters to get bounds on the mass of top quark. It was before top quark was discovered. So that was part of my work, kind of part of a duty, so to speak. So I worked with my advisor was Leif O'Koon, who was basically a phenomenologist, an expert in weak interactions. He wrote several very good textbooks on the electromagnetic theory, on physics of elementary particles. But apart from one paper which I wrote with him on those bounds, we didn't work much. but he kind of protected me from, I mean, I guess he was my protector

in many respects. I mean, we could get into that. Life in Moscow, so it was Soviet Union and just post-Soviet Union. It was very different from what it is now. So the very different from what it is now. So the expectations from young physicists were different and there were different obstacles and different challenges. My math advisor was Vladimir Arnold. I was just going to his lectures in Moscow State University, even though I studied at a different university. He didn't know I was not a student there. So I just attended his seminar, his lectures, and he gave me… I mean, he gave every year, he would give a list of problems to the participants of his seminar. If you wanted, you could work on them. And so I worked on one of these problems and it was very useful for me because I learned about topology, about many, many things.

But he actually gave me some advice, which I guess I used. I mean, he gave me lots of advices, but one of them was to always try to find all the parameters in your problem and always try to take them to extremes. That's the advice which you can take outside the science as well. So your physics supervisor was David Gross? In Princeton, yes. When I got to Princeton, my supervisor was David Gross? In Princeton, yes. When I got to Princeton, my advisor was David Gross. But again, when I was a graduate student, David was kind of going through a complicated

phase in his personal life. And he basically let me do whatever I wanted, which was the best case scenario for a good student. Good students should not ask problems from their advisors. They should find problems and maybe if they have some difficulty in solving them, they should come to the advisor and ask for advice. So David gave me, when I was a student, he gave me some practical advices, but we started working together already later when I already had the job actually. Which was actually the best, also the best time because I got to work with time because I got to work with happy and confident David. So he already was in Santa Barbara and I got him interested in non-commutative geometry.

So we found interesting solutions of gauge theory of non-commutative space. We found non-commutative monopoles and found that they actually carry the physical string attached to them. So the Dirac string of magnetic monopole of Dirac, the string of Dirac monopole, which is kind of imaginary object on a noncommutative space becomes physical. So these are sometimes called Gross-Nikrasov strings. Right, right. So, what's something else David Gross taught you about physics that wasn't in any textbooks,

something that you still carry with you today? He always told me to believe in myself and to be confident and that don't get intimidated by either intimidating people. So if people tell you that you should not work on something because it's wrong or because it's morally wrong or because it's whatever reasons, if you feel that it's important, work on that. Was there something in particular that you were insecure about, that you were coming to him saying, I'm not sure if I should work on this?

And then it was that moment that he gave you that advice? Well, it was, I think it was before one of the string conferences. Incidentally, it was a string conference in Paris. It was before one of the string conferences. Incidentally, I think it was a string conference in Paris. I think I was competing, working on similar topics with a group of prominent physicists. You felt in competition with them. like you felt there was a rivalry. Yeah. Yeah.

So I was, I was doubting whether I should maybe I should change the topic of my presentation that strings and speak about something else. And because they paper didn't come out, my paper didn't come out, so we didn't know exactly who had what. David told me that, don't worry, just keep your poker face. He had some stories when he was coming to a conference and somebody gave a talk saying that maybe it was Lenny Susskind, so David and Lenny had some kind of rivalry, saying that probably it was about instantons.

Instantons are irrelevant in some problem or irrelevant, I don't remember exactly. He had the result, which he could confirm with the models and examples, which showed precisely the opposite. So it was a very happy coincidence. It made a good show in the sense that people make strong claims and then one claim after another, and then there's some discussion. And of course, David was right, so it's his story. So he told me that you should just aim for be yourself, believe in yourself, and present what you have, and maybe the other party doesn't have as strong a result as you think.

And he was right. So that's the right thing to do. So I've had another one of your collaborators on, Edward Frankel, and I'll post that podcast on screen and in the description. He was on several times actually, and it's coming up again to talk about the geometric Langlands. What is it about Edward, Frankl, that you admire? You all both seem to get along mathematically, but also as people, so you can speak about both. Well, we are... Well, what I admire about Edward, so he's...

First of all, he's charming. He's very intelligent, very knowledgeable, but also... I would say that for a person of his charm, he actually has a big heart, which is very rare. Sometimes people use the charm to their advantage and eventually they become cynical. Edward is the opposite. So he's evolving, he's learning and he's interested in things I'm interested in. So it's always, I mean, I always learn something from him and it's always not just the pleasure, it's a challenge and the pleasure to be in communication with him.

So mathematically now, what is it that you admire? So you admire his heart, sure, at a personal level? Yeah. So he combines kind of a deep knowledge and understanding of very abstract things. And also, but he also knows how to connect them to things which kind of I understand as a physicist. So he can speak to me both as a physicist and as a mathematician. So, and to some extent. Yes, okay. So earlier you also mentioned physical intuition and I want to touch on this because, well,

physical intuition can take you so far. Do you find that it hinders at some point? Do you find that, well, it helps in general and if it depends then what does it depend on? Well, on my path, I was not always guided by physical intuition. So either, I would say I'm guided either by mathematical intuition or by physical intuition. So there is always some kind of some sense of beauty in both worlds and they sometimes complement each other, sometimes they are compatible. Well, of course, historical famous examples where they are in contradiction and then mathematical intuition,

for example, for Dirac was more important physical intuition and it led to a change in our physical understanding, like with the concept of antiparticles, with positron and things like that. But of course, physical intuition sometimes could be misleading because we are limited by our experience and the models we study. So again, the good example would be the invention of quantum mechanics. You have to really step outside, step out of your, you know, mind box to embrace the quantum intuition. You cannot understand it using you know, everyday experience. We get used to it, but to truly understand quantum mechanics, I don't know what kind

of meditation you need for that. I haven't found it. I have a very good friend, film director Ilya Khryzhanovsky, who wanted to make a quantum film where you could experience quantum reality through a multitude of universes you live in. Some people say that he succeeded to some extent. Some people say he made a step. We keep talking about this with him, whether maybe animation could be good media for explaining what quantum world is. So here's a question. I was just at dinner last night with some people and it's commonly said that when we measure we observe something real and we don't see a superposition and in part that's the

measurement problem. And then I just asked, well, what would it look like if you observed a superposition? And the people were pausing it, because it's not exactly that you see something here and here, but it's half transparent. No, we observe effects of a superposition, because I think the issue is that the devices we measure with are kind of classical devices. And the way the brain will analyze this is a classical brain.

So to really understand superposition as a quantum, you have to be a quantum entity to measure superposition. So Greg Moore, which I want to talk about as well, he described something called physical math and that's in contrast with mathematical physics. Now I know that you didn't come up with that distinction, but do you see yourself as more of a physical mathematician or more mathematical physicist? I think of myself, so I think of myself more of, so I'm a physicist than a mathematician, but I'm equally interested in mathematics as I'm interested in physics. For me, mathematical beauty is as important

as physical beauty. Physics beauty, not physical beauty. Physical beauty is also important, but it's different. Let's talk about mathematical beauty. Physical beauty is something for another time, but give a taste for what it is to find something mathematically beautiful for people who haven't experienced that. FanDuel Casino's exclusive live dealer studio has your chance at the number one feeling winning, which meets even the 27th best feeling saying I do. Who wants this last parachute? I do. Enjoy the number one feeling winning in an exciting live dealer studio,

exclusively on FanDuel Casino, where winning is undefeated. 19 plus and physically located in Ontario. Gambling problem? Call 1-866-531-2600 or visit connectsontario.ca. Please play responsibly. you Well, of course, there are kind of professionals who come up with examples of mathematical constructions for laymen, which are kind of beautiful. For me, mathematical beauty is what unites distinct parts of mathematics. For example, the fact that you can think of prime numbers, prime integers as of points on some space. So for me this is a beautiful concept.

In what sense? Well, it's in the sense that integers are analogous to the ring of polynomials. So the ring of integers, you can add, you can multiply, you cannot always divide, but some integers are divisible by other integers. So just like polynomials, you can add them, you can multiply them. Some polynomials are divisible by other polynomials. So if a polynomial vanishes at a point, let's say a polynomial of x vanishes at point a,

it means that it's divisible by x minus a. So in that sense, prime numbers are what the other integers might be divisible by. So these are the analogs of those minimal polynomials x minus a. So you can develop some kind of geometric intuition about integers starting from that observation. Okay. You have a lecture titled Natural Language, Geometry, and Physics. Yes. What I want to know is what's the relationship between human language and then the mathematical

language that is used to describe the universe. So I'm very much impressed that I'm amazed by the fact that human language, so natural language in the human sense, evolves. So it's not a static object. It's a kind of a dynamical system. And as such, I was proposing to study it as a physical system like a growing crystal or melting crystal. In my work on Instant Dots, which I reviewed for you, the concept of emerging geometry appears. The problem from gauge theory became a problem of enumerating some combinatorial structures, partitions, which you can visualize as young diagrams,

or you can visualize as some kind of paths with wiggles, wiggly paths. And the enumeration of those objects, you can also view as a search for what's called equilibrium distribution in the ensemble of dynamically changing shapes. So for one young diagram, you can go to another one by changing a few wiggles. So you can assign the probability for those wiggles, for wiggles appearing or disappearing, so some kind of transition probabilities or transition amplitudes. You could ask what if those wiggles were analogous to the words or the sentences in the natural

language. The way the language changes in time could be maybe assigned some transition amplitudes, transition probabilities, and maybe there is some emergent geometry which emerges as the most probable configuration. So maybe to the natural language, certain geometric shape is associated just like to this random partitions in my incident calculus, the Zyberkutn curve, the geometric object was associated. So language is much more complicated than the ensemble of young diagrams. So the geometric shape which is associated to language probably is more complicated than

the two-dimensional surface, but maybe it's less complicated than the world itself. So I was proposing in this lecture to embark on the statistical analysis of the language, not in the way it's done in the large language models, but in doing some kind of time sequence analysis of language. So to study how it changes from, you know, century to century, maybe from year to year. But also the pun in this title was that the geometry is the language of nature, of physics. So that's a natural language in that sense.

Also physics is. Have you followed up this work? Not quite. Not yet. But in the back of your mind, are you developing conjectures or you've just abandoned it and it was a fun activity at the time? No, no, no, I just so I have as I told you

This is very hard problem. So you don't you don't attack it head-on. So I'm I'm developing models Which will kind of you know surround this problem and then I will let that interesting when the time comes so I'm developing other models for for simpler systems, simpler than languages. So speaking of Young diagrams, and I don't want to get us, it's a more ground that'll take us quite some time to get to, but there is gauge origami. And if I understand that correctly,

that has to do with counting Young diagram configurations associated with tetrahedra, whose edges are colored by vector spaces? So we should fast forward to a few years later. Okay. So 2003, with Andrey Okunkov, we understood how to evaluate asymptotics of the partition function, but then it was understood that the whole partition function is a very meaningful and interesting object because it contains information about strings, black holes, local Kaleib Yals.

Also thinking about this partition function, people came up with the notion of the so-called refined topological string. So this whole thing was worth studying. And so one way to approach this problem is to say maybe the Zabir-Kutnik curve, which is the object which captures the limit shape, the asymptotic of the partition function, can be somehow quantized or deformed. It had to be deformed by two parameters, by two deformations, to capture the structure of the whole partition function.

This led to the notion of QQ characters, which is maybe not the most fortunate name, but it has a reason. This name didn't come from nowhere. So it's a notion. It's not observable. So it's a way to measure the partitions. So it's a device. So the device with expectation value, first of all, contains information about the shapes of the random young diagrams and so on, but on the other hand, has some

analytic properties allowing to maybe compute it or to write some differential equation on it. So I realized that one can define some kind of gauge theory problem. It's best done in the context of non-commutative gauge theory, where gauge fields live on a tricky spacetime, which has different parts. So it's four-dimensional, but it's not like four smooth dimensions. It's four dimensions, four dimensions with coordinates x, y, z, t, and then four dimensions with coordinates, I don't know, u, v, w, s.

Sure. Okay. So, then, once I understood that it works and it has good properties and it actually is behind this observable, which I call QQCharacter for the reason, which maybe I will explain later, I asked myself, this is where the mathematical intuition kicks in. So, if you have two transverse four planes inside the 8-dimensional space, why not adding other 4-dimensional planes? And so the total number of 4-dimensional planes you can have is 6, and so they would be in correspondence with the edges of a tetrahedron. And the axis of the four complex dimensional space would be in correspondence either with

vertices or with the faces of a tetrahedron, depending on how you think about it. So a tetrahedron is just an object which helps in keeping track of all the moving parts of this construction. But it's really just a gauge theory which lives on a particular singular spacetime. The reason it's interesting to study is because from the point of view of the observer who lives on just only one of six sheets of this complicated structure, you are just exploring all possible local or semi-local observables of gauge theories. You have observables inserted at the point, you have observables which are inserted along a two-dimensional plane or another two-dimensional

plane. And it's constrained to be six because of the rotational symmetry. Of course, if you don't have any symmetry in question, you can place things any way you like. It will be unruly. But if you insist on things being invariant under rotations in now 10-dimensional spacetime, you will be bound by placing things along the coordinate planes. And so this is what this gauge origami picture is emerging. It's called gauge origami because it looks like you're folding paper at different folds. Right. Is this at all related to the positive geometries of NEMA or the Amplitohydron?

Not as far as I know. But in my experience, things are rather related than unrelated. So whenever somebody claims that here I come, at least in my experience, every time a person comes up and says that, here I have a construction which deals with the same objects, but it has nothing to do with the work of Nikita, I later prove he's wrong. OK. What's something that you've learned as a lesson from collaborating with Edward Witten.

And also I'd like of affinity to rigor. He told me on many occasions that this desire to be rigorous is sometimes very constraining. Sometimes it really stops you from doing something because you cannot proceed in a rigorous manner. But I found that one can be rigorous in a different field. So our stumbling blocks were kind of complementing each other. In our work, we were trying to combine something I learned with something he learned. He was working with Kapustin on the Geishner approach to gematric long lengths, I was working on my partition functions. I was certain it was the same thing. So these were very similar. The domain, the realm where we were discussing,

where we were discussing the subject of our studies, the same subject. It was certain that there was a way to relate our viewpoints. But I had my stumbling blocks and he had his stumbling blocks. It just turned out that somehow they were at the right places. I could advance where he couldn't and he could advance where I couldn't. And so I guess the main lesson from Edward, which unfortunately I still to this day cannot quite embrace, is that one should work on some problem and then move on and then maybe come back to it. I seem to get stuck on things for a much longer time than he does. He manages to learn about many, many things and to advance many things and to be useful

in many things because he doesn't get stuck in the problem. And by getting stuck in the problem, you mean that if you're not making progress, you still continue to think about it, whereas with Edward, he will transition? Probably, yes. I mean, it's not that I'm continuing to think about the problem, not because I'm kind of stupid, that I don't understand that it's least nowhere It's because I feel like there is a progress to be made and or I just cannot not do this It's it's one of this case one of these things when you do what you cannot not to do like it's an obsession

Like this obsession. Yes, like an obsession Okay, so what did you learn from Greg? Greg Moore? Persevere. So if you persevere. Ah, the opposite. Okay, got it. So Greg is close to be in that sense. Yes. You can get obsessed with things and it's okay. Yeah. Do you have a specific example in mind with you working together with him? Yes. We had a...

So remember I told you that before attack enzyme recruitment problem, one had to learn how to compute certain integrals. So we were playing with integrals over like hypercalary manifolds, hypercalary quotients using all kinds of a covariant localization, all kinds of tricks. And at some point, so this was my, what is called, miraculous year. So this 1998, when I met Albert Schwartz and Santa Barbara. Also in Santa Barbara, I met at some point, Greg Moore and Saf Setti. I don't know if you know Saf Setti, he's a professor at Chicago. So Saf was presenting his work with Sturde, I think, on the computation of Witten index for the quantum mechanics of D0 brains. So D0 brains are the interesting particles which are defined as Z-brains. So they are particles in type 2 A string theory.

as Z-brains. They are particles in type IIa string theory, but secretly, the secret life is that they are also black holes and also gravitons of 11-dimensional supergravity. The main conjecture of Witten of 1995 was that the strong coupling limit of type IIa string theory The reason why was that the strong coupling limit of type IIa string theory is 11-dimensional M theory, which contains 11-dimensional supergravity in its low energy limit. For this conjecture to every number of D0 brains. And so, it's a question about supersymmetric quantum mechanics. This is a question about conventional mathematical physics, I should say. And so, Sath was discussing the case of of two zero brains, so just two particles. And he showed that to compute with an index, one had to compute certain integral over matrices,

supersymmetric matrices. And Greg and I, we were in the audience. And I remember after this talk, I told Greg that this is one of the examples of the integrals we should be able to compute because this space he's integrating over happens to be, among other things, also a happy-color space. And so we quickly set out to do the calculation using our tricks. I quickly confirmed that for two particles it produced the answer that SAF was giving. But for three particles, it was already a complicated multiple residue thing, so it

looked hopeless. But Greg, with his perseverance, he set out, I don't know how many hours he spent, but he carefully analyzed all these residues, and it came out to give the answer which we wanted. it. So, then, of course, once you know it works, then you can be clever about it and find more scientific ways of proving what you want to prove. But you really need this confirmation. And so this was just one example of what Greg, with his ability to do two-de-four calculations, Perseverance and Pustas. And so we were able to prove the conjecture of Michael Green and uh, God probably which

Which in the in the Thanks to the work of satan's turn actually implies The fact that for any number of the zero brains they form a bound state So in that way we save them theory Hi everyone, hope you're enjoying today's episode if you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here.

Hit subscribe and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L dot org, KurtJaimungal dot org. What do you want to be known for? Oh. What do you want to be known for? Other than saving M-Theory. I'm putting my physics hat. I'd like to contrast some of my calculations, some of my predictions with actual experiment.

Right now it seems that the most promising venue for high-energy physicists to make predictable predictions is condensed metaphysics. So with my current student, Yugo, we just published a paper where we applied a clever mathematical technique from young-mills theory to the physics of two-dimensional graphene. So maybe when people finally discover superconductivity, high-testing superconductivity in graphene, maybe they will use some of our constructions. So I would be happy to be known for that. Now, speaking of your student, what advice do you have for researchers young and old? Be passionate about what you do.

That's the best advice. Many years ago, like 20-something years ago, I gave an interview to my former university, to my undergraduate school. They asked me, what advice do you give to current students? My advice at the time was don't overburn. Don't burn yourself. Don't do too much. Don't burn out.

Don't burn yourself. So don't do too much. Don't burn out. Right. But I think these days the attitude of, you know, modern students is different. So my advice is go the opposite way. Be passionate about what you do. Burn yourself, whatever you want. Burn yourself whatever you want. Burn yourself. So are you suggesting that, I don't know how long ago that was, a decade or two decades ago?

Two decades ago. Yeah, okay. Are you suggesting that back then the problem was over passion and it would be to their detriment and you're saying now there's maybe some apathy? saying now there's maybe some apathy? Maybe people, well, I don't want to put it that negatively. I just want to say, I think people now know a little bit better how to take care of themselves. Now in my time, it was a really typical thing that people would commit suicide because they were not successful. I mean not because they would overwork themselves.

I think people are now much more aware of those things. And maybe there is a little bit of apathy too. So personally speaking, how do you stay strong during the bad times? Well, I guess I was lucky not to be in too dark times, but I do get stressed a lot. Physical exercise is one way to deal with it, so I exercise a lot. There are lots of ways to take care of yourself, which are important. So breath work, meditation, yoga, gym, be in nature, bicycle, skydiving, see other people, travel, see the world, spend time with the native people, just all kinds of things. Well, speaking of spending time, thank you for spending so much time with me.

You're welcome. It was a pleasure. It's three hours at this point, over three hours now. I appreciate it. Thank you. Thank you. Hi there, Kurt here. If you'd like more content from Theories of Everything and the very best listening experience,

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