 Thank you very much. So I have very few slides, so I should probably start with some stories of my interactions with Maxim. I first met Maxim at the Gelfand seminar in Moscow. And unfortunately, I believe half of the audience was also present at that seminar. So whatever I will say, you probably know, but the other half probably wasn't present, so I will say anyway. So it was a seminar, you know, this typical Gelfand seminar, which starts very late and then there are maybe two talks scheduled and one never ends and the second never starts. So that was a seminar where we were supposed to learn about Vasilyev invariance from Vasilyev and about what is now known as Konsevich model of topological gravity from Maxim. And somehow both talks started in some order, I don't remember which order, and then Gelfand immediately interrupted Maxim saying that, well, the math you're talking about is okay, but you will never become a great mathematician because your voice is not loud enough. And then there was a series of interruptions of that sort where every comma and every symbol had to be explained and then at some point Gelfand asked, and what are these letters tau i in your formula mean? And Maxim said, well, this is just some apparatus in topological gravity. And apparently, the word gravity has such an effect on mathematicians that that was completely clear to Gelfand, so he didn't ask any further questions. So then, well, fast forward, I was very privileged and honored to work in the same institute with Maxim and one of the ideas of his, which was a great temptation for me to follow and to try to imitate, was this idea of the construction of the compactification of the modular space of what we feel is called Walsh's instantons in string theory. Yes, we still have trying to figure it out. And so I always wanted to find this similar construction in the case of gauge theory instantons in four dimensions. And of course, one of the difficulties is that in Maxim's construction, gravity plays an important role because he was studying the holomorphic maps of Riemann surfaces together with the modular Riemann surfaces. So this is what in physics language means that it has a signal model coupled to gravity. And for the Marshall gauge theory, we normally try to avoid coupling to gravity because that's, this is always the next step. So at some point, we had, we were with Maxim at the airport after another conference which was related to Gelfand, was Gelfand 90th birthday in Boston. And so we had a few hours to kill and so we had a few beers and decided that there is a construction of a proper compactification of gauge instantons in four dimensions. And by tradition, this was never written up and we were still planning to work on it. So what I'm going to talk about is related to all these topics which I mentioned so far but probably it will not be obvious from the presentation. So let me now proceed with my goal. Let me remind you what are the Dyson-Schwinger equations. These are the equations on correlation functions in some, in quantum field theory which, naively you think about these equations follow from the invariance of the integral under the small deformations of the integration contour. So you make a change of variables and then as a result, so you make a change of variables, declare that the integral doesn't change and the result you derive an identity that the correlation function of these operators with the insertion of the operator which is the variation of the action can be expressed as a sum of the correlators of the fewer operators. So here you have n plus 1 operator and here you have n operators and one of them is hopefully even simpler than the original ones. So whether these equations are useful or not depends on your luck. So if you can choose nice observables and sometimes you choose a limit of your theory, this limit could be a classical limit or it could be some more sophisticated limit like a planar limit. Then the Dyson-Schwinger equations form a closed system. It could be a closed system of algebraic equations or differential equations, but at least it's a system of equations which then you can try to solve using ordinary mathematics. So for example, in a classical limit when h bar goes to zero, while the right-hand side naively goes to zero and so you get a statement that the operator corresponding to the variation of the action is zero, which means that you have reproduced the classical equations of motion which we view as closed system of equations. So Dyson-Schwinger equations are in a sense a statement that in quantum theory if you know how to solve classical equations of motion then you basically know how to solve quantum theory. Of course this is in practice it's not, it's rarely the case but it's a principle. An example of how this works, I mean it's not really a workable example, it's really a dream, but it's a dream which driven many people to consider many interesting things. In gauge theory where your set of fields is a space of, is a set of, your fields are connections and the action, the simplest action in the Young-Mills section which is the L2 norm of the curvature of the connection. An observable which you can consider is not really a local observable which is in a certain point but rather something which is associated to a loop and the representation of the group, the group here is un. So the observable is just a trace in the representation of the monodomy of the connection along the loop and one particular observable which is the normalized expectation value of the Wilson loop in the fundamental representation and the end-dimensional fundamental representation. So Misha came just the right moment because that's the equation which he could understand or argue that it is wrong, that if you take a limit, so it's more sophisticated than the classical limit, it's a limit when the number n, number of colors in physics language goes to infinity and the coupling constant goes to zero so that the product which is called Hooft coupling is kept finite in this limit, the Dyson-Schwinger equation becomes a version of, it's nonlinear diffusion equation in the loop space so it's some kind of loop space Laplacian which is rather tricky to define but it's a second order, well it looks like a second order but in fact it's a first order differential equation, the differential operator in the loop space produces a product of these same, of these observables which is supported on the loops which have self-intersections. So if your loop gamma happens to have a self-intersection then you get the product of loops corresponding to the parts of this loop. So it's a closed system of, so it's a system of closed equations and the dream of gazing duality from there for the last, I think, 30 years was to make sense of these equations and try to derive some representation of solutions. So this, another example which really works and because it's actually a finite dimensional system, it's a matrix model, it's not the kind of matrix model which Maxime was talking about at El Gelfand's seminar and there is actually a story about the relation of his model to that model which I will skip. So here the space of fields is just one matrix N by N matrix, Hermitian matrix if you like and the action is, it's a trace of a polynomial function, some polynomial of degree p which doesn't have to be prime and the observable is what's called resolvent, so it's a trace of 1 x scalar minus phi, phi, this matrix again divided by N. And then in the limit, in the planar limit, when h bar goes to 0 and N goes to infinity with the product fixed, the Dessen-Schwinger equations become an algebraic equation, in fact it's an equation of the hyper elliptic curve, so some physicists know what it is, where y is the expectation value of that observable, of that resolvent shifted by the derivative of the potential and g is a degree p minus 2 polynomial, which is to be determined from something which I will skip. So this is the example where the equations really work and so the solution is presented in the form of a curve and then you can study systematic expansion in h bar around this curve. Now, so this was a standard story, now I want to generalize and go beyond what was discussed. So let me remind you that the path integral in quantum field theory typically involves summation, so in addition to the integral, there's also a summation. It's a summation over topological sectors and for example in the gauge theory, this would be the summation over the sectors of fixed instant on charge. So this is the integral expression for the instant on charge. And so formally you have an infinite sum of integrals over different spaces, the spaces of connections on the bundle of a particular topology. So I would like to find a non-perturbative version of Dyson Schringer equations, which would involve insertions of variation of the action with respect to not only small deformations of the contour, not only small shifts of the variables, but also large deformations which would take you from one instant on sector, for example, to another. So this delta A, it's a small variation of a connection almost everywhere in space time except for a small region of a neighborhood of a point where the instance on charge is supported. So this is like adding a point like instant on. It is actually very hard to make this well defined and precise, but I will present an evidence that this is actually work, that it actually works so such relations exist. So let me just make a historical remark that this consistency between the perturbative expansion and so the kind of equations which you get from the perturbative small variations of the field variable and non-perturbative expansion is was used by Marek of Schiffman and Zaharoff in the exact computations of beta functions in supersymmetric theories. As a principle, this is not something really new. So this is experimental theoretical physics, so I'm going to find a set of interesting examples and from that hopefully one would extend this idea to more general cases. But for the moment I'm just playing with the theories which I know best. So these are gauge theories and SIGU models. Maybe it's a good point to tell you another story about Maxim. Now that I mentioned SIGU models, it's a bit redundant because it refers to the story which I already mentioned, but nevertheless. So at this first Gelfand seminar where I met Maxim, after the seminar Gelfand assigned his senior and junior pupils a series of lectures and seminars to students who just came to the seminar because they didn't know anything and I was among these people. And so there were some seminars on statistical physics, on representation theory and other things and one seminar was called String Theory which was given by an algebraist who actually I think didn't know about string theory. But as I will not mention his name, but he said something which was that at the first seminar he said that, well, did you hear the talk by Maxim? Did you get this impression that once he approaches a problem that after that there is no sense of thinking about this problem because the problem is completely solved and there is nothing you can add to this? So this is why I will not talk about SIGU models today. So I will talk about supersymmetric gauge theories and SIGU models in the back of my mind but I will not say about them. And more precisely I will talk about N equals 2 supersymmetric four-dimensional gauge theories and a particular class of gauge theories. So let me remind you that supersymmetric gauge theories with extended supersymmetry N equals 2 have two kinds of fields. So they have so-called vector multiplets which contain gauge fields and a complex scalar and two fermions all in the joint representation of the gauge group and the so-called hypermultiplets. These are matter fields, matter representations which involve two complex scalars in conjugate representations of the gauge group and two fermions. And the particular class of theories which I will consider will be the theories for which the gauge group is a product of unitary groups and the representation is a sum of some number of bifindamental representations. So these are representations which are tens of products of the defining N-dimensional representation of one factor and it's conjugated for another factor, maybe the same factor. And some number of fundamental representations. So this is N stands for the defining N-dimensional representation of the gauge group factor u and i and mi bar is a multiplicity space. So you may have some number of copies of these fields. So this is the same in the more physical language. And mathematically these theories are called Quiver theories and the data which you need to specify to define such a theory is in the random graph and so the vertices of this graph label the gauge group factors and the arrows label the bifindamental multiplets and then you have an assignment of vector spaces, sorry, the color spaces and the flavor spaces to the vertices. So if you have a vector space mi assigned to the vertex i then it means that you have dimension of mi, fundamental hyper-multiplex at this vertex. The theory has parameters which are the complexified gauge couplings. At each vertex i you have a gauge coupling g squared, g i squared and there is a theta angle which is convenient to combine into this complex combination which is the point in the upper half plane and sometimes it is convenient to think about the elliptic curve which is for which tau is the modular parameter and sometimes it's convenient to introduce the node which is the exponential of this complexified coupling. Now in quantum field theory the coupling constants are typically not the parameters because they usually depend on the scale at which you measure these couplings but you can impose the condition that these couplings actually make sense at the very high energy in the ultraviolet and actually that turns out to be an interesting condition which in physical terms it's an example of a physical problem whose solution has an ad classification. So it turns out that this condition implies that your graph is either the finite ad-dinking graph or an affine ad-dinking graph. And so it means that in this Quiver theory somehow hidden is the group which is the ad-simply group or the katsubudi group which is by no means present it's not an explicit symmetry of the theory you see it in the combinatorics of Lagrangian but we shall see that somehow this symmetry and its quantum deformations like the Yangian or quantum affine algebra and something else which I don't know is actually present as a symmetry which is visible as a guiding as a principle organizing the Dyson-Schwinger equations which I'm going to present. Now to complete the discussion of gauge theories which I want to discuss I should also mention that the parameters of the theory also involve the masses of the matter fields and so there is a choice of the mass for the fundamental hypermultiplets which can be organized in the complex matrix and there are masses of bifinlemental hypermultiplets which are complex numbers associated with the edges of the Quiver so in fact these masses can be viewed as one co-chain on the graph. Now to make this equation which equations non-perturbative Dyson-Schwinger equations really mathematical statement I need to deform my theories so that the partition function and the correlation functions of interest would become finite expressions, finite sums more precisely. So I will subject my theory to two deformations so one deformation is so-called omega deformation it's a deformation which shifts the scalar field phi from the Lagrangian by a covariant derivative in the direction of the vector field in space-time which generates rotations. So you have two independent commuting rotations in four dimensions and so you take them with parameters epsilon 1 and epsilon 2 and this will be the deformation parameters of this omega deformation. The term omega stands for the standard notation for the Lorentz generators, parameters of Lorentz rotation in textbooks which is usually not by omega so this is why I call this omega. So these parameters epsilon are parameters of dimension of mass and this deformation breaks translational invariance of the theory down to the rotations which commute with these rotations. So the idea is that if we can solve the theory with this deformation then we can recover the original theory where epsilon goes to zero. But the theory with this deformation at least some correlation functions are easier to compute. Another deformation which is also interesting and useful and which effectively is responsible for why I worked at IHS for a long time is the non-community deformation where you replace the space-time, R4, this is the Euclidean space-time by non-community of space so it's an algebra, essentially it's a Heisenberg algebra very naively you just say that the coordinates in this new space do not commute and the commutators is a constant, constant matrix theta. So it's an algebraic deformation in a sense of a Lagrangian. You make the coordinates in space-time and therefore the fields of your theory operators in some Hilbert space which is the representation of this algebra. The simplest representation of this algebra is of course the space of states of two harmonic oscillators. Again, if you solve the theory with this deformation then by the limit theta going to zero you can recover the original theory. So it is actually known that once you make these deformations the resulting theory can be viewed as some kind of matrix model where instead of the gauge field you introduce the operator x hat so that the original curvature is related in a simple fashion to the commutator of these matrices and the field phi of x, the scalar field in that joint representation which belongs to the vector multiplet becomes an operator phi hat which is, so it's an operator in the Hilbert space which is more or less the original field phi where you substitute instead of the arguments x hat, the operators with some ordering prescription and then you add a certain quadratic expression which is the generator of rotations in the space of oscillators. So as I said before the crucial, I mean the success of the Dyson-Schwinger equations and the generalizations depends on the good choice of observable. So the observable which I want to introduce is, well, it will be a function of one variable x and naively this is just the characteristic polynomial of a joint field phi. So it's just determinant x minus phi. So if phi had its eigenvalues a this will be just a product, just a polynomial. However, the precise definition is tricky and so it actually involves the ratio of infinite dimensional determinants. So remember we passed from the field phi to an operator phi hat in the Hilbert space and so this is the determinant in the Hilbert space. So it turns out that the proper definition is the ratio of these determinants and as you can see this is a function with a priori complicated singularities in the variable x. So it turns out to be a rational function in each instanton sector. It's a rational function of degree n. So for large x it behaves as this polynomial of degree n but then it has corrections and negative degrees in x. Now for the Quiver gauge theory where we have several gauge factors we will have several such observables so we just take the same definition for each field phi i hat. So the main claim is that there exist certain Laurent polynomials or Laurent series for fine Quivers in this observables y sub j of x shifted by some linear combination of the masses and the epsilon and the mega-information parameters which starts as y i and then corrections which involve various monomials in this variable in this y's with coefficients which are the products of the exponentiated gauge couplings and the mass polynomials such that its expectation value is actually polynomial in the variable x. Sorry for the font. So you see what's the trivial here so y a priori has poles it has many poles in the variable x so the higher instant on charge you go to the higher would be the number of poles but this combination of y's is such that these poles will cancel between different instanton sectors and so that means that you have some relation between the path integrals which take place at different instanton charges so this is like a discrete version defined difference version of the derivative of the action which is inserted under the correlation function so I will call these variables these observables, these combinations x i i the fundamental QQ characters it will be clear why I call them that if you take the limit epsilon 1 epsilon 2 goes to 0 so we go back to the ordinary Young Nils theory super unknown theory then these expressions do become the fundamental characters of the group G sub gamma which I evaluated on a particular group element G of x which is constructed out of these y's so in this limit all the shifts of the argument disappear so you take a product of the vertices of the quiver y corresponding to the vertex raised to the power which is the core root of the corresponding AD Cosmodia Algebra fine dimensional or infinite dimensional and then the the complexified coupling exponentiated coupling and the metapolynomial to the power negative co-weight and so because of these co-weights which are rational multiples of core roots the result actually does not belong does not quite belong to the simply group but belongs to what's called the conformal extension so it's a central extension by c star and sometimes by c star cross c star so it's some extension of the group but you can evaluate this element you can take this element of this conformal group and substitute into the trace in a fundamental representation corresponding to the i-th fundamental weight so this is going to be a finite or maybe infinite series in this monomials in y and the claim is that in this limit epsilon 1, epsilon 2 goes to 0 the expectation value of this expression is a polynomial in x in fact in this limit it becomes really an algebraic equation on y so you get a map so you get a map from you get one parametric family of conjugacy classes in the group c g gamma such that the the corresponding class functions, conjugacy invariant functions are polynomial so you have a rational map rational curve in the space of conjugacy classes on the group so you get the curve which you can leave to a curve in the maximal torus of the group g which is a cover of the space of conjugacy classes and that gives you what is known as a Zebra-Quitten curve so that's one use of this Dyson-Lewis-Schwinger equations if you take the limit when only one of these epsilon parameters goes to 0 then these expressions psi i become the fundamental q characters of the Yangian of the Lie algebra g sub gamma which were constructed by night and if you do a slightly different generalization of the story which I presented you take the five-dimensional theory compactified on a circle then these expressions become the q characters of the quantum of fine algebra which were introduced algebraically by Frankl and Rache-Tichin and again the claim is that you now get one parametric family of the conjugacy classes suitably which you suitably define conjugacy classes in the Yangian or in quantum of fine algebra such that the fundamental q characters are polynomials in x so you have a rational map now to discuss the case of general epsilons which is kind of interesting because it's sort of the double quantum deformation of the classical theory of characters and conjugacy classes and also to write to explain how I write the general expression for this q-q character I need to use Nakajima's quiver varieties which are associated with the same quiver gamma and the two-dimension vectors w and v so this variety is actually hypercalic quotient which here I will write as a holomorphic symplectic reduction with a quotient taken in the GIT sense or vector space so it's a vector space which is again constructed using the quiver data you have the space of homes between the vector spaces assigned to the ends of each edge from the source to the target and the homes from the vector space assigned to between two types of vector space vector vertices v's and w's you take its contention bundle so that's the holomorphic symplectic manifold and the group GLVI which naturally acts in the spaces preserves the holomorphic symplectic form so you can make the holomorphic symplectic reduction there are some stability parameters which I will skip and the result is the quiver variety physically this quiver variety takes branch of for example three-dimensional n equals 4 gauge theory with the gauge group which is a product of group UVI and with WI fundamental hyper multiplets for each UVI factor in the gauge group and the bifendamentals again determined by the edges of the graph and here is the formula so the general formula is that you take the integral of the quiver variety of certain characteristic classes so one of these classes is very simple it's the it's a covariant Euler class of a tangent bundle I'm afraid I switched it to change the W and V parameters so that's a simple class and another class is Trichier this is where the Y variables, Y observables of the gauge theory will enter so you take the ratio of these Y observables shifted by the chain roots of certain complexes of this quiver variety so these are so called tautological complexes if somebody asks me a question later I will explain what they are so it's a finite for a finite quiver sum over the dimensions of the V spaces so these are summed over and W's are fixed so this is they enter the the parameters of this generalized character QQ character and in a fine case the sum is a priori infinite but it's so the dimensions belong to a cone so it's a sum where at each degree in each variable QI you have a finite number of terms so it's a k plus minus k plus k minus kappa so over this space there is a complex of shifts so I take it as a quiver and chain character you get some number well locally you can represent this complex by two term complex so think of it as a difference of two bundles and so these bundles have some rank which I don't know could vary so k kappa plus labels the chair roots of the of the term on degree one and kappa minus labels the chair roots in the term of degree zero I think I have time so if you have a vertex I in this in the quiver you have some number of edges which this is actually unrealistic peak well it could be realistic for the affine d4 case so you have some edges which enter and some edges which exit the vertex so you have a complex when you use the maps so you have in the definition of the quiverite so we have this space of homes between the vector spaces the e be the particular operator and then because the quotation bundle there is an operator acting in the opposite direction so so you can compose so this is my complex so you use whatever you have at your disposal you have maps b e from which so b if e is the edge which whose source is i so it's this is this edge you can apply b e to the vector space v i and it will take you to the space which is the target which is at the target of the edge and that's one of the linear maps which enter the definition and the same for the other arrows and the point is that the moment map equation is such that the composition of these maps is zero so you have a complex canonical associated to each vertex so it is this complex so the chain roots of this complex that the quverian chain roots of this complex can be used here so it's a general formula but it's too abstract and scary so let me give you examples where it becomes something very simple so let's consider to be specific the theories with the single gauge group factor so u n gauge theories and there are two cases the a1 case where the quver is just just a point there is another case a0 hat the point with with a single edge with a single loop so in the a1 case this fundamental qq character is very simple it's a sum of two terms one you shift the y observable by epsilon 1 epsilon 2 plus epsilon 2 and another you invert it so this was so this is the formula for the fundamental qq character in the a1 case there is only one fundamental character because there is only one vertex but the general formula involves also the choice of there is a choice of a vector space w and some parameters which are the quverian chain roots of this vector space so these are w complex numbers u1 uw and so the general formula is a little bit more complicated and I think it didn't fit my oh no it is yes so it involves products so it's a sum over all partitions of the set 1 through w into two disjoint sets and so it involves the ingredients like in this formula y shifted and y inverse which is not shifted but there is a pre-factor which uses independently epsilon 1 epsilon 2 so this is where the qq character differs from the q character so this is why it's really a deformation and the claim is that the expectation values of all such observables in n equals to theory are polynomials in x so it's an infinite set of nontrivial equations and the second example is this a0 hat theory which is known also as n equals to star theory it's a theory with mass if I join hyper-multiplet it has one parameter which I call mu the mass of the adjoint hyper-multiplet it's what I used to be called m sub e the mass associated with the edge so even though physically it seems that asymptotically conformal theory which is the theory with fundamental hyper-multiplets and the theory with adjoint hyper-multiplet should not be very different the expression for the qq characters is drastically different and the reason is that this theory has this as a hidden symmetry it's a Youngian of SL2 and here it's a Youngian of the Cosmoodya u1 Cosmoodya so here's the formula it involves it's a sum of all partitions all young diagrams so if not for this complicated factors this sum will give you the dedicated function which you saw in the previous lecture except it was raised to the power 24 so here it's it's a kind of a dressed version of that formula which incidentally if you specify parameters can produce a dedicated function raised to interesting complex powers anyway so here it's a sum of all partitions and then you have the product over the boxes in the on the boundary of partition on the boundary of Young diagram of the partition this Young diagram of my partition then I use color check this is how far we went from the times described by Ludmille so this is the boxes the squares in them so this is partition lambda this is what I call delta plus of lambda so these are the squares which can be added to the partition the squares which can be removed from the partition from the Young diagram so I call them delta minus of lambda so you have the product where the terms with the red boxes are numerator and the ones which can be removed are denominator and there is some pre-factor, rational pre-factor so these are all the shifts are defined in terms of the arms, legs, hooks and contents of the boxes now so the applications again the main claim is that the expectation value of this is variable and there are also the versions with W's which I will not write because it will take too much time incidentally the Quiver variety which I had to integrate over to get this formula here it's a Hilbert scheme of points on C2 so it's a Hilbert scheme of points but it serves to write expressions for theories of any rank and for physicists maybe I should say something so this is the picture of R4 where the my gauge theory leaves and so there are in particular there is a rotational symmetry with parameters epsilon1 and epsilon2 now the Hilbert scheme of points is also kind of modulate space but if you look carefully at the weights of the rotational symmetries acting on C2 in which this Hilbert scheme of points leaves they actually live in a transferential space and so here the weights are mu and minus mu minus epsilon1 minus epsilon2 so if you like so if you think if you think about the string realization of the gauge theory and the gauge theory leaves on the stack of let's say D3 brains then there is an auxiliary gauge theory which will produce this Quiver variety which leaves in the transfer space so what are the applications while you can derive actually and rigorously prove some of the statements of the DPSCFT correspondence which is a principle which is not a statement, it's a principle that the correlation functions of current observables in four dimensional supersonic theories are homomorphic blocks and form factors of some conformal field theories and the massive integral deformations into dimensions so it's a trivial correspondence between different dimensions in different dimensions of space time so as an example, one can prove for example that the instanton partition functions of the Quiver theories of A type so these are the theories for which the Quiver diagram is just that or an end goal for special choices of masses obey the equations of homologic for the decouplings of null vectors in representations of erasora which is the prediction of Degaiot and Tachikala and so this is the mathematical proof of that you can also show that the partition functions the instanton partition functions of the surface defects where it turns out that this Y observables which I defined fractionalize so out of one observable you get to define several and so it turns out that this partition functions obey the Kndelich's homologic of type equations and so again this is just a mathematical proof so as a conclusion on speculation, I will just ask a few questions so we know that this A.D. symmetry even though it is not possible not in any simple way possible to see in Gage's theory as in quantum field theory nevertheless it is visible in the in string realization of the theory because that's an enhanced Gage's symmetry in type 2A description in type 2B description so the question is what is the meaning of the Youngian deformation of this of the symmetry G gamma in string theory, is there any role which could play in others again so the gravity now comes back so far I discussed the theories on rigid spacetime but it is tempting to speculate that there must be also Dyson-Schwinger equations which involve changing the topology of spacetime and so maybe one could derive a nonlinear version of Miller-Dewitt equation on the wave function of the universe which would be much more useful than the linear equation which doesn't fix much and maybe it could be used as a symmetry principle in the string vacuum and there are many mathematical questions even at the level of the equations which are described so far one is to find a version of these equations in the context of topological strings so there is an overlap of the problems which are described in the language of Gage theory and the problems which can be defined in the context of topological strings of the type of noncompact collabiaus so there must be a version of these QQ characters which would serve all collabiaus all Tori collabiaus another question is to derelate the QQ characters which have two parameters on equal footing epsilon 1, epsilon 2 in the Jungian case and the exponents in the quantum affine case to the QQ characters C deformations of QQ characters styled by Nakajima and Franklin Hernandez so conjecturally these are the same things but so far the definitions which I saw looked very much different and the last is my acknowledgement of my thanks to all my collaborators who helped me learn these things over the many years of research question is it possible to say that if medical is made in a way like maybe but I'm not sure so one thing which I should say, which I know which I probably should explain better is that in defining these correlation functions so these correlation functions are integrals over other kinds of Q varieties not Nakajima so the adding of point like instanton which I was talking about is an operation on these varieties but the statement that there is a particular combination of Y observables which I insert here which is an integral over Nakajima variety and it's a combined thing which has these nice properties of being a paranormal function means that there is an extension of Nakajima correspondences which he constructs on these spaces to the product spaces so maybe in the language of this picture so you see Nakajima varieties are more or less modular spaces of framed instantons torsion free shifts and some kind of version on this space and the quiver instanton varieties which I call the other quiver varieties live on this space so from the direct geometric point of view what you study, you study shifts well this is in a fine case you study shifts on the on the four four complex dimensional so this is a four-dimensional variety complex which is a product of C2 or you can compactify it to Cp2 and the Norbey fold where gamma is a subgroup of discrete subgroup of SO2 and so you study shifts which are supported on this cross so they're supported here and there even though it's a singular variety somehow it seems that there is a there is a good so the whatever I'm trying to say apparently there is a virtual fundamental cycle in the modular space of of such shifts and so this computation which involves the product so this product of Nakajima varieties which come from this axis and instanton which are which depend on Nm and other numbers which I didn't specify in this talk so this disjoint union of products of modular spaces it's a boundary of the modular space of shifts on this cross so it is on this space that one needs to develop the the version of HECI correspondences and so on and I don't know what's the picture for the case of finite quivers so this is for a fine quivers and it must be something of finite which should be simpler but I don't know what it is very trivial question so this PT character of this the character is a double obtained HECI algebra or something like that I think they defined this using the representation theory of quantity algebras for them they are completely different parameters yes so that's why it's a bit untrivial to establish the correspondence because for me there are two parameters on equal footing and of course there is Daha somewhere here but I don't know but this Q and T are different parameters so that also test the parameter but we know that there are two parameters which are on equal footing