 Okay, so thank you so much for your invitation first of all Trieste and also to ICTP and but today I'm going to talk about Integral integrals, Integravidity versus the San Juano few cities, but Gatesville and of course there are various different differences, because there are so many differences in the literature about Okay Then turn it on right Okay, it's better. Yeah, hopefully I can you can hear so So there's a relation between Integral ability and Gates series and there are various proposals in the literature But part of when people use the Integral ability to try to solve the Gates series It's something that there is a Gates series is complicated things But maybe you specialize to some particular sector or you take some special Gates series topological his theory, etc And then there is an and then there is some integral structure there So you try to take advantage of that to solve the Gates series for example So that's that's roughly in this direction and trying to use the Integral ability to solve Gates series Which is a very interesting thing Except that mostly what I'm going to talk about is the opposite direction So namely, I'm going to use the Gates series or point of view series as a tool to understand Whatever is known about various integral models in the literature so that's the direction I want to go and Well, let's see. So then for the aspect of integral models. I want to understand well Today, I want I would say that I want to understand minimally the Yambach's equation, which I call YBG with the spectral parameter Right, so this is what I want to understand. Well, there are other ingredients But I think it's often the case that one of the this Yambach's equation is the definition of Integral ability Although there are several different definitions for integral models, but this I think is one of the canonical ones So I want to understand this particular equations Well, so what I understand means that well, sometimes you might want to construct new solutions but also sometimes people find solutions for certain cases, but But people haven't found a solution in some cases So we want to explain the pattern of what is known about the Yambach's equations in the literature So that these are the things I want to do are starting with Gates series Now here I emphasize the fact that I'm going to think about the Yambach's equation with a spectral parameter And that that's important because that's really a crucial part of the story So It's likely that most people are very familiar with the Yambach's equation itself But just remind you what Yambach's equation was So Yambach's equations, so there is some operator R So it's the endomorphism of some vector space V So it has in a sense it has four indices say IJKL for example IJKL if you choose the basis of this vector space Say it's a big vector space is going to be in some representation of some group for example say SU2 spin J representation so this is a vector space V and And then so this is the operator and then I consider the three tensor products of vector space V times V times V and And then on this space I could define the R matrix Let's see so I could so this R itself is the endomorphism of V times V but for example I could take R12 which is the R acting on the first two times the identity on the third component So this is the endomorphism V times V times V and Similarly, I could define one three two three and and the celebrate to the Yambach's equation says that That we have this equation Well, it's important. So let's write here so that big everybody can see one you sorry, okay, I'm going to write this afterwards and R23 is equal to R23 R13 R12 So this is the identity so this is the element of the endomorphism so you can take a product compose it and And this is the equation Now so well this is so it's something in some cases people are really interested in this this type of equation But I need a little bit extra ingredient Which is a spectro parameter. Oh, there's a color choke. So I'm tempted to use it. So In my case what happens is that there is a one parameter? familiar of our metrics There is a matrix is a functional parameter and this is the power spectro parameter I'd like it Z for example sometimes or you So and then there is a The equation here well in general actually depends on the two parameters But here imposing condition that it depends on only on the difference. So that's one parameter You care about such details, but anyway for the most typical case. It's just one parameter now Then then there is a parameter here you U plus B and B Then these are you for a speed You so that's an interesting equation Where there's a particular pattern in the argument? and so this is the n-box location with the spectro parameter and it's well so one interesting question is whether you can find a solution to this and If you don't know anything about the integral models that the most brute force way I would do It's just like the older components of the matrix for example say for example if I jk runs over two indices for example I that has two to force power 16 Well, let's see anyway, there's many components So and then light on the components and just write on the question and try to solve it That's the most nice thing to do except that then you immediately realize that it's not easy to find a solution Because this is over constraint solution Well, for example, if this index i jk etc runs on one to n This gives order n to the six power equations But our matrix itself. It's not only four indices are to the four n to the force power for example So it's highly over constraint. So it's very hard to find a solution and if you do a linear algebra for example unless there is some miracle and there's some linear Dependency between these equations you won't find a solution but that miracle happens and in that interesting question why that miracle happens and The question is to understand that miracle from gay series Well, of course, this is not the only approach people try to take advantage of symmetry for example, one approach is to take advantage of symmetry and People use the quantum groups and other symmetries Features something has come to but but I'm working a few series. I want to understand hopefully everything in gay series Now, why is this equation important? because it's related with Existence of conserved charges which is another characterization of integral models So suppose that I have these are metrics Then I could define a statistical mechanical model Well here I define a classical statistical mechanical model defined on the two-dimensional lattice. So so there is a two-dimensional plane and There is a statistical lattice and well, for example for the simplest case, let's take the the square lattice and Let's also assume that the that it's a Torah for example periodic boundary condition, sorry, this is supposed to be on the boundary for example, so the rest is supposed to be a lattice for example and then you can define a statistical partition function by associating a factor of r to each factor So what do you do is that you associate this vector problem? Well, okay first of all dependent on a spectro parameter So you could have to associate for example the same spectro parameter you and Let's see for example. I could associate the same spectro parameter you here, etc and and then Once you see this intersection You associate this our matrix here which has four in this seat and then when you when you connect these Four-valent vertices you're going to sum over the corresponding indices. So namely if you have this for example, I jkl for example I will say that there is a Boltzmann weight I jkl But now suppose that there is a Okay, pqrs for example, and then I'm going to also associate pqrs But when we connect these things together I'm going to make a identification that s equals to j So this s is replaced by j. So take a product and and sum over the j So each time so you it's not everything you associate the our matrix to each vertex And you're going to connect everything together and each time you connect it you sum over the corresponding index And by repeating that you obtain something so eventually you contract all the indices and you define the partition function Yes, in this case, it's a torus and if you want to you can also try to have the boundary It's like a fixed boundary condition, etc on the boundary Well in that in those cases you have to make sure that Integral ability is preserved on the boundary boundary condition and there is a reflection You know, for example K matrix. There is a similar equation involving you have to take into account Effective boundary and then there is a boundary generalization of the unboxed equations for example. So So this defines so to the two-dimensional statistical lattice model once you have this statistical mechanical model So this is just a definition but what's crucial is here is that this yambak equation guarantees the guarantees that this The commute commute. Well, let's see. Should I explain this? Okay. Well, maybe I just state the fact So from here, so this is the partition function But for example, you can instead of computing everything but vertex you can try to first compute this part and then take And then combine everything together to form the whole two-dimensional lattice. So namely, you might want the first compute this thing name a single layer Well because of periodic boundary condition, it's it's identified So this is a transfer matrix and then suppose that these are Spectre parameter here. So this is known as a transfer matrix and the partition function then the power of This thing where this L is the length in this direction and the crucial point is the yambak equation means that these two transfer matrices commute to transfer matrices with different the spectral parameters commute and hence you can expand to you in you to And because everything commutes for arbitrary values you and we you can expand and the coefficient should also mutually commute so if you expand like Tt and un for example, then they should be should be should mutually commute So that's another characterization of integral models that they are mutually commuting charges So for this reason what's crucial is that you need a spectra parameter u to do this expansion If you don't have if you have a solution yambak equation without this u you do not necessarily have this conclusion because you simply cannot expand So this is the well known Well stuck in the In the integral of all those but the reason I spend some time explaining this is it's actually They are there although there are a lot of literature About trying to use gay series to understand some aspect of integral models There has been a long-standing question of how to understand spectra parameter in gay series Well, I go to explain a little bit more later, but before coming to details Well, first of all we have a yambak state question with a spectra parameter and then there are several different approaches to this and I have tried several different things over the past five years or so and What I'm going to talk about today is one approach Which uses a four-dimensional gay series Well, so this is somewhat like a Chan-Simon's like series, but in four dimensions and this approach has been Well first proposed by Costero in I think 13. It is a very interesting paper and so we are trying to take advantage of the full power of his understanding and So there's what I'm going to talk about this based mostly in work in progress Which Kevin Costero and Robert Diagraff And everything so that's what I'm going to talk about And well before we thought talking about that let's also mention that there are several different other things I have tried Another approach to this In fact, that's what I talked about here in this exact same room three years ago But I know that there isn't too much overlap, so I'm not going to assume anything and and also Well, I'm not writing a review about this whole stuff, which is supposed to be very pedagogical So if you're interested, please have a look once it comes out. But anyway, so the another approach is here I'm going to use the four-dimensional But not necessarily four-dimensional use is the other idea is to use supersymmetric Fever gauge theory and this approach I call the gauge gauge YB because it's a relation between And back to the equation of the gauge theory and Well, I intentionally wrote the suji gauge theory somewhat vaguely because there are actually several different versions You can do with this The original story I worked on was for the n equal 1 for example But you can also go to 3d and then 2d for example 2 comma 2 for example and Right, so that's and then well what's interesting about this story is that since well first of all the main idea Here is that there's a n-buster equation So what is the counterpart of that here? Well, the simple answer is that it corresponds to some duality which I call the n-buster duality so it's a duality among Among a quantum few series suji quiver quiver gauge theory So the n-buster equation from automotive duality But this is very nice because once you promote it to a duality you can compute various different partition function for example And then you can obtain various different answers for example It's just that one answer but sometimes for example in the case difficult case of work zone is that original study working on S1 times S3 But they are also supersymmetric localization function Which is sometimes I go a computer with the Francesco and also the two money sugar So I also computed this observables and then you obtain different solutions for the answer equation so it's actually you could obtain new solutions and the new solution is a It's supposed to contain all the known solution of the star triangle relation With the positive force my weight and it's a crazy generalization of the ising model with many parameters continuous mean discrete means And and it involves for example well function wise this involves the some special function or the elliptic function and You can take various in degeneration limits. There is a quantum that all the folk come assemble There you can go to gamma functions. Well the gamma function case. It's a little bit. So that's okay Yeah, sure. There is a expression involving gamma function theta functions so there are various special functions appear and And so there's a very nice structure here well, so I'm very fascinated by the story except that By talking with people I realize that this is not necessarily the type of the integral models people mostly encounter So namely these are solutions typically are not quasi classical So what is the first place I cross it means the question classical means that there is some parameter expansion parameter like h bar for this our matrix So if you have this our matrix are you for example, but suppose it has an expansion parameter h bar and Then suppose that it starts identity and first order. It's let's write is a small e r Plus for the h bar square So in some cases there is a nice expansion parameter around which the surface identity and then expand it's one to the first order and a second order, etc and If there's such a parameter the solution is called the quasi classical for example well, but many of the solutions are not great like classical whereas Many people in the literature encounter class of great like classical solutions So I'm not completely producing some of many of the things people are the most standard stuff people are discussing Now here in this approach I'm going to what Well, not this but these are particularly useful for seeing the can see quasi classical solutions And in fact, it's useful to for to see the classification of the face I classical solutions as we will see Or maybe I'm going too slow But any idea why well From the viewpoint of integral model for example, I mean from the viewpoint of integral models is very hard to see that Because it looks like a completely different integral model and unless you know that underline gaster It's very hard to see but except that it might be a different answer of the different type Which is like maybe there is an underlying geometrical structures Algebraic structure like a quantum group might be the same for example for example these these are associated as you know with the scratching algebra for example, and Well, I don't really know this is three modes gamma case but But given aspects grinding already you can try to change the representations for example So maybe it might be the changing representation might correspond to changing the geometry or maybe we are identifying the news Even new structure for example each form of ours. They might be new algebraic structure That'll be more fascinating, but I don't necessarily know because I Definitely to see that that the quantum group or if the group that structure here, but not necessarily directly So far in this approach. Yeah Yeah, yeah Well, so you're talking about a one-by-order for this key for example Well, sorry, but failed it doesn't necessarily have this are for example Yeah, yeah, I think so so I do I don't know but I do suspect that this might be a new algebraic structure underline So if you know if so this is at risk can be defined for SLN and it has to the formation parameter So these are continuous parameters like a Q of the quantum group plus integer are for example Yes, so that's a discrete parameter and maybe it might be a new a completely new algebraic structure That would be very fascinating and to some extent we already know the our metrics and in some simple cases Given the arm it if you can define algebra easily, but that's our metrics of complicated our metrics whose indices are continuous parameters, for example, so Standard method doesn't necessarily work. So but anyway, I think there is a lot to understand even mathematically About because these are concrete solutions it goes through quantum field theory, but the resultant solution is very concrete and And well, just I should say that what the spectra parameter is so in this approach vector parameter becomes in this approach as we will see a position by geometric opposition on surface surface, which I like see and Here in this case, it's actually related to some you are symmetry in super symmetry and And so then we there is a you are symmetry in the you IR could mix with the free bus in the trees So there is mixing and there's a parameter space and and that type of things. Yeah, it's related with the spectra Well, if I'm sorry No, you are symmetry in the IR. I should have said right is related to spectra parameter And well, so there are some indication questions about how to relate this and have some thoughts But I won't say anything today and Well, of course you are symmetry typical in the brain realization is a rotation in the transverse direction. So that's also geometrical so they might be relations and In fact, since I'm I started saying this there is another there is another thing I tried Which am I like a because it's a sort of straight a different particle. So What another approach is that try to use the scattering amplitude of Prana scattering amplitude of for the nico 4 and 3d heavy gem theory nico 6 heavy gem theory So in this type of approach whenever there is a for example, well, that's exactly the case in ABGM Well, there is a line I really regarded the crossing of particles for example gruelons and and then And then there's some interesting spectrum from there is a nice structure in particular the our matrix The one consequence from this is that our matrix is has written it as an integral as grass mania integral Because people are trying to write this scattering amplitude in terms of the grass mania So interestingly some of the our matrices for a young young for PSE 2-3-4 or it'd be 2, 2-3-6, so 6-3-2, 2, 2 has nice grass mania formula for example although in this case the unfortunately the manual spectra parameter is like a deforming the helicity and The physical meaning isn't completely clear so Let's try to describe the theory Kostero's theory and Kostero goes through the goes through a series chain of arguments starting with 40 nico 1 supermails And then do the twist keeping a master fermions except that after all The resulting thing that for this crucial is a very simple Well, let's consider four dimensions Well first let's light on the action and then try to explain and then this is the sigma times C and And then there is a DZ Which shall Simon's A? So this is a very simple action So let me explain the notation. So first of all first of all what's crucial is that For the geometry is always of the form demon surface times demon surface So this is where the spectra car leaves. So I said the geometrical surface, but this actually see see a lot of deer So this is why the spectra parameter leaves and this is another surface Which I could take for example for T2 for example So this is the surface where integral lattice model leaves and let's take the coordinates to be XY for the first sigma and then there is a complex parameter which I was writing Z here So this is a complex parameter. So say this like the real part in the modernized part T minus T and C to have a lot use that but the Z Z and Z bar So in particular, I have assumed that this is a complex manifold so there's a complex structure here and Well, it's actually turns out that the series top logical in so So we need a homomorphic structure here on this demon surface In fact already here I already need it up and here the series top logic top logical along this I didn't use any metric here like for writing down this action now This itself is already actually very interesting because because typically in Chan-Simon theory You wanted to do the Chan-Simon say let's start with a 3d Chan-Simon theory where people are more familiar with And in that case you wanted to define a 3d Chan-Simon theory on three arbitrary three manifold And that was actually crucial for understanding not about it because there is a 3d covariance So the original motivation of wisdom was to there's a Jones polynomial, but that's defined by projections and Somehow not serious could see that it's independent of the way its projection is done But that becomes completely trivial in Chan-Simon's formulation and that was the power of the 3d covariance So that's what's every very crucial for application of the node series Here I'm using a different thing. So I'm now I don't necessarily have the covariance is three manifold or four manifold in this case And I just restrict the geometry to this case If I that's a little bit of trick Which by inside is very simple, but So you need to change the mindset a little bit to in order to discuss integral models with our Yonbach's equation with suspector parameter So that as compared with discussion or not invariance, and then so what's this a what a chance a is Also, there is funny is that this a Is a connection of course But we actually don't use the some of the components of this Connection is the bar Bar so no easy Well, no easy doesn't mean that we we are fixing the case So that is equal to zero that would break the case symmetry But a lot of what I'm doing is that in the body of a z is unspecified and I'm not really using that to write down this action Now the chance I was a is the standard char-Simon's term action chance of action a which the a plus cubic cubic term so if you like you can integrate the parts and I think there is a factor to I guess like a Z FH a FHF so this is by integral of parts This is like a set angle But the set angle depends on the complex parameter Z And also for example the parameter here H bar one of H bar low to H bar So this H bar obviously plays a role of the front constant But it's not necessary quantized so in the 3d char-Simon series was quantized that was a little And that was because well one way to explain that but in fact we did similar operation like this So in that case, there's a char-Simon's term on the defined on the 3-fold 3-manifold But for that you need a polarization of the gauge view. It doesn't necessarily exist So you do into a part if you can do the integral parts if the 3-manifold is the boundary of the 4-manifold and to write it in the boundary of 4-manifold But it depends on the choice of the 4-manifold so that's why this label was quantized But here all the fields are already defined on the 4-manifold So that's this Lagrangian. Is there anything else I should explain? Right, DZ, char-Simon's Right, so I'm going to Consider a slightly generalized version of this momentarily, but this is the simplest case So it's extremely simple action. Now the question is What does it help? Well Just like ordinary char-Simon series We attempted to compute observables and the good observable is the Wilson line So in the case of the compact group, the usual char-Simon series The Wilson line gives nothing but it, so we attempted to do similar things But first of all we have this product geometry So we have to think about which direction the Wilson line spreads Now we have to be careful with the fact that there isn't very easy So in the usual way to define the Wilson line you have to do the parallel transport using the gauge connection to To compute the horonomy and then take a trace However, there isn't very no az so you cannot really transport in the direction of z So what you want to do, at least I'm going to do, is to consider Wilson line here on this plane Let's take this to be T2 for example, elliptic curve So remember this is supposed to be where the lattice model lives So depending on what kind of boundary conditions you want for the statistical mechanical model You might have to think about different surfaces Say it might be a disc if you want a boundary condition, then you have to think about a boundary condition of this But let's say you have a torus for example, so that the result in statistical mechanical model is because periodic boundary conditions And let's do the similar, well and then what I want is to exactly the same picture which I wrote 10-20 minutes ago Which was this picture, but now the meanings are slightly different, well first of maybe I should orient these So these are the Wilson lines, so each of these becomes a Wilson line So this is a sigma which is a topological feature, along which the theory is topological So I'm not going topological plane Now there was a parameter, well let's see first of all if it's a line Then it should be a particular point on this surface So the whole geometry is this time the surface C, well the curve C Excepted in this particular case This surface C is taken to be a complex plane as this Z, Z is a complex coordinate for example Well I'm going to comment on the generalizations momentarily But this is the simplest case So there is an infinity But and then it has a fixed point So for example this used to have a spectra parameter for example Let's see parameters like Z Say let's say w for example, and these are located At point z and w, well you can try to change the position if you like But these are located at different positions So the idea is pretty simple so there was an unknown parameter But that was turned into a coordinate of the extra dimension And this z and w should play the role of the spectra parameter Well except that there could be various objections to this So well okay so at least a statement one might want is that I consider the expectation value of these views on lines Representing the statistical lattice and then compute the expectation value And then that reproduces the statistical partition function of the integral of models that's the type of things But there are many things, well first of all why it's integrable But even before coming to that it's not even clear why this is the statistical mechanical model in the sense I may explain So namely in general Well in the statistical mechanical model I mentioned everything comes from the local interaction at the vertex So namely there isn't a long-range interaction Between the variables here and between the variable there So so you have to make sure that whatever is you're computing here After factorizes into local contribution from here Once you know that it becomes closer to the statistical mechanical model I was talking about now Let's see Now now there is one peculiar and not yet another peculiar feature of this theory Um which is that this theory So of course the feature tentatively called costero theory very interesting. This is power counting Unrelomalizable Well because if you don't have this this in three dimensions that's what the usual chance I must have but you have this extra coordinate dz So well depends on how you normalize this But if you keep the canonical dimension for the gate field here is a dimension full parameter here This makes the story power theory power counting unrenormalize non-renormalizable Which looks bizarre However, well this is a nice theory And in particular due to the equation of motion all the counter terms which you can think of Gauge invariant counter terms actually banished because the the equation motion is like basically f equals to zero So, uh, so it's actually doesn't much too too much trouble that the theory is not power counting unrelomalizable And and kevin costero has an amazing story of trying to Define use use the battery in very costly formalism to define nearly mathematical rigorously part of a different of use theory And he has shown that this theory is Is uh, it's finite except that please don't ask me why because some of the arguments I still don't understand Yeah, but anyway, but anyway, you can already see that the usual problem associated the counter term It's it's not there and also well if you're really skeptic about About this for example, there's actually a realization of this starting for four-dimensional n equal one super ml And and try to give twist the theory giving a master formula on the center and that will give a uv completion to this theory So if you like I could rewrite on that and uh, uh, and that that itself should it's fine But anyway, the reason I said that this power counting unrelomalizable is that the theory Uh becomes ir free So the long distance physics is trivial So suppose that I have this Complicated with the wisdom lines and then try to do the perturbation for in the usual rules of quantum field theory And what I do is that while they are grueling here and grueling here and grueling here, etc Uh, I have to think about all sorts of grueling exchange And there there might be loop diagrams, etc Which you have to think about well except that the fact that it's a theory is uh Trivial in the long distance means that you really don't have to think about uh, the grueling exchange From from here to there, which is far away So this means that the non-trivial effect Should come from the local intersection Uh between uh between these two useful lines Right. So here on this on this surface it really it really crosses So there is a grueling exchange And you go to higher loop order. They are more complicated loop diagrams, etc That will more complicated story But the still the fact that there is a local contribution associated to this vertex Is fine So that explains why this partition function which in principle everything is Grewed together it while entangled together in a very complicated manner Factorizes into local contributions Uh from this vertex So this is a very interesting explanation In fact in the other story of supersymmetrical theory I talked about Similar localization come from supersymmetrical localization bosons and fermions Cancelations, etc. So one loop is up. So that's that that's the explanation there But here we have a different explanation. Sorry, that's a comment. Maybe I shouldn't Um So Well now Once you have this thing Then then you can try to compute it simply just do the loop computation one loop What really will one loop to loop To uh to compute our matrix and uh and then well we call that there is these are associated located at the point z and w So that should give the our matrix And it turns out this only depends on the difference Well, which is not too surprising here because there is a translation symmetry So in this case the only it depends only on the difference. So if you do the loop computations That should keep this one But but it's still hard problem as we know in one of the few series because loop computations hard So the first thing we should do is to try the simplest Namely to do the three level Uh, so which is this diagram? So what is the three level answer? So at three level so it's the first leading order in h by expansion So so namely if you have this r h bar of z minus w But then I might like this identity plus h bar r z minus w And and what is this? Or are itself Well, so that that's going to be the correlator of two Wilson lines Yes at an angle here. Yes, right. So namely, right. So right. So the point is that here There are a lot of different observables and they might be focused each other But we can forget about the interactions among them just concentrate on the So it's a sort of factorization of the correlation function of the Wilson lines So what is this going to be? Well, you still have to do the Do the integral you write down the propagator and then do the integral So there is a rule on and there are positions here and there you have to integrate over the order possible positions but except that But you you don't don't Well, but here let me explain the strategy different ways. So, namely if you have this structure Or maybe do I want here? Let's see ij Okay, either it's fine But let's see. Okay. Well, it should be fine. Yeah, so sure for example So let's let's try to compute this and suppose that there is an index ij ijk for example and and we knew the law from from the firm diagram Which is that whenever there is a vertex there should be a structure constant So Uh, so if there is a if I double this dummy index by a for example, then what you should have is that There is a structure constant ij i k a Well, maybe I should write it here so that the correspondence is super clear And then here I have a well already say I anti-symmetrized everything for example things like this So the answer at least without doing much complicated computations for the whole computation that's not complicated, but Oh, oh, sorry. Yeah, sorry. Yeah, okay. I didn't say explicitly, but let's take a gauge group g right And in fact the point is that it's it's general yes Oh, yes, that's right. Yeah, uh, well, well, okay. Sorry. Sorry. Sorry. Yeah, okay So maybe I shouldn't use the notation a a or anything right? Yeah. Yeah. Yeah. Yeah, that's right. That's right And then you know Yeah, okay. Yeah, okay, right. So So these are in some representation. So this is in the presentation r and the presentation r prime, for example Uh, there could be different representation, for example. Yes a is that joint, right? And then what you obtain is that So this r Thanks, but thank you for asking. I didn't if I'll say explicitly. Yes So you can easily see that they should take the form of the a ta for example ta prime ij for example ij. Let's see Yeah, uh, sorry, ik I guess for a ta is the representative for the generators For the so a is the joint index and ij, etc. Are the indices of representation Well, this could be the same representation, but I know the straight in different a general expression Where the two representations are different So there should be such such groups here if after and then then well then it turns out that This is supplemented by a very simple factor z minus w Well, at least it's it's okay dimension wise as I said earlier because the theory is unreliable This theory h bar has a dimension length So at least it cancels the dimension and so this is a very simple answer So in fact it turns out the answer is exactly this Well, that requires computation, but this is very easy now It turns out this is The arm, uh, this is the what we call the rational solution well so here Well, okay, see here the construction It looks by construction. Well, the answer is out of this speech. I'm coming to but by construction because we could do the Patabasic expansion h bar So the arm matrix is quasi classical and this expansion are Is this a classical arm matrix? Let's call the classical arm matrix And uh, in particular this classical arm matrix satisfies some equation. Well, of course, uh, there is a full quantum Yonbach's equation So if you expand in h y if you expand this way And then take the whole term of all h square There will be a relation And the relation is that r12 u And r3 v Uh, okay, maybe I to save to save time. This is this type of the sum u plus v Plus, uh, r2, uh, 1, 3 Uh, 1, uh, 1, uh, 2, 3 So this is the Yonbach's equation, but uh, h bar limit of Yonbach's equation Uh, known as the classical Yonbach's equation And people have found the simple solution of this classical Yonbach's equation is this one So we have reproduced this, uh, this, uh, classical arm matrix It's a very interesting mathematics. So this is about the classical arm matrix But here's a very interesting mathematical result due to Dreamfield Saying that, uh, if you know the position, right, so, right, basically if you know, uh, this classical arm matrix Then, uh, by using the Yonbach's equation You actually know, so, so maybe you could suppose that you have, uh, two classical arm matrices Sorry Suppose that I have two quantum arm matrices here Uh, such as my Yonbach's equation And let's do the expansion and then obtain two classical arm matrices But suppose that you find that the two classical arm matrices are the same and then, uh And it turns out that the the full quantum arm matrix is actually the same So, um So this reproduced a classical arm matrix, but if you could resort to his general statement Uh, we actually know about the full quantum, uh, arm matrix Which is interesting as a statement in part of the quantum future because otherwise you have to do perturbation all the way over Uh, but if you take advantage of this structure, uh, the full structure Uh, full part of the series can be computed, uh, from the full arm matrix Which is computed by universal arm matrix Yeah, yeah, that's right. That's right. Yes Well, non-part of the corrections, uh Uh, I see well Yeah Yeah Yeah, right. So it's a finite series, right Yeah, I don't think there are any instantones and things like that. So Yeah But uh, I don't think there are instantones, etc So I think so so that's the physical argument But I believe that if you look into the argument of drink or probably it's it might be somewhere there That that that I'm not so sure maybe some people in the audience might know Yeah, so this is uh, well So, uh, at least we have already derived this, uh, simplest, uh, classical arm matrix Uh, this was Essentially, they are already this was they are already in the Costello's paper from 2013 But here, uh, we are using a more elementary approach, which is easier to understand for a quantum future No, okay that that I don't know right Yeah, so that that that requires to understand who part of part of the city, right? So part of the correction to this for example, and that seems hard Yeah, yeah, I think so, right Yeah, there is a prediction for part of so I think you're at least you can yeah We are just lazy, but you can go to high loop one loop to loop, etc And then this should match for example that that's a very yeah, that's very convincing check Perhaps I should do at least one loop for example Ha ha ha ha Right, and you find the graduates that do the two loop Anyway, I'm joking Sorry, yes Well, yeah, it should be at least order by order Order by order. Yeah. Yeah, that's right. That's right. Yes, and uh, right Well, sorry, but in fact, right, so sorry But maybe I should say that I realize that I haven't told you the most crucial part which is that uh, well, okay So you already explained explain the integral of all the r matrix, but why is the theory? Why do we obtain the solution to Yanbach's equation? Sorry, I haven't said that Which is the very most important part. Sorry. I almost forgot that that that's the problem of doing the blackboard talk I forget what I'm supposed to talk about and Right so So what is the well, so Yanbach's equation graphically means that uh, that there are three different lines, for example and I could change the related position like this for example And usually Well, you have to go through discontinuities coming from here to there Because it's a two-dimensional picture and obviously there is a point where everything There is a three-point intersection which is singular But first of all here, we have the spectral parameters z w and z w for example and z w So although in this plane the stopological plane, it looks like that they are on the Well, they might finally change it for example. There is a triple intersection But actually on this on this plane On the other plane for a morphic plane Where the spectral parameter leaves But first of all they are at the different points So even when it looks like that everything coming together and everything is singular. First of all, they are still far away In the uh, extra dimensional direction And besides uh, but the theory here in distance along this direction is topological So this is the topological plane So it actually shouldn't matter Exactly how these lines are written So the topological invariance topological The theory is the topological on this plane together with the fact that they are separated In the in the complex plane, sorry, expect the c The curve for the spectral parameter So these two facts explain why uh, if you compute these two, but two Useful lines, they have the same expectation value And so and then this was the yambas equation So this is somewhat similar to the Similar argument in knot series. So in the memory in the case of knot series There are three lines and they change the rated position that's right. It might for three And the usual argument is that that there's 3d covariance and the series topological So you can change the position without then doing changing anything and that was here But here you don't have a 3d covariance, but uh, we have the Topological stills have the topological invariance on the two plane And and the cost we pay is that we don't distinguish between over crossing and under crossing for example So in knot theory, we worry about whether it is over crossing under crossing that changes the knot, which is very crucial But here it's a plane. So I don't have any distinction So I lose some information of the knot, but at the compensation I obtain the spectral parameter So there is some intersection between the yambas integral model with the spectral parameter and the knot theory Oh, oh, yes, because the time is almost over, right? But yes, but let me just mention Well, uh, uh, so far I only reproduced it. So maybe some people might need to support it But let me let me, uh, yeah spend two three two minutes maybe Explaining what the general result is. But in fact, I we believe that the This uh, framework is very general. We can change very groups, et cetera representation, so hopefully we could reproduce Quite a bit of the aspects of the integral models And at least one one is one result. For example, I can tell you in the two minutes. I guess is that There is a classification of integral, a classic our metrics are classification due to berabian dreamfield And but the precise theorem Is a little bit more complicated But uh, roughly speaking the statement is that classic our matrix comes with three types rational trigonometric What this means is that in practice our matrix reading some polynomial in z Or it was z and x, for example Yeah, but there's a polynomial Or trigonometric sine and cosine And here in this elliptic case, we have theta functions, for example The our matrix is written in terms of this function, for example And and correspondingly the spectra parameter Leaves on c Leaves on c star Uh, lived on the elliptic curve Now there's a counterpart of this Now I think I only have one minute, but I could tell you one minute, which is that Uh, we could generalize the story a little bit. Well, okay, so far I was writing d z with shan simons a But that in fact that implicit assume that that was c, for example Complex plane the sigma was a complex plane and that was why I was writing d z with a differential z See it's a holomorphic. Yeah, holomorphic for it. Oh, sorry. Sorry. Yeah, sorry. Yeah, I was writing the wrong thing Right, so this was used to be c Sorry, yes, thank you But I could take c to be a different uh, different manifold For example, I could take to be c star Or elliptic curve, for example And then like take some differential, for example, in this case, there is a simple differential d z In this c star case, there is a simple differential d z over z And in this case, there is a differential d z where these are obvious canonical coordinates here and I use this In the definition of the action Well, so so this is the one differential But well, of course, then then you can say that why don't you take a hydrogen theme on surface, et cetera Well, now the point is that This is not the complete classification of general r matrix, but not classical r matrix So, namely, this is a situation for, oh, sorry, maybe I'm going over one minute. Sorry. Sorry. But let me finish one minute So this is a situation for you can take a classical limit. So h bar goes to zero is applicable So here I have h bar. So, namely, this is a parameter So I could always take h bar to zero and discuss classical limit However, the subtlety is that there might be zeros, for example of w, for example Omega If there's a zero of omega, it's like ascending h bar to infinity So the semi-classical argument here doesn't apply. So it's an intrinsically quantum theory at least near that point And it it might be fine as a quantum theory But it doesn't necessarily have a part of the expansion I was talking about so it doesn't fit this classification So, uh, so I want a differential Uh, but no zeros and possibly and it is the mathematical result That if you want the differential, uh, globally defined metamorphic differential on the surface With no zeros, but possibly poles, these are only three possibilities So number of zeros minus number of poles is the two g minus two And so in this case, for example, the pole, there are two poles that order of two poles at infinity In this case, they order one pole at two points. So in this case, that's number two, for example And in this case, number zero, for example And that number goes to minus two, minus four, et cetera, you go to high genus even surface So, uh, so you can convince yourself that this is the only possibility And that that matches nicely with the classical R matrix, for example Now the dream, okay, so maybe I should finish, but the dream fuel service is more, for example You cannot really find the solution to Well, other than an case in the elliptic cases, for example And there is a counterpart of that here that the bundle module has parameter, et cetera But anyway, so this is one small thing and But we are working on other aspects and hopefully we could explain Various results in the integral models from this four-dimensional gaze theory approach