 Okay, thank you. Yeah, so my talk will be kind of mathematical illustration to physicist's talk, to Marklas and Sergei. And I will, so the whole perspective will be kind of, on resurgence will be some kind of the current left shot theory, which was already mentioned. And I will start with this kind of simplest instance of resurgence, when I integrate exponent of polynomial, or rational function. Now, for example, and yeah, in fact, it will be because I kind of will model a field theory, I'll say that's kind of zero dimensional Q of t. For example, I integrate exponent minus S of X divided by H bar, the X. S is a polynomial, let's say, in one variable. It will be my action. And I integrate over certain chain of integration, which is called gamma. And the main theme of my talk will be rotating Planck constant. So H bar will be non-zero complex number, no longer positive guy. And let's do it's kind of a recase. So the polynomial is X cube over three minus X in order to get simple derivative. And what are possible contours of integration? If argument of H bar is zero, so it means that H bar is positive number, say real number. Then there are three direction at infinity when action goes to plus infinity, like X cube to positive real numbers. And you get argument zero, argument two pi over three and argument four pi over three, gets three direction. And you have kind of three possible chains of integration, but only two independent because some of them is equal to zero. So you get two interesting integrals for each H bar to study. And that's one way, but there are also left side symbols. In this case, it will be just steepest descent pass. To draw left side symbols we'll start with critical point of the action. So what a critical point? Now you solve equation S prime as X over zero. And because I was smart to choosing this one over three, you see that solution is this, you have two solutions, X one is equal to minus one, X two is equal to plus one with two critical points. And then you get critical values, S alpha is equal to S of X alpha. So you get just two critical values, S one is equal to plus two-third, S two is equal to minus two-third. And these are elements of this, what's really called Braille plane, values of my function S, the points S one and S two. And now what we do, if H bar is not real, or we define left side symbol, this symbol gamma alpha H bar, where alpha is one two, one of these two critical points, is pre-image of the ray, which would be like this or like this, of ray S alpha plus H bar. And it's positive numbers. So if you draw a ray in direction argument of H bar, straight ray, under map, map S from C, this kind of X variable, to C, which will be this Braille plane. And if you take, you see that the map is doubly roomified at this point, and consider pullback, you get upstairs, you get two, actually copy of R. It will be two copies of real ray. And this will be two domains of integrations, and then we define integral E alpha of H bar, E alpha is one two, one of my critical points. It will be the integral of exponent of the X of this symbol. And it's defined up to sign because one should orient this line. There are two ways to orient it. There's no preferred choice of priority. And when H bar goes to zero, this fixed argument, which is still not real, it has a synthetic expansion, E alpha H bar will go roughly the following. First it will be, this will be a plus minus. Then they get exponent of this value of critical points, minus S alpha divided by H bar, which will be minus very big real number, minus one, oh, it will be some number. Then multiply by square root of two pi H bar. Then divide by maybe derivative of function at point X alpha to power one half. And then we get series of static with one. And this will be divergent series. It will be former power series, which is divergent. We want to kind of resume this series. Yeah, in principle, also one can see these things. It's E alpha H bar is integral over S belonging to this S alpha plus H bar, plus exponent minus, so H bar times some algebraic function. One can write it on the grid, actually one form, maybe DS. And one form function is S goes to DX over DS and invert S as a map. So it will be three-valued algebraic function. Okay, so that's very simple game. So we have this nice synthetic expansion and let's me remove kind of universal term. From this expression, so define G alpha of H bar. It will be exponent plus S alpha divided by H bar and multiplied by two pi H bar to power minus one half. I will not treat these things times E alpha H H. It's certain former power series in H bar, starting with something. So we get two functions. Each function, in fact, it's onus function, and onus function when H bar is not real line. So we get analytic functions, we get two analytic functions on upper half plane and on low half plane, and which are C infinity up to boundary. So the analytic inside and continues to C infinity functions everywhere and the stealth expansion will be exactly this series. Yeah, so we get two functions in upper half plane, two function in low half plane and how they behave if you go through the lines. We get some jumps and we have jump along one ray when argument of H bar is equal to zero, E is the following. You look what happens with this integration cycles. If you look, turns this ray very close to the rate, then gamma one, if you rotate it doesn't change, but gamma two goes to gamma two plus gamma one minus because it wasn't very precise about sense. And similar for low plane, but what does it tells me about this rescaled integrals because they rescaled, what happens is that J, let's say one H bar, if you cross the line goes to J one, but J two goes to J two and integrate and rate exponent minus this S one minus S two of each bar, which will be very big positive number, real number. So this things will be very, very small multiplied by J one. And similar if you go to this thing, but now gamma two will go to gamma two and gamma one go to gamma one plus gamma two. You get different metrics. So what goes on? I claims that what we do, we glue some holomorphic vector bundle on complex plane with H bar parameter. It will be kind of trivial outside kind of C plus C and this basis corresponding to my critical points outside arrays, outside two arrays. And if you cross the array, what will be the transformation? We apply certain cross, let's say a left ray. We apply transformation, which is this linear operator, which can be also sorted as a following query. We take exponent kind of diagonal metrics of exponent of S alpha divided by J bar, alpha is one two, multiply by metrics one, one, zero, one. What happens with cycles integration and conjugate by diagonal inverse power. So we get over diagonal term will be exponentially small and similar, I think we go on the other way. Yeah, so we make some bundles glued from trivial bundles and certain transformation long cuts and all this thing says that this G one, G two form holomorphic section of the glued bundle. It says nothing more. And usually people then say that how this makes resumption makes sense of form of power series to promote it to actual net function to barrel resumption, but one can do it from different perspective already on this terms. I don't think about Borrel transformation transform at all. I can ask the following. Yeah, so this Borrel summation, if you get something series N, A, U, N, then we're making U series sum over N factorial, Z to the power N, for example, and make some analytic continuation and Laplace transform, and we want to get actual function. How it's applicable to this situation? Yeah, first remark that this series G alpha, they do not depend on the choice of sector of upper half plane, lower half plane, just the same. Why it happens? This guys I see infinity, but when we change by something which has trivial Taylor expansion zero, exponent one over x, it has trivial expansion. So what happens is G one, G two formal a certain series independent on sectors. And here we get actual holomorphic function let me explain versus, yeah, or maybe I can continue as a board. So the kind of a very simple proposal, how to make resumption of this formal power series to promote to this function, it just looks on the structure, it gives you the answer. So can first solve Riemann Hilbert problem. It means that kind of glue bundle, according to this two transformations along craze. And what does it mean glue bundles? So you glue some bundles and you make holomorphic trivialization. You can do it one way or another. And what you found, find holomorphic trivialization. So it means that you get certain function G of h bar let's maybe some G to value function for h bar, not real lines and such a jump of this thing along this, each ray will be given exactly by left to right multiplication by this stuff. So we found this in some way. And the main point, it's for the things we really need this critical values and this integer matrices. Nothing else, need to know only critical values and this integer matrices, like in this case. Yeah, so we get certain G value function, but then if you expand and form power series, it doesn't depends on the choice of things. You get certain from this universal procedure, certain invertible metrics with values in this coefficients in formal power series. It's independent on the choice of sector. And then that's by definition, we see that G formal multiplied by G1 formal will be E's analytic function in each bar because it will produce this holomorphic sections in this holomorphic trivialization. Yeah, it's convergent series. Yeah, so you do some calculation with the universal problem, apply to the form of series, multiply, you get series which is convergent. Now you can evaluate for, let's say, sufficient small values of each bar and now you apply to this a priori given construction back and you get actual value of your intervals. Yeah, so it's a very simple way to calculate this thing without borrel transform. Yeah, so that's and it's kind of equivalent to calculation with borrel transform and this radial stuff and so on. But I kind of find it mathematically more appealing because you don't really need a fine coordinates in each bar, yeah, so you just can play in different situation. Yeah, the same story works in for high dimensional integrals. Yeah, if you have some algebraic variety of complex dimension N, let's say capital N, for example, could be Cn and some polynomial and as this polynomial map and we get some polynomial volume element, you want to integrate these six and you apply maybe some algebraic volume form on manifold. Yeah, again, one can do left shift symbols but left shift symbols will not pull back, it will be the following things. Left shift symbol. I actually assume that my critical values are all more nice, not the more complicated stories Sergei considered, kind of really isolated. I think left shift symbols will be gradient trajectories for real part of S divided by each bar starting from critical points and then you'll see that it is a kind of vibration N minus one dimensional spheres and projects to this ray, which I draw on the plane, exactly. And to draw gradient trajectory, you use some scalar metric and such it does matter in good situation, which metric you choose. So you get the things and the same story happens, you get some upper triangular matrices with integer matrices and the whole story repeats. The integral we'll have, again we'll grow like exponents critical value divided by each bar and multiplied by two pi h bar to power N over two and then we get some series. Yeah, so from this perspective, what you really know, you should know all critical values and certain this interest element of the upper triangular matrices, which will be number of gradient flow for this one point to another. Okay, so that's finite dimensional picture and now we'll try to see what goes on in infinite dimension in field theory. Yeah, so we do pass integral. Yeah, so my space will be infinite dimensional. I do not like it is very big x and it will be set of points which will be cx, which is a map from interval zero one, kind of time, bleeding time maybe, with coordinate t to, yeah, so I'll start with first not to give you an example, hyperbolic plane. You can take any remaining manifold and also assume that fixed point, the end point goes to point zero of the two given points on the hyperbolic plane zero x one and all things depends on the distance because of homogeneity, so it will be some real number. Yeah, so I get this infinite dimensional manifold and action functional will be the following. It's usual kinetic energy, okay. Yeah, so we get functional, some infinite dimension manifold, but now the whole philosophy is just to complexify everything. So we can see the x-complexified in the space of maps zero one, but to complexified hyperbolic space and I use this realization hyperboloid, two h2-complexified, which will be just collection of whatever z1, z2, z3 in c-cube, such that z1 square minus z2 squared minus z3 squared equal to one kind of complex sphere and plus the same boundary condition. Now, so the functional expand to holomorphic function, infinite dimensional manifold story and now we want to calculate integral over maybe x real. So what is this integral? One of the mechanics teaches us this is the heat kernel. The claim is I'll use kind of heat kernel and I use quantum mechanical notation, but the pre-tobials go from the bar, exponent h bar, h x1, where h is Hamiltonian, which is minus one half of Laplacian and half of the plane. And you can see the heat kernel, the time is h bar and connect to x zero x one. Actually this is a bit funny, why h bar appears in power one? In integral it was in power minus one, yeah? But the reason is the following. If you look on the action, the action is in time of homogeneity degree minus one. So it means that if you're a scale, things divide by h bar, it means your time multiplied by h bar. So it means that you map short, if you short interval and you get exactly this thing. Yeah, so you get heat kernel and people worked on this for centuries, they know what is this heat kernel and classical formula, you can look, I don't know, even in Wikipedia maybe. This says that it's something like square root divided by two pi h bar to power three half, exponent maybe minus one, half of two. And now there's a main integral which appear here, which I'll write in the following way. I go from s zero equal to l square over two, and l is the distance between two plus infinity, exponent minus s over h bar, multiplied by cosine hyperbolic square root of two s minus cosine hyperbolic of l power one half times ds. Yeah, so you get this formula and already this formula tells you everything about what are matrices, what are critical values and so on. We get certain algebraic functions of square root of some polynomial of infinite degree, I would say. Yeah, some. And what you get? You get function of s, which is ramified multiplication points. Which is the same as complex critical values of action on xc, eventually, are numbers sn, when this guy vanishes, and numbers are the following form, l plus two pi i n squared over two n is integer. What does it mean? It's actually very funny. You see that hyperbolic space, it's like sphere of radius minus one. Yeah, if you were on a usual sphere, if you get two points, you can see the geodesics, and can have infinitely many geodesics going around. We add two pi times radius. But here it's immediate, and you can see that it's a little bit different. Two pi times radius. But here it's imaginary radius, it's radius two pi i. So we add to a real number, this imaginary number, guys. In what we get here, we get points on, some integer points on parabola and c, kind of, this will be l squared over two, n equal to zero, but it's, this is the critical value, which really we see in the classical station, and this are imaginary things which you see under complication. And from this story, you see that one get many integrals. One get not only this integral, but kind of take pull back, doesn't this thing draw a horizontal race from other points. And then we can rotate the story. If we rotate, from this formula one can see immediately all infinite by infinite integer matrices, which go to the case. And in fact, the integral alpha is equal to n is integer. If you get these integrals, before division by these things, we see that they grow like exponent of this critical value as n by h bar, but multiplied by h bar to power minus one half, in some series. And this one half says that it's morally like finite dimension integral of manifold of dimension minus one. And I think it's kind of zeter regularized value of dimension, which we can see in this story. Yeah, I think it's actually, it's very simple story. And I don't know, maybe it's new. I have to ask specialist and I made my calculation myself. And here we're kind of in lucky situation, we can solve everything explicitly. But if you consider geodesics on kind of not on more complicated algebraic varieties, you don't know what the prediction will be the same, the same structure. And in particular, we can kind of have this decomposition of one integral goes to some of another integrals and in more generation, we don't have this luxury to have finite dimensional integral presentations. Yeah, so it's asked to the question to really construct measures on this left-hand symbols, complex value measures. It's like in the usual probability so people know Brownian motion. It's actual honest measure, which gives a heat kernel. But here to the completely different sub varieties of semi-infinite dimension and it should be kind of complex value densities because it's a whole thing, it's Christ was a proof. Yeah, so it's one kind of baby example when you see infinitely many critical values. Yeah, so this whole story has some generalization to the case of one forms, which are closed one forms. Yeah, first I'll start with finite dimensional case. When you have this again algebraic variety of complex numbers and instead of action and function we'll get algebraic one form, which is closed and one can go to universal cover and universal cover this is put back with this form will be differential of some action, certain holomorphic function. It's no longer polynomial and then one can repeat the story. So we get symbols and one get also this regular S functions G alpha H bar when they divide by leading term and these things independent on the choice of lift because you add to function just a constant and they are normalized by this constant. Yeah, so you get finitely many, yeah, for example, if consider zeros of the form assumes it's again generic kind of Morse zeros finitely many of them. So get only finitely many formal power series. Get finite number of elements formal power series in each bar. That's just stupid copies. Yeah, so it's kind of baby finite dimensional example of this channel assignment theory, which Sergei considered and I will go to kind of the most basic example in imagine sterling formula. Yeah, so what do I mean here? This variety is C star and form is the X minus the X over X, which we can read as differential of X minus log X on universal cover. Okay, so downstairs we have only one critical point, zero one and critical values will be zero one, but also we get the scopes and maybe CN is will be one plus two pi IN. We add the period of your form. But let's do kind of first real integral. If each bar is positive number, then the symbol is only one critical points or only one symbol is just positive three. It's kind of from zero infinity. So the critical point at one and this two gradient trajectory goes both to zero infinity. Okay, and the integral which we have here is, I will not put index here. IH is equal to integral from zero to infinity, exponent, you have to write this minus X of each bar, the X symbol store and modifying it will be the following. It will be exponent of one over each bar to kill critical value. Then we'll apply by two pi IH bar to power one half. It's the dimension contribution times A0 of H bar and then you see that it's one over square of two pi exponent of one over each bar. This is exactly leading term in sterling formula for factorial and so we get some certain series. And following Sergey's practice, I'll draw first coefficients. Yeah, these are not Bernoulli numbers because Bernoulli numbers appears to be an expansion of logarithm. These are completely messy numbers. It's not the one, which one. Maybe I'll write it again. Some of them H and this series. What does the Borel transform? K of A to the zeta. This should be nice functions with infinite analytic continuation. And what is this function? It is the following function. Claim, it's given by the following integrals. One over two one half and two pi IH and makes certain contour integral. Contour around one. The x, x minus x minus zeta minus one to power one half. Yeah, so you can see the contour but it should be not too small because this guy has ramification pointed two points near point one. Yeah, this thing goes like something like quadratic function near one. And it has two ramification points. This contour should surround this two ramification points so it's completely one-valued function. Yeah, so you get this thing and how one can treat it. Yeah, so my suggestions, it will be some kind of infinite dimensional algebraic geometry, a bit similar to what I have for heat kernel. Namely, for any zeta, which is not in two pi IZ, and this two pi IZ, it's difference between critical values. I define a curve, I can find infinite genus curve. Zeta in sitting in C times C with some coordinates z1 and z2. And the equation is the following. It's exponent z1 minus z1 minus zeta minus one times z2 squared is equal to one. Yeah, it's essentially a kind of consider. One was z1 will be kind of logarithm of X and z2 will be denominator in the formula. And so you get this curve of infinite genus. It's ramified in some kind of infinitely many points, roughly rank in some arithmetic progression. And we have one form and to integrate over one chain classes in each one of this curve. One form, which is z2 times differential of e to z1. Yeah, just mimic this expression. Yeah, so you get a family of infinite genus curves, depending on parameter zeta. And when zeta is two pi IZ integers, this curve degenerates. So you get some non-trivial, monodrome, which is easy to describe. And what I integrated, so eventually it's immediately proved that this thing has infinite analytic continuation because they can follow this cycle along any path. Yeah, so that's kind of a point of view from this classical resurgence when we want to make Laplace transform. And if I don't want Laplace transform, I want to kind of glue bundles. The whole story, it's even simpler. It's absolutely became almost cartology. The whole story, I have this function J, yeah? And maybe take G plus is equal to H bar is equal to G of H bar. But where H belongs to C minus negative ray. And to divide G minus H bar, this will be function, this will be one over G of minus H bar, H belongs to C minus positive ray. So I get one function on this domain and one function on this domain. But in fact, I can restrict them to G plus, I can restrict to right-hand plane and G minus is, it will be C infinity, and analytic inside. And G minus restricted to right-hand planes again is C infinity. And now we can compare how they, what's this jump along the ray. And the claim is the following. It's G minus on positive ray which goes up is equal to G plus, multiply by one minus exponent minus two pi I over H bar. And G minus restricted to R, are negative to plus by one minus exponent of plus two pi I over H bar. So here now forgetting about Borel transform whatsoever. I say that what I glue here, it's even simple situation. I consider three will run one bundle on C. I make two cuts and along the cuts I apply this transformation which are very close to identity and has trivial Taylor coefficients. And this may describe this analytic properties, what kind of resurgent properties of this series J, from power series J, which through Borel transform is kind of more tricky. You get some, in Borel transform you get this kernel and kernel gives multi-valued function or function on universal cover odds of this theorem if I just all integers. Yeah, but this is kind of much more clean description. And here, what's the comparison with old story? We have matrices which are differ from identity matrix by exponential small terms, but now we have not of diagonal terms but those on diagonal terms. And these things can be explained in the following ways. So my variety is C star and have my one critical points x equal to one. And in the case of functions, what I was wondering how many gradient lines go from one point to another. But here it's kind of gradient lines on universal cover or gradient lines of this, and I get essentially two interesting gradient lines going to itself one direction, one way up or going up a direction. So it appears multiplicity actually minus one by some orientation. So it's here to get kind of coefficients minus one, which is integer. And here to be length of the integral form of geodesic. So it explains all kind of mystery of gamma functions at once, yeah, this picture. Yeah, so I will not talk about, yeah, so the natural guess what goes on the chain semi-surry, it's kind of very similar to gamma function story, yeah. So those are different details, but positive, yeah, yeah. Yeah, okay, no, what's wrong, I don't understand. Minus H belongs to where? Minus H belongs to this guy. But it's a different function. It's function on different domains. In fact, this also comes from the analysis of the quantum mechanical problem of the harmonic oscillation. Yes, yes, no, it's related, yeah, yeah, definitely, yeah. It's, yeah, actually that's exactly, no, not exactly, where I will go now, quantum mechanics. That's kind of my point, it's will be complexified quantum mechanics. Now, so what will be this, my variety to be already complex, I don't pick any real parameters here, will be a space of maps from interval to some complex algebraic symplectic manifold, symplectic form is also algebraic, like cotangent bundle, we see star cross star. We see, and said that if we should put some boundary condition, zero and one are algebraic, say Lagrangian and submanifold. So on this guy, I have a closed one form. I just integrate form omega along the path to get one form. And, yeah, so if I want to write in some coordinates, if let's say Pi, QI are local coordinates on M, conjugate coordinates, then you write kind of the action, it's defined up to constant, is equal to integral over sum over Pi DQI of DT, kind of first order actual functional. And it's actually not the most general action which you can write in maybe kind of boundary terms, but also this L zero and one can give you some boundary terms, but what one can add here one can also integrate H of Pi QT DT. Yeah, it's a different term and it's heat kernel when this Hamiltonian is really present, it's quadratic in my mentor and Hamiltonian which gives you geodesic flow. But now I'll consider case when there's really no such data, I have some pure geometric data, just two Lagrangian submanifold, nothing else. And then we should get some kind of number or all the story. So critical points, if you consider this kind of the most basic action, are constant maps to intersection points, to some zero one to some intersection point P alpha. And then one can try to repeat the whole game. We get infinite dimension manifold, complex manifold with closed one form. We get this critical points and interesting gradient flow. What are gradient lines? The gradient lines, if you get some point P alpha one, P alpha two, will be pseudo-halomorphic disks, will be pass in space of pass. These boundary conditions will be pseudo-halomorphic disks. Then you write Cauchy-Riemann equation for some almost complex structure, which is not original complex structure, it will be different almost structure, kind of compatible with, it's a different one, independent on each bar. Yeah, for example, you can choose hyper-colorimetric, but it's a story, it's very soft, you don't really need integrability condition. Hologmorphic symplectic structure is what? Is the complex symplectic structure? Hologmorphic symplectic structure. Algebraic homomorphic, it's not Caillir form. It's just G square equal minus identity. No, no, but here I get pseudo-halomorphic disk for some different story, because you need some Caillir metric and if you analyze it and take real flow for real part of the section, you get certain different almost complex structure. And in kind of stocks race, the things when you bump from one critical point to another, critical, another line will mean that one over each bar, integral of form over some such disk is positive real number, that gives a condition on each bar. So what this integral should morally calculate? Yeah, if manifold is cotangent bundle to some, all right. What is H bar here? Sorry? What is H bar here? H bar is a complex number. Complex number. Yeah, yeah, this is the condition of argument of H bar, so it should be argument of this interval. So if M is cotangent bundle, suppose L0 is arbitrary. So cotangent bundle is fibered by cotangent space. But suppose L1 is cotangent space at some given point. So this will be kind of L1. L0 it will be L1. This integral, what it should be morally give because L0 should give like family of demodules, holonomic demodules or differential equations depending on H bar and which converge to some spectral variety L0 in classical limit. And this thing should calculate solution with the right way. If you consider intersection point, corresponding to this left symbol should, this thing should be solution with right double AQB asymptotics. The value of solution point Y. So the interval should be solution is correct. Double AQB asymptotic at point Y. I'll write some system of equations and see the solution. Yeah, one can try to look also all the story. And for example, one can try to move point Y and see this resurgence, how depends on point Y. And then there's some kind of interesting phenomena will happen here. In general, if you do finite dimension interval and start to move parameters, since the gradient number of gradient lines will jump according to usual kind of Picard-Lefstra's formula and what happens in usual station, if you have two gradient lines, they can gradient trajectory one point to third point, third point then for special values of H bar, the secal lines you get in U gradient line which goes from first to three. And what goes on in this geometry? So if you have count this homomorphic disks in real co-dimension one, it can split in two homomorphic disks. It's exactly like gradient trajectory splits to two gradient trajectory or will be completely new phenomena. So it will get develop some disks on L zero or develop disk on L one. Yeah, so you get something new phenomena which has no analogs in finite dimensional situation. So something wrong goes with your number of gradient lines. I have a couple of minutes to explain what goes on. Yeah, so there is a problem with individual Lagrangian with L zero and L one when you get this homomorphic disks. It has nothing to do with pass integral. It's some kind of Lagrangian in zero or one. It's not in correction, both of them. It's just this individual guy, something wrong goes on. How to understand what happens if you can see the fundamental group of this space? It's a pretty big space of pass. It's roughly get a contribution of fundamental group where one end can appear and where another end can appear. In this station, L one is simply connect but L zero could be have huge homomorphic group and maybe also pi two of M itself. Yeah, that's a rough picture. And that's actually the origin of the story. That has big fundamental group. It has many, for example, rank one local system. So let's return to the case. Case manifold form with this finite dimensional case. You have this finite dimensional case with one form and what one can try to see that this get a homology of X maybe, sorry to be a bit with the risky topology. It's really important to say this way with algebraic forms and with differential D plus one over H bar eta. Yeah, this integral is actually calculated. The volume form can be considered as close class in this case and integration cycles are functional in this case. This kind of leftist symbols gives basis in dual space. When considered symbols in universal cover that's something for which this expression one can integrate. So get a basis and when you cross the stock's raises basis goes to some linear combination of other elements of the basis. But one can twist the whole story by considering some torus which will be home from fundamental group of my manifold to C star. It will be C star to up to finite thing to first beta number. It will be finite dimensional torus and for any point on this torus we get corresponding local system and we can put things with coefficients in the local system. Now with regular singularity. So it's the whole story. So you get not only one store on only one this calculus, but depending on a torus. All this resurgence story with gluing bundles will be kind of holomorphic family of such guys on a torus. Get holomorphic family over some torus, complex torus. Of resurgence picture in this case when I glue things along race not through Barrett transform. Yeah, so it looks at the same one can do an infinite dimension. So we'll get space of one dimension rank one local systems. It's again C star to some finite number and this effect says use the following. In kind of quantum field series there will be certain normalization. This space will be modified. It will be replaced. The space of rank one local system will be a complex variety, but it will be not a torus. It will be not even a billion group. Yeah, so it means that all this twisting parameters it forms some abstract complex variety which actually cluster varieties in various situations which is closed but not equal to a torus. Namely what goes on? If you consider a generic argument of H bar this will be identified at least some part of it with a torus. But if you go through H bar go through some the stocks ray, certain stocks ray. What you get? You get automorphism of the group ring of the fundamental group. I'll finish in a few minutes. If H bar argument of H bar crosses ray which is argument of the integral form of the disk such as the boundary of this belongs to let's say L zero or L one. You change, you may apply, automorphism of group ring of the fundamental group of L zero say, corresponding to L one. And this is a wall crossing which was young started and it's actually explained by Gaiotov-Muronetsky through some gauge theory but I claim that this purely geometric story have nothing to do with peculiarity. The story depending on small parameter. Yeah, for example, there's a very simple example. Suppose my Lagrangian manifold, let's say zero, it's S one cross S one. Or it contains S one cross S one, it's because I should be elliptic curve. Yeah, because it's complex curve, yeah. And suppose I have some holomorphic disk with this, which is boundary on some thing. Then if I cross appropriate ray, the argument of the integral of two forms of the disk, the transformation will go to the following. If I get C stars cross C star, goes to something like Z one, Z two, multiply by one plus exponent and integral of eight bar of this disk, D multiplied by Z one. So it's some nonlinear change of rank one, space of rank one local system. And this is 4D wall crossing of Gaito-Murnetsky. So it's before going to this pass integral, it's already in each individual manifold, you get kind of identification of Toray. Yeah, kind of the most basic example if consider Lagrangian manifold sitting in Cartagena bound to C, which is some PQ plus XQ plus some whatever, some kind of symbol of Hamiltonian with cubic potential. Then the standard picture is that what you get, you get five stocks raise and you get nonlinear Riemann-Hilbert problem. Namely, you want to map, you want to find two functions like Z one, Z two here, Z two, Z three here, Z three, Z four here, Z four, Z five here, and here Z five, Z one. You have two functions in each race, which are C infinity up to the boundary, up to boundary in each sector. And when you cross the sector, for example, you go from Z one, Z two, to Z two, Z three. So Z two go to Z two, and Z three will be exactly the same formula. It will be Z one times one plus exponent of corresponding constant in the sector of each bar, which will be totally real, times Z two. Yeah, so you get, now you glue nonlinear manifold and consider homomorphic sections. This is basic kind of fordival crossing in gauge theory origins. Again, you solve the Sriman-Hilbert problem abstractly, and then we can add also some vector bundles here. For example, it's only some parameter spaces for integrals, and for manifold, for things you should do something like this, G plus gain exponent, something of which bar, maybe G two times Z one, something like this. Now glue vector bundles, again using exponential terms and monomials in Z. This will be 2D4D wall crossing, and the conjecture, this gets some big class of the similarity co-options, which give formal power series expansion. In this formal power series expansion, conjecture will be resurgent in Borrel summation way. Yeah, so that's a picture, thank you. Any questions? So what is nonlinear manifold here? It's kind of, you make gluing, it's manifold. Nonlinear means that we glue by some nonlinear change of coordinates. Yeah, I think this exponential term comes from some incident, perhaps? Yes, yes, yeah, it will be this area, so this homomorphic disks, yeah. Is there a version of this where sort of disks have spheres and without boundaries and lagrangians? Yeah, it's interesting story because it's all the story, it's only genus zero, only disks, but definitely high genus should contribute to the second quantization of the story, which I don't know. You have this example with the hyperbolic plane. Yes. You have this third point lying on the perimeter. Yeah. What are the intersections of the terminals in this plane? It's very easy to calculate from this double cover story. It will be plus minus one, alternating. And almost every division. No, plus minus one, I suppose, yeah. Yeah. Of the diagonal terms. Of diagonal terms, yeah, it's completely controllable. The story, yeah.