So maybe I'll start. So there was some small left over from lecture, from second lecture, which I want to explain.

It's a comparison between BETI and the RAM, isomorphism, and a wall crossing structure for the case of one-forms, not functions, but two-forms.

So what is the setup? I have, as usual, smooth algebraic variety over complex numbers. I have a divisor with normal crossing in x. So more like I can see the chains with boundary on d. That's the point. And then I can see the closed one-form in x.

So I have closed algebraic one-form. And I want to search some kind of BETI in the RAM cohomology. And before, I think it lets me choose some computation, which one can easily show to exist.

Algebraic computation, x bar, which is smooth. And said that complement to x contains some divisor.

It's a divisor which consists of three parts.

It's, again, normal crossing divisor. Horizontal, logarithmic, and vertical, I explained in a second. And said that d total, which is the closure of d, union with these three things, is, again,

normal closing divisor. So in x bar, I have four types of boundaries. And the property as a following. Form alpha extends to kind of open part of horizontal part.

I think it's, so the horizontal place where 2dh open. Open part of the horizontal. Or equivalently, I just write that form

and do it by the same letter. Alpha will be considered as a form on x bar minus. Logarithmic and vertical part. And near these things, the form is not defined.

It should have the following kind of analytic topology. Form will look the following. Alpha will be some regular form without any impulse.

Plus some linear combination of d log z i. The product of z i is equal to 0 is equation for logarithmic part of my form, of my divisor. And c i are just some non-zero complex numbers.

And plus second part will be vertical part. And I think in terms of function, it will be product of some other variables. d j to some power n j, multiplied by 1 plus 1.

Where a product of j z j is equal to 0 is equation for d vertical. And n j are strictly positive integer numbers. Yeah. So one can make such picture to infinity.

Then one can consider a coherent shift on x bar, which I think it will be vector bundle. Omega f, which will be defined as the following.

You can see the shift of forms, which are log forms on x bar relative to all things.

And vanishing on d bar. Vanishing means it's this complex. And such that omega, which alpha belongs to the same story.

It's kind of extension of the complex, which alpha, yeah. And this is nice coherent shift. And one can prove the theorem. Kind of essentially closed sub-bar result for case

when device is empty of, which is not maybe logarithmic, when you have only vertical part in this case.

If you consider a hypercomology of x bar with a risky topology, and with this shift omega f, and with differential h bar plus h alpha, it's closed under both differentials,

then it does not jump at all for all h bar. And can one also put constant in front of the alpha or not?

No, if there's no logarithmic part, one can also put constant along the alpha and also get it to zero. If there's no logarithmic part, it's an additional story, which I will not talk now. If d log is empty, then one can put two constants.

Which cannot be simultaneously equal to zero. Could be simultaneously equal to zero. Yes, yes, yeah. Equal to rank of commode of x are risky. This omega f as well, yeah. One can put both constants to be zero

if the logarithmic part is trivial. Yes, it's the sixth generation, which happens. But I will not talk about this story. And I will not explain to you the proof of the theorem.

Or maybe I will explain in some simple case what the main idea is. Let's consider this is a really simple case, and d total is empty. So x is in x bar. We have just closed variety and closed homework in one form.

Why is this rank of commode doesn't jump? Or maybe it should be a little bit empty. No, maybe I removed it in the case of one form.

Let's explain this situation. What we calculate? In this case, we have a line bundle, which is O, with flat connection, d plus alpha. Let's put h bar equal to 1.

We have a flat connection on line bundle. And here we calculate this Dirac commode of this line bundle, commode of x with flat connection using Dirac model. And then there is a well-known saying that story kind of

should be Simpson and others, yes, that there exists unique up to constant, because it's a simple thing, harmonic metric on this line bundle respect

to flat connection. And then you can identify Dirac homology with Higgs homology. So it means that it gives you some line bundle, another line bundle, holomorphic line bundle, E with Higgs field,

which belongs to sections of x and endomorphism E times 1 form. And then you consider different homology. We'll consider Delbo homology, which

will be hyper-homology of x, as analytic or as risky. Then consider E times 1 forms with differential acting by Higgs field. And then there are homology equal to this thing.

The reason is essentially if you consider d bar plus phi, and here you get d plus alpha operator. And take commutant, of course we

get the same identity as the usual Hodge theory. Now that's a proof. No, that's a general fact for harmonic metric. But what is harmonic metric here for this local system?

Harmonic metric is very simple. It's constant metric. And norm of 1 at any point is equal to 1. It's constant metric in this trivialization. Why it's harmonic? Then consider flat trivialization. We will locally write A is differential of some function

f locally. And then the metric in this trivialization will be metric exponent of f squared. And then consider logarithm of this guy. It's kind of one by one matrix, Hermitian matrix. Depending on point on my manifold,

we consider logarithm of these things, take d d bar, we get definitely 0. So it's harmonic. So what is the class in the commutator? Anti-commutator. What is the condition of harmonic that you? Yeah, there is some notion of harmonicity

for metrics on flat bundles. And in particular, for rank 1 bundle, it means that if you consider flat sections, consider logarithm of norm, you get a pretty harmonic function. But it's also for arbitrary rank. You can see for local system, for arbitrary rank

on K-linear manifold, one can see the same notion of harmonic metrics. And then this is general things that the rank homology, this case is just local system, one can associate it by some formula, some holomorphic bundle plus Higgs field. Another bundle, yeah. You have another bundle, and such a kind of Higgs

homology equal to the rank homology because of this identity for Laplacians. But the identity could go things with different bundles. For different bundles, yeah. You can identify as C infinity bundles. Yeah, which shows that they come all just the same, yeah.

It's kind of generalization of usual Hodge theory. Yes, yes, yeah. It's some compact manifold here. But here, what happens? So we get this another bundle and some Higgs field. Then we can make calculations, and first of all, Higgs field,

because it's a line bundle. It's the same as holomorphic one form, because endomorphism of line bundles is constant. And phi is, I think, it's equal to maybe plus minus. Maybe there's some constant. But form alpha may be divided by 2, something like this.

Yeah, so it's proportional to form alpha. Line bundle is not trivial. But line bundle, one can also identify because it will be line bundle of rank 0 or degree 0. So it will have unitary connections, u1 connection, flat u1 connection. And it will be a kind of holomorphic bundle

associated with flat u1 connection. In flat u1 connection, you take something like i times maybe imaginary part of alpha and maybe again divided by 2. You get closed purely imaginary one form up to some constant.

Yeah, so you get some bundle with some flat connection. But the bundle is, you see, it's not trivial. How do we proceed? When we calculate this cohomology, it will be, in analytic topology, it will be concentrated. The complexity will be acyclic outside of 0's of phi locally.

So it's direct sum of contribution of neighborhoods of 0's of alpha. And then 0's of alpha, called z, it will be some union of some connected components.

And this is Delboe cohomology, take direct sum over b, consider hyper-commulgic of some neighborhood. What did you want in the last line? No, in the last line of the holomorphic and then somewhere of the logarithm? Holomorphic bundle associate with a flat u1 connection

given by a trivial bundle with this coverage connection form. So now you have considered neighborhood of this zb and you repeat the same story. You get e and some x and phi. But my form is on zb, locally is exact.

It vanishes on zb. It can be represented as a differential sum function in zb. The same with imaginary part. So you see that near zb, this flat connection is trivial.

Monogram is trivial. So e is identified with o, this trivial bundle, in some canonical way. OK, so you see that you can replace by usual story

and you get commodious equations as forms. So you just multiply by alpha. OK, so that's a Excuse me, the general case, you also use a harmonic metric? Yeah, the same harmonic metric, but then one should use Michizuke's theory of analysis

how one should do this irregular singularities and so on. So that's the thing. So you get this cohomology is do not jump for h-bar. And I claim it's not Durham cohomology

which we're interested in, it's slightly different from when we calculate integrals, we integrate something like Durham cohomology, depending on h-bar and alpha. So it will be something like hyper-commodule of x without compactifications. You can see the forms with hd plus alpha.

You can see the open part. And you can see the commodule on open part. You compare this closed part to take direct image and you see that something will go wrong. But only for problem, it's different only

for special values of h-bar which are union of i gets ci, where ci are this non-zero numbers. And consider ci divided by positive integers.

For this, for such values of planck constant, you don't get, isomorphism gets something, some small problem. I want to compare this Durham cohomology with BT cohomology.

And for BT, I just essentially already wrote you what I need. I have slightly different.

Yeah, it's not isomorphic to Durham cohomology of open variety. Maybe I'll put minus, maybe I'll put, I think, like this.

Forms vanish on d. And yeah, that's what we're interested in, integrals. And it's not isomorphic for this exceptional values of h-bar. OK, so now we write z union of alpha is, OK, just repeat.

z is zeros of alpha. So you are in the general case, or you are simplifying assumption on the d-bar as well? No, no, no. In the general case. So now consider zeros of alpha. So this theorem doesn't apply to sub-alpha?

Yeah, it's, sorry? You're referring to the theorem about the d-bar? Yeah, because in sub-alpha, it's just no logarithmic divisors. Yes, it's, yeah, it's, yeah. Residues, yeah, it's good, yeah.

Residues, yeah. Yeah, it's zeros of alpha. It is a compact subset because of our picture of what? In x-bar minus d-log plus d-vertical. It's really compact.

And then we can write as a union of connected components. This would be, and as I just already told you,

near each, for any b, near zb in analytic topology, there exists unique function fb, holomorphic function. So that fb restricted maybe to reduce divisor is 0,

to reduce sub-scheme 0. And d of b is equal to alpha. Yeah, so you locally represent as a critical point of some absolute canonical function.

And then I define b-terrealization, which depends on my component of critical point set and Planck constant, which will be, I'll write you formal definition. You get R gamma of zb. And then you take, if you don't know this notation,

then it will help. You get shift of vanishing cycles, you divide by function of b-t divided by h-bar. And then of what? So consider, yeah, so first you

consider a constant shift on x minus all divisor

restricted to the neighborhood. Then you do what? You extend by factorial to d. Yeah, so it's responsible for chains with boundary,

a commutative pair. But then you should extend to still some larger story. X embedded to x-bar minus d-loc and d-vertical. Now extend by star.

Yeah, so it's roughly, if you don't have this infinity, you can see the commutative shift of vanishing cycles. Then you take care of the k of infinity.

I'll say conjecture, because I didn't check, should be theorem, I didn't check all details,

is the following. That one should have comparison isomorphism between beta and the RAM cohomology. Namely for h-bar, which doesn't lie to some countable union of race in C.

Namely, race again by conditions that argument of h-bar is equal to argument of integral of alpha of some chain,

where gamma is a pass in where, ah, sorry, I just was maybe a little bit, I

can say, not just zeros of alpha, but zeros of alpha and of all its restrictions to intersection of components

of d-bar and d-horizontal. Sorry, yeah, because I have many, many strata and I want to assist vanishing all strata as well.

Yeah, the rest is the same. Yeah, so now I have some kind of bad direction for h-bar. Namely, I can see the passing x-bar minus d-log and d-vertical connecting point with ends

in belong to this z, some bad points. And consider arguments of such things when integrals are not zero. So I've got countably many homological class,

get countably many numbers, which could be everywhere dense on the plane. And then the plane that's for h-bar, which is kind of not bad, you get canonical isomorphism comparison, isomorphism between what? You consider this hypercomology of exactly these things,

which maybe denotes kind of h, kind of maybe modified from h-bar of x alpha, just like this.

This slightly modified thing, so it's the wrong version,

will be isomorphic to the direct sum of all connected components of this guys, beta, g-bar, tangent by c.

But here will be some interesting point here. I want to put two different Planck constants here. Maybe I put here kind of h-prime, and here it's h-bar. What is h-modified?

H-modified, it's this definition, this hypercomology with this nice complex. Because there are comology jump, and if you don't do modified for this bad values of h-bar.

But this comology do not jump in h-bar. They all stay the same, because it looks only local contribution for each critical points. So you get these things, and what are h-h-prime? It should be kind of holomorphic in h-prime,

and h-prime should belong to the certain disk, kind of open disk, whose 0 is in direction h-bar. So yeah, so it looks kind of bizarre things.

Maybe I can say that if you go to inverse variable, so maybe we can first write formulas. It means that argument for h-bar minus argument. So this is an isomorphism, what's the state? So no, it's statement's not isomorphism.

It's statement and hidden construction. What is hidden construction? That's an isomorphism. It's an integration of a left shift symbols. For all h-primes, it belongs to something, you have isomorphism. Yes, one can have some isomorphism,

which is holomorphic, yeah. And h-bar should be sufficiently close argument, and real part of h-bar, h-prime inverse times h-bar. Bigger than some constant. So maybe one can go better to inverse variables,

and say get h-prime inverse belongs to half plane, which goes in direction h-bar inverse. So what are you given, h?

I have two numbers. I get h is not on bad race, but h-bar is some small deformation of h-bar's small deformation, and h-bar should be sufficiently small. H-prime, h-prime, h-prime, yeah.

Make inversion, you from the circle get half plane. You can see the inverse of the numbers belonging to the circle, whose boundary contains zero, get half plane. Yeah, so it's a pretty abstract story.

Why do we need h-bar, h-prime? Because the integrals will still converge. Now, I want to say that integrals, which I defined, have analytic continuation to some decision to launch domains.

Yeah, actually, what is really nice in this formulation is that this bet exceptional sets are completely disappear from the formulation. I can modify the rank homology a little bit using complication. And global beta homology are not

equal to this direct sum exactly with the same exception. So the whole thing is kind of cancel, and one have very clean result. And now I'll just show you a kind of basic example. One can really go to the very end and understand everything.

So x is C star, this coordinate. So x, small x. Form alpha will be 1 over x minus 1 times dx. And this boundary divisor will be zero.

So we're interested in this connection. And of course, the computation have only one complication, Cp1. And one point will be, the logarithmic will be zero. Because form has a logarithmic poles at zero

and has a high order pole to infinity. And maybe there's no horizontal part. So we get this notification. And a generator of this homology of each modified

whatever, x alpha, is form dx over x. That's called volume form.

It's easy to see that it's generator. So a set of zeros of my forms is just one point. In this point, x is equal to 1. Form vanishes exactly at one point.

So in this case, kamalja are one dimensional. And if h-bar is positive real number, then the generator of its kind of Morse point.

And you get only one leftmost symbol. And leftmost symbol will be positive x, a real part of real line sitting in x. And on this line, it contains some things

which goes to 0.1. And on this symbol, we'll get a function. f1 of x, which is equal to log x plus 1 minus x.

What is nice about this function? This function at this point 1, which belongs to z, is equal to 0. And differential of my form is form alpha. Differential of functions form alpha. So it's normalized the function to have critical value 0 at my point.

And now what I integrate, just by definition, I get e, again, modified in a different sense. As of first lecture, maybe get kind of e plus for h-bar. It will be the general prescription of the following.

You take square root of 2 pi h-bar. And 1 is the dimension of my manifold. So it's this universal formula.

Then we integrate from 0 to plus infinity. Exponent of 1 over h-bar f1 of x. Now function, what's the graph field of this function? I have this x from 0, 1 to infinity. And function f will have the graph like this.

So it will go to minus infinity in both sense. So you integrate these things, multiply by volume form. This is exactly volume form. So you have this integral. And then you can calculate it very easily.

It's gamma s divided by square root of 2 pi s to the power s minus 1 half, exponent of minus s. And it is equal to 1 plus some asymptotic series in s. Sorry, s is just denoted by h-bar inverse.

So you get a kind of stealing formula. And so this function, it appears in comparison

isomorphism between Betty and Iran. And what is nice, it's kind of compatible with this picture. This function extends to invertible function

in c minus and negative in h-bar plane.

So I get function defined only for positive real number. But by this comparison, isomorphism should go to some bigger disk. But in fact, the disk will be half plane. And then when I start to rotate it, because my walls will be only here, I get exactly the complement to negative number.

So I get automatically explanation why this thing is invertible from this picture. But then one can go to have different domains. Now consider function e negative of h-bar, which is 1.

You now consider h-bar negative number. Again, draw a symbol and calculate integral. And if you calculate integral, you get things related to these things. It's function on complementary picture.

And now what are bad directions? This is what's a bad directions. Bad directions is imaginary axis.

Yeah, isomorphism fails, yeah. Because you integrate alpha, you can see the periods of my form alpha.

And you have only one interesting integral, which is 2 pi i, because residue is 1. So this is a kind of bad directions. Kind of stock's raise here. Not stock's raise, imaginary stock's raise here.

I don't have things. It's er plus and er minus. And I have one function which defines on the one half what extends to larger things. Another function defines on the other half. Another thing we consider jump.

If h-bar belongs to er plus, then what we see is that i times r plus, if h-bar belongs to these things,

then you have one function from the left hand side to another from the right hand side. And they differ by the following things that something like a minus is equal to a plus minus exponent of 2 pi i over h-bar divided by i plus.

And if h-bar belongs to minus negative part of imaginary axis, then i minus is equal to i plus minus exponent of minus 2 pi i over a i plus.

And here again, one can understand everything. So first we get this kind of minus sign. And this minus sign means the following. X is c star. X is a cylinder topologically.

It has one zero for my point. And there exists essentially one loop connected to each other with some gradient line. And this multiplicity, this thing means it's n plus minus gamma 0 is equal to minus 1.

So we get one loop, but the sign is equal to minus 1, I think. So it's integer numbers. We cannot see it's in the formula. It's minus 1, but it's integer. And what is exponent? Exponent, it's integral 2 pi i. It's 2 pi i or minus 2 pi i.

2 pi i is integral over gamma 0 of alpha. And minus 2 pi is integral of minus gamma 0 of alpha. Yeah, so one can understand all terms in this formula through this comparison isomorphism. Yeah, so if you consider this, yeah,

so my original left shift symbol was something like this. But then it start to rotate. Kind of you rotate the argument of h bar, start to rotate, and you get the same things plus union of the circle. And this expresses this formula. Yeah, so all the story explains nicely

all properties of gamma functions, which we know from general principles. Yeah, so OK. Now, yeah, and what follows, I will simplify notations.

Forget about these divisors. Ignore that they exist, just to make notation easier.

So what happens in general? I have closed one form on some variety x. Then for each non-zero Planck constant, I have two isomorphic things.

Yeah, I have h modified the RAM, x bar, x alpha. And it's isomorphic to this other stuff. Some of Betty. You can get this comparison.

But these things, it's a holomorphic bundle in h bar. So you can see, but it doesn't have any natural flat connection. No natural connection.

Unlike for the case of function, we can identify, but here we get really local systems. If we go to Betty's description, local system is different monodromes. There's no reasons to identify cohomology. They have just the same rank, but they're not isomorphic for that connection. And similarly, if we get a families,

we consider some holomorphic family, x depending on some parameter u, alpha depending on parameter u, where u belongs to some parameter space. Then we get holomorphic bundle on c star, h bar,

and your parameter space. No flat connection again. But still one can make a situation when you get some part of flat connection. I say that my family is isomonodromic.

If on a total space, if you choose some global one form,

it's kind of total space of a family, which fibered to u, this fibers of u, this would be x, u.

Get form which is closed globally. And said that alpha u, for any u, is restriction of alpha global to the fiber. In this case, if you fix h bar,

we get kind of the same periods for one form. We get isomonodromic connection. And so we should expect Gauss-Mannian connection in direction of u.

And what we get, we get a family of flat connections on parameter space, which holomorphically depends on h bar. So in h bar will be still no connections.

So connection will be in this product space, only in this direction. Sorry? Ah, OK. This is a definition of? What definition? Definition of isomonodromic. Isomonodromic, it's actually not a property, it's a data.

It's a choice of global one form, which is closed on the total family. Because the restriction to fiber gives our forms.

So the Gauss-Mannian connection is on the type of homology you discussed?

On this homology, depending on parameters, yeah. You can see that. But then we have this comparison isomorphism in wall crossing structure. It describes jumps, yeah, in comparison isomorphism.

Because I have some bad h bar when

I don't have this isomorphism, so it's an open dense subset. And this isomorphism jumps. And in particular, so isomorphism between one, between sum of beta and pi naught of z, u.

And z, u is quick two points of set. And consider, again, this homology of this z, beta, u, and h bar, tensoring C. It's identified maybe in two different ways if you go through two sides of the wall.

This h dirham is modified with the notations I have. It's from two sides of the wall here. And then I get automorphism of the space, of the space.

And the whole thing is described using formalism of wall crossing structure, which I explained last time. We need some gradient algebra, local system of graded Lie algebras on, again, on my parameter space.

So the Lie algebra will be dependent on h bar and u.

And graded in gradient lattice depends only on u. It will not depend on h bar. What will be the gradient lattice? Yeah, that's kind of a tricky story.

It will be a mixture of two things.

Just maybe a better notation.

z, u will be zeros of r for u. I want to put up index because low index is denoted by some connected component.

So the gradient is the following. If I have a path, just before going forward, I have this thing, which is, let's m denote by number of connected components. It will be something isomorphic to zm minus 1, this thing.

And it's root lattice of a series a, a minus 1. It's like I explained for the case of function, you get. You should think of them as homologic classes of that would be a good thing.

Yeah, it's in quotes two endpoints of the path. So if you have a path connected to components, you get some vectors either. If you connect components with b1 and b2,

the path then should be vector, or if contains with itself, it will be zero vector. And gamma 0 u, it's the homologic class of the path. I forget the endpoints. You can see the homologic class of the path.

And it looks a little bit excessive grading. Because if you know homologic class of the path, then consider the boundary will be h0 of this 0.

You get elements here. But I want to kind of keep it separately, because it will include the case of form, and it will work nicer with this situation. So the central charge, h bar will be 1 over h bar of zu.

Where zu maps from its kind of trivial will be trivial on root lattice. And given by integral, and on this gamma 0 u is given by integral of form alpha, which is well defined.

OK, so we get Lie algebra Swiss graded by things. It's actually a local system, again, some open part of u when a number of connected components stays the same.

And what will be the Lie algebra?

This Lie algebra will be g h bar u.

It will be block diagonal matrices, kind of endomorphisms of direct sum of this h beta, as in Russian numbers.

So it's kind of matrix algebra. We consider the block matrices, because we have direct sum decomposition. Translating by seriesing, maybe just modelled by torsion here as well. Sorry. This group I want to modelled by torsion.

Kind of group ring of this lattice. So this Lie algebra, it's matrix valued functions

on a multidimensional torus. Obviously graded, because each block belongs to some grading in root lattice, and by some monomial here.

One can repeat this story, what is a wall-crossing structure. Namely, walls will be points in this parameter space. When the central church of some element of graded lattice is positive real number, and then you

should have such elements sitting in case of a subgroup on this wall. So it's like a strategic condition. But here is a big difference, what I explained last talk. Last talk, Lie algebra was finite dimensional.

Now it's infinite dimensional. But each graded component is finite dimensional, it's algebra is infinite dimensional. I lost a little bit. What was the role of this definition, capital ZUH bar? Where does it appear in this? You wrote something which is like a functional.

Ah, it's a functional ZUH bar. I have a local system of Lie algebras, and I get a local system of homomorphism of lattices to C. That's it. I just get graded lattice to C. Maps to C in some continuous way, depending on.

Yes, and how do you grade the endomorphism of the direct sum? Endomorphism of graded sum, you grade by A minus 1 lattice, home from one guy to another. It will be difference between two base vectors. Yeah, so it's kind of such things.

Yeah, but you see this algebra is infinite dimensional, and one should take some care about it. So this wall crossing, I define what some properties of wall crossing structure on specific kind of slice. I consider C star multiplied by some specific point.

If I restrict the wall crossing structure, what I get, I get kind of maybe infinitely many rays,

where I get, should put my onto your transformations, and should get elements of some Lie algebras. So wall crossing on such things,

just before going on, if you consider wall crossing on the C star times, I think it's given by, it's just collection. It's a collection of elements of alpha.

And this h bar is 0 of various components. It's just a collection of these elements of all possible elements in my gradient lattice, such that ZU h, ZU whatever of gamma is not equal to 0.

Because in this situation, I don't have any associativity property on this kind of copy of R2 minus 0. My walls will be infinitely many rays,

and the rays will be arguments of such numbers. And there's no constraints whatsoever. And the sixth definition at wall crossing structure on this C star times U0 has support property.

If there exists quadratic form on what? On my gradient lattice times R, such that restriction

of quadratic form to the kernel of my map,

which is typically co-dimension two subspace, is strictly negative definite for any gamma such that alpha gamma is non-zero, the quadratic form of gamma

is strictly positive. Where is U0? U0, yeah. So we'll get, we'll just make a break soon.

So what, like imagine if gradient lattice is rank of 3.

Or maybe gamma 0, because in part it's really relevant. Rank of gamma U is 3. So get map from Z3 to C. And so you can imagine integer points in R3, objects to plane.

And this support property means roughly the following. You get point 0, and you consider some kind of cone here, which surrounds the kernel of projection Z. And the support of my collection of elements

are such that only elements of lattice, which lies in this circle part of the complement. Is it infinitely many small gamma, or? Oh, infinitely many. For all, yeah, for all. All, yeah. So it lies in kind of this discrete set.

Yeah. Why is this condition? Yeah, it's in various situations, because this wall-causing story appears to various stuff. That one can use using some differential geometry. For example, here, imagine like X is some Keller manifold. And we're interested in gradient lines for real,

for interesting gradient lines of real part of exponent 1 in my knowledge bar alpha. You can see this, make its vector field, the real vector field, consider gradient lines.

And then as a claim that this homology class, if you bound the length of such a gradient line, you can bound the integral of any form, any closed form, because of some compactness. And to eventually translate, you see that if you bound the length, you have only finally many choices

and get some uniform picture like this. Yeah, so it's some very root, some estimate. It holds in real life. And yeah, so it's one condition. And the second definition, it's wall-crossing structure of the same story is analytic if it has support property.

And moreover, norm of A gamma, which is a matrix, essentially,

it's monomial, it's a matrix, is bounded by C1 times exponent of C2 times norm of gamma. Or you can replace by support property of norm of 0 gamma.

Because support property means it's essentially one bound, essentially equivalent. And this bound, it's more difficult. It, again, should fall from differential geometry. It's roughly said that number of gradient lines should grow exponentially. That I don't know how to prove directly.

But it's some kind of backdoor one can argue that it holds. And maybe if we can break it, I just

want to go to essential part. Yeah, so it looks like a stupid estimate, support and growth. The claims are falling. Analytic wall-crossing structures on the same thing

can be describing kind of coordinate-free language in the following grades. It can be defined in kind of infinitely many ways, depending on some discrete choice, by the following data. So we get matrix-valued functions on C star to power

n. So let's invert C star to power n, on which this function is called in a way, to some toric variety. And suppose this toric variety contains a chain of CP1s,

chain of CP1, which are toric orbits. So it will be fixed points, and it will be one-dimensional toric orbits. And then you get some chain which is called C.

This will be some singular curve. We take some analytic neighborhood of this chain. And I consider a bit sloppy. I consider vector bundle. Maybe it contains a kind of boundary divisor.

It still has some divisor with normal crossing around. And what I consider is a vector bundle.

3-infinity intersection, U. This is UC. It's analytic neighborhood. And consider a vector bundle, what's called E, on UC. Essentially, there are lights on this divisor,

whose restriction d-infinity, it's a complement to the open orbit. Said that restriction to this d-infinity is identified with some local system, which we know

a priori, because it's homotopic equivalent fundamental group of a neighborhood is Z. So we should describe what is monodromy. We have this betic homology depending

on the alpha on connected components. But also depends on h-bar. And h-bars, they form a local system. So this local system should be homotopically C star

is number 2S1. It's neighborhood of this thing has also fundamental group. So what does it look like? This analytical crossing structure, which is a collection of these elements which support properties and the growth, is the same as a holomorphic bundle on the neighborhood of this chain

of CP1 and some sort of a variety, with some trivialization. Yeah, yeah, with some, yeah.

So yeah, so it's a kind of geometrization of this notion.

This vector bundle should be identified with local system, which come from this direct sum story. No, no, no, of all crossing structures is this Lie algebra. I have this local system of Lie algebras.

And yeah, yeah, so it's an all concrete story. Maybe I think it's time to make some small break here. If we consider this intersection device, I have my D infinity intersect with Qc. It's homotopic equivalent to my curve C,

which is just union of spheres, and it's up to two cells. It's homotopic equivalent to S1, which is homotopic equivalent to C star.

And on C star, we get local system. Yeah, so it's up to two cells. But this may be more complicated than the graph, right? No, no, no, no, no. I said that it's a chain of toric CP1s. So I got a cyclic in the wheel.

I have a wheel, a closed chain, yeah. So cool. What's a chain, huh? Yeah, yeah, yeah, so it's a description of local system is a bit complicated. But why all this happens? Let me just give you some root explanation.

So I have this a minus 1 right, but I have a map from whatever is U is integral form to C, which is R2. I have a map from lattice, which is kind of Zn to R2.

I have an additive map. Then consider dual maps from R2 star, which is again R2, to what? To Zn dual times R, which is Rn.

I have a dual map. And this can be thought as, assume that it's embedding. So what happens in dual space, I have a real plane.

Kind of R2 sitting in Rn. Now what I do, I cover my plane, but I have also cells lattice structure. I cover my plane by convex polyhedral cones.

I say that it's R2, which lacks in the union of convex rational cones. And there are some conditions, but roughly it's a picture that it should be kind of, you divide by many small sectors,

and you cover by small cones around. And if you get such a situation, you get a fan. And if you get a fan, you get a toric variety.

This will be my toric variety Y. It depends on the choice of my convex rational cone. This toric variety will contain naturally a chain of P1, because this top degree dimensional cones will correspond to points in toric to zero dimensional orbits.

And co-dimensional phases correspond to CP1. So automatically, at infinity, you get a wheel of CP1's. But now, you do the following. But this wheel is not a wheel, right?

It's more complicated. No, no, no, no. Fixed points only correspond to O. I don't have any, it will be not complete toric variety. It will be some kind of quasi affine, whatever. Because CP1 responds to a base, right? No, CP1's correspond to co-dimensional phases. In toric varieties, you get a fan, open cones

correspond to fixed points, and kind of dimensions opposite. And CP1's correspond to co-dimensional phases, so they touch each other. And it contains a union of CP1. But when the T1 is accepted, there will be mainly CP1's intersecting the same point? No, no, no, CP1's correspond to maybe just other CP1's.

For each point, you get two CP1's. It will be left and right petal. I have, kind of, really, essentially, two dimensional picture around. And then we'll just draw, kind of, get something like this. This corresponds somehow to the intersections

of two copies of CP1, like if you have an edge, right? Yeah, it will correspond to CP1, yeah. Its infinity will be CP1, kind of transversally. Yeah, so roughly speaking, one can think the following. C star to power n maps by logarithm of norms to Rn.

And if you consider paths in C star to n, which correspond to paths which go along this whole open domain, it will converge to one point in toric variety. But if you go in the middle, the limit will be undetermined to the point in CP1. So the path just gives a group? Yeah, yeah. So you get the toric variety. And now there's some kind of easy calculation.

You get this curve. And we consider formal neighborhood of curve. It's kind of formal scheme. And consider hypercommodules of this formal neighborhood with coefficients of the shift of functions

which vanish on the divisor at infinity, or vanish on the divisor. You consider hypercommodules, you consider hypercommodules, no, commodules with this shift of ideals. And they claim it's 0 if i is not equal to 1.

And if in H1 has basis, kind of topological basis, corresponding to integer points, non-zero integer points in gamma supported on something

like this, this complement. So you can see the kind of dual cones. Take union. And you get exactly H1, and no other comology around. Maybe I'll just give you some simple example.

Suppose my lattice is really take care of vectors in gamma satisfying support property for certain quadratic form. Oh, I'm sorry. In integer points.

A point in gamma satisfying support property. So it's really easy calculation with comology. I can show maybe just really stupid example. Suppose my lattice is z. And just two dimensional, so I don't really have this projection.

But I still need a fan. And my fan will be, this will be my fan in z2. You get just four domains. So the variety is, the corresponding variety is cp1 cross cp1.

Kind of 0 infinity. Support property means that it belongs to some set which may be bounded by quadratic form. Yeah, so this quadratic form was a little bit eclectic.

So there exists a constant. Is it something like what you have there, the condition? Yes, on quadratic form, yeah. On the quadratic form, but what is the gamma? You want a point in gamma, not a quadratic. No, each one has a basis corresponding to set of points in gamma, which

are kind of possible candidates by the support property when a gamma is not 0. That is a q gamma is positive. Yeah, yeah. Possible, OK. At some point you divide by the Planck constant. No, no, no, there's no Planck constant.

I discussed this. No, you see, it's the isomorphism of local system. It's a local system on s1, which is magnified with a local system which depends on a. Oh, maybe you're right. Because maybe divide by Planck constant, Planck constant will be argument. Planck constant will be zu of gamma.

Let's click this. Your vector in my gamma is, my gamma is the vector in my lattice. You can see the vectors in my lattice, for which the support

property. And consider, no, no, here I don't have anything. No, no, here it seems clear. OK, yeah. And yeah, one can make simple calculation. For example, yeah, just want to jump between two things.

I have a fence of varieties CP1. I can see the kind of neighborhood over this view of CP1, which is not a fine surface. It's not a fine surface. And even neighborhood. And the only functions are constant. And the functions vanishing on these things have only first homology.

If I calculate kind of using some check covering, and essentially what happens in lattice vectors, you can see the maybe integer points in this half plane, this half plane, this half plane, and this half plane. And remove, it will be one part of check complex,

and others will be octants. And then we get almost zero points here. So something similar happens in higher dimension. Of course, here we have a lot of choice. We can make a smaller fan here. And on rule space we get a larger support. And eventually we can cover a support property by this thing.

So there will be some ambiguity. So it's not a unique description, but there are many ways to describe in this space. And you compare two different ways by some blow ups. Yeah, so you kind of replace. And this is kind of a basic thing from which all follows.

Because if you consider the deformation theory for this equation, we get vector bundle in neighborhood. But to realize somewhere, but now we start to deform it. Because it's deformed by something trivial on the divisor. So the matrix-valued function vanishes on the divisor. And deformation theory is given by H1,

this level of tangent space. And see that you get exactly the same data as for these elements of my matrices. It corresponds to monomials belonging to such a domain. And then one can make it really globally. And this convergence, this gross condition

that there was some analytic property implies that it can go from power series to actual neighborhood, 21 correspondence. OK, so you get some kind of bizarre analytic spaces which are neighborhood wheels of CP1s. And now if we move point and want to satisfy support

property, we have this associativity laws for wall crossing when things go together. Yeah, it's all complicated business. And then associativity law for wall crossing structure

is equivalent to the following thing. This bundle stays the same. You see some e stay locally the same. But what we change? We change in bednik of R2 to Rn when

we identify the things with this wall crossing structure. But analytic coverage stays the same. But then if you go to different domain, we can use different fields to make some blow ups. But that's roughly the picture of what happens.

Now, so all this complicated wall think are embedded to some kind of nice analytic language. You get holomorphic bundle in some open variety. Kind of a local system, just a multiplication of the tables.

Now I'll just be very brief. Now we want to jump to infinite dimensions. Some people were not present on the last talk,

but I cannot really help with this. Now, what I explained, for some infinite dimensional varieties, one can try to calculate with holomorphic conforms.

One can try to define all the structure without definition of what a global dioramic homology and so on, using just this wall crossing business. And the basic example was the following. I have a holomorphic symplectic manifold. And I have two complex Lagrangian submanifolds.

Again, holomorphic Lagrangian submanifolds. And my space will be infinite dimensional. It will be space of paths, these endpoints on my Lagrangians.

And one form is closed one form on x. Infinity is given to integral of two form over the path, in some obvious sense. It's a closed one form.

There's also some issue of orientation and so on. And eventually, we should get homology depending on h-bar, which is holomorphic in h-bar C star,

using these walls. And if one has isomonodromic deformation, one should expect a flat connection. You get some parameter space, a whole thing is a parameter, and get flat connection in U, holomorphically

depending on h-bar. OK. So that's what one should expect, naively. What is capital H-bar? So what we use, we use vanishing cycles for intersection points.

And then we should use whole crossing structure, coming from path in path space. Kind of gradient lines, which we say is holomorphic disks, and all this complicated story. And what is the example of isomonodromic deformation? Should have a holomorphic bundle.

Vector bundle on C star, yeah. How is it constructed using the?

All these walls, yeah, which you have. Constructing using whole crossing structures, and all this business. Because critical points of zeros of h-bar, zeros of infinity, are constant maps identified with its finite dimensional space.

Iso, then it's called Iso? If we have isomonodromic deformation, yeah. And what is the example of isomonodromic deformation? Is the following. If my manifold is cotangent bundle, then omega is differential of Liouville form. L0 not exact, but L1 will be, L1 depending on a point,

will be cotangent space to the point, and x and u will be x. And parameter space will be, so we have cotangent space

to x. We'll have a zero and intersect with some vertical one x and start to move this L1. So then you should have a bundle on C star cross u? Yeah, I should get a bundle on C star u, on a bundle, should have bundle on C star cross x,

which is holomorphic here and flat here. Some kind of holomorphic float homology. Yeah, it's all this, very nice, and we want to try to see how it will work.

And then we'll have a very interesting trouble. Because in the definition of all crossing structure, consider something like gradient lines for this one form, which will be holomorphic disks, pseudo-holomorphic disks. So we're interested in general in pseudo-holomorphic disk

related maybe L0 and maybe L1, maybe depending on point u. And we have this holomorphic disks. And they are in general isolated, but when one parameter family gets some boundary, and boundary space of disks has the disk

can degenerate to sequence of two disks, which is responsible in a whole crossing picture to the rule, if you remember, something like you have kind of number of the disk is jumped by these things.

You get kind of product here, which correspond to product in this picture of two compositions. But if you get some new phenomenon, which is you never see in finite dimensions, new phenomenon,

boundary over space of disks, if you want to prove that whole associative things, will not hold. Also contains also another type of boundaries.

And you could have such degenerate disks can appear. Yeah, from boundary, yeah.

Real bubbles from boundary, yeah. Yeah, so it's trouble. So you get a trouble comes individually from L0 or from L1

because it means that you have a holomorphic disk with a boundary on L0. Yeah, it never happens in two situations for L. So there's no disks with a holomorphic disk with a boundary to the holomorphic disk

with a boundary on L and strictly positive area. In two situations, if L is exact, so in my last lecture, I explained the case of exact L

and then one form was differential of a function. When L is exact, so it means that we express two forms differential of one form and one form on L is differential of some function. M is exact. Yeah, it means that M is exact and L is exact.

It's some structure. It's not a property. It means that M is exact and so on. So why is the integral 0? If considered integral of the omega of the disk

is integral over d omega of any disk is integral of omega of the boundary of the disk is integral of the f of the disk and is 0 by Stock's theorem.

But area is a following situation. Area is a real part of 1 over h bar of integral omega of the disk. Yes, we get a contradiction. For any pseudo holomorphic disks, the area should be strictly positive, but it's 0.

So it's one case, but there is another case which is also funny. Second case is when M is cotangent bundle, L is a graph of closed form.

Then the integral gains 0, but by different reasons. Because you get integral of disk form omega, the beginning is the same as integral of d eta of omega of a disk is integral of eta of a boundary disk.

It's integral to the alpha over, you get projection from cotangent bundle to x itself. You get projection of a projection of dd.

It's integral of d alpha or projection of d. And this is 0, because d alpha is 0. The end is different. The beginning of the argument is the same,

but the ending is slightly different. So there's a completely different story, which explains that for the case of one form, one can go to this infinite dimensional story or finite dimensional story. It will be the same business. So I claim this is kind of things with disk and so on. It will be the same as finite dimensional integral in my story.

But in general, except of these two exceptions, except this kind of closed form or exact thing, we should get this trouble. And what's the meaning of this story?

So what goes on here? How to understand the wall-crossing structure? It looks like everything will be broken.

But this explanation, what happens? It claims we will get a wall-crossing structure in a larger Lie algebra.

We have this previous Lie algebra, but we have now a larger Lie algebra. So before we have this guy, before we see the trouble, we have this Lie algebra. But after the trouble, I think we should understand we have different Lie algebra,

kind of matrix value, new Lie algebra. It's kind of old Lie algebra, and we take semidirect product with vector fields on torus.

These are matrix-valued functions on a torus. And these are vector fields on a torus. You can see the vector fields. So new gamma-graded component will be something like this.

We take y to power gamma. And you take maybe yi times du dy. It will be basis of new graded components of the larger Lie algebra.

It's again graded by letters, essentially by the same letters. And so it's kind of what is going on. And I explained before you had kind of theorem

how to identify a wall-crossing structure on a C star times point as a vector bundle. So before, analytic wall-crossing structure on C star across a point was the same as a holomorphic bundle on some neighborhood

of a chain of CP ones, plus some trivialization or identification on some divisor.

But now I think one can kind of repeat the whole story. This analytical crossing structure is largely algebra. It will be the same as holomorphic bundle on something else.

Not on a neighborhood of a chain of Euler-Torek varieties, but just some complex manifold which contains a wheel of CP ones and have some stratification and maybe some trivialization. And just in the first order, coincide the historic variety.

Then you modify by some cross-vector filter. You change coordinates in your different patches and glue a new manifold. Yeah, so it's in some sense one can say that before we have holomorphic bundles,

we have only gauge theory, but now we have gravity, kind of diffeomorphism group. Yeah, so it's mixture of two. You see modified, it's a germ of complex analytic variety

which contains wheel of CP ones and it's stratified like Toric variety. OK. And maybe normal bundles to strata are trivialized to this CP one to keep a kind of first derivative.

And the rest will be the same. Yeah, so you get some kind of, and if you move holomorphic structure, you locally do not change this guy. Now that's essentially what Gajote-Murnetsky kind of discovered. It was something called 4D, as it's called,

2D-4D wall crossing structure. And 4D was for diffeomorphisms or maybe symplectomorphisms, in fact, in real life.

And here it was previous things with matrix valued functions. For this, as you can see, it's Chekote-Vaf, something. Yeah, so it looks like it's trouble.

These extra disks are eventually encoded to some little bit larger Lie algebra. Yeah, in fact, one can check it formally. Yeah, but this can give us some kind

of rough explanation of where it happens

and why we should have these changes of coordinates appearing from nowhere.

If you get m omega and h bar, then we can associate a real symplectic manifold. It will be m, a real part of 1 over h bar omega.

And add something called b field, which will be just closed, purely imaginary form, which will be i times imaginary part of 1 over h bar omega. So we get symplectic manifold plus purely imaginary closed form.

And in such a situation, again, one should expect something like Foucault category. What roughly is this Foucault category? Objects are Lagrangian sub-manifolds in m,

omega real, Lagrangian with respect to real form, plus bundles with connection on Lagrangian manifold. Such a curvature of the connection will be b field multiplied by identity operator.

Yeah, that's a rough definition of the object. And homes are intersection points. Or intersection points, maybe take homes within bundles.

And then it's also the a infinity category. Again, consider some holomorphic disks. Take some holomorphic disks and integrate this. You get sum of a disk, integral of a disk, omega plus b.

You have real symplectic manifold with respect to form. Then we can try to make this Foucault category. And naively, it looks like a kind of holomorphic family of categories in h bar of categories.

Yeah, that's a rough point. But now let's consider L0, L1 are holomorphic Lagrangians. So now what is this sum of e to the minus? Sum of all holomorphic disks.

It will be contribution of some infinite terms. Yeah, it's a really long story if you don't know it. It's a bit too late to explain right now. You get this trouble.

Now we have two holomorphic Lagrangian manifolds. And just before going on, I want to say the following. In fact, here in this Foucault categories,

one can get a trouble. L and e connection is not an object. If there is something m0, if some m0 is present, m0 is responsible for holomorphic disks.

If there exists a holomorphic disk for something, another complex search compatible with the real. Sided boundary of the disk belongs to L. And area is positive. Such things destroy objects in Foucault category.

It's a well-known thing that's holomorphic disk bounded. But pseudo-holomorphic disk for some kind of another structure. Maybe it's some kind of j-holomorphic disks. And in illustration, we start with complex symplectic manifolds. Say for simplicity, it's hypercalor manifold.

The fact is, it's easy to see that such holomorphic disks, because the structure depends on h-bar, do not exist for almost all h-bar. For h-bar, argument h-bar not equal to argument

over integral of two form over some kind of class of the disk belongs to relative pi 2 and L-holomorphic. For L-holomorphic, this E will be flat bundle.

There is no such disk. So it means that one can put as object of a category, arbitrary holomorphic Lagrangian, arbitrary local system. It will be not abstracted object of Foucault category.

Now if one have two guys, and you try to calculate HOMs between another thing,

you get two objects of Foucault category. When you want to calculate HOMs, you should calculate things coming from intersection, but also the differential coming from M1, from holomorphic disks. And again, holomorphic disks will not

exist except for countably many bad directions for the same reasons. So it looks as for generic direction of h-bar. You get very nice things. You get a list of object of categories. You know HOMs identified with local vanishing cycles.

But then if you cross the wall, something bad should happen. Category depends holomorphically. And nobody says that if you have your object, if you cross the wall, you get the same object given by the same data. So it means if you consider rank one local systems,

you should kind of change of coordinates. So it means that you introduce outside of also introduce coordinates in space of some domain space of objects. But now you change coordinates. And this will be different reasons which come from this story.

And eventually, and also with HOMs, you get these jumps with identification. So it's also kind of morally follows from consideration in Foucault category where you get diffeomorphisms. And eventually, it gives kind of a new definition of what is new definition of a resurgent functions.

I'll just try to do, kind of try to explain it in most named term.

Yes, so you get, you see this hourly algorithm is built from two pieces. You get a change of coordinates and matrices. And what happens is if all the resurgent functions, you get some function in one variable, divergent functions.

Yeah, so there are kind of two steps in Riemann-Hilbert problem. First, we should kind of even imagine we have kind of finitely many rays.

And we get some variable y1, yn. And the first problem is the following. Find functions y over h bar, where h bar doesn't belong to stock's rays. And satisfying some conditions will cross along the way.

The typical conditions are following. Suppose you get two functions y1, y2. Here it's kind of minus and here it's plus. And suppose, for example, y2 minus is equal to y2 plus. But y1 minus is equal to y1 minus times 1 plus x

point 1 over h bar, y2 minus. So we have functions on maybe two functions here, two functions here.

And if you consider jumps, given by such formulas. It's a typical wall-crossing transformation y1 minus. It's kind of exponential decaying guy in h bar. Yeah, so you get equations like this. So y1 plus is not equal to 1 plus.

Oh, 1 plus, yeah, sorry. Yes, this is a little bit kind of non-linear part of jump formula. But then we get jump formula for integrals. It's how we parameterize local system. But we also should parameterize integrals. And we get something like yy minus is equal to yyj,

with some different indices. You get something like whatever, yj plus. Maybe it's plus again, 1 h bar. Again, multiply by x point 1 h bar times yk plus something like this. Then we get secondary wall-crossing formulas

for matrix part of my Lie algebra. And if you consider solutions in this guy, h bar will be function dependent only on sectors.

But asymptotic expansion and h bar will have kind of finite limit. So you get some form of power series expansion. This form of power series expansion doesn't depend

on the stock sector, because we change by things with zero tail coefficients. So we get some universal divergent series, some class of divergent series in one variable. And I think it follows from all those geometries that you can prove that they are Borelli summable, if you make a Laplacian form.

So we make something like c over nj over n factorial. Maybe some dual variable, some kind of t to the power n. It will have endless analytic continuation with countably many singularities. So what people expect for resurgent functions. But this I want to expect.

I think the product we have is yet to extract it just for this geometric description. We get analytic variety, which is neighborhood of chain of CP1s, and analytically vector bundle in neighborhood. And then the series will follow from nothing. Which data will it depend on?

Yeah, it should depend on, essentially, this deformation of complex structure in neighborhood of a real CP1, and deformation of vector bundles in this neighborhood. So it will be some coordinate free, in a sense, description. OK, thank you.