 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. So as you know, interval systems is a big subject in mathematical physics and you also have gauge theories. 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? And then I get a bunch of matrix elements of the R matrix. 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 this H bar is a plant constant, which is proportional to the omega deformation parameter, which I call epsilon. 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.