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Nina Holden: He's going to tell us about the Young Mills flagship from the physics perspective.

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Nina Holden: And.

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Courant Events Right: Hi, everyone. So, I changed the…

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Courant Events Right: somewhat the title of my talk after I read more carefully the email from organizers.

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Courant Events Right: who asked me to give a gentle introduction in the theory of Yamul's flux tubes. So, original title was basically about this point

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Courant Events Right: core of the talk, but… but that would not be gentle, no matter how hard I drive. So, I should say it's a bit hard for me to really estimate what is the,

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Courant Events Right: what is obvious to people, what is not obvious to people. So, I did… I did not try to make my talk rigorous, but I did try to make it intuitive, and I hope understandable to a broader audience.

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Courant Events Right: So please, interrupt and ask if something is unclear, especially if you're not from that field and you haven't thought about young mules.

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Courant Events Right: Recently. So, the goal of this talk would be to, explain the role of, confining flux tubes, or strings, I will be using these two terms interchangeably, the role that they play in confining, gauge theories, like Young Mills.

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Courant Events Right: And basically review what we understand about, these, strings.

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Courant Events Right: So in more details, the outline will be that I start with a really general introduction, because I don't think we had talks about young meals at the workshop yet, and I will introduce

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Courant Events Right: What these confining flux tubes are.

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Courant Events Right: And then I will talk about what we call, effective field theory approach, or effective string theory, as we call it sometimes, and that is mostly work done during my PhD across the street from here, when the physics department was still on this side of Broadway, even closer than when it is now.

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Courant Events Right: And then I will switch to discussing Lattice approach. So I'm not an expert on Lattice, but I'm sort of a user of Lattice field theory, but okay, I think I can give

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Courant Events Right: try to give a bit of an overview, and then I will switch to this fourth part. So after these three parts, we'll sort of formulate some issues, some obstacles that we ran into on this research direction, and then in this fourth part, I will talk about some very recent and ongoing work

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Courant Events Right: Of, of putting young males on hyperbolic space, which, we, we hope can help us to overcome this.

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Courant Events Right: This obstacle.

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Courant Events Right: Okay, so, Young Mill's theory is the theory of, of gauge fields, right? So gauge fields are vectors in, spacetime, and they live in the, algebra of some, Lieg group.

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Courant Events Right: So I will, I will discuss SUM here, and then from this, gauge field, we form a connection, and importantly, this connection has a term that is nonlinear in the fields, and this nonlinearity is controlled by G that we call, Young-Meil's coupling.

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Courant Events Right: And, okay, this theory can be defined in any number of dimensions. I will talk about dimensions 3 and 4, total spacetime dimensions 3 and 4. See, 2Dang Niels is also an interesting example, much more solvable, and it has some special features, but I will restrict in this talk to 3 and 4.

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Courant Events Right: So, from this connection, we form an action.

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Courant Events Right: Okay, and with this action, we can at least formally define a path integral that allows us to

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Courant Events Right: Questions? It's not a connection, it's field strength. Sorry, yeah, field strength, yeah, Karosha, yeah.

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Courant Events Right: Apologies. the, the, then, then, and then, observables.

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Courant Events Right: in, in this theory, in particular, expectation values of, certain operators, that, okay, we call them gauges invariant operators, but, but let's discuss some examples. The main example that will play a role here is, so-called Wilson loop, where I take, some, path-ordered exponent.

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Courant Events Right: Of, of my gauge field, over some… for example, over some closed loop.

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Courant Events Right: That is, that is wrong. Well, this arrow should go down, I don't know why it's supposed to be a closed loop that is oriented. Or we can also consider some local operators, for example, the Lagrangian density itself is a nice operator that we can insert here, and in principle, calculate correlation functions.

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Courant Events Right: Any questions at this point?

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Courant Events Right: Okay, good. Maybe one thing I say, so I will… I will mostly talk about, Euclidean theory, although I will have some, sort of, I will get some inspiration also from Lorentzian version of the theory. I hope it will be… I hope it will be clear, but… but most of the talk, I try to formulate in Euclidean signature, because probably it's a bit more.

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Courant Events Right: Okay, so let's get… let's target some basic intuition into dynamics of this, Young Mills fields.

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Courant Events Right: So first, let's sort of go back to what all of us know, which is electromagnetism, or Maxwell theory, and this basically happens when I replace this non-abelian group with just U1.

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Courant Events Right: So, for U1 theory, we all know that if we put, like, an electron and positron, or, minus and plus charges, the field lines, they will kind of spread out.

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Courant Events Right: of everyone's pace, and then this, the potential energy between these two charges will be, Coulomb low, that is equal for is 1 over R.

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Courant Events Right: So, for this non-Abelian, generalization of, of Young Milfields, which is, sorry, non-abilian generalization of electrodynamics, which is basically Young Mills, something different happens.

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Courant Events Right: If we put these two analogs of sources, so this, something we call quarks, which are basically analog of electrons.

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Courant Events Right: Charges for these young millfields.

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Courant Events Right: These field lines, because of that non-Abelian term, and the non-Gaussian term in our action, in our measure, these field lines tend to attract to each other, and they form what we call flux tubes.

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Courant Events Right: Okay, so instead of this field being spread out everywhere in space, you find a field configuration that is what we call confined to some string-like object of some typical widths, and this width is gonna be called LS, which is the string length, and the string length, inverse of the string length, is some energy scale, which is a dynamical scale of

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Courant Events Right: of QCD, or Young Mills theory, okay? We call it lambda QCD because quantum homodynamics basically has, the same features.

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Courant Events Right: Okay, so it's completely not obvious why this happens just by staring at the action, right? But we know that it happens.

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Courant Events Right: We have abundant experimental and numerical evidence that that happens. So, sort of, historically, people first realized that something like this would be happening by staring at the spectrum of mesons, some particles.

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Courant Events Right: that people discovered colliders. Basically, all you need to see here is that there is a dependence of

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Courant Events Right: mass square of the particle on angular momentum, and it is very close to linear for many particles, and sort of the simplest explanation for having a linear dependence between mass squared and angular momentum is that if your object is some kind of long string or rod, because then mass goes linearly in length, and angular momentum is quadratic in the length, hence the relation.

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Courant Events Right: But, okay, you could come up maybe with some other explanations, but then, when computers appeared, people went and actually set up these, numerical simulations.

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Courant Events Right: my largest gauge series that I will review in the third part of the talk, and indeed, we see this, string-like object forming, okay? So these are these quarks, and then you see very clearly, this, this guy, and this guy is more like a proton, but anyway. But why and how they form, that, that remains, sort of not understood.

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Courant Events Right: even… okay, there are some toy models for how this is happening, but really, it's fair to say we don't really have any analytic understanding for why this is happening. And this is more or less the point of this research direction, is to try to figure out how to get some intuition.

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Courant Events Right: So let me say that… that… so I talk about strings, so at fixed time I have a string, but this is a surface in spacetime, so that I try to indicate here. So if at some moment of time I have a string, and then in spacetime it spans a 2D surface that I refer to as a world sheet, and then if we go to Euclidean signature, which we will, so the strings correspond to two-dimensional membranes in Euclidean theory.

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Courant Events Right: Okay, so I want to make sure that no one is confused that I talk about strings, but really, from Euclidean point of view, this is two-dimensional surfaces.

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Courant Events Right: Okay.

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Courant Events Right: Good. Let me review one more, important, idea, which goes back to Koft, that is, large n or topological expansion of my, Young Mills theory. So…

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Courant Events Right: the… we… we go… we take our action, and then if I expand it in powers of fields, schematically, I have a Gaussian term, and then I have a cubic and quartic term that are suppressed by powers of my coupling constant, and now let me perform sort of a formal perturbative expansion, in this coupling constants.

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Courant Events Right: And then I will, basically, for a moment, we can ignore that our fields depend on space and time, we can just treat them as matrices. Remember, there's some matrices, that, that, that, that, for example, form an adjoint representation of, of,

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Courant Events Right: Yamil's theory, so then I will need to contract indices of these matrices, and I'll start getting some factors of n coming from the traces, and then you can see if I have, for example, at order G to the 4, two diagrams like this.

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Courant Events Right: So I contract my… so this double line notation, it corresponds to contracting two indices of the matrices, and every time I have a closed loop, I have a trace, okay? So you see that here I have two more traces than here, and that is because this diagram is planar, and this diagram is not planar. So this is very similar to topological recursion that one gets in matrix integrals.

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Courant Events Right: In matrix models.

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Courant Events Right: although sort of precise analogy is a bit delicate, but the idea is here. The main point is that when I take M to infinity, so I take a rank of my gauge group to be very large, these planar diagrams

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Courant Events Right: with the same power of the coupling constant, they dominate over non-planar diagas. And in fact, for most of my talk, I will simplify my life by going to this large end limit, where I can ignore some non-planar corrections.

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Courant Events Right: So, next idea, it is very… it is very vague. No one was able to make it precise in Young-Mill's theory, but it was made precise in some other theories, okay? And we believe that more, this is true. So… so it's good to have this… to spell out this idea for us to develop intuition, okay? So, when coupling becomes… if we take coupling order 1, and I will discuss in a moment what kind of

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Courant Events Right: values of couplings we should take. But if we take coupling order 1, then basically, at each order in N, I should re-sum all possible diagrams that are planned, right? Because they all contribute the same.

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Courant Events Right: And then the idea is that these diagrams become more and more dense, and they form some kind of surfaces, and these surfaces are nothing but the world sheets of my strings, okay?

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Courant Events Right: So that is an intuitive picture, but we very strongly believe that that is what is happening.

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Courant Events Right: or, did I say… well, they, they become more and more complicated, and they… no, these diagrams, yeah, these diagrams live, in, in spacetime. We can think of, sort of, position space, perturbation theory, so they just live in, in D-dimensional, whatever, in this, whatever the Daniel's theory lives, so they are boundaries, okay, do I have a… yeah, maybe if…

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Courant Events Right: maybe if I go… if I go to the next slide, it becomes a little bit more concrete. So let me… let me try to evaluate expectation value of my Wilson line that, for example, I take as a circle here, this red line. Okay? So then, it will comprise of diagrams of boundaries of these diagrams, they will… they will all be fixed.

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Courant Events Right: will repeat into this Wilson line, but then in the middle, I would have all this complicated diagram that all of them are planar, and they sort of form a surface. Yes?

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Courant Events Right: Wait, what the fuck?

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Courant Events Right: Fine, fine, you're good.

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Courant Events Right: Well, that's the population on both sides of the students?

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Courant Events Right: No, no, okay, let's count the number of loops, because n, power of n, comes from a loop, right?

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Courant Events Right: 1, 2, 3, 4 loops, because loop is a contraction of indices, you get delta IJ. And here, I have… this is all one loop, you see?

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Courant Events Right: Yeah, thank you. So here I have just two loops, and here I have four loops, that I have a relative factor of n squared.

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Courant Events Right: Yes? Which is observable.

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Courant Events Right: Yeah, this is just, sort of this, here I'm computing just, sort of, roughly a vacuum, whatever. Yes, but then on the next picture, I'm computing expectation value of a Wilson loop, and these are these diagrams where, sort of, the outer line can be replaced with a Wilson loop. For example, a Wilson loop on a fundamental. Yeah, but genetic…

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Courant Events Right: if I had some external legs, they could connect to operate, like, a generic piece of a diagram.

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Courant Events Right: Yes? Sorry? Not… yes, yes, I can take some scaling limit, yes, but Yes.

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Courant Events Right: In more details, thank you, but I wanted to… simplified it.

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Courant Events Right: So… so for the rest of the talk, okay, so that's what I was explaining, that if, for example, if I compute the expectation value of some circular Wilson, a Wilson line of any shape, I effectively get some surface, and the point is that the surface is the same picture as this… so you can think of a Wilson line as, at a given moment of time, as I have my quark and anti-quark, and the fact that there was this flux tube

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Courant Events Right: forming, in between the quarks is the same picture as the surface forming from my diagrams, okay?

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Courant Events Right: So that is, that is the intuition here.

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Courant Events Right: you said that the surface forms already for finite N, but then you said one state, so… Well, it's… the idea is that if I have a finite N, I would need to include also some diagrams like this, they are not too strongly suppressed, right? And then these diagrams will form some surfaces with handles, surfaces of non-trivial topology. That's why it is the same as topology

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Courant Events Right: expansion, the matrix integral, I think of expansion 1 over n as expansion genius of, sort of, effective surfaces that I need to consider. Yes? Yes.

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Courant Events Right: Yeah, I'll get there, I'll get there.

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Courant Events Right: Does anything change in the voltage include the quarks? Well, then, at infinite N, first of all, quarks also do not matter. At finite N, quarks, they make some holes in my surfaces.

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Courant Events Right: And yeah, let's… life will be hard enough without quirks and at large end, but…

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Courant Events Right: Okay, good. So basically, that's what motivates us to study of a theory of a single flux tube. So if I take this n to infinity limit, the idea is that all sort of observables, say this expectation where you fill some light, they're dominated by a single string existing at any moment of time, of my Euclidean time, so… so these flux tubes, or strings, are the most basic

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Courant Events Right: objects in confining theory. So if we sort of understand a single flux tube, theorem of a single flux tube, then we're making some progress towards understanding confinement. Sort of, if we cannot understand even the theory of a single flux tube, there's sort of no hope to understand confinement. That's the slogan for today's talk.

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Courant Events Right: So, let me review one more important feature, which is an idea, which is asymptotic freedom.

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Courant Events Right: So, so far, I was a bit agnostic about the size of a coupling g I was formally expanding it, but what is this G? And it turns out that the answer depends on the distance, or momentum, that, in Fourier space that we consider. So, actually, both in four dimensions and in 3 dimensions, G is always effectively

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Courant Events Right: large at long distances. So I had this distance scale, which is the size of my… width of my string, and when I go to distances, if I can see the, say, surfaces that are larger than this distance scale, so they really look like surfaces, their thickness is small compared to the widths.

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Courant Events Right: effectively, this Yamil scouting becomes large. Even if I try to set it to be very, very small in my original definition of the action, somehow it gets what we call renormalized, and it becomes large.

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Courant Events Right: And converse is also true. If I go to very short distances, or very, very large mementa, then my effective coupling becomes small, although this last, last… while first thing is sort of precise, this second statement is hard to make precise, because C is non-linear, so two very large mementa can always conspire and make a very small momenta.

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Courant Events Right: And that actually causes lots of problems in real QCD calculations, but I will actually try to make this statement precise in the last part of the talk.

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Courant Events Right: precise at the physics level, of course. Okay, good. That was the… that was the introduction. Any more questions?

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Courant Events Right: So then let me move on to this first part, very important for physicists, intuition, effective field theory approach.

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Courant Events Right: So, naively, as we just said, situation is sort of the hardest at very long distances, because our coupling is always very strong, and sort of it's hopeless to do anything. However, it happens that that's not the case. Instead, there is a different weekly coupled perturbative description that shows up for very, very long strings.

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Courant Events Right: So first, let's consider a situation where somehow we set up our gauge field so that there is a single, straight, very, very long string, okay, kind of stretching across some spatial directions.

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Courant Events Right: So, the claim is that its fluctuations, meaning deviations from this exactly straight form, can be described by a new two-dimensional quantum field theory whose fields are coordinates of the embedding of my surface into my space.

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Courant Events Right: So I call these fields, gallstone bosons, for, for a good reason, but we don't have to go into this. So I have D-2 of them, right, for example, stringos on 1 and 0 directions, and then I have from 2 to D minus 1,

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Courant Events Right: D minus 2 gallstone bosons.

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Courant Events Right: So, then I… I can form out of them this sort of object, I can complete them in a D-dimensional, vector by just adding what I call world sheet coordinates, so tau and sigma, they will be parameterizing, this, world sheet surface, okay, and then Xi's, depending on tau is sigma, they will parameterize deviation from that surface.

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Courant Events Right: So, so kind of, this is just geometry, but the claim is more than that. The claim is that these coordinates, they, I can actually describe them, as a two-dimensional quantum field theory.

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Courant Events Right: And I have the rules for writing down the action for this two-dimensional quantum field theory based on symmetries.

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Courant Events Right: So this, action, okay, rules, is that you first make some object that is an induced metric, then you make extrinsic curvature, you make a Riemann tensor, and then you write all different variant terms. Okay, that, that, that is.

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Courant Events Right: Okay, maybe, maybe a bit technical, but the punchline is that we have the rules for writing down this theory. Let me… the leading term, maybe, maybe that's important. Realize this leading term is the area term. So to leading order, and, and leading order in what? Leading order in,

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Courant Events Right: at low energies, or at long distances. So, when I look at this action.

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Courant Events Right: There is sort of a trivial term, which is a constant, and then I have a Gaussian, free massless field.

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Courant Events Right: Okay, so that's just the relativistic kinetic term for… sorry, not 1, but D minus 2 fields, for example, two in four dimensions, and 1 in 3 dimensions. And then I have some corrections, and corrections, they all have higher powers of my string lengths, but also higher powers of derivative effect on field. So this is sort of a fancy version of Taylor series, okay, or multiple expansion, or whatever you call it.

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Courant Events Right: But the idea is that if I go on very low energies at very long distances, all these nonlinear terms are suppressed, and I have, again, a perturbation theory, because my measure is sort of close to Gaussian.

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Courant Events Right: Okay, so that is an important point, that I have an expansion where my young mules theory is sort of the more strongly coupled, I have a new description, gets better and better the longer I take my string.

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Courant Events Right: Okay, let's maybe clarify a little bit one point, because you all know that Young-Mil's theory has a gap it confines, so why do I talk about some gapless modes, right? So that is an important point to avoid confusion.

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Courant Events Right: So, this is the energy scale, alright? And, okay, this is zero, so here I have some gap, and this gap is actually of order of my string tension.

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Courant Events Right: But then, on top of it, I can see that, okay, there are many states, I skip them, but then there is this state, which is a single long string, and actually its energy is very large, so R is the length of the string, it actually goes to infinity when I take it large, and then on top of this state.

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Courant Events Right: this is where I have this, gapless mode. So this is what I study, and the idea is that, this, there are some solid symmetry arguments, involving some one-form symmetries, that, that this sector is sort of a super selection sector, so I can just study this, ignoring all this gap excitations that are much lighter. So this is what I'm doing from the kind of spectral point of view, so I'm having a theory for

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Courant Events Right: Very light, or gapless in the infinite volume limit, modes.

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Courant Events Right: Okay. And, and okay, so that was sort of just classical action, but now again, I can put this classical action, and it is formally defined a path integral, where now I have a path integral over, coordinates of my surface, and I weight them with this effective, effective string action.

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Courant Events Right: Is this, is this discussion clear?

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Courant Events Right: Yes? That's it.

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Courant Events Right: It's just, it's just, I wanted to say that it's fully consistent to focus on these light states with respect to this… maybe, let me phrase it this way. This single string state is a ground state in a sector with certain quantum numbers.

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Courant Events Right: Well… Sorry, the question is, how do I… how do I identify this sector of a single long string? Okay, then we need to invoke something that's called center symmetry. So there is a one-form global symmetry in Young Mills, when you put it on some compact manifold, which is basically a holonomy of SUN transformations, and one preview of SUN transformations wind around the circle.

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Courant Events Right: And then the claim is that the single long string winding around my spatial circle, or equivalently going from one

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Courant Events Right: part to the other of the entire, spacetime is the ground state, in a sector with the,

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Courant Events Right: center, with the charge 1 under this, center symmetry, under the ZN symmetry. Okay, that's a bit…

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Courant Events Right: Excellent.

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Courant Events Right: Usually, for a character system, you explain everything.

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Courant Events Right: Good, good, yeah, good. So, so we… Yeah, so the question was that, that if you go somewhere high up, then you should see some chaotic, Wigner-like distribution of these energies. And that is not…

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Courant Events Right: true, it will become true somewhere here, but I claim that near this special state, I have very ordered

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Courant Events Right: spectrum, actually very close to free theory. And I will show you, but you're asking the right question. It's more or less what I was trying to say. But that is a non-trivial point. There's the whole point of effective theory. We regain control

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Courant Events Right: for some energy band, where we have our effective theory description. But eventually, this effective theory description breaks down, so if I go another…

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Courant Events Right: LS-1 above here, if I go somewhere here.

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Courant Events Right: Then, all these dots, dots, dots in my actions start to matter, and I lose control again.

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Courant Events Right: Statistics of the energy we get.

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Courant Events Right: We're getting there, yeah. We're getting there, that's the way to do it, okay. The question was if there is a relation between these energies and what we get from Young Mills, but that's exactly the point of the talk, so I will not answer it now. Okay, good. But I want to say that what's important is that this path integral, even for physicists, is not

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Courant Events Right: defined non-perturbatively as this, right? So I want to make this distinction that the Young Mills, Lagrangian, for me, it is something well-defined, something I can… it's hard to make calculations, but in principle, it defines my theory with any precision. While this object does not, I need to supplement it with something to say what all these dots, dots, dots are, or have some alternative descriptions. That is an important point. But it's still useful, because if

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Courant Events Right: usefulness is if I want to make a calculation with a given precision, I need a given number of coefficients, for example, like this coefficient alpha is the leading non-universal coefficient. But if I want to go to higher precision, I need to know more and more terms in my action, which was not the case for Young Mills, because there was only one term, and that's it.

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Courant Events Right: Good. So I want to emphasize this particular theory of fluctuating surfaces.

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Courant Events Right: Some similar theories of locating surfaces are integrable and can be exactly solved sometimes. So examples include, say, critical bosonic string, some flux tubes and some supersymmetric versions of Jangil's theory, some two-dimensional gravity theory, so these are some relatives of these theories, where you can make some progress. But it is a particular, rather complicated, non-integrable theory of classic

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Courant Events Right: Of two-dimensional surfaces.

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Courant Events Right: Two-dimensional gravity, if you want. So, it's actually…

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Courant Events Right: some fact that's not so trivial, but was shown, that Yamil's

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Courant Events Right: this theory of the theory is classically integrable, but quantum mechanically, it cannot be integrable. And this classical integrability actually plays an important role.

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Courant Events Right: So, now, let me… let me consider something I've already… yes, sorry.

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Courant Events Right: This said that the action was not known, actually. There are many terms that aren't known, so how do we know this is classical interval.

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Courant Events Right: What's good.

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Courant Events Right: Yeah, good question. So, how do I… yeah, sorry, I should repeat the question. Yeah, so the question was how I said it's classically integral, but then I said that I also don't know the action. And the answer to this is actually, again, it's my fault not being careful with terminology. Classical theory is just whatever comes from this area term, which is an infinite set of terms.

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Courant Events Right: But they are all fixed.

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Courant Events Right: And this term is that even though I write it in the classical action, actually, for me, it's equal to what I would call a two-loop

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Courant Events Right: term, because two-loop calculation with these vertices gives me the same order as, sort of, T-level insertion of these sort of vertices, okay? So this is the terminology, what I call classical is just whatever comes from this series where you have one derivative per field.

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Courant Events Right: Thank you.

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Courant Events Right: Other… yes. Yes, yes, but… and this is something that happens to be integrable at quantum level, if you are in 26 dimensions, but… but not in 4 dimensions, sorry, 3 dimensions.

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Courant Events Right: But why do you call interior of Young Mills String? Because this is just something

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Courant Events Right: No, no, good. I'm not, but, okay, maybe I should embed these articles. So, there is the theory of young mill string, okay? And it corresponds to some particular, let's simplify… corresponds to some particular choice of coefficients here, okay? I do not know what that theory is, but it's kind of unique, and it exists.

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Courant Events Right: And I know that that particular theory is not integral, right? But the point here is that instead of starting this full theory, I study some approximation to it, which I call effective strength.

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Courant Events Right: Is that point clear?

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Courant Events Right: Let me try to make… maybe if I explain this slide, become a little bit more concrete, my discussion. What can I actually calculate in this theory? Okay, I've got some formal action, okay, let's try to compute something, and also answer Denise's question, how it compares to some calculation Young Mills. So now let's consider Yang Mill's theory on the cylinders. So far, I was sort of on just R4, now let me consider… let me compoxify one direction.

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Courant Events Right: on some circle of radios much bigger than my string scale, okay? And then, what I will consider, I will consider a single string that winds

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Courant Events Right: around this circle, all sort of in Euclidean is some membrane, right, some, this band that winds my, my cylinder. And now, what happens is that my, my theory here.

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Courant Events Right: this theory, it also lives on a circle now.

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Courant Events Right: So, now I can sort of go from the action for a second to a Hamiltonian picture and calculate energy spectrum of that theory on this,

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Courant Events Right: On this circle. And the claim is that my effective theory allows me to calculate this energy spectrum, so n label some energy level, and this, this, each energy level is a function of the length, and I can, expand it as a series in inverse power of the length.

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Courant Events Right: With the CNs are dimensionless numbers that are calculable.

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Courant Events Right: Okay?

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Courant Events Right: And the… so we can calculate this up to… up to rather high order, and moreover, there are some, some kind of interesting resumation techniques related to, to, to… so after yesterday's talk on, on, form factors, I was tempted to put some, things related to, terminatic beta and that, and, and, and this approximate integrability, but I…

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Courant Events Right: Restrain myself, and, and, so, so, so, but the claim is that we can, we can have some useful calculations of these energy levels.

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Courant Events Right: Yeah, but… Yeah, but you can do some approximate integrability, I'll tell you later.

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Courant Events Right: there are some… some kind of curious things, and then there is some… there is some toy model, set of integral theories that shows up here that are kind of related to this string, and then… and then there's some interesting corrections to this form factor business, but this will take us far, okay? Instead, let's discuss something more mind, yes.

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Courant Events Right: So, how does the momentum along the string enters into this EL? Yes, so… so momentum… momentum is roughly, like, N over R, or whatever. Momentum is order 1 over R,

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Courant Events Right: So, I think of it, again, kind of… there is some quasi-particle description here, and I think of these little wiggles, right? These are gallstones, each of them have some fixed momentum that are for a 1 over R. Some order one number divided by R, and which state I consider exactly depends on how many gallstones I have.

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Courant Events Right: But then, of course, the number is not conserved, they start to mix, etc, etc. But roughly speaking, typical momento is sort of 1 over R.

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Courant Events Right: Okay.

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Courant Events Right: Is that…

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Courant Events Right: So this R is not the curvature from earlier the R, right? No, no, no, no. Yeah, sorry, yeah, sorry is the command, yeah, R is radius, yeah, for… Really computing…

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Courant Events Right: without all of the curvature trucks? No, no, no, they are… they… they're included here. Some of them, this high order, the idea that the first… I forget, but I think the first time the curvature term appears is 1 of R to the 7, but then you… you can still… you include it, yes, yes.

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Courant Events Right: So the first few terms that are universal do not depend on these curvature terms, but then you add curvature terms, and they contribute here, and they're important. I will show in a moment.

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Courant Events Right: you cannot… yeah, you… the point is that there is no meaningful way of setting coefficient of these terms to zero. There is no prefer… I mean, in quantum field theory, they get sort of renormalized. There is no meaningful way to set them to zero in this theory. That's the point.

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Courant Events Right: Moreover, okay, maybe explain the next part that will also answer a little bit your question, okay? But the short answer is that there's no meaningful way, and it's a kind of non-zero here. So the simplest time problem

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Courant Events Right: well, mathematicians, maybe, that you could make out of this with the curvature. Yeah, yeah, there are these, this kind of… this… these are some toy models for mathematicians that, that, that we can discuss. They're useful, but, but they do not capture all the features of this theory.

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Courant Events Right: which essentially behaves on some particular choice of these high curvature terms, and kind of the goal here is essentially to figure out what they are.

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Courant Events Right: And I'll try to get there in the remaining 20 minutes. I don't know how much time I have. But let me do very… let's discuss a bit Lattice Gauge theory, maybe it's good for people to…

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Courant Events Right: To review, so, okay, I'll try to be a bit quick, so I consider some Euclidean lattice with project boundary conditions, some torus, and on each link, I define, a variable, that is a group element, okay? So they live on these, links.

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Courant Events Right: And then as a next step, I form a plaquette. So plaquette is a product of four group elements, so it's labeled by a point at which plaquette starts, and manure the two directions in which plaquette faces.

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Courant Events Right: Okay, so I think picture is self-explanatory, I have this product that corresponds to the product of these four link variables, and then I form an action of Lati's Yamil's theory, which is the sum over all plaquettes, all the positions, all the orientations, and then I take a trace in the fundamental representation.

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Courant Events Right: Okay?

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Courant Events Right: So there is some lattice action, and then I take a continuum limit. So by continuum limit, I take the lattice spacing to zero, and I actually need to take the lattice gauge coupling that shows up here also to zero, but keeping some effective coupling at some physical length scale fixed.

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Courant Events Right: And the claim is that, morally, this largest gauge reaction becomes the continuum of Jan Mills action, and that basically comes from the fact that this plaquette sort of implements a parallel transport around the circle and corresponds to the gauge field curvature to leading order in

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Courant Events Right: and small, A. So, more concretely… yeah, yes, sorry?

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Courant Events Right: It's kind of like a Wilson loop, but… Yeah, little tiny Wilson loop. But when it's put in the action, you take the real part, whereas when people consider Wilson loops as observables.

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Courant Events Right: What doesn't take the real part? Is it important to also include the imaginary part? Yeah, well, I think that's just to make the action, real, yeah, so there is… to actually match the young mills action, I think you need to…

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Courant Events Right: take the real part here. For the observable. No, for the Wilson lines, you keep the… you can… Is it important to keep also the imaginary part for the observables?

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Courant Events Right: Yes, I think so, because, well, because the moral of the… where does Wilson Line come from? Wilson Line comes from the, basically, it's a phase for a charged particle that travels on some trajectory.

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Courant Events Right: Alright, so the idea is that if you have some charged particles, say, in Maxwell theory, right, and it travels on some line, it keeps accumulating a phase.

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Courant Events Right: proportional to, sort of, integral of a gauge field along its trajectory. So that's why, that's why in the observable, you want to keep an, an I, and an expectation value actually becomes exponentially small when theory confines what that is.

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Courant Events Right: So, it's exactly… I'm coming to Wilson Lines.

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Courant Events Right: Didn't rewrite the defini… but, okay, in the, yeah, in the Wilson line, okay, as was just asked, we simply take a product of, of plaquettes.

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Courant Events Right: Along, some, contour, and, and that becomes, again, in the continuum, the Wilson line operator that I had before. In particular, we can take, we can choose,

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Courant Events Right: one of our circles of the lattice to be much smaller than the others, and then insert the Wilson line, a conjugate Wilson line, and calculate this correlation function.

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Courant Events Right: Okay? And the idea is that this, on the lattice, this is just finite integral, right? There's a finite-dimensional integral, which is convergent for any lattice spacing.

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Courant Events Right: And then, in the, in the limit, A going to 0, you just numerically see that, that, that the correlation function converges to something sensible, okay? So it may be hard to prove

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Courant Events Right: That this limit exists. But numerically, we just see that this limit, exists.

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Courant Events Right: So we believe that Jamil's theory is well defined under the largest gauge theory in the continuum, at least for observables of this sort.

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Courant Events Right: Produces, the, the continuum answers.

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Courant Events Right: So, next question is, okay, how do we compare these results to our effective string theory, right? That was basically, basically, basically the question. That's something that's calculated, okay, numerically, maybe, but from first principles and Young Mills theory.

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Courant Events Right: Okay? And I want to compare it to what I calculated in my effective string theory.

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Courant Events Right: And here, I use a little bit of Hamiltonian intuition, right? I can decompose any correlation function, I can insert a full set of states, so if I sort of, for a second, go to kind of real-time quantum mechanical description, I can insert a full set of states.

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Courant Events Right: in between my operators, and then the correlation function can be decomposed as some coefficients that depend on which Wilson lines are inserted, but then times these exponents.

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Courant Events Right: of energy and then minus tau, so in real time, those would oscillate in Euclidean, those, of course, all decay, right? So this, this set of numbers, EN of R, it's the spectrum of, sort of, single string states.

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Courant Events Right: of, Young-Mille's theory on S1, times, okay, let's say T2, but let's say this T2 is much larger than my, than the circle that I want to keep a small circle, a finite circle in the, in the continuum. And again, at large N, I can ignore, interactions of, of this string with some, handles and with some other, other

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Courant Events Right: states that are kind of little strings living somewhere else, and that comes from my planar diagrams argument, and it is also seen numerically that that is what's happening. And then this spectrum, EN of R, is the same spectrum as my effective string theory has, okay? So these are these numbers that I was computing. Okay, so that is my theoretical prediction.

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Courant Events Right: Now, let's see, does it work or it doesn't work, right? Because I have some calculation, and I have some numerical data, so let's see.

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Courant Events Right: Okay, so this is, 4D SU3N mules. By the way, one thing you should actually, again, numerically, SU3 already very close to SU infinity, so these calculations were done for a different n, and that was checked, but okay, I'm showing you… of course, data becomes harder and harder when n is larger numerically, but…

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Courant Events Right: But, but anyway, kind of, never mind. So this is the ground state energy.

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Courant Events Right: For example, and this, again, there will be many plots, I don't want you to process all of them, but what you should see that the error bars are really small, and this theory curve, so these different lines are sort of different orders in my calculations, and I do have a couple of parameters with which I feed.

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Courant Events Right: But these curves do look, very well. So, basically, on the left, there is some…

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Courant Events Right: lower-order calculations, they don't fit quite well, and then we improve, and then you see especially this red line, so there's different colors at different states, different quantum numbers. They do work, they, they, they do work, pretty well.

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Courant Events Right: Siri?

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Courant Events Right: Yeah, so you're… so you should also see an error… look at error bars, right? So error bars show one sigma deviation. So basically, if the error bar is so large.

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Courant Events Right: So the probability that this point is here is, like, roughly 30%, something. So, on the other hand, if you look at this tiny point.

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Courant Events Right: then the probability that is here is, like, 10 to the minus 10, okay? So you should not just look at the points, you should also look at these bars, which are error bars, but yeah. Can you just say a few words, like, what are the main features of this curve that you should pay attention to?

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Courant Events Right: What is this curve which is almost flaring? Yeah… Yeah, you're asking very advanced questions for the time that is left. I wanted to stay a bit impressionistic, but roughly speaking, roughly speaking, and no, it's the right question. Roughly speaking, curves that go up.

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Courant Events Right: They correspond to these massless particles.

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Courant Events Right: Basically, what we see here is, like, 1 over R behavior for… for energy being equal to momenta, and momenta being 1 over R, and when you see this flat thing, this is something I did not mention, that corresponds to having an additional massive particle that lives on the string world sheet, but, yeah, okay. Yes, you're probably okay.

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Courant Events Right: So, yes, but Slava is completely right, we need to… okay, I'm hiding many things under the carpet here, that's right, but it was actually the most…

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Courant Events Right: One of the most interesting discoveries in this business so far is that we realize that you also need to add some massive particle

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Courant Events Right: that lives on the surface. So the surface is not just a theory of coordinates, but it also has some sort of massive excitation.

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Courant Events Right: that, whose geometric origin is not very clear, but actually origin in Yang Mill's theory by now, I think, is more or less clear, but it's true, okay? In a longer version of this talk, I discuss it a lot, but okay, this is what's called world sheet axion, and say on this plot, all these energy levels.

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Courant Events Right: They correspond to various excitations of this, third field.

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Courant Events Right: So I have two fields that correspond to coordinates, and I have the third field that calls an axion, and an energy that is… that needs to be added to make the theory work. But there's just basically, like, two more parameters of the axion mass and any field, many curves. Okay, happy to discuss it in more details. I think that…

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Courant Events Right: That will take me a bit far away. Maybe, maybe this, again, for someone who is especially, say, an integrability person, maybe used to think in terms of S-metrics, as opposed to.

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Courant Events Right: energy level, so that's something we can calculate also and compare today to the S-metrics of these excitations. And okay, this is a three-dimensional theory, again, three dimensions, there is even more data, and it all works.

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Courant Events Right: Surya?

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Courant Events Right: You, you worried about these guys?

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Courant Events Right: Okay, this is some data processing, okay. This is done carefully, and there's a change of variables, and and okay, there's error bar, there's some direction which error bar is largest, and that's what's indicated.

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Courant Events Right: Let's not… let's not get hung up on this. The point is that, we have a non-perturbative definition.

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Courant Events Right: which is Lattice, we have two analytic perturbative descriptions.

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Courant Events Right: which is, original Yamil's action, which works well at short distances, And then we have…

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Courant Events Right: effective string theory description, which works for a single string, but okay, it works for very long distances. But we do not have a universal description, and sort of the problem here is that it's…

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Courant Events Right: unclear how to connect the two at this moment of the talk. This is what I… this is what I… what I started with.

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Courant Events Right: And… and this lattice is a way to make sense of the Samuel's action on perturbative, but okay, it's a computational tool, not an analytic tool, and we kind of want to… we want to do better.

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Courant Events Right: Yes.

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Courant Events Right: So there is a… at the lattice, at least, the stroke up in the extension, which allows for, you know, to prove… Yeah. Yeah, but…

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Courant Events Right: Very good. You see, the problem with this is that, as you remember, I said that actually to get to the continuum limit, I need to take lattice coupling to zero, where this expansion that you're talking about, it breaks down. And the belief is that there is a phase transition in between the two.

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Courant Events Right: So, short answer is I think nobody knows, so it's not obvious that these surfaces that you get in the strong coupling expansion are connected to these ones. It may be the case, but I don't think we have very good

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Courant Events Right: arguments for this to be the case. People try to explore this, but… It's not obvious.

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Courant Events Right: We just have to recover rotational bias, which is not present in the strong government expansion.

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Courant Events Right: In particular, and then there is… there is a phase transition. I… So… so let me… let me say maybe a little bit along these lines, but… but… but, okay, why don't we just…

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Courant Events Right: make a size of my circle very small, right? Because naively you want to say, okay, let's make circles smaller and smaller, and then I'll go to short-distance physics, and I'll match the perturbative Yamil's description.

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Courant Events Right: But that does not work. Like, if you were Catholics looking at my plots, none of my plots were going all the way to R equals 0. All of them were sort of terminating somewhere around R equal LS. And there's a good reason for this, because if you look at the ground state energy and you continue it a little bit, actually it develops a singularity at some finite radius of R, and that is a signal of deconfinement.

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Courant Events Right: Phase transition.

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Courant Events Right: So basically, the theory, when I try to put Gang Mules on smaller and smaller circles, eventually, theory goes to a completely different phase, and my string description completely breaks down.

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Courant Events Right: Okay?

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Courant Events Right: Very good. And then in the remaining, I don't know how much time I have, I will… 9 minutes, okay, good, that is enough. So basically, that is the problem, right? So we have these two descriptions.

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Courant Events Right: We want to understand better how one emerges from the other, how the string description emerges from the other, and ideally present a full, complete theory of these 2D surfaces.

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Courant Events Right: And as an idea, what we want to do now is to put Janielson on hyperbolic space.

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Courant Events Right: Instead of putting it on the tors.

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Courant Events Right: So hyperbolic space, physicists call it Euclidean anti-decidral space, that's the same thing. That's very… yesterday, thankfully, it was mentioned, so I can be… I can be quick. So it is a hyperboloid, okay, in, in D plus 1-dimensional Minkowski space.

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Courant Events Right: Let's say, defined by this equation. So there are, there are various useful models of hyperbolic space, for example, hyperbolic plane, with a metric like this.

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Courant Events Right: where the boundary, it's important that hyperbolic plane has a boundary, at least conformal boundary, which here is a plane.

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Courant Events Right: Or I can use this global, what's called global, under this little space, where it looks like a solid cylinder, okay? So before, I was drawing a cylinder, it was a hollow cylinder, now it is a solid cylinder, and again, boundary is now a sort of a hollow cylinder.

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Courant Events Right: cylinder. So, isometry group is SO1 comma D, and so it's the same, same means it's the same number of isometries, as RD. So, hyperbolic space is maximally symmetric. When I put theory on this, I don't break any symmetries, I just deform the symmetries. That is important.

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Courant Events Right: Hyperbolic space has both infinite volume, and it has a boundary. This is also important.

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Courant Events Right: Now, let me take a little deviation from, from Yamil's theory, a little detour, and just talk very quickly about genetic properties of quantum field theory and antidecitor space. Because I think this is… this is sort of a much more, much broader tool than just studying confinement.

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Courant Events Right: So, SO1 committee is also a conformal group in one dimension lower.

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Courant Events Right: And that means that correlation functions of my quantum field theory of operators that are placed on the boundary, they behave as operators in conformal field theory of one dimension lower. And in particular, there is a relation between dimensions of operators in conformal theory and masses of my particles.

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Courant Events Right: In, in the bulk.

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Courant Events Right: So, for those of you familiar with the DSCFC or holography, that rings a bell, but the point here that there is nothing, nothing, nothing from holography is important here. And however, this theory, there is a catch. So, this theory has no stress tensor. So it's not like a local conformal filter, in particular, if boundary is two-dimensional, there will be not Verasaurus symmetry, but there is always a global conformal symmetry.

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Courant Events Right: And there is, very importantly, conversion-operator product expansion. So many of the things, axioms from conformal field theories, they do carry on to this setup of quantum filters and other distributor space.

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Courant Events Right: So, spectrum of deltas, of dimensions of CFT, that's the spectrum of the global Hamiltonian, spectrum of translations with respect to the star variable, and it is discrete.

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Courant Events Right: as infinite volume. At the same time, these boundary correlators are sort of on-shell observables, LIS metrics, they don't depend on field radiation, etc, etc. And very importantly, for confinement, the strong infrared effects, they're regulated at the distances for the ADS scale, despite the fact that the volume is infinite.

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Courant Events Right: Just very quickly, there is many, many interesting recent works on quantum field theory on anti-decidra space. In particular, you can reformulate QFT as just a set of ordinary differential equations.

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Courant Events Right: as some of my, colleagues at APFL have done. And then, yes, regarding to… okay, I was hoping to anticipate your question, but go on.

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Courant Events Right: Well, you can, but you don't get a local current on the boundary, so you still get… it generates a symmetry, but there is no local current.

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Courant Events Right: You can write… you still have some word identities, but but you don't get a protected operator on the boundary.

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Courant Events Right: No, no, no, but in a CFT, without any supersymmetry, stress sensor is protected, right? It has dimension D. Here, you don't have a protected operator.

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Courant Events Right: You get some operator of spin 2, but it has generic dimension. Nevertheless, you get kind of a topological line that implements,

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Courant Events Right: The symmetries for you.

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Courant Events Right: But I wanted to sort of relate it a bit to the question Nikita asked yesterday, is that conformal field theory on ADS is… if I might take my theory conformal, then it is a CFC on a half space.

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Courant Events Right: But on top of it, you can get some interesting, new phases of theories if you put, if you put it on ADS space.

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Courant Events Right: Related to discussion, yesterday, okay.

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Courant Events Right: Just advertise this paper. Let me, let me, let me in the remaining 3 minutes, go back to my, to my young mules story. Okay, so that will be very impressionistic.

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Courant Events Right: I apologize for that, but okay, but we'll spend more time on discussing basic things. Just bullet points, okay? So, I get a new dimensional coupling now, which is my QCD scale, in the units of ADS radius. At large value, I go back to flat space, that's what I care about.

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Courant Events Right: And its small radius theory is weakly coupled, so young wheels, all, low, weak, strongly coupled modes get con… Get, get cut off.

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Courant Events Right: And it's important, also, to use Neumann boundary conditions for gauge fields.

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Courant Events Right: So, now I have my young meals in the bulk, I have some formally defined, conformal theory on the boundary, and the last ingredient that I need to add is I add my flux tube, okay? So now I have a Wilson line that lives on the boundary.

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Courant Events Right: And it creates a flux tube that spans two-dimensional anti-decitor space, sort of going radially through my anti-decitor bulk. And the crucial point here is that an antidisitor space

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Courant Events Right: confinement, according to my definition, which is formation of these flux tubes, it appears already at weak coupling, because the field lines, even though they don't interact with themselves at weak coupling, they get sort of confined to each other by ADS.

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Courant Events Right: potential.

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Courant Events Right: So even at weak coupling, I get some objects that I can study, and that when I increase the radius.

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Courant Events Right: What we believe continuously, becomes the confining flux tube.

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Courant Events Right: So, this… field…

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Courant Events Right: world sheet field X, that was a member of my two-dimensional quantum field theory, sort of the variable in my two-dimensional quantum field theory becomes some particular operator in this boundary conformal field theory, and what I used to call energies on a circle, now the analogy is the dimensions, or energies, of the global ADS Hamiltonian.

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Courant Events Right: as a function of ADS radius. So this is ADS radius, and that was the radius of the circle. And now, this thing is the following. I can now go to weaker… small radius, and do calculations in perturbative Young-Neil's theory, and I can go to large radius.

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Courant Events Right: and do calculations in my effective string theory. So the novelty is that I can continuously interpolate between my effective string theory calculations on this side, using whatever this action, and the weakly coupled Young Mills calculation from first principles.

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Courant Events Right: Let me… okay, that's how calculations look like. It's how every slide would look like if that was a regular talk. Maybe just one thing that it's not completely… there is… there is this long formula that shows something how these calculations are done. So this is some three-loop calculations in effective field theory. In ADS, it would be prohibitively complicated by direct calculation.

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Courant Events Right: Instead, you use something called residentiality antsatz. You come up with some set of generalized poly logarithms that you argue should be multiplied by some rational functions

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Courant Events Right: And then you bootstrap your answer, and this appearance of this Zeta of 3, this Riemann Zeta of 3, for example, is not a coincidence. The idea is that we know at which organ perturbation theory which Riemann Zeta functions will be showing up, so there is some probably interesting number theory going on.

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Courant Events Right: Anyway, that… okay, that's really the… the punchline, instead of staring at these terrible formulas, just to visualize the result.

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Courant Events Right: I show some Paderi summations. So I have some calculations at weak coupling, have some calculations at strong coupling, and then I assume that this is some analytic function of the coupling, so I just do what's called double-sided Padre series, and I kind of try to predict what the answer would be in the middle. So that is not something rigorous, that more or less to guide the AI and to give some impression of where

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Courant Events Right: where we stand. And the belief is that, okay, there are some ways to argue why, I can explain that these curves for any value of the radius, they sort of converge to a few percent everywhere in the middle, and the belief is that the further we do the calculations, the better they converge. Okay, so these are my conclusions.

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Courant Events Right: So I said that flux tubes are very important for confinement.

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Courant Events Right: We approached them in four different ways, right? So on this cylinder, we used effective string theory, and we used lattice, and then we once went to, hyperbolic space.

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Courant Events Right: And again, there we did effective string theory, although very quickly, and then we also did perturbative young Niels theory. And this last approach allows us to sort of match

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Courant Events Right: directly, at least in terms of this interpolation, the world sheet degrees of freedom, and the gauge theory. Degrees of freedom, as sort of I try to indicate with this plot.

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Courant Events Right: And this was the plot for the cylinder, which worked nice at large radius, but we sort of did not know what to do at small radius, while this plot goes all the way to zero.

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Courant Events Right: And sort of the vision here is that for, at least for large angular mills, single flux tube is some well-defined two-dimensional theory, some, some theory of surfaces.

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Courant Events Right: And… and the question is, is there some intrinsic two-dimensional definition? Because so far, I can only define it by embedding in some much bigger theory, like four-dimensional Young Mills, but that looks like an overkill, because I only look at a small sector of that full theory.

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Courant Events Right: But that, to me, the fundamental question here is that, is there a way to define this two-dimensional theory in some intrinsic two-dimensional terms without using the full

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Courant Events Right: set of degrees of freedom of young or serious set.

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Courant Events Right: That's it.

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Courant Events Right: Even though I'm Christianistic, I think many of these ideas connect the goals to our operation.

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Courant Events Right: We think that there's a certain number of points that, like, a bit of… What?

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Courant Events Right: What's… Aspect of all of these modules. Particularly recommend to try to study.

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Courant Events Right: Well, good question, I mean,

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Courant Events Right: I think that… I think there are… yeah, there are many connections to mathematics, but to me, the most, interesting for us are these… these recent developments, right? This connection to… so we start getting some weakly coupled, so, weekly coupled in terms of Jangil's fields, calculation of emergence.

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Courant Events Right: Of these, of these surfaces, right? So, so that's… that's what I try to… the big question here is to… to try to understand, maybe to try to prove rigorously, right, why…

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Courant Events Right: That these surfaces make sense as isolated objects.

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Courant Events Right: But, okay, this is not, this is not a very, a very, you know, a simple and concrete mathematical question, but there's an idea that is, to me, that is the most important, okay. As some intermediate steps, okay, say this, as I said, this…

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Courant Events Right: these things, I mean, for sure, there is… well, people… similar calculations that people do for amplitudes and flat space, they definitely have some connections to mathematics, some cluster algebras, and things like this, things from number theory, and probably that can all be

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Courant Events Right: also use here, and probably help us a lot to do calculations to… to high order, okay? Then, as I mentioned, we also do some, some Betancas calculations that I didn't talk to, but okay, that connects to this integrability story. So I think there are a few.

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Courant Events Right: Well, any question, how about you?

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Courant Events Right: So, I'm trying to understand why you expect there to be no phase transition on… in the ADS radius, because I'm thinking if the radius is very small, you're basically just

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Courant Events Right: Or that's a…

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Courant Events Right: This is very big, the theory will think for a long time, it's on Euclidean space, so… Yeah, yeah, no, that is a good question, and okay, first thing is that if you put Dirichlet boundary conditions.

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Courant Events Right: There is certainly a phase transition, and that's also people's study. The claim is that there is no phase transition with, with Norman boundary conditions.

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Courant Events Right: And there is no proof so far. More or less, what we are… we're doing this calculation is that everything is consistent so far, with there being, like, a smooth interpolation.

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Courant Events Right: And intuitive picture is what I said, that you get all the features of your strongly coupled theory, you kind of seeds for all the features of your strongly coupled theory, for example, this Wilson line formation, because no, it's not some kind of order parameter. But anyway, there is an interesting question. Can you define some kind of order parameter for confinement in this ADS setting?

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Courant Events Right: Nobody knows. Maybe also a question for a mathematician or something, because I will try attempting to make the statement rigorous, that there is no phase transition, right? That, I think, is another interesting thing. Can you rigorously prove that there is no phase transition as a function of radius?

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Courant Events Right: And so far, this is a conjecture, and it's an intuitive picture that you have the seeds for confinement. And maybe one more thing is that you see that even though you go from weak coupling to strong coupling, these anomalous dimensions, they all change by order 1.

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Courant Events Right: So the picture… good analogy here is, like, epsilon expansion will sufficient point. So you go from 4 dimensions to 3 dimensions, and you could have said, okay, maybe there is something crazy going on, but what's going on is, like, you get some auto-one anomalous dimensions, and if you, you know, compute too high in a foreign perturbation theory, you get some

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Courant Events Right: decent precision after some, you know, parallel estimation. So that's roughly the vision here, that you get all ingredients already at recoupling.

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Courant Events Right: Let's just… one more question a bit from this side, and then we… Very straightforward.

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Courant Events Right: But, maybe a more conceptual question. So, on one hand, you keep talking about, flux tubes. On the other hand, you work in periodic mills. So, what do the strings represent for you? Are they bubbles, or…

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Courant Events Right: well, there are also blue balls, but these, these objects, in some sense, I… when I introduce these Wilson lines, it's like I do have non-dynamical, quirks. Like, very heavy, if you want, non-dynamical quarks, it's sores for me.

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Courant Events Right: this train.

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Courant Events Right: Okay, good. Okay, Michael, yeah, sure. We're good.

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Courant Events Right: You too much, yeah.

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Courant Events Right: Okay, some of the stochastic geometry at play can be glemed by looking at the simplified setup of lattice gauge Z2H theory.

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Courant Events Right: And in that case, a reasonable observable is equal to the probability that your system of random floating surfaces

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Courant Events Right: no boundary, plus a boundary imposed in the traditional surveys to the prescribed boundary response.

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Courant Events Right: No delete.

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Courant Events Right: It's just fine.

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Courant Events Right: And for that, you can actually… first of all, you can…

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Courant Events Right: And he also told in that case that there is a non-bin-based condition.

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Courant Events Right: Addition from the surface of the perimeter of the conviction.

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Courant Events Right: location.

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Courant Events Right: I don't think we should see in that case to be difficult.

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Courant Events Right: So that's geometric a random packet.

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Courant Events Right: The tools are around our minds, but the should be some kind of information-pack model, and you need the application, the general breathing model.

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Courant Events Right: the environment.

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Courant Events Right: Good.

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Courant Events Right: there's a lot of insight that came about this capacity currently awaits a lot of sense.

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Courant Events Right: Yeah, I, I completely agree. And the…

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Courant Events Right: The, sort of, the problem with this

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Courant Events Right: It's also true that for Z2 gauge theory, there's some sort of effective spin that we can write down. The problem is that that surface

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Courant Events Right: is not an isolated two-dimensional theory, because there is no analog of this large element for C2 gauge theory, meaning that I start going to have some surface.

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Courant Events Right: And then, if it fluctuates, more and more, it starts emitting things.

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Courant Events Right: into the bulk of gauge theory. Okay, that is the physicist picture, and just kind of the last thing that there is no well-defined two-dimensional theory, isolated from the bulk, that defines the surface.

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Courant Events Right: If there was some discrete gauge theory, which can analyze all this large N… Yeah, yeah, yeah.

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Courant Events Right: That's on any platform out there.

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Courant Events Right: Yes, exactly.

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Courant Events Right: Exactly. In senior wage theory is not, and sort of in generic angular series also not.

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Courant Events Right: But in this large and unmil theory, so that was a crucial point, there exists a theory of single surface, so that is my claim or conjecture, and why I motivate to study is because theory is simpler than to understand, you know, this general bulbic geometry. And then, once I understand that, we can do some sort of expansion.

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Courant Events Right: to include this more complicated geometres. That's kind of a program, but a good one does not have to do it. The thing you bring it out, we use it, we definitely think about digital gauge theory, but it's kind of simpler, but also harder in some sense.

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Courant Events Right: good old braids.