 Lecture is extension of Riemann Hilbert correspondence, h bar equal to zero. That will be in the following setup, x omega, which will be algebraic symplectic manifold with C, which is embedded as open dense symplectic leaf to some Poisson variety. And I assume that Poisson variety is compact, algebraic, and the complement is normal crossing the divisor. Be sure that quantization exists. And also you choose a class. No, no, no. What I assume? That's it. Okay. Let's read. I don't know. It guarantees there exists some shift of categories. Yeah. And I have a certain twist class, which in real life always zero. And then as we discussed, there should be some hypothetical Riemann Hilbert correspondence. We should get to okay categories, depending on the parameter h bar, which is okay category of x. Endowed with a symplectic form, real part of each bar and a B field will be a class of imaginary part of omega h bar plus B zero. The class in this thing belongs to h2x and maybe C mod 2 pi Z. And then it's really hypothetically for h bar, small but nonzero, should be equal to certain category of objects in quantized, and I will not repeat all my excuses, category of x bar. Said that the restriction to the boundary is zero. It will be a little compact supporting the bar. Okay. Yeah. No, we can discuss. It's in process for hours. You have a category on the left. On the right-hand side, it's not totally sure. It's a category, but yeah. You said objects. Then just change. Yeah. Category, yeah. Category of objects. Category. Full subcategory of objects consisting of this. Okay. And let's call something like DQ models also depending on h bar. Now, or yeah, so it's in a sense, it's kind of a model and equivalent to quantized B model on the same manifold, not on mirror dual. It's not mirror symmetry, but in other equivalents of categories of A and B type. Yeah. And now the category on the right-hand side. We have category, in fact, we have categories of various rings over C of h bar, h inverse, and just a little bit more space. So we can consider journals of analytic functions, journal of meromorphic functions, and journal of functions, kind of an epsilon goes to zero, or of the disk analytic functions on a small punctured disk, which could have essential singularity. Yeah. And what we have equivalents, it's equivalents over these big categories defined over this big ring. But right-hand side categories defined, in fact, have a model over there. And that's also included in its kind of formal version. And this is what formal deformation quantization gives its actual object, which we have in deformation quantization. And in particular we get, we kind of have limiting category when h bar is equal to zero, is some full subcategory in coherent shifts with compact support, and also on x, and also twisted by this B zero. So we should put actual job. This can be called congenitalization of Higgs bundles. If x is a Cartesian bundle to some smooth variety, smooth compact variety, say, then Higgs bundles are, can be interpreted as coherent shifts on this Cartesian bundle with compact support, which is ramified cover of Y. Higgs shifts. No, Higgs bundles. Higgs bundles are... Because they are perfect, but the support... No, no, if it projects to the base variety it will be actually vector bundle, but upstairs it's, I don't know, that's the definition of Higgs bundle. No, no, it should be compactly supported, in some special cases. Yes, yes, yes. Yeah, so the conclusion, this family of categories is H extends to H equal to zero, and that's pretty non-trivial procedure, but in the description of this category we use not series and H bar, but series expand one of H bar. So how we can make a meaning with H bar equal to zero is not clear, and yeah, so the goal of my today lecture is to describe extension of this family of categories to H bar equal to zero, kind of not using the Riemann-Hebert correspondence, but entirely in terms of Foucaille categories. At least some families of objects want to, depending on H bar, we want to say that H is limited, it's some object in some limiting category. Yeah, so to do so we first fix smooth compact algebraic Lagrangian, and I will use the approach of two Foucaille categories as starting a singular guy, but we'll start with smooth guy and with some local system and try to interpret this object on a small neighborhood of L compact in X, yeah, no, no, I don't know, can you read, yeah, it's compact in X, yeah. Okay. I will, it's my lecture, I will later go to, yeah, that's okay, yeah, so, so what we do? So we have this compact X and we take small neighborhood, and I explain to you this philosophy that to construct object Foucaille category we should start maybe with singular Lagrangian manifold here at smooth and forget about complex structure at all, so it will be real Lagrangian manifold and real symplectic manifold L is Lagrangian with respect to real part for any given H bar. Yeah, so it will be real Lagrangian and real symplectic manifold and then first you should have kind of local Foucaille category which will be finite dimensional modules of certain algebra, finite dimensional DG modules over certain algebra AL and this algebra in the smooth case is twisted because of our twist B0 version of chains of the following DG algebra you can see the chains of a base loops of monoid of base loops in L and pick some base point, it's a topological monoid to take chains I will explain in a minute this is DG algebra graded in degrees less than 0 and H0 of this algebra is, it has only non-positive converters and 0 is a group ring of the fundamental group so the twist will be by class in second commodule with this E1 coefficients by two things, it will be restriction of class B0 to L B0 is for second commodule class in ambient variety can restrict to L and also add first Stiefel-Whitney, Stiefel-Whitney class of L which is belongs in commodule description Z mod 2Z and Z mod 2Z is naturally embedded in R mod 2 by Z Don't you want to take integers now? What? No, no, it's here one can take how to see this twisted models so the kind of basic example if X is a cotangent bundle to some smooth variety Y and B0 is comes from the following, it's responsible for possibility to divide first-chain class of the tangent bundle to Y to itself and it's kind of two torsion class and how can one see this? It's considered image of what? I can see the first-chain class of the tangent bundle to Y and maybe take pullback I is projection from X to Y multiply tensor by minus 1 under the map H2 XZ times sorry, R maybe multiply by pi yeah, pi is 314 it's another pi to map to H2 you have this obvious map so and this first Schiffel-Wittnig class second Schiffel-Wittnig class it's again responsible for first-chain class of L by 2 and then what we conclude is that this twisted local system for co-cycles given by this image of this map is the same as a local system on the total space on C star bundle over L which is equal to canonical bundle times pullback of a minus 0 section whose monodromy, whose holonomy along kind of fiber C star is equal to minus identity yeah, that's an elementary description what is this twisted local system in this case and even in the most special case if let's say Y is a curve and L projects to Y it has only double ramification points this got tangent bundle, LCT coincident has only double ramification point then the same thing you can say that is local systems on L minus set of ramification points is holonomy equal to minus identity around each ramification point so are the finite numbers? the finite algebraic curve has finite number of ramification points so that's a category and in general it should be deformed by holomorphic disks with boundary on L so in principle this category of L mode should be deformed by holomorphic disks should be holomorphic disks such that boundary goes together and the beauty of the complex situation is that for generic H bar there's no such holomorphic disks at all for generic value of argument of H bar except some countable set of holomorphic disks for appropriate almost complex structures so what is the explanation? we can introduce almost complex structures on X compatible with the symplectic form of a special type coming from almost quaternionic structure on X omega is a choice of three is collection of three non-degenerate forms non-degenerate such that omega is equal to omega 1 plus i times omega 2 and point-wise this any tangent space with all these three forms isomorphic to standard quaternionic space quaternionic space with three standard forms standard by linear forms the space of choices is contractable so we should just choose omega 3 in fact omega 3 will be not a symplectic form it will be just two forms space of choices is contractable because structure group which point-wise is a symplectic group with complex coefficients is a homotopic equivalent to unitary grouping in quaternionic coefficients it's maximal complex subgroup okay so let's choose any quaternionic structure and it gives us automatically almost complex structure for any h-bar so if we fix non-zero complex number then real part of omega of h-bar and write h is polar coordinates is equal to this I think and then this we have almost complex structure can naturally from this almost quaternionic geometry and then you have pseudo-holographic curve on any pseudo-holographic curve we'll have vanishing of two forms omega 3 kind of you have all the three forms from 3-dimensional space and forms which are orthogonal to from plane orthogonal to this guy we'll vanish on this curve so you get vanishing of omega 3 and both form vanish and in particular this guy vanish omega 3 this guy vanish which is imaginary part omega divided by h-bar and so the conclusion is that if you have any this homomorphic disks that integral of the disk of pullback of form omega is non-negative real number in fact a positive real number if phi is not constant map this is in fact a homological condition one should know exactly what is a disk not only it's a homology class the integral of omega gives a map from certain finite rank letters second homology of x is per L the coefficients to C it's a lattice of finite rank image and image is some countable subset and if arc h-bar is not equal to arc integral omega of some less beta beta belongs gamma and this map is usually denoted by central charge so there are countably many bad rays outside of this bad rays you'll have a very nice situation where there's no pseudo homomorphic disks so the categories shouldn't be corrected and we should get a fully faithful some fully faithful embedding from this category of twisted local system systems L into the 4k category of x is the rest you chose h-bar equals to r to the power of theta so it is kind of multiball function h-bar now theta is a real number defined after 2 pi z it's kind of polar coordinate it's any complex number it can be uniquely written just a small remark when I discuss 4k categories in general there was one of the troubles which I kind of skip it's about the smaller space of disks has kind of wrong dimension have larger dimension one should be dropped something and this was some kind of controversy about this but in fact there was a close approach by John Pardon a few years ago which explained that one doesn't need to perturb anything at least on the nose and interpret virtual fundamental class of this maybe horrible spaces of maps as a co-homology class with very dualizing shifts so a kind of series of shifts gives some solution without this trouble so one don't have to worry that in principle defined category should move all things somehow it's not really trouble okay so we get maybe countably many arrays and also we get uncountably many arrays and for this each uncountable all good arrays we get a full embedding but what happens if you cross the stock's ray it can be called this by analogy with differential equations like stock's rays we get a different embedding just let's embed kind of a little bit on the left and on the right we get two different embeddings but they are very close to each other and if you try to think for example water model space of objects on one category and another category it should be kind of like open embedding so you embed the same thing to the same that means that you apply some automorphism then we should get an automorphism of this category of category A and models with coefficients in a certain ring and the ring will be maybe Jan was right maybe I should at least I'll explain what it means with coefficients in this ring what is gamma theta gamma theta is elements in gamma beta it's a monoid driven by zero and classes with strictly positive area sorry gamma zero that's it and gamma zero plus is a cone actually it contains this cone and it will be the following it will be zero in union of kind of finite sums of classes of holomorphic disk which we enter which can appear in I claim that what we get here will be a strict convex cone in gamma sigma so it means that we get some closed cone strictly contained zero in some half space and then we get arbitrary infinite sum and get some kind of virtual form of power series ring why you get this boundness in fact the stiffness will be this cone can be chosen in sense uniformly when you rotate sigma this is kind of the following reasoning choose the basis omega i and alpha i of representing second homology with real coefficients the omega i are two forms on x the omega i is equal to zero and alpha i are one forms on l and omega i restricted to l is equal to the alpha i that's how you write representatives of relative homologous class so the homologous final dimension choose some basis and it infinity also should be careful with infinity let's assume that omegas are less than constant and also choose some remaining metric which is compatible with symplectic form kind of rough sense which I explain when I spoke about categories all two forms are kind of bounded by omega and alphas are also bounded because l is compact and then if you integrate over disk pullback of omega i plus integrate over boundary of the disk pullback of alpha what you get you get something bounded by area of the disk remaining area plus another constant times the length of the boundary and there is a kind of grom of inequality that's for pseudo-holomorphic disks this length of the boundary is bounded by constant by the area so this is actually one term and what it will mean what is again we call support property which means if we choose norm on r for beta in minus 0 pseudo-holomorphic disk exists implies that norm of beta is in some constant universal constant per norm of 0 beta if you for example imagine that gamma is c3 so you can r3 you get a map from some dimension space map to plane r2 which is c mxz and here one should draw kind of complement to two cones and this domain projects and if you consider pullback of any array you get strictly convex cone and the support or possible classes of beta will stay in this outside area do you mean formally with respect to the angle? the angle, yeah so you get certain automorphism and automorphism may be denoted t3 tθ as we make kind of rough picture tθ is acting on h0 of modules now this category has t structure because algebra is negatively graded and the heart of t structure will be twisted local system in normal sense so it's something which makes from one local system another local system and I'll draw a picture in the case of dimension 1 which is dimension 1 so l is complex curve and suppose I have this red pseudoholomorphic disc with boundary on l with certain argument theta then it gives an automorphism of the group ring of fundamental group depending on small parameter namely a group weight the automorphism is like this suppose you get some element of the fundamental group some part of the path crosses this loop then it goes to you replace this guy by small correction the same element plus another element of the fundamental group with weight which is close to zero or a more invariant way to say it if you have a local system ignore this twisting on l then you get new local system first cut along this red line divided by 2 and glue back not identity map by identity plus the same thing times a column along this red line starting from a given point it will be locally constant system of identification and they get new local system that's kind of only an A1 level approximation but it gives the right idea and in high dimensions disks will be not isolated but will have dimension if dimension is bigger than 1 then expected virtual dimension real dimensional space of disks will be 2 and minus 2 and this will be some real hyper surfaces which are fallated by circles kind of naively one can repeat the same procedure intersect by path with hyper surface and go along circle and go in force so that's I want to stress that all this disks and or families of disks are not totally canonical it depends on choice of almost quadratic structure but the automorphism is just one automorphism doesn't depends on anything so actual disks depend on almost quadratic structure but automorphisms automorphism is absolutely canonical now there is here convergent questions on which don't have good transfer now for simplicity just look only only at the sigma s automorphisms model maybe inner of algebra in L and we ignore ignore twists let's say just automorphism of the group ring of fundamental group first of all we choose some generators and then the automorphism t theta should be act on generators like this g i n plus infinite sum certain g belonging to pi 1 beta in h2z the boundary map of beta is abelinization of g belonging to h1 so we get certain looks and the automorphism will be certain infinite sum times exponent minus 1 over h bar omega or the same as beta it will be along the rate will be real numbers very large real numbers so it will be very small and like by g and this I interpret as element of this group ring of fundamental group actually it's form of power series and this monoid it's not really into it's kind of image of the element and map element of monoid to this exponential expressions okay so you get certain infinite sums t theta is canonical so why you choose l not as base point no just to speak about fundamental group I don't know to speak about fundamental group we need a base point it's yeah it's just convenience it's really not essential yeah you leave to the yeah kind of leave this out to automorphism to some automorphism to have this form it's better to speak about terms of group points but to be more lengthy the action you create when you cross the wall yeah yeah this automorphism of the group ring yeah so you're trying to make it to realize it as a group but it's actually something higher or yeah it's actually not a group it's dg algebra not only pi 1 but yeah then one can yeah it will be very heavy notations I just want to say basic I think yeah what we expect first we expect this coefficients are certain integers there's really no good explanation for this but it's experimental fact that in all examples which we know integers negative yeah no there's nothing positive because it's kind of arbitrary things and b to belongs actually to gamma plus yeah this c i d d some intersection and this is this coefficients should they themselves be some intersection I don't know yeah I don't know what one can try to think what is topological interpretation of this going to your picture yeah it's like intersection with numbers yeah but it's generator goes to some infinite summit if it's whole pictures complicated I don't know what to say precisely then it's again this anti as a parametric inequalities it's lengths of g in a group sense less than constant times whatever normal or the same is normal b because we put this support condition but the crucial it's Gromov as a parametric inequality kind of formulated here but the crucial thing that this numbers should grow that more than exponentially that's maybe a little bit too optimistic but it's analogous to this question about categories when we get exponential growth the constant in the exponential is positive or the constant in the exponential I think it should be positive because they say integer numbers negative integer numbers some constants so we get this automorphism of formal power series ring but by this condition you'll be able to substitute finite value of h bar on the ray and you get actual map you get certain conversion expression but for finite value of h bar it will be no longer automorphism it will be just some map which will have different radius converges for different elements of a fundamental group the next situation would expect maybe this column to be just a ray a single ray so you're just a single sum and these coefficients so the exponentials get just power yeah, yeah, sorry? I don't know what to say for non smooth cell I will later speak about non smooth cell I have some on my own plan or whatever so in this case one can ask about non linear Riemann-Hilbert problem this Riemann-Hilbert it's not the same as Riemann-Hilbert correspondence Riemann-Hilbert correspondence we can see the trivial bundle over some germ of 0 of a disk with fiber roughly m into the modulus stack space of let's say finite dimensional local systems of given rank now what I do I make infinitely many cuts draw this pieces of rays in the disk make cuts along stock's rays and modify just try to glue new bundle just if I cross the ray I apply the automorphism T sigma acting on modulus space T sigma as function of h bar and glue new space and we want to find holomorphic, the question is to describe holomorphic sections of new non linear bundle how to say it mathematically for each generic which is not on stock's ray you should have a map from a holomorphic map new seta from certain small domain and I think the right domain is draw a small circle tangent to 0 and center has argument theta from this domain to m which is extends as c infinity to 0 so you get actual value of this map at point 0 certain map just I didn't finish satisfying the following constraints if sigma let's say generic 1 is less sigma generic 2 than sigma generic 1 plus pi so you have two direction sigma 1 sigma 2 then the product then m sigma 2 restricted to this common some common part is equal to mu sigma 1 times the product order product of all T sigma are special stocks directions so this are maps one should figure out make a little bit smaller domain on which all things have sense by convergence constraints and this will be polluted on this common convergence domain now so it's pretty complicated I think but in fact it's equivalent to a finite amount of data you should choose at least three of them so that all angles will be less than pi over 2 and it's enough to have this map associated to small disks for finding to many of them with this co-cycle constraint so this positive answers to the convergence questions are included these things, yeah this will be converted for small so now I'll try to formulate the conjecture it will be this extension so this will be my extension of Riemann here but to a bar equal to 0 which say that in germ of holomorphic section of this modified nonlinear bundle should give in germ of at each bar equal to 0 of the family of holomorphic objects holonomic object of family holonomic objects the sense of context in each bar and such that over these things I get flat over CH bar deformation quantization modules and classical limit classical limit I get a vector bundle on L one should be a little bit more careful for example here if you consider germ of sections let's say this is given value at 0 that means that here you can fix some initial twist local system and here's vector bundle plus certain constraint on one half of first derivative in each bar maybe I'll explain after the break what here goes on so it's a specify certain vector bundle and but it's value at H back to 0 but also specify somehow also some part of first jet data and we'll make some 5 minutes break details the case of local systems which we have a twisted local systems on L of rank 1 so we get a twisted kind of torus without a single base point twisted it will be torsor over we'll discuss more of this part of it is now questions are allowed or not what is the right twisted local system sorry my handwriting became systems on L of rank 1 you can ignore this and it will be torsor over each one L c star which will be home so we fix a point in what is written in the law of a pick at a torus twisted there and we fix a point it will be a value of section or equal to 0 and this remain hybrid problem what I do mean by this vector bundle or maybe twisted vector bundle this half of derivatives again go to this favorite example x is t star y y will be compact so it's hybrid to y and L will be compact curve in it will be smooth also compact curve so this correspond to case of d models without single derivatives and again twist in fact for curves the first-gen class is always dv is L by 2 2g minus 2 is dv is L by 2 but it's kind of extra fact to ignore is because we really have structure not just vanishing to 0 and suppose this map L to y has a degree k so it means that we answer the question about JL k so on A side of my picture I have twisted it will be rank 1 local systems on L minus formification points again I assume that only double formification for the map yeah so we have local system with monodrama equal to minus 1 around formification points this is a torsor over c star to power gL with gL is a genus of this curve L and number of formification points it's twice genus of L minus genus of C genus of y so that's A side and what is B side you can see the group g which is gLk and consider Hitching system so what does it mean we get more to the space of Higgs bundles for a group JL and it will contain this open part these things which we are interested in maybe deformed L prime will be some curve another smooth curve which could coincide with L and metallic L prime will be a line bundle smooth curve in cotangent bundle of genus same genus and L prime is a line bundle let's say isomorphism class of line bundle on L prime of degree this half of this number of formification points and this can project to Hitching base without bundle maybe with multiplicities the projection is forgetting line bundle so the fiber is dorsal over abelian variety it's all L prime L is fixed for me from the very beginning it's kind of varying curve not very far from L just a second it's Hitching base and this open domain all smooth curve is line bundles now as well known this moduli space of Higgs bundles is moduli space of Higgs bundles includes twister family so it will be familiar for varieties depending on H bar which is m0 and mh will have some flat deformation for H bar non-equal to 0 mh will be moduli space of H bar connections it's in the differential period with H bar in one key vector bundles it's amorphic to what seems to be called deramoduli space of stack for all H bar non-equal to 0 mh and we get some flat family of varieties so now what we do we get this family of varieties we make a blow up of the total space at the following locus it will be fiber of the Hitching local system over point L so all possible line bundles from L it's a billion variety it's a part of the special fiber and we make blow up sitting in mh when we make a blow up we get exceptional divisor we get exceptional divisor and now we interested in a section of blown up vibration which goes to a point of exceptional divisor and what is this exceptional divisor exceptional divisor it's fibered over this torsor it's a billion variety of dimension G which is homomorphic to S1 and fiber is a fine space which is parallel to to the conormal bundle to this locus base of Hitching system in the fiber to the tangent space which is tangent space to the base of Hitching system so it's R to the same number but it's complex variety and it's easy to see that in fact the L homomorphic is analytic space and the homomorphism is canonical to this twisted torus of local systems which is torsor over C star to the same number so you get it's roughly a Riemann-Hilbert correspondence we can see the line bundles this lead connection on a curve with the same as representation of fundamental group analytically but this is not algebraic map but still canonical identification and if we fix a point here we fix a point in this blown up divisor and if you think what does it mean it means that we are interesting in section of original bundle and we fix kind of half of first derivative among all of this lead bar which is basically involved description that's how correspondence of even if you want to identify parameters here and there at least in case of rank one if you fix value you fix something on B model size so now I just say couple worded in practice if you want to study concrete example cell is almost never compact it's not compact as I explained with at least in case of curves one can achieve compactness by making some additional blow ups but let's make some kind of really messy symplectic manifolds and messy B model and in principle it's not desirable to make this new blow ups there is a kind of situation which is pretty common kind of general type for which one can still work first I'll give some general definition a point on a Poisson manifold with open dense symplectic leaf is called logarithmic if there exist local analytic coordinates in this point set it inverse to this Poisson structure gamma omega is equal to some if you get such form and divisor when symplectic form is pulse is given this equation for 1 to m where m is some number between 0 and m and one definition second definition L city connects bar gamma L bar is called I don't know adapted to Poisson structure if L bar is smooth compact in locus of logarithmic form points which is open locus transversal to divisor and Lagrangian in open part tell me the closure of the intersection can you repeat what is open locus of this set of points which are logarithmic form and open subset essentially by definition L should lie entirely in this open subset we are going to work with smooth Lagrangian I will explain because later there will be some troubles with non-smooth one troubles on the whole and it looks that for such things it's in a sense one can work without making blobs and will be the same more or less story automorphism of these fundamental groups now for a complete dimension of L is equal to 1 L is a curve this automorphism T-sitter which I talked about is acting on the ring of L it should preserve or conjugacy classes of small loops around punctures that's kind of clear from geometry because we modify for some compact loops they do not intersect loops around punctures in particular for this rank one systems we are going to make automorphism of this twisted torus which will be this twisted torus in genus 2g and maybe you get something like n punctures this twisted torus is c star 2 or 2g plus n minus 1 and it will be holonomic along punctures and this automorphism T-signal acts trivially on this quotient so it's exonofibers in some way so that's the foundation structure for the semper is preserved there are fixed conjugacy classes in the simplified clip preserved yes it's some non-linear map from a torus to itself depending on formal parameters so there's a big question how calculate this automorphism T-signal in the case y is equal to 1 this should be done by spectral networks it could be in principle not compact I can now consider station with irregular singularities whatever in the continental space can be considered as multi-valued kind of meromorphic one form on curve y because locally it looks like a graph of one form one form is automatically closed so we can see this like some finite collection say i where i lies in a finite set local system of finite set also with some singularities and this finite sets will be fibres you get many one forms this get a more Nitzky-Netzky spectral networks for given theta draw trees typically there will be three valentries these end points will be ramification points p-protection of ramification points p-protection L2 some finite subset and each edge for each edge I choose two branches of my multivariate one form say 2 and edge will be on edge imaginary part of say i1 of edge bar vanishes so if you have locally two hallmark forms difference will be hallmark form divide by edge bar get things and take imaginary part you get some fallation edge should go along this fallation and real part will be positive and each vertex inner vertex usually it's a three valent vertices you get edges are labeled by three pairs of indices and eventually it's kind of degree three vertices and degree one vertices you took two two branches which are coincided branches say 1 say 2 say 1 at the end point is equal to psi 2 at the end point point why these things correspond to holomorphic disks yeah these are nice objects they don't depend on any choice of what almost quaternionic structure the idea is the following choose kind of family one parameter family of quaternionic structures such as curves homomorphic curves question will force you to be close to the following objects when you have the disk will map to the following disk limiting disk will project to the tree and fiber at end point will be a point fiber at end point on a edge will be an interval it will be straight interval in cotangent fiber connecting point c i 1 of y and c i 2 of y this intersection point with Lagrangian variety and at triple point you should add triangle fiber at 3 valence vertex is a straight triangle now this will be one dimensional story in principle could draw tree in high dimensional situation as well but there are many troubles with all this in practice it's kind of work one can calculate and calculate this automorphism of group ring of fundamental group whatever this non-linear tori what I just want to briefly mention that there is a different approach again if you want to calculate on computer many many this coefficients which I mentioned it's automorphism of tori so some different and it's just if I can found the spectral networks which people can so can you write down the formula for the automorphism in terms of summation it's all pretty complicated let's say some so if we simply study again rank one system on Lagrangian varieties actually not really relevant again for the case axis cotangent bundle again one can think about terms of curves prescription works does really need that the case is a curve so for any h bar one equal to zero one get some beta model space which depends on h bar which will be algebraic variety defined over integers and it form a local system on c star this h bar coordinates why it's a local system now for example one can try to think of some irregular data the local system appears when you have irregular data suppose y is a curve and you describe irregular data by some the stocks diagrams around each point and you describe what is a local system it's some local system plus filtrations blah blah blah but it gets something defined over integers but now when it draws the stock's length it's if you multiply this stocks parameter by h bar the whole thing starts to rotate and h bar goes around this whole thing kind of rotates and goes to itself so it means that you get a variety with automorphism and the automorphism first in a regular case it's trivial automorphism in many cases it's a finite order but in general it's infinite order automorphism so you get maybe one variety one algebraic variety plus automorphism and of infinite order it's important maybe I'll write this t with notation a loop around h bar equal to zero this automorphism this variety say in practice in beta h bar equal to zero in practice log-collabia variety and one can speak about logarithmic compactifications when form element has pulse of first order and the next date it's kind of a bit lousy you can see the finite chain of p1s logarithmic compactification by some normal crossing divisor point p equal to be zero p1 and I assume that all cp1s are strata in natural stratification for the normal crossing strata in the natural stratification suppose I have a chain of cp1s and finally and assume that t around h equal to zero which defines an open part of my variety extends to identification of a neighborhood of fp0 and neighborhood of finite point now so I get a variety of automorphism and then what we can do neighborhood of the whole chain and glue to ends by these automorphisms it will be neighborhood of this chain with glued endpoints so I want to stress that this guy is no longer algebraic variety it's not even a piece of algebraic variety because you glue by some kind of automorphism of infinite order so since it's kind of similar to in kind of paddock series is fontan-fark curve we're going to take an annulus and glue one end of annulus and another end of annulus by map x goes to x to power p by some kind of it's even non invertible map so I get this bizarre guy and what will be further piece of geometry here the assumption is the following if you have a device with normal crossing and we get zero dimensional stratum, normal crossing divisor in some manifold of dimension that's a capital N dimension manifold so we get zero point stratum then we get automatically a lattice of rank N namely the dimension stratum will be my example one of points P i you can see the h1 of neighborhood of P y of P i intersecting to the complement to divisor to my divisor D so I get something homotopic and it's first homologous rank N lattice now if I get one dimensional stratum if I get one dimensional stratum which is complement to zero dimension stratum is c star so cp1 minus c star we also get a lattice of rank N namely what we do we can see the neighborhood not of point but over circle we generate homotopy group that's the neighborhood of point of S1 and of generators of P1 and repeat the same procedure so we get rank N lattice for divisor normal crossing also to special one dimensional stratum and this all lattices are identified with each other we get identification along the chain gamma P0 is the isomorphic to lattice of the first cp1 gamma P1 gamma second cp1 finally we get to the last point and the assumption that this isomorphism is the one which is given by my automorphism because automorphism identifies the whole things and the holonymy along the chain is equal to isomorphism induced by my monogram map so it means that all lattices are identified and all lattices will be identified with gamma star where gamma is second homology of so here we get some kind of ordinary symmetry situation because this H2 is very close to H1 of L so you see that the first homology of one variety is due to the first homology of something else X, L, yeah it's on all this picture X is a cotangent bundle to Y and L is sitting in Y so what goes on we get a new variety with M tilde it's analytic variety with normal crossing divisor and chain of cp1s now closed chain of cp1s inserted at first order it at this chain it coincides with the toric variety the toric variety the torus is gamma dual tensoring c star so we get this situation and central charge which was mapped from gamma to C I again assume that it gives epimorphism of gamma to R2 R2 equal to C so it's kind of full rank this by duality gives an embedding from R2 embedding to gamma star and then R now so we get this picture and the claim that this analytic data under certain some convexity constraints gives you a bunch of automorphism of t-ceta over tori analytic space plus central charge gives a collection of nonlinear maps and in fact the whole story can be sort of at the end of the day the pathology of the fukai categories blah blah blah but why eventually we go to this situation and why we get this t-c this how do you get the data? it will be maps from arguments will be just suppose we have this analytic state we get some analytic variety this tor section then we have kind of new question which will be eventually equivalent to nonlinear Riemann Hilbert problem which will be the following to describe maps from punctured disk m tilde and the picture is the following we get this chain of Cp1s in a sense of toric geometry we do the following for each 0 dimensional stratum I associate point Pi gives a convex simplitial cone integer simplitial cone and for each edge I have a common phase between two cones so what I get it's a bit hard to draw again imagine in three dimensions you get some simplitial cones which attach one to each other and form a certain figure which is supposed to be similar when I wrote the support property so you get some nonconvex cone union of convex 1 to m and this will contain R2 the image of this I think by central charge so this thing will contain some plane now what goes on this plane will be the same as C plane for variable hbar and we divide hbar in different sectors finally many sectors and if hbar states in one sector it means that here we approach one point of this Cp1 if hbar states in a sector then my map will be very close to one point and in each sector the map will be the following first we choose local coordinates identify the historic variety hbar goes to point in local coordinates in the torus and point in the torus is the following beta goes to exponent zbeta over hbar multiplied by some analytic function analytic map in the sense of tori which is extending to C infinity on the boundary boundary of my sector that's another question to study this map really for asymptotic we need to only note the central charge so how it's related to Riemann-Hilbert problem the story is the following first of all if we consider this analytic space in the chain of Cp1 then we can consider its formal completion at the chain we get some formal scheme and the classification group of these formal schemes the classification problem of these formal schemes it turns out to be very easy it's given by exactly data of this automorphism t-sitter the same number of parameters if you consider the formal scheme then it's really one-to-one correspondence to all this family of automorphisms but without any convergence conditions and if there is convergence conditions then we get this actual variety and if you consider the sex solution of Riemann-Hilbert problem which I explained to you by infinitely many rays it's exactly equivalent to this analytic question so it's just different languages for the same story there is a very delicate question where this convergence and existence of holomorphic drum is the same story in one direction the implication is obvious in other we don't know in fact but it seems to be if you get this kind of full-crossing structure this bunch of automorphisms with exponential bounds then it can glue analytic space but this opposite question if you have analytic space when you get it formally will be this convergent series this seems to be a pretty tough question which we don't know but even without convergence it seems to be as good as well so we have replacement of Riemann-Hilbert problem even if its answer will be negative for this question so it's something which one can do explicitly in principle again on computer and why it corresponds to a real-life situation it's kind of obvious when we consider equation depending on small parameter then we get and consider very small circle in when h bar equal to some constant in this parameter space it's mapped to some curve in Mbeta and this curve will be the following so if you get this circle each interval will grow very close to each point and logarithmic coordinates will be straight line and will go very fast on this so what is small became very big and what is big became very small so this type of behavior and Tiffon try to think on this question this space of the sections will be like sections of some new analytic vibration of h bar so it's implicitly describes you a fiber at zero finish for today