 Так что, мы сделаем несколько курсов здесь на очень-очень много вопросов, и мы увидим много-много субъектов, и это связано с курсами, которые я делал 2 года назад, но это большая разработка, и мы увидим, что это такое. Итак, мы поговорим о квантизации, и я просто начнём с унитраконтизации, и с квантовыми механиками. Да, это что-то, что я хочу сказать, это действительно не преследовательность, это какой-то симпатичный статус, это очень файзий, так что конструкция не очень каноничная, так что мы начнём с реальной, как и синфинитией симпляктики манифолта, и, как говорится в квантизации, к этому манифолту мы должны насочить комплекс хилбрдспейс Х, который зависит от манифолта и симпляктики структура, и лагранжен субманифолт должен дать примерно 1 вектор, и если мы посмотрим в экземплеи, в деталях, в принципе, лагранжен субманифолт должен быть довольно расширенный, если мы добавим extra data, примерно 1 должен быть половина дентстики, и субманифолта зависит от манифолта, и, может быть, форма субманифолта будет внутрь лагранжена субманифолта, которая дентстика, так что это будет позитивный номер. Это такая ровная птичка, и, как говорится, это основные экземплии. Первый экземпль в экземплее, р-дименция 2D, и омега это 1 over hbar times standard form, hbar будет небольшой параметр, все зависит от, да, так что в секретаре можно попробовать увидеть эти омега, это 1 over hbar, какая-то стандартная форма, и в общем-то, тогда хбт-спейс это L2 of rd, это координат x, и L2, опять-таки, это не функция, но дентстика, координат субманифолта и x-координат, или можно выносить их в y-координат, и у нас фурер-трофон, и в фурер-трофоне фурер-трофон должен быть экспонентом, может быть, выносить сентябрами, выносить, выносить, выносить, или выносить сентябрами. И если лагмажный манифолт, то это граф, то это деференция, Function, a function is from r dx to r, real-valid function, then, and also gets this half density, c of l rho will be exponent of i f over h bar, of x over h bar, now i is square root of minus 1, because I don't, this is, and then you can see the, I project my Lagrange manifold to r dx, and take direct image of this half density. Yeah, that's, you get some function. Yeah, so it's how this all looks in non-compact case, and in compact case, I'll just say this one example, x will be sphere, and now h bar will be 1 over integer, so it will be some quantization condition, not any, like constant in a compact case, works, and then omega will be, will be 1 over h bar omega 0, so omega 0 is area form on sphere divided by 2, so we see that interval of omega is equal to 2 pi n, and corresponding Hilbert space is c n, and Lagrange manifold, again, not every Lagrange manifold will be quantized, should be, again, some quantization condition, it's called Bohr-Zommerfeldt, quantization condition on Lagrangian, I'll explain it in a second. Yeah, so, so you have a sphere, and you choose Norsen, Sausson-Paul, and you draw, is Lagrangian various circles, and you draw capital N circles. The quantization condition is that integral of omega over upper half or lower half of the circle belongs to pi 2, 2 pi z, so if you divide by pi, you get odd number, total areas, and you get capital N circles, and this will be basis of Hilbert space, yeah. So, that's a rough picture of quantization. If my manifold has complex structure J, so it is Keller, complex structure compatible simple electric structure, then one can be a little bit more concrete. Then the Hilbert space you define as kind of space of holomorphic section of certain holomorphic bundle, which I'll write like this. Kx is a canonical bundle, it is a complex dimension, so it's canonical bundle, yeah, I'll write square root, which is not always makes sense, but this product should make sense as an actual line bundle, and 2 pi first-gen class of E should be equal to class of omega, so this relation between this line bundle and this like this, and so that's the definition, let's see what was an example, yeah, I just want to write that the scalar product is a following, if you get maybe 2 sections, you define scalar product as integral over manifold, you consider scalar product pointwise, so for this I need metrization of both bundles, and to multiply by volume element, yeah, so that's the definition of scalar product. Now what goes in complex case, which is easy, yeah, suppose my manifold is complex structure just Cp1, then E will be of N, N is the same there, and canonical plus power one-half is O minus 1, and then one can see that eventually that gamma of Cp1, if you make terms of product of N minus 1, is one dimension, is N dimensional, yeah, and non-compact case is kind of similar, suppose my space is just C2 power D, and E will be, and also Kx to power one-half was trivial line bundle, but metrization on E is given by exponent minus sum over Zi squared, 1 to D, metrization of section 1 will be given this function, so dd bar of this thing will be give flat scalar metric, and on E is K to one-half is trivial, and then one get kind of description of the same Hilbert space in different ways, which is various interpolates isomorphism, so the same Hilbert space, which you saw on the left board, which was L2 of Rd, can be realized as space of holomorphic functions on Cd, set at integral of f squared, this point is minus sum of Zi squared is finite, and this will be L2 norm, yeah, so the same space one can see as this Fourier transform functions on real variables of holomorphic functions, okay, yeah, so that how one can materialize this, this correspondence at least for Hilbert space using scalar geometry, not any complex manifold, not any symplectic manifold has complex structure, one can use almost scalar structure, so it will be no condition of integrability, and instead of global sections, consider kernel of derecuperator, which will be derecuperator in strong magnetic field, I want to say, so in case this derecuperator is d bar plus d bar cross, and takes from sections, now C infinity sections of what, I have the same bundle in one-half, and multiply by d-bar forms, now consider even part of d-bar forms, and mapping to the same thing is omega-odder, yeah, so the derecuperator in scalar cases d-bar plus d-bar, and it exists for non-material complex structure as well, and then you see that if omega is one over each bar of some reference form, then dimension of h will be the same as, for h-bar cross to zero will be the same as index, and this is equal to integral over 1 over d factorial omega-zero times 2 h-bar to power d, plus smaller terms, yeah, so this, so if you choose this almost scalar structure, you get some construction of this Hilbert spaces, finite dimension is compact. Now I'll go to next things, so I assume that my, like in all these examples, I have small parameter h-bar, and I have now real valued function, I will associate operator h-bar to h-bar, which will be self-adjoint operator. I do it by the following formula. If I have two sections and I want to calculate matrix coefficient of my operator, my definition will be the dune by the following integral. I just substitute to this formula value of f at point x. So it gives, obviously, some Hermitian form and gives self-adjoint operator. In particular, just by definition, if you get function one, this corresponding operator will be identity operator. So that makes a following claim. In scalar case, it's, I think, theorem of Borderman, but in non-keler case, it should be true, just for families of Dirac operator, is the following. If h-bar goes to zero, then one have the following isymtotic expansion. So if you take two functions f and g, consider operator's case point to them, put to this product, it will be asymptotically equal to the following. It is a product of two operators here. The claimant will be close to quantization of another function. It will be fg plus h-bar. And in fact, it will be sum of n greater than zero hn bn fg hat, where what? So, first of all, the leading term is a product. And in general, for any n, bn is like from c infinity of x. c infinity of x, t of x actually be complex valued, is a bideferential operator of order nn. So, it takes most n derivatives in each argument. Then we get bn1 anything is equal to bn anything one equal to zero for n at least one. So, it has no constant term. And then there will be some kind of reality condition. So, bn fg-bar is equal to bn fg are real functions. And finally b1 of fg minus b1 of gf is equal to i h-bar f force one bracket. So, what do you get? Let's forget about this last thing. This thing is called star product. It can be defined in any station. It can speak about differential or poly-differential operators. It could be algebraic varieties, smooth varieties. You get a family of bideferential operators but this is only essentially one axiom of the star product that if you consider sum of bn fg times h to the power n gives associative product x complex coefficients in form of power series in j bar these coefficients in functions. So, we understand these things to see if h linearity would be c bar. The associativity is kind of obviously follows when these things come from the quantization because product of operators is associative and it's this kind of homomorphisms more or less faithful. But one can classify, ask the formal question and it was the idea to study general star product which was by Bain-Flatau-Fronsdell-Nicholas-Röschen-Sternheimer many years ago. So, it's a kind of big program and I will talk about it in a minute. Just one small remark. Kind of from this perspective of star product it became kind of clear why kind of question why this symplectic manifold relates to complex Hilbert space. So, we can see the algebra of symmetries of symplectic manifold. It's more or less the same as functions up to some finite dimension. I think it's more or less real valued functions. This Poisson bracket, which is given by omega inverse. So, this is the algebra of functions and the algebras of symmetries of Hilbert space are the same as i times self-adjoint operators, the group of unitary group. And now you see that if you're in a leading term if you see that functions give essentially all self-adjoint operators then the bracket will be close to the bracket here. So, we see that the algebras have kind of same size, maybe both infinite dimensional, but not isomorphic. And because they are not isomorphic there could be no canonical construction. That's the main trouble here. So, how star product look like in natural examples? You can see the algebra, because it's kind of pure algebra equation I can start with algebra of polynomials in case a field of characteristic zero, complex numbers and m could be 2D. For example, if you think about flat space and let's gamma will be constant by vector field, but not neither symmetric nor skew symmetric. This could be not necessary skew symmetric. Just get some matrix. Then define bn of fg, defined as 1 over n factorial and take sum over i1 up to in, g1 up to gn. This one can immediately check that it gives this star product, it gives associative product and what is basic example if m is even number ij is equal to delta ij plus 1. So, matrix gamma is 0, 0, 0 and here identity on third block. Then what's the algebra you get? You can see the function and we separate variables into groups, dn to d, and attach to this f hat will be operator in, replace this variables by hd on the right. We get differential operators. So, it means it's hxd plus k goes to hd. We get isomorphism of algebra of polynomials with differential operators and this product, gamma product gives just the usual product of differential operators. So, the algebra is what's called the variable. Algebra is differential operators in the dimensional space with hbar put into the game. One can make another matrix, if gamma is skew symmetric for example, here we get 1 half times identity minus 1 half times identity just skew symmetries. The original matrix, one can get something which is called to substitute this formula, it's called moyale product and the main advantage of this moyale product it is sp to dk invariant unlike this product of differential operators it's completely invariant and if k is my ground field this r then formally one can write moyale product it's given by some kernel which is a bit similar to Fourier transform exponent of i over hbar and you integrate area of symplectic form of the straight triangles with vertices x and z. It's kind of analytic expression but if you make a synthetic expansion hbar equal to 0 you get some poly differential operators which are exactly the same moyale product. And this is invariant it's kind of very important it's Hermann-Vey algebra but also get Andree-Vey representation that symplectic group of g r acts on automorphism of this algebraic moyale product by the way it's the same Veyle algebra algebra and this maps to up to characterization of unitary operators of L2RD set projectively in RDA yeah yeah, so this there are two looks like two different product but these are isomorphic algebras and I just want to say make my small definition what is the gauge equivalence between between two star product product prime product double prime so it's given by some sequence operators and another sequence by differential operators the gauge equivalent is is a sequence of operators with pn is let's say some c infinity x or whatever algebraic have differential operator of order at most n p0 is identity and pn of 1 is equal to 0 for n greater than 1 yeah, by the way to this condition it's not really necessary but it's kind of shows that the algebra has very clean unit element which is 1 anyhow it will have some unit element but just convenience and and such that the following such if consider p which is sum over pn hn or h this this sum we get operator which is invertible this gives isomorphisms of algebra algebras transform one product to another product okay so what so gauge equivalence gives the same isomorphic algebras and what a basic tool here there is certain construction by Fyadosov which I will not discuss given any non-flat connection on tangent bundle on symplectic manifold he produces certain explicit star product the basic idea is the following locally manifold looks like a vector space it's a tangent space on the tangent space you can use moyale product which is comminable and then it uses connection to glue this moyale products and second thing if one changes connection connection then get gauge equivalence product again by certain explicit gauge equivalence so it means that you get canonical equivalence class of star products and the theorem one can also classify all star products with given given x omega kind of complex valued star products up to gauge equivalence I naturally want one correspondence as a correspondence it's called something like deline invariant what will be invariant with star product it will be certain commulgy class in one over h bar and consider the second commulgy of x with coefficients in series in h bar such that this class is equal to omega divided by i h bar the things plus 1 and in particular this construction the class is really equal to omega divided by h bar no correction terms it's kind of rescaling family so now I'll go to some kind of story so the conclusion you get many many choices you get many many isomorphic algebras but what one can of course if you have different isomorphic algebras is the categories of modules are equivalent but then one can analyze it's kind of a bit more complicated story the claim categories of modules of quantized algebras this algebras may be denoted by h bar which is like c infinity of x star product a canonically identified for example you have explicit gauge equivalence but then if you go about 3 gauge equivalences if you got 3 things that composition will be not identity but turns out it will be conjugation by some element in the algebra it will be in an automorphism it satisfies some additional the canonical identification of categories so that's for any symplectic manuals if you dose of constructions which depends on all connections eventually get just one category on which for example symplectomorphism group acts so symplectomorphism group of x acts on this category c now so what we see we have kind of 3 levels of abstractions we can get some abelian category maybe depending on h bar and formally sense of form power series then we have certain algebra a h it should be associative algebra also depending on h bar and formally and then we get Hilbert space also depending on h bar maybe not formally we are using this quantization parameter h bar quantized quantization constraint and how these things are related this relation here is like this this category is something like h modules of the algebra and a h a h is endomorphism of some projective object generator which is a h itself if you have category of modules you can see the one dimensional one consider its endomorphism group and get opposite algebra so that algebra is kind of category with a choice of some projective object and the relation here is kind of essentially algebra acts on h on Hilbert space epimorphically so that's there are three three different things and what I can say about this so there was a paper about maybe eight years ago by Gouk of Inviton it contains some very remarkable proposal and it was never materialized mathematically so it's not yet realized even physically it's not yet realized what data you need from a symplectic meaningful to get canonical construction of Hilbert space something is missing because symmetry groups are not the same and the additional necessary data maybe sufficient additional data for x omega giving everything here in particular h should be the following it so x should be totally real sub-manifold kind of fixed points of anti-halomorphic evolution on halomorphic symplectic manifold and some halomorphic manifold with halomorphic to form symplectic form and this thing should be kind of big maybe complete if you get real manifolds always have some small simple small complex manifold neighborhood it's not a big deal it's kind of canonical thing but to have really big neighborhood they claim that it's the data which somehow fix the Hilbert space it's very mysterious there's no really rigorous formulation here and in fact what I have to add is some small extra topological data on xc which I will not talk about there was actually strange coincidence of ZIP proposal and something which I proposed together with Konell several years ago so it's related to the following to the conjecture of Alexei Belov Konell and myself and the conjecture is almost proven now the conjecture is the following we start just a case of affine space that automorphism of in algebraic sense of c2n c2d with standard symplectic form is canonically isomorphic to some huge group of polynomial symplectomorphisms to the group of automorphism of V-algebra using Myel product we see that this thing is Myel product gives invariance equivalence under sp 2dc under symplectic affine transformation but the claim it works for nonlinear polynomial transformations so get some canonical way so there are several reasons for this conjecture I will explain in the end of my today lectures some original reasons but this Belov Konell came to this conjecture it was using characteristic p and the proof used characteristic p almost proof which exists now used characteristic p it should be different proof and it works for real numbers we can make real numbers here that matter so these things should work over any field of characteristic zero and then I have I can make kind of companion conjecture which is related to Govvitan proposal of cononicity of the situation conjecture 2 then this automorphism of V-algebra this is the same group should act to projective group of unitary transformation of Schwarz space so functions this rapid decay which is invariant to the free transform which is pre-Hilbert space sitting in L2 of RD and so this group should act V-algebra acts on Schwarz space it's a denser space and conjecture that the section of symmetry group goes to symmetry of this representation so it will be generalization of various representations kind of so various representations is action by automorphism of space this conjecture was kind of very paradoxical for example if you look algebraic limits of some algebraic varieties if you bound degrees of polynomials and Lie-algebras are not isomorphic it's exactly these functions but the groups somehow a bit smaller and they don't feel this not isomorphism of Lie-algebras okay I think it's kind of confirmation of Govvitan proposals when you see that complex manifold maybe algebraic complex the symmetries of this simplification will be small enough to really act on a Hilbert space so now after this introduction to star products let's see what is the goal of my course I will describe something different not unitary quantization and not Govvitan proposal but quantization of complex algebraic symplectic manifolds yeah, so these objects which should kind of appear in this proposal but without anti-halomorphic evolution and the main story here is a little bit two different canonical not fuzzy as we have in this unitary world constructions kind of A model and B model construction of what of categories so unfortunately the whole thing it's kind of really goes to some most abstract story some numbers, vector spaces need some work so this basic example if my manifold is cotangent bundle to some smooth algebraic variety then A model category is representation of fundamental group one of version of A model will be representation of fundamental group of Y not necessary finite dimensional one can say it's kind of module so the group ring and it contains kind of finite dimensional representations small part objects small possible things and B model will be dY modules dY is differential operators global dY modules it's think and it contains as part let's say holonomic modules and contains algebraic vector bundle plus flat connection with regular singularities and Riemann-Hilbert correspondence said that these two small pieces of large categories are equivalent and equivalence is kind of transcendental, it's not algebraic you can do different algebraic calculation here and there it's kind of analytic not algebraic and I will try to say it's a bit hard to formulate all things precisely here that these things will be very very general for general complex simplectical break variety I'll have two categories of different nature and some kind of holonomic modules will be equivalent categories essentially the ultimate Riemann-Hilbert correspondence ok so now maybe this break about maybe 5-10 minutes ok so I'll continue so as explained I want to quantize complex varieties so I'll now talk about star products on complex analytic first simplectic manifolds and because I don't know don't use any reality conditions no anymore so new hbar will be i times old hbar kind of this and bar and physicists will be related by I and hbar is complex yeah it's just complex formal parameter yeah just there's really no need to write square root of minus 1 anymore so first I'll give a definition a of dq algebras dq is deformation quantization on X on X is given by choose choose some covering and assume that all intersections are Stein homomorphically convex some balls actually I'll give brief in terms of concrete terms first on each ual ualpha we fix some star product depending on on index alpha on algebra functions sequence of B differential operators with the same axiomatics which I had before then on each double intersection I should have gauge equivalence a sequence of differential operators p a n alphabeta is differential operator from the intersection itself order n and satisfying the same condition and axiomatics will be as a following and this star product is associative star product and this is identify star products on intersection and axiomatic on triple intersection on any triple intersection so you get something open domain on double intersection and on triple intersection we get 3 star product we get 3 isomorphisms and the axiom is what I get here is identity so the composed identity isomorphism and so what the whole data and what it gives and gives a шеф of algebras of a formal power series I don't know maybe called something like AH and plus isomorphism and go to which algebras should be free typologically free cah modules and in case of generators will be identified it will be part of the data with OX as Schiff of algebras of complex numbers ok, so that's and it's obvious how to this notion of kind of gauchic equivalence of such things you get the same Schiff's and isomorphism given again by some sequences of poly differential operators yeah I think it's it is hard to classify not unlike in in the real case it's there was this classification result which shows it's this formal pass and second homology and in complex case it's more the reason why by the way I just forgot to say that a social product or alpha and we consider this Poisson bracket appears in the first term it's a constraint yeah, in principle one can forget about this constraint about star product and to get just some Poisson structure on your manifold not necessarily a symplectic one but even in a symplectic case it's hard to classify the reason locally all star products in a symplectic case are equivalent to each other so it means that if you give a non-star product then classification is given by H1 of X with coefficients on certain shift of groups will be and groups will be automorphisms of AH given by poly differential by gauchic equivalences which gives identity automatically in this case modular H bar because modular H bar will identify with functions and this orthomorphism group it's very of this gauchic equivalence it's actually in a symplectic case it's very easy to because the shift of group is the same as shift of invertible elements in this algebra modular center and this is non-Abelian group so it's kind of hard to make deformation series with non-Abelian group no clean answer and the idea is to replace kind of modify a little bit the question so we get this automorphism maybe this shift of AH it's a shift of groups replace these things replace by kind of automorphism of category instead of automorphism of algebra consider automorphism of category which is something called crossed module it's like two-step complex of non-Abelian groups this will place an ecological degree 0 and degree minus 1 and here we get inverse and if you substitute this number you see that it's the same as C of H bar inverse placed in degree minus 1 and it's completely Abelian homological degree 1 and it's completely Abelian so it means that one should kind of modify a little bit the question what is the modification it's some very simple word definition is an algebraoid over some commutative ring R is an R linear category which is non-empty and all objects are isomorphic so it's pretty this category is many isomorphic objects it can replace by category is one object if you have category is one object you get algebra over R if one object you get the same as associative algebra over R but to have different object you get isomorphic algebras since algebras are not canonically isomorphic you choose some isomorphism so it's little modification so it's what is similar to it's similar to notion of group void imagine like you have a connected topological space then for each point you get fundamental group and this group are not canonically isomorphic you choose a pass between two points so you get a category where all objects are isomorphic and each object has automorphism all morphisms are invertible and this is called group void instead of group we have algebras called algebraic and if you have algebra you get category of modules and if you have algebra it gives a billion category of modules of it which are functors and if you choose one object it will be equivalent to R modules to A modules A is endomorphism of this category of any given object and see all this is in symplectic case or it's just pass on no no no this now this story it's could be done in pass on this thing as well yeah but even in symplectic case I claim the story is complicated now so you get so the idea is that kind of modify this story one can make kind of parallel definition in shifts of maybe algebroids I want to make a shift of algebroids using this star products it will be slightly different story so again I get on each open set I get star product on each intersection I get gauge equivalence and now maybe just put it in other colors modification algebroids now this axiom will be replaced to some additional data on each triple intersection this composition which I have between three algebras composition is not equal to identity but equal to adjoint action by certain element which I said have to add to the game it's some new data f alpha beta gamma f alpha beta gamma it belongs to algebro functions u alpha u beta u gamma this product say product prime and it should be invertible element so I get material element I'll get axiom will be the following on each quadruple intersection it's kind of a bit messy to rate what I get so one can denote elements intersections vertices of triangle then for each phase one gets certain invertible element and plenty of isomorphisms and product of four elements f these things is equal to one instead of before it was compositional to identity now I have corrections go to next comological level yeah it's next cacycle story and its modification and then it gives the following it gives a of algebroids of c h bar and in particular shape of categories of modules and plus equivalence model h bar is yeah so it's it's this kind of more tricky object but description of parameters it's much easier to be essentially the same as real story yeah so what is nice here we get this shift of categories we get global object so we can speak we get certain categories something like modules of deformation quantization also snow algebra around and now I just want to say this theorem so the first thing it's in this algebroids it's it's so general notion we can start not necessarily with Poisson's break just any Poisson's not symplectic structure any maybe degenerate Poisson's structure and then it's kind of hard result result it's kind of formality in my formality theorem in deformation quantization that any for any complex Poisson's structure you get canonical shift of algebroids deformation quantization sense it's like in fiduciary rescales Poisson's structure yeah just rescale Poisson's structure and make this linear family yeah it's it's kind of black box and depends on the choice of dream associator in fact several constructions Gallo group action Gallo group Grotting-Dick-Dick-Middle group acts on such constructions so for example the way to associate to Poisson's structure a star product in sense not there's some hidden universal parameters one can modify it in one way to another yeah so it's in the form of within an available but what do you mean by black box is it a proof or the it's a proof but it's kind of it's theoretical from couples things you cannot really calculate because Feynman diagrams are not known and yeah so it's a theoretical result yeah yeah not at all yeah but anyhow in general Poisson in general Poisson situation if one get some star product and this maybe dq algebra it one can twist some simple way any shift of algebra it any shift of dq algebra it by element in h2x c of a bar by this second just in simple electric case all various shifts of algebra it's obtained by for the canonical one by twists or really parameterized by twists all shifts dq algebra vists applied to the canonical canonical shift there is a 2 and 3 are coming from this construction 2 and 3 no no no no this is notion of algebra it's some kind of nice story but which replace deformation theory by this simple guy that's it okay this comes from this no no no it's a theorem I don't know I didn't explain where it's come from and how it's proven just for your information it's not a theorem to prove yeah yeah so this is a black box but in fact 2 and 3 in this theorem can be described independently in simple electric case using kind of fidosov construction yeah it was written in paper by in papers by Kashiwara and Shapiro but they are not not alone I also I remember period when fidosov construction arises I also kind of made it for myself yeah so it's pretty easy yeah so one shouldn't go through this deformation criterion the snow doesn't depend on the whole story for simple electric case it doesn't depend on on associator all these horrors are completely irrelevant Will you explain where is the twist? twist no no I will not explain yeah but yeah so the answer will be essentially the same as in the real case in the real case I say that this classification of star protocol gauchic is h2 of rx with form power series in h bar and here it's kind of the same story as it was essentially twist is the following you choose your cacycle on triple intersection get constant function at least it's really really easy if you get certain c alpha beta gamma which is in a constant c hr star you modify unfortunately erase my thing c alpha beta gamma c alpha beta gamma multiplied by c yeah that's that's a twist in terms of effect and the fact it's cacycle guarantees the difference is okay yeah so twist it's really straight forward in terms of this covering sorry you said that it's not connected that the stories are connected to cacycle to cacycle which cacycle the one you called 5 yeah it's connected yeah twist acts on this yeah okay yeah so what is the role of twists in fact this again one can make the it's a bit longer story I suppose we get a cotangent bundle with this standard form so we get by all this story we get a shift of algebraids and in fact it will be shift of algebras but the claim it is claim it is not equal to the naive dq modules yeah so you have coordinates x i on maybe x i on on y and consider expressions h, di and x shift of algebras it's not this naive which you exist on cotangent bundle so don't get differential operators or micro differential operators and the twist is twist parameter is a certain class in the second commode which class it will be a certain commode class on y and which class on y you can see the pi i sorry this number i still appears in the game of this first-gen class of canonical bundle of y so again we get kind of half canonical bundle here which sits in h2 yeah this element sits in h2 maybe x c model 2 pi i z and it's two torsion class so twice the sinkers will be equal to zero it's two torsion class which can be maybe sorted you consider some four-dimensional algebras of y and c mod 2 pi i h2 of x c star which sits to h2 of x yeah so so it's two torsion element it's actually zero if sorry because square root of line bundle it doesn't have to be exist in the given class of h2 c star plus minus 1 and you can make certain h2 for this sorry c1 of canonical bundle of y so what why it's not 2 pi i why it's square root yeah it's yeah it's zero if there exists square root of canonical bundle in the cargo of y and what is the origin of square root it's some kind of calculation but let's me explain basic idea the whole story was based on moyel product fiddles quantization using glunt moyel product and let's do suppose we get some moyel product we get maybe variables xd and y1 by d which before I denote by consider moyel product and consider expression f which is assume that f is linear in y so it should correspond to vector field by quantization yeah let's just do two variables and consider xn y for any n again what will be differential operator corresponding to this function the moyel product it's given by kind of symmetrized product and if you see what is symmetrized is equal to x1 dx i put also each bar equal to 1 for simplicity plus x in this one dx x you average over all possible ways to put differential operation in the middle because my hat will be dx and now we calculate it what is it we get xn into dx and plus 0 plus 1 is n divided by n plus 1 xn minus 1 and what is this guy is equal to n times n plus 1 over 2 so we see that this thing is equal to xn dx plus 1 half n xn minus 1 which is xn dx plus 1 half of divergence of the vector field on the line and this is this one half I kind of explain one another so that's the basic calculation behind how I get this square root of so this is small kind of problem with this twist which appears all the time and I will not go let's assume that we can learn this twist story so this little modification of what maybe one can see now let's proceed we get a shift of categories of modules over this algebra dp algebra and suppose we have a global object we say it is it is called holonomic if it is locally kind of flat or free logically free over power series which deform something over h bar and coherent shift over o x at h bar equal to zero so we get deformation of coherent shift and in this case kind of simple claim dimension of support of this model m whatever m f m model h m is at least d which is half of complex dimension of x and support is always is always quasi-tropic and model is called holonomic if the support and support is Lagrangian which means it has the smallest possible dimension at any point so this is kind of this holonomic models which eventually appear in this Riemann-Hilbert correspondence correspond to Lagrangian subvariety maybe some vector bundles on them of this story with the star products I explained that locally they all look the same if you know symplectic structures they are all morphic to each other and there is something similar for holonomic modules let's l will be smooth analytic kind of analytic Lagrangian at manifold then locally there exist unique up to isomorphism holonomic decimodial said that its classical limit is as a morphic to functions on Lagrangian submanifold that's really easy calculation you just first construct this model and see that it has no deformations local coordinates again we introduce some coordinates i this Lagrangian submanifold will be like this given by this equation then algebra what do I denote at least germ of the shift of algebras is the folic I can see the analytic functions in x1, xd and also hd1 hdn not necessarily polynomial put hbar and this module and ml is function one variable and what is the automorphism group of this module locally it just multiplication by constants then if my twist parameter is equal to 0 one can study what are local objects on Lagrangian manifolds which are locally as a morphic to this guy and turns out that global objects on l locally as a morphic to ml again first commode of some shift but it will be twisted story it will be line bundle on l may be formally dependent on parameters plus flat connection on what line bundle dependent on parameter and flat connection also dependent on parameter on classical tensor canonical l to power minus one half so this line bundle will be not of degree 0 but of degree half canonical class and what does it mean in principle there is no no chairs of square root it's equivalent to flat connection on m classical square root multiplied by KL inverse this will be honest a line bundle and you choose a flat connection so that's the case when the twist is nonzero and if x is a cotangent bundle and we use twist by K y to power one half if you want to get differential operators so if you want kind of d u modules in this case it's always reduced to the linear in y situation linear you had linear differential operators x to the power n times y you took z no it was for my previous situation yeah no it eventually reduced to this action of vector fields yeah yes yes yes because in case of cotangent bundle these symmetries only functions in vector fields yeah yeah but differential operators anything I just want to say that you get flat connection on m classical multiplied by KL minus one half and times pullback of K y to power plus one half this one can analyze yeah so it looks kind of pretty abstract but in fact it's sync which is very well known in differential equations it's kind of w kb it's called vorosshiv I think something like this some people called vorosshiv parameters yeah that's let me tell me what does mean in real case real life suppose let's say y is a curve y you get cotangent bundle to y this fiber is by vertical fibers and then you get your sub variety L which is usually called spectral curve and assumes that L is smooth and projects to y with only double no no no just one equation just one equation depending on parameters it's not integral system so it's you get such a picture and when you leave to some modules you get kind of d h bar y module module holonomic module depending on parameter on h bar for example we write some equations something like h dx square plus blah blah blah some vfx things like this and then classical limit you go to hyperrhythmic curve we get this story if you get this equation depending on small parameter then you get this various ramification points alpha and sum over r alpha is exactly difference between s divisor on L is difference canonical class of L minus pullback of canonical class of y and it's even number it's divisible by 2 we are lucky for the curve it's divisible by 2 but in principle nobody guaranteed it and all this stuff will be the same as representation of fundamental group of L minus the set of ramification points 2 c of h bar star such that if you go around any ramification point you go to element minus 1 yeah and how it is arising real life what is this local system you get yeah so you get this spectral curve and this relation and this dq module outside of ramification points is as a more to standard module because you write this locally locally L is graph of df for certain function function is defined up to constant and you can see the standard dq module you can see the exponent of f over h bar multiplied by locally it's shift of modules analytic functions in x and h bar so you can guess what is the module in coordinates in this way and it doesn't make if you change f to constant you get canonical standard module and the set of isomorphism will be torsor over h bar and then if you go around you get some representation from the middle group which is kind of very well known procedure in differential equations and you get to differential equations you get for any loop you get invertible element so it's I just want to say that it's all this abstract categorical nonsense with twist that eventually goes to very concrete geometry and now I'll just say finish with you verse about non-formal quantization so it's before it's essentially everything I want to tell about kind of formal quantization non-formal so the story that's here we'll have no shifts so it's pretty pretty pretty different story I have algebraic symplectic manifolds or some field k it's not always big quantized non-perturbatory what is the what's the problem this is a billion category or algebras whatever modules locally can do something but this is a problem when x is not compact problem at infinity of x this kind of no control for hn terms n goes to infinity just kind of maybe suppose x is a fine space and suppose it really deforms whole algebra then we get something like x2d we get product on this guy but the problem if you consider the coefficient of high high power of h bar there will be polynomials of high high degree in x and this should be somehow controlled and in order to control we have at infinity one should choose a compactification and suppose it's Poisson compactification and x will be open symplectic leaf then this deformation quantization story says it's basically fine definition of kind of good category good dq modules will be coherent on dq modules over h bar and here one should use deformation quantization for Poisson manifolds in full force and more doubt by those supported at infinity and then you get something more reasonable but I think maybe I better stop now and continue next time