All right, thank you. So first of all, I'd like to thank the organizers for giving me this opportunity to present my work. So I'm going to talk, well, this is, I'm going to mainly explain my work with Kevin Costela, which appeared in August last year, but it's also based on a series of work papers I've written with various people, not many various people, yeah. So then as the title explains, I will talk about three things, string theory, gauge theory, interval system.  It's another big area of research and there's a huge overlapping region here. And finally, we have string theory which we don't know much about, but it's certainly intersect with the other two areas. So my talk lies in this intersection. Okay, so, and especially I will talk about quantum interval systems. There are two famous quantum interval systems which turn out to be equivalent. One is the XYZ spin chain, which is a one-dimensional quantum mechanical system with spins either taking up or down and they align on a line and they interact with nearest neighbors, okay? And or equivalently, I will talk about the 8-photox model, which is a two-dimensional lattice model. So here I have drawn lattice on a torus and you have spins located on the edges of the lattice, okay, and they interact with neighbors around vertices. So it's been observed that these two famous quantum interval systems arise from various quantum field theories. So I listed five of these, five examples of such appearances, but I will probably don't have time to explain all of them. So I will focus mostly on these highlighted ones. So the first one is from 2D3D40 gauge theories with four supersymmetries. So that's something people often call 2D necrosis of Schville corresponds by referring to the 2D case, but it also has an analog in 3D and 40, okay? And the second instance in which these interval systems appear from quantum field theories, 40, 50, 60 gauge theories with eight supersymmetries. So that's something people may call 4D version of necrosis of Schville, okay? And we also have, more recently, people have observed that from 3D gauge theories with eight supersymmetries was structures of spin chains appear by these people. Balimo-Dimofti-Coyote and Braver-Mann-Finkerberg come to Kodera-Nakajima-Webster weeks, so it takes one line to write the names. Also, we also have a little bit more complicated 4D theories with n equals one supersymmetries for supercharges and in which such structures of interval systems have been found. And finally, I will, and this is the main character of my talk, I will talk about 4D version of Schramsheim's theory, which was constructed by Costello some time ago, five years ago, and which was also the subject of studies studied by Costello, Whitton, and Amazaki. And also, that's the main theory we, I analyzed in my paper with Kevin, okay? So the question is given these, all these five instances in which the same spin chains or the last largest model appears, the question is the following. So why does a single quantum interval system, so remember those two interval systems are really equivalent. So there's a single quantum interval system appearing from multiple quantum theory setups. So this is a big question we wanted to understand. And all right, so this is a physics talk, so I'm sorry about that, so I'm going to confuse many people. But so we find a physics answer. So the physics answer is that, well these appearances are, well these quantum theory setups are actually different descriptions of the same, one and same physical system related by dualities in string theory. So once you embed those quantum theories into string theory, and you can use string dualities and you can relate one another. So there's only one system we are talking about really. So that's why the same quantum interval system appears. Okay, so this is the question, this is the answer, this is the main message I'd like to convey in today's talk. So here's the outline of my talk. So I just explained the motivation for our work. And then I will, next I will explain how integral will last models. So for example, the 8-vitics model I talked about arises from something I call partially topological quantum theory. And then I will explain how string theory realizes such a structure of partially topological string, quantum theory. And then I will apply, start applying string dualities to such string theory realization which give rise to various string theory setups which can be interpreted as various quantum theory setups which are the items in the list I showed you. Okay, and then finally I will summarize and give an outlook. Okay, so that's the plan of the talk. So far, so good. So if there's no question, I'll explain how integral lattice models appears from partially topological quantum theories. Okay, so consider first, consider two-dimensional TKFT and suppose it has line operators L alpha. So alpha is just an index distinguishing different line operators. So I put the theory on a torus. So here's my torus and take a number of line operators and wrap them around various one cycles of torus and make a lattice. Okay, so here I have five line operators making a two by three lattice. Okay, the problem we are interested in is to compute the correlation function of this configuration of line operators in this theory. So how do we do that? So one way of doing this is to divide the torus into square pieces. Okay, well it doesn't show really. So here's actually there's a square. All right, so each piece, each square piece looks like this. So you have a little square and two intersecting segments of line operators. Here I have L alpha, L beta intersecting in the middle. All right, and because of topological invariance of theory, well it doesn't matter how exactly the shapes of these squares are. Okay, only the topology models. So now the idea is to compute the path integral first on each piece and then glue the results together. So now in order to compute the path integral on this piece, first of all, we have four corners and we have to specify boundary conditions on the corners. So which I label A, B and C in DA. Okay, so for the purpose of visualizing, for the ease of visualization, let's deform this picture slightly into this picture. Okay, by this picture I really mean this one. But here the corner is cut out into an arc-like shape. But I distinguish the corner from the sides by drawing the corner with double lines. So I have this picture, but if you look at this picture, it just looks like two open strings coming from left and bottom and they scatter off to top and right. Okay, it just looks like that. And over each open string, so for example, you have an open string here propagating to the right, all right, so carry is actually a particle in a middle, say particle of type alpha and this particle sweeps out the wall line of this particle, it's this line. Okay, so wall line of particle is a line operator. Okay, and furthermore, we have open strings and which end on brains. Okay, open strings, that's the end on brains and brains are labeled by these A, B and C, D, these labels. So we have this picture. So technically I think what we're doing is we're dealing with two dimensional open closed topological point of view theory with line defects. So now let's do pass integral on this picture. What do we get? So let's say what we have four boundaries, I mean four open strings, two initial open strings and two final open strings. And let's say what Hubert spaces of these strings are like V, A, B, alpha for this one. Okay, so it's the space of states of an open string with brains A and B with particle alpha attaching in the middle. So we have four of these. If you do the path integral on this piece, what you get is a scattering amplitude. So which is a linear operator, which I call the R matrix, or alpha beta A, B, C, D, which is a linear map from the initial state space that's V, A, B, comma alpha tensored V, B, C, beta into final states. So that's this time set, okay? Is this clear? So what do you mean by path integral of this picture? Partition function. Yeah, partition function, Z applied to this. Right, so in any case this is just a matrix and so you can talk about matrix elements once you specify states in these spaces which I denote by I, J, and K, L inside circles. Okay, so by this picture I mean this matrix, particular matrix element. So this is the result of the path integral on this picture on a single piece. Now what we want to reconstruct the whole torus, what we have to do is we glue these pieces back together. So single piece looks like that. If you collect all these pieces and glue them back together, we get this picture, okay? So now to glue them together, what we have to do is the following. So first of all, we want to pick boundary conditions on the corners, so which means we choose A, B, C, D and those labels on each double line circles. And then second, I will, and then I will also choose states of open strings in each single line circles, okay?  What I do is I take the product of all R matrix elements and then I finally sum over all possible configurations of boundary conditions or brains and also over all possible configurations of states. Is this clear? Yeah, this is just a glowing of TQFT, right? So, but if you look at this procedure, this is nothing but the definition of the partition function of lattice model in statistical mechanics, all right? In which, so, spin sites, so spins are located on these single line and double line circles and spin variables are labels of states or brains over which we sum over later. And the Boltzmann weight, so that's the E to the minus KB energy. It's just the R matrix element. Was there some strange point of intersection of two solid lines? Also, that particles can change state, type of strings can change, and they'll feel old. Yeah, I'm assuming such things doesn't. Yeah, so it's slightly symmetric that some vertices we should've done. Yeah, right, that's serious. I'm assuming there's nothing like that going on. So, for the purpose of this talk, they'll become clear in the next slide, I think, why I'm assuming that. By the way, so, this is just the partition function of a lattice model. So, in conclusion, I can say that if you have a two-dimensional two-dimensional quantum theory, put it on the torus, wrap line operators and make a lattice and compute a correlation function, then that's equal to the partition function of a lattice model, two-dimensional lattice model in statistical mechanics, okay? So, this just follows from the structure of the biological quantum theory with line operators. All right, now, so what's gonna be really interesting is when there's really a hidden extra dimension in the theory. So, suppose the theory is not really two-dimensional, but really a higher-dimensional theory on a torus times some additional space C. So, what I'm doing is I take a higher-dimensional theory, I compactify that on this extra dimension C, I get a two-dimensional theory and I'm assuming that's topological quantum theory, okay? And, now, suppose it's a case, then, and suppose this higher-dimensional theory has line operators which descend to the line operators I was talking about. Then, two things have, well, this implies two things. The first, each line operator carries a continuous parameter, which I call the spectra parameter of the line, your alpha, which is nothing but the location of that line in C, okay? So, it gives you naturally a continuous parameter. Second, I can define something called the transform matrix by this picture. So, this is a alpha horizontal line making a loop in the horizontal direction and intersected by segments of vertical lines, okay? So, this picture defines an endomorphism in the space of states on a circle intersected by these line operators, okay? So, it gives initial state, I have some state here. After the action of the transform matrix gives you another state, which lives here. Okay, so if you define transform matrices by this picture and then take two such transform matrices and then they commute, they have to commute if the theory really is a higher-dimensional. The reason being, because we have a two-dimensional torsion invariance on the torus, which is, say, well, torus is this screen, say, right? So, by using that torsion invariance, I can move each line operator, horizontal line operator up and down and I try, say, I take the bottom line of horizontal line and move it up and try to pass it over to the top and try to get this picture. Naively, if you only have two-dimensional torsion quantum theory, it doesn't allow you to do that because topology changes. But because we really have a higher, well, extra-dimension C, so suppose that's the direction perpendicular to the screen. So, and generically, these horizontal lines are located at different positions in this extra dimension. So, when you try to go from this picture to that picture, there's nothing singular going on, okay? They miss each other in the extra dimensions. So, this equality holds. So, the existence of extra-dimension C implies these two things, which in turn implies the lattice model we obtain is integrable because these two things imply that there's a series of commuting conserved charges which I obtain by simply expanding the transform matrix in the spectral parameter, okay? So, the lattice model is what we obtain from this topological quantum, 2D topological quantum theory with extra-dimension C. It's not just random lattice model, but it's an integrable lattice model. And by the same reasoning, you can also show, we easily show that the R matrix I defined satisfies the Young-Bach's equation. Yes? Do you mean that our matrix has a spectral dependence and it's a spectral, Young-Bach's equation with a spectral parameter or do you mean the constant parameter? No, no, it depends on, it depends on spectral parameters, assuming that the higher-dimension theory is not topological on C. So, this theory is topological on T2 but not on C, so that I get a non-trivial dependence on the spectral parameter. So, how do you recover the, you started with a constant R matrix in the previous discussion, you had a constant R matrix? No, I didn't assume anything there. I was assuming, well, I was assuming that you can just define this R matrix and then... I was just thinking of... I was just thinking of parameters, yeah. Right, so, yes. So, you can include in this alpha or beta these labels, the spectral parameter, if you want. I see, and you never asserted that it's satisfying Young-Bach's equation. No, no, not here, not here. All right, so, so, this is the conclusion. Correlation function of lattice of line operators in a topological, 2D topological 1P theory with extra dimension C in which the theory depends non-topologically is equal to the partition function of an integrable lattice model, okay? So, this is a beautiful observation by Kevin. Which I extended slightly in my paper. So, yeah, so this really explains where the spectral parameters in integrable lattice models come from, geometrically. So, it's very nice. All right, so, are there any questions so far? So, this is just some abstract argument which I'd like to make concrete using string theory, okay? Yup. From your discussion, you assume that you are finding better or different. So, do you have any factors or instructions? Yeah, I mean, when, yeah, when spectral parameters can take certain value, the R matrix can diverge. So, I'm assuming that the situation is generics to avoid such divergence. So, I'd like to explain that there's a string theory setup which realizes this two-dimensional topological warm field theory with extra dimensions naturally, okay? So, I'm going to, I'm diving to string theory and using brains and that kind of stuff, so. All right, so, be prepared. So, consider a type two B string theory and take a stack of N-defi brains, which looks like that, and put them on a full manifold M times C which lies in, which is, so M is actually zero section of T star M and C C. So, the 10-dimensional space time is T star M times C. All right, and it turns out that for some fraction of supersymmetry to be preserved, this C, which is a complex curve, has to be flat. So, the only possible choices are whether C is a complex plane or a cylinder or an elliptic curve. We have these three cases. But in any case, we know that on D-fi brains, if we have N of them, we, on the D-fi brains, we get six-dimensional maximally supersymmetric amuse theory with gauge symmetry, U-N, or if you decouple, well, if you throw away the decoupled U-1, center of mass U-1, we get G, well, gauge group, S-U-N, all right? And it turns out that, well, because of this particular background geometry, this six-dimensional theory turns out to be topologically twisted along M, all right? And it turns out that if you analyze what this topological twist does, well, you see that it makes the theory topological on M for manifold M, but homomorphic on C. So here I get a six-dimensional topological homomorphic theory on M times C. All right, so this is, well, and in order to connect this six-dimensional topological homomorphic theory, we have to get somehow two-dimensional topological homomorphic theory, all right? Because I was talking about two-dimensional topological theory with extra dimensions, which we can take to be C. All right, so in order to get such two-dimensional topological homomorphic theory with extra homomorphic direction C, what I do is something called the omega deformation. All right, so up to now M was some full manifold, but let's specialize to the case where M is R two times torus, all right? And there's a background field in string theory called Raman-Raman forms, and especially there's Raman-Raman two form, and we give it a particular background value, which I'm not going to explain in detail, but I will do that. Then this actually introduces something called the omega deformation to the six-dimensional theory, and this breaks the topological invariance on the full manifold M down to topological invariance on just this part, T two. And therefore we get a topological homomorphic theory on T two times C, which is perfect because that's exactly the kind of structure we were looking for, okay? So now what's missing is line operators. We have to insert line operators in this topological homomorphic theory on T two times C, which we can do using open strings. So suppose you have a semi-infinite open string ending on the stack of D five brains. The endpoint inside the D five war volume theory acts like a charged particle, infinitely heavy charged particle, and as the open string moves, well, it's part line of the endpoint gives you a creative wisdom line in this six-dimensional theory, okay? Which is, which actually sits at the origin of R two in R two times Taurus. So wisdom line in the six-dimensional space time lives at the origin of R two, but some one cycle K in curve K in Taurus, and it also sits at some point on this homomorphic curve C. Okay, so we can create a wisdom line like this. So, well, and this wisdom line actually carries the, is in the vector representation of the gauge group because there are N choices of D five brains for each open string to end on end. So that corresponds to the vector representation, all right? So because now we have 2D topological component phi theory, component phi theory with extra homomorphic direction C with line operators, and so we can do what I explained before and get an integrable lattice model, all right? So that's what we did in our paper. So what integrable lattice model? Yes? Can you move it? So the wisdom line operator is supported on a closed corner or open? Let's assume it's closed because yeah, it's in a Taurus and it makes, it's a closed curve in the Taurus. It's a closed curve, so there is a string which comes in and opens and ends on a closed curve? Yeah. For the coming, okay. Right. So let me see. So first of all, we have to identify, so we have to do two things. We have to identify what kind of topological homomorphic theory in 4D on T2 times C we get by this procedure and what kind of lattice model we get from that theory with wisdom lines, all right? So omega deformation actually simplifies some theory to a theory in dimension lower by two. So it's some technical fact which has been known for quite some time. So because we are applying this omega deformation to six-dimensional theory on R2 times T2 times C, the deformation reduces the six-day theory to a 4D theory on T2 times C and this 4D theory turns out to be this 4D transform theory of Costello, all right? And... Your construction C could be the elliptic curve? Yes. So that theory was introduced in my PG thesis in 1996 when I was shown to be anomalous. Uh-huh, I didn't know that. I didn't know that, okay, so yeah. Last accommodated anomaly. Uh-huh, okay, I didn't know that. And what forces your open string to follow this loop K? Open strings don't do what you like. What do you mean? I mean, the fact rate, the going line. How do you force the line operator to be along the curves which you want? Oh, I can't do that? I mean, what we normally mean by, let's say, D5 brain, and that's Neumann conditions along the directions of D5 with more volume, and they're usually in the transversal, so the open string goes everywhere along D5 and sits in the position of the D5 at the transversal direction, but it's not fixed on a specific contour. Okay, so, for all you can explain that in detail later. So, but anyway, so we get a Fourier transform theory, which is whose action is like this. So we have one over H bar, and Lagrangian is very strange. DZ wedge transforms free form. All right, so this DZ, holomorphic complex coordinate on C, okay?  And we can calculate the R matrix in perturbation theory in H bar, all right? So you can just do that, and we have three cases. So the first two cases, we have a C equals complex plane or a cylinder. In those cases, we get rational six-barrex model or trigonometric six-barrex model, and which are equivalent to XXX and XYXXZ spin chains. So if you take the, if you look at the conserved charges of these models, they include Hamiltonians for these spin chains. Okay? And for the elliptic case, we get not XYZ spin chain as one may naively expect. What you get is something called Felder's elliptic dynamical R matrix, which carries an extra parameter lambda, which is in the dual carton of the gauge algebra. And what this extra parameter is associated with the degrees of freedom living on faces of the lattice. But that parameter is absent for these cases. All right, and, but this elliptic dynamical R matrix is related to the more well-known Baxus elliptic R matrix for Advaidex model by just conjugation by some operator. And this Advaidex model is equivalent to the XYZ spin chain. So that's what you get. So now let me, yep. So what is, is there any physical reason for just sticking yourself to the car? So, because, right, because- He even suffers late work. No, it doesn't- It's one of four, it's not managed conformity. Yes, that's one reason. But more fundamentally, because we have, we started with six dimensional theory on the four manifold M times C and I, the string theory set up construction only logically twist the theory along the M direction for manifold direction. And it doesn't do that to the directions of C, okay? So M can be any four manifold C, but C has to be flat in order for some supersmithries to be preserved. So C has to be flat. And that restricts the choices of C, okay? So because we have embedded everything into string theory, now you can apply a string dualities. All right, so first chain of dualities, well, there are infinitely many chains of dualities you can think of, but one interesting chain of dualities you can apply to this setup is S duality and then T duality in the direction, horizontal direction of torus, right? So I start with D five brains with fundamental strings ending going in either horizontal or vertical directions in the lattice, and then the run-and-run to form field, okay? Now I apply S duality, which turns D five brains into NS five brains, F ones fundamental strings into D one brains and C two into B field B. And then I apply T duality and it's five remains as NS five actually because it wraps around the direction to which I apply T duality and D ones going in the horizontal direction of the torus now becomes D zero brain, D one vertical D ones become D two brains and B field remains as B field. But anyway, so that's what I get. So suppose for simplicity, let's take big M to be two. So I only have two NS five brains, which are these, and then I have D zero and D two brains. I have D two brains coming from somewhere ending on either of the NS five brains like that. And let's say, well, the number of D two brains ending on NS five, the second NS five brains is big M, okay? The total number of D two brains is small M. And I also have D zero brain coming from the orthogonal direction and ending on one of the D five brains NS five brains. All right, so this big M is something called the magnum number of the spin chain which just counts the number of up spins in the spin chain cell, right? So NS five two corresponds to spin up and NS five one corresponds to spin down. So I get this picture after applying dualities, right? Now, let's forget about D zero first, okay? And then I get, I have this picture and let's take consider the rational case and let's turn off the B field for simplicity. Then this picture, this NS five D two brain system is a very well known one. It realizes two dimensional N equals four comma four gauge theory with U M gauge group with N matter fields. So U M gauge group comes from this part of D two brains. I have M of them and M hypermultiplex M matter fields comes from this region of D two brains, right? So I get this theory which is described by this quiver. And now Z alphas or locations of D two brains in the in the vertical direction, vertical direction is the Z direction, okay? So they are parameters determining which determine twisted masses mass mass parameters for these matter fields, okay? And now if you turn on B field, what it does is that it gives additional masses to various fields and that breaks N equals four comma four surface symmetry down to two contours. So it breaks half of the surface matrix, all right? And let me mention that generally if you look at this picture, the theory you see that theory is in the Higgs phase because NS five brains are in generically located at different X eight coordinates, all right? So if you know what I'm talking about then you see that's the case. And for the alpha all equal and when there's no B field, it's known that in the fixed phase this theory flows through a topological M model in topological M model whose target space is the cotangent bundle of the grass money on M comma N, okay? And if you have more NS five brains then what the target space becomes just cotangent bundle of various partial flag manifolds. So that's what NS five brains and D two brains create. Now I insert D zero brain. So remember D zero brain, sorry, so D zero brain what it does is actually it creates a local operator in this M model, okay? So and because D zero brain has a continuous parameter this parameter Z its location and you can expand this local operator operator in this Z parameter and you get a bunch of local operators which generate the color ring, the local algebra of local operators of the M model which is the quantum common g ring of the target space, okay? Now, but for generic values of the alphas and the B field this brain is replaced by the reverend version. On the other hand, if you remember where the D zero came from it came from a horizontal, a wisdom line in the horizontal direction of the lattice model, all right? So D zero, well, inserting the D zero brain represents the action of a transform matrix like this. So if you insert the D zero brain it inserts in the lattice model a horizontal line which is transform matrix. And therefore from these two pictures you conclude that the equivalent quantum common g of the T star g or dross mania M common M is really the algebra of conserved charges in the xxx pin chain of length M but you specify the magna number big M. What is the notation means? This is q by c star N. So this comes from glN flavor symmetry acting on. So this comes from glN and this comes from additional u one which comes from the omega deformation parameter. Your notation is slightly misleading. Small N is large and big M is small. Well, I wanted to use big M because that looks like the standard notation for magna number. These are not giant magna. Yeah, well, I agree, it's a little awkward. But anyway, so this is just a statement of 2D necrosis of Schuessville that corresponds for this particular theory. So brain and what's nice is that brain construction provides a concrete realization of the transform matrix, this T operator which is not as far as I understand what I'm aware. This has not been understood geometrically very well but from using this picture we can try to understand this T operator in terms of the quantum geometry of the target space. That can be nice. So now, so is there any question on this spot? Okay, so now what you can do is something more. So instead of D2s, what we can try to, what we can replace D2s by D4 brains to get this kind of picture. So we have KD4 brains here and each D4 brain comes from left and on one of the NS5 brains. And then there's a slight displacement in the Z direction and then they go off to infinity to the right. Okay, so you can consider this kind of brain picture. So then this brain picture, D4 and NS5 brains part of this picture realizes instead of 2D theory, 40 theory. So what we get from this picture is a 40N equals two supersymmetric case theory which is described by this linear quiver. So that's what we get if I use D4 brains instead of D2 brains. So now, so if you remember how I applied the Geoledys, I applied S Geoledys, T Geoledys and then before. So we can do the reverse Geoledys transformations. Then you'll see that these D4 brains in the original brain setup are D3 brains. And so picture looks like that. So this is a torus of the lattice, okay? And which is just stuck of ND5 brains actually. And D3 comes from transverse direction that hits NS5 brain, so which is this part. D3 coming from this direction hits the D5 brains. And then there's some region in which D3 and D5 make a form of bound state. And then D3 brain leaves the D5's and moves on into the board. So I get these strips of colored regions in the D5 brain theory, or in 40 Chan-San theory, these are strips of self-operators, okay? So we have strips of self-operators in the vertical direction of the lattice model, but we can think of these each strip as just a thick line operator. And you can identify what kind of representation these thick line operators carry. Then you actually find that they are vinyl modules of SLN. And in any case, that's what we get for the vertical lines. Now I insert the zero brain, which creates a horizontal line operator, okay? So I get the corresponding transfer matrix by doing that. And this produces a transfer matrix of something called non-compact XXX spin chain. By non-compact, I mean vertical lines or carry infinite dimensional representations. So I get a spin chain whose spins take values in infinite dimensional representations. And this is indeed what people have observed in this particular linear quiver theory. People have observed that the structure of non-compact spin chains appear. So now we can talk about, up to now, I have talked only about closed spin chains, but what about open spin chains? So we can also do that. All right, so if you remember, in order to arrive at this kind of brain setup, I applied at some point T-duality in the horizontal direction. So that's, but if that direction is not periodic, so in order to get an open spin chain, then I can no longer apply T-duality in that direction. But I can still apply S-duality, which preceded the T-duality. If I just do that without applying the T-duality, what I end up with is this setup. Okay, instead of D-4s, I get D-3s. Instead of D-0, I have D-1, okay? But if you look at the D-3 and NS-5 part of this system, instead of 4D theories, it realizes 3D theory. And it says, so this D-3 and NS-5 part of the setup realizes 3D and equals 4 surface-mercic linear quiver theory, which is the dimension reduction of the 4D linear quiver theory I just explained. All right, so, and indeed, so the prediction from this brain picture is that the structure of an open XXX spin chain should appear in this 3D and equals 4 linear quiver theory. And open spin chain has a much larger symmetry, underlying symmetry than the closed spin chain. And so in this case, what I get is the young gen. And indeed, it's been observed that young gen appears from this particular theory. So open spin chains can be understood from the same point of view. All right, so you can do something more. For example, if you take C to be the elliptic curve, and then you can apply T-duality along both directions of the torus elliptic curve, then D1, D3, and S5 brain system, I just mentioned, becomes D3, D5, and S5 brain system. And D5 and S5 make bond states, make bond states like that. So you can consider this kind of setup. And what I, what you get by doing that is something called brain tiling, okay? And it's, and it creates, realizes a particular, it's some four-dimensional supersymmetric gaze theory on R2 times the elliptic curve. And then D1 becomes D3. D3's create subs operators in this 40 theory on R2 times E. But anyways, so if you have 40 theory and compute the partition function on this geometry, what you get is something called supersymmetric index. And I'm computing how, well, and you can study how an assertion of a subs operator affects the supersymmetric index of this theory. And indeed you find that a structure of an interval system appears from subs operator in this theory, like by looking at supersymmetric index. What is the notation? Notation here, you mean? Yeah. It's a, so you have two directions in E. As you go around either direction, R2 is rotated by either by the real part or imaginary part of the epsilon. So it's a twisted product between R2 and E. At least it's not a product, it's a fiber bundle product. Yeah, okay. Fiber product, really? Fiber product, fiber. That's what I mean by, that's what I mean by, yeah, some non-trivial vibrations. That's what I mean by a twisted product. All right, so let me just summarize what I have done. So I explained how integrable lattice models can be constructed from line operators in partially topological morphine theory in two plus N dimensions, in which, so this two, so the theory is topological in two directions, but non-topological in the additional N directions. Okay, and I told you that such a series naturally actually arises from string theory, okay? And then once you realize such a structure in string theory, you can apply gel lattice in string theory which allow us to connect it to various other quantum theory setups in which interoperability has been found. Now, so where can we go from here? So if you have, because you can consider, well, infinitely many different combinations of gel lattice in string theory, all right? And the prediction is for each chain of gel lattice, you can extract some quantum theory setup in which you should be able to find similar integrable system structure, right? So you should be able to find something like necrosophosphidic correspondence for infinitely many different gel lattice frames. So you can do a lot, you can have a lot of fun here. Right now, in this talk I explained how the transform matrix T operators arise from Wilson lines in Fortician's M theory. It turns out that if you use tough lines instead of Wilson lines, then what it creates is something called Q operators which are also important operators in spin chains and lattice models. So this is working in progress with Kevin and David Guy Otto, okay? And there's some relation to something called chiral algebra's 40 theories. Because just because in this lattice model we have holomorphic directions, okay? And you can consider the algebra of line operators. Then if you just, if you look at the whole, they make a chiral algebra because we have a holomorphic dependence in that direction. That's also a work in progress with G1O. Now, at the end of the, in the last part of my talk, I briefly mentioned the name brain tiling. And there I said that I was computing certain supersymmetric index in the presence of subs operators and I find structure of interval systems. But there are actually many different versions of supersymmetric indices. You can compute in that theory. And for each version, the same story can be applied. And by doing that, you can expect to get new solutions of the Young-Bakso equation by just changing the geometry on which you compute the index. Okay, so that's all that's working in progress with Kevin and Masahita Yamazaki. And finally, the big question people may want to ask is that, all right, so x-spin chains appeared in the context of ADS-CFT correspondence. And so it's a vast subject which may or may not be connected to this, but if it is, it would be wonderful. So I guess that's all I wanted to say today. Thank you. Any questions for Jean? Well, your constructions were limited to the A series, right? Yes. So to understand something about the ADS-CFT, you need to extend this to supervision. Which you have paper on. And we believe there's some related, some version of this construction can produce super case, but we haven't worked that out yet. Do you think you can, well, does it give you a new way to study the spectrum of suspensions? Or a relation from that? That's a good question. That's a good question. Maybe, yes, maybe no. I don't feel I'm entitled to make a definite answer here.


You know what I would like to do first is in a way pay tribute to our grandmothers and our mothers which have gone a long way to where we are now which is way too far away from where we want us to be, our daughters and granddaughters to be. So we cannot turn our back to what has been done. That was our President Christine Lagarde. On the occasion of International Women's Day, she joined Dr. Ngozi Okonjo-Iwela, Director-General of the World Trade Organization in Geneva, Switzerland, to talk about women in the economy and to share their own experiences as female leaders. You're listening to the ECB podcast, bringing you insights into the world of economics and central banking. My name is Katie Ranger. This is Eleni Gyokos, who moderated the conversation. A warm welcome to all of you. I'm Eleni Gyokos. I'm an anchor and correspondent at CNN. Delighted to be here at the World Trade Organization in Geneva. As we celebrate International Women's Day, it is not lost on us just how much work has been done, how much work still needs to be done. And we meet at a very distressing time globally. There's trouble brewing in so many places. We just have to look. And when it comes to issues with the economy, with climate, with anything that has to do with the macroeconomic environment, women are disproportionately impacted. And we must not become complacent. We must continue to maintain, to gain and propel women in the workplace, in institutions, and ensure that there is education for the girl child that is absolutely vital. Two women that will continue to fight these battles for us at the very top are Dr. Ngozi O'Conjoella, the director general of the World Trade Organization, Madame Christine Lagarde, the president of the European Central Bank. They join me now. I'm delighted to have both of you with us. Thank you so much. Dr. Ngozi, I want to start with you. I was really just ruminating over how far we've come and there's spots of real hope. And then you look at numbers that have changed dramatically specifically since the pandemic. If you had to draw up a balance sheet right now of where we are with equality and women's growth, what would you say? What is your view? Well, Lenny, thank you and thank you for joining us. It's wonderful to have Christine Lagarde with us here today. I would say that, look, although we've made some progress, the recent crisis that we've experienced from the pandemic to the war in Ukraine with the attendant energy, high energy and food prices with the existential threat of climate change, I think we've been set back. And actually some of the numbers bear this out. I think in 2020, before the pandemic really deepened, the World Economic Forum had estimated the amount of time it would take to get gender equality or gender parity to be 100 years. After given all these crises, the latest estimate is 132. So we've been set back 32 years already, 100 years is so difficult to imagine that my granddaughter's granddaughter would be there and we may still not have gender equality, but now it's 132 years. The World Bank has estimated that 70 million people have fallen into absolute poverty because of this. And the Oxfam is saying that 47 million people in poverty are women and girls. So when you look at all of this in terms of impact, I feel we've been set back and it's going to take us another decade or more to even catch up to where we were before. Or more. And Madame Lagarde, here's the reality. Gender equality is fragile. It is fragile and it can be derailed. Can we catch up on those 30 years that we've regressed? You know what I would like to do first is in a way pay tribute to our grandmothers and our mothers, which have gone a long way to where we are now, which is way too far away from where we want us to be, our daughters and granddaughters to be. So we cannot turn our back to what has been done. My mother, for instance, could not open a bank account without my father's authorization. That's just a minimum. And when you look at the level of education in many economies, girls had zero access, limited access. There is progress, not where we want it to be. And girls are still discriminated compared with men or boys. For the advanced economies, I'll tell you how I describe it. I think women, particularly since the COVID hit, have been multitasking with limited assistance. They've been overworked and underpaid. They've been pretty strong entrepreneurial, but very poorly financed. So you can see that there is a series of gaps that need to be addressed and need to be redressed. So for me, it boils down to four Ls that I keep repeating myself. So for those of you who have heard me mention the four Ls, bear with me, but repetition is part of the game. Law, there are still many countries that some of you represent here that is not equal, where we still have discrimination against women. So that's the first thing that needs to be fixed. Second, learning. Girls have to be educated, need to be given access to education, and special efforts need to be made in that respect. That's not the case everywhere in the world, far from it. Third, labor. Access to the job market and equal pay for the same job. Even in the most advanced economies, if you take the Nordic countries out, because they are probably best in class, but if you take those countries out, the gap between a man and a woman is still anywhere between 20 and 14 percent for the same job. So that needs to be addressed as well. And the final point, which I made in my remarks, is leadership. We still have too few women in leadership positions, and they can do the job. They can do it better sometimes. Lada, you don't actually need to whisper that. You got me thinking, because you're talking about the generational gap. My mother only went to school until primary. She couldn't go to high school. My grandmother completely in the rural village.  And it's truly illuminating to think that. But you said something in your remarks, which I wanted to ask about anyway. Two women in a 26 ECB governing body. Should I ask what your experience is like or what the men's experience is like? What happens if you change that figure? What would happen? I'm not suggesting that we switch the other way around with 26 women and two men. I know this was the position of RBG at the Supreme Court when she was asked, how do we redress the situation? And what do you regard as equality? And she said nine women at the Supreme Court to make up for the past. So this is not my suggestion. And I have said once, the picture would look different if Lehman Brothers had been Lehman's sisters. Okay? I still believe in that big way. Dr. Ngozi, I mean, it's fascinating. You've done a lot of firsts in terms of ministerial positions, both of you. And of course, heading up these very traditional institutions. And it's been incredible to see you at the helm. What have you had to do to change the structure, the ethos, the way that women are firstly viewed within the organization institution, and also when it comes to trade policies? Has it been top of the agenda priority? Yes, Eleni, absolutely. And it's still a work in progress. So two years here, it's a work in progress. I think I pay tribute to my predecessors. But one of the things I discovered when I came is internally, we didn't have agenda policy. There was none. I asked about it, and they said there had been one in the making for seven years. But it had never been made. So the first thing I try to focus on internally is how can we have an organization of this size and importance we don't have within our organization the secretariat agenda policy. So we worked on it really fast. We set a deadline by which we must have it. And I think everybody knows I'm very results focused. I have to have a deadline. I have to push my results. We got it. Before we went for Christmas, we got agenda policy. So we have a agenda framework now. And we have the elements of how to redress some of the issues concerning women within. And beyond that, looking more at diversity and equity. So beyond different gender, there are other aspects of diversity. So that's one. Second, I want to say that when I came, I also thought just what Christine said, we need to have, look at the balance of senior management to begin with. All through the organization, but let me at least begin where I can. And I'm very proud that for the first time in the senior management team of the WTO, we were able to get four fantastic deputies of which 50% are fantastic women. And they're sitting here in the audience. It was a competition. They didn't walk into the job because they were women. All I did was insist that any shortlist must have at least one woman on it. I didn't say which woman. And when we did a fair competition, the two women were at the top. And they're sitting right here. We have Annabel Gonzalez and Angela Ellis. I want to do a shout out for the men too, because they too did their competition. So I'm not trying to say. And one of the men is, DDJ Zhang is representing us in Doha. But DDJ Pao Gam is here, who was also, he worked with Christine, by the way. He has two bosses on the podium. So, sorry, just one last point. So we did that. So we have 50%. But we just looked at some numbers. Anusha gave me, Anusha is the head of our gender unit over there. She gave me some numbers. Among our directors, we have, this is also part of senior management. We have 20, 20. It was a 22% female when I arrived. We are now at 43%. So we've doubled that. And we've doubled it through competition. I want to repeat this. All I said was there must be women on the shortlist. And I'm going to try to use that to drive the way we do business. Can I just comment on, because I think Ngozi is making such an important point, it goes down to the drilling and the knots and bolts of how things are done. Because quite often you say, we need equality, 50%, 50%. But then you start drilling into the numbers and you disaggregate and you see by segments and you understand better the nature of discrimination. When I was finance minister, I was coming to that. I think yes and no. Deliberate in the sense that it requires a bit of an effort. You know, when I was finance minister in France, I had various big partly state-owned companies. And I called all the CEOs of those companies once a year to ask them how the business was doing, how the strategy was and blah, blah, blah. And I always said, what about your boat composition? What is it like? Men, invariably it was predominantly men in many occasions, only men. And I said, how can that be? I said, oh, I can't find women. I can't. I'm really trying hard. And I said, well, you know what? Why don't you try another year? And maybe next year you will have found some women. They came back the year later, still hadn't found any women. But in the meantime, I had prepared my own list. So I pulled out my list with about 20 names of women who were competent, qualified, if given a chance, would say yes and so on and so forth. And that changed the picture. It was not in and of itself sufficient. But you have to really go into the details. And what Ngozi said, which is I want at least one woman in the shortlist. That is the trick. Because if a woman is given a chance, she will thrive. Because she has worked much harder than many others to get to where she is. And it's speaking about getting a deliberate pipeline of leaders of women within organizations and institutions, which seemed to be one of the most important steps in the value chain in getting women to the top, to the top of their game. Yeah, I really want to agree with Christine in that. It takes time. And you have to do it in a way that, OK, it doesn't make men feel like you're doing things without merit or that they are being left behind. But invariably, there are just some innate built-in. I don't know whether they are structural. Sometimes I say they are structural. Sometimes systemic biases that are there. We don't even recognize they're there. And so in order to work against those, you have to insist. And you also have to have that pipeline. And sometimes it means looking at the policies of an organization. And we are beginning to do that here, too. Sometimes policies look very innocuous, like they're right. But if the policies keep giving you the same answer, the same time, which is not delivering the change you want, then it means you need to look at them. And query is at least the right policy. So systems, policies, institutional approach, if you want to change things. So it's not that when Ngozi Okonjiwala leaves, then things go back to where they were. That's not what I'm looking for. I want systemic and structural change so that things can continue. And it can be fair for both men and women. That is what we are looking for in the organization. Is everyone on board with that ethos and philosophy? Any friction we need to know about? Well, let me say that we have some ways to go. The feedback I get from some of the women is that not everybody is on board. Let's just be frank about it. It's not because people are bad or men are bad. No, we have wonderful, productive, and fantastic male colleagues. So sometimes in built-in biases can just be there and you don't even know. So we have to really work hard. It will take time. It's two years. We are making some differences, but it will take us time. For women of color, we are talking about women. I just want to give a shout out for women of color in the world because everything you find that is difficult, it's doubly difficult for them. So that's another thing we need to work against. People don't...  Especially if you're dressed the way I'm dressed. The first thing is they don't think you have anything intelligent to say. You're right. And they don't... They immediately, innately feel a woman of color has less to say, is less intelligent. And if she's dressed like an African, oh my God, she doesn't look like us. But this is me. It's who I am and I love it. We love it. You know, when you were running for DG, the whole of Africa, we were all behind you and one of the things that I was talking with all my friends is we could not wait for you to walk in the halls of the WTO with your incredible clothes and shaking things up. I just want to end up by saying, look, don't let it bother you, I've never had. I love it. I've never had them underestimate you. Yes. It's good to be underestimated because then you just take them by surprise. So we want to know what is in the book that Madame Lagarde was saying. Yes, the book on Christine is a star. I'm not advertising for you to go and buy it, but it's good, right? For women and leadership, real life, real lessons. And interviewing so many incredible women, you know, what is the message when you have to engage in these topics, Madame Lagarde? Because I think that women, the new generation of leaders, are looking to both of you for advice on how to deal with things, how to go up against some of the tasks that often feel Sisyphean. Do exactly what Ngozi said. Be yourself, dress the way you feel comfortable, happy and proud of, and just be in the room. I think that, you know, I've seen Ngozi in so many situations, and it's not just because of her history and the reputation. There is a way of taking control of a room that you can all have. It's an issue of confidence. For those of you who are interested, by the way, there is a book written by Kathy Kay and Claire Shipman called The Confidence Code, which really identifies very clearly where we are short on confidence and why we are short on confidence. I am still short on confidence, I tell you. And I remember discussing that with Angela Merkel and she also told me she was short on confidence, which is why we generally super prepared a file, a meeting, we are briefed to the extreme, we know inside out all the details and the numbers and all that, and we spent the whole night before the meeting that we have the following day working and studying. We do that much more so than colleagues. We do. Look at yourself, think about it. And that's to make up for this confidence shortfall that we have, that many of us have, maybe you don't. It comes with age, but it's there. And I think that if you identify the issue, if you appreciate that you have to work on it, it helps going forward. Now the second thing that I would say and that I learned from the coach on the national team that I was on when I was 17, is when the going gets tough, greet your teeth and smile. Now try to do the same thing. Greet your teeth and smile at the same time, good luck. But what I mean, what she meant by that and what I mean by that when I say that to particularly young women is it will be difficult. There will be obstacles. You have to face them. You have to just get over it and keep going, keep going. Some people say, don't give up. I say, greet your teeth, smile, get on with it, just go. Because we have that strength. Each and every one of us in this room, men and women alike. And we just have to draw on that to overcome the difficulties that will come our way. No question about it. I want to do a shout out, as Ngozi did, for all women and women of colors in particular. There is one avenue of training and education where there are so few women that they are given an avenue to join institutions. And that is economics. Try to find women who have degrees and high level education in macroeconomics, in monetary related economics, very, very few.  So you have to pick your battle. You have to choose your field. And then you can excel. That's what I would say. Dr. Ngozi, I want to weigh in a little deeper into the issue of what women of color experience. And I think this is absolutely vital. And it's absolutely striking. When I travel across the African continent and I meet the most incredible women that are bearing the burden of running a home are probably the trader in the family. They're involved in informal trading. They are planting so that they can eat. And yet they're excluded from the economic realities. If you then propel this person to get a proper education and then eventually get into the workspace, then they also deal with more issues. How are you thinking about tackling this through policy or through mindset change? Or what is that change that needs to occur? Well, I think it's through both policy and mindset. And through those who are in positions of power, actually taking steps to try to change things. First of all, I believe in evidence base. So the first thing I say to people is there are so many studies. We don't have to prove anything again. And I want to repeat that because studies have undoubtedly shown that empowering women is smart economics, is smart for the family, it is smart for the community, for the country. And women plow so much of their resources back into their family and their community. They educate their children better, faster, so that this is good for the community. So the first is evidence base. So I tend to try to quote what there is to show people that we are not just saying it. It's not a good thing to say, it's actual fact. Two is when you're in a policymaking position for goodness sakes, try to do something with the position you're in. And when I was finance minister, you know, we keep saying we have to do this, we have to do that. And then it struck me, well, you're finance minister. You know, you have the budget, you can convince the president that we should do more for women using this instrument. So you have to act at a policy level. So what were the things that we tried to do, for example? We started a program called G-Win, Growing Women and Girls, trying to invest more in women. And an example of this was saying to my colleagues in the cabinet, if you can produce additional results for women, you will get additional budget increment. So we set aside a small part in the budget. So for instance, Aki Adeshinu, who is now the president of AFDB, got additional budget because he was doing the e-wallet, you know, trying to get women access to pesticides and fertiliser faster. And he was able to include two million more women on this e-wallet. We gave him additional budget for the health sector, when they did more results, we gave them budget. So we used the budget as an instrument. I remember the president announcing this when he was presenting the budget in parliament. And of course then, you know, there was a bit of an opera from the men, but in the end it prevailed. And this produced results at a macro level. So I believe in using your policy instruments to do that. Supporting women entrepreneurs through the U-Win program, we also try to do that. And then on a micro level, I also believe in, I by the way, I believe that if you change the life of one person through your action, that is powerful. Don't think that it always has to be at a global level or at a big level. Even one person whose life you can change by mentoring, by assisting, so you also try at a micro level to do what you can to change the lives of other people and I've tried to do that. And you've done a lot of that and we appreciate the work that you're doing on the ground.  I've got the tools and the policy at my disposal. But do you need women in these roles to make these big decisions on this? Or men going to be making these decisions as well? Because you went through the numbers, right? It's of the percentage of men or women in multinational institutions. Mostly men. Too many, yeah. Do we, I mean, we need the women in these roles? Or else these decisions won't happen? These efforts won't happen on the ground? You know, I don't think that it is deliberate on the part of men to ignore the women, but it's just, sometimes it doesn't cross their desk, it doesn't cross their mind. And I remember when I was finance minister too, there was a clear issue of Crédit à la Consommation, so consumers' credit, which invariably targeted women. And it's a sad reality, but the financial literacy of women is certainly smaller than that of men. And there were wonderful targets and praise by some credit institutions. So we had to put in place a policy in order to protect the women that were the targets of those advertising campaign, of those marketing campaign by credit institutions, to warn them about the consequences of taking too many consumers' credit at a time and end up in actual bankruptcy situation. Now, I had a really tough time convincing my treasury officials that it was necessary, it was helpful, it was needed for the stability of finances, not just for those women, but of the financial system at large. It took a while, and I can assure you, and I still have former colleagues of those days who say, oh yes, I remember, your bankruptcy and Crédit à la Consommation reform, oh, la, la. But it really was helpful and it put those credit institutions on their guards that they couldn't just have those very deceptive and not properly informed campaigns and decisions as a result. So that is one area that colleagues, males had not experienced because they were not confronted to it. And that's where you can add a new dimension. I feel like a whole bunch of education in this space. Can I just jump in on the men? You know, I... Please. On the issue of the men, no, I say to women, we shouldn't also just be talking to each other. It's not, because we can't do it alone. And I want to say that men can do a lot. There's a role. Two of the presidents I served in Nigeria were very, were good in terms of getting women in cabinet. The first president, Thomas Angel, increased the number of women in cabinet to 16%. President Goodlock Jonathan doubled that number to 33%. He not only doubled it, he gave the women the most important posts. Finance, defense, aviation, education, petroleum. He did that. I'm not saying that everybody performed marvelously, but his intent was to show that it's not just about the women being in cabinet, but also they can do the crucial jobs. And he wasn't that popular for it. So men can act. And men have a role. Men, in my life, many women have been fantastic mentors to me. But so have men. And some of the chances I got came from men who believed in me. And I have to say that there was my former boss, Moin Kureshi, who was managing director. There was Jim Wolfenson, who was president of the World Bank. There was Bob Zelik. Some of my bosses just had so much confidence in me that I could waltz around the place. They thought I could do whatever. So that really helped. So men can do that. And my father, I always joke to people, my father had two daughters and five sons. And I know my brothers, if they hear this, it may say you're maybe exaggerating, but my sister and I believed he expected so much of the girls. He thought we could do anything. And we always used to joke that he expected less of the boys and more of the girls. So men can be critical. That's all I'm saying. They do have a role. So please. Your father's story is the confidence point that I made earlier. He gave you that confidence. He entrusted you with what he thought you could achieve. And you did. And you exceeded his expectations. Confidence factor was there. I don't know what your mom did. She must have done quite a bit, too. My mom did a lot. I have to give her a shout out. She's still alive. When I think about the current global economic environment, I get very scared about where we are. I really, it keeps me up at night. I perhaps think too much of global issues, but I wonder how this is going to affect the next generation of girls. I worry about what's happening in Afghanistan. It concerns me to see what's happening in Iran. It concerns me, there's so many different tensions and wars playing out globally. Does this also keep you up at night? Why am I just crazy? No, no, you're not crazy. You're not crazy and it's beautiful because it's not even your direct job when it's ours to worry about the economy and to do what we can to improve the situation. As president of the European Central Bank, my job is rather limited but critically important and it is price stability, which means fighting inflation, which has been generated by the energy crisis, instrumented by the terrible war against Ukraine. And sometimes I ask myself, am I really helping? And I know that those that are the prime victims of high inflation are the underprivileged, are the vulnerable, are the lowest paid, are the women. They are the first and prime victims of that situation. So we have to do what we have to do and we will restore that price stability and we'll do whatever it takes to do that. But yes, it keeps us all up at night because it's not a pretty situation and the geopolitical, you mentioned a few countries, but there are others as well where it's hurting, where it's bad and where it's not just the act of gods that we will see because of climate change, that will happen. It's act of men. Thank you, yes, act of men. Yeah, and I, please, Dr. Ngozi. Yeah, I was going to say, you're not crazy. I want to repeat what Christine said. The geopolitical tensions do keep me up at night, sometimes, and there's so much, it's increasing the tension by the day. And of course, it's impact on the multilateral trading system and on trade is palpable. I think it's leading to more protectionism, which is what we don't want to see. We have seen that the crisis we've had have pointed out certain vulnerabilities in the trading system. We've seen the vulnerability of supply chains that we need to build global resilience. And we've seen the concentration of manufacturing of certain products, which is certainly not resilient for the world. So the crisis has shown vulnerabilities, but my issue is, what is the response to this? I would hope the response is not more protectionism because there are actually countries and people who need the multilateral trading system to work. Japan imports 60% of its food, 60%. It's a net food importer. This is a developed country. If the multilateral trading system doesn't work, how do some people feed themselves? So that worries me. If we put more trade restrictions on goods, as we did at some point during the pandemic, then what happens is people won't get access to supplies. I'm proud that members of this organization, after the initial response with those kinds of trade restrictions, did lower them and goods did move around. We had PPEs, we had other supplies that moved around and trade played its role. So I'm just saying, let us not forget that the multilateral trading system has delivered for 75 years. It's helped us keep peace until now. It's delivered goods. Let's not throw out the baby. Let's not go protectionist. Let's rebuild it. Let's reimagine it so it can be used as a tool to build global resilience. We call it re-globalization at the WTO. Use it to bring in those poor people in rich countries who got left behind. Use it to bring in poor countries who are not part of the first wave of globalization. That's what we would like to see. Mr. Lagarde, you want to jump in here? Well, I can only agree with what Ngozi has said. And if you look around and ask yourself who has benefited from globalization, the first instance response that generally comes up is China because of the hundreds of millions of people who were lifted out of poverty. But looking closer to home at the moment, Europe. Europe has been a prime example of free movement of goods, capital people, and has benefited enormously from that free movement of good and is currently still exporting about 30% of what its economy produces. So it's not just the open economy in Sweden, in Finland, or other places. It's the entire European bloc that actually has the benefit of this trade and this movement of goods around the world. We cannot let that go. And I don't want to go back to Montesquieu, but it's... Is it fragile right now? Of course it is fragile. Of course it is fragile. You have this sort of potential Western versus Eastern bloc. You have these geopolitical tensions. You have war. You have sanctions right, left, and centre. So, and protectionist measures that are on the rise. Yes, it is fragile and it needs to be secured. And I often wonder, I'm giving you insight into my thoughts now, what I think about at night. What if there were more women at the negotiating table? For peace talks. Should I be naughty? Yes, please. I'm going to be naughty and say that a lot of the tensions right now are being caused by men. And that is true. It is a lot of the tensions and the war and so on. It's like your Lehman brothers and Lehman sisters thing. You know, I'm not doing this to say that, you know, women don't have their issues and they couldn't also get it, but just look at it. I just have this slight feeling that maybe if more women were involved, there would be less ego. The amount of tension might not be so high. You know, people might be willing to talk to each other a little bit more before we make that very difficult step that might not work for the world. So I'm just praying, I think about it every day because the consequences will be so dire for this organisation and for our trading system. You know, so I think bringing women in might help. And that hasn't happened yet at scale. Well, they are not the ones making these decisions or causing the tensions from whether it's from east or from west or from north or south. They are not the ones. I think by the way, there are studies on the ability of women to participate in settlement or peace negotiations and it's documented now that those settlements and those peace agreements are actually reached more easily when women are at the table and more to the point are implemented more frequently and better. I think because we bring to the table this practical sense which strips out a lot of the inflated ego that is unnecessary. Am I being naughty now? If you were invited to engage in talks, would you agree? Yes. Peace talk? Absolutely, yeah. Absolutely, and why would I agree? Not because, I'm sure for Christine, it's not because I think, you know, it's because I worry about the world. And if there's anything you can bring to the table that would help de-escalate the situation and make people see each other not as evil, you know? It would help. I always say we need to know where we have to have strategic cooperation and where we have to have strategic competition. And both can coexist side by side. Right now we're seeing strategic competition on steroids. Can I, you know, it's, I didn't know you were going to ask this question, but you saw that instantly we both said yes. I think it implies in the way we operate exactly what we are saying, somebody with a large ego, man or woman, doesn't matter, would probably say, mm, what are the likelihood of failure? Am I going to come out of that with success? Will I be given the credit, blah, blah, blah, blah, blah? All these things, risk analysis, pros and cons, benefits. We don't do that. We just say yes, if it can help, we'll do it. If we fail, so be it. And I think that that attitude, you know, I didn't know you would say yes, I didn't know I would say yes, but that's... I didn't know I was going to ask the question, I just took a chance. Okay, we're going to start wrapping up. And I want to give you an opportunity to give final thoughts. We've got incredible faces. Yeah, I have to say, I haven't gone to a lot of conferences where I see mostly women in the audience, so it's fantastic to see this. Final thoughts, how you're feeling about the future? What the next step is for you? What the future of my child will be, my girl child? What can she look forward to? Well, you know for someone like me who is an optimist by nature, and who never says never, I'm not feeling so good these days. I'm really, really concerned that we're sleepwalking into potential world conflict. And my focus is what can I do about it? So as to save us, we see what is happening. People are dying. I lived through a war for three years, where from the time I was 12 till I was 15. So war to me is not an academic issue. I lived through it, it didn't want me a day. Sometimes not eating, running from place to place. My parents lost everything. So it's not academic. When I see the pictures, it hurts. So my question to myself is what can you control? What can you do? And trying to see whether this organization, what we can do, we're supporting members to try to help with food prices, to try to help with trade, to keep it open, to whatever we can do to be supportive of a more peaceful world where people have enough to eat and a place to sleep. That's what I worry about, the fundamentals for ordinary people. And then just praying that we won't get into a world where my granddaughter will have to witness the world that I witnessed when I was growing up. Thank God.   And that to me means a lot because it means that despite everything that I feel just like Ngozi, I have ultimate faith in each and every one of us. And I have seen many instances, circumstances where one single individual, one single woman, one single man, can actually change the map, can change the scene. So each and every one of us we can do something where we are. And sometimes it goes beyond us and there is a higher call and you have to just move and say yes, as we just did when you asked the question. I mean, for how many years was there no agreement at all at the WTO? And I'm not putting the blame on any of the predecessors of Ngozi. But suddenly somebody gets in, gets stuck. Once the data is prepared to cooperate, once to lay a ground where people will actually talk to each other and something gets agreed. That's what I have inside me that gives me faith that of course we can improve the situation if we put our mind to it and if we believe a little bit up there as well as I do. But that inside each and every one of us there is enough good to take us further into something that will be nice for my granddaughter, your grandchildren and yours one day because don't worry, that is something which happens all the time, the passing of time. And then I have to have energy to look after my grandchild. But do you think Dr. Ngozi that if there were more women in leadership roles, some of these negative scenarios that you are worried about might be averted, could be averted, will be averted? Well, it's a big sweep to just generalize and say yes. But if women of a certain character were there, yes, I think we might. So I don't want to give the impression that just any woman walks into a job and all is well. That's not what I'm saying. But yes, there are enough women leaders that I see in the world today who have what it takes. And I believe that if they were in those roles, things might be different. Thank you. Round of applause, ladies and gentlemen. Women and leadership is the book. Cannot wait to get my hands on that. I've read a little bit. Thank you to all of you for joining us today. A big thank you to the director general of the World Trade Organization, Dr. Ngozi Okon-Joyweila and Madame Christine Lagarde, the president of the European Central Bank. Thank you so much. Well, that brings us to the end of this very special episode for International Women's Day, a truly fascinating conversation between Dr. Ngozi Okon-Joyweila, director general of the World Trade Organization and our president, Christine Lagarde. The talk was moderated by Eleni Gyokos. Check out the show notes for additional material on this topic. You've been listening to the ECB podcast with Katie Ranger. If you like what you've heard, please subscribe and leave us a review. Until next time, thanks for listening.