Semi-infinite topology of the complexified path integral
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Kontsevich, Maxim
I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.
 Okay, thank you. So my talk will be kind of a mathematical illustration to physicist's talk to Mark Wilson-Sergay. And I will, so the whole perspective we will get on the results will be some kind of the current left-hand theory which was already mentioned. And I will start with this kind of simplest instance of the resurgence when I integrate exponent of polynomial or rational function. Now, for example, and in fact, it will be because I kind of model a field theory. I'll say that it's kind of zero dimensional 50. For example, I integrate exponent minus s of x divided by h bar dx. S is a polynomial, let's say in one variable. I do the 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 its kind of a-recase. So the polynomial is x cubed over 3 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 directions at infinity when action goes to plus infinity, like x cubed to positive real numbers and you get argument zero, argument 2 pi over 3, and argument 4 pi over 3, gets three direction and you have kind of three possible chains of integration, but only two are 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 of zero. And because I was smart to choosing this one over three, you see that solution is this. You have two solutions, x1 is equal to minus one, x2 is equal to plus one, you get 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, s1 is equal to plus to third, s2 is equal to minus to third. And these are elements of this, what's your g called? Braille plane, values of my function, s is a point s1 and s2. And now what we do, if h bar is not real, we define left side symbol, this symbol gamma alpha h bar, where alpha is 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 as 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 omified 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, the alpha is one two, one critical point, 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, we go roughly the following. First of all, we get 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, if minus one, 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 formal power series, which is a very interesting way, kind of resuméed this series. Yeah, in principle, also one can see this thing, it's e alpha h bar is integral over s belonging to this s alpha plus h bar plus, exponent minus, so which bar times some algebraic function. One can write it on the grid, actually one form times maybe ds, and one form function is s goes to dx over ds, and invert s is a map, so it will be three-valued algebraic function. Okay, so that's very simple game. So we have this nice asymptotic expansion, and let's me remove a universal term from this expression. So define g alpha of h bar, it will be exponent plus s alpha divide h bar, and multiplied by two pi h bar minus one half. I will not treat these things as e alpha h. It's certain formal power series in h bar, starting with something. So get two functions. Each function, in fact, it's on this function, and on this function, h bar is not real line. So we get two analytic functions, on upper half plane, and on low half plane, and which are sin infinity up to boundary. So the analytic inside, and continues to sin infinity functions everywhere and the still expansion will be exactly this series. Now, so we get two functions, an upper half plane, two function, low half plane, and how they behave if you go through the lines. Get some jumps, and we have jump along one raven argument of h bar is equal to zero, is the following. You look what happens with this integration cycles. If you look, turns this array 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 over two bar which will be very big positive number, real number. So this things will be very, very small multiplied by J one. Similar if you go to these things, 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 claim that what we do, we glue some holomorphic vector bundle, on complex plane with h bar parameter. It will be trivial outside C plus C, and this basis corresponding to my critical points, outside to raise, and if you cross the ray, 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 way. We take diagonal matrix of exponent of S alpha divided by J bar, alpha is one two, multiplied by matrix 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. Similar thing you go on the other way. So we make some bundles glued from trivial bundles, and certain transformation long cuts, and all this thing says that this G1, G2 form holomorphic section of the glued bundle. It says nothing more, and usually people then say that how this makes resumption make sense of form of power series to promote it to actual net function do Barrelli resumption, but one can do it from different perspective already in this terms. I don't think about Barrelli transformation transform at all. I can ask the following. So this Barrelli summation, if you get something series N, H1, then we're making new series sum over N factorial, Z to bar 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? 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 guy says infinity, but when we change by something which has trivial Taylor expansion 0, exponent 1 over x, it has trivial expansion. So what happens is G1, G2 formal certain series independent on sectors. Here we get actual halomorphic function. Let me explain. Maybe I can continue as a board. So the 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 glue bundle, according to this two transformations along trace and what does it mean glue bundles? You glue some bundles and you make halomorphic trivialization, you can do it one way or another and what you found, find halomorphic trivialization. So it means that you get certain function G of h bar. Let's maybe some G2 valid function for h bar, not real lines and such as the jump of this thing along this, each ray will be given exactly by left or right multiplication by this stuff. So we found this in some way and the main point, it's for this thing we really need this critical values and this integer matrices, nothing else. You need to know only critical values and this integer matrices, like in this case. So we get certain G-valued function, but then if you expand and form power series, it doesn't depend on the choice of six, you get certain from this universal procedure, certain invertible matrix with values in this coefficients in formal power series. It's independent on the choice of sector. Then thus 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. It's convergent series. So you do some calculation with 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 h bar. Now you apply to this appropriate 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 population with Borrel transform and this radial stuff and so on. But I kind of found it mathematically more appealing because you don't really need a fine coordinates in each bar. Yeah, so we just can play in different situation. Yeah, so 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 S, this polynomial map, and we get some polynomial volume element, you want to integrate this six and you apply maybe some algebraic volume form on manifold. 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 will assume that my critical values are all more some nice, not more complicated, stories Sergei considered can really isolate it. I think left-shift symbols will be gradient trajectories for real part of S divided by each bar, starting from critical points. Then you'll see that it is a kind of vibration m minus one dimensional sphere and projects to this ray which I draw on the plane, exactly. For gradient trajectory, use some scalar metric and such it does in good situation which metric you choose. So you get the things and the same story happens, you get some upper triangular matrices, integer matrices and the whole story repeats. The integral will have, again, will grow like expanse critical value divided by each bar and multiplied by 2 pi i each bar to power n over 2, and then we get some series. From this perspective, what you really know, you should know all critical values and certain, this entries element of the sub-parangular 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. So we do pass integral. Yeah, so my space will be infinite dimensional. I denote like it is very big x and it will be set of points which will be cx which is a map from interval 0, 1, kind of time, leading time maybe, this coordinate t to, yeah, so I'll start with first one trivial example, hyperbolic plane. You can took any remaining manifold and also assume that fixed point, the end point goes to some point 0, x1 is equal to x1, will be two given points on the hyperbolic plane 0, x1, and all six depends on the distance because of homogeneity. So this will be some real number. Yeah, so we get this infinite dimensional manifold and action functional will be the following. It's usual kinetic energy. 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, 0, 1, but to complexified hyperbolic space and I use this realization hyperboloid to H2-complexified, which will be just collection of whatever, Z1, Z2, Z3, in C-cube, such that Z1 square minus Z2 squared minus 3 squared equal to 1, complex sphere and plus the same boundary condition. Now, so the functional expends to holomorphic function infinite dimensional manifold story. Now, we want to calculate integral over x real. So what does this integral? One mechanics teaches us this is the heat kernel. The claim is I'll use heat kernel and use quantum mechanical notation but the pretty obvious 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 x0x1. Actually, this is a bit funny why H bar appears in power one, in integral it was in power minus one here, 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 scale things divide by H bar, it means that time multiplied by H bar. So it means that you map short interval and you get exactly this thing. So you get heat kernel and people worked on this for centuries, you know what is this heat kernel and classical formula. You can look, I don't know, even Wikipedia maybe. This says that it's something like square root divided by H bar to power three half, exponent minus one, two. Now, this is the main integral which appear here, which are written the following way. I go from S0 equal to L square over two, 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 and S. So you get this formula. Already this formula tells you everything about what a matrices, what a critical variance, so on. We get certain algebraic functions of square root of some polynomial of infinity degree, I would say. Yeah, some. What you get? You get function of S, which is ramified notification points, which is the same as complex critical values of action on XC. Eventually, our numbers SN, when this guy vanishes and numbers are the following form. L plus two pi IN squared over two, the N is an integer. What does it mean? It's actually very funny. You see that hyperbolic space, it's like sphere of radius minus one. If you were on the usual sphere, if you get two points, we can see the geodesics. We can have infinitely many geodesics going around. We add two pi times radius. But here it's imaginary radius two pi i. So we add to real number this imaginary number guys. In what we get here, we get points on some integer points on parabola. Let's see. It will be L squared over two and equal to zero. This is a critical value, which really we see in the classical station. This are imaginary things which you see under complication. From this story, you see that one get many integrals. One get not only this integral, but take pullback, doesn't this think draw horizontal rays from other points. 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. 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 and h bar. But multiplied by h bar to power minus one half. This one half says that it's morally like finite dimension integral of manifold of dimension minus one. I think it's the regularized value of dimension which we can see in this story. I think it's actually it's very simple story. I don't know, maybe it's new. I have to ask specialist and I made my calculation myself. Here we're in lucky situation, we can solve everything explicitly. But if you consider geodesics on not on more complicated algebraic varieties, you don't know the result, but the prediction will be the same structure. Particularly, we can have decomposition of one integral goes to some of another integrals. In more general situation, we don't have this luxury to have finite dimensional integral presentations. It's asked the question to really construct measures on this left-hand symbols, complex value measures. It's like in usual probability theory, people know Brownian motion, it's actual honest measure which gives a heat kernel. But here it will be completely different subvarieties of semi-infinite dimension and should become complex value densities. Because it's a whole thing, it's Christ was a proof. So, one example, when you see infinitely many critical values. So, this whole story has some generalization to the case of one forms. It should closed one forms. First, I'll start with finite dimensional case. Then you have this again, algebraic variety of complex numbers. Instead of functional, we get algebraic one form which is closed. One can go to universal cover. And universal cover, this pullback of this form will be differential of some action. Certain holomorphic function. It's no longer polynomial. Then one can repeat the story. We get symbols and one get also this regularize function, 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 normalize by this constant. Yeah, so you get infinitely many, yeah, for example, if consider zeros of the form and assume that again generic kind of more zeros, so you get infinitely many of them. So, you get only finitely many formal power series. You get finite number of elements, formal power series in each bar. That's just stupid copies. Yeah, so it's kind of baby finite dimension example with channel Simon's theory which Sergei considered. Okay. And I will go to the most basic example in, if one can imagine sterling formula. Yeah, so what do I mean here? This variety is C star and form is dx minus dx 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 a scopes and maybe Cn is will be one plus 2 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. The symbol is just positive ray. It's going from zero infinity. So the critical point at one and this two gradient trajectory goes 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, log x of each bar, the x, simple 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 2 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 root of 2 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 Sergei's practice, I'll draw first coefficients. These are not Bernoulli numbers because Bernoulli numbers appears in expansion of logarithm. These are completely messy numbers. Maybe I'll write it again. 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. It's given by the following integrals. One over two one half and two pi IH. And makes certain contour integral. Around one, the x minus zeta minus one to power one half. So you can see the contour. It should be not too small because this guy has a ramification point at two points near point one. This thing goes like quadratic function near one. And it has two ramifications points. And this contour should surround this two ramification points. So it's completely one valued function. So you get this thing and how one can treat it. So my suggestion is it will be some kind of infinite dimensional geometry. It's similar to what I have for heat kernel. Namely, for any zeta, which is not in 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 kind of consider one of us. 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 ranking 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. Just mimic this expression. So you get a family of infinite genus curves depending on parameter zeta. And when zeta is two pi integers, this curve degenerates. So I 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 you can follow this cycle along any path. Yeah, so that's kind of point of view from 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 is even simpler. It's absolutely became almost cartology. The whole story. I have this function J. 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 is will be function to be one over J of minus H bar. 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 to 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 the jump along along the ray and the claims the following is G minus on positive ray which goes up is equal to G plus. Multiply by one minus exponent minus 2 pi I over H bar and G minus strictly to our negative plus played by one minus exponent of plus 2 pi. So here now forgetting about Borrel transform what's her I say that what I glue here it's even simplification. I consider trivial rank one bundle on on C I make two cuts and along the cuts applies 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 rubital transform is kind of more trinkets you get some in in a better transform you get this kernel and kernel gives multi valued function or function on universal cover of this. See her me fighters all integers. Yeah but this is kind of much more clean description and here what's the comparison with old story we have we have matrices which are different from identity matrix by exponential small terms but now we have not on diagonal not of diagonal terms and this things can be explained in the following cases so my variety 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 grad grad 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 the direction so it's a piece of 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 one 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 what goes on the chain semi-surry. It's kind of very similar to gamma function story. Yeah so different details but. Sorry. Positive yeah. Yeah okay no what's wrong I don't understand minus it. My message belong to this guy. It's a two different it's functional on different domains. In fact you this also comes from the analysis of the quantum mechanical problem of the harmonic. Yes yes you know it's related yeah yeah definitely yeah. Yeah actually that's exactly not not exactly where I will go now quantum mechanics. Yeah my point is will be complexified quantum mechanics. So 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. Complex algebraic symplectic manifold symplectic form is also break like cotangent bundle we see star cross star we see and and said that. If you put some boundary condition. Zero one algebraic say Lagrangian sub manifold. Yeah so on this guy have a closed one form. I just integrate. For omega. Along the path to get one form. And. Yeah so if I can want to write in some coordinates if it's a PI QI. Like coordinates on M. Contribute coordinates then you write kind of the action. It's defined up to constant is equal to integral over some over PI. DQI 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 L01 and give you some boundary terms but what one can add here and can also integrate H of PQT DT. It's it's a different. To arm and hit kernel when the when this Hamiltonian it's really present it's quadratic in my mentor and the Hamiltonian which gives you the basic flow but now I consider case. And there's really no such data I have some pure geometric data just to Lagrangian sub manifold nothing else. And then 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. And then one can try to repeat the whole game we get infinite emission manifold complex manifold with closed one form. We get. This critical points and interesting ingredient flow. What are gradient lines. The gradient lines. If you get some point PR1 PR2. I will be pseudo homomorphic disks will be passing space in space of pass is boundary conditions will be pseudo homomorphic disks. When you write cashier equation for some almost complex structure. Which is not original complex structure to be different almost structure. And kind of compatible with. Yeah, from can choose hypercalametic but it's this story it's very soft to don't really need integrability condition. Structure is what is the complex structure homomorphic symplectic structure algebraic homomorphic. And the scalar form. No, no, but here get pseudo homomorphic for some different story because you need some killer metric. And if you analyze it and take a real flow for real part of the section, you get certain different almost complex structure. And. And kind of stocks race, the things when you you bump from one critical point to another. Another line means that one of each bar integral of form of some. Such disk is is positive real number that gives a condition on each bar. So what's what this integral should morally calculate. Yeah, if. Many of us is cotton and bundle to some right. Sorry, what is age bar here. Each bar is number complex number. Yeah, yeah, this is the condition of argument of each bar. So it should be argument of this interval. So if M is cotton and bundle suppose L zero is arbitrary. So cotton and bundles fibered by cotton and space, but suppose L one is cotton and space at some given point. So this will be one. It is your it will be one. This integral. What it should be morally give because L zero should give like family of. D modules, the modules or differential equations, depending on each bar. And which converge to some spectral variety L zero in classical limit. And. And the things should calculate solution with the right way to consider intersection point. Responding to this left symbol should this thing should be solution with right double a kb asymptotic. The value of solution point y. So the integral should be. Solution is correct. So kb asymptotic at point y. System of questions and see the solution. Yeah, it's 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 is some kind of interesting phenomenon will happen here. So in general, if you do find a dimension integral and start to move parameters since the gradient number of gradient lines will jump according to usual. The car left as formula. And what happens in usual station. If you have two gradient lines, they can. Gradient structure one point to third point, third point then. For special values of each bar and the secret lines you get in you gradient line. And what what goes on in this geometry. So if you have. Count this homophobic pseudo homophobic discs in real co dimension one. It can split in. Two homophobic discs exactly like gradient trajectory splits to two gradient trajectory or will be completely new phenomena. So it will get. Get develop some discs on zero or develop this on one. Yeah, so get something. Some new phenomenon which has no analogs and finite dimensional situation. So something wrong goes with your number of gradient lines. Just have a couple of minutes to explain. What is your. Goes on. Yeah, so there's a problem. This individual Lagrangian manifold with L zero and L one when you get this homophobic discs. It has nothing to do with now this pass integral. It's some kind of Lagrangian 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 what happens if consider fundamental group of. This space. It's pretty big. A space of pass. It's roughly. Get a contribution 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. And maybe also pi two of M itself. That's rough rough picture. And that's actually the origin of the story that's it 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. We have this finite dimensional case is one form. And what one can try to see that. This. You get. Comology of X. Maybe I'm sorry to be bit. Is the risk it apology. It's really important to say this way. This algebraic forms and with differential D plus. One of H bar. Eta. Yeah, this integral which calculated. The volume form can be considered as close class in this case and integration cycles are functional. So this case. This kind of lecture symbols. Gives basis in dual space. When considering this universal cover. That's something for which this expression one can integrate. So get a basis and when across the strokes raises basis goes to some linear combination of other elements of the basis. But. One can twist the whole story. By considering some. Taurus. It should be home from fundamental group of my manifold. To see star to the C star to up to financing to first beta number. It will find a dimension towards and for any point on this Taurus 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 get not only one store on only one. This calculus but depending on the torus. All this resurgence story with gluing bundles will be kind of Hallomorff family of such guys on on a torus. Get Hallomorff family. Over some torus. Complex torus. Of resurgence picture in this case when a glue sinks along race. Not not 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 against the start of some finite number. And this effect says use the following in kind of in quantum field series will be certain normalization space will be modified. For it will be replaced the space of rank local system will be complex variety but it will be not at all so 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 the torus. But if you go through each bar. Go through some stocks ray. Certain stocks ray. What you get you get automorphism of the group ring of the fundamental group. Yeah is it. I'll finish in few minutes. If H bar argument if you bar crosses. Ray. Which is argument of the integral form of the disk such a boundary of this belongs to. Let's say L zero or L one. You change may apply. Automorphism of group ring of the fundamental group of L zero say. And this is a wall crossing which was young started and this. It's actually explained by Gaiotov Moronezki. Through some gauge theory but acclaim this purely geometric story have nothing to do with. The particularity. The story depending on small parameter. Yeah for example. Very simple example suppose my Lagrangian manifold. Let's say it's S one cross S one. Or it contains S one cross S one. It's because. I should be a lipti curve. It's complex curve. And suppose I have some homomorphic disk. This is which is boundary on some thing. Then if I cross appropriate ray is the argument of the integral of two forms of this. That is it. The transformation will go to phone if I get stars cross star. Goes to something like Z one. Two multiplied by one plus exponent. And so to go away bar. Of this disk. The multiplied by. Zero zero one so it's it's some nonlinear change of rank one local space of run run one local system. And this is for D wall crossing of Gaiotov Moronezki. Before going to this pass integral it's already in each individual manifold to get kind of identification of Toray. Yes, kind of the most basic example if consider a grand manifold sitting in Cartagena bound to C, which is some pq plus xq plus some. The integral of part of Hamiltonian with cubic potential, then the standard picture is that what you get, you get five stocks race, and you get nonlinear. Riemann Hilbert problem. Namely, you want to map. You want to find two functions like the ones that you hear the two, the three here. For here's the four, the five here. Here's the five one. Two functions in each race, which are seen finitely up to the boundary. The boundary. In each sector. And. When you cross the sector, for example, go from the one to two. To the two to three. The two and the three will be exactly the same formula to be the one times one plus exponent of corresponding constant in the sector, which bar which will be totally real times. The two. Yeah, so you get a now a glue nonlinear manifold and consider how Morphic sections. This is basic kind of fordival crossing in in gauge theory origins. Again, you solve the Srimann Hilbert problem abstractly and then I can add also some vector bundles here. For example, if you it's only some parameter spaces for for integral send for manifold for things you could do something like this G plus gain exploring something called which bar. Maybe G two times the one something like this. You, you, you know, glue vector bundles again using exponential terms and monomials in the. This will be 2d4d will crossing and the conjecture that this you get some big class of the personality cobertures which give former power series expansion in this form power series expansion conjecture will be resurgent in Borrell summation way. So that's that's a picture. Thank you. Any questions. So what is not linear manifold here. It's kind of you make gluing. It's manifold nonlinear means it's glue by some nonlinear change of coordinates. This is this is this area of this disc. Is there a version of this where the discs have spheres and without boundaries and the grunge. It's interesting story because it's all the stories only genus zero only discs but definitely high genus should contribute to some second quantization of the story. You have this example with the hyperbolic plane. Yes. You had this point line. Yeah. What are the intersections of the tables in this case. It's very easy to calculate from this double cover story. It will be plus minus one. No plus minus one. It's minus one I suppose. Yeah. Of diagonal terms. Yeah. It's completely controllable. Any questions. Thank you again.
 Okay, thank you.  So my talk will be kind of a mathematical illustration to  physicist's talk to Mark Wilson-Sergay.  And I will, so the whole perspective we will get on  the results will be some kind of  the current left-hand theory which was already mentioned.  And I will start with this kind of simplest instance of  the resurgence when I integrate  exponent of polynomial or rational function.  Now, for example, and in fact,  it will be because I kind of model a field theory.  I'll say that it's kind of zero dimensional 50.  For example, I integrate exponent minus s of x divided by h bar dx.  S is a polynomial, let's say in one variable.  I do the 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 its kind of a-recase.  So the polynomial is x cubed over 3 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 directions at infinity when action goes to plus infinity,  like x cubed to positive real numbers and you get argument zero,  argument 2 pi over 3,  and argument 4 pi over 3,  gets three direction and you have kind of three possible chains of integration,  but only two are 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 of zero.  And because I was smart to choosing this one over three,  you see that solution is this.  You have two solutions, x1 is equal to minus one,  x2 is equal to plus one,  you get 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,  s1 is equal to plus to third,  s2 is equal to minus to third.  And these are elements of this,  what's your g called?  Braille plane, values of my function,  s is a point s1 and s2.  And now what we do,  if h bar is not real,  we define left side symbol,  this symbol gamma alpha h bar,  where alpha is 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 as 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  omified 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,  the alpha is one two,  one critical point,  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,  we go roughly the following.  First of all, we get 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,  if minus one,  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 formal power series,  which is a very interesting way,  kind of resuméed this series.  Yeah, in principle,  also one can see this thing,  it's e alpha h bar is integral over s belonging to  this s alpha plus h bar plus,  exponent minus,  so which bar times some algebraic function.  One can write it on the grid,  actually one form times maybe ds,  and one form function is s goes to dx over ds,  and invert s is a map,  so it will be three-valued algebraic function.  Okay, so that's very simple game.  So we have this nice asymptotic expansion,  and let's me remove  a universal term from this expression.  So define g alpha of h bar,  it will be exponent plus s alpha divide h bar,  and multiplied by two pi h bar minus one half.  I will not treat these things as e alpha h.  It's certain formal power series in h bar,  starting with something.  So get two functions.  Each function, in fact,  it's on this function, and on this function,  h bar is not real line.  So we get two analytic functions,  on upper half plane,  and on low half plane,  and which are sin infinity up to boundary.  So the analytic inside,  and continues to sin infinity functions  everywhere and the still expansion  will be exactly this series.  Now, so we get two functions,  an upper half plane, two function,  low half plane, and how they behave if you go through the lines.  Get some jumps, and we have  jump along one raven argument of h bar is equal to zero,  is the following.  You look what happens with this integration cycles.  If you look, turns this array 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 over  two bar which will be very big positive number, real number.  So this things will be very, very small multiplied by J one.  Similar if you go to these things,  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 claim that what we do,  we glue some holomorphic vector bundle,  on complex plane with h bar parameter.  It will be trivial outside C plus C,  and this basis corresponding to my critical points,  outside to raise, and if you cross the ray,  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 way.  We take diagonal matrix of exponent of S alpha divided by J bar,  alpha is one two,  multiplied by matrix 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.  Similar thing you go on the other way.  So we make some bundles glued from trivial bundles,  and certain transformation long cuts,  and all this thing says that this G1, G2 form  holomorphic section of the glued bundle.  It says nothing more,  and usually people then say that how this  makes resumption make sense of  form of power series to promote it to  actual net function do Barrelli resumption,  but one can do it from different perspective already in this terms.  I don't think about Barrelli transformation transform at all.  I can ask the following. So this Barrelli summation,  if you get something series N, H1,  then we're making new series sum over N factorial,  Z to bar 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?  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 guy says infinity,  but when we change by something which has  trivial Taylor expansion 0,  exponent 1 over x, it has trivial expansion.  So what happens is G1,  G2 formal certain series  independent on sectors.  Here we get actual halomorphic function.  Let me explain.  Maybe I can continue as a board.  So the 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 glue bundle,  according to this two transformations along  trace and what does it mean glue bundles?  You glue some bundles and you make  halomorphic trivialization, you can do it one way or  another and what you found,  find halomorphic trivialization.  So it means that you get  certain function G of h bar.  Let's maybe some G2 valid function for h bar,  not real lines and such as the jump of  this thing along this,  each ray will be given exactly by  left or right multiplication by this stuff.  So we found this in some way and the main point,  it's for this thing we really need  this critical values and this integer matrices, nothing else.  You need to know only critical values and this integer matrices,  like in this case.  So we get certain G-valued function,  but then if you expand and form power series,  it doesn't depend on the choice of six,  you get certain from this universal procedure,  certain invertible matrix with  values in this coefficients in formal power series.  It's independent on the choice of sector.  Then thus 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.  It's convergent series.  So you do some calculation with  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 h bar.  Now you apply to this appropriate 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  population with Borrel transform and this radial stuff and so on.  But I kind of found it mathematically  more appealing because you don't really  need a fine coordinates in each bar.  Yeah, so we just can play in different situation.  Yeah, so 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 S,  this polynomial map,  and we get some polynomial volume element,  you want to integrate this six and you apply maybe  some algebraic volume form on manifold.  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  will assume that my critical values are all more  some nice, not more complicated,  stories Sergei considered can really isolate it.  I think left-shift symbols will be gradient trajectories for  real part of S divided by each bar,  starting from critical points.  Then you'll see that it is a kind of  vibration m minus one dimensional sphere and projects  to this ray which I draw on the plane, exactly.  For gradient trajectory,  use some scalar metric and such it  does in good situation which metric you choose.  So you get the things and the same story happens,  you get some upper triangular matrices,  integer matrices and the whole story repeats.  The integral will have,  again, will grow like expanse critical value divided by each bar  and multiplied by 2 pi i each bar to power n over 2,  and then we get some series.  From this perspective, what you really know,  you should know all critical values and certain,  this entries element of the sub-parangular 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.  So we do pass integral.  Yeah, so my space will be infinite dimensional.  I denote like it is very big x and it will be set of points which  will be cx which is a map from interval 0, 1,  kind of time, leading time maybe,  this coordinate t to, yeah,  so I'll start with first one trivial example, hyperbolic plane.  You can took any remaining manifold and also assume that fixed point,  the end point goes to  some point 0, x1 is equal to x1,  will be two given points on the hyperbolic plane 0, x1,  and all six depends on the distance because of homogeneity.  So this will be some real number.  Yeah, so we get this infinite dimensional manifold and  action functional will be the following.  It's usual kinetic energy.  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,  0, 1, but to  complexified hyperbolic space and I use  this realization hyperboloid to H2-complexified,  which will be just collection of whatever,  Z1, Z2, Z3, in C-cube,  such that Z1 square minus Z2 squared minus 3 squared equal to 1,  complex sphere and plus the same boundary condition.  Now, so the functional expends to  holomorphic function infinite dimensional manifold story.  Now, we want to calculate integral over x real.  So what does this integral?  One mechanics teaches us this is the heat kernel.  The claim is I'll use heat kernel and use quantum  mechanical notation but the pretty obvious 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 x0x1.  Actually, this is a bit funny why H bar appears in power one,  in integral it was in power minus one here,  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 scale things divide by H bar,  it means that time multiplied by H bar.  So it means that you map short interval and you get exactly this thing.  So you get heat kernel and people worked on this for centuries,  you know what is this heat kernel and classical formula.  You can look, I don't know,  even Wikipedia maybe.  This says that it's something like square root divided by H bar to power three half,  exponent minus one, two.  Now, this is the main integral which appear here,  which are written the following way.  I go from S0 equal to L square over two,  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 and S. So you get this formula.  Already this formula tells you everything about what a matrices,  what a critical variance,  so on. We get certain algebraic functions of square root of  some polynomial of infinity degree, I would say.  Yeah, some.  What you get?  You get function of S,  which is ramified  notification points,  which is the same as complex critical values of action on XC.  Eventually, our numbers SN,  when this guy vanishes and numbers are the following form.  L plus two pi IN squared over two,  the N is an integer.  What does it mean? It's actually very funny.  You see that hyperbolic space,  it's like sphere of radius minus one.  If you were on the usual sphere,  if you get two points, we can see the geodesics.  We can have infinitely many geodesics going around.  We add two pi times radius.  But here it's imaginary radius two pi i.  So we add to real number this imaginary number guys.  In what we get here,  we get points on some integer points on parabola.  Let's see. It will be L squared  over two and equal to zero.  This is a critical value,  which really we see in the classical station.  This are imaginary things which you see under  complication. From this story,  you see that one get many integrals.  One get not only this integral,  but take pullback,  doesn't this think draw  horizontal rays from other points.  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.  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 and h bar.  But multiplied by h bar to power minus one half.  This one half says that it's morally like  finite dimension integral of manifold of dimension minus one.  I think it's  the regularized value of dimension which we can see in this story.  I think it's actually it's very simple story.  I don't know, maybe it's new.  I have to ask specialist and I made my calculation myself.  Here we're in lucky situation,  we can solve everything explicitly.  But if you consider geodesics on  not on more complicated algebraic varieties,  you don't know the result,  but the prediction will be the same structure.  Particularly, we can have  decomposition of one integral goes to some of another integrals.  In more general situation,  we don't have this luxury to have  finite dimensional integral presentations.  It's asked the question to really  construct measures on this left-hand symbols,  complex value measures.  It's like in usual probability theory,  people know Brownian motion,  it's actual honest measure which gives a heat kernel.  But here it will be completely different subvarieties  of semi-infinite dimension and should  become complex value densities.  Because it's a whole thing,  it's Christ was a proof.  So, one example,  when you see infinitely many critical values.  So, this whole story has some  generalization to the case of one forms.  It should closed one forms.  First, I'll start with finite dimensional case.  Then you have this again,  algebraic variety of complex numbers.  Instead of functional,  we get algebraic one form which is closed.  One can go to universal cover.  And universal cover,  this pullback of this form will be differential of some action.  Certain holomorphic function.  It's no longer polynomial.  Then one can repeat the story.  We get symbols and one get also this regularize function,  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 normalize by this constant.  Yeah, so you get infinitely many,  yeah, for example, if consider zeros of the form  and assume that again generic kind of more zeros,  so you get infinitely many of them.  So, you get only finitely many formal power series.  You get finite number of elements,  formal power series in each bar.  That's just stupid copies.  Yeah, so it's kind of baby finite dimension example  with channel Simon's theory which Sergei considered.  Okay.  And I will go to the most basic example in,  if one can imagine sterling formula.  Yeah, so what do I mean here?  This variety is C star and form is dx minus dx 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 a scopes and maybe Cn is will be one plus 2 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.  The symbol is just positive ray.  It's going from zero infinity.  So the critical point at one and this two gradient trajectory goes 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,  log x of each bar, the x, simple 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 2 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 root of 2 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 Sergei's practice,  I'll draw first coefficients.  These are not Bernoulli numbers because Bernoulli numbers  appears in expansion of logarithm.  These are completely messy numbers.  Maybe I'll write it again.  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.  It's given by the following integrals.  One over two one half and two pi IH.  And makes certain contour integral.  Around one, the x minus zeta minus one to power one half.  So you can see the contour.  It should be not too small because this guy has a ramification point at two points near point one.  This thing goes like quadratic function near one.  And it has two ramifications points.  And this contour should surround this two ramification points.  So it's completely one valued function.  So you get this thing and how one can treat it.  So my suggestion is it will be some kind of infinite dimensional geometry.  It's similar to what I have for heat kernel.  Namely, for any zeta, which is not in 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 kind of consider one of us.  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 ranking 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.  Just mimic this expression.  So you get a family of infinite genus curves depending on parameter zeta.  And when zeta is two pi integers, this curve degenerates.  So I 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 you can follow this cycle along any path.  Yeah, so that's kind of point of view from 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 is even simpler.  It's absolutely became almost cartology.  The whole story.  I have this function J.  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 is will be function to be one over J of minus H bar.  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 to 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 the jump along along the ray and the claims the following is G minus on positive ray which goes up is equal to G plus.  Multiply by one minus exponent minus 2 pi I over H bar and G minus strictly to our negative plus played by one minus exponent of plus 2 pi.  So here now forgetting about Borrel transform what's her I say that what I glue here it's even simplification.  I consider trivial rank one bundle on on C I make two cuts and along the cuts applies 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 rubital transform is kind of more trinkets you get some in in a better transform  you get this kernel and kernel gives multi valued function or function on universal cover of this.  See her me fighters all integers. Yeah but this is kind of much more clean description and here what's the comparison with old story we have we have matrices which are different from identity matrix by exponential small terms but now we have not on diagonal not of diagonal terms  and this things can be explained in the following cases so my variety 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 grad grad 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 the direction so it's a piece of 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 one 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 what goes on the chain semi-surry. It's kind of very similar to gamma function story. Yeah so different details but.  Sorry. Positive yeah.  Yeah okay no what's wrong I don't understand minus it.  My message belong to this guy. It's a two different it's functional on different domains.  In fact you this also comes from the analysis of the quantum mechanical problem of the harmonic.  Yes yes you know it's related yeah yeah definitely yeah.  Yeah actually that's exactly not not exactly where I will go now quantum mechanics.  Yeah my point is will be complexified quantum mechanics.  So 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.  Complex algebraic symplectic manifold symplectic form is also break like cotangent bundle we see star cross star we see and and said that.  If you put some boundary condition. Zero one algebraic say Lagrangian sub manifold.  Yeah so on this guy have a closed one form.  I just integrate.  For omega.  Along the path to get one form.  And. Yeah so if I can want to write in some coordinates if it's a PI QI.  Like coordinates on M.  Contribute coordinates then you write kind of the action.  It's defined up to constant is equal to integral over some over PI.  DQI 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 L01 and give you some boundary terms but what one can add here and can also integrate H of PQT DT.  It's it's a different.  To arm and hit kernel when the when this Hamiltonian it's really present it's quadratic in my mentor and the Hamiltonian which gives you the basic flow but now I consider case.  And there's really no such data I have some pure geometric data just to Lagrangian sub manifold nothing else.  And then 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.  And then one can try to repeat the whole game we get infinite emission manifold complex manifold with closed one form.  We get.  This critical points and interesting ingredient flow.  What are gradient lines.  The gradient lines.  If you get some point PR1 PR2.  I will be pseudo homomorphic disks will be passing space in space of pass is boundary conditions will be pseudo homomorphic disks.  When you write cashier equation for some almost complex structure.  Which is not original complex structure to be different almost structure.  And kind of compatible with.  Yeah, from can choose hypercalametic but it's this story it's very soft to don't really need integrability condition.  Structure is what is the complex structure homomorphic symplectic structure algebraic homomorphic.  And the scalar form.  No, no, but here get pseudo homomorphic for some different story because you need some killer metric.  And if you analyze it and take a real flow for real part of the section, you get certain different almost complex structure.  And.  And kind of stocks race, the things when you you bump from one critical point to another.  Another line means that one of each bar integral of form of some.  Such disk is is positive real number that gives a condition on each bar.  So what's what this integral should morally calculate.  Yeah, if.  Many of us is cotton and bundle to some right.  Sorry, what is age bar here.  Each bar is number complex number.  Yeah, yeah, this is the condition of argument of each bar.  So it should be argument of this interval.  So if M is cotton and bundle suppose L zero is arbitrary.  So cotton and bundles fibered by cotton and space, but suppose L one is cotton and space at some given point.  So this will be one.  It is your it will be one.  This integral.  What it should be morally give because L zero should give like family of.  D modules, the modules or differential equations, depending on each bar.  And which converge to some spectral variety L zero in classical limit.  And.  And the things should calculate solution with the right way to consider intersection point.  Responding to this left symbol should this thing should be solution with right double a kb asymptotic.  The value of solution point y.  So the integral should be.  Solution is correct.  So kb asymptotic at point y.  System of questions and see the solution.  Yeah, it's 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 is some kind of interesting phenomenon will happen here.  So in general, if you do find a dimension integral and start to move parameters since the gradient number of gradient lines will jump according to usual.  The car left as formula.  And what happens in usual station.  If you have two gradient lines, they can.  Gradient structure one point to third point, third point then.  For special values of each bar and the secret lines you get in you gradient line.  And what what goes on in this geometry.  So if you have.  Count this homophobic pseudo homophobic discs in real co dimension one.  It can split in.  Two homophobic discs exactly like gradient trajectory splits to two gradient trajectory or will be completely new phenomena.  So it will get.  Get develop some discs on zero or develop this on one.  Yeah, so get something.  Some new phenomenon which has no analogs and finite dimensional situation.  So something wrong goes with your number of gradient lines.  Just have a couple of minutes to explain.  What is your.  Goes on.  Yeah, so there's a problem.  This individual Lagrangian manifold with L zero and L one when you get this homophobic discs.  It has nothing to do with now this pass integral.  It's some kind of Lagrangian 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 what happens if consider fundamental group of.  This space.  It's pretty big.  A space of pass.  It's roughly.  Get a contribution 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.  And maybe also pi two of M itself.  That's rough rough picture.  And that's actually the origin of the story that's it 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.  We have this finite dimensional case is one form.  And what one can try to see that.  This.  You get.  Comology of X.  Maybe I'm sorry to be bit.  Is the risk it apology.  It's really important to say this way.  This algebraic forms and with differential D plus.  One of H bar.  Eta.  Yeah, this integral which calculated.  The volume form can be considered as close class in this case and integration cycles are functional.  So this case.  This kind of lecture symbols.  Gives basis in dual space.  When considering this universal cover.  That's something for which this expression one can integrate.  So get a basis and when across the strokes raises basis goes to some linear combination of other elements of the basis.  But.  One can twist the whole story.  By considering some.  Taurus.  It should be home from fundamental group of my manifold.  To see star to the C star to up to financing to first beta number.  It will find a dimension towards and for any point on this Taurus 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 get not only one store on only one.  This calculus but depending on the torus.  All this resurgence story with gluing bundles will be kind of Hallomorff family of such guys on on a torus.  Get Hallomorff family.  Over some torus.  Complex torus.  Of resurgence picture in this case when a glue sinks along race.  Not not 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 against the start of some finite number.  And this effect says use the following in kind of in quantum field series will be certain normalization space will be modified.  For it will be replaced the space of rank local system will be complex variety but it will be not at all so 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 the torus.  But if you go through each bar.  Go through some stocks ray.  Certain stocks ray.  What you get you get automorphism of the group ring of the fundamental group.  Yeah is it.  I'll finish in few minutes.  If H bar argument if you bar crosses.  Ray.  Which is argument of the integral form of the disk such a boundary of this belongs to.  Let's say L zero or L one.  You change may apply.  Automorphism of group ring of the fundamental group of L zero say.  And this is a wall crossing which was young started and this.  It's actually explained by Gaiotov Moronezki.  Through some gauge theory but acclaim this purely geometric story have nothing to do with.  The particularity.  The story depending on small parameter.  Yeah for example.  Very simple example suppose my Lagrangian manifold.  Let's say it's S one cross S one.  Or it contains S one cross S one.  It's because.  I should be a lipti curve.  It's complex curve.  And suppose I have some homomorphic disk.  This is which is boundary on some thing.  Then if I cross appropriate ray is the argument of the integral of two forms of this.  That is it.  The transformation will go to phone if I get stars cross star.  Goes to something like Z one.  Two multiplied by one plus exponent.  And so to go away bar.  Of this disk.  The multiplied by.  Zero zero one so it's it's some nonlinear change of rank one local space of run run one local system.  And this is for D wall crossing of Gaiotov Moronezki.  Before going to this pass integral it's already in each individual manifold to get kind of identification of Toray.  Yes, kind of the most basic example if consider a grand manifold sitting in Cartagena bound to C, which is some pq plus xq plus some.  The integral of part of Hamiltonian with cubic potential, then the standard picture is that what you get, you get five stocks race, and you get nonlinear.  Riemann Hilbert problem.  Namely, you want to map.  You want to find two functions like the ones that you hear the two, the three here.  For here's the four, the five here.  Here's the five one.  Two functions in each race, which are seen finitely up to the boundary.  The boundary.  In each sector.  And.  When you cross the sector, for example, go from the one to two.  To the two to three.  The two and the three will be exactly the same formula to be the one times one plus exponent of corresponding constant in the sector, which bar which will be totally real times.  The two.  Yeah, so you get a now a glue nonlinear manifold and consider how Morphic sections.  This is basic kind of fordival crossing in in gauge theory origins.  Again, you solve the Srimann Hilbert problem abstractly and then I can add also some vector bundles here.  For example, if you it's only some parameter spaces for for integral send for manifold for things you could do something like this G plus gain exploring something called which bar.  Maybe G two times the one something like this.  You, you, you know, glue vector bundles again using exponential terms and monomials in the.  This will be 2d4d will crossing and the conjecture that this you get some big class of the personality cobertures which give former power series expansion in this form power series expansion conjecture will be resurgent in Borrell summation way.  So that's that's a picture. Thank you.  Any questions.  So what is not linear manifold here.  It's kind of you make gluing.  It's manifold nonlinear means it's glue by some nonlinear change of coordinates.  This is this is this area of this disc.  Is there a version of this where the discs have spheres and without boundaries and the grunge.  It's interesting story because it's all the stories only genus zero only discs but definitely high genus should contribute to some second quantization of the story.  You have this example with the hyperbolic plane.  Yes.  You had this point line.  Yeah.  What are the intersections of the tables in this case.  It's very easy to calculate from this double cover story.  It will be plus minus one.  No plus minus one.  It's minus one I suppose. Yeah.  Of diagonal terms. Yeah. It's completely controllable.  Any questions.  Thank you again.
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