 Okay, so I'd like to thank other organizers for invitation and give me a chance to talk here. So I will talk on Coulomb branch of supersymmetric case-related feature I'm working on with whatever I want to think about for several years. So we like to be able to develop papers on the subject. Sometimes we start our colleagues. So in these works, we try to give a mathematical rigorous definitions on what physicists call Coulomb branch of the gauge theory. So the first talk is about the motivation and some basic idea of the definition of our, basic idea behind our mathematical definition of the Coulomb branch. So the input data is the following. So G is a complex reductive group, say general linear group. And sometimes we also consider the maximal compact subgroup. So for GRN, the unitary group. And I take M, so this is a symplectic representation of G. So when you consider maximal compact subgroup, this is a quaternion representation of the maximal compact subgroup. So symplectic representation mean that, so this is a usual representation and also M is a symplectic vector space which is preserved by the group G. So this particular example, which in fact later I will assume this assumption. So if N is a complex representation of G, then you just take direct sum with its dual turn. It has a natural symplectic form. And this is a particular example of a symplectic representation. Then, phis is assigned to the dot data for the N equal to supersymmetric gauge theory. And then also by compasification of bias one, so you get 3DN equal to 4. So this is a quantum field theory. And so mathematically it's not clear what it is about. There's no rigorous definition. But anyway, so given G and M, you consider connections on the principal GC bundles. And so this is basically connection, G C connection plus spinus with values in M. So since M is a representation of G or GC, so you can consider the associated vector bundle for the given GC bundle and take tensor product with spinner bundle and you consider the section. So those are fields. And you introduce certain Lagrangian, take a path integral and you consider quantum field theory. Those are the quantum field theory which Phis is considered. Then from them, they assign the so-called Higgs branch and also Coulomb branches. So this is a hyperkeler manifold with C star action, Y star action preserving the linear metric but rotating. Complex action, in fact it's SU2 action, SU2 action. So in my lecture, we only see the S1 action in SU2 action, possibly with singularities. And the Higgs branch is easy to define. So Higgs branch is, this is hyperkeler quotient of M by G, G over GC. So I use the following notation, M triple slash G. So this is a simplectic reduction. So you have moment map and take affine algebra geometric quotient. Mu inverse zero is affine variety and take GIT quotient or it is also different. So if you consider hyperkeler moment, so this is this funny, fancy mu is hyperkeler moment. And the Coulomb branch is more difficult to define. And at least, so together with Brevan, think about we define the Coulomb branches, affine algebraic variety, simplectic structure, homework simplectic structure on regular locus, together with, oh, by the way, so I should say that this SU2 action is induced from SU2 action on M given by the cotonic multiplication. SU2 is considered as a few, it's a kinetic number with absolute value one. So you have multiplication and descent to the quotient of space. So I would like to explain this. Yeah, blackboard is too small, so I need to erase. I hope this is readable, is it readable from the back? Ah, I even don't have eraser somewhere. Ah, this one. So eraser depends on the country, so each country have different erasers, so. I believe Japan has most advanced eraser. So if you have a chance to visit us, test, you will see and we have eraser cleaner that is really good industry. So anyway, this is what I want to say. Ah, and the assumption maybe, when M, as I said M is M. And also, in fact, it is interesting that so by-product of our construction, so which seems to be not recognized in physics until recent years. So this is quantization. So in general, if X is a fine algebraic variety with form-action-practical structure, or maybe Poisson structure in more general. So it means, in particular, the coordinate ring of X is Poisson bracket with the usual properties of the Poisson bracket, right between two. Then we say H bar. So this is a algebra over polynomial in one variable with variable H bar is a quantization of X. This is by definition. So if you make H bar to be zero, so which I mean, you just take this quotient. So this is isomorphic to coordinate ring of the variety. And the second, so if you take function f, take f and g, then the Poisson bracket fg is recovered by the following formula. So you take lift over f and g, and take commutator. Then at H bar equal to zero, this becomes commutative ring. So if you take commutator, this is divisible by H bar. So this is equal to Poisson maximum up to maybe this. Modular H. So this is a usual way to define Poisson bracket. If you have a non-commutative deformation of commutative algebra, then you define Poisson bracket in this formula. So you get Poisson algebra. And conversely, if you have Poisson algebra, then you ask whether this, this kind of non-commutative deformation exists or not. And then when you find such non-commutative algebra, it is called that it is a quantization mix. And for Higgs branch, at least when m is n. So example, maybe I can get in here. So if m is n plus n star, then you consider differential operator H bar, H more precisely H bar differential operator, ring of H bar differential operator on n. So you take half Lagrangian subspecial and you consider differential. H bar differential operator is something like H bar times differential. Then you have the commutation relation with variable X. And if you specialize H by called zero, then you get simple active form like this. And so this is a quantization. This is a quantization of just c of n plus n star. Or maybe just n plus n, variety n plus n. And Higgs branch is a reduction. And there's, so reduction is basically algebra procedure. Procedure, so first you take the level set of the moment and then take invariant part. So you have maybe, yeah, this is a little too small. Maybe I just erase this. Then you have the so-called quantum Hamiltonian reduction. Well, it's not so difficult for the set term. I don't recall it. So basically you take the level set of the moment and take invariant. Then you get quantization. So this is for Higgs branch. And so our definition is automatically gives also the quantization in this sense. In fact, that is the way how we define the symplectic structure. So we introduce quantization of the chromatic and retainance and undefine the format which I just erased gives a symplectic form on this Coulomb branch. So I hope this is, somehow it becomes a little bit clear how we approach to this problem. So other than just trying to define the variety itself. So we think that the ring of function on the variety is more fundamental. So maybe I forget to mention that I didn't know the Coulomb branch by m, mc. Coordinate ring of mc. So this is mc, as I said, mc is affine algebraic variety and we consider polynomial functions on mc. So this is c of mc. And in fact mc, mc will be recovered. So if you use construction, standard construction algebraically. So once you have the commutative ring with several nice properties, then you can recover the variety as it's spectrum. So you consider space of idls and put some top-loads and consider functions. I will not recall that this is standard. So for affine algebraic variety it is the same as defining the affine variety and also this commutative ring. So I rather construct this commutative ring. Commutative ring. And in fact this is called chiral ring in physics or sometimes called ring of monopolar operators. So the element of this ring is called monopolar operator. So this is an operator acting on the Hilbert space in canton field. So how you should consider this monopolar operators. So take a point and it's neighborhood, small neighborhood in R3. So I cannot write the three-dimensional picture but this is picturing in 3D and I have a point. And the boundary, boundary is small two-sphere. Then so if you have canton field theory, then canton field theory gives a partition function which is a number assigned for three manifolds given by pass integral over the space of all fields. And if you have this kind of boundary, three manifold with boundary, then you consider the space of boundary conditions. And this function depends on the boundary condition. So it gives element in the space of function on this boundary condition. So from this, for physicists assign the canton field space. So I hope, so you are familiar with at your Seagal's axiomatic approach to topological canton field theory. So for three manifolds, you assign numbers and for two manifolds, you assign the Hilbert space. So this is a boundary condition. Then you observe that this should be commutative ring. And this is chiraling. And element in here H is monopropate. So why it is commutative ring and how you consider the multiplication. So this is, I mean, if you are familiar with TKFT, then maybe you are hard about two-dimensional TKFTs. The one-dimensional smaller version. So if you consider the pair of pants, so you get multiplication. And in fact, the TKFT, two-dimensional TKFT is nothing but Robin is algebra. We have the same for here. So you consider, instead of pair of pants, I consider the following picture. So I have two points and I consider two spheres surrounding these two points. And this is a three manifold with boundary. So I did not this by X. So boundary of X is S2 and S2 and S2. We have three boundaries. Two are in, orientations are different from in two spheres and out that two. So then the partition function for X should give homomorphism from Z of S2, 10th of Z of S2 to Z of S2. So as I said, this is H, 10th, right? So this is multiplication. And I hope it is clear that this is commutative because so I write two points and I take two surrounding, two spheres, but so this is a three-dimensional picture. So there's no particular ordering for this point and this one among this point and this point. So you can continuously change these two points and somehow this multiplication should be invariant and this can change. So it is commutative. And also associativity is also clear if you consider the picture which you add one more point. So from this consideration, this H should be commutative ring. And this is chiral ring. So in order to define the Coulombs, as I said, Coulombs can be recovered from this chiral ring by spectrum. So it is enough to define. Yeah, in fact, in 90s people study this kind of problem in at least in specific situation. Somehow this approach sometimes works well, but sometimes not because of some technical issue which I will mention afterwards. So this is the nano-sensory, for example, nano-sensory or maybe it's three-dimensional by the version is cast-sensory. So first people consider, so I return back the original input. So you maybe, for here, you better to consider the maximal complex. So you have compact re-group and it's cotonic representation. Then, so you can consider the generalized cyber-wicketing equation. So you can consider partial differential equation in four-dimension or three-dimension. But I just consider the three-dimension case. So it is the equation for pairs A. So this is the G-connection, G-G-connection. Yeah, you must be a little bit more careful for what makes sense to write down the equation. If you are familiar with cyber-wicketing equations, it will be clear how I actually collected it, but somehow I just give you a schematic where you can understand the equation. So S is a m-valued, value spinor. Then you consider the equation that S is cubed by Dirac operator. And also, so you consider the hyper-keton moment map for S is equal to F of A. So mu of S is roughly half-valued in. So hyper-keton moment map is map from M to the algebra of G, Dirac process, tensile X R3. But R3 is an imaginary part of the cotonium and that is identified as the algebra for SP. And for this, you must replace this by one-homes. So once S is appropriately understood, then mu of S has values in one-homes with values in the dual of the real algebra. So maybe this is what you study. So Carbacher lives in the same space. Carbacher is G star by one-homes. So this equation doesn't make sense. As I said, I cheat a little bit, but schematically the equation is correct. At least if all the bundles are trivial, locally this is, my equation is correct. So anyway, this is some partial differential equation and if you take quotient by gauge group, this is elliptic and you just count the partition function G for M, X3, is number of solutions. So for example, the simplest case, Caston theory is a case when G, C, S, U, two and M is equal to zero. So in this case, there's no first equation and the second equation just means Carbacher is zero. So you consider flat connection and you just count flat, flat connection. That is basically definition of Caston invariant. Then for three manifest, so this is closed case. And if you have two manifolds, G of S2 is roughly homology of modular space of solutions of 2D version, 2D reduction. For the case of Caston invariant, so as I said, the equation is just flat, means flat connections. Solution of the equation just means flat connection and the 2D version is just the same, the flat connection on two manifolds. And it is the famous space, I mean it has singularities, but this is also identified with modular space of rank 2, C1, zero are stable bundles. And it is a projective variety and you consider this homology. And if you have X with, so three manifold with boundary, so three manifold with boundary, sigma then you have GX3 in G of sigma 2. So this is a cycle in modular space given by boundary condition. So you consider the, so it depends on the equation, but so basically you consider some sigma. So you consider the modular space varying all the boundary condition and take boundary value gives some cycle. So this is a heuristic approach too. And under some assumption X and sigma, people succeed to realize this idea. So basically singularities of modular spaces. But somehow, so this assumption is too restrictive and because of this, I mean this is partial, only partial success, not full success. And somehow this approach is not so much studied afterwards. And also in this approach usually you get, and also, as I said, so we should take sigma to be S2. So in fact, S2 is somehow because of this issue. So if people don't consider this case. Then the solution, all solutions, all the solutions, all the solutions all solutions of star are basically reducible. So if you're familiar with cyber-wittern theory, then S2 has a positive color culture. Then from this equation you derive as vanish. So it is a reducible solution. That means that this is the modular space is completely singular. So usually people study the modular space which has mild similarity. So you have irreducible, generically you have irreducible connections, irreducible solutions. And in some smaller dimension you might have reducible. But somehow by dimension counting argument for homology or something, you just avoid those singularities. That is a way people justify this approach. So in this approach, this sigma S2 is the worst case. You cannot apply this kind of. And also, so in these most approach, I mean if this modular space is some reasonable space, usually maybe reasonable is not good, but some usual variety. So this is mostly smooth and might have some smaller singularities for smaller dimension. Then the homology group is finite dimension. Z of S2 would be finite dimension. Then also you must define multiplication on this so that this is commutative. But then anyway, so this, because this is finite dimension, this is zero dimensional variety. I mean still algebraic geometry can still study some non-trivial zero dimensional variety. But this is not correct, not correct Coulomb branch. Coulomb branch is, maybe I didn't say, the Coulomb branch, the dimension of Coulomb branch is twice of the length of the group. So this is completely wrong if this would be a zero dimension. So somehow it means that you must really work on this singularities on the modular space. Any question? So our approach. So we use, we replace S2 by R, Raviolo, maybe I, so here in Italy I should, I must remember, but these are incorrect words. So this is a singular version of Ravioli. So you write Ravioli in this way. So this is the union of, so D is a formal disc and we check the times is the punctured form of this, D minus then. So in fact, as I initiate in the motivation, I mentioned that these two words are very small. So you consider the point of singularities. So you don't really don't consider S2, but so this, I wrote this picture, but in fact, I don't really consider the few features defined over the outside of these boards. So in fact, I consider the few features has singularity at this point. This is more correct intuition for monopole operator. So somehow S2 is not completely correct. So I should consider this S2 is very, very small. Otherwise it's not correct. And this is more complete, more correct picture. And use more algebraic, geometric language. So in fact, this is fortunately for language. People has developed corresponding things in geometric equations. This is the affine glass money. So this is the correct modular space. So I will, tomorrow I will give some more detailed discussion for the affine glass money. So there's a several ways to understand it. One is, so you consider K, K is a ring of formal lower power, field of formal lower power series and consider O, so this is formal power series. So spec of K is D times spec of O is D. And affine glass money is defined as G of K divided by G of O. This is one way, but also this is space of polynomial maps. There's a topological approach to affine glass. Polynomial map from S1 to maximum compact group and divide by adjoint action, action from light multiplication. Or this is polynomial map from S1 to GC, which sends F of one to time, unit, unit element in the group. So this is a kind of space which topologist has been studied for many years. So this is the case when you don't have the matters. So maybe I didn't mention that what is matters. So the spinors with virus in the representation, M is called mattress. And this is the case when there is no mattress. So we should consider that. Instead of the genuine modular space on S2, we should consider affine glass money as modular space of the equation on S2. So this M is called zero, should be replaced. So this is an infinite dimensional variety. So the modular space is usually finite dimensional because of the local nature. This is infinite dimension. So this is good because we really want to have very big commutative ring as homology of some space. So finite dimensional space is not good space to have this infinite dimensional vector space as a homology group. And for when M is N plus N star, we consider the space R. So this is the space which we introduced in our definition. So R is, or maybe R with GO action. So R divided by G of O. So this is some stack, infinite dimensional stack. It is a modular stack of bundles, G bundles, and the various sections on this labiola. So more precisely, what does it mean? So we have modularity of P1, S1. So this is a bundle and section on upper, upper disk. And also P2, S2 on D. And gluing by P1, restrict to puncture disk to P2, respecting S1 and S2. So this is a modular space we will consider. So then we claim that H star of G of O. In fact, so because this is a global portion to stack, so this is nothing but equivalent homology of this space R. So R is defined by assign, so R is assuming that P2 is trivial bundle. In fact, because of this identification, so once S1 is given, then S2 is automatically defined, automatically introduced by this isomorphism. So you don't need to specialize it. So anyway, I will send the more detailed tomorrow. So anyways, so I have considered this equivalent homology, and this has convolution product. So basically by the same rule, which same way as intuition. So instead of balls, two balls surrounding two singularities, so we consider three copiers of disk glued over the puncture disk. Then you have projection PIJ, and you define P1, 3. So the convolution product is defined in the usual way. So this defined the product. So this is a little bit, I cheat a little bit because in order to pull back homology class, you must have the Poincare duality, but R is not the smooth. So you must be a little bit careful, but still nevertheless, you can define the convolution products the last way. So I will not explain that part. So anyway, so you can define commutative multiplication on here, and main cell, so that H star, your R with star is a commutative ring, and finite regenerate, it has several good properties. Hence we can define the Coulomb branch as spectral. And quantization, this is automatic. So you consider sister acting on disk, or the R acting by loop rotation. So you have disk and origin, and you consider loop rotation. So it acts on the everything or the space which we consider. So you consider S, S1 equivalent, C star equivalent. So this is A by H bar, and this is the quantization. So this is basically our definition. And so far we checked, so this is somehow a little bit abstract definition, but somehow we have some tool to define, to describe MC more concretely, in particular for quiver, so the quiver basically, we explicitly determine what MC, in some cases, at least. Then it is the same as for what physicists computed in very different way. So because, so far no counter example that this definition is not correct are found. So we believe this is really correct definition. And in fact, as I hope that our intuition is motivated from physics, so it must, it should be correct reading. I hope that I still have some time to discuss example. So I start with a very trivial example. So I take G to be C star and N, N is zero, M equals also zero. Ah, maybe I should emphasize that my representation could be zero representation. And in that case, the affine glass, I only consider affine glass mania, not the space R. Then affine glass mania of G. So as I said, this is, there are several ways, G of K divided by G of O, but it is also map S1 to G of C divided by G of C. Polynomial, but there's no, I mean, polynomial map, but there's no, how to say, so this is, this is also S1. So if you want to have polynomial map S1 to S1 with this, then it is just G of N, so N is integers. So this is bunch of points. So you consider corresponding homology group of this. But as I said, this is a quotient of stuff. So we should consider G divided by G of O. So at each point, of course, this group actually, this is the point. So this is just, and also you just replace G of N. So this is a kind of parabolic subgroup. So you can ignore this all part. So it is well known that the requirement of homology of point is polynomial in one variable. So this I did not by C of double. So double is a generator for this equivalent homology. But I need to distinguish, so this is N, so for each N, I have this one. So I do this by R times, R N. So R N is fundamental class, this point, G of N. So this is the description of this homology group as a vector space. So then I need to describe the multiplication. And in fact, for this, it is well known multiplication, this is a group. So you have font-reagging products on this homology group. So more concretely, so if you consider Z N and Z M. So this is mapped to G of N plus N. Point one is multiplication on the target. So it means R N multiplied R M is just equal to R N plus N. So combine this H star of G of O divided by G is C of W and R of one and R of minus one. So these are generated by those three elements. And R one times R minus one is equal to R of zero. So this is the unit. So this is a commuter tripping, and if you take a spectrum, so this is the C cross C star. C corresponds to the first variable and the C star corresponds to all three. And this is the R of C cross S one, and this is the R of C cross S one. And this is a flat hypercalamanic order you have. And in fact, I mean, people, we've finished already, so this is a two-trivial case, so physicists don't consider this example seriously, but so in this case, this is a free theory, so you get Coulomb branch is something trivial, and indeed Coulomb branch is flat space. Okay, maybe I finish by saying the second example. So second example is you just take n to be one-dimensional vector space. So then the space is almost the same. So R is the pairs of the affine glass mania and together with sections. So you have P one S one and P two S two. So in this case, P two is trivial. And S one is, so anyways, this is a trivial one though, so S one is just formal power series. And in this setting, phi is given by G to phi n. So R is triple, so G of n and S of one, and S of one, not triple pairs. So P one is a trivial bundle, phi is in this one. And the condition is G to the n of S one is still in C of z. So because I assume S one, S two are both a section defined over the disk, and phi is a identification outside the origin. So S one, S two must coincide outside origin by phi. That means if I multiply G of n to S one, then I get S two. But S two is defined over the whole disk. That is the condition. So it means that S one, if n is positive, then this is automatic. S one is polynomial, and if you multiply G of n, this is polynomial, and if n is negative, then it means S one is, in fact, originally S one must be divisible by minus of n. So somehow, maybe I don't picture. So the picture is, so you have points, and n equals zero, we have O, and we have O. For positive part, we have O, but for negative part, n equals minus one, we have G, O, and we have G to the square of O. So the picture is different for positive part and negative. So because of this, multiplication is different. So if R prime n is fundamental class over the fiber over Z of n, then prime R of one times prime R of minus one. So if you multiply the positive one and negative one, then nothing will change, but if you multiply positive and negative one, then it is no longer unit. So this is equal to W. So this is a generator form. So I get H star of G of O divided by R, X and Y. So X is equal to prime of R one, Y equals plus one, and the equation is X, Y is equal to W. But W is just given by X, Y, so this is polynomial in two values, X, Y. So its spectrum is C2, and this again has a flat, flat metric. And in fact, this is, in this case, physics says the Coulomb branch of this case. This is called QCD in physics literature, and this is a flat, flat R4. So this is the physics answer, and it matches this. Sorry for all the time, I say I stop now here today.