And I was lucky to met Maxim long ago. So we met the very first Monday of September of 1980, when Maxim just become 16 like a week ago, maybe like 10 days ago. And at age 15, he entered Moscow State University.

At age 16, turned 16, he came. There as a freshman, he came to Gelfand Seminar. So that's where we met. Actually, I heard about Maxim as about this great high school kid two years before that, but didn't see him. And so we were talking ever since we

get to the same point of space time. And it was kind of incredible joy of my mathematical life to be able to talk to Maxim and to be able to discuss what happens in mathematics, what is mathematics, and how to do mathematics with him and what's going on. And I should say that there's many things you all know,

Maxim. But among other things, it was great that Maxim always thinks about things in a very simple way and a very non-technical way. And so when you have a chance to talk to him, so you get the picture in the best possible way. And he explains this picture using

the minimal number of words and sometimes even less than that. And on the other hand, I mean, it's his universality. So again, we all know that. Excuse me, this is for you all. I kind of imagine when I was flying here that Maxim met three great mathematicians, Oller, Thurston,

and Grotendy at the kind of beginning of his career. And he would come to them and say, I have some simple story to tell you. So he would tell the story. And each of these mathematicians would think that, yeah, this is a guy who picked up my style. This is my style mathematician. But then notice that these three mathematicians probably

would never talk to each other. So that's kind of universality of Maxim. And this universality was present essentially from the very beginning. I cannot say like 34 years ago because he didn't know maybe too much mathematics. But definitely in a few years, so his approach to mathematics was kind of universal. There was no divisions on branches of mathematics,

no geometry, no algebra. So it was all together and attempt to try to understand the simplest things, simplest terms. And at that time, some citation from Maxim, I remember if somebody approached him, me for example, asked some kind of question, which wasn't quite the kind of special Maxim said,

I'm specialist in general questions of mathematics. And so you remember that? This somehow disappeared later on as far as I understand. I mean, you no longer say this. So this universality and simplicity

was from the very beginning, so somehow it was kind of internally in him from the very beginning. And again, so I'm very happy that I had a chance to somehow to see how Maxim grows and talk to him. And so I can only hope that for the next 34 years, I would be able to understand what you're talking about.

OK, sorry. Let me start now the talk. So I'm talking about which quantum field theory.

So I'll start with analogy. So well, so let's suppose that we have M, which is topological threefold. Then let's suppose that we have two loops here. A and B, which are homological trivial.

I mean, sorry, I'm writing. The class of A and the class of B equals 0 in H1 of M. So homological trivial. Then we can talk about linking number.

We can think about the linking number. Let's denote this like AB. So this is some integer. And A and B. And so this in the 1, 1, H2, 1, H1 of T, 1, and A.

I mean, let's suppose just homological sphere, or just say the same. Yeah. So and then let's go into this next one. So this is actually the simplest correlator in Chern-Simons. Like Maxim was considering this 20 years ago. This is the simplest correlator in a TQFT.

And then you can talk about Chern-Simons as Maxim did and develop the whole theory. So this is topology. Now let's look at, I would say, analysis arithmetic.

You have a Riemann surface. Then let's assume that you have two divisors here. And let's assume that their degrees are 0. And so the picture is like that.

B, you have A. And then there is a notion of a Green function, g of xy. Let me remind you what is a Green function. Adjust the solution of the differential equation, d d bar of g of xy equals delta function of the diagonal.

So diagonal. Yes, it's a real value function which solves this differential equation. It's the same, because I'm right. There's a thing which makes it like this, but you don't see domains and such. So it's a Green function. I tried to use less number of words, but not that.

So it's a Green function. You're right. It's a real value function. Actually, it's a current which generally solves this equation. So here is a harmonic representative of the class as a diagonal. Otherwise, it doesn't have solution. And it was a great idea of Parshin and Arakilov, so it was suggested by Parshin to Arakilov,

to say that if you consider this Green function and evaluate it in these two divisors, then it's independent on the, it slightly depends on the choices here, like choice of mathematics. And this is a completely well-defined function, real value function. And the Arakilov said that this should be considered as an intersection number at infinity.

I don't want to elaborate further to that, because then you have to talk about algebraic curves defined over rational numbers and about height pairing. And then this will enter as a key. It's a function, it's a number.

It's a number, yes. It's a number. It's a real number. It's a function. It's a real number. Maybe I should say this is like G of a0, b0 minus G of a0, b1, and so on. It's a number. What's the problem with this?

OK, you know, the green function shows particularly, so it's 0 and false here and there, right? No, no. You take green function, it's a function of two variables, x and y. So for every pair of points on the curve, you have a number. Now let's suppose that you have two points which you counted as signs plus and minus. And then you extend this function by linearity.

So you get a formula like that. It's still a number. And so Arakilov suggested that this number is a kind of intersection. It's a part of hyperion, but it's kind of intersection index of these two divisors in the sense which is a little mysterious and the whole Arakilov series coming from here. Now let's do this analogy a little better,

just a little better. So we want to say that this G of a, b is actually a linking number. It's a Hodge linking number. Again, it's not clear what it means. It just works. But still the picture, the mental picture

which we have about this is the following. We imagine that we have some space x which is 3D. Now it's 3D. It still comes from our surface in a certain way. And then the space is fibrate. And the fibers are actual Riemann surfaces

over some space which is one dimensional, 1D. And so this is a classifying space of some group, a pro-algebraic group I'm going to talk about, Hodge-Galo group. And so it's not clear yet what this is. But the point is that whatever it is, it's like a bouquet of circles. In particular, there is one map of this single circle

to here. And so now talking about linking numbers, any point x in the original curve gives rise to a section. Let's call S sub x of this imaginary vibration.

So again, we don't quite say what the section is. But then we kind of postulate that if you take sections, these sections evaluated as a and as b, then their linking number in this 3D manifold supposed to be is defined as a Green function which

has been defined before. So it's just an analogy. But it explains what I'm going to do. So what I'm going to do, I'm going to take as a motivation the story which is going on for dimensional manifolds. And I'm going to develop something like that, but different, quite different, which is going for complex algebraic manifolds, where

the Green function will be the simplest possible thing which I get out of this. So the goal, I want to define this quantum Hodge field

theory such a way that, first of all, it should provide another language for the usual Hodge theory. And secondly, it is kind of Hodge version.

It can be viewed as a Hodge version of Chern-Simons on odd-dimensional manifolds. All right. So before, and again, the last thing. So it's not going to be some discussions about this field. We will be talking about some concrete things which

come in of this. And so we define by, we define as correlators. Now, before I go to that, I still want to make this analogy more precise. A little more precise. And so I need to explain what the Hodge-Galer group is.

And so I need to say what the Hodge theory tells us. So the next topic is, what is the Hodge-Galer group? Yes? Is there any relation between this linking number

and the way you define it as a correlator on the Riemann surface and something like WZW models, which would be called a work group? I didn't, Sergei, let me try not to go into this. Because I didn't even define anything yet. So let me get this vector first. Yeah, so otherwise, I will run out of time very quickly.

So let me start from some definitions, which many people know, that a weight n pure Hodge real Hodge structure is a very simple data. It's a vector space, V over real numbers.

And filtration on the complexification of this vector space, VC, such that this vector space, VC after that, is presented as a sum of its components, Fp of VC intersect

Fq of VC bar, where p plus q equals to n. This is a weight n Hodge structure. And they form categories, category, and the category of pure Hodge structures

is equivalent to the category of representations of group. I mean, algebraic geometries will kill me unless I write this thing. What it really means, this algebraic group whose complex points are just c star cross c star.

And it's over real, so the complex conjugation acts by interchanging the two. So it's basically a representation of c star cross c star with some reality conditions. That's why you get this p and q numbers. And then this is just a pure Hodge structure. That's what we find in the cohomology of algebraic

varieties, which are compact and smooth. And then there was Deling, who said that there is a thing which is more general, works for any varieties. It's a mixed real Hodge structure. And so according to him, this is just a vector space V over R. And it has two filtrations.

The one filtration on the same vector space is going up. It's called the weight filtration. And another filtration is going down. And it is on the complexification of this vector space. So this is a Hodge filtration. So it's linear algebra data.

And the condition was that if you consider the root wn of this V R, this is pure R Hodge structure of weight equal to n, for any n. And then it was a very surprising theorem,

if you think about it, that this guy's form an abelian category, proved by Deling. And then using this fact to gather his observation, that there is a fiber functor. So let's call this category of real mixed Hodge structures. And again, Deling proves this is an abelian category.

This is not at all obvious. And I mean, I don't know how Deling get to this idea. So I mean, from some argument, from some ideology of motives and so on. But I mean, it's not that easy to see this right away. So there is an obvious functor, just as a vector spaces over real numbers, which

are assigned to this Hodge structure as the original vector space. Now we are in business because as Tanakh and Groton teach us, we can take the aftermorphisms of this tensor functor. And this aftermorphisms form a group, a pro-algebraic group, which is called G Hodge. So this is the definition of the Hodge group.

And there is a structure theorem, which says that this group looks as follows, that this group G Hodge projects down to the group GM C over R, the one which was introduced there. And the kernel of this map is a pro-unipotent group, U Hodge.

And how we see that this is true, there is a functor. If you just consider the associate graded for the weight filtration from real mixed Hodge structures, by definition of what mixed Hodge structure is, you're going to pure Hodge structures over R.

And then, of course, there is a functor back. And this means that there are two maps between the corresponding groups, between the Galois group of this category and Galois group of that category. And this means that not only we go this way, but there is a section here. So, OK, now one can summarize this whole discussion

by saying the following, that this second functor taking associate graded for the weight filtration is the equivalence between the category of R mixed Hodge structures and the category of LiU or just U Hodge

models in the category of pure Hodge structures. It's a little, maybe a little complicated way to say that the original category is category representation of G Hodge. But you can phrase it in this way using the semi-direct structure of G Hodge.

Those are just representations, category representations of the unipotent part, but in slightly different, richer categories and vector spaces. All right, why we need this? Let's go back to our linking numbers. So now I can say more precisely what all this analogy is supposed to mean, this one.

So if you just take first dimensional homology of this curve X minus A modulo B with real coefficients, then this is, as Deline tells us, it's a real mixed Hodge structure.

And therefore, it has to be understood as a representation of this Hodge group. And so first of all, we have to take associate graded for the weight filtration. And it has three pieces. Two of them are very easy, just one dimensional vector spaces in degree, I mean, of weight 0 and minus 1 minus 1. And in the middle stays a homology of the compact curve.

And then, so this, just this guy is element in this category, in the category of pure Hodge structures by definition. Now, if you want to have a mixed Hodge structure, we need to act by some unipotently algebra here, which makes weights go down. And the only non-trivial operator,

which we're going to see, is this operator GAB. So the little theorem is that this Green function, which I thought about as an intersection number, or RKL intersection number, or as a linking number, there's nothing else but precise description of this mixed Hodge structure by a single operator. Oh, and then one can explain, I mean, more now just

what this has to do with linking number, take an exercise. I'm not going to talk about this. So I assume that you have a family of circuits of an actual circle. And then exercise, define linking numbers the same way. So you can do this as this kind of monotony.

So what this operator means, I want to emphasize this again, that over this, oh, here, it's good. Over this imaginary, I mean, not quite imaginary. Now, if you define this group, so this classifying space, which topologically book here circles, and there is a single circle here. And so what we're doing, we're kind of taking monotony around the circle, and we get to this Green function.

So that's what it means. That's another way to interpret this. OK, now, after this kind of easy analogies, let me go to a little more sophisticated analogies, which grow from here. And so again, we run analogy between arithmetic

and geometry, or analysis. So then the next step, we have Galois world, and we have a Hodge world. And let's suppose that we have some field k, and let me even assume it's embedded to c. Then here, we have some algebraic variety over k.

And here, we have some algebraic variety over c. And here, we have the Galois group, the absolute Galois group. Now here, we already know that we have an analog of that. So what is this analog?

So the point is that the Galois group acts on the cohomology, the Talcohomology, no matter what this is, of this variety. And so if you want to have analog of that, that's what Deling provides to us. He says that the cohomology of this manifold,

I mean variety of complex numbers with real coefficients, is a mixed Hodge structure over R. So this is Deling. And this means, as I just explained, that the Hodge-Galois group acts on this vector space.

But this is not how Deling constructs this Hodge structure. He does this linear algebra and uses linear algebra definitions. He never used D'Hoch. But if you go to arithmetic, the story is much finer because the same Galois group acts everywhere. I mean, if you ask a question, how did we get acting Galois

group on something like a Talcohomology, and the Grothendieck says that it acts on et al's side. And therefore, it acts on categories of et al's sheaves.

And therefore, because we can take constant sheaf, we come from here to the fact that it acts on the cohomology because this is just the x group between the simplest et al's sheaves. And because Galois groups act everywhere, it acts there as well. Now when you go to the Hodge story, nothing like that happens so far.

And there is no et al's side, no et al's side, basically. And so it was a dream of Bailynson. I can put it like Bailynson's dream, to have something like Hodge's side.

But he didn't quite say what it could be, but just said it's very unfortunate that we have et al's side here and nothing there. And so the proposal, which is the main idea of the story I'm talking about, is that one have at least some approximation to this Hodge side,

and it's given by this Hodge quantum field theory. And it generalized Green functions and so on. So that's a Cal proposal.

Let's assume, first of all, give yourself some data. Let's assume we have a map of complex algebraic varieties. Let's call this map p. So we want to do things relatively. And then the proposal is that there exists some structure, you can call it open string structure, where?

On, first of all, the derived category of all holonomic d-models on x, on original x. I draw here x, Carly. Now, again, as stated so far, it's

not clear what it means, just a slogan. But in particular, it includes, as a data which we kind of can construct and observe, some correlator functions, which I call Hodge correlators. The base plays a role, but it does

play a role in the properties of this situation. And actually, OK. So there is a Hodge correlators between holonomic d-models, between reducible holonomic d-models. And let me explain, actually, now more carefully

what this means. So first of all, you start, what does it mean to have open string theory data or something like that? So you start with a topological surface, which have holes. And in addition to holes, it has some marked points on the boundary, and it's oriented.

And the marked points are considered modular isotopy. So then, actually, it's convenient to shrink holes which do not have marked points to puncture.

So you might have puncture here, which also kind of consider red, and call special points. So I have special points. It's a puncture. I mean, the thing is no point. Go on.

Let's not discuss this. So special points means there's red points and this one. So it's marked points and punctures. And so what do you want to do? We want to assign to special points objects, actually, irreducible objects of our category. Let's say it's db.hold, but it could be something else.

And then the geometry says is that between two consecutive objects, there is orientation. So there is an arc which go in between two objects. So you can have some point s here and some point t here.

And then there is an arc which go in according to orientation of the surface. And then we want to assign to this arc home of as, the objects. Let's call this object as. So to s, we assign this between as and at. And people recognize that that's, I mean, I call this Open String Theorem.

I'm not sure how I can call this. But that's what happens a lot in, for example, the subject of paper of Greg Moore and Graham Segal, I mean, and Maxime and Kevin Costello working station, which like that. So that happens quite often, such setup.

And so now what I wanted to do, I want to say that these correlators. So we should have some correlators. They assigned to this data, to the surface, which is decorated by some objects.

And morphism told this data. So we should have Hodge correlators, which are something which lives on the base. So the point is that we should have Hodge correlators, which are assigned to s and some data for your main category. And this guy lives on b.

S is a topological input. It's a decorated surface. So this s, this is decorated surface. Yeah, that's a good point. And then it's not only decorated, but it's inhabited by some objects of some category who have some social relationships like this.

They have homes to neighbors, which live to the right. And so to this data, we associate something on b. And that's in the form of structure or is it topological, topological or even surface? No, this is topological. But the input is algebraic variety over b. And what you do is a puncture.

And what you do is some sort of a puncture. So I'm not going to do this full strength, of course. But I'm going to give some examples. So for example, who is our green function on a curve? So this will be as a correlator, which

is assigned to this picture. So you have a disk. And on the disk, you have delta function at x. This is little x. Delta function at y. Then you have, I mean, in our station, which I considered, I just have a trivial local system, constant local system here. And then as soon as you have four objects like that, constant shift maps to delta functions in obvious way.

And dual delta functions maps uniquely to constant shifts. So there is a data like that. And so you can ask, if you believe to what I said, you can ask a question. What's going to be a correlator of that? And the answer is this is a green function. It turns out to be. Now you can ask question even more generally.

What will happen if I do this like delta x? Green function is better than structure. So I started with a curve here. I started with a brick variety, which now I make a curve. Yes, now I am slightly cheating here, but I said that I considered delta function x and delta function y. And I explained that to talk about green function,

I need to have a little bit more of data. But if I replace delta x and delta y by this delta of divisors, then the statement is completely accurate. And I can make sense of this. I just don't want to proceed there. But more generally, you can have now x any compact, smooth, algebraic variety over c.

And you can have any two irreducible local systems. Let's call them L and M. They're irreducible local systems on x. And then, again, there are trivial morphisms here,

there are obvious morphisms this way, and there are obvious morphisms the other way. So this diagram gives you immediately some guide for which you can take the correlator. And then, again, I can postulate that this will be the green function, but now defined depending on this pair of local systems.

And I need to define what this is. So that's my next goal. So I need to define what this guy is. And then you will see what the ambiguities and so on. So before I do that, let me be a little bit more

specific about correlators of whom I want to consider in the lecture, or I want to define the lecture. So I want to consider the graded category, which

are called, probably not me, Carlos, the category of harmonic bundles of x. And so the objects of this category are semi-simple local systems on x.

And the homes are defined as, so if you take home between local systems L and M in this category, this is just the x group between these two

local systems in the category of ships on x. So it's a graded vector space. And why actually I consider this strange gadget? Because there's a very deep work 20 years ago. So Hitchen, Donaldson, and Simpson,

I mean, if you combine what they were saying, then you get the following result. If you have L, which let's say is a simple local system on x,

then Hitchen, Donaldson, so they proved that there exists unique harmonic after constant harmonic metric on L. And then Simpson, so he took this harmonic metric

and he worked out the classical Hodge theory package. As Carlos was saying, there were some revolutions. And so this revolution was to realize

that you can run Hodge theory on arbitrary local system, not necessarily on variation of Hodge structures or something like that. And so you can do that. And in particular, this provides the usual setup, d, d bar, and d, d bar lemma, and harmonic forms. So all this you can do on the Dirac complex of x

with coefficients in irreducible local system. Now, after that, I can tell you what the green currents are in this setup.

So the green functions, again, are solution of the differential equation. The d, d bar of some current equals the delta function. But we need to be just a little more careful

by setting this. So green currents. So given L and M as before, as on the right-hand side, on some variety x, which has dimension over complex numbers

n, there exists a current, not uniquely defined, which one can call the green current, G L M, which is a current on the n minus 1, n minus 1 distributions on x

cross x, with coefficients in a local system, which is basically home from L to M, tensor home from M to L. The only problem is that this local system lives on x.

So you have to pull it to the square. And you have to pull this to the square. Now you get local system in the square. So you take distributions at local system. And it satisfies the famous equations that 1 over 2 pi i d, d bar of this phi, of this G L M,

equals the delta function of the diagonal, which may be defined because we take delta function with values in some vector bundle, but there is a natural constant shift sitting there, because it's just home L to M from M to L, minus a harmonic representative of the diagonal. Now the cohomology class of this is 0.

And so d, d bar lemma and Carlos Simpson tells us that there is a solution. And we take the solution. It's not unique, defined up to d or d bar of something. But it exists. And this is what I want to put as something which corresponds to this diagram. And you see now that, again, it has these two local systems. But it actually correlates, or it involves other two points.

And it makes this whole thing somehow run over the base, which is x cross x. It's no longer a function of the base, it's distribution. And actually, I mean, it's a generalized form. It's a differential form, degree n minus 1 minus 1, with generalized coefficients. But still, it lives on the base. And that's what the correlator tells you.

I still want to present this in a little bit more picture way. So how do you think about this Green function? So again, we have L, one local system, M, another local system. We have x cross x, which somehow I think about as these

two points. And I have homes from L to M, and homes, I mean, our homes from M to L. And so this whole picture is a kind of image for this Green function. And I want to emphasize that this picture is symmetric. So you can rotate it by 180 degrees.

You will not notice the difference. You might arrange these things in such a way. You'll get exactly the same kernel, Green function. All right, now let's run now the construction using this Green function. So as I said, we want to associate correlators

to arbitrary surfaces of arbitrary genus. And so we have to start with something.

So we take some surface, so something which looks like that. It has points on the boundary. And we put some local systems at these points, L1, L2, L3, and so on. This is inside of the surface. Unfortunately, this is just a disk. But it could be something more complicated.

I just don't want to go into drawing of this thing. So let's just stick to the simplest picture. And then you need to take some ideal triangulation of that. So this means that you draw some diagonals inside.

And so in general, you take a decorated surface S and pick an ideal triangulation. Let's call T of this decorated surface.

So now, I also need some other gadget that appears from nowhere, the twister plane, C2, with coordinates z and w. This is the twister plane, the same one everybody talking about when talking about twisters.

And then when I have an edge, let's say I have this edge, E, internal edge. I want to put to this edge a Green function, the following conversion of this Green function. So I call D C E of GE.

And this is by definition the following. So you take differential one form on this twister plane and plus some linear differential operator, ZD bar. So you apply this to the Green current you have by theory of harmonic bundles.

And then you also multiply it by something that looks innocent, some formal element which has degree plus 1, which corresponds to this edge. I'm not going to explain right away at least what the mean has. But then if you count, you will see that this whole fellow has degree 2n.

Because the form has degree 2n minus 1, then you have one form here or differential operator, degree now 2n minus, the form has degree 2n minus 2. Then you have degree 1 differential operator and some other degree 1 element. Now then what you do, you cook up the product of these guys.

So you take these guys which are assigned to the edges, to all internal edges. And it doesn't matter in which product you multiply them because they're even. Then you take a trace of this.

I will explain this in a second. But you also need to multiply this by some harmonic forms which you put on the external edges. And that's important on its own. So we have green forms here. And so here we have some harmonic forms. Let's call this alpha 1, 2, or just

called alpha f assigned to the edge f. So we put some harmonic forms here, which represent some classes formed from here to here. And we take the product of all of them somehow. And then what we get, we get a differential form

with singularities. That's one problem. But another thing is that it takes values in some very complicated vector bundle, this connection. And so we kill the vector bundle by taking traces. It's a little technical thing to do. I'll tell you in a second what this is. And then at the end of the day, you just get a differential form with generalized coefficients.

Now, who are these traces are? I can say it here. If you have a triangle in this triangulation, let's call T, then you have like L1, L2, and L3 sitting here. And you take home from the previous to the next one.

Three homes appear here on the sides. And there is a natural trace map to see. So this is a trace map over the triangle. And then it takes a protocol with all triangles. And that's the form which you call up on the base.

So where it sits? It's a form on what? Yes, I'll explain in seconds again. Yes, yes, yes, all triangles. T under triangle, is that the tau?

So T is tau. That is a T. This is T. This is T. So what I'm doing, I'm saying that I have this green current sitting here, harmonic form sitting here. So leaving some vector bundles. But then these vector bundles, as you can see.

Can you? Did I erase this? Sorry. Did I erase this? So, oh no, it's there. So you associate these homes to kind of oriented in two sides, say just here. So I'm going like this way. And one side is here, oriented here. Then you can shrink all these triangles,

apply trace in each of the triangles, you kill the coefficients. All right, so this is not yet what it's supposed to be. The last step, we need to take summation over all triangulations, equivalence class of triangulations.

And so the last step of this definition is this. So the Hodge correlator of this harmonic classes is defined as the following. So you take this form, kappa T, assigned to a single triangulation.

You take the sum of these forms over with the weight over all triangulations, equivalence class of triangulation, of all triangulation, as much class of triangulations. Then it takes sum over all surfaces, isomorphism class of decorated surfaces,

which you can somehow put using the same data on the boundary. And then all this you integrate or rate sits. It sits on x over b raised to power given by the triangles of this particular triangulation, t.

And the point is that what you integrate is a current. So you can run the integral. And so at the end of the day, you get something. What is this something? So what this integral means, again, so you have originally have differential form which lives on x to triangles of t.

Oh, sorry, x of, yes, triangles of t. And then when you integrate it down, you get just x over b. You get just to the base. So you have a bunch of varieties which sit over the base.

And you take the fiber product. And products also for grid functions are fiber products from the base, yeah? Yes, yes, yes, yes, yes. Yes, I'm sorry, yeah, it's a fiber product, yes. Yes, yes, yes. So you get something from the base. Now the question is, what is this on the base? And in principle, the construction tells you all the algebra behind. But algebra is nice. And let me work it out.

So what is on the? First of all, once again, so this is a key point. So let me kind of summarize what we get. So we start with a single decorated surface. And we put some inhabitants, some object

on this decorated surface. They have homes from one to the next one. And we represent these forms by harmonic forms. And then we do something with the inside of the surface, create some kind of current using green functions, integrate, get something on the B. Actually, not on the B. We get something which lives on B cross C2, twister line.

Because if you look at the definition of the green form, the green form becomes a differential zero or one form along the C2 variables as well. And that's very important. So we get something which lives here. Now, what is the properties of this something? So let me just say the format.

First of all, I consider this construction as a multilinear function of this harmonic classes. And I can do this just by saying, it's a kind of formal remark, I can put on each edge Casimir of F, which is element in home from L1 to L2,

which sits on this edge, to the dual space to this home. And so my harmonic classes sits here. And so I can take just the identity map, which sits here, integrate all these classes. And then on the output, I get something which lives on the dual to the home spaces. So if I proceed with this, then I get some class.

So I get a single class with this Hodge correlator class, which looks like, as it lives in the sum derang complex on B, tensor forms of degree less or equal than 1 on C2. And then it lives in some linear algebra setup,

which is related to this harmonic category. And so now the question is what this guy is, where it lives. And it's very easy to see what it is. Because when we're talking about this construction, so we take a surface.

And so we put some objects on the surface. And then after the dualization, what I put here, I put here dual to homes to each of the edges. And so I basically have tensor product, cyclic tensor product of the homology groups here, dual to homes. And then take symmetric product over all holes which I have.

So this is what the construction gives you. And I denote it just by some letter. So that's kind of huge linear algebra guy, which you assign to your category. So I will define in a second. But the main claim about this is

going to be the following, that this guy has a BV algebra structure. And therefore, if you take this guy shifted by minus 1,

then this is a differential graded Lie algebra structure. And so I get differential one form, which lives in some DGLI, some universal DGLI, which relates to a category. So now the main claim, which I maybe put here, tells you what is a single property which

this correlator satisfies. I will tell the definition of this a little later. I just want to give you somehow the setup. So the key property is that if you consider this guy,

let me put here little 0's. So I take sum over connected surfaces. I mean, if you don't follow, it's fine. Just want to make the correct statement. So the main theorem is that there is a differential here because it's a BV algebra.

This is not just a BV algebra. I mean, this is a variation of BV algebras because you have a base. And all this base, you have your variety sitting. And each of them, you have these categories. And to each of these categories, you assign some BV algebra. And all together, the whole variation, which is a variation of pure Twistor structures. Or let's talk about Hodge structures,

just restrict to Hodge local systems, so Hodge structures. But in particular, you have a connection, which I denote by D. And so now the statement is that if you consider this D plus G0, this is just a DG connection on this B cross C2.

And this is the first claim. That's already a claim because you have lots of forms here and some complicated graded object here, differential grade Lie algebra. And so all together, it still sits in degree 1. And the second property is that if you take this connection, it's not flat.

But if you restrict it to the line where Z plus W equals to 2, and this is flat. And that's the main property of this construction. So you get flat connection with values. Can you remind who is Z and W? There is a C squared, which appears from nowhere. It's still in the board above you.

It's on our head here. Z and W are the coordinates. Yeah, yeah. Yes, thank you, Don. So you have Z and W, which was part of the definition of this kind of modified Green function. So the modified Green function was a function which was a form which depends on the Z and W. And the key property of the object which I get is that not that it's flat. It's not true.

That it becomes flat when you restrict to this twister line. Now, let me reformulate this again, postponing discussion of what this SC is in a different way. So here's a different way to say this. So just from linear, it needs a little discussion

related to hoge structures. So so far, I just said that I produce flat connection. But now I claim that this flat connection has some hoge meaning. And here's what it means. So first of all, given this is some kind of result.

Given a complex of variations, let me assume that the pure hoge structures, real hoge structures, let's call them M, M on B.

I can define a functor which takes this M and makes a complex out of this CH star of B with coefficients in this variation. There is some construction. But the properties of this construction, which I'm going to use as a following,

that first of all, the meaning of this CH is this, that this CH B M is nothing. It calculates. It's quasi-isomorphic to R home in the category of R mixed hoge structures from R of 0 to this M.

So you can just say this is hoge cohomology, or people call it absolute hoge cohomology, of this variation. So it's a complex. Now there is a product structure in this complexes. So if you have two variations, N1 and M2, then you can take

CH, I'll skip B, and CH of M2. And there is a map to CH of M1 tensor M2. And this map is commutative and associative on the nose.

This is its main property. Not up to something, but on the nose. Therefore, and the last, only two blackboards here.

So the last property is this, that if you take any element which lives in CH1 of this fellow on B with coefficients in endomorphism

of a certain variation, and you assume that it satisfies Maurer-Kartan equation, then this produces a variation of mixed hoge structures on B.

That's called M, which mixes the original one, which means that the associate gradient for the weight filtration of this guy is the original variation M. And so basically it says that if you start with some variation of pure guys,

and you want to get something mixed, all you need to do is produce some Maurer-Kartan element in this complex C1, which calculates absolute cohomology group. Not very surprising. The surprising thing is that there exists such complexes which are commutative associative. But then it's kind of natural.

But then if you do have such a machine, then you can say that if you started with any kind of linear algebra data, for example, you started from a variation of B-V algebras. Like in our case, it was SC of category of harmonic bundles.

Then you apply this machine, CH, and you produce another B-V algebra. This was a variation of B-V algebras, but now you produce another B-V algebra, just CH of r x B. And this is just a B-V algebra.

So this is a variation over B, and this is just an algebra of complex numbers. So now I can formulate this theorem in a different way. I can just say that this is equivalent to another theorem, theorem two, which says that the total Hodge correlator

class, which I can denote by G. First of all, this is exponent of the one which corresponds to the connected surfaces.

Secondly, it lives in CH0 of this SC of harmonic bundles. And finally, most importantly, it satisfies the quantum master equation.

So there is a total differential on the thing which I didn't actually define yet, and so it kills this element. So this is a B-V differential.

As I said, I didn't define yet the B-V algebra structure, but if you believe me, it exists in variations, and there is this B-V algebra, this is a B-V algebra, so there's a differential there. And this is just equivalent for those people who know the formalism, the B-V formalism, immediately recognize that this is the same as to say

that if you just consider this G0 and shift it by minus 1, it satisfies the Mauro-Kartan equation with respect to this product which I was talking about there. Notice this is Mauro-Kartan in a quite, quite non-trivial B-V algebra.

You have to start with this linear algebra game on your category, and then you apply CH function to this. So you get this B-V algebra, and so you get Mauro-Kartan elements there, all solution of quantum master equation, all flat connections, they're all the same. Now, what is it good for? So how do we get back to the goal which

was originally formulated? Can I just ask, when you say B-V algebra, do you mean B-V infinity algebra or B-V? B-V algebra, differential graded algebra with Laplace operator, B-V Laplace operator, differential, et cetera. So that comes from a very careful choice of

Oh, it's the specific choice of C harmonic that allows you to do the same thing. So it's a choice of harmonic form or alternate? No, no, no. I mean, this is a formal construction. You take any category, like graded category, you can run the construction. You don't need any assumptions. No, but your representative depends on

So first of all, the definition of this B-V algebra is quite universal, but the properties that you get solution of, you get elements which always satisfy the quantum master equation,

but if you choose a different representative, it will be not exactly this one, but the one which differs by the B-V differential. So it's cohomology class is uniquely defined. So let's just summarize what we have. So first of all, as I promised, we should have something when the base is just a point. And so if the base is just a point, then

I claim that what we get is just a Lie algebra map. I'll explain why, but you get a Lie algebra map from Lie U-Hodge to this B-V algebra, B-V Lie algebra connected of R X to get the

Lie algebra shifted by minus one. So this is just, I mean, it's just a reformulation, one more reformulation of the properties of the Hodge-Galo group and what was done.

But now you can say that this is kind of quantum situation. So this is of a point, but this is like G quantum. You use all surfaces to do this. But if you decided to go only to surfaces which is on the disk, so then you get G classical. So this relates only

to the disk disappoints. Then you go to something much smaller, which is the cyclic homology complex of this category, tensor of a fundamental glass of your manifold shifted by two, as far as I remember. And there is an actual projection from here to here.

And so now this coach-correlated construction, it gives you this map, it gives you this map. But now the classical one I can interpret because you can say, this is the last thing which I'm going to say, but let me still do it. There's some kind of formal arguments

familiar to the people who work with this algebra, that if you take CC of the category,

let's say tensor is H2n, then this maps to H0, this maps to A infinity funters, from your category to itself. Again, I'm explaining this in a very loose way,

because I don't explain you why, I'm just saying that that's true. So this gadget, H0, this maps to infinity funters on your category, and that's where your E algebra goes. So this means that it goes here to this funters, A infinity funters on your category.

While on the other hand, there is a part, a piece of derived category of holomorphic de-models, which consists of smooth de-models, de-smooth of X. It's de-models whose complexes

whose cohomology are just smooth local systems. And you can say that this is just funters from, let's say, bar construction of this, applied to this category, to vector spaces. No matter what this means, it's just that you have your category and then you take something like DG funters from this category somewhere and you get, you recover your category.

So again, forget about details. The point is that you start with this nice category which Simpson considered harmonic bundles, and by formality, it knows and recovers the whole de-smooth on X, in a funtorial way. It's just a triangulate envelope of this.

And so if you have anything which acts by infinity transformations here, and that's what our Lee Hoare is doing, it acts by infinity transformations here, this just means that it acts here. And that's all. That's what was promised. So I promised that in

the end of the day, when B is a point, so B is a point, we are going to... Actually, did I ever say this? I'm sorry. Oh, I didn't. Sorry, I need to do this. I need to say

what the output is. I thought I did. Just one more minute. This tells you what you get in the end, what's the structure. So this is a summary of what we get. So what

this quantum Hodge field theory tells you. So when you're on the disk, the classical Hodge field theory, which is a disk situation, it's by definition what this is, this gives

you the Hodge side. And what this means, how I understand this, this means that I get functorial homotopy action of, let's say, Lee Hoare's homotopy.

Which, I'm not talking about C star cross C star, by A infinity equivalences of the big category. Now I'm talking about what it should be in general. And this is what was hoped for, because I started with some manifold X, which was

complex manifold, and I was saying that the dream is that there is some manifold, if this has dimension N over C, there should be something of real dimension, 2N plus 1, which somehow is fibered over B over Hodge group, in particular B over U Hodge. And this means that in some sense, this group have to act here. But it doesn't.

It doesn't act on this. There is no spatializer, and there is no action. So what is happening is that instead of this picture, we take this thing. So we take this G Hodge, or U Hodge subgroup, X by A infinity transformations on this DB of homotopy.

And that's what we keep in mind when we say that there should be some kind of picture like that. That's the arithmetic side. This is like gadgets for arithmetic side, and what replaces the action of the Hodge-Gullar group on the category. But this is just a disk sector, and this is actually only the case when B is a point.

Now you can ask what happens if B is not a point for any B. I'm not going to elaborate with this. I don't have time, but oh, I have to stop. Just a sec. So for any, I get some language to develop Hodge theory of the base. But now the last word is that now if you do the real quantum Hodge field theory,

and the correlators, we have the correlators, this is for a master equation, so this should go to the deformation of the case when B is a point. Let's go to the deformation of this picture. Thank you.

Could you say something about the whether or no issues with ultraviolet singularities? It would be better if they were. I didn't see them, so the integral seems to be convergent. It would be better if the, but there's, I mean, I check and check many times that

over the disk there's absolutely nothing, and I think I checked that there is nothing in general, and I wish I have them. But did you try to start with this law of diagonals? Yeah, yeah, there are arguments, I mean, why? I know, I know, I know. I'm saying that I think I proved that there are none of

them, but I don't think I have them, because the story can be run as a parallel to Chern-Simons, and somehow it's true there, it's true here. I mean, I think a related question, so like, you know, in my scenes, proof of formality theorem, there was no counter-terms, but you have to work very hard to show that

the master equation holds. Yeah, so here's the fact that you have a distribution, this is where the story sits, this is that you have the current, that you integrate the current. This is a delicate statement, so that's where the convergence of the integral sits, and I was saying that if I just sit over the point, so I think I'm very confident about this,

but on the other hand, if there's a slightly different story, it would be just nicer, that's all. So in your point 2, the deformed situation, it's more about, say, what's the deformed DB whole of X, for instance?

Well, I better not, because it's, first of all, there is this deformation which you guys consider, and I thought I'm going to run to that, but I don't quite see this, so what I'm saying here is that I produce this total green class,

and where is this picture? I think I arranged this. So there is this big BVA algebra, and it projects to the small BVA algebra, and when it projects to the small BVA algebra, the data which you have here is precisely the action of the Hodge-Guller group. Now when you lift it to the, the action of the Hodge-Guller group on this D-smooth on the category holonomic D-models, now the big BVA algebra is the

deformation of the small one, and so it's supposed to act, and it acts on many things, and what it acts on, you can say this is deformation of what you had before, so this kind of argument, but I am unable to, even for the case of the curve, I am unable to get, to see directly that this is a quantization of several local systems.

It's expected, but it's not clear. I mean, you can see it's a deformation, but I can't, I don't see non-commutative deformation.