 I'm very happy to be here giving this talk to this audience. I will be talking about some works I did with Gerald Dunn in 2012, also in new lights of mixed anomalies where they stand. This is really an admixture of two methods. One of them is resurgence semi-classics. And the other one is mixed anomalies. And it's an idea to try to understand quantum field theory non-perturbatively as much as possible. So here is the motivation that we sort of put as a goal to ourselves in 2012. It's a, so there is this very famous question which one can perhaps attribute to Dyson or to Hooft, can we, whether we can make sense out of quantum field theory or whether there's a continuum definition of quantum field theory? Of course, all of us know that these are very difficult questions. I will just make these remarks due to Mike Douglas, given in string met 2011, says that we can construct Euclidean functional integral, there is no difficulty defining Gaussian integrations, so and so. And these developing this PRST and BV formulations. But the real difficulty, he quotes, is that QFT is still not rigorous, is that standard perturbation theory only provides only an asymptotic divergent expansion. Of course, this audience is not unhappy with this statement. Rather, they are happy because it is, the series is asymptotic, so we can do something interesting with it, we can do something useful, okay? So this was partially our motivation. By the way, during my talk, if you have questions, please feel free to ask any time you would like to do so. And so and in the last many years, a number of people got interested with this kind of question in the context of quantum field theory, string theory. And a number of these folks are in the audience. You know, I've been particularly influenced by the work of, of course, Ekaal primarily and also Marino Skiapa in the context of string theory, matrix models, which are quite interesting. Also, the talks, many talks by Konsevich and his work with Soybulman, and other people who are not in the audience. Of course, we all know that the main theme of this kind of ideas is, is sourced by Ekaal's resurgence theory. And this is applied very usefully by Pham, De La Bair, Voross, and others in Houston to quantum mechanics. So the main promise of the resurgence theory that all of us know quite well is that it is a way to combine perturbation theory with non-perturbative methods. And it's a way to extract information about one concerning the other. This is the main goal, okay? And so our physical motivation, as I said, whether the question of whether there's a non-perturbative continuum definition of quantum field theory. And here, I do not mean something formal. I've been personally very much influenced by lattice field theory. And lattice field theory is not only something formal, it is something of practical utility. I really mean a continuum definition, not in a formal sense, but also in a practical sense, that's something that we can calculate things. And the more modest question is that whether we can make semi-classical methods exact, and when is this possible? And in this context, I would like to say a few things later on, on the connection between resurgence explanation and left-shed symbols that Maxim commented this morning. So, and but I want to make this remark about something. When I was much younger, I read Coleman's lectures on instantons. And one statement there that bothered me tremendously was the fact that these saddles are of measure zero in either pet integral or ordinary integrals. I was deeply disturbed by this thing. And nowadays, at least in many integrals that we can deal with semi-classical methods, or by using Picard left-shed theory, we think that the effect of the saddles is not measure zero, but rather measure one. So we can map many integrals exactly to some of left-shed symbols. So, you know, if you ask this kind of questions, you immediately find yourself in a realm of a number of interesting questions in quantum field theory. One of them is something called IRR normal on puzzle. I will describe this later. This is really the way that perturbation theory diverges. How it diverges? And what is it controlled by? And what fixes the divergence of the perturbation theory? The other one that is not very well understood in the context of quantum field theory and quantum mechanics. In quantum mechanics, it's much better understood. The multi-instanton calculus not just saddles, but in quantum mechanics and quantum field theory, we also have correlated saddles. Or we can call them critical points at infinity in the firm's old language. That the symbol is not necessarily Gaussian. And how do we work with this thing, precisely? And also the thing that one other thing that I will mention briefly is the meaning of non-instanton saddles in quantum field theory. There are many quantum field theories which has saddles different from the instantons, okay? Now, probably I shouldn't say much about mathematical motivation. There were two excellent talks this morning. As I said, resurgence is the new idea due to Professor Ekal. And it is a unification of perturbation theory with non-perturbative sectors. And it has been studied rather deeply in the context of ODE's, nonlinear ODE's, integrations, and so on and so on. And as we all know, in most cases, perturbation theory is divergent asymptotic expansion. So the idea is to elevate this series expansion into something called trans-series, a beast which involves not only expansion parameter, a series in the expansion parameter, but an expansion in terms of exponential. Of one over exponential minus one over coupling or logarithms or multi-exponential and things like that. And try to construct a trans-series that is well defined under analytic continuation in such a way that it incorporates Stokes phenomena and left-shed-timbled decomposition as precisely as possible. So this thing has the promise that perturbative and non-perturbative sectors in a given theory are deeply entwined. So this gives a different perspective, sort of philosophical shift, at least in the context of quantum field theory. Is that semi-classical potential, semi-classical expansion has the potential of being exact in some cases, okay? Now, here is the outline of my talk. I will again describe some problems in quantum field theory, then I will describe resurgence in simple examples in quantum mechanics. And then tell you the difference between a typical quantum mechanical problem and a typical quantum field theory problem. Namely, the existence of something called infrared norm alone. And so I will tell you some hints about connection. It was already mentioned in today's this morning's talk about left-shed-timbled decomposition and resurgent trans-series. Of course, these are all tied up with the complexification of the path integrals, as well as it's also tied up as mentioned this morning with the WKB expansion, or sometimes called exact WKB or complex WKB. Now, so as a quantum field theorist, you are in some sense mentally forced to this program. There is really no escape in the standard discussion of the quantum field theory. So I will mention an example here, but the example can be literally anything. I took some sigma model whose target space is the complex projective space, CPN. For this particular model, large n methods are very successful, but many problems are unresolved at finite n. You can take another two-dimensional sigma model or whatever problems I will tell you are also existent in the context of, for example, quantum chromodynamics in four dimensions, okay? So the first problem is, as I said, perturbation theory is asymptotic, even after regularization and renormalization. This is, of course, everybody in this audience know this, but it is useful to mention. So one question is, is there a meaning in perturbation theory? And in such type of theories, for example, CPN, it is classically scaling variant. There are instanton solutions. Usually, at least in the past, in the 70s, people talked about something called dilute instanton gas approximation. But this approximation is only valid in theories with scales. For example, in quantum mechanics, where instanton has a characteristic size. The saddle in the problem is characteristic size. So in theories which are classically scaling variant, instanton can be any size and dilute instanton gas approximation assumes that the typical separation between saddles in the Euclidean space time is much larger than the size of the saddle. But since the large size of the instanton can be anything, literally, there is not quite a meaning to this approximation. In Coleman's lecture, this is called infrared embarrassment because we do not know what happens with the large size instantons. So in the past, primarily by Affleck and others, it was suggested that one can solve this issue with the scale of the instanton by putting an infrared regularization to the quantum field theory, by putting quantum field theory in a box. Anyways, that's a very nice way to think about quantum field theory with some IR regulator, with some UV regulator, to make it mathematically a better-defined thing. But whenever you do that, whenever you put the theory on a circle, historically, people always talk about very often thermal compactifications. And usually, if you have some interesting confining theory, it usually goes to another phase. It doesn't remain in the regime that you would like to study the theory. So there is also some incompatibility of the large-gen expansion with the dilute instanton gas. It is whether these two things are incompatible because we trust large-gen expansions, but we do not trust the dilute instanton gas in this context. And finally, I will explain later, in most of these theories, there is IR renormal on ambiguity. This is a pathology put forward by Tuhut many years ago, 30 or 40, oh God. Yeah, I cannot sum up. Subtract 79, and I will come to that later on. Now, if you would like to understand these quantum field theories better, of course, we need useful tools, new useful tools. And from the mathematics side of the story, the new useful tool for us is resurgence theory and trans-series in their various incarnations. Because it tells us that as long as we are given a partition function, we can view many of the constructs that comes from a quantum field theory as some class of analyzable functions in the language of Eqal. And there is also a geometric side of this story that has been discussed this morning. This is the complex Morse theory or Picard-Levschetz theory. This is an attempt to find the cycles attached to the critical points. Now, from the physics side, there are also two new inputs. One of them is we learn better how to do semi-classical analysis in a more reliable way. And this goes in combination with this idea of analytic continuity. And today, I will tell you that the idea of adiabatic continuity, it is intrinsically tied up with mixed anomalies. So I will make that as precise as possible. This is sort of new perspective compared to our work from 2012. So here is a simpler question. The question is, can we make sense out of the semi-classical expansion? All of us know that semi-classical expansion is a trans-series expansion. It has a perturbative part. It has multi-instanton contribution. I omitted the logs momentarily there. And there are exponentials which are instanton factors, saddle factors. And there are also fluctuations around those. And so all of the series that appears in this trans-series expansion are very generically asymptotic. So it has already been mentioned, if we have a divergent asymptotic expansion, we do construct a Borel transform. And from this Borel transform, we can do Laplace integrals. This Borel transform is a function of the Borel plane variable t. And this Laplace transform would give you something meaningful. This is the standard physical definition. Of course, for this Laplace integral, it depends on the orientation in the Borel plane. So but this is at least something that physicists use. But it's a much more general beast than that, as we know now. So this thing, this Borel resummed version, would be quite meaningful if this Laplace transform did not have any singularities in the direction of the integration. But usually there are singularities. This is the Borel plane, as I showed t, Borel t plane. There are singularities, either poles or cuts in the Borel plane. And whenever we are doing these summations, we have to go either above or below these singularities. We can define left and right Borel resummation. And these two things differ from each other by some purely imaginary parts. And very typically, in both quantum mechanics, as well as in quantum field theory, this purely imaginary part, ambiguous imaginary part, has the form of an instanton. What physicists would call an instanton, but generally, a saddle point contribution. And you can look to the discontinuity of these left and right Borel resummation. And the discontinuity is given in terms of a sum over these exponential factors. So now this is the ambiguity of perturbation theory. We took something which was divergent and mapped it to something which is ambiguous, left, right, ambiguous. Now, at least in the context of quantum mechanics, this thing, this how to fix this pathology, is understood in the early 80s by Bogomolians in Houston. Now, they did it first in the context of double well. And if you look to take time direction infinites, there are instanton, anti-instanton, instanton, anti-instanton. If you look to the typical textbook, you will see a picture like that. But in reality, apart from these instantons and anti-instantons, which has some characteristic size, OK, there are also correlated pairs like instanton, anti-instantons, or instanton, instantons. The characteristic size of these objects, these correlated events, is much larger than the instanton, but much smaller than inter-instanton separation. So when you are doing a semi-classical analysis, you have to take these things into account properly. So now, there is a way to calculate the amplitude for these events, these instanton, instanton, or instanton, anti-instanton events. And so whenever you have these two objects, the separation between them is no longer a moduli. Now, the interaction depends on the separation, but it is a quasi-moduli. So you have to integrate over that. This is actually, in the modern language, this is part of the left-shed symbol connecting these two things. Now, at least for the instanton, instanton, there is no fundamental difficulty. It's just an integration. You do it, you do get some meaningful results. However, if you try to do this integration over the quasi-zero mode for instanton, anti-instanton, at first, you do get a meaningless result. And it is meaningful to get a meaningless result, a posteriori. The reason is following. So if you actually, first, let me tell you what Bogomolnians in your standard, if you take the coupling constant to negative coupling, for example, okay, I am describing what they did. And due to integration, the integration becomes meaningful and you can do it. And then continue back to positive coupling that you would like to calculate, you get two-fold ambiguous results. So the reason at the very first place you get something nonsensical is because you are trying to calculate something manifestly ambiguous for real coupling constant, okay? So it turns out that this event, correlated instanton, anti-instanton event, has a nice real parts and also purely imaginary ambiguous complex parts, okay? Now, of course, we know this in retrospect quite well because we are on a stocks line and it had to be so, okay? Now, here is the magic that occurred in quantum mechanics and discovered by Bogomolnians in Houston is that imaginary parts, coming, imaginary ambiguous part, coming from perturbation theory, and imaginary ambiguous part, coming from the multi-instanton calculus, cancel each other precisely. To be more precise, Bogomolnians in Houston show this at leading order, you know, just exponential factors and no fluctuations around them, but this statement works out at higher orders as well, okay? This is the first glance of Borelacal summability in the context of this particular pet integral for quantum mechanics. This can be either double well potential or periodic potential, it doesn't really matter, okay? Now, something I showed, Gerald and I showed was that, so there is a particular rate of divergence in perturbation theory. It's not only n factorial, but there is an n minus one factorial there, n minus two factorial so and so forth, and all of them has these coefficients, but if you go on and continue, imaginary part of the instanton-anti-instanton amplitude, you see that there are fluctuations, quantum fluctuations around instanton-anti-instanton and they will give you a series in G square and they come with these coefficients of A sub n, G square G to four, G is my expansion parameter so and so forth. You see that if you do Borelacal summation of the first thing up there, first asymptotic series up there, it actually cancels all the sub-leading terms down there as well. This is true in typical quantum mechanical systems with instantons. It works, we showed this for example, double well as well as periodic potential for example, and there is also an appropriate factor coming from the instanton action. Now, one manifestation of resurgence is that fluctuations about an instanton-anti-instanton saddle are determined by those of the vacuum saddle, but how about the fluctuations around perturbative saddle versus instanton saddle? Now, because these are in principle, these can be different in different topological sectors when you write down a pet integral. There, I use this quote from Eqal from early 80s, I think the quote is that resurgent functions display at each order of their singular points, a behavior closely related to their behavior at the origin, perturbation theory, lose the speaking, this correction, resurrect or surge up in a slightly different guys as it works at their singularities, okay? So, one question is whether there is a relation between perturbation theory around perturbative saddle as well as the instanton. And here, I want to first make a remark, actually the answer to this question was understood long time ago in an almost unknown and unappreciated work of Alvarez and Casares. It's a very important work, in my opinion, but I think none of the experts in the field knew about it. We saw at least one paper of Casares on that. And indeed, you can put this machinery into work. You can calculate expansion around the instanton, look at this resurgence line for arbitrary levels and for arbitrary level numbers and at desired order. You can put this to a Mathematica file and it gives you something, at any desired order you get a result, 20 order, 30 order, it doesn't matter. This is something coming from, you can, the interesting thing is there is a precise relation. I will not tell you this relation because I want to go to quantum field theory. There is a relation, precise relation, very simple relation between perturbation theory around perturbative saddle and perturbation theory around non-perturbative saddles. And this morning there was a question about, how can we show this by using Feynman graphs and things like that? With Feynman graphs, this calculation is actually extremely painful because even perturbative level, you do green functions in the background of an instanton, which is not quite easy. Even the leading order result are difficult. People actually calculated this, Turbinair, Shuriak et al. and others calculated this at Trilu Porter and the agreement is at seven decimal places. It's quite impressive, okay? But as I said, I would like to move to quantum field theory but this is the state of affairs at least in quantum mechanics. So why is this happening? Probably I should skip this slide because it's fairly elementary. But here is an ordinary integral. Let me mention at least four students, okay? So here is an ordinary integral. We can deal with it with the steepest descent. We can deal with it exactly for this particle integral. It's probably some Bessel. And we can also deal with it by using critical points and left shift symbols. So this integral has just two critical points. One of them is at zero, the other is probably at pi over two and attached to each cycle, there is a, to each critical point, there's a symbol. It's actually a mathematical identity, exact identity to write this down just before the stock slide as a sum and, you know, or as a difference. So you can write it in these two manners. You see that cycle jumps, but the coefficients which is integer in this particular case also jumps. In such a way that there are two jumps in the problem such that physical entity that you are calculating, namely the integral, does actually not jump, okay? This is also, the very same spirit is also present in many applications of the wall crossing. Okay? Now, of course the ambiguity in the left, right, Borel resumption is actually same thing as the ambiguity for the integral over the left shift symbol J zero. Exactly at the stock slide, this integration contour jumps. You see that the tails on these integral J zero cycle, the tails flip orientation. This is the imaginary ambiguity that we are getting plus, minus, I times exponential of something, okay? So in this sense, left, right, Borel resumption is in some sense left, right, left shift symbol integration. Okay? Now, probably I should also say that in order to determine these cycles, we do write down a complex fight version of the gradient flow equation. I wrote it up there, starting with the action of the zero dimensional theory and these gradient flow equations happens to be an instanton equation for one higher dimension. Here I am in zero dimension for this integral and the gradient flow equations are this and this is actually instanton equation in holomorphic supersymmetric quantum mechanics. Okay? Now, now, so the story in quantum mechanics is actually rather happy story. So there are still many unknowns for many systems, but we understand great deal of the precise relation between perturbative vacuum and instantons as well as perturbative vacuum and instanton and anti-instanton sectors. Now, the question is whether this mechanism that works on nicely in the context of quantum mechanics can work in the context of quantum field theory and untrivial quantum field theory. And the famous answer is no. The famous answer on R4 is due to Hooft from 79 and in 84 there is a very nice paper by François David which says that in the de-compactification limit we cannot get, we cannot cure these ambiguities in the de-compactification limit by semi-classical methods. Okay? Now, let me describe briefly why this doesn't work and physicists know this in general. So if you, of course in CPN, as I said, there is instanton, there are instantons and in some way let's say you calculated instanton, anti-instanton correlated event and this gives you a singularity in the Borel plane but this is a singularity which is very far from the origin. Okay? This red one is some instanton and anti-instanton singularity. Okay? And things like that I think were put forward by Lipatov. Now, actually if you look to Borel transform it has many more singularities. If you are in CPN model of order and singularities which are far more relevant than instanton singularity. Okay? But we do not know any semi-classical object which can accommodate, which can fix these pathologies. Okay? So this is the problem. These singularities are famously called IR-Renormalon by Tuhoft, the ones on the positive axis on the Borel plane. And there are no known semi-classical configurations which can take, which can fix these pathologies of the perturbation theory. Okay? Now, this is a real problem in QFD. There are some phenomenological approaches to this to have to deal with this but there is no satisfactory microscopic approach. Okay? So this is some thing I stole from one of the papers of Parisi. So he says there are new singularities which cannot be controlled by using semi-classical methods. But one thing we started to understand fairly recently is the following. What happens if you push a strongly coupled theory to a weak coupling regime without generating any phase transition? Namely, by keeping its character. And I will make this precise in a mathematical sense. I will write down sort of a version of the partition function, not the usual thermal partition function but the new partition function similar to index-like objects and supersymmetric theories. But this is possible. In most quantum field theories, you can push quantum field theory into weak coupling semi-classical regime without changing the ground state structure. Now, so what happens then in these theories? We know this very famously. For example, in supersymmetric theory, right, the way that with an index is calculated, it's just using by compactification. And compactification preserves quite a bit of the character for the index it preserves, you know, it's independent of the compactification. Now, as I said, a traditional way of compactification that has been done in the literature of quantum field theory, usually compactification radius is viewed as an inverse temperature. And very large circle is close to zero temperature, very small temperature is close to very high temperatures. And very typically, there are either phase transitions or rapid crossovers. So there is some sort of singularity. And this idea of continuity or adiabatic continuity is an attempt to avoid all sorts of phase transitions in by any means possible. Is it possible? Yes. I wrote here the Witten index. And this thing is actually, if you compactify it on say R3 times S1, it doesn't matter R times S1, it is independent of L. It is pure number. There is no non-analysis in it. So it tells you that as an idea, this is possible. The fact that in non-supersymmetric theories, such compactifications are possible has been people started to understand these things starting in 2007. I think I did this with Larry Yaffe around that time, but then we started to understand in many other theories. Okay, so now let me consider CPN models. This N is a complex field. N, modular square is one. It is the CPN field. One can do some decomposition into some angular part, into some radial part, this pi coordinate and theta coordinate. And one can do something interesting, actually. One can define this theory. Let me write this down. Is there a chalk? Tables, I'm sorry, okay. So this M field, x1, comma x2 plus L, you can impose the following boundary condition, omega ij and j, x1, comma x2, where this omega matrix is something of the form, one exponential to pi i divided by N, so and so exponential i to pi N minus one divided by N. This is something which lives in the global symmetry of the theory. So it is for CPN, we have a global symmetry as UN, more precisely it is PSUN, okay. You can use such a boundary condition, okay. So what does it tell us? It sort of forces the theory Okay, let me talk over this picture. If you do a regular compactification, all of us know that the modes in quantum field theory split up to usual Kaluzak line modes. And Kaluzak line modes are quantized in units of the two pi over L. Whenever you do this kind of twisted compactification, so the modes happen to be quantized in much smaller units. It is two pi over NLs, okay. So what do you mean by compactification? It's always on circles or? It's on circles. It's on circles that you can afford to mark these modes. Yeah. So you start with a theory say on R to the D and you consider this on R to the D minus one times S1. Moreover, when you consider, for example, usual partition function, right, Sd of eta, there is a trace, exponential minus eta h. Trace always mean in the pet integral formulation, there is a circle. And beta is the radius of that circle, okay. In the pet integral language. And you assume the supersymmetry is unbroken? For this construction, supersymmetry is immaterial. How do you say that this trace is under compactification is the same number? If supersymmetry is broken, how can you say? Okay, in supersymmetric theory, if supersymmetry is not broken, then either broken or not broken, this is independent, in supersymmetric theory, this would be minus one to the F, and this would be independent of beta. And now in non-supersymmetric theory, to find the counterpart of this thing is non-trivial. And actually, I wrote the formula a bit later, so sorry about that. Here it is, oh, can you see it? Let me just write it. For example, in CPN, this boundary condition corresponds to the following partition function. Z of beta, trace exponential minus beta h. And it is i two pi k divided by n, q sub k, product over k. And this q sub k are generators of the charges associated with the SUN symmetry, okay? This is not a regular partition function, but it's a graded partition function. The thing it does, just say, the thing it does is that it takes the states in the Hilbert space, for example, some representation, and it creates them. It gives them, it does exactly what supersymmetry does. In supersymmetry, you have both n and fermi, and you assign plus one to both states, minus one to the fermi states. And here, this thing, if you have some many states here, it assign phases, one exponential two pi i over n, exponential four pi i over n. And in this partition function, actually, in this partition function, this kind of, this class of graded partition function, you can actually construct things which are continuous as a function of beta. Okay, but this actually doesn't matter. You know, in the SUZE or non-SUZE case, can we consider these generators as KGV or KP hierarchy? Because I know that, in principle, in the non-SUZE case, there is a possibility to construct the center part. In this, the question would be, these operators would correspond them to what? In this case, these are some generators of the SUN flavor symmetry in this system. And the example of the consort charges. Consort charges, yeah, yeah. These are, these are consort charges. Which commutes with the Hamiltonian, exactly. It's possible, because for the non-SUZE system, there is a... There is no, there is no problem. Trace is always over the whole Hilbert space, yeah. And these are these charges commute with the Hamiltonian, and it doesn't matter if I deal with the supersymmetric theory or non-suppersymmetry, and I can construct such objects. Now, this thing, this twist, from the pet integral point of view, corresponds to making Kaluzak line splitting much smaller at the three-level. Now, this is an interesting thing, actually. Because you are on a finite circle, L can be very small, but you see there is something very funny going on. If you take M to infinity, this thing becomes a continuum. In quantum field theory, this is something very important, especially in lattice field theory. This is something known as Eguchi-Kawaii, reduction of volume independence. And we were, when we were writing this, Gerald and I were putting these boundary conditions, we were very much influenced from that idea. And this Hamiltonian interpretation came a bit later, okay? Now, actually, we wrote this kind of boundary condition literally by intuition. But very recently, this has been made much more rigorous. Now, I will state some facts in the case of CPN, but they are more generally true, okay? To be more precise for CPN model, for example, the global symmetry is not precisely SUN. It is SUN mod ZN. The reason for this is that in the CPN, we have U1 gauge redundancy. So, and ZN subgroup of PSUN is also part of U1 gauge redundancy. So, because of that, the global symmetry is really SUN mod ZN. And it turns out that on R2, there is a mixed anomaly between this PSUN and charge conjugation symmetry. Okay? There is a mixed anomaly. This is due to Komargotsky, Cyberg, and others, okay? And now, what does mixed anomaly means? It means that if you gauge PSUN, you lose charge conjugation symmetry. This has some implications for quantum field theory. It tells you that in a quantum field theory is the vacuum of the quantum, only a finite number of possibilities are possible. Either one of these symmetry is broken or the infrared theory is CFD, or you know, you have some TQFT supplementing the ground state of theory. You know, there has to be some TQFT, okay? Now, how do we see that? Actually, if you just think the topological term in this theory, so it is, of course, quantized. It is just quantized in units of integer. This tells you that theta angle is topi periodic. So now, if you gauge PSUN, the topological term is modified like that. This A is a combined PSUN field. I will not describe this procedure, but the topological term becomes quantized in units of one over n times an integer. This is, in mathematics, this is, I think, the second stifle with me class, okay? Then this tells you that theta angle is topi and periodic. So theta equal pi is no longer special and you lose the charge conjugation symmetry. The thing that is quite interesting is the following. That is quite interesting is the following. If you do a normal compactification of this theory by thermal boundary condition, it turns out that you cannot preserve this mixed anomaly with normal compactification. You can preserve the mixed anomaly if and only if you impose this particular boundary condition which corresponds to a background field for the PSUN, okay? It is really an if and only if statement. So you can preserve the character of the theory, ground state structure of the theory with compactification only if you use these boundary conditions. In this sense, these boundary conditions is unique, okay? This is most clearly understood in a work by Yuya Tanizaki and his collaborators. And this is not only special for this theory. If you take, for example, supersymmetric version of this theory, there is a mixed, in that case, there is again a PSUN symmetry, but there's also a Z2 and discrete KIDL symmetry and there is a mixed anomaly between the two. And actually a very beautiful way to calculate written index is to do that in that theory and just exactly this twisted boundary condition, you can do this calculation and you can preserve the mixed anomaly at arbitrary compactification. I emphasize that no other boundary condition respects this mixed anomaly structure, okay? So the story is quite general. If there is a mixed anomaly, this kind of twisted boundary condition can be used to guarantee the persistence of the anomaly, okay? However, these boundary conditions are also present even in the case there are no anomalies. You can use them and preserve adiabatic continuity as much as you can. Even in CPN model, this mixed anomaly is only a tetraical pi, but you can use the boundary condition at tetraical zero as well. Or you can do the same thing in principal KIDL model, which is a matrix model on group manifold or on the Sumino-Vitton theory. Doesn't matter. Yes, question. Yeah, is this for individual M, yeah? Yes. Controversy's large N expansion. Large N works in general quite well, actually. This gives many things consistent with the canonical large N expansion. This is something that checked at the time. And as I said, you know, I made a picture here. So for example, in the case of the first one is the Hilbert space picture. What's going on the Hilbert space? The second one is the pet integral picture and the last one is the anomaly picture. So what happens is that if, for example, in CPN model, gauge invariant states are in the adjoint representation or singlet representation of SUN because the genuine symmetry is really PSUN. There is nothing in the fundamental representation of SUN. There cannot be. It is not gauge invariant, a state like that. So, and this thing, this twisted, this graded partition function takes these states and attached to each one of them different phases. So when you do the states sum, there is immense amount of cancellation between states. And so much so that when you go to small circle, normally every state would contribute. You know, all the words of the Hilbert space would be there. But if you do this particular compactification, only a handful of states, one, two, sometimes, if you take the larger limit, it is literally one state contributing and it is the ground state, okay? So, and because of this structure, this kind of object preserve adiabatic continuity. So, now let us turn back to our story about non-perturbative settles in the problem because at least when we reduce to quantum mechanics in this way, we should be able to deal with interesting topological configuration. Configurations, here what happens, okay? This is a picture of the instantons. The first one, this is for Cp2, so N equal three. The first one is the regular thermal compactification. Take an instanton and change its size moduli. You see that one lump there or spike, there, which looks like spike, but it is rather smooth. It sort of spreads out and it becomes just one lump. Its action is whatever it is, it's independent of that. But if you put this non-trivial flavor halonomy, this background, what happens is that this is N equal three, so the usual instanton splits up to N objects, N settles, N different settles. This is a fractionalization of the usual BPSD instanton settles to some fractional objects, okay? So this is for SU3, it splits up to three, but in general it splits up to N. This will have some very interesting implication because the action of these configurations will be proportional to one over N. Now, actually you can classify these settles. They happen to be these tunneling events. They are in correspondence with the root system of the Lie algebra. Maybe this is core roots, I don't know. You can write the erection. If you choose this particular background, the erection is one over N of the instanton action. And you can show that the 2D instanton action can be written as a product of these fractional instanton actions. And here this K sub V is the dual coccether label for each. So you see maybe dual cut label, I'm sorry. Coccether number is some of them. Dual coccether number is just some of them. So this splits in a Lie algebraic way, these objects, okay? Now, the interesting thing is this dual coccether number is related to the beta function of gauge theory. For all sigma models, this is true, okay? So these fractionalization is very important for at least IRR normal or problem. You can go to these quantum mechanics. You can calculate mass gap. There is an energy splitting between ground state and first excited state. You find something which is exponential four pi over lambda. Remember that lambda is G square N. Okay, remember that the usual instanton would give exponential minus four pi divided by G square, which would be exponentially smaller than this. And it does something very interesting, okay? So in particular concerning your question, for example, the usual instanton factor in this theory is four pi over G square. If you work with the Tohoft limits by keeping lambda G square N fixed, this is proportional to exponential minus N. This is the irrelevance of the usual instanton in the usual larger limit. But you see here that these things are really coming with exponential four pi G square N, and this is finite in the larger limit. So it's perfectly nicely compatible with the larger limits. And the result is also, by the way, compatible with the mass gap that's obtained in the larger limit that's discussed in the textbooks. Now, of course, these one instantons, these fractional instantons, as in the case of quantum mechanics, is not the whole story. You can write correlated events. You can write instanton and instanton correlated events. So the separation between them is a quasi-modular, modular again, quasi-zero mode, and you can find the amplitudes. So there are two kinds of, in some sense, two events. One of them is you tunnel in one way and you come back or you tunnel in one way, you come back in the other direction. And the second thing that I wrote there, you tunnel and come back to the same place, the counterpart of the usual instanton and instanton in quantum mechanics is very important. If you calculate its amplitude by using either this method of Bogomolni's ingestan or in the modern language by doing the appropriate Timbal integration over the quasi-zero mode direction, you do get this correlated amplitude. And you see that it is two-fold ambiguous. Of course, this would tell you that there is something manifestly wrong with this semi-classical expansion because you are trying to calculate something meaningful and you are getting something two-fold ambiguous and not so meaningful. In the past, at least 70s, 80s, people cared with this kind of things, at least in the context of quantum mechanics, few people cared, like Bogomolni and Zingestan, but I do not think that it is appreciated in its widest generality. Even if you look to Polyako model, this issue is even in Polyako model, but he didn't write anything about it, you know? So, and these things are important and interesting because, for example, one can do perturbation theory in the reduced quantum mechanics and this is done very nicely in a, again, virtually unknown paper by Mike Stone and Reeve, and you can derive perturbation theory. By the way, there is a Mathematica program now. Out there, it's available on the Wolfram Library. It is written by Tin Suleiman Pasic, he is at your neighboring institution here, you give it to it and it calculates these growths up to 500 orders in an hour. If you want 200 orders, it is 10 minutes. Okay, maybe less, I don't know. So, you, indeed, you do, for example, you can do left and right borel resummation and you do get this ambiguity and, again, by this borelical resummation, these ambiguities cancel. However, there is something magical that happened at this stage. That is, although we are in quantum mechanics and the cancellation is something that looks manifestly quantum mechanical, something magical happened. That magical thing is, actually, this was an ambiguity that I called IR-Renormalon. This beta zero there is one over n. It is closer to origin compared to usual instanton and anti-instanton by a factor of n. So, this is exactly the location of the two hofts IR-Renormalon. This is quite interesting. I should say that, at least in CPM, in CPM, these points matches exactly. But, in general, it didn't need to be so, you know, there may be some numerical differences. For example, in pure young meals, you do get one over n, but you do not get 11 thirds, you know, from the, that appears in the beta function. So, it is a little bit numerical accident here to be strictly honest about it. But it is also exactly the location of this IR-Renormalon. So, in 79, to ask this question, can we make sense out of QCD or QFT in general? And the reason he thought that this was quite unlikely was because he thought that it wasn't quite possible to understand these IR-Renormalons by semi-classical means. And I think this is really at least one way of making the theory calculable and seeing that these two of the idea that IR-Renormalon is there and it cancels with something is actually correct. Sort of a realization of an old dream. Just, this is a side remark. For example, there are other saddles, for example, non-self-truel saddles. If you look to principle chiral model, you know, there are there. Ulenbek, Kerun Ulenbek actually did an explicit construction for them. She called them unitons. And in physics literature, they were quite understood, not understood at all. But if you insist on this kind of boundary conditions, these twisted boundary conditions, you can show that these unitons fractionates. Again, very similar story. You understand great deal of their structure. Of course, principle chiral model is also very interesting because if you add the Vesumino term, it becomes the Vesumino-Witton model. You can understand the theory quite precisely by using compactification and by looking to anomalies and semi-classics. Now, this is the sort of emerging picture, at least in semi-classical domain. This is some sort of resurgence triangle. There is a perturbative vacuum. And underneath that, there are things like instanton, enta-instanton with topological charge zero, but finite action. And whenever there is a pathology up there, it is fixed by the thing which is down there. And there are other topological sector. For example, these fractional instantons and fluctuations around it. And they are also tied up in some way with the perturbative vacuum, at least in the context of quantum mechanics. So at least in the regime where semi-classics is reliable, this has a potential to give a more or less complete non-perturbative description of the QFDA. I think it has such a prospect. We are still working on this kind of problems. I was hoping to report on some progress in Schwinger model, but I got confused, so I will not report on that. But yeah. So the idea is that every pathology within the theory fixes with another thing which looks again ambiguous within the same theory. And this is very in spirit, this is the idea of median resummation in resurgence theory that Jean-Écald put forward and also very nicely reviewed, and also generalized in the work of Ricardo and others. Now there was in this morning some remarks in resurgence theory concerning path integrals. So this is a very active topic. It is the complexification of the field space and finding the critical points and associated symbols. In general, it's a very hard and interesting problem. And in this context, I just want to mention one interesting problem. So at least when we were dealing with this kind of saddles, we also encountered some very weird looking saddles. Because when you complexify field space, you do not only get nice real solutions, but you do also get some interesting solutions which may be complex, sometimes multi-valued and singular. And in the past usually if you have something even discontinuous, it is rejected in the context of the path integration because they were just thinking in the context of real path integrals. And of course in that case, they would cause infinite action. But it turns out that in the newer version of the story, the space of saddles is much richer and much interesting and things that were not to be appropriate in the past turns out to be appropriate and have finite action configuration. One such thing is following, one such example. This is for example, supersymmetric version of the double well. You know, nice example, just supersymmetric version. You can actually integrate out fermions exactly. You can go to Hamiltonian constructs, grade the Hamiltonian plus minus in the bosonic Hilbert space, fermionic Hilbert space. And in those spaces, this potential looks like a tilted double well and tilting is just only of order of h bar, it's small. Now, if you look to this system, you would think that, you know, you can ask, is there a non-perturbative saddle contributing to the path integrals? In the system, first of all, I should say that supersymmetry is broken, okay? There is a non-perturbative ground state energy, positive ground state energy. But if you just look, invert this potential as I did there and look to a solution, you know, in the space of real paths, there is no finite action solution. I mean, if a ball starts to roll down from that hilltop, it will roll down and overshoot. It will go to infinity, it will not come back. But if you write down complex fight version of the gradient flow equation, so this was, for example, let me, let this be with the field del phi over del u, this depends on time and gradient flow time minus del s bar over del phi bar, again, t and u. And if you write down, on the right hand side, you can write down the complex fight version, holomorphic version of the equations, and there is actually a solution. It goes there and if the energy was lower than e2 there, it would go, roll to that direction and turn back from the real turning point. But up there, what happens is that there is a solution. It goes there, hits a turning point in the complex domain, and comes back, okay? This is sort of a complex solution. Now, this first plot here is the bounce at the level e2. It comes, it starts from there, rolls up there and comes back. But if you, the other one, this thing, as I said, is a complex object. It has both real and imaginary parts. Black part is the real part and red part is the imaginary part. It does something interesting. And on the way, you see that it has an imaginary part also. And if you calculate the action, this imaginary part turns out to be quite meaningful. And it actually gives a purely iPI contribution to the action. A topological angle, but it is sort of hidden in the left shed symbol on the left shed cycle. So actually, this thing, this overall sign, guarantees that if this sign was not there, semi-classic would tell you that ground state energy is negative, but that wouldn't be consistent with supersymmetry because supersymmetry tells us that ground state energy is positive, same definite. And this complexity of the solution and the fact that imaginary part is iPI turns out to be quite important to understand full dynamics non-perturbatively. I will just mention one other example then, stop. This is an example which we have periodic potential, cosine potential. Now, if you integrate out fermions, you can again write down h plus h minus Hamiltonians in the bosonic and fermionic Hilbert space. And the potential looks like this. So sort of, there is a gap here of order of h bar, but if you are looking to the ground state properties, naively, you may think that there is only one saddle and it contributes the tunneling from here to there. And this would actually give a positive ground state energy, but it is known that in this system, the supersymmetry is in fact not broken. And the resolution of that is that there is another saddle which sort of goes halfway, hits a complex turning point and comes back. And the action and property, many properties of these two saddles are actually exactly the same. The only difference is their phases. So the summation of these two gives you zero. This is the semi-classical reason why the ground state energy stays zero in this theory. This is sort of interesting, because if you were to ask this question to a traditional quantum field theorist, even some of the best quantum field theorists give this potential and ask ground state energy. They would tell you that because of this, ah, sorry, because of this thing, ground state energy would be negative, they would tell you, but the contribution of this is positive, the complex one, and the reality it is zero. So there is this kind of very delicate cancellation. Now, one very interesting thing is this solution, the real solution is completely smooth. You know, it is nice and smooth, but the other solution is very bizarre. You know, if you move a little bit away from the stock slide, this smoothens out, but if you are exactly on the stock slide, this actually real part is a discontinuity, but so does the imaginary part blows up. And this turns out to be still finite action, because, you know, this, the kinetic term in the holomorphic variable, z dot square is, if you just look to the real part of these ah, ah, complex fight quantum mechanics, it is x dot square minus y dot square. So the kinetic term is not a positive definite sign in terms of these variables. So some, ah, something that blows up here cancels with something that blows up there, and you can show this quite precisely by regulating this, by moving just a little bit off the stock slide. So indeed these solutions are there, and they contribute, and they do very interesting things. Now, ah, let me, ah, summarize. So I think these ideas, semi-classics, semi-classics, resurgence semi-classics, you know, mixed anomalies, adiabatic continuity can be used, ah, to our great advantage to learn about non-perturbative aspects of quantum field theories, either asymptotically free or otherwise, ah, and maybe, ah, even people, some people started discussing this kind of things. It may have some useful region of overlap with lattice field theory, some things can be tested, and the space of shadows now is actually richer than we anticipated maybe 10 years ago, and there are many interesting things. So I think I should stop here. Questions? Yeah, one comment. I think that the necessity of complex shadows was already quite established by boroughs and company. I mean, of course, your examples are a little bit more dramatic because not included in the symmetry and so on, but in the famous paper by Valia and Parisian boroughs, they really made clear that if you don't include the shadows, you will never get the right. I totally agree with you. It is my omission. Sorry about that. Yeah. One more substantial question. So typically, normal shadows are set to subsequence of diagrams but diverse factorially, right? Right. So when you do this adiabatic compactification, what happens to these diagrams? Because I think there is a paper by team, right? Yeah, there is a paper by team. They have to not diverge anymore. So in a sense, you cut this renormal singularity from the very beginning. It transforms to something else, probably. So we do not know the details of how this adiabatic continuity works. But the fact that these, you know, indeed the class of diagrams that used to be called renormal on diagrams, on R4, on R2, when you compactify them, they are not contributing in the way that they used to do on R4. They only do so if you take N equal infinity limit but N equal infinity is same as the compactification. So my best guess is that they really transmute, you know, in this effective quantum mechanics because of this holonomy background, you are really inducing a potential on these manifolds, okay? So I think this thing is transmuting to that. But this is an outstanding question and I think extremely hard question. I wish I knew the answer and I wish somebody, you know, can answer that question in, you know, near future. It's a very important question and I don't understand how that happens precisely. You would expect that as L goes to infinity, it would go smoothly from this description to the standard R2 description. Okay. This is actually the talk that I plan to give but I couldn't finish it completely. But I will tell you one very nice example, okay? So at least in one theory, we were able to calculate an observable for any compactification reduced. I will only plot that. This is actually the work by Saash. I think I wrote it, but N-VIP. And they consider Schwinger model in two-dimension, one plus one. And they look to chiral condensate. And actually, on R times S1, beta, you can do this for this model as well as any generalization of this. There are also generalizations of this Schwinger model that has been discussed recently. You know, on R times S1, we can calculate these condensate on small circle by instanton methods. And it gives us something like that, okay? But reliability of semi-classics breaks down here. Don't take that part literally. And now this is condensate. This is compactification reduced. And you can take the exact result and plot it as a function of L. And the result is this, okay? Now, if you take the exact result, you know, which is... Here, the result was an instanton effect. It is some number divided by some coupling times beta. And this e-beta is the expansion parameter. If you take the exact result, there is an expansion of that in this form, okay? And you can plot this. And it is actually that, okay? But we didn't calculate this perturbative thing exactly. We just realized that from the exact result you can deduce such an... And there is no other factor. So this is one example where, in the context of quantum field theory, one thing that you calculate by semi-classical methods interpolates, when you take the full fluctuations around this thing, interpolate to the result on R2. So for many things, many things that we did by using compactification, we actually expect this kind of behavior. One reason underlying this is this idea of Eguchi-Kawaii large and volume independence. Because in many of these theories, if we take large N limit, we can actually prove this constancy in the large N up to one over N corrections.