Зохар Комаргодский - Linear defects in QFT

[00:00:09]Yeah.

[00:00:10]Yes, yeah. No, you have to start the translation so we can start.

[00:00:31]Okay.

[00:00:59]from the last couple of years.

[00:01:02]I've been very interested in this subject for the last 4 years or so. So, we started looking into it around 2002.

[00:01:11]And um Uh there is a lot of parallel works. So, most recently with Fielder, Papo, and Brandon.

[00:01:21]They were both in Stony Brook. I have some slightly earlier work with the group of Maryland and condensed matter, my son Amber Cashley, Chris Fetchesin, and my student Siwei Zhong.

[00:01:34]And then um a little bit earlier work with Cuomo, who is now professor in Sissa, Mark Mezei, who is now in Oxford, and Aviad Livne Moshe, who is still here at Stony Brook.

[00:01:47]Okay, so this is more or less uh I'm going to try to summarize some of the main points of this the over the last couple of years.

[00:02:00]So, historically uh the subject of defects in quantum field theory was very, very important historically.

[00:02:09]So, in fact, Wilson uh invent Well, Wilson discovered the renormalization group by trying to solve the Kondo problem.

[00:02:18]So, he just tried to solve the Kondo problem and he discovered this whole idea of renormalization.

[00:02:23]And the Kondo problem is a So, it's is a very special type of defect in quantum field theory. It's a line defect um in the metal.

[00:02:32]In fact, people later discovered that the Kondo problem is exactly solvable and that also opened the way to integrability in many body systems. That was done by Nathan Andrei, Itzhak Wiegmann.

[00:02:46]And of course, uh the subject of conformal symmetry uh came from the Kondo problem in partially from the Kondo problem because they discovered this over-screened Kondo defects.

[00:02:57]So, the subject of line defects in two dimensions has been historically very important.

[00:03:02]And actually almost nothing is known uh well much less is known about extended operators, extended observables um or line also line defects in higher dimensions. So, in 2 + 1, 3 + 1 like the extended quantum observables or extended operators or defects have been much less looked into in higher dimensions because it's much more difficult.

[00:03:32]So, I'm just going to show you some uh results from the last couple of years about this subject.

[00:03:40]So, the setup is going to be the following. It's a very It's a simplified setup, but it's still very interesting for uh experiment and some simulations and also for theory.

[00:03:51]So, the setup is very simple. We have some gapless theory in this space-time dimensions. It's some conformal theory in the dimension in this space-time dimensions.

[00:04:01]And then we have a one-dimensional defect. So, the defect could be oriented in time, then you can think about it as an impurity just modifies the Hamiltonian like in the Kondo problem, or the defect could be oriented in space.

[00:04:14]In that case, you can think about it as an extended quantum operator.

[00:04:19]Okay. So, we have this uh setup and uh for instance, in condensed matter, we could think about some bulk gapless system and we add some impurity. Okay? And the impurity couples to the bulk.

[00:04:34]Maybe there are some operators.

[00:04:36]Maybe some new degrees of freedom sit here the impurity and these degrees of freedom couple to the bulk and they could couple via some kind of Heisenberg-type interaction or something else. Okay?

[00:04:48]This is the rough uh setup.

[00:04:50]And people have looked into very many, many problems in this class of problems.

[00:04:56]Uh there are Wilson loops, which are diagnosis of confinement, Hoof loops, uh symmetry defects, symmetry-protected topological uh phases of defects, anyons, spinning fields, many, many, it's a zoo of various possibilities. I'll uh give you some examples later. I want to first show you the theory of uh this extended observables.

[00:05:21]So, today I will uh tell you about general results about renormalization group on extended operators and then I give you three examples. I'll give you three examples where uh we can have nice applications of the general theoretical results.

[00:05:40]Okay?

[00:05:41]Uh are there any questions about the introduction?

[00:05:45]Just stop me if there is any any comment or question.

[00:05:51]Okay, now let's discuss the setup mathematically.

[00:05:57]We discuss a line defect in a D-dimensional in a D space-time dimensional conformal field theory.

[00:06:06]Now, a line defect has to break some of the conformal symmetries. So, the full system has some big conformal group.

[00:06:14]But the line defect has to break it to a subgroup. And so, if the line defect is conformal, so if the line defect is tuned to a conformal fixed point, then the line defect preserves essentially SL2R.

[00:06:28]And SLD-1 is a group of transverse rotations around the defect. So, if you go back to the picture, we have conformal symmetry on the line, SL2R, and transverse rotations.

[00:06:43]This is the setup of a conformal line defect. Now, a line defect doesn't have to be conformal. Even if the bulk is conformal, there could be some renormalization group flow on the line.

[00:06:54]And then the symmetry that's preserved on the line is my is smaller. It only has time translations and transverse rotations.

[00:07:02]Okay? So, this is the general case. This is the conformal case.

[00:07:07]Okay.

[00:07:08]Now, I will tell you first about a very interesting observable that you can define, which we sometimes call quantum defect entropy.

[00:07:18]So, what is it trying to solve?

[00:07:21]When let me just go back to the Kondo example, I'll try to explain the motivation for this quantum defect entropy. It turns out to be a very important object. So, when we couple some impurity to a magnet, we can ask how many degrees of freedom does the impurity support?

[00:07:39]So, we can try to ask what is the zero zero temperature entropy of this impurity. How many degrees of freedom does it host?

[00:07:48]And this has to be defined non-perturbatively. Of Of course, it's easy to define if the defect is decoupled. It's just the number of quantum mechanical degrees of freedom on the line.

[00:07:59]But I'll give you a definition that's non-perturbative and it has very interesting properties.

[00:08:04]So, the definition is to take your line defect, make it into a circle, instead of an infinite line, make it into a circle.

[00:08:13]Calculate the expectation value of the circle.

[00:08:16]That's the logarithm of the expectation value of the circle.

[00:08:19]And then apply the differential operator 1 - R d by d R, where R is the radius of the circle. I forgot to put it in the picture.

[00:08:28]R is the radius of the circle.

[00:08:31]It turns out that this is a completely scheme-independent calculation. So, even you might think that there are divergences in this calculations.

[00:08:41]But once you apply this differential operator, all the divergences cancel out.

[00:08:45]And what you obtain is a kind of a non-perturbative definition for defect entropy, quantum defect entropy, that we call sometimes little s or log g. These are just two notations for this object. Little s or log g.

[00:09:01]And this gives you a a way to count the number of degrees of freedom on any extended operator, extended line operator, in any conformal field theory in d dimensions.

[00:09:12]Okay? Now, the reason that this observable is interesting is because it will obey rigorous results about the renormalization group flow.

[00:09:20]Very similar to the results of Zamolodchikov about c-function. It will turn out that this quantum defect entropy obeys similar results.

[00:09:30]But this is true in any dimension, not just in two dimensions. Here, the CFT is in any number of dimensions, D.

[00:09:39]>> Sorry, can I ask who who who's running here? I mean, the radius run runs the scale or what changes?

[00:09:48]>> excellent question. So, there are two situations. One is that you already took your extended operator to be at the fixed point.

[00:09:56]So, your extended operator is already this, uh the preserving this symmetry. In that case, this is independent of R and it's just a number. It's the quantum entropy of a conformal defect. Okay. If on the other hand your there is some running on the line defect, there is some RG evolution on the running uh RG evolution on the line defect, then this becomes a function of R, a non-trivial function of R, and I will now you show you the outline of a proof that this function is monotonic, like Zamolodchikov's function.

[00:10:31]So, I will show you that if there is renormalization group on a extended operator, it turns out that this is a monotonic function of R.

[00:10:40]>> Mhm. But but but is it possible to have cusps on this line defect? Usually there are additional renormalization if there are some cusps.

[00:10:49]>> Yeah.

[00:10:51]Um yeah, there is a whole theory of what happens if it's not a circle, but it has some cusp.

[00:10:58]Uh I was not planning to talk about it today, but there is a whole theory of how to deal with cusp singularities.

[00:11:04]>> Mhm.

[00:11:05]>> Okay. So, what is the intuition of making a circle of this extended operator? The intuition was of course in quantum mechanics to quantum degrees of freedom to count degrees of freedom in quantum mechanics, we make the space-time into a circle, well, time into a circle in quantum mechanics, and that just gives us a trace.

[00:11:22]Here, we're making a circle, but the bulk is not empty. The bulk is some gapless CFT.

[00:11:28]So, it's some kind of a way to count the number of degrees of freedom on this extended operator even if the bulk is not empty.

[00:11:37]Okay. So, this is this observable uh it's scheme independent observable.

[00:11:43]It's intrinsic.

[00:11:45]And uh it has two names. One is quantum entropy. Sometimes people call it S.

[00:11:50]Sometimes people call it log G.

[00:11:54]Okay.

[00:11:55]Now, it's very hard to measure this in experiment because in experiment you cannot take the Kondo impurity and make space-time into a circle. So, to measure this in an experiment, you have to do something a little bit more sophisticated. Maybe I'll go into that later today.

[00:12:12]So, it's difficult to measure in experiment, but in principle it's well defined.

[00:12:17]Another thing is that in the literature on Chern-Simons theory, people often do that. They take Wilson lines and make them into a circle. That defines a quantum dimension of an anyon.

[00:12:28]So, if you look at the literature on Chern-Simons theory, this is a very common thing. People calculate these expectation values and they call them quantum dimensions of anyons.

[00:12:37]But the quantum dimension of anyon has a very interesting constraint for a unitary Chern-Simons theory that this G has to be bigger than one.

[00:12:46]And therefore log G has to be positive and therefore S has to be positive.

[00:12:51]That's a property on topological in topological theories.

[00:12:55]Today, I'll show you that when the bulk is actually gapless, like the bulk is massless boson or massless fermion, this is doesn't have to be true.

[00:13:04]Sometimes the actual number of degrees of freedom on this kind of extended operators is actually negative.

[00:13:10]What does it mean to have a negative number of degrees of freedom? It means that kind of this line defect kind of pushes away stuff from the bulk.

[00:13:19]That's possible when the bulk is gapless.

[00:13:22]So, in our case, G bigger than one will not always be true. There will be example where G is smaller than one.

[00:13:28]Unlike topological theories.

[00:13:31]Okay, now there is another thing that you need to know about what are the operators.

[00:13:37]So, first of all, there are bulk operators. These are the ordinary operators in conformal field theory that we are familiar with.

[00:13:44]But also, now there are operators that live only on the line defect.

[00:13:48]I'll show you a picture in the next slide.

[00:13:50]And these two sets of operators have nothing to do with each other.

[00:13:54]So, there is a nice picture here that the ChatGPT gave me.

[00:13:59]So, there are operators that live in the bulk.

[00:14:01]And then there is the line defect.

[00:14:03]And then there are operators that live on the defect. And in principle, these two sets of operators have nothing to do with each other.

[00:14:10]Except that there is a a new kind of operator product expansion where you can take a bulk operator and expand it in terms of defect operators.

[00:14:20]So, there is a formal Taylor series where you take a bulk operator and do a Taylor series expansion in terms of defect operators.

[00:14:28]Okay, so when we look at critical phenomena in the presence of line defects, we have to calculate critical exponents in the bulk, which we know from statistical mechanics. But now we have a new set of critical exponents.

[00:14:41]These are critical exponents that live exactly on the defect.

[00:14:45]Okay, they're specific to the defect.

[00:14:47]They They correspond to the you know, like susceptibility in the presence of a defect. All these kind of things will be characterized by new critical exponents, which are different from the bulk critical exponents.

[00:15:02]Okay, so this is the general setup.

[00:15:05]And also, there is the notion of screen screening screening, uh which means that you may put some line defect, but maybe at long distances it becomes completely trivial.

[00:15:18]At long distances it could just be the unit unit operator.

[00:15:21]So, then we say that the line defect is transparent or trivial at long distances. Okay? So, once one central question in this business is what are the space of conformal line defects, non-trivial conformal line defects, which are not trivial at long distances.

[00:15:39]Okay.

[00:15:40]Now, I want to go back to the definition of this quantum entropy and show you what kind of theorem you can prove about it.

[00:15:48]Okay.

[00:15:50]So, imagine that there is a renormalization group flow on a line defect. Let's say we have a line defect in the ultraviolet, at short distances, and we deform it by some relevant perturbation.

[00:16:00]This is a deformation with some defect operator.

[00:16:03]And then we have a renormalization group flow on the line defect. And as I said, there is a scheme-independent quantity that you can calculate, which is the quantum defect entropy.

[00:16:14]And it becomes a non-trivial function of R, exactly like we just discussed. It becomes a non-trivial function of R, and at short distances, it probes the ultraviolet number of degrees of freedom. Sorry, it's the first line here. That's the short distance limit, R going to zero.

[00:16:31]And the long distance limit, it probes the number of effective degrees of freedom on the defect at long distances.

[00:16:37]So, there is some function that interpolates between these two numbers.

[00:16:41]And what you can prove is that this function is monotonically decreasing.

[00:16:47]So, a basic result in this field that is not trivial to prove, but okay, I'll I'll tell you what the general idea is.

[00:16:57]It's kind of It's a little bit more complicated than Zamolodchikov's proof.

[00:17:01]Is that you try to calculate the gradient of this quantum defect entropy, and you find a way to express it as a two-point function, which is manifestly positive. This is just energy energy correlator on the defect.

[00:17:16]And then, it's multiplied by some kernel, which is 1 minus cosine, which is also non-negative function.

[00:17:23]So, therefore, you obtain that the gradient is non-negative, sorry, not positive. The gradient is negative, and therefore, the number of degrees of freedom on extended operators or line defects cannot increase. This is a rigorous result that's valid in any RG renormalization group flow on line defects, and I'll show you some applications today.

[00:17:46]Are there any questions about the uh statement here?

[00:17:50]>> So, the unitarity is assumed, right?

[00:17:53]>> Yes, we're very very good. We assume unitarity because we assume that the two-point >> Yes, yeah.

[00:18:00]>> of energy energy operators on the defect is positive, non-negative.

[00:18:06]So, I'll show Yeah.

[00:18:08]>> When we consider defect, usually it's necessary to have a kind of matching between organization group flow on the defect and in the bulk. What is the analog of this matching in this case?

[00:18:22]>> Yeah, so here, the whole lecture today, the whole talk today, is in the setup that the bulk is already tuned to fixed point, and the renormalization group is only happening on the line.

[00:18:35]This is the setup today.

[00:18:37]Uh this is a very helpful way to separate the two phenomena because they're independent of each other.

[00:18:43]You can have renormalization group flow on the line, and the bulk is already tuned to a fixed point, you know? Like somebody tuned the couplings in the bulk to fixed point, and the renormalization group flow happens on the line.

[00:18:54]This is the setup that we have renormalization group flow on the line.

[00:19:00]This is a very good simplifying situation that allows you to separate the two phenomena.

[00:19:05]>> Mhm.

[00:19:07]But but there are some situations, for instance, when we consider some surface defects, so usually some it's necessary to consider matching of beta function in on the defect and in the bulk. And so here we have no such matching at all, yeah? I see.

[00:19:26]>> Yeah, here the bulk is CFT. It has no scale.

[00:19:30]And there is some beta function on the defect.

[00:19:34]>> Mhm.

[00:19:35]>> And the discovery here is that in Zamolodchikov's case the central charge was important to track the renormalization group flow in the bulk.

[00:19:44]Here it turns out that it's quantum entropy.

[00:19:47]This S plays a similar role and it's monotonically decreasing.

[00:19:54]>> Mhm.

[00:19:54]>> And I'll show you one interesting application of this today for magnets.

[00:20:01]Three-dimensional magnets.

[00:20:03]Just ferromagnets.

[00:20:05]Okay?

[00:20:09]Next.

[00:20:10]So there are similar results about Kondo by Affleck, Ludwig, Friedan, Konechny.

[00:20:18]This is a generalization to any number of dimensions. Here we're assuming a line defect in any number of dimensions, including three dimensions, four dimensions.

[00:20:26]And you can think about this result as a generalization of the classic results by Affleck, Ludwig, Friedan, Konechny.

[00:20:36]Okay? That's the context.

[00:20:39]Also, you can wonder if there is some connection to information theory.

[00:20:44]In the case of the central charge of Zamolodchikov Casini and Huerta showed that the central charge theorem of Zamolodchikov is closely connected to some theorems in quantum information theory, which are the subadditivity of the entropy of the entanglement entropy.

[00:20:59]Here it's more complicated. It turns out that the entanglement entropy of an impurity is not quite G. It's a little more complicated, but still Cassini, Salazar Landea and Toroba were able last year uh to understand the connection of this theorem that I told you with quantum information. It's more complicated than just the subadditivity of the entanglement entropy in the presence of line defects. It's a more complicated story, but it still works.

[00:21:29]Okay.

[00:21:30]How do How is this theorem proven?

[00:21:33]Well, I don't have to explain the full proof, but basically, um you look carefully at the word identities on the defect, and um it's The proof is really complicated, but basically you try to look at careful You try This is the line defect, which is a circle, and you wrap it like you put it inside some kind of donut, where the donut is a particular integral of the bulk energy momentum tensor. So, you use the fact that the bulk is tuned to a fixed point, and then you wrap the line defect with a very particular donut, and then you make the donut very big, and you can obtain a lot of identities.

[00:22:16]You can obtain a lot of identities for correlation functions of energy, energy, energy, energy, energy on the line defects, and one of those identities turns out to be exactly this gradient formula that I showed you.

[00:22:29]I don't know what the other identities are, or whether they're useful, but one obtains many, many new identities.

[00:22:37]Okay, so that's roughly speaking how the proof goes. I don't have time to explain it in detail.

[00:22:42]>> Uh sorry. Uh uh and what will happen if you consider this the torus knot, for instance, not a circle, but uh torus knotting in this case?

[00:22:53]>> Uh well, you're asking what if I didn't take the circle here?

[00:22:57]>> Yeah, yeah.

[00:22:58]Just consider the knot knot that uh situation, and for the torus case it's just torus not >> Uh, yeah. I don't think anybody looked into it.

[00:23:11]I I don't know if there is a interesting result. That that case that I showed you leads to an interesting result because it gives inequality.

[00:23:19]I don't know if you can obtain something in other cases.

[00:23:23]Mhm.

[00:23:24]Yeah. And I'll tell you one last general theorem about line defects.

[00:23:31]For local operators uh, a very useful thing is to ask how they transform under symmetries, of course. Yeah? Because local operators are in representations of the symmetry of the system.

[00:23:45]So, you can ask what about extended operators like line defects? Are line defects in representations of the symmetry?

[00:23:52]And it turns out that the answer is no.

[00:23:54]No. Line defects are not in a representation of the symmetry, but they're in a representation of a fractionalized They're They're a fractionalized representation of the symmetry. What does the word fractionalized mean?

[00:24:06]It means that if you act with the symmetry on line defects, you might get a central extension.

[00:24:13]So, only a central extension of the symmetry acts on line defects. This is this phase. It's the cocycle of the central extension.

[00:24:20]So, unlike local operators, the extended operators are in represent are in more complicated representations of the symmetry. They might involve some cocycles.

[00:24:32]And that also leads to very important constraints on the renormalization group.

[00:24:36]Because the renormalization group cannot change these cocycles. And that leads to some constraints on long distance dynamics. Even in the quantum problem, people have found new applications for the long distance dynamics using these cocycles.

[00:24:50]I will probably not have a lot of time to explain this today, but I just wanted to say that this is another non-perturbative constraint on extended operators having to do with their funny representation theory under global symmetries.

[00:25:10]Okay?

[00:25:11]So, uh well, just because before we begin to study examples, I will say that when you have an extended operator like a line defect, basically the question is is it screened? Is it just completely trivial at long distances and you don't see that there was an impurity or is it a conformal non-trivial fixed point? These are the two extreme cases.

[00:25:36]In between, it could be that the line defects become topological and various other things. Okay? So, there is like there are all these possibilities for long distance dynamics and well, that's what we care about, right?

[00:25:51]Long distance limit. So, uh when I we discuss some examples now, that's what you have to keep in mind that uh we want to understand the long distance limit of some interesting line defects.

[00:26:06]Okay. So, now let's go to examples.

[00:26:11]Uh I have I'll skip four slides now, which are not very important.

[00:26:17]I'll go right into example. So, you know, like the simplest example above two dimensions is the ON model.

[00:26:25]Wilson-Fisher model because we can do many calculations and it's a nice way to warm up to and also it's very important for experiments, so that's a very nice example that this was the first example we looked at uh 3 years 3 and 1/2 years ago.

[00:26:42]So, we we just tuned the bulk to the critical point of the ON model, that's the Wilson-Fisher fixed point.

[00:26:49]And imagine here just imagine we're in three dimensions. So, two plus one dimensions, three dimen- three classical dimensions.

[00:26:57]And you can implement it in statistical physics on a computer very easily.

[00:27:01]And the line defect is very simple. We just turn on a magnetic field.

[00:27:05]But usually we turn on a magnetic field everywhere in space, right? And we study the phase diagram of these magnets as a function of the magnetic field. Here, we will just turn on a magnetic field at the origin.

[00:27:17]So, you see there is no we just integrate over time.

[00:27:20]And this is phi one as a function of time at x equals to zero. I should have written here, comma uh Sorry, I should have written that.

[00:27:29]I'll put it in the chat.

[00:27:34]It should be phi one as a function of t at x equals to zero.

[00:27:48]Okay, so we're studying the phases of a magnet when we turn on a magnetic field not everywhere in space.

[00:27:54]This problem of turning a magnetic field everywhere in space has been beaten to death. It was already studied by Landau probably 80 years ago if not more. Here, we're discussing a new problem where we turn on a magnetic field in some region of space.

[00:28:11]So, let me just quickly go back to the picture at the beginning.

[00:28:15]So, we're just turning on a magnetic field in some region. Let's say over this couple of sites or one site or or eight sites, it doesn't matter. We just turn on a localized magnetic field and we will try to discuss the phase diagram of this.

[00:28:30]What is the long distance limit?

[00:28:32]Okay? So, you're asking what what is the long distance limit if we're turning on a magnetic field in some localized region of space.

[00:28:42]Okay? This is a nice question.

[00:28:45]So, here this is actually a very nice application of the gradient theorem about the gradient of the quantum entropy.

[00:28:52]If you Let me show you how it works in this example.

[00:28:59]In this example uh at very short distances, the magnetic field goes to zero because the magnetic field is a relevant is a relevant perturbation. at short distances. So at short distances, log of g the first line here is just zero. Because at short distances, there is no impurity.

[00:29:21]The long distance limit is very complicated. We cannot solve it exactly, but from the gradient theorem, we learn that the long distance limit has to have a negative log g.

[00:29:31]And therefore, the long distance limit cannot be trivial.

[00:29:35]So one immediate conclusion here one immediate conclusion that follows from the g theorem is that the long distance limit cannot be trivial.

[00:29:44]And it cannot be gapped, okay? So the long distance limit cannot make this line this magnetic field go away. Most likely it becomes a conformal field theory.

[00:29:54]Okay, and people have started looking into it with Monte Carlo. And for instance, this with Monte Carlo, you can try to look in this into this problem.

[00:30:04]Now I'll tell you what can you do analytically. I'll show you some analytic results.

[00:30:09]Uh first of all, you can do something very simple.

[00:30:12]You can just do perturbation theory. a la epsilon expansion.

[00:30:17]And in the epsilon expansion, you see that indeed this magnetic field goes to a conformal line operator.

[00:30:22]So it's an interesting conformal line operator.

[00:30:25]And this is valid near 3.99 dimensions, like in Wilson-Fisher. But what about the interesting case of three space-time dimensions, 2 + 1?

[00:30:40]So in 2 + 1 dimensions, it turns out that there is a very nice large N expansion.

[00:30:46]You can already see that there is a very nice large N limit in the epsilon expansion, so they commute.

[00:30:52]But you can set you can directly go to 2+1 dimensions and perform a large N expansion.

[00:31:01]Let me just explain how it works.

[00:31:03]So, in the epsilon expansion is very obvious. You just follow your nose. It's perturbation theory. But more generally, you can do a large N expansion directly in three classical dimensions. So, we're discussing this defect, okay? We're trying to renormalize this line operator where we're integrating phi one, which is the magnetic field in the direction one.

[00:31:27]We're trying to discuss the renormalization of this exponential operator.

[00:31:31]And we're claiming that it has to be a fixed point. There has to be a fixed point of the renormalization group.

[00:31:38]Indeed, it turns out that there is an interesting Hooft coupling in this problem. So, if you take the value of the magnetic field and divide it by square root of N, where N is the ON N of the ON model, it turns out that this is a nice double scaling limit. And in these conventions, lambda will flow to some fixed point. And our job is to find the properties of this fixed point.

[00:32:00]Find the quantum entropy, find the dimensions, and then compare to experiment.

[00:32:07]And it turns out that in the large N limit, indeed, there is a classical saddle point. This problem has a classical saddle point. The problem of localized magnetic field leads to a new classical saddle point, and you can use it to calculate exactly many many observables in the large N limit. In the 't Hooft limit, basically.

[00:32:25]The this problem has kind of a 't Hooft limit.

[00:32:29]So, people have not yet done that completely. We don't have the full renormalization group flow in the large end limit, but people were able to obtain the properties of the fixed point uh in the large end limit. So, in the large end limit, it turns out that you can do some coordinate transformation, map this problem to ADS2 times a circle.

[00:32:56]In D equals 3, this becomes a circle.

[00:32:59]And then you can do a Hubbard-Stratonovich transformation on ADS2 times S1.

[00:33:04]And you can calculate the properties of this line defect by minimizing this non-local action, which is a Schwinger-Dyson type type action. So, you have a trace log of some differential operator plus the corrections from the curvature.

[00:33:20]And then there is some boundary to boundary boundary propagator from the magnetic field.

[00:33:27]And it turns out that this problem is just exactly solvable in the large end limit. You can find the classical saddle points. You can even study fluctuations.

[00:33:35]And you can determine completely the infrared limit of this localized magnetic field.

[00:33:40]Uh let me show you the answer.

[00:33:43]Uh well, let me just give first we there was a lot of progress recently on doing Schwinger-Dyson equations in ADS2.

[00:33:52]So, this kind of this kind of trace log heat kernels in ADS2 were calculated recently by Carmona, Pietro. Yes.

[00:34:00]>> Carmona. Kind of lost to what where it is to when did it appear, basically?

[00:34:07]>> Yeah.

[00:34:09]Yeah, let me go back to explain that.

[00:34:13]Yeah, I'm sorry that I sped up a little bit. I'll explain where this came up from. So, originally the problem is a straight line that represents a constant magnetic field at the fixed point in the infrared uh in the CFT.

[00:34:26]Now, here you can do calculations in flat space can be a little bit difficult. So, there is a small trick to do a coordinate transformation to switch to a new calculation in ADS2. Let me show you how it's done.

[00:34:41]So, I'll go to the chat.

[00:34:45]So, a you write the metric of flat space in a the following convention.

[00:34:53]You say that there is a coordinate a Z Let's say a coordinate T along the line.

[00:34:59]And then you put radial coordinates.

[00:35:03]You put radial coordinates a a transverse.

[00:35:09]So, you take this metric.

[00:35:11]This metric you just have T is along the line and the transverse directions are in polar coordinates.

[00:35:18]Okay? And now you do a conformal transformation where you pull out an r squared. So, you just write it as r squared times dt squared plus the plus the r squared divided by r squared.

[00:35:36]You just write it like that.

[00:35:40]a You write it like this.

[00:35:46]Okay? And then you use the fact that the you assume that the magnetic field on the line has already flowed to the fixed point and the bulk is already the fixed point. So, therefore the system has wild invariance, conformal invariance, and you can get rid of the prefactor.

[00:36:02]And that is And then you obtain exactly ADS2 times S1 because you can get rid of the prefactor. So, we identify the first two terms are as ADS2 and the last term as S1.

[00:36:16]And this is just a trick that allows you to do calculations in ADS2 times S1 instead of in flat space.

[00:36:25]We assume that the magnetic field got to the fixed point, then we do a coordinate transformation to ADS2 * S1, and then we solve the Schrdinger-Dyson equations in ADS2 * S1.

[00:36:35]I don't know how to do it otherwise.

[00:36:36]It's just that technically the most easy way to do it.

[00:36:40]>> I'm mainly doing calculations in flat space should be easier than in curved space, no? I mean >> Yes, yes, 100%. But you see, in flat space uh we have to tell the equations that there is a magnetic field along the line defect, right?

[00:36:57]So, we're not just doing calculations in ordinary flat space. We also have some singularity on the line defect. Now, what what does the coordinate transformation achieve?

[00:37:07]What the coordinate transformation achieves is that it pulls If you look at the coordinate transformation that I just wrote, it puts the line defect at the boundary of a Poincare disk.

[00:37:18]It's at the boundary of the Poincare disk.

[00:37:20]>> Mhm. Mhm.

[00:37:21]>> And there is, you know, 30 years of literature of how to quantize systems where there is some funny boundary conditions at the boundary of the Poincare disk because of the ADS/CFT correspondence, people developed a lot of technology of how to deal with Green's functions with some boundary conditions.

[00:37:39]>> So, you're just saying that the theory with line defect in flat space is equivalent to ADS2 theory with some special boundary conditions.

[00:37:47]>> Only if the line defect is conformal, precisely.

[00:37:50]>> Uh yeah.

[00:37:52]>> So, there is a caveat here, exactly what I said which I I kind of said very quickly.

[00:37:57]I expect that this whole problem of flowing from zero magnetic field at short distances to the fixed point should be solvable in the large N limit, but I don't know how to solve it.

[00:38:09]Instead, we only have a solution of the fixed point itself at long distances because only at the fixed point we can transform to ADS2 * S1.

[00:38:20]>> Mhm.

[00:38:22]>> And then we can use this whole technology of how to write green functions, boundary to boundary, bulk to bulk. And there is a lot of literature on the heat kernel in ADS2. So, we can just leverage You know, you encounter some very funny special functions and people have written identities for them.

[00:38:39]So, there is just a lot of technology to do that.

[00:38:43]And that was our idea. Our main idea was that this problem with the magnetic field should be solvable in the 't Hooft limit, which is large N, basically.

[00:38:52]And then we just leverage this technology to solve it.

[00:38:56]I'll show you the solution now.

[00:39:01]Okay?

[00:39:02]>> Okay.

[00:39:05]>> Any other questions about what the question is?

[00:39:09]Or Okay.

[00:39:12]By the way, this little s is the Hubbard-Stratonovich field.

[00:39:17]>> Mhm.

[00:39:18]>> So, is there a very interesting uh review or reference, if you want to learn about this ADS2 technology, one loop corrections, heat kernel calculations, is this Carmi Di Pietro Komatsu paper.

[00:39:33]We used it very heavily, because they found some very non-trivial identities between hyper geometric functions that we used.

[00:39:42]And this is what you find after a lot of work. It's a lot of work.

[00:39:46]Uh and this is the final result. We were able to calculate exactly the large N predictions of this problem of localized magnetic field. Indeed, it flows to a fixed point.

[00:39:56]And there are two main points that you can qualitative points. One is we were able to calculate exactly the quantum entropy.

[00:40:04]And it's negative. I told you it's also scaling with N, and it's negative. I told you that for any AdS topological theories, it's always positive. But here we see that it could be negative, and it could be arbitrarily negative, in fact.

[00:40:20]Another thing is the scaling dimension of the magnetic field.

[00:40:24]At the UV, this is relevant, and the magnetic field starts to grow towards strong coupling.

[00:40:29]But, as you reach the fixed point, you just settle at the fixed point. So, the perturbation by the magnetic field becomes now irrelevant.

[00:40:37]And, indeed, you obtain a completely stable fixed point. This fixed point is stable. It has no perturbations on the line defect that can take it away.

[00:40:45]And, you get 1.5. So, in the ultraviolet, the scaling dimension is 0.5 on the line defect.

[00:40:53]The This is a defect operator.

[00:40:55]Uh at the infrared, it's 1.5. So, it becomes completely irrelevant.

[00:41:01]And, indeed, this is a stable fixed point.

[00:41:04]Uh so, what's the bottom line? The bottom line is that the claim is that this operator becomes a conformal line operator in the Wilson-Fisher fixed point.

[00:41:13]Uh there is a lot of work on the Wilson-Fisher fixed point. Probably, since it's 6 on Zuber, people worked on it. But, this is a new fact that there is a line operator. It becomes conformal. We can calculate many properties of it.

[00:41:27]And, these are two. Now, let's compare it to simulations and experiment.

[00:41:33]So, there are no experiments at infinite N.

[00:41:37]These are infinite N calculations. But, there are some corrections. We don't know these corrections.

[00:41:43]But, we now have simulations at N equals to 1, which is the Ising case. The Ising magnet, it's the simplest thing.

[00:41:51]And, well, the infinite N result suggests that it should be 1.5, and you get 1.6.

[00:41:58]I think it's good enough.

[00:41:59]Uh and for the G function, uh the quantum dimension was also measured already.

[00:42:05]Uh well, the infinite N result is that it's minus 0.15 if you plug N equals to 1.

[00:42:11]And, in the Ising you find minus 0.22.

[00:42:15]Okay, it's good enough. I don't know.

[00:42:18]>> What what sorry, what what kind of experiment is that kind of what is the >> This is an exact quantum simulation.

[00:42:25]It's a quantum simulation. It's not an experiment.

[00:42:28]>> I mean but it's kind of lattice lattice version of the theory what >> No, it's not a lattice version of the theory. So what people have done to perform this simulation is that well, here we're discussing the problem of putting localized magnetic field in Ising or some Wilson-Fisher fixed point.

[00:42:48]So what people do is that they find they take a system of electrons in a magnetic field and this has various quantum Hall states.

[00:42:57]But sometimes a phase transition between two quantum Hall states can be a Wilson-Fisher transition.

[00:43:03]So the simulation that was done by these two people that are Heh and Zhou is that they found a system of electrons that are moving in a constant magnetic field for which the phase transition is in the Ising universality class between two quantum Hall states.

[00:43:19]So they managed so they just reproduce the Ising universality class by a quantum Hall transition.

[00:43:24]And in that framework it's easy for them to put a constant magnetic field and to check the critical exponents on the line defect.

[00:43:32]And that's how they obtained these predictions. You see they're quite precise. It's like two digit almost yeah, one well, 10 to the minus two error roughly speaking.

[00:43:44]So it's a very good numerical technique that was invented recently.

[00:43:52]Mhm.

[00:43:53]Okay. So we So there is a lot of evidence now that indeed the renormalization group flows for this localized magnetic field problem, there is a fixed point.

[00:44:04]A large n calculations roughly agree with experiment. Well, with simulations.

[00:44:09]And also they roughly agree with Monte Carlo simulations, which are classical classical Wilson-Fisher simulations, not quantum.

[00:44:19]Okay.

[00:44:20]Any other questions?

[00:44:21]>> Well, is it possible to consider the opposite problem when we consider constant magnetic field with some hole in magnetic field?

[00:44:32]Just just absent absent of magnetic field in a very small region.

[00:44:37]>> Yes, I actually looked at in this problem. I have a I have an appendix in the same paper. No, no, it's not in the same paper. Where was that? Yeah, I looked at this problem some years ago, 2 years ago, and in fact that problem is also also solvable. There is a renormalization group flow that makes the magnetic field locally grow back to the magnetic field in the ambient environment.

[00:45:02]So, in that sense, that problem is becomes a little bit trivial because the normalization group flow makes that special point look indistinguishable from all the other points. Here it's different. Here at this special point there is a conformal line defect and you can measure correlators and so on.

[00:45:19]So, yeah, it turns out that your variant turns out to be a little bit simpler.

[00:45:26]Okay, now I'll show you some cases which are not well understood.

[00:45:31]So, this line operator is just an exponential of a scalar field.

[00:45:36]Let me show you back. It's an exponential of a scalar field in the O(N) model.

[00:45:41]Now, even in the O(N) model, there are variants of this problem that we don't understand at all, and these are non-Abelian exponentiations of the scalar field.

[00:45:52]So, it's not Wilson lines. We're not even discussing Wilson lines yet. It's still magnets. Let's consider the O(3) magnet.

[00:45:59]Just O(3) magnet is a standard classical magnet.

[00:46:03]In the standard classical magnet, we have three fields, phi1, phi2, phi3.

[00:46:08]And we can couple them to a qubit.

[00:46:12]So, we just take a special point.

[00:46:14]Instead of putting magnetic field through that point, we take a special point, remove that point, and put a qubit.

[00:46:20]Or more generally, we put a spin S representation, spin S representation of SU2.

[00:46:26]And these are the operators of this Q or the act on this qubit, on the spin S representation. It's called qubit sometimes.

[00:46:34]And we just couple the bulk fields to this qubit.

[00:46:38]So, it's a little similar to the problem of constant magnetic field. If you just put the local spin, let's say, pointing up. If you just dis- like you force the local spin to point at up, it becomes the same as the problem of constant magnetic field.

[00:46:53]But if you allow the local spin to fluctuate, uh this problem leads to a non-Abelian line operator already in Wilson-Fisher model. It's not a gauge theory.

[00:47:03]So, it's an exponentiation of the Wilson line of the Wilson-Fisher fields coupled to some matrices of SU2 in a spin S representation.

[00:47:12]And here again, there is a renormalization group flow, and here much less is known.

[00:47:18]There is a little bit of quantum simulations, a little bit of classical simulations, but we don't understand the phase diagram. I'll just tell you a little bit about what's known about it.

[00:47:28]This is like uh This problem is like boson Kondo. You can think about it of boson Kondo. This coupling reminds you of the coupling of the spin to the current for the Kondo problem. Here we're coupling the spin of the localized spin S representation to the bulk order parameter. Okay.

[00:47:50]So, it's some kind of bosonic version of the Kondo problem.

[00:47:54]It's very easy to realize experimentally. We just put a spin S in a quantum antiferromagnet.

[00:48:00]So, I'm sure there will be some results about it in the near future.

[00:48:04]But, here is what we can do with field theory at the moment.

[00:48:08]Of course, there is some epsilon expansion.

[00:48:11]Epsilon expansion turns out to be quite complicated in this example.

[00:48:15]But, there is a fixed point. One finds that there is a fixed point at very large impurity spin. So, at least when the impurity spin is huge, there is a fixed point.

[00:48:25]And also in three space-time dimensions, there is some sort of effective field theory and some approximate calculations at large spin.

[00:48:34]It looks like it flows to a fixed point.

[00:48:37]And there are also some recent theorems saying that maybe it's a fixed point all the way to S equals a half.

[00:48:44]But, we don't have the like numerical data, very strong numerical data yet.

[00:48:49]Okay, this is just the this problem is just very difficult because of the non-Abelian nature of this line operator.

[00:48:56]Uh we don't know how to solve it.

[00:49:02]Okay, I just also want to make comments about Wilson lines. Wilson lines are probably the most famous types of defects.

[00:49:10]In a Wilson line, we do something similar.

[00:49:12]We couple the gauge field to the spin operators.

[00:49:15]Let's say in SU2 gauge theory.

[00:49:18]We take spin operators and couple the gauge field to the spin operators. So, it's very similar to what I just told you in the uh in the case of the O(3) Wilson-Fisher fixed point. Here, we are doing it with the gauge field. And here again, there is a lot of literature, but actually the situation is more complicated because if this representation of the Wilson line is very large, there is a Schwinger effect.

[00:49:45]So, if you take a Wilson line and you put a probe charge in a very large representation, it creates a very strong electric field.

[00:49:53]This is like the electric field that it creates and S is the spin or the charge of the local charge. It creates a very strong electric field and there is a Schwinger effect. So there is some instability.

[00:50:08]And here again, that means that perhaps some Wilson lines actually don't exist or maybe they're screened.

[00:50:17]And we try to solve this problem recently in the billion gauge theory.

[00:50:22]And here are the results. Let me explain the results in the billion gauge theory.

[00:50:27]We just put a charge Q in a billion gauge theory and we ask what are the fixed points of this line operator.

[00:50:35]So if the charge Q is very small, let's say we just put charge one Wilson line or charge two Wilson line.

[00:50:42]Actually there are two fixed points. So the ordinary Wilson line is what is here, but there is another fixed point that we discovered.

[00:50:51]So there are actually two conformal Wilson line operators.

[00:50:55]Then when the charge increases, you get the Schwinger phenomenon and you lose the fixed point completely.

[00:51:01]So even in a billion gauge theory, if you put a charged defect of very large charge, there is no fixed point anymore. There is some instability and we simulated this instability on the computer and what happens is that the electric field decays exponentially and there is some scalar condensate that also decays.

[00:51:22]So you're putting a probe charge of some a large probe charge with little Q which is too large, it creates a Schwinger effect around it and there is screening and it flows to a trivial line defect. So this is an example of a renormalization group flow on a line defect that goes from conformal in the UV to trivial in the infrared.

[00:51:44]Okay?

[00:51:45]So uh even the understanding of which fixed points actually exist for line operators like that, which Wilson lines actually exist and how many fixed points are there is something that people are investigating and there is a paper from just 2 days ago by this group Artico, Meneghelli, Savian, and Trallis and they found lots of new fixed points of in some simple SU2 gauge theories.

[00:52:12]Because of this phenomenon, it's a very non-trivial subject. Even the classical Wilson line story turns out to be a very non-trivial subject of which way, okay?

[00:52:22]So, here I can stop for questions and that's more or less all I wanted to tell you. Thank you.

[00:52:35]>> And >> [clears throat] >> And when we consider talk about the Schwinger effect, you means the Well, usually when we have very We have large charge, you have a kind of instability due to the larger than 137.

[00:52:56]>> Yes, it's exactly This was discovered in 1949 or something.

[00:53:01]>> So, you mean you mean just this phenomenon. So, it's not the usual Schwinger but the phenomenon of over over charge situation.

[00:53:10]>> Yeah, well, let me just explain. There is something that people missed in that story. So, if your charge is smaller than 137, you get fixed points. This Wilson line flows to an interesting conformal defect. If it's above, you have a runaway behavior.

[00:53:27]There is no fixed point.

[00:53:28]There are two things that people missed.

[00:53:30]One is that even before you get to 137, there are multiple possible fixed points.

[00:53:37]And some of the And this instability may exist for for of these fixed points, too.

[00:53:42]And the second thing is that when you cross this instability, there is like an exponential scale that's generated. So, if you look after this instability happens, there is some cloud that screens the charge. But look at the distance scale here. It's 10 to the 20.

[00:53:58]So, the cloud has some It's like in confinement problem. You get an exponentially bigger distance scale than the size of the atom.

[00:54:08]So, there is some kind of dimensional transmutation.

[00:54:12]You get I mean, the scale at which you screen this atom is 10 to the 15, 10 to the 20 times bigger than the size of the atom.

[00:54:21]And before you reach this instability, this is just a simple case of a billion gauge theory. There are many many fixed points. The story seems very complicated. And some of them also have this kind of instability.

[00:54:31]So, it is true that uh the fact that once you get to 137 was uh of course known for many years, but the people actually missed a lot of the details uh of what happens before and also how this instability kicks in.

[00:54:49]Yeah, but it's the same thing.

[00:54:52]>> Mhm.

[00:54:53]But but but for instance, when you consider is the huge charge in graphene, you have indeed a kind of clouds of uh special states around It's a kind of uh you feel of states around uh the uh heavy charge uh in graphene.

[00:55:12]Is it analog of this situation? So, uh in that case, it's possible to recognize uh the this additional uh structure due to cyclic organization flow.

[00:55:25]>> Yes.

[00:55:26]Yes, it's a It's basically the same Yeah, you can show that once you go to this unstable branch, you get you get some kind of cyclic renormalization group flow with the famous states.

[00:55:39]>> Mhm.

[00:55:41]>> What has not been observed in graphene yet, that's why we wrote this paper, what has not been observed in graphene yet is that there are many fixed points even before you reach the instability.

[00:55:51]And also graphene people have not observed that there is an exponential bigger scale.

[00:55:56]Why did they did not observe that because in graphene you need to put some dielectric material to change the to tune the instability to the point that maybe lithium will hit that instability. And then this should be observable, but they haven't yet observed that there is a dimensional transmutation.

[00:56:15]So there are small details that have not been seen yet in graphene, but it's the same physics.

[00:56:20]>> Mhm. So in indeed you have a kind of cyclic renormalization group flow in this situation, yeah?

[00:56:28]>> Yes, yes, but people in graphene have not observed that what happens is that two fixed points You see what happens here is that before you hit 137 there are two fixed points. Then they annihilate to one. And then they go to the complex plane.

[00:56:41]There are no more fixed.

[00:56:42]People have not This leads to dimensional transmutation. In fact, that people believe that that's what happens in QCD.

[00:56:49]People have not seen that in graphene yet. They did not see the formation of this exponentially big scale.

[00:56:56]Sometimes people call it Miransky scaling.

[00:56:59]>> Yeah, yeah.

[00:57:00]I I I have in mind just Miransky scaling, yeah.

[00:57:03]>> Yeah. So Miransky Really? So what we wrote What we said in the paper is that this phenomenon of the screening at the 137 should obey Miransky scaling.

[00:57:13]This has not been seen in graphene yet.

[00:57:17]>> Mhm.

[00:57:24]As far as I remember, there are some papers by Levitov and and Katsnelson, they just tried to relate the phenomena of scale if you must taste with Miransky scaling.

[00:57:38]Maybe I am wrong.

[00:57:41]>> I think what what sorry what Levitt of did is they did some calculations here when you're already unstable. What they said is that there are some holes in the Fermi surface and you need to fill them up.

[00:57:56]Yeah.

[00:57:57]Yeah.

[00:57:58]>> Another question, if you consider the truth truth line instead of Wilson line.

[00:58:07]You expect the similar situation similar fixed points and and the whole and the whole duality came line between Wilson line and I mean as duality story.

[00:58:21]>> Okay, that's an excellent question. Let me explain what you find. If you look at this diagram and you assume that you go to strong coupling, that means that you are increasing the value of e squared by 2 pi.

[00:58:32]So you go from 100 from 137 to 50s, 40, 10.

[00:58:38]If you look at this diagram, you would naively conclude that a strong coupling there are no Wilson lines. All the Wilson lines are screened.

[00:58:45]By S duality, that means that Hooft lines should not exist. Hooft lines should be all screened.

[00:58:51]At weak coupling by S duality.

[00:58:54]And indeed, we did a calculation for what happens to Hooft lines in SU2 gauge theory.

[00:59:01]And the conclusion is that there are no there are no Hooft lines at weak coupling in SU2 gauge theory. All of them disappear.

[00:59:07]But it's not because of this phenomenon.

[00:59:09]There is something else that happens.

[00:59:11]A Hooft line creates a very strong creates a magnetic field.

[00:59:16]But in SU2 gauge theory, there are W bosons.

[00:59:20]And you know, there is this Callan-Rubakov effect that fermions can reach the core of the monopole.

[00:59:27]It turns out the W bosons have a crazy behavior.

[00:59:30]They just jump in and they they just singular.

[00:59:34]So, fermions are kind of borderline because they can jump in and come out.

[00:59:39]But, W bosons behave super radiantly.

[00:59:42]They're completely unstable around the Hooft lines.

[00:59:45]So, we did this calculation in SU(2) gauge theory at weak coupling and the W bosons are perturbatively unstable around a Hooft line. And in fact, all the Hooft lines are screened.

[00:59:56]And there is a similar diagram that is not published for how the W bosons just kill the Hooft line.

[01:00:05]And it's compatible with S-duality because that means that at weak coupling there are no 't Hooft lines and at strong coupling there are no Wilson lines.

[01:00:14]>> Mhm.

[01:00:16]So, this this is the analog of instability of vector bosons for for instance, resonances in the magnetic field. So, >> Exactly. It's also analogous [clears throat] It's also analogous to the instability of the standard model to constant magnetic field that was discovered by Ambjørn and Nielsen.

[01:00:35]>> Mhm.

[01:00:35]>> Ambjørn and Nielsen. Ambjørn and Nielsen.

[01:00:38]>> Mhm.

[01:00:40]>> Yes. So, 't Hooft lines are in 't Hooft lines just don't exist at weak coupling in non-Abelian gauge theory.

[01:00:47]>> And the And the same about dyonic loops, yeah?

[01:00:52]>> Well, I haven't studied that. I don't know for sure. I haven't looked into that. It could be that there is a balance between the dyonic charge and the magnetic charge.

[01:01:02]I actually don't know. It's a good question.

[01:01:05]I don't know if dyonic lines exist or not. I have to look into it.

[01:01:10]It's not obvious to me.

[01:01:12]>> Mhm.

[01:01:14]>> Also, S-duality doesn't imply that they should disappear. They might exist.

[01:01:21]Yeah, I actually don't know. I've never looked into that. But they because the Wilson lines they repel electric charge.

[01:01:30]Well, depends on what the charge. Yeah, they they attract one charge and repel another charge.

[01:01:36]And so okay, yeah, so you're saying that there will always be W bosons that will be much more enhanced in unstable. Yeah.

[01:01:44]It makes sense. Yeah.

[01:01:45]Yeah, most likely dions don't exist, but I haven't looked into it.

[01:01:48]>> Mhm.

[01:01:52]And is it possible to discuss the S duality for the your entropy?

[01:01:59]Or is >> Um, yeah, it's possible.

[01:02:03]Yeah, actually you could try for instance, there are exact results in supersymmetric theories.

[01:02:09]>> Mhm.

[01:02:10]>> And you could try to calculate the entropy at weak coupling and strong coupling using localization. I don't know if people have tried to do that.

[01:02:18]>> Mhm.

[01:02:19]>> But actually you can localize this observable.

[01:02:21]There this observable which is a circular line defect has can be you can do supersymmetric localization for it like Pestana.

[01:02:29]>> Mhm.

[01:02:31]>> Uh, but I haven't tried that. Yeah.

[01:02:41]>> [clears throat] >> Okay.

[01:02:43]>> All right, going back to the beginning of the talk. I was wondering uh, in this setup uh, what will happen if we introduce multiple parallel line defects together?

[01:02:54]Especially if you bring two or more parallel line together to undergo a defect fusion. How the total entropy changes or behave?

[01:03:03]>> Oh, that's yeah. I think a lot of people are looking into it now. There is a recent paper by uh, you. There is a recent paper by E Fam Wang.

[01:03:13]I can spell it if you I in the chat. Do you want me to speak?

[01:03:16]>> I know, I know, I know him.

[01:03:18]Yeah. So, people are trying to look into that. I'll I'll say one thing that is very interesting to me, but which is that you know, for local operators, there is an operator product expansion. So, you just take two local operators and you do a Taylor expansion. Kind of some kind of Taylor expansion.

[01:03:39]It turns out that for defects, the singularity the leading singularity in the expansion is exponential. So, it's not algebraic.

[01:03:47]That's a very interesting qualitative fact.

[01:03:50]I I can write it down.

[01:03:52]So, if you take two line defects L1 and L2 and you do a Taylor expansion, the leading singular singularity is one by the distance.

[01:04:03]So, it's exponential of one by the distance.

[01:04:06]It could be negative one by the distance or positive one by the distance. So, it could be exponentially smaller, exponentially larger. It depends on the model.

[01:04:14]And it depends on L1, L2, and L3.

[01:04:18]So, you get a very different structure from the standard operator product expansion.

[01:04:24]And how the G function behaves under this kind of operation is good question.

[01:04:28]I don't know if something is known about it.

[01:04:32]>> Uh-huh. Okay. Okay, thank you.

[01:04:36]>> Yeah, I don't know if something is known.

[01:04:43]And yeah.

[01:04:45]Is there Is there any other questions or >> Oh, I I also want to ask like can it possible to happen like an intersection of line defect together?

[01:05:04]>> Intersection of two line defects?

[01:05:07]>> Yes, yes, yes. Is it possible to happen something like that?

[01:05:10]>> Yeah. Yeah, I don't know if anybody looked into it, but it's possible for sure.

[01:05:16]>> Yeah.

[01:05:18]In that case, the G function also behave differently completely from what >> Yeah, for instance, in a classical magnet, you can have a line of you can have this localized magnetic field on one line and a localized magnetic field on a transverse line and they could intersect and there could be some critical behavior coming from the intersection.

[01:05:38]Uh I don't know if anything is known about it at all.

[01:05:41]>> Okay, thank you.

[01:05:51]>> Okay.

[01:05:52]I I see I see no more questions. Oh.