 Okay, thanks for the opportunity to talk here. So I'm gonna talk about some general gravitation theory of Dabaha-Gajibra. I've been working for almost more than three years and the draft keep growing. And I just tell you only a tiny part of my project because of the time limit. But I'll be here until the end of February so we can discuss if you have questions. So the goal of my talk is to just show some evidence between the relationship between I call A-brain category, a modular space of law connection of the one-spark-shot torus of S2C with certain simpler form. I'm going to explain is some, my conjecture is so which some relation between the representation category of Dabaha-Gajibra of rank one, of rank two. Okay, to tell you the story, so let me begin by theorem by Oblenkoff. Tells that the deformation quantization of the coordinate ring of modular space of law connection character variety of one-spark-shot torus of S2C in holomorphic, in holomorphic, holomorphic in holomorphic in complex structure, complex structure J. I call it J coming from the S2C. So the coming from this really good group is equivalent is isomorphic is the sphericata. I call it SHW dot is sphericata. So I'm going to explain what it means. So the modular space of law connection of one-spark-shot torus can be written the hypersurface in C3 in such a way that as a cubic surface, two plus T square plus T minus square, whether T is a monotony or a holonomy as a one-spark-shot, the puncture on the torus. So the holonomy is really D takes a value in SL2C. And so the X and the Y are holomorphic in complex, I call it J coming from this S2C. And if you do the deformation quantization, what you would end up is the following algebra. So the X, Y, and Q combinator will give you Q minus Q inverse of C and everything C click, Y, Z. I'm not going to write down everything, but just X and Z, X, Q combinator is Y, and also, so no longer X and Y are no longer commutative. And X square plus Q minus Q inverse of Y square plus Q Z square, Q one-half, X, Y, Z, Q one-half plus Q minus one-half square plus Q one-half T minus Q minus one-half T inverse square. So this algebra is exactly the quantization, deformation quantization of this hyper surface. And so this is actually called a spherical Dabafake algebra and the relation to the Dabafake algebra is indeed given here. So I don't have a time, so just you can just read in this board. And I just want to study this representation theory of this algebra in terms of geometry of the homogeneous space of black connection. So to study first before going to the geometry, I just want to briefly review the representation of spherical Dabafake. So there is a polynomial so-called representation which sends the spherical Dabafake to the endomorphism of ring of symmetry functions of rank one, CQT is a functional field with variable Q and T and X plus X inverse is the variables and X sends to X plus X inverse and Y sends to the T, X minus T inverse, X inverse, X minus X inverse and the shift operator and T inverse X minus T, X inverse, T minus T inverse, W inverse. So shift operator acts on X, I just multiply by QX and Z is, similarly you get the difference operator but so the generator is just coming from the relation. So I'm not gonna write. So this is called McDonald's difference operator, difference operator. So for instance, so this ring of symmetry function is spanned by McDonald's polynomial which is the eigenfunction of the McDonald's difference operator under the polynomial representation which whose action can be written as follows. Since this is rank one, this L is an integer which levels the number of boxes of young tableau of one row. And furthermore, so to study the representation theory, so you just define so called the raising or lowering operator so which increase the number of boxes or decreases number of boxes. So this R is a raising, L is a lowering and so on, P and S. And so where the L is defined by X plus Q L, T, Q minus inverse L, T inverse, Z and L L, lowering operator is X plus Q L, T, Z. And another polynomial representation, the act of McDonald's polynomial as follows. P L is equal to some number one minus Q three, two L inverse T minus two, P L plus one and the polynomial representation lowering operator. So it just increases the number of boxes and the lowering operator decrease the number of boxes Q to L, one minus Q to D to L minus one, T to D four divided by one minus Q to L minus one, T square P L minus one. So therefore, so lowering operator decrease the number of boxes. And so in the representation theory, the first thing is to study the finite dimensional representation. Finite dimensional representation arises when this lowering operator annihilates the McDonald's polynomial. So therefore the numerator here becomes zero and you can just quotient idea generated by this new vector and you get the finite dimensional representation. So I just want to classify when the finite dimensional representation appears. So the only denominator, sorry, numerator vanishes the finite dimensional representation appears. So the condition is when, so they just want to remind later. So when Q is root of unity, so this factor vanishes and also when T square is equal to Q three minus odd number to avoid the denominator vanishes. So you just, I just consider odd power here and the T square is minus Q to D minus N. So all the capital N, L, small N are integers. And then the second, the second, the third will vanish the second factor. So when under these conditions, so the finite dimensional representation appears. And then the goal is to see how this finite dimensional representation appear from the geometry. So that's the goal of my talk. To see the geometry, so let's look at, recall that the, not to be a harsh correspondence which states as follows, module is a flat connection of T two point of SL to C is the homomorphic to the module is a hex bundle of one puncture torus with SU two where the, so this is module is a hex bundle. So where the gauge connection at the, there is a puncture, so you have to impose the singularity which whose behavior, so the gauge connection, this is the hex field, is one half beta, sorry, I shouldn't say, beta plus I gamma DZ over Z. So I just consider the puncture and you just consider local corner Z and the setter is the angular variable around the punctures. So this is the ramification, the same ramification for hex bundle and it will go to the, here you have a parameter T, T is related by exponential, sorry, so the alpha and beta gamma, beta gamma are carton torus of SL to C, SL to C. So therefore T square ramification is indeed related to pi gamma plus I alpha. So that's the relation between the monotony of the puncture and the flat connection and the tameramification of the hex side. So this is not a big harsh correspondence and since this is hyper-genome manifold, so we have, I take I, J, K, so the I is a complex structure coming from Riemann surface, T two, point and J coming from character variety or SL to C. Okay, so that's the standard weight. And then homomorphic syntax of form, which is omega K plus I omega I is written by I times two pi dx, which dy, motor to Z minus x, y, x, y are written there. Okay, so moreover, so if you have the not a harsh correspondence one can use the heating vibration, so the geometry schematically can be written as follows. So you have a mh to the alpha in space, which is a parameter at trace of phi square, phi is a Higgs field. So it's a complete interval system, so it's a generic fiber, so this is a heating vibration. So the generic fiber are, in this case, rank one, so the generic fiber, I call it both F, is just two torres. However, it's elliptic vibration have a degeneration at the origin, so it's called a global nipotent cone. Global nipotent cone, sorry, I think the color doesn't work. The global nipotent cone, so the topology changes and the global nipotent cone look like a fine D for singularity. It's the elliptic vibration, the singular fiber is classified as codire, so therefore it fits into codire classification I zero star, and the center guard is the usual bungee. So when alpha, beta, gamma are nipotent, or if you turn off the time ramification, so these four divisor shrinks to the point, and you get the original D-Pero case T2 mod Z2, but if you turn on the ramifications, so you have an exception of D by the D1, D2, D3, and D4. So this is a geometry of the imaginary space of block connections. And to approach the geometric orientation theory of the, of the alpha-pank algebra, so I use the 2D model, two-dimensional model whose target is the character variety or huge motion space. So, so the, I use the trick, the physics approach is the 2D model, so this is the approach by Goukou-Fwitten. So the, so the trick is to use a big brain, so trick is to use big airbrain. So usually airbrains are the Lagrangian inside a simplex manifold, but so the, the Kapustin-Witten and Goukou-Fwitten introduce so-called canonical quiescentic brain, which I call the BCC, which is a line bundle over the, this target space, modular space of block connections. And a Fuss curvature is satisfied following condition F plus B is equal to a real part of the holomorphic simplex form, omega j, and also omega, omega is, I just consider 2D model, so which is the remand surface map to the modular space of block connections, and omega is a simplex form in a model. Omega is imaginary part of one over h bar, shifted, so scaled by scale omega j, holomorphic simplex form. And the proposal, physics proposal is that the, if you consider the algebra of the, the open string, whose boundary is BCC, specified BCC, is that the, will give you deformation quantization of the target space. Which is the Sprecher-Dacher, according to the Oblenkoff theorem I raised just now. So could you just say, is B a choice you wanna make, or more for me? Oh, this B field, sorry, in physics 2D model can have some B field in the, yes, yes, yes. And the parameter matching is as follows, so the Sprecher-Dacher has Q and T, small Q and T parameter. Small Q is, here is identified 2 pi i h bar, and T is a ramification of the modular space of block connections. And also to the A model, you can have a bunch of A brain, whose support is usually Lagrangian. And you can consider the open string between the BCC boundary condition and the Lagrangian support boundary condition, and you can consider joining, you get another BCC BL plane. So therefore, so homo BCC BCC acts on the strip, another strip, so this will give you the module of the Sprecher-Dacher. So therefore, so given an A brain, you have corresponding the module. So here's the object in the A brain category is the A brain BL over there, and given BL, you have a corresponding representation theory, so that's why, so there's a conjecture between A brain category and the representation category, Sprecher-Dacher. And I just give you the concrete example, so this dimension of BL is just dimension of this home space of BCC, so it's a hubris space in physics language, it's given by the Groton-Dick Riemann-Lachformler, CH, BCC. So BCC is the line bundle, and also brain, usually A brain carries the local system or flow connection, inter flow connection, so you can consider the inverse bundle over the A brain and top class of the support L. So this is the Hilsberg-Riemann-Lachformler. So usually this is the general quantization story. So therefore, when L is a compact Lagrangian brain, so dimension is the finite dimensional. So therefore, if you look at the geometry of the Hitchi module space of module of the flux connection, the compact A brain provides a finite dimensional representation. And since we saw that from polynomial representation, we have a candidate, and we have a candidate for compact Lagrangian brain coming from Hitchi vibration we can compare. That's what I'm going to do the rest of 10 minutes. So the Hitchi vibration is a complete integral of a system, so the genetic fiber holomorphic in complex structure I, but it's Lagrangian, the other complex structure. So if you look at the genetic fiber, genetic fiber of the Hitchi vibration, I call both F, it's always the I, J, K, always the holomorphic in complex I in the language physics is a B brain, and Lagrangian in the other complex structure, okay. So therefore, so you have to tune, so this is always one of the H bar, sine, theta, omega I plus cosine theta, omega K, where I just take H, it's H, the absolute value, e to the I theta, theta is the angle, okay. However, so since genetic fiber is always B A brain, so you have to tune, so in order for the genetic fiber to the Lagrangian, you have to tune the theta always equal to zero, otherwise it makes the mixture between, seems like it forms a mixture between I and K, so it cannot be Lagrangian. So therefore, so in order for the genetic fiber to be Lagrangian, H, so the theta has to be zero, so this is the condition. And also if you can compute the dimension of the home space BCC, I call it, let's call the BC, B bold F, B bold F, so you will get the one over H bar, okay. So since, if you use the Groton-Degren-Riemann block form, this has to be integer. So therefore, so if you look at the parameter matching between Q and H bar, this will give you that the Q has to be a root of unity, pi I of some number, capital N, N is integer. So which is the one I have written down here, okay. So therefore, so the genetic fiber will give you first condition and is the final dimensional presentation. Another compact, evenness, okay, okay, I will tell you. So evenness comes from the following fact. If you look at the fiber, so during the class relation of homology, genetic fiber can be written in terms of two times bungee union with I, one to four dI, exceptional divisor. So therefore, so this is two and the volume is four, so it has to be even, okay, okay. And also if you look at the exceptional divisor dI, so dI is the, let's call the brain supported on dI. So this becomes Lagrangian for genic omega x when the alpha plus i gamma, so which is the ramification specified for Higgs bundle is proportional to IH bar. So proportional means it's a real. When this is condition satisfied, so the dI is Lagrangian with respect to omega, omega here, for genetic omega, for genetic H bar. And if you compute, so this dimension of BCC B and also which is equal to C minus one half is integer where the C, C is actually relation between Q and T, so this is called the central charge. And if you look at this relation, so this condition will give you T square is equal to Q to the minus two K minus one, okay. So this to L, sorry, where L is integer. So therefore, so this brain supported accession divisor will give you the second finite dimensional representation. And furthermore, at the end, so you can ask the same question for Banji. I don't think so. So brain supported Banji. So this Banji becomes Lagrangian when this ramification satisfied the following condition. One half to alpha I divided by I over H bar is equal to, is proportional to I H bar, which is real. And if you compute dimension home, BCC B Banji, so this is equal to two C minus one over two H bar minus one, which is integer. So then, so you just use this condition and it just plug into E to the two pi I H bar, and so if you use this condition, you will get this relation. So this relation will give you the third condition. So therefore, the Banji will give you the third condition. So therefore, there's a one to one correspondence between finite dimensional representation of a spherical data and a compact A brain coming from hitching vibration. So that's the story. And furthermore, if you use the so-called fiber class relation, somehow, sometimes brain supported direct fiber can hits into the nepotent cone and then it splits into the two part of the nepotent, global nepotent cone. So the module for fiber is the extension of the sum Banji plus expression revised D1 plus D2 is splits into Banji plus D3 plus D4 by using fiber class relation. And for instance, at some certain condition you can have the extension module. So that can be understood that the brain is decayed to the two splits in the two Lagrangian brain when you hit in the global nepotent cone. So therefore, you can explicitly see how the representation appear coming from this hitching vibration. Thank you very much. Any questions? Maybe I'll ask a question since the various speakers mentioned the affine Youngians. Could you mention how this algebra relates to affine Youngians? Okay, so yes, I can explain to you. So this is the story of the, K-theorics story of the, the story of the morning sessions. And if you consider, in particular, so this Freckard Dacher can be understood as the algebra line operator in 40 n equals two star theory. And if you take, so 40 n equals two star theory are released on circle S1, you get a 3D n equals four theory, four star theory. And this, so the, so this Coulomb branch of 40 n equals two star theory can be identified as the module, so the flag connection I talked about. And the module's Coulomb branch of the three n equals four star theory, if you do the deformation quantization in the similar story, will give you the Freckard Dacher, but it's trigonometric version, trigonometric version. So the Kuiber diagram, 3D n equals four is, for instance, like SU2, which is gauge group in the Zildar joint. If you try to look at the Youngian version, is Youngian version indeed, you have to add one more, one more hypers, four. Then the Coulomb branch of this theory, and if you do the deformation quantization, will give you a spherical Dacher of rational chain algebra. Spherical rational chain algebra. So this is indeed the Youngian version of the Davao-Hecki algebra. So it's kind of a Li-Aziba version of the Davao-Hecki algebra. And it's indeed a Huber scheme of points. Okay, yes, and your case is not rational, it's not trigonometric, but it's like elliptic? Yeah, elliptic version. Yes, it's like elliptic toroidal. Yes, yes, yes, yes. If you take the rank goes to infinity, you get the elliptic whole algebra, GL1-Troidal. Okay, that's good. So if you take instead of one puncture, let's say n punctures, Yes. The n type is just increase the rank, rank. SL2 goes to SLN, or you just wrap the n and five rings. If you take n punctures. You get some different algebra line operator of classes theory specified by this n punctured torus. Right, which is a quiver with vertices. Yes, yes, so I think it becomes like this way, right? Like the quiver with this part. Here with vertices, the quantization of its column branch gives, if you do it like in the elliptic version, it gives a toroidal elliptic algebra of GLn, where n is the number of vertices. Oh, I see, I didn't know that. Maybe. In a piper of, Okay. Okay, so then maybe I should, there must be some real duality between them, Okay. Oh, okay, maybe I will learn from you during this time. Okay. I think I should stop it maybe.