OK, thank you. Yeah, so my talk will be kind of a mathematical illustration to his talk, to Marcus and Sergey.

And I will hold perspective on this. This will be some kind of Pecora-Lefchitz theory, which was already mentioned. And I will start with this kind of simplest instance of resurgence, when we integrate exponent

of polynomial rational function. For example, and in fact it will be, because I kind of will model a field theory, I'll say it's kind of zero-dimensional Q of t.

For example, I integrate exponent minus s of x divided by h-bar dx. s is a polynomial, let's say, in one variable. It will be my action. And I integrate over a certain chain of integration,

which we call gamma. And the main theme of my talk will be rotating Planck constant. So h-bar will be a non-zero complex number, no longer a positive guy.

And let's do its kind of airy case. So the polynomial is x cubed over three minus x in order to get a 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, 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 two pi over three, and argument four pi over three, you get three directions. 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-shot symbols.

In this case, it will be just steepest descent path. To draw left-shot 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 the solution is this. You have two solutions, x one is equal to minus one, x two is equal to plus one. You have 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 function S, the points S one and S two. And now what we do,

if h-bar is not real, or we define left-shot symbol, this timbal gamma alpha h-bar, where alpha is one-two,

one of these two critical points, is re-image of the ray, which would be like this or like this, of ray S alpha plus h-bar

and its positive numbers. So if you draw ray in direction argument of h-bar, straight ray, under 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 rheumified at this point, and consider pullback, you get upstairs, you get actually a 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, where alpha is one-two, one of my critical points, it will be the integral of exponent of dx of this timbal,

and it's defined up to sign, because one should orient this line, there are two ways to orient it, there is no preferred choice a priori. 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 of all, this ambiguity 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,

or 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 with static with one. And this will be divergent series,

it will be formal power series, which is divergent, we want to kind of resummate 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 h-bar, times the sum algebraic function, one can write it on the great,

it's actually one form case, maybe ds, and one 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 me remove a 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 multiply by two pi h-bar to power minus one-half. I will not treat these things, times e-alpha h-bar. It's certain form of power series in h-bar, starting with something.

So we get two functions. Each function, in fact it's honest function, and honest function in 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 they are analytic inside,

and continues to C-infinity functions everywhere, and this expansion will be exactly this series. Yeah, so we get two functions in upper half-plane, two functions in lower half-plane, and how they behave if you go through the lines, get some jumps,

and we have to jump along one ray, when argument of h-bar is equal to zero, is the following. You look what happens with this integration cycles. Well, here look, turns this ray very close to the right, 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 signs, and similar for low-plane, but what does it tell me about this rescaled intervals?

Because I rescaled, what happens is that j, let's say, j one h-bar, if you cross the line, goes to j one, but j two goes to j two,

and I integrate, integrate exponent minus this s one minus s two h-bar, which will be a very big positive number, real number, so these things will be very, very small, multiply by j one.

And similar, 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 a different matrix. So, what goes on?

I claim that what we do, we glue some holomorphic vector bundle, holomorphic vector bundle on complex plane with h-bar parameter,

which will be kind of trivial outside kind of C plus C, and this base is 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 array, we apply transformation, which is this linear operator, which can be also sort as a following way. We take exponent,

kind of diagonal matrix of exponent of s i s alpha divided by h-bar, alpha is one two, multiply by matrix one one zero one, what happens with cycles integration, and conjugate by diagonal inverse power,

so we get all diagonal term will be exponentially small. And similar thing could go on the other array. Yeah, so we get, we make some bundles glued from trivial bundles

and some transformation long cuts, and all this thing says that this g one g two form holomorphic section of the glued bundle.

I think more, and usually people then say that how this makes resummation

makes sense of form of power series to promote it to actual analytic function, do a Borel transformation, but one can do it from kind of different perspective already on these terms. I don't think about Borel transformation transform at all. I can ask the following.

Yeah, so this is Borel summation. If you get certain series, n a to n, then we're making new series, sum over n 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?

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, we change by something which has trivial Taylor expansion zero,

exponent one over x has trivial expansion. So what happens? This g1, g2 formal, a certain series independent on sectors.

And here we get actual holomorphic function. Let me explain versus maybe continuing as a board.

So it's a kind of very simple proposal, how to make resumation of this formal power series to promote to this function. It just looks on the structure, it gives you the answer. So it can first solve Riemann-Hilbert problem.

It means that kind of glue bundle according to these two transformations along grace. And what does it mean glue bundles? 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, this may be some g2, 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 the things we really need this critical values and this integer matrices. Nothing else.

We need to know only critical values and this integer matrices, like in this case. So we get a certain g-valued function.

But then if we expand and form power series, it doesn't depend on the choice of things. You get certain, from this universal procedure, certain invertible matrix with various in coefficients and 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 is analytic function in h-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 this form of series multiply, you get series which is convergent. Now you can evaluate for, let's say, sufficient small values of h-bar. And now you apply to this operator given construction back,

and you get actual value of your intervals. So it's a very simple way to calculate this thing without Borel transform.

And it's kind of equivalent to population with Borel 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 h-bar, so you just can play in different situation.

Yeah, the same story works 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 is a 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.

Now, again, one can do Lefscher symbols, but Lefscher symbols will not pull back. It will be the following things. Lefscher symbol.

Actually, I assume that my critical values are all more nice, not the more complicated story, Sergei considered, kind of really isolated. I think Lefscher symbols will be gradient trajectories for real part of s divided by h-bar, starting from critical points.

And then you'll see that it's a kind of vibration m minus one dimensional spheres and projects to this ray which I draw on the plane, exactly. And for draw gradient trajectory, use some scalar metric,

and essentially it doesn't matter in good situation which metric you choose. So, the same story happens, you get some upper triangular matrices, with integer matrices, and the whole story repeats. The integral will have, again, will grow, like expand critical value divided by h-bar

and multiplied by two pi i h-bar to power n over two, and then we get some series.

So, from this perspective, what you really know, you should know, you should know all critical values and certain, this entries element of the upper triangular matrices, which will be number of gradient flow for this one point to another.

Okay, so let's find a 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 denote like this very big X, and it will be set of points which will be, seek X, 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 non-trivial example, hyperbolic plane. You can took any remaining manifold, and also assume that fixed point, the end point goes to

point 0, X0, will be two given points on the hyperbolic plane, X0, X1, and

the whole thing depends on the distance, because homogeneity will be some real number. Yeah, so get this infinite dimensional manifold, and actual functional will be the following.

It's usual kinetic energy. Yeah, so we get function with some infinite dimensional 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 two complexified hyperbolic space, and I use this realization hyperboloid to complexify it,

which will be just collection of whatever, Z1, Z2, Z3 in C cube, such that Z1 squared minus Z2 squared minus Z3 squared equal to 1, kind of complex sphere, and plus the same boundary condition.

Now, so the function will expand to holomorphic function in infinite dimensional manifold story, and now we want to calculate integral over maybe X real.

So what is this integral?

One of the mechanics teach us this is the heat kernel. They claim it's... I will use kind of heat kernel, and I use quantum

mechanical notation, but it's pretty obvious. I go from the bar, exponent is h bar h X1, where h is Hamiltonian,

which is minus one half of Laplacian and half of the water plane. And you can see the heat kernel, the time is h bar and connect over X0 X1. 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 homogeneous degree minus one. So it means that if you rescale 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.

So you get heat kernel, and people worked on this for centuries, and now what is this heat kernel? And classical formula, you can look, I don't know, even in Wikipedia maybe,

it says that it's something like square root divided by h bar to power three half, exponent maybe minus one, two, and now this is the main integral which appear here,

which I'll write in the following way. I go from S0 equal to L square over two, where L is the distance between, to 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. 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, like square root of some polynomial of infinite degree, I would say. And what do you get?

You get function of S, which is ramified ramification points, which is the same as complex critical values of function 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, where 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 consider geodesics. When you have infinitely many geodesics going around, we add two pi times radius.

But here, it's imaginary radius is radius two pi i. So we add a real number, this imaginary guys, and what we get here, we get points on, some integer points on parabola, and see, and this will be square root of two, n equal to zero.

But this is a critical value, which really we see in the classical station, and these are imaginary things which you see under simplification. And from this story, you see that one get many integrals, one get not only this integral, but can take pullback. Doesn't this thing draw a horizontal raise from other points?

And then we can rotate the story. 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 this integrals, before division by these things, we see that they grow like exponent of this critical value, as n minus h-bar, but multiplied by h-bar to power minus one-half.

And this one-half says that it's morally like finite dimensional integral of manifold of dimension minus one.

And I think it's kind of zeta-regularized value of dimension, which we can see in this story. Now, I think it's actually, it's a 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 unlucky situation, we can solve everything explicitly. But if you consider geodesics on more complicated algebraic varieties, you don't know the results, but the prediction will be the same structure.

In particular, we can kind of have this decomposition of one integral goes to some of another integrals. And in more general station, we don't have this luxury to have finite dimensional integral presentations. Yeah, so it's asks the equation to really construct measures on the left-hand symbols,

complex value measures. It's like in usual probabilities here, people know Brownian motion. It's actual honest measure, which gives a heat kernel, but here it will be completely different sub-varieties of semi-infinite dimension. And it should be kind of complex value densities, because it's a whole thing that's cries to the 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 get algebraic one-form, which is closed.

And now, one can go to universal cover, and on universal cover, this is full back of this form,

will be differential of some action, certain holomorphic function. It's no longer polynomial. And then one can repeat the story. We get symbols and one get also this regularize functions

the alpha h-bar when I divide by leading term. And these things are independent on the choice of lift,

because you add to function just a constant, and normalize this constant. Yeah, so you get finitely many. Yeah, for example, you consider zeros of the form and assume that again generic kind of more zeros,

finitely many of them. So you get only finitely many form of power series. You get finite number of elements form power series in h-bar. Those are just stupid copies.

Yeah, so it's a kind of baby finite dimensional example of this Chernon-Simon theory, which Sergei considered. Okay, and I will go to the most basic example

in one can imagine Stirling formula.

Yeah, so what do I mean here? This variety is C*. 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, 0, 1.

And critical values will be 0, 1. But also we get these copies. And maybe Cn, it will be 1 plus 2 pi i n. We add the period of your form.

But let's do kind of first real integral. If h-bar is positive number, then the symbol is only one critical point. So only one symbol. Symbol is just positive ray.

It's coming 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. My h is equal to integral from zero to infinity exponent. You have to write this minus x over h-bar, dx, simple story.

And modified thing, it will be the following. It will be exponent of 1 over h-bar to kill critical value, then multiply by 2 pi i h-bar to the power one-half.

It's the dimension contribution times a0 a of h-bar. And then you see that it's 1 over square root of 2 pi exponent of 1 over h-bar.

This is exactly leading term in Stirling formula for factorial. And so we get some certain series. And following Sergei's practice, I will draw the first two coefficients.

Yeah, these are not Bernoulli numbers, because Bernoulli numbers appears in expansion of logarithm. These are completely messy numbers.

Maybe write it again, sum of a, m, h, n, this series.

What is the Borel transform, k of a to zeta?

This should be a nice function with infinite analytic continuation. And what is this function? It is the following function. Claim, it's given by the following integrals, 1 over 2 one-half and 2 pi i, and makes certain contour integral,

contour around 1 dx, x minus log x minus zeta minus 1 to the power one-half.

Yeah, so you can see the contour, but it should be not too small,

because this guy has a ramification point at two points near point one. Yeah, this thing grows like something like quadratic function near one, and it has two ramification points. This contour should surround these two ramification points,

so it's completely one-valued function. Yeah, so we get this thing and how one can treat it. Yeah, so my suggestion, 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 2 pi i, z,

and this 2 pi i, z, 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 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 we integrate over one chain, classes in each one of this curve,

one form which is z2 times differential e to z1. I just mimic this expression. Yeah, so you get a family of infinite genus curves, depending on parameter zeta. And when z2 is 2 pi a integer, 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 it can follow these cycles along any path.

Yeah, so that's kind of a 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, it's even simpler. It's absolutely became almost a tautology, the whole story.

I have this function j, yeah. And maybe take g plus is equal to h bar is equal to j 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 1 over j 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 infinity. And now we can compare how they,

what's the jump along the ray, and claim the phonics. g minus on positive ray which goes up is equal to g plus multiplied by 1 minus exponent of minus 2 pi i over h bar.

And g minus restricted to r are negative, 2 plus multiplied by 1 minus exponent of plus 2 pi i over h bar.

So here now, forgetting about Borel transform whatsoever, I say that what I glue here, it's even a simple situation. I consider trivial rank one bundle on c, I make two cuts, and the longer cuts apply this transformation which are very close to identity

and has trivial Taylor coefficients. And this describes these analytic properties, what are 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 multivalued function, or function on universal cover of this c rheumified as all integers. But this is kind of much more clean description.

And here, what's the comparison with old story? We have matrices which are different from identity matrix by exponentially small terms, but now we have not of diagonal terms, but on diagonal terms. And these things can be explained in the following ways.

So my variety is c star, and have my one critical point 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 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 there 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 geodesics. So it explains all kind of mystery of gamma functions at once. Yeah, this picture. Yeah, so I will not talk about...

So the natural guess, what goes on in Chen-Samasuri, it's kind of very similar to gamma function story. Yeah, so those are different details, but...

Positive, yeah. Okay, no, what's wrong, I don't understand. Minus h belongs to...

Minus h belongs to this guy? But it's a different function. It's a function on different domains. In fact, this also comes from the analysis of the

quantum mechanical problem of the harmonic oscillator. Yes, yes, no, it's related, yeah, yeah, definitely, yeah. Yeah, actually, that's exactly... No, not exactly where I will go now, quantum mechanics. No, that's kind of my point.

It will be complexified quantum mechanics. So what will be my variety? It will be already complex.

I don't pick any real parameters here. It will be a space of maps interval to some complex algebraic symplectic manifold.

Symplectic form is also algebraic, like cotangent bundle, or C star cross C star. We see, and said that it should put some boundary condition.

0, L1 are algebraic, say, Lagrangian submanifold. So on this guy, if I closed one form, I just integrate form omega,

and along the path I get one form. And yeah, so if I want to write in some coordinates,

if let's say pi, qi are coordinates on m, conjugate coordinates, then you write kind of the action, 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, maybe kind of boundary terms, but also this L0, L1 can give you some boundary terms,

but what one can add here, one can also integrate h of p, q, t, dt. Yeah, it's a different... And this hit 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 a 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, two 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 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-holomorphic disks. It will be pass and space of pass.

These boundary conditions will be pseudo-holomorphic disks. Then we 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, depending on h bar. Yeah, for example, we can choose hypercalor metric, but as a story, it's very soft. We don't really need integrability condition. Like, synthetic structure is what is the complex synthetic structure? Holomorphic, symplectic structure.

Algebraic, holomorphic. It's not scalar form. It's just G squared for minus identity. No, no. But here, I get pseudo-homomorphic for some different story, because you need some scalar 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 in kind of stock's race, these things when you bump from one critical point to another, another line will mean that one over h bar integral form of some such disk

is positive real number. That gives a condition on h bar. So what this integral should morally calculate?

If manifold is cotangent bundled to some variety of... Is h bar here? Sorry? What is h bar here? H bar is a number, complex number. Yeah, yeah. This is a condition of argument of h bar, so it should be argument of this integral. So if M is cotangent bundled, 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 d-modules, holonomic d-modules 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, if you consider intersection point, it's pointing to this Lefchitz symbol, this thing should be solution with right WKB asymptotics.

The value of solution point y. So the integral should be solution is correct, WKB asymptotic at point y. You write some system of equations and see the solution.

Yeah, one can try to look at all the story. And for example, one can try to move point y and see this resurgence how it depends on point y. And then there is some kind of interesting phenomena will happen here. Now, in general, if you do find a dimension integral

and start to move parameters, since the number of gradient lines will jump according to usual Piccadilly-Lefchitz formula. And what happens in usual station, if you have two gradient lines, they can gradient trajectory one point to third point, third point and for special values of h bar,

you get a new gradient line, which goes from first to three. And what goes on in this geometry? So if you have count this pseudo-holomorphic disks in real co-dimension one,

it can split in two holomorphic disks. It's exactly like gradient trajectory splits to two gradient trajectory or will be completely new phenomena. So it will get develop some disk on L0 or develop a disk on L1.

Yeah, so you get something, some 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.

Goes on. Yeah, so there is a problem with individual Lagrangian manifold with L0 and L1 when we get these holomorphic disks. It has nothing to do now with a pass integral. It's some kind of...

Lagrangian in 0 or 1. It's not interaction in both of them, it's just with individual guys something wrong goes on. How to understand what happens? You can see the fundamental group of this space. It's a pretty big space of paths. It's roughly get a

contribution fundamental group where one end can appear, another end can appear. In this station L1 is simply connected, but L0 could be a huge group and maybe also pi 2 of n itself. That's rough picture. And that's

actually the origin of the story. 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 finite dimensional case. We have this finite dimensional case is one form and what one can try to see that this we get a cohomology of X maybe

I'm sorry to be a bit risky topology it's really important to say this way algebraic forms and with differential d plus 1 over h bar eta. Yeah, as this

integral which you calculated the volume form can be considered as close cost in this case and integration cycles are functional for this case. This kind of leftmost timbals gives basis in dual space. Even consider

timbals in universal cover. This is something for which this expression one can integrate. So get a basis and then across the stocks 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 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 the

torus all this resurgence story with gluing bundles will be kind of holomorphic family of such guys on 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 bary-transform yeah so it looks at the dimension rank one local systems it's against C star to some finite number

and this effect says use the following in kind of in quantum field series there'll be certain romanization the space will be modified for it will be replaced the space of rank local system will be a complex variety but it

will be not at all so not even a billion group yeah so it means that all this listing parameters it's from 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 h-bar goes through some the stock 3 certain stocks 3 what you get

you get automorphism of the group ring of the fundamental group finishing

fuel minutes yeah if h-bar argument if you bar crosses array which is

argument of the integral form of the disk such a bundle of disk belongs to let's say l0 or l1 you change you may apply automorphism of group ring of

the fundamental group of l0 say corresponding l1 and this is a wall crossing which was young started and this it's actually explained by

Kyoto Muranesky and through some gauge theory but I claim this purely geometric story have nothing to do with peculiarity the story depending on small parameter yeah for example this is very simple example suppose my

Lagrangian manifold let's say here it's s1 crosses 1 or it contains s1 crosses 1 it's because I should be elliptic curve yeah because complex curve here and suppose I have some holomorphic disk this is which with

bunder on some think then if I cross appropriate ray the argument of the integral of two form of the dicks the transformation will go the following if I get C star cross C star goes to something like z1 z2 multiplied by 1

plus exponent and integral of h-bar of this disk d multiplied by z1 so

it's with some nonlinear change of space of rank one local system and this is 4d wall crossing of go to more nets case so it's before going to this if pass integrals already if we in each individual manifold to get kind of identification of tori yes kind of the most basic example if you consider a

cube plus something a symbol of part of Hamiltonian with cubic potential then

the standard picture is it what to 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 z1 z2 here z2 z3 here this reason for here z4 z5

here because if I see one 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 and go from z1 z2 to z2 z3 so z2 go to z2 and z3 will be exactly the same formula to busy one times 1 plus exponent of corresponding constant in the sector which bar which will be totally real

times z2 yeah so get now a blue nonlinear manifold and consider holomorphic sections this is basic in 4d evil crossing in gauge theory origins again you solve this Riemann Hilbert problem abstractly and then I

can add also some vector bundles here for example if you it's only self-parameter spaces for for intervals and for manifold for things you can good should do something like this G plus gain exponent something called which bar can be g2 times z1 something like this you you you know

glue vector bundles again using exponential terms and monomials in z this will be 2d 4d wall crossing and as a conjecture this you get some big class of this analytic objects which give formal power series expansion in

conjecture will be a resurgent in Borel summation way yeah so that's that's a picture thank you it's kind of glue it's manifold non-linear it means it's

glue by some nonlinear change of coordinates yes yes yes it will be this

areas of this holomorphic discs yeah it's interesting story because it's all all the story it's only G is zero only discs but definitely hygenus should

it's very easy to calculate from this double cover story it will be plus minus

one no plus minus one I suppose yeah yeah of diagonal terms yeah it's completely controllable