 Окей, спасибо большое. Я рад, что это последний спекарь. Подожди, это конференция. Так что, это фотография из Википедии. Так что, может быть, это первая фотография, которую вы найдете из Samsung, если вы гуглили. И эта фотография была с ИЧС. ИЧС был commonplace between me and Samsung В первый раз мы встретили, в 2006 году я был студентом в Принстон. Я пришел к ИЧС для студентов, пгс-студентов, для исследований, и встретил Samsung в первый раз. Мы ходили в форс, и говорили о том, что компонентаризация, структура H3 в Калаби-Яу, quantization of B-модель и так далее. И я был очень впечатлён, как Дипа Сотс, Samsung, что-то мне показал. И с того, как мы научились много раз, и главная тема моего current research, Gage theory, Centro integrable systems, это потому, что центральный тупик от школы, от которых мы пришли. Ну, я был тоже в Ленинграде, в фадеевской школе, Gage theory, Centro integrable systems, это продолжает мой исследователь. Итак, мы говорим сегодня о Several papers, я will summarize the key results from those, they made more recently in collaboration with the people, with the postdocs Rubin Ferasik, Alexander Tsumbalik and Chris Elliott and those are references. So, the topic of today's lecture would be a group version of a Hitchin system. And to start, we'll just recall what the Hitchin system is. So, maybe most of people are familiar but just a reminder. So, for a complex variety X and an effective divisor D of complex variety and complexative group G, G-Hicks bundle would be a pair P-Phi, where P is the principal G bundle and Phi called Hicks field or Hitchin field is a section of their agent valued bundle valued in the leal algebra, usually taken tensor with canonical line bundle on X with a singularity Cd. So, that's the usual thing. And then there is a theorem by Tachin, Mukai and others that there is an algebraically integrable system on the modular space of meromorphic Hicks bundles on a curve, on a Hitchin system. So, this touched physics in many contexts. So, a Hitchin system on a complex curve X that plays a prominent role since, you know, after the geometric Langlands program from Baylinson-Dreamfield and linked to n equals 4 supersymmetric gauge theory by Kapustin and Tvitten. And then there is a construction of the compactification of the 6d2,0 self-dual theory on X, viewed as 4dn equals 2 theory of Gallota and Diber-Kvitten system associated to that. After quantization, this Hitchin system or the integrable system of Hicks bundles, it relates to N'Krasov-Schittashvili limit of of the 6d2,0 self-dual theory compactified to the 4dn equals 2 theory, compactified on the curve on which Hitchin system is supported. And after further hyper-color rotation and quantization, or as in our slang we refer like in the case of two epsilon's, it relates to the total theory in correlation functions determined by the W-algebra. So, there is a hero here with like two ways of making the system quantum in a sense or deformation and quantization. There are two parameters which control like the classical story going to one way quantum or then going to the field theory quantum in a sense. So, then to move on and see how you can make that story like group like instead of the algebra like, let's recall an abstract Higgs bundle. So, there have been given a definition of abstract Higgs bundle, which is helpful for this work and it goes to Danagin Geitzgeri. So, to say it, reformulate and think about an ordinary Hitchin system on a kerfax with singularity as an example of an abstract Higgs bundles which will be valued in the canonical London line bundle Kx on D. Now, an abstract Higgs bundles would be pair p, c where p is the principal g bundle on x and c is the sub bundle of the agent bundle. And each fiber of c consists simply of centralizes in g of regular elements of the group g. So, then there would be an equivalence between an abstract g Higgs bundle and an abstract spectral data where spectral data consist of an abstract so-called abstract camera cover x tilde and this usually transformed on x. So, this is Danagin Geitzgeri construction and this construction then can be specialized to a valued Higgs bundle. Higgs bundle valued not in the canonical line bundle but instead on something else. So, concretely we can replace the space of values the usual canonical line bundle Kx by a family y of groups fiber over x. And then the Higgs field phi becomes a section of this thing, of the tensor product of the family y of groups and c which was mentioned on the previous slide. And then one can get let's see if I can close this one. What does this product mean if y is not effectable? No, we think about that as an abelian group a bundle of abelian groups fibers over x a bundle. So, then one can associate to this abstract spectral data a model space of Higgs sub g of y slash y over x. So, a canonical example of this fibrations studied mostly by works of Danagin was the case when y itself an algebraic variety which is fibered over base variety x and generic fiber is an elliptic curve. So, then you can consider model space of g bundles on y and this model space of g bundles on y is isomorphic to the model space of g Higgs bundles on x with values in the elliptic fibrations y over x. So, this construction of Danagin. So, in particular if y itself is two-dimensional algebraically integrable system fibered over x with elliptic fibers then the result in space bun g over y or g Higgs bundles of y over x is an algebraic integrable system. So, this is called also in slang like algebraic integrable system elliptical elliptical in vertical directions elliptical in momentum directions. So, now we can have vertical here or here in the classification of the previous page like we can consider the three cases for the structure of the vertical group of the vertical fiber namely if you take one-dimensional abelian group then it could be either non-degeneric elliptic curve or you can degenerate it by nodal degeneration to a multi-political group C star a nodal degeneration or you can first degenerate it to an additive group or just see a complex line additive group or that's the cut degeneration of elliptic curve, right? So, then I found the following quotation from the lectures of Rob Danagi in 2003 book. So, he says that Hitchin system is a modular space of bundles on arbitrary curve x bundles with arbitrary structure group with values in line bundle which is a group variety of rx with fiber g additive and then the modular space of g bundles on elliptic fibrations involves Hitchin bundles with structure group g and with values in elliptic fibrations so that's what was well studied by Danagi and other people and then he asked this question is there an interesting geometric interpretation in the remaining trigonometric case where there is no answer so it was open at that moment but apparently like many aspects of the answer to this question were known before just not connected for some reason together So, this is the question mark in the middle case where the fiber is multiplicative group while when the fiber is additive group it's the usual Hitchin system when the fiber is elliptic group then it's the modular space and then the modular space where the fiber is elliptic fibrations of rx So, the purpose of this talk would be essentially to clarify the geometrical aspects of this question mark So, we will describe and answer on some of Danagi question what is the multiplicative Hitchin models and how it connects to other topics in physics and mathematics So, we'll focus concretely here on horizontal horizontal-rational case when the base x is c or one point complication of c So concretely by x we'll take p1-infinity or well, maybe I'm mixing notations I'll be mixing notations in slides sometimes I denote by xc-point-infinity or sometimes by xp1 Usually, it will be clear from the context So, let's say that x is p1 and a pair with a section of canonical bundle with a degree 2 pole at infinity in the usual coordinates of the form dx and then we'll fix the divisor So, divisor which will be valued in account of the dominant co- weights of the co-weight lattice So, it means that to each point supporting the divisor we'll denote a co-weight by omega i check an element of the co-weight lattice lambda So, then a multiplicative g-hicks bundle on x which is framed at infinity with singularities in a co-weight value divisor would be pair p, comma g where p is the principle g-bundle on x with framing at infinity and g Now, unlike unlike the usual Hitching field, it's a section of a joint group valued bundle So, the difference with the Hitching definition is simply like little letter A is replaced by capital A So, which means that it's valued in groups rather than the algebras and the definition also singularities changed correspondingly So, namely, near each singularity where we have a specified co-weight Their singularity of the group value field is described by following You take, like the diagonal part or the carton part to be their image under this co-weight embedding the image of local coordinate near xi under this co-weight action multiplied from the left and from the right by analytical functions, group valued functions in that patch And we also fixed the framing bundle p and the value of the Hitching field at infinity So, that would be the module space of framed g-Hitching bundles So, that's the definition to continue, we need to get it There are questions about the definition Good Okay, so maybe some mathematicians will like another connection to the specification of the weights or singularities, namely they are canonically linked So, indeed if you take a formal neighborhood of which puncture xi then restriction of multiplicity of Hitching field g of x defines simply an element of the log group you expand in the local coordinate zx-xi so g double around parenthesis of z And this element is well defined after the joint action And also the singularity class by regular elements So, consequently the singularity class of an element of the log group would be a coset or like double orbit in the affine-grassemanian So, that's g of around parenthesis z quotient from the left and from the right of g double of g regular series in z So, orbits in affine-grassemanian are in canonical bijection of dominant covates of g And so fixing a degree of puncture is the same as fixing dominant covates at each xi So, so this spaces the modular space of multiplicative g-hicks bundles have been or their version of their particular versions in some particular cases have been considered before sometimes under the name of modular g pairs and some of earlier works include Batachin Artunaforov Medvedev Cherkis Kapustin Bredin Chernakov Dolgoshev Levin Alshanevsky Zotov Hurtubis Markman very extensive paper of the elliptic case and then Franklin Go and Boutier worked on with similar definitions for the Languines program Now, we'll take extra more conditions. Namely, we take Kalabiya condition on the base on X So, while the structure of multiplicative g-hicks bundle makes sense for any curve X, the modular space that we'll study, the modular space of multiplicative g-hicks bundles will carry canonical symplectic structure only in the very special situation. Namely, when we can equip the curve for the base X by a non-degenerate section of the canonical line bundle This is possible if X is a flat curve So, we'll take X to be C, C star or elliptic curve So, in the case of C or C star, this section develops a pole at infinity but not zero. What is the definition of non-degenerate section? So, that it doesn't develop zero So, in this case, one can equip the modular space of this multiplicative g-hicks bundle with a symplectic form like the other cases of hygenus. So, this is difference with the story of Hitchin systems. Hitchin systems are interesting to study for curves of arbitrary genus but here, we'll get a nice algebraic integrable system for nice symplectic structures on the modular space of multiplicative g-hicks bundles only when X is of genus 0 or 1, not hygenus So, in the case when X is an elliptic curve it was studied in the work of Hurtubis and Markman So, now let's go back a few tens of years and recall what's rational Poisson-Li group is and what's rational R-matrix which goes through many works on integrable systems, and stuff, right? So, let K of X denote the field of rational functions on X and G be the infinite-dimensional d-group valued in this field of rational functions and G sub 1 would denote sub-group of elements which are framed at infinity so that the value at infinity is 1 So, then this loop group carries a structure of Poisson, namely Poisson-Li group where this client Poisson structure defined by the rational R-matrix with this famous kernel 1 over X1 minus X2 where omega is the quadratic Casimir and this is back to the famous classical works from 80s about integrable systems So, quantization of this classical R-matrix leads to quantum rational R-matrix and definition of Yang-Yan etc. Yes This is your white vibration from the beginning which you haven't shown in three cases This constitutes like three probably in equivalent integrable systems But now you are talking about rational brackets Right So, there are horizontal and vertical cases We have... Now you are talking only about horizontal Yes, I'm talking about when it's rational horizontally So, usually also in some slang or jargon People say about integrable systems which are trigonometric rational elliptic in horizontal or like X-direction or trigonometric rational elliptic in the vertical or momentum direction So, our convention is that for this talk that we'll take it to be trigonometric in the vertical direction and the principalities of what is said here applies to all cases on X But, for concreteness in this lecture I'll take X to be P1 Okay So, this is a reminder about the scaling bracket which maybe more than half audience here doesn't need, but might want to look at it So, what's the scaling bracket? So, if you have an algebraic function on from G from this finite-dimensional group G we can consider the evaluation functions evaluation functions associated to a point X on a curve X which is just composition of the evaluation map and the value of this group sorry, composition of the evaluation map and that one So, and then the R matrix defines possible bracket on the evaluation functions by the formulas of the evaluation right gradients So, if the left gradient is defined by action from the left and the right gradient by action from the right of the vector fields then the Poisson bracket of two evaluation functions at points X1 and X2 is given by in the product of the left gradients at these points minus inner product of the right gradients In particular, if phi is a joint-invariant evaluation function then the left gradients right gradients coincide so consequently the Poisson bracket would be zero So, from a joint-invariant function one can construct a computing Hamiltonian and then the projection to the torus divided by the while group by the characters or a joint-invariant function would be isotropic for the Poisson structure So, one of the things I can concretely demonstrate today is the following statement So, the sub-variety namely the modular space of multiplicative Higgs bundles is a sub-variety in this infinite-dimensional Poisson group and namely it is a symplectic leaf and it is a symplectic leaf with respect to this kinetic Poisson structure defined concretely and consequently this modular space of multiplicative Higgs bundles it carries canonical halomorphic symplectic structure coming from this rational armature type and later in the talk we'll relate that symplectic structure to other definitions or other perspective on this modular space Yes In the previous slide the torus, the construction, how do you come is this TX inverse modular YW Yes, so what's the question? I didn't understand how the torus comes Yes You consider a joint-invariant function and then you have a projection to the conjugacy classes of elements in the group which is the torus mod the wild group Okay, so let's try to prove these statements we'll demonstrate the key steps here So, first of all let's see how can we identify the tangent space at the point on the modular space of multiplicative Higgs bundles So, to take a tangent space what can we do to a group element we can multiply it or deform it from the left and from the right and let's say that this deformation denoted by XI left and XI right and I will use the convenient notations for physicists thinking about all of that taken in some faithful matrix notations and here G is a group element XI is the algebra element but this multiplication means in a faithful matrix representation so if you like just think about them as matrices with a formula like that and then there is of course equivalence relation between you can conjugate and transform XI left to XI right with the same result so the equivalence relation on these pairs is that XI left to XI right is equivalent to XI left to XI right plus this deformation where you have a joint action on XI where a joint action is GXI G-1 So, geometrical way to say that is that away from the divisor the pair which defines the deformation is a section of the quotient bundle defined by this equivalence relation and then there are the usual gauge transformations infinitesimally parameterized by the sections of GP shifted by 1 so consequently the deformation complex of the multiplicative G X bundles is the following complex where this denotes their gauge deformations and A is a quotient defined here by equivalence class of left and right pairs so now let's consider a tangent space at some point at some particular configuration of the Higgs field and compute the dimension of the tangent space so at least to see some to feel what it looks like so let's say by the equivalence on this space left and right deformation pairs we can move everything to the left component right and then let's take the following assumption assume that G of X is generic near each singularity XI so let's assume that G of X is regular semi-simple when X goes to XI and in this case this map, a giant of G of X is diagonalizable near XI and then let X dependent carton subalgebra H sub X be a centralizer of that element and we'll assume that there is a limit of that centralizer as X approaches to singularity so in this case let's split in the usual way the algebra to the carton and it's a Borel positive and negative and let E alpha sub X would be generators of the root spaces so then we can see that the tangent space in the equivalence frame XI left XI left 0 is generated by the sections of this form where we have a sum over those root subspaces of the early algebra which are evaluated by the co-weight defining singularity with a positive number and then the deformations are parameterized by the terms of X minus XI to the contraction between the root and co-weight and arbitrary coefficients so this is concrete like coordinate choice to parameterize locally deformations and then assuming that we are in frame where every co-weight is dominant the total dimension of the deformation space would be just the sum over these local dimensions so that would be the sum over all the singularities and the sum over all positive roots and in a product between or a relation by co-weight and the root over which we sum and so we come to the formula that the total dimension of this modular space of multiplicity of his bundles is the evaluation of the total co-weight the half sum over the positive roots so that's the dimension other questions at this point ok so so now we can handle the dimension and we can feel what the tangent space is so we are like in a situation to study the symplectic form so let's come let's see what the symplectic form of objective Higgs bundle is and how come that it agrees with the Sklannian rational armatrix so we want to show that it's a symplectic with respect to the Poisson structure so a tangent space to a symplectic leaf locally at a point or in a Poisson variety and would be the image of the Poisson map from the co-tension bundle and the Jacobi identity would guarantee us that the distribution of the subvarieties of those images in the tangent subspace is integrable so the usual convention is shaded to a local function called Hamiltonian by physicist in the neighborhood of a pointed Poisson variety Hamiltonian vector field that's the action by the Poisson map on the differential of the phi so then the condition that the symplectic structure on S on a subvariety agrees with the Poisson structure would be the relation between the symplectic form evaluated on the vector fields associated to the Hamiltonian phi and psi and Poisson structure evaluated on the differential of this Hamiltonian so we want to show we want to give formula for the symplectic structure on the module space of multiplicative Higgs bundles and verify that for that formula this relation holds so but before that would be step 2 and the step 1 we also need to show that the image of pi, the image of the Poisson structure is the tangent space to the multiplicative Higgs bundles so I'll start by writing concretely the formula for the symplectic form on the module space of Higgs bundles question? so remember that we've parameterized the deformation of the multiplicative Higgs bundles but by pairs we have left-right modular equivalence relation and here is a formula for the symplectic form on the module space of Higgs bundles I'm just presenting you a formula and then we'll show that this formula agrees with the rational R matrix so the symplectic form is indeed induced on the sub-variety of the module space of Higgs bundles from the Sklianian Poisson structure on the infinidimensional group so what's the formula? you take here a sum over all neighborhoods of singularities and close these singularities by little disks and integrate over those disks this combination where 0 denotes excuse me, here somewhere okay so so we have disks ui and by 0 we denote a patch of x where all disks are subtracted like everything else and decompose xi into their local versions near each disk i and in everything else is denoted by 0 by subscript 0 so then the symplectic form has this formula so first let's notice that what's written here is well defined namely we need to show that under the change of the equivalence frame in the second argument when primes are shifted to some agent transformation their result doesn't change okay so let's shift then the deformation will be computed by this and then by integration by the contour integration we can see and moving by stocks the sum over d ui to the boundary and then integrating over the u0 we'll get integration of the regular part of the regular integrand here and that's 0 so omega would be invariant under equivalence in the second argument then we can also show that the previous formula is anti-symmetric so by suitable choice of equivalence frame set xi0 or prime to 0 and in this frame well maybe I'll just skip it because it's quite straightforward but anyways by doing this manipulations to save time we'll see that omega is anti-symmetric so then maybe a little bit more interesting calculation is to demonstrate actually agreement that the symplectic form that we've defined concretely before it comes from this client Poisson structure so we need to show that omega evaluated on those vector fields is equal to the Poisson bracket between the Hamiltonians which generate those vector fields so we'll choose for the equivalence frames for the vector fields in this form so xi of Hamiltonian psi x1 would be presented in this form and the other one would be presented in this form by setting to 0 their second component and then we'll evaluate the symplectic form of defined by concrete formula and this is a counter integral which involves like propagator 1 or w-x1 x1 extra insertion points and we integrate over w over all those boundaries of disks and then by taking the residues it's straightforward to see that the result is indeed the combination defined by this client armatrix so that's Poisson bracket between evolution functions psi and phi this component is consistent with a different patch dUi is the contour around each local patch and the sum of words is the same as the contour over the regular okay, so that concludes the proof but now let's just summarize the statement, what we had we had infinite dimensional Poisson league group with the client armatrix that g of k of x and I outlined the steps of the proof of the key statement that the symplectic leaves in the infinite dimensional Poisson league groups are exactly the module space of multiplicative Higgs bundles so the symplectic leaves are classified by picking a divisor on p1 valued in the co-weight lattice so you pick points attached to each point and element of the co-weight lattice dominant cone and that would be your symplectic leaf so, yes this requires an additional choice of performance about the you embed co-weight into tangent to the group at unity as far as I remember right? so co-weight for me is a map from the multiplicative group from c star to the maximum torus of g a map from c star or from u1 to the torus of g you embed the co-weight lattice into the tangent to the group no, we don't need to embed it no so in terms of like concrete parameters so what is the space of parameters for the symplectic leaves the space of parameters consists of the following first complex parameters namely the positions of the points of the singularities and discrete parameters and the choice of the co-weight at each singularity so the choice of the cartanza group inside the group no, no, no everything else is equivalent okay so that's one punchline to carry out of this from this talk that we know what are the symplectic leaves and those symplectic leaves are modulus space of multiplicative keys bundles and that's maybe important or illuminating to people who like geometric representation theory and studying representation theory by Kirillov in terms of the symplectic leaves so this quantization of this Poisson lig group or this rational Poisson lig group yields to Youngian then to this symplectic leaves one can attach the modulate of Youngian and understand representation theory of Youngian in terms of symplectic leaves okay, so we are good on that now let's give another connection to periodic monopoles what it came from yes point regarding here is it known or is it possible suppose if I consider some deformation of the explanatory bracket deformation of the explanatory bracket, okay I want to extend the hypocycles and get a minor right position so there is deformations or modifications of the symplectic correspondence that's a great question, I don't know I don't know I didn't look at it maybe, interesting question how much people suggest a comment found even more generalization I see, I see, I think okay, I think it's a nice question to explore, I didn't do any work about deformation okay, so we've had various motivation from physics and sort of physical argument why the same story could be captured by the module space of monopoles now let's give a concrete and precise statements like switching of context is a little bit so different in the sense that all the work on the previous slides we'll take the context of algebraic geometry we'll start from varieties but now we'll go to the context of nonlinear partial differential equations which describe the space of monopoles of course there is relation by two but we'll see what concretely we'll need so we'll take the following definition let x would be identified well, the x from the previous pages complex line C we'll identify it with R2 with flat Yve collision metric and we'll take M to be flat 3-dimensional equation manifold a product of X and S1 and we'll lift the divisor D the collection of points on X with values in covariates to divisor D tilde on M by attaching to each point some value in a circle in the vertical direction of the circle so the circle coordinate will be called T and the horizontal coordinate would be X as usual as before and G sub C would be denote a compact group associated to the complex reductive group G so then P sub C would be principal GC bundle on the 3-dimensional real-dimensional space equipped with the smooth connection A and the smooth Lie algebra value scalar field now it's Lie algebra value again V valued Higgs field, which was before so that's a mathematical definition for for the field content to say what Bogomolna equations are and then a monopole on M with Dirac singularities would be a configuration of A, phi which satisfy this Bogomolna equation or that's a reduction of self-dual equations to 3-dimension and the singularities are again specified by saying what is the co-weight near each singularity and here it's very transparent so since by definition co-weight is embedding a map from U1 to the maximum torus then we say that a monopole satisfies this singularity condition just when the configuration of the monopole near each singularity is the image of the unit of the standard unit Dirac monopole under this map so it sends the co-weight since unit Dirac monopole to a torus or the conjugacy class of the torus near each singularity so that's the traditional O ok, so in one direction the map from the monopoles to multiplicative Higgs bundles is very straightforward namely if you have a monopole next time S1 we can simply restrict the fields A, phi to a horizontal slice any point on S excuse me which doesn't contain singularities and then 0,1 part of the horizontal connection will determine structure of homomorphic G bundle on X so that homomorphic G bundle is a part of the definition of the multiplicative Higgs bundle and then we need the Higgs field so what will be the Higgs field? well monopole equations they imply in particular like one of the if you break the self-dual equations into the complex equations real equations then the complex equation could be written in this form and that implies that in the trivialization let's say in which del bar A is del bar the homonymy, the vertical homonymy of this complexified connection over the circle is homomorphic on X minus the divisor and so we'll have a homonymy morphism written by Kapustin and Cherkis from the modular space of monopoles to this modular space of multiplicative Higgs bundles so then we can define that the space of the polystable multiplicative Higgs bundles would be the image of the homonymy map in the modular space of multiplicative Higgs bundles and there would be version of Donaldson Ulinberg-Yauer or Kobayashi Hitchin correspondence which provide algebraic distribution of the stability condition on the modular space of Higgs bundles and the idea of this correspondence is to build the reverse map to the one which sends monopoles to Higgs so this reverse map would find S1 prime invariant harmonic Hermitian matrix on the extended four-dimensional space X times 1 prime where the zero circle is used to convert bogamon equations to self-dual Yang-Mann equations invariant around the circle and the Harmonian Hermitian matrix will equip in the solutions to the structure of instant solutions of the self-dual Yang-Mann equations so after the work of Donaldson and Consequent works a convenient way to build such harmonic matrix by running gradient descent on the Yang-Mills functional and it's quite straightforward to show well it's to show that the functional excuse me non-trivial part of this analysis is to show that when this flow continues it actually reaches the harmonic metric in the limit of the infinite time and for the compact complex surfaces the proof was given by Donaldson and in Simpson the heat flow was extended to non-compact surfaces with a certain assumption namely particular that assumption was an assumption of finite volume which doesn't apply to the present situation then further using essentially Simpson method Charbon and Hurtebes in 2008 they extended the proof to the case of monopoles still on a compact space total space X where X is compactified to a finite volume but they allowed singularities so they could handle the analysis which required in the proof of this flow to harmonic matrix near singularities and then again 10 years later Mochizuki in 2017 so just 2 years ago finally relaxed the finite assumption of Simpson and consequently from his analysis there is a proof that Donaldson-Olebek Yao Kobayashi-Hitching correspondence exist from multiplicative polystable bundle on P1 to singular monopoles on R2 times S1 so this is this line of works added more necessary analysis to the original ideas of Donaldson's for the heat flow over the Young Mills equations so now the space of singular monopoles on this 3-dimensional flat space R2 times S1 it comes with a natural hypercolostructure induced from the canonical hypercolostructure of the space of fields and the monopole equation treated as hypercolormap so hypercolostructure can be thought as twister family usually denoted with parameter by zeta of 2,0 symplectic form so we'll denote them to omega sub zeta,R so given the Donaldson-Olebek Yao Kobayashi-Hitching map we can pull back the hypercolostructure omega to the module space of polystable bundle and then define the omega zeta the symplectic form the module space of Higgs bundles is a pullback so what we want to show the few steps in the next slides is that the halomorphic symplectic structure on the module space of Higgs bundles induced by this vertical hypercolostructure on the module space of monopoles sometimes called like symplectic structure I in the slang of Hitching systems is equal to that symplectic structure defined in the previous part of the talk by this cloning rationalR matrix so in this way we'll connect the monopoles to rationalR matrix very completely by direct computation so let's do that so let's say that A would be the gauge field of the monopole and decompose it in terms of x and x bar x is the complex coordinate on x and here we extended the 3-dimensional case to the 4-dimensional case by just adding the circle along which the fields are invariant so direction in this circle corresponds to the scalar field phi of the monopoles so y is s plus i t plus another direction so the complex part of Bogan-Wollin equations is 0 comma 2 flat curvature equation that's the commutator of the bar vanishes and then we can write the 2 comma 0 halomorphic form defined by the variations of the monopole fields by this formula where that would be which product of the variations of A and A prime which is the halomorphic 2 comma 0 form dx dy on x times y this is the same as writing the variations of 0 comma 1 or the bar parts of the connections because 1 comma 0 parts are projected out by waging with 2 comma 0 form so this is canonical formula for the symplectic structure then the modulus space of monopoles and our goal is to show that it coincides with the formula we've written down before for the symplectic structure of the modulus space of hex bundles which also shown before coincides with this clanning formula let's do this computation quickly so we have that the connections are self-duals so that means f 0 2 form f 0 comma 2 is 0 then for a local variation we can have a potentials so that the local variation delta of the connection is del bar of b so b are local potentials we cannot of course do it globally around like each chart i and so there would be corresponding bi local potential but by local gauge transformation we can ensure that if we pick a horizontal slice s equals 0 we can trivialize there so when restricted to those slices at s equals 0 we have just the canonical del bar variation now since this bi is this potentials generate local gauge transformations we can compute the variation of the group Higgs fields which was the monogamy of a bar dy bar over the vertical circle and those left and right vector fields are values of this potentials at 0 or 2 pi r we can present s1 as interval from 0 to 2 pi r between 0 and 2 pi r so now let's integrate this formula where the variations of the gauge connections are replaced by del bar of two potentials and for that it will be necessary to integrate twice by parts or apply twice stocks formula to connect that to formula B without del bars and one can do it carefully by decomposition of by cutting the 3-dimensional spacing to the cylinders the vertical cylinders built on top of each patch ui and then first you integrate to the boundary of the cylinder so then formula will involve B without del bar and another B with del bar and then using agreement on the boundary between the variations since this is variation of the gauge field one can move with the computation further I'm moving maybe fast but just to give the logic flow and then one can integrate another one by stock theorem reducing to the boundary and get in the final result that the omega form the symplectic form the module space of monopoles evaluated on the variations represented by the del bar of those potentials in terms exactly in the form that we had for the module space of Higgs bundles where xi is the value of B at zero or at the 2 pi r so so this concludes that concludes illustration of of the proof that there is a symplectomorphism between the module space of monopoles in the vertical hypercal structure in the module space of Higgs bundles so there are been very connected works of course I mentioned that the key is by Gerasimov-Harchov-Lebedev and Ablesin about Youngian and Contai single monopoles on R3 and by Karmitzer-Webster-Wiegs-Yakobi on Youngian slices and affine grass mania and by Braverman-Finkilbert-Nakajima on Coulomb branches and slices and affine grass mania so let me have a well I should finish soon let me say in a few words epsilon twist of the story you take the following you deform the module space of monopoles by considering them not on a straight like rectangular product between X and S1 but on a twisted product of epsilon so that X is moved by epsilon when you move it around the circle and then the result in module space is still hypercalametric and used from the flat Yiff-Klidian structure and we want to connect it to like in the previous slides so the appropriate version of the module space of Higgs bundles becomes the module space of epsilon difference connections namely the Higgs the usual group-valued Higgs field which could be treated could be treated as a epsilon difference connection so this is this is difference version and the picture becomes like a difference version of the usual case of the Hitchin system where the module space of flat connections on X viewed in the complex structure J so this is the complex structure J version of that case there is a there is a version for the Steinberg section which will be here like an allegro of Hitchin section or constant section for the case of Hitchin systems and and from that section one can construct the sub-variety of epsilon difference operas so this well I'm skipping because need to finish but just to give keywords again of operas it comes as a mirror of the canonical choisotropic brain quantizing the module space of Higgs bundles in the case in the usual case of Higgs bundles but in the case of their group-valued Higgs bundles we see that there is similar statement holds and so when you do quasi-digitalization of the epsilon difference operas you get bit ansatz like equations quantizing integrable system on the module space of multiplicity for group-valued Higgs bundles and the connected works goes to Frankyler Hitchin and Seva Stianov and later developments with Samson and Nikita connecting to the gauge theory to the supersymmetric gauge theory in 4 dimensions in omega backgrounds and furthermore second quantization or like if you do first the deformation of this operas and then quantize it then you get the difference W algebra or whose currents are also known as quantized QQ characters or in these notations in the current notation, epsilon, comma, H bar characters okay, maybe a few concluding remarks is that which I didn't explain but people know that the module space of this multiplicity Higgs bundles of G is of AD type is identified with Coulomb branches of N equals 2 AD quiver gauge theories but what was shown before sorry, my talk was about general case not necessarily AD then SPZ duality on the fibers of integrable system of this multiplicity of Higgs bundles it provides geometric view on the Q geometric length described by Frankyler Hitchin Frankyler Okonkov Well, we sort of explained that quantization of M-Higgs in the vertical complex structure provides Youngian's modules and various versions of Speedchase and Betanzatz H quantization of Steinbeer section leads to W algebras and then it relates to the vertex algebras at the corner and also Cocha of Konsevich and Sobelmann and maybe one more point is that you can do the quantization of this modular Higgs multiplicity of Higgs by the path integral of the form like PDQ the inverse of the symplectic form and that leads to semi-halomorphic Chain Simon theory on X times S1 times R the first section which I can see in Nikrasov's 96 PhD thesis and then the theory was developed independently by Costella and further concrete computations done in Costella, Witton, Yamazaki and this is very parallel to the relation between quantization of the flat connections on X and the usual Chain Simons on X times R1 like in the Witton work but making like a halomorphic version of the real coordinate Ok, so that's I conclude here with the Samsung Thank you Are there any questions or comments? So, is it the Bogomolny equation the modular space of solution is it SU2 modular space or U1 modular space you are considering? Compactly group Any compactly group So, but there is a result by Donelson that says that modular space of solution to this Bogomolny equation for SUT actually and also this rational function which is fixed at infinity or zero at infinity is fibred by the circle over this modular space of solution of this Bogomolny equation So, this Are you referring to R3? Sorry equations on R3 or on which place? Ok, so here we consider R2 times S1 a different setup Ok Yes Is there a good point in the last slide? Yes Is there an elliptic analog Yeah, is there an elliptic analog with bungee? Is there an elliptic analog with bungee? Well, yes I think it should be halomorphic chair assignments in three-dimensional complex space Like here this chair assignments Well, the short answer is yes but I'm not sure if it's written down somewhere but to me it seems there is a straightforward way to replace that circle which is here by elliptic group and instead of studying this semi-halomorphic chair assignments just consider the chair assignments theory on complex three-dimensional Yes I'm sure if I missed it was mentioned in the talk but still Do you have some analog of the spectral description of this space like the spectral curve and the line bundle on it Yes So the spectral description is the cameral curve We have concrete formulas in paper with Nikita from 2012 What is the final formulation? What is the final formulation? There we have each point of the syntax manifold kitchens can be described as a curve in some surface right? and the line bundle on it Right So here it would be slightly more involved maybe I will write a couple more formulas So let's say that you have x is our base curve So in the Hitchin case you would take t star to x Here we don't take t star to x We assume already that x is flat and we'll take x g the maximal torus of g but for j this should be In case of group j everyone can go to surface Yes one can go right right Excuse me Let me So let me just write it concretely Так как я беру here rank G and write. So the equations here, which define the curve here are given by writing a collection of the characters, where I run over fundamental representation of G. GFX — это группа, в которой есть философия, и вы выигрываете их к фиксированным функциям, скажем, TIFX. Ну, полиномиал или рациональные функции с фиксированными структурами — это устройство. Так что, вместо структуры оригинального группы, вместо структуры в спектрул-кюрфе, он вырезает камерал-кюрф в высокомерситете. Вы можете его заменить по спектруль-кюрфу в легкости, если группа есть GL, а потом можно вырезать детерминатор, который подбавляет всю информацию об этих характерах. Но это более генеральная конструкция, так что вы вырезаете эту кюрфу, так что это камерал-кюрф в 1-й классе, ронг-g, дименциал-спейс, и на этом камерал-кюрфе, опять-таки, есть линия-бандалов, которые находятся в структуре конструкции Донаги Гайсгори. Это было важным, что на поверхности, где спектруль-кюрф вырезает, это дименциально, и, поэтому, это фасон. И здесь я не могу понять, как структура в спектруль-кюрфе можно вырезать из-за этой камеры, потому что там была только часть генеральной конструкции, симплятия структура в модууме спектруль-кюрфов, на поверхности фасона. Но здесь, я имею в виду, это не необходимо для нашего аргумента. Мы конструктировали симплятию формы в модууме спектруль-кюрфов, в зависимости от камерал-кюрфа конструкции. Мы конструктировали их прямо из-за фасона. Мы конструктировали, чтобы показать, что модууме спектруль-кюрфов в модууме спектруль-кюрфов есть, на самом деле, симплятия структура в пауэсонной группе. Мы пришли из-за этого конструкции, без реферии к конструкции, которые вы увидели. Мы поднялись до конструкции, но мастер-фиггер продолжается.