 Thank you very much. It's a great honor for me to give a talk on this workshop celebrating Kodaka Shibara's birthday. So, Kodaka Shibara developed many important theories, but I cannot, so I can produce only new spaces, new varieties, and today I also send my recent construction of new syntactic varieties and its contentation. So, this is based on the joint work with Braver and Finkaba. So, the paper appeared just this, today. Okay, I first reviewed mathematical definition in Coulomb branch. So, Coulomb branch is originally for serving mathematical physics, but we try, so then the definition is, in mathematical point of view, it's almost clear and you cannot justify the physical definition, I believe, but we give some alternative mathematical reverse definition. So, let us take reductive complex data, reductive group. Then I consider the affine plus manual. So, this is modular space. So, this is a pair of bundles and trivialization. So, P is a G-bundle over the risk and phi is a trivialization except the origin. So, D is a formal risk and D is the puncture risk. And you have the action of G-o. So, I did not do this by all. And so, you have the action of G-o, affine plus manual. And so, I will later use geometric satake correspondence to respond when I just recall. So, you consider the category of power sheets, G-o equivalent power sheets on affine plus manual. And then you can define convolution product. So, then it becomes a monoidal category. Then, this monoidal category is equivalent to the category of representation of Rangran's DR group. And this convolution plus corresponds to the tensor product. This is Rangran's DR group. So, G is also we take N. So, this is a finite momentum representation G. So, this Coulomb branch is defined for a given pairs of group and its representation. Then, I consider a variety, which is a kind of certain space which lives over affine plus manual. So, this is a modular space of triples P and phi and S. So, P and phi is the same as affine plus manual. So, this is a section of associated bundle. So, you consider, so since you have representation G, so you can consider vector bundle twisted according to this P. And then, you import this condition. So, you send section by phi, the distribution. Then, phi is a ration just, just, triplication outside the origin. So, phi of S may not be regular at the origin. But, we require this to be N of O. So, this is regular at 0. So, this is a closed condition. If you don't put this condition, then it is a vector bundle over affine plus manual. But, because of this condition, this is just a closed sub-variety in the affine plus manual. And, do you have an action of G of O? Or R? This is good to wait. Maybe I'm confused how to use it. So, this is fine. So, we consider G of O is equivalent to Borelmua homology of R. And, I mean, I don't explain how a compilation product is defined, but you have considered some diagram. For the left end is a product of affine plus manual, and the right end is affine plus manual. Single copy of affine plus manual, and you pull back and push forward. So, on this variety R, you have similar diagram, and you can define a compilation product. So, this is a ring. And, in fact, because, I mean, you already see in this geometric set, okay. So, this tensor product is, of course, commutative. So, this is again, basically the same reason, this is commutative. So, we just define Coulomb branch, MC. So, this, as I said, this depends on MC of G and N as its specter. So, we show that this is finitely generated, and this variety is affine algebraic scheme, but in fact, this is affine algebraic variety, no normal affine algebraic variety. And, so, this, this construction, for various choice of G and N, you get many, many spaces. And, this includes many interesting spaces. For example, slice is in affine glass manual. It's rank of G. Demention is equal to twice, twice. Slice is in affine glass manual. So, this appears, if you choose G and N from those, you take quibagase, quibagase. I will explain later for very special example of finite type. And, but even for affine, if you take quibagase or your affine type, then you get so-called Chex bo variety. So, this is a new kind of variety, which was introduced to Chex several years ago. And, this, this variety has a natural quantization. Yeah, that I will explain now. So, you consider the C star action on the disk. So, you consider the equivalent homology of larger space. So, this C star action also affine glass manual, and R, space R. Then, so, this, this becomes non-commutative. And, you can consider this to be a deformation, non-commutative deformation of the Coulomb branch. So, deformation parameter H bar appears like this. You consider this equivalent homogenous additional C star. So, this is a polynomial in one variable. So, this, this is non-commutative deformation or quantization. Which object in the blackboard is called the Coulomb branch? MC. So, it's just upstairs. MC. MC, this spectrum, this commutative ring. This, this is usually called the quantized Coulomb branch. So, then, because of this, so, if you have F and G functions on the Coulomb branch, you can define Poisson bracket. So, first you lift it here, and then you take commutator. And, this, this vanishes at H bar equal to zero. So, you can divide by H bar. And, then, you specialize to H bar equal to zero. So, this is a definition, usual definition of Poisson bracket, once you have this non-commutative deformation. And, you can also show that this, this, this gives, this is coming from symplectic, symplectic form on regular locus, the Coulomb branch. Coulomb branch is in general singular, and we conjecture that this singularity is so-called symplectic singularity. In the sense of volume, but so far we don't have proof of this statement. And, there are, I mean, in some case we can construct a resolution singularity, but that's, that's not so often. And, we do have an example which is known to be, that, which is known to not having a symplectic resolution. So, you really have some singular varieties. Okay, so this is just to review any question. So, when you say, symplectic singularity, do you mean? So, this is in the sense of volume. So, if you take, there's some resolution, not, so you, you can pull back this symplectic form defined on regular locus. Then, it extends, but not necessarily non-degenerate. So, if it is non-degenerate, this is a symplectic resolution. So, symplectic singularity says that the existence of such resolution. And you, I mean, there are some, some, some general theory about this structure of variety having symplectic singularities. So, you can use some, some result. For example, the number of symplectic leaves are finite. So, it's a kind of nice space. Yeah, maybe in this example, so you can directly check the number of symplectic leaves to be finite. So, I'm not sure that this is useful or not, but I think it's a, it's a little optimistic, but it's a natural conjecture. Okay, now I just start new construction. So, from this definition, you have map from space R to to affine glass mania, just for getting section. So, instead of considering Brouwer-Hommology of this space, you just consider objects, sheeps, sheeps on affine glass mania. So, I denote this by a. So, this depends again by g and n. So, this is pi star omega r. So, this is a deriving complex on r. So, you can recover the, so, this is, I take everything in the equivalent category. So, pi is not proper. Pi is, I mean, so, the pi, pi, pi is in fact infinite dimension. So, infinite dimension vector space. So, you must be, but it is, you can approximate when, I mean, I don't give a precise definition of this homologics, but you can always approximate it by finite dimension of space in practice, if you fix the degree. So, fiber is a vector space. And, but the dimension of the vector space changes if you change the point in the finalize mania. So, this is a complex. So, this is not, I mean, you must take in the completion, but basically this is complex on the affine glass mania. And, if you take homology of the affine glass mania, then you recover this the original. So, in that sense, you can, you can, instead of working on this, you can consider this properties of this shift A. Then, you can show the following. So, this is the first method. In fact, this is not so difficult. So, A is a commutative ring object in D. That means, you have multiplication. So, as I said, on the category of fiber shifts, geo equivalent of affine glass mania, you consider the gene geometric, you consider the convolution product. That extends to this category. So, this is a convolution product A itself in this. Then, you have a morphism to A. And, you also have a unit. So, this is 1. So, this is skyscrapership at the unit. Maybe I should say that this skyscrapership is a unit for this tensor monoidal category. And, satisfies this usual property. So, this unit under associativity and commutativity. So, commutativity means that, for example, commutativity means, so, you have, so, this in geometric sattake, in the proof of geometric sattake, you construct commutative constraint from here to here. Maybe I need a little bit. Yeah, I cannot really distinguish the two A's, but, so, you swap two factors. And, there's natural isomorphism between them. And, you have map, maybe 1, 2. So, this is commutative diagram. This is commutative. So, the derived category is of what kind of, which kind of coefficients? I cannot hear. For the derived category, you take which kind of shapes? Constructive shapes. Constructive shapes. Geo equivalent constructives. Yeah, complex. Yeah, complex. Yeah, you can work on by integers probably, but we are not sure that what kind of space we get. Is that an E2, I mean, is that an E2, okay, you know? That kind of thing, I don't know. Yeah, in fact, I said that the proof is easy, but, in fact, our proof is so far not satisfactory, and we at some point, some, that argument there is. So, observation. So, there are other examples of commutative ring object. Yeah, in fact, there are not so many, as far as I know. So, one is, of course, unit. Unit is 2-3, but another one, important one is this, what we call regular shape. So, this is, this corresponds under geometric satake to, to regular representation. I think it's a great G check. Ah, let me see. Yeah, maybe you are right. Regular representation. Yes. So, we want to have objects in our fine glass veneer. So, it's really confusing which, we should put the check. So, natural question is AR appears as A of G prime for some G prime and N. And in fact, yes. So, this is a very, two different construction. I mean, this is not really construction. So, we have to make this construction and we just ask this old example can be constructed in our new way. What? Yeah, yeah, G, the group must be changed. So, you need also luck to map it to the same glass veneer. Yeah, same glass veneer, but G, yeah, it's a little bit complicated, more complicated, I will explain. In fact, the answer is yes. I'm after this modification for type A. So, G prime has a map to G, but G prime is much bigger. I mean, this is not, in general, not pervasive. So, this is just complex. It's okay. So, but it's not in the direct category of pervasive. So, it can be in the direct category of representation of the complex. Not, not, it's, I think it's a little bit more complicated. It is a variant, derived category, which is much more complicated. So, it's not just derived category of pervasive. It's just a, it's a, it's a, it's not a project with grand, derived category, which is considered, which you can also describe on the little side, but it's, yeah, there's some discussion by Bezo Kashnik, think about, but the object, yeah, this category is much more complicated. So, you don't know, you don't know what it becomes, is there some category? I don't know. Yeah, we don't know. No, it's absence is the whole point that they don't know this. Otherwise, if we construct this common branch, it's completely algebraic. And what do they want to say? So, so, I know, I mean, at least after there's some modification of this construction, we hope to get this regularity for classical groups constructing similar, but, but we are sure that the exceptional groups for regularity, for exceptions, you cannot construct by any choice of G prime Y. So, this is really new construct, new, new, combining. And once you have this commutative ring object, you can take a tensor product, that is also commutative. I will explain it. The column branch is simply a very small dimension for a place rank, yeah? Uh-huh. And suddenly, is it, is this group of, the influence, don't group on it, what, what, what do you think? It's an object, what? I don't quite understand the question. So, this is object, A, A, A, are you asking about A? Because the spectrum of this, I think you get some variety of section G check, yeah? No, no, there's no action of G check. G check only acts on the model of the previous chief, and this is a model of something, well, it's not, it's equivalent to a model, which is, I already take a problem for me, no, no action on G engine. Okay, so, so, now from now A is just commutative ring object. Then, as I said, so, then you can h star of G of O, O equal G of A. So, this, in the original case, you recover the, the model, so this is commutative ring again, and also you can slightly modify, you consider x of skyscraper shift at the origin between A, then this is also commutative ring. So, in the original case, I mean, this becomes trivial, but in general, if you consider some case, I mean, for example, sorry, I will consider, let me see, maybe I need to expand it a little bit. What? Is it graded commutative ring? That is, is it graded commutative? Graded commutative ring, yes, yes. But there's no, no auto-homology group, so it's really commutative, not super commutative. And so, if you take this regular shift, so this is studied many years ago, like hop, there's a crack new and Ginsberg into 2004. So, this, maybe I start with this example. So, spec of x group of one A. So, this is near potential for the Lagrange-Dier group, and the spectrum of h star of G of O, F, R. So, G check times G divided by double. So, this is basically constant size. So, you get, let me see, this, yeah, you get some symplectic varieties, and from our point of view, you can consider this to be construction of symplectic variety using regular shifts, but of course, in this particular case, you get some old friends. But it always says something, I mean, this kind of construction might be important in, because near-potent comm are very important in geometric presentation theory. So, this kind of construction may give us important varieties in geometric presentation theory. So, I slightly generalize modified construction. So, this is the notion of flavor symmetry group, which is just used quite intensively. So, we consider larger space G children arc theme. So, any representation larger group which contains G as a normal subgroup, and I take GF to be quotient. So, this is called flavor symmetry group in physics literature. Then, we have pi. So, I consider the larger group. So, this is pi. So, group of G children. But since G of F is a quotient of G children, you have a map to group G of. So, I denote the composition by pi. Pi children. So, you can just mimic pi star of omega of RG children. So, if I take this to be A. So, then this is the object in G of GF of Grassmanium G of F. And this is again commutative ring object. So, in order to recover, as I said, for type A you can recover this regular sieve. So, we take the following construction. So, this is a query of type A. So, n minus 1, n minus 2, and 2, and 1. So, this means G. So, maybe I put out G is product of GLI up to this circular vertex. I want n minus 1 and G children is basically you add GLN or additional one times G. But you divide by all scalars, diagonal scalars. And n, n is just all linear maps corresponding to ROTS. So, if you know the definition of quiva varieties. So, you consider n direct sum of n and its real space. Then you have a symplectic form and you just make symplectic reduction by this group G. Then you get nilpotent cone of type A of GLN. And so, this is some miracle guess. It doesn't happen usually, but so, Coulomb branch of this quiva gauge, sorry, coincides with this quiva variety. So, this we proved in earlier paper. So, this Coulomb branch for this example is also n. And in fact, we can make a refinement of this. So, this above construction, pi children star of omega RG children. So, this is you get object in the flavor symmetry group, the PGLN. So, this is regular shift. So, this we proved in this paper, appeared in this morning. So, in the remaining time, I will explain the application of this to the understanding Coulomb branch of star shaped quiva, right? Coulomb branch of quiva gauge. So, we combine, we take several copies of this example and then combine at the center at the vertex n. So, this is the star shaped quiva gauge, sorry. So, you can have l legs, l legs. And Philippe explained that, so, in this case also, G is product of all GLs and divide by again by central scale. So, Philippe explained that corresponding quiva varieties are open subset in the space of flat connections on S2 minus L points and with some monodomy fixed around this punctures. Maybe I just said character variety. No, in this example, no. And this correspond to, as I said, regular near-potent orbit and it correspond to taking semi-simple, regular semi-simple conjugacy class at f, at those points. Then, so, you can const, you can first make n this central vertex to be flavor vertex. I mean, you don't take the linear group and you just take push forward and to this affine-glass manual for this central vertex. Then, you just, from this theorem, you just see this, this a of G and n is just i of delta shrink of a r to a r. So, you take l copies, right? So, this is basically Q-ness for me. So, as a corollary, so, this spectrum, Coulomb branch, is the variety which Ginsburg and Kazdan introduced recently. So, this is one application. And so, Ginsburg's Kazdan's paper is still in preparation, so maybe nobody knows their definition. So, I must recall what it is. But basically, this is the right-hand side. You just take right-hand side for any group. This is the spectrum. So, this is, I mean, they use some algebraic construction, but by a lot of, I forget his name, but some student of Ginsburg saw their construction is same as this. So, this is for generality. And in this case, as I said, AR is something, I mean, by geometric circuit. Again, this is a regular representation. So, everything you can do by explicit construction. But, so, once you have, as I said, a regular, so, this commutative ring of, you don't need to take the same one. So, you can take any, any, any commutative ring of which you can make the same construction. So, you get really much more examples of this kind of construction. And, so, the remaining time is, maybe I should explain this. The motivation of Ginsburg has done construction and why it may be something to do with star-shaped quivers. So, that is what I want to explain. So, that is based on the conjecture of Mu and Tachikawa. In fact, this is, so, this is a physical fact. And they just translate it into a statement, mathematical statement. And so, for them, it's not really conjecture. So, from physics, physics perspective, so, they expect, there's a functor from two Buddhism categories, two Oromax and Plectic varieties, category of Oromax and Plectic varieties, this group action. So, to S1, so, if you, you first take complex reductive group, that is first fixed. And, so, this construction gives long runs dual. So, I just, I take G, G check, corresponding to S1. And, so, for sigma, so, this is some lemur. So, this is topological one. So, some, some, too many for this boundary. Then, it associates M sigma. So, this is Oromax and Plectic action. And, you have an action of G check of number of boundaries. And, what is the composition of homomorphism? So, if you have sigma1 and sigma2, then, so, this is sigma. M of sigma is a simplectic reduction of the diagonal. So, at each boundary, you have an action of G. So, you, you, you, you, so, this sigma, one, sigma2 have a common boundary. So, you have a corresponding group. You take the simplectic reduction by this group. And, there is some normalization condition. Before that, the first, the first one is G check. Yes. Thanks. So, if you consider the cylinder. So, this is the identity in this Buddhism category. So, this should correspond to identity object in this category. That is just the quotient bundle of D check. And, so, if you take this disc, then, it should correspond to G cross constant slice. And, so, from this action, so, once you have this three boundary, then, by, by this gluing, you, any, I mean, two manifold can be constructed by gluing. So, by this action, gluing action, so, this is what I call gluing. You can construct variety for anyone. But, so, we don't know what it is. There's no problem of isomorphism or, or, or your actions. Only have a symmetric group of actions. And, I, I don't know. You don't, you don't put this, and there's no symmetric group of acts. Your variety at sigma, everything is canonical. You can make everything canonical. When you do the composition of cobaltism, it's, ah, so, this is non-trivial statement. So, you, you, you have, so, this is what you want. So, you can divide this or this, and you must get the same isomorph. Yeah, that I don't, I'm not quite sure. And, in fact, this is based, even for, for, for g, g check is sl2, then, this m of this, this one. Maybe I just write m3. m3 is just c2, tensor c2, tensor c2. So, this is easy, but this, this, this checking of this is non-trivial statement. And, g check, ah, for sl3, then m3 is, ah, minimal nil potential for e6. Minimal nil potential. So, this, this we proved. And, so, you, you see this e3, if you remember, this, this is coming from the star-shaped quivers. So, when sl3, for sl3, we consider this, this star-shaped quiver. So, you, you see this e6 link integral from this, and you can really show that this is minimal nil potential for e6. But, this is some non-trivial. From the definition, I don't, it's not so clear. So, anyways, I just explained the physical intuition why this, this, this, this construction, for type 8 is something to do this quiver geyser, star-shaped quiver geyser. So, if you have some supersymmetric quantum field theory in three dimensions. So, you can assign the so-called Coulomb branch, which I just explained in the case of geyser. But, you can also consider the Higgs branch, which for the geyser, this is, this is m, mc of gn for geyser. And, the Higgs branch is just n plus n star divided by g, for geyser. But, physicists recently consider another construction, different construction of supersymmetric quantum field theory. So, they start with 6d, some mysterious theory in 6d, and you're compacted by Riemann surface. Riemann surface, or this, this is too many times this one. Then, you get some supersymmetric quantum field theory in 3d. So, this is example. Geyser is another example. And then, from this, you can get Coulomb branch and Higgs branch. So, expectation is, if you consider the Coulomb branch of this theory. So, this is open subset of character variety. And, Higgs branch is this, just I described, m of gn. And, this gluing property is also consequence of this construction. But, this 6d theory is something mysterious, so nobody quite understands. And, physicists say this in the category of this supersymmetric quantum field theory, you have involution called 3d millisymmetry, which says if you have a theory A, which have Coulomb branch and Higgs branch, and if you apply millisymmetry, you get another theory whose Coulomb branch and Higgs branch is swapped. So, as I said, if I start taking the Quieberg Geyser, Starship Quieberg Geyser, it's Higgs branch which I just explained. And, Higgs branch is this open subset of character variety. So, if you just make change, then you get this correct answer. So, Starship Quieberg Geyser theory. And, this 6d, compact with s2 minus n points times s1. So, this is 3d millisymmetry. And, so the 3d millisymmetry is some new phenomena. But, people somehow started to understand some relation between the, some mysterious relation between the quantization of both sides. So, I think, so it may say something about some quantization of character varieties. That is, I think, a very interesting problem to study. Thank you for your attention. So, medically, the duality is implemented as what? As a symplectic duality. But, I mean, what does it mean? Well, but mathematically, so this, this category of supersymmetric quantum fields in 3d is very difficult to make real. But, in this example where you know, you know the mirror duals. So, they just naively say that in this 3d millisymmetry, Coulomb branch and Higgs branch are swapped. That is only, I mean, physical definition of this 3d. The quantum level as well, yeah? Quantum level, I mean, if you, ah, let me see. So, if you, you, you have two, it's something similar to this usual mirror symmetry. In the usual mirror symmetry, you have two u1s and you exchange two u1s. And, in this 3d manager supersymmetric field, you have two SL2s and you swap this. It, it, it's all homomorphic, like, categories of coherent shifts on one or exchange with categories one. Not something like that. So, so, at this moment we only understand the, the statement in symplectic duality. I mean, you can see, symplectic duality, you don't need a 3d mirror symmetry, in fact. So, you can say it's a relation between the Coulomb branch and Higgs branch. But, you have both Coulomb branch and Higgs branch have natural quantization. And, those, and you have, ah, a notion of category O's. And, then, there was category A, C, U, D, R, D, R to each other. That is a statement of a symplectic reality. Maybe just, you, you, you started to speak about Gisburg-Karstner variety and then said it should be related to this moral, uh, conjecture. But, you didn't explain how, what is the relation between Gisburg-Karstner? Ah, so, this, this satisfies, this is just- M3, okay, yeah. M, M, R, yeah, so, this is just, I mean, I, I just only consider the case of, uh, sigma, this, uh, two sphere, but it can be generalized arbitrary, arbitrary surface. So, it's just sphere with L branches, yeah. So, on the Gisburg-Karstner, Kazdan proved that this growing, most important growing property is, is satisfied. So, is there an expectation, uh, uh, what happens to your 2D, uh, sphere theory, if you're quantized, if you're on place M3 by, uh, its quantization, does, is there a three-dimensional theory or something, is that, I mean, physically, is there something like that, or? M3, I'm not. Because M3, M3 comes as a, as a, whatever, as a Higgs or Coulomb branch. M3 is a Coulomb branch, uh, from this, I mean, you, you must, otherwise you must start with from this. And this should, this should have a, a quantization, uh? Other countries, what do you mean by other countries? There's one, I mean, uh, so there's, uh, upgrade of this, this construction. So, instead of a simplistic variety, you can construct the vertex operator algebra. And, uh, so this simplistic variety can be, uh, reconstructed from this, uh, vertex, algebra associated, associated variety. And I heard that Aracawa recently succeeded to construct vertex. I mean, the quantization, so you, your construction of, of the Coulomb branch comes also with a quantization? Yeah, this I explained. So, but can you, what, what if you, M3 is not the, if you replace M3 by it, by the quantization, I mean, what, what should you get? That is some new, I mean, even in the classical level, this is a new variety, not known before. Except two examples. What, can you make a field theory by replacing M3 by the quantized, uh, M3, by the quantized version? I, I don't quite understand what you mean by. If you replace simplistic varieties, uh, by, uh, by quantizations, uh, is there, is there an expectation that this also, that there is a field theory like that? Like, I mean, you could upgrade from dimension two. I mean, this, again, that, uh, you, you just replace this, uh, symplectic deduction by quantum Hamiltonian deduction, symplectic, I mean, in no, I mean, deformation. I think if you take a group, you can, uh, naturally associate a 2D theory, but with a quantum group, you can get a 3D theory, topological theory. So is there something similar here? That's my question. I see. That kind of thing, I don't know. Other questions? Okay, so thank you.