categories. So in principle this should be the following. We have a real kind of

C-infinity symplectic manifold and something called B-field, which is one

can think it's like took a chain representing class B in H2 of X. Then the

thing should give, I'll tell you what are problems, an infinity category or complex numbers. And in fact will be Kalabi-Yau category. Kalabi-Yau dimension

Kalabi-Yau dimension n, when n is half of real dimension of X. So that's

principle. And what is a rough idea? It's first approximation objects in category

F of X. Objects of this category will be, say, compact Lagrangian sub-manifolds and

Hamiltonian isotope you should give, if you apply Hamiltonian isotope, it should be the same object, isomorphic object. Then home spaces between two Lagrangian,

home complex between two Lagrangian manifolds should be equal to C to the set of intersection points. If intersection is transversal and by this Hamiltonian isotope one can always achieve it. And then

you can see the A-infinity structure for any key. If you want to

write L-infinity composition, there should be six maps, it's tensor, it's tensor coefficient, it depends on the choice of intersection point, should be equal

to, if it means that you fix intersection points of L0, L1, L1, L2 and so on, it will be sum over all maps of holomorphic disks, pseudo-holomorphic disks X, such as

the boundary of this disk is union of Lagrangian manifold and it meets the intersection exactly at this point, and exponent, let's say, minus integral of

omega over disk, plus roughly I times integral over B over disk, pure imaginary number defined up to 2 pi IZ. So maybe we should put some plus-minus. What is

pseudo-holomorphic disks? There's one small definition, if you have almost complex structure on X is compatible with omega, if omega of xi, j xi is bigger

than zero for any real tangent vector, and zero real tangent vector. Now, so it guarantees that for any holomorphic, pseudo-holomorphic curve, the

integral for omega will be positive number, if it's an trivial curve. Yeah, so that's rough idea and there are several problems here. So problem number one, this thing

should be z-graded, this complex should be z-graded, and also we should orient the space of disk should be zero-dimensional and we should orient

the space of disks, orientation of, maybe virtual orientation of space of disks,

in order to get some fundamental class. So the second problem is, we're going to consider the area of the disk, depends on the homology class of the disk, given class of the disk, which is in H2 of x and union

of Li from 0 to k, using the general coefficient. We want to have finiteness

of the space of disks, and we need in particular compactness. We need a compactness, so there's a compactness issue, because I didn't say that manifold is compact, it should be not compact manifold. The third story, even for

given homology class, we get some integer number in front of this story, or maybe rational number, but then there's a question, why the series is converges, if it's made over all degrees of the disks.

Convergence of series, and first point is that presumably I should add more objects, one can put like local system on Lagrangian manifolds, and I'll also discuss it,

need more objects. For example, we want the schedule to be triangulated and approved some triangulated. Yeah, so there are four different issues, and I'll briefly

tell you what is the state of art about one. Orientation and grading. First note is the following. You can see the symplectic group, which is structure group of a tangent bundle. It contains unitary group, and this is homotopic

group. Unitary group is a maximal compact group here. Hence one can speak about its homotopic group to G, L, N, C. So there are three groups, which are

homotopic group, and particularly one can speak about chain classes of tangent bundle. And the condition, which is necessary to make the grading cases two

times first-chain class, dx is equal to zero in H2Xz. This is a condition, and more precisely, one needs not only the equality, but some additional data, as to the equality, and data is the following. Let's consider a bundle over X.

It is compact fiber. If you pick point X, the fiber text is a set of Lagrangian subspaces in the tangent space. So it's a certain guy, and let's

assume that N is at least one. This space is connected, and its

fundamental group is Z. One of these things is Z. So if you kill all high homotopic group adding cells, you get homotopically circle. And this circle is twice times the circle coming from canonical bundle. This guy has a

two-fold covering, oriented Lagrangian manifolds, and one can easily see that if you put almost complex structure, you get the circle is the same as this one, and the data will be the following. The data will be a map

from, let's call it something like arg, mod by Z. It will be a map from the total space of this guy to arg mod by Z, such that if you restrict its

on every fiber, you get a map to a circle, and it represents a generator of H1,

each fiber, and it gives a generator of the first homology of Lagrangian grass many on the fiber. It's generator of H first, of pullback of the

fundamental class. So how it arises in practice? If X is scalar

manifold, and we get section XKX squared, which does not vanish as any point,

so we get at each point volume form and define it up to sine, hallmark form defined up to sine. Then we get a map, namely you take square root of this guy,

which is defined now up to sine, restrict to Lagrangian, and X will be Lagrangian subspace,

and X, and take argument of this guy, and it's defined, argument is usually defined up to 2 pi Z, but because up to sine defines up to 2 pi Z, up to pi Z.

So we get this function. So that's things responsible for your manifold, and now what about Lagrangians? This object of 4K categories should be not arbitrary object.

If L X is Lagrangian, then L also maps to this total space, because at each point of Lagrangian you'll get its tangent space. So you get a lift to this L X, this vibration,

and then we get this map arc mod pi Z to R mod pi pi Z, and then we get R,

and you get a map from Lagrangian to a circle, and this map gives a class, you get a class mu belonging in H1 L Z, because you get a map from a circle,

and what we needed, we need mu equal to zero, it's called Maslow's index, for each loop we get certain integers, so we should get this function to be equal to zero in first cohomology, and additional data, grading on L is a choice of a lift from L to R,

a choice of a map, which produces a composition in the argument.

And then it's easy to see that for two graded Lagrangian manifolds, if intersection is transversal, for graded L1, L2, and intersection is transversal,

you get a map, index, kind of Z grading, you can assign a certain Z grading to each intersection point, and if you interchange L1, L2, then index goes to n minus index.

So it will be Calabi-Yau duality, so the same intersection point, you can see this element of complex of homes in one direction or another direction, and they have opposite, with some called n, indices and get Calabi-Yau duality.

Okay, so that's the kind of standard thing which one should do about Z grading,

Z graded things, you should consider kind of Calabi-Yau varieties to get in a symplectic sense. Now, the question, I still didn't finish the story, one need orient disks, then this Lagrangian manifold should be,

it could be in principle not orient, it should be spin manifold in the sense that second Stiefel-Whitney class is zero. This I discuss first in the case, in the case when B field is equal zero.

And this is Stiefel-Whitney class. And more precisely, one need data, one need not equal to zero.

For example, this canonical representation of Stiefel-Whitney class, you choose some cell decomposition, arbitrary cell decomposition, and this Stiefel-Whitney class is represented by the k-chain,

which is sum of all co-dimension two cells in the barycentric subdivision. You get some cell decomposition, now you make this barycentric subdivision,

and then this equality one should, like this equality of

two c one equal to zero should promote to some data, namely, L is compact, L is compact, L is compact, yeah. And then one needed to write as a boundary of certain one chain,

these coefficients in Z mode two. This will be like a spin structure, or a chain of spin structure on your manifold. Then,

this is if you can see Lagrange manifold with such things and you get some well-defined orientations. In general, if B is not equal to zero, it's some bit longer story,

then presumably what one needs is that B restricted to L should be a torsion element in this common group and then

what will happen is that one should consider not L itself, but not just L, but local systems, kind of twisted by Job, whose class in H2

maybe kind of C star containing is i times B plus a Stiefel-Wiecny class, image of Stiefel-Wiecny, image of Stiefel-Wiecny class.

We can interpret Z mode two as plus minus one or pi z of mod pi z in this Z mode two, Z mode two embedded two. Okay, yeah, so that settles the equation one.

But what about compactness? It's of disks with given area.

Suppose we get the symplectic manifold, which is not necessarily compact, but it's, I have to say, it's called paracompact. It contains a dense countable subset because C infinity manifolds in principle could be very big.

Countable to infinity, yeah, X countable to infinity is admissible. And if for any compact subset in X and any positive number A, real number, there exists

k2 in X, said that k1 sits in interior of k2, also compact, and almost complex structure

G on X, say, compatible with omega, such that for any map from the two-dimensional disk

to X, which is J-halomorphic, such that f of boundary belongs to k1 and integral of pullback of omega of disk is less than a implies that

f of the whole disk belongs to k2. So it's essentially kind of a form of the condition, which is more or less tautologically what is needed to have,

at least for some almost complex structure, you have this property. Yeah, so it's kind of almost tautological condition, which what I need to have a compactness, even existence, I don't really fix one. And the claim, if I'm not confused with limits,

in this case, one get a canonical A infinity category, which doesn't depend on the choice

of almost structure, over Novikov field, which is by definition is the following. It's

a set of infinite sums, ci t to ai, formal sums, countable sum, said that ci are complex numbers, ai are real numbers, and a limiting point of ai is plus infinity,

and t is just a variable. I eventually want to put a t equal to e to minus one, but I'll put it as a formal variable.

And that's it. I think the idea, I think, behind these things is this guy called Groman. He proposed something similar, but a bit more strong. I think it's kind of really maximal

generality, one can define this Foucault category at least over Novikov field. The reason is the following. If you want to count these disks,

suppose we get two Lagrangian manifolds, l1, l2, or several Lagrangian manifolds, lk. And you want to study some compositions, so you want to map holomorphic disks to their union.

And if you bound the area, then it sits in some compact space, you can put some k2, so the whole thing's up to a given area, bounded, sitting in this. Or maybe we can say something, just a second. Maybe this almost complex structure,

maybe I should be a little bit more precise here, almost complex structure on x minus k1,

such that for any almost complex structure, the j is for any j, such that j restricted to x minus k1 is j prime. So you can see the holomorphic disks, which inside satisfy some uncontrolled Cauchy-Riemann equation, but outside will be controlled. And eventually what you go on,

you just kind of, sorry? Is it a version of Hartreeft's theorem or some weakening? No, no, no, it's just condition. It's just condition, which is a priori, it's not clear how to check at all. But if condition, you can check, it can exhaust for larger and larger area by different compact set, and put one complex structure here,

another complex structure here. So it's a completely formal argument with limits, nothing deep here, maybe guarantees the whole thing exists, and then eventually it will be also unique. So it's pretty empty constraint, but then one can kind of guarantee it by various means.

There's no boundary condition. In this story, there's nothing about Lagrangian manifolds at all. Yeah, but in the compactness, you do have boundary condition.

Yes, yes, no, but it's kind of a bit stronger than, I forget, it sits. So what does it mean? There are no Lagrangians at all? No, no, no, no, it's about disks. There's no disks with bounded area which go to infinity. No, I need Lagrangian manifold to be compact. My Lagrangian manifold.

No, no, there's no Lagrangian manifold here, but I say that it sits in the union of compact guys, it sits in some compact. That's it. Sir?

If I replace a line by the totally real, not Lagrangian, but just totally real. It will be also the same, you see. So there are various sufficient conditions,

various kind of sufficient conditions, some criteria when the whole thing satisfies.

So, one is when X, omega, has a contact type at infinity. And what does it mean? It means that there exists a vector field, a real vector field, and a compact subset,

a subset, such that

latency of omega is equal to omega on X minus K. And then the following, the boundary of K is smooth hypersurface in X, and so you get this K, and this vector field, this boundary C,

looks outwards, directed, is directed outwards, let's see, and

moreover, X minus interior of K is vector field, forget about symplectic form, as symplectic form as well, is isomorphic to C times positive for A, and vector field is

coordinate t, and vector field dudt. So it will be just kind of cylindrical end. And then automatically these things will be contact manifold. Now that's a very typical

situation, and it's when it holds, okay, if X is a cotangent bundle to Y, where Y is interior

of a smooth compact manifold. Now then one can easily see that one gets this contact boundary.

So it's one good situation. The second good situation is kind of more general, so one implies two, so two gives a more general class of examples. When there exists a compact subset,

almost complex structure, almost complex structure, and the function H from X to, let's say, positive numbers, which is proper, and said that, or maybe I don't even need

complex subset, there exists almost complex structure and H, said that H restricted to the pre-image of some tail, is

polyureth subharmonic. In what sense? It means that they've considered a map, a germ of a map, of a G holomorphic map from a disk to this H minus one of this open domain.

Then the pullback of H is a polyureth harmonic function, Laplace, Laplace and the pullback is non-negative.

So it also guarantees that if you stay in some, up to some level set of your function, boundary of your disk, it cannot go outside by maximum principle,

because then they'll get a point on a boundary when the value of H will be, get the local maximum, which contradicts this condition. Yeah, this is a more stronger condition. In fact, people construct a long time ago,

if you get contact type boundaries and construct at infinity this plural function, so these things will be really strictly positive for any non-trivial disk. So you get something like approximation of calorimetric. And this is pretty complicated differential geometric data, and there are reasonable

examples, like in photovoltaic categories, when it's really this condition, it's not the one. But the third condition, it's of different nature, it's also very useful,

there exists complete remaining metric g on x satisfying three inequalities.

So the curvature of the metric is uniformly bounded. The injectivity radius is also uniformly bounded, so there are local pieces, which looks like very close to Euclidean balls. And the third condition is that both omega and omega inverse,

it's a section of the exterior power of cotangent bundles of tangent bundles, are bounded with respect to metric g, are bounded also uniformly. And then it guarantees really exactly this thing, it's not that for each area one can find,

this disk can go too far. And the reason is the following, if you have this compact set k1, and you have a disk, then first of all the integral of to form and integral of

and remaining volume will be bound by constant. So it means that this remaining volume is bounded, remaining area is bounded. And now if the thing is very long,

so the damage is very big and area is bounded, then at least at some point it should be very close to, in kind of Gromov-Kozol distance to an interval, it should be kind of one dimensional object. And then,

so it will be kind of very long cylinder, injectivity radius says that it's big, so it like sits in Euclidean space. But then this long and since cylinders are not calomorphic, do not satisfy Cauchy-Riemann equation badly. So that's a rough idea.

And it's really different criteria, so it's not related to one and two in any way.

And this kind of fundamental equation for my future story will be the following. Suppose XωC is a halomorphic symplectic manifold and assume that it is embedded,

is open, dense open symplectic leaf to compact Poisson algebraic, say Poisson right X bar.

The equation is in the situation X and I take real part of ωC, which will be automatically real symplectic form admissible to infinity. So this is the situation which I need for

quantization and information quantization sense and this is in Foucault sense. So it's true for many examples. Yeah, first it's definitely true if X is cotangent bundle,

explain it in real case, it's okay, or if X is c cross to c cross, this kind of standard form to some power n, product of copies of standard multiplicative torus. And also it's true if X is,

if dimension of X is a surface and X bar minus X is a divisor, not necessary with normal crossing.

So in surface case, my story is the following. If it's a divisor, one can easily check, this could be not normal crossing, you can make blow-ups to make its normal crossing, it will be still Poisson divisor, you can reduce the equation to normal crossing. In the normal crossing case, the third criterion works very well because

essentially you can have poles of order one and then you get kind of like cylindrical ends or pole of higher order, you get very open ends and it's very easy to construct this metric. It's a very rough picture. Also harmonic function and I don't know how to get here,

third criterion number three. Third criterion is applicable. And I think it's kind of not true, at least not on the nose true. In the simplest example, consider R4 minus zero and it takes standard form.

At infinity the behavior is okay, but near zero it's wrong and it's completely parallel to kind of first example of non-compactifiable hallmark symplectic method C2 minus zero.

So those are good indications that are all the same story. Okay, so this is all about compactness and third point is convergence of series. Yeah, I think it's

yeah, that's a really main stumbling block in the whole story for 25 years.

It's still open question. So what we use now is kind of surrogate solution, work over non-archimedean field of former series.

So what really we need, we need some tool to go from this form of things to actual objects.

We need a tool which controls number of holomorphic disks of area less than a. We want it to be growth at most exponentially in area.

So what is this situation here? There are some cases where we kind of

understand a little bit more about Foucault categories from homological mirror symmetry, like Green-Tik, three-fold and so on. And in this example, it's clear that the radius of convergence is strictly less than one. Yeah, typically you should expect to get

integer numbers and radius should be one because coefficients of size one. But in Green-Tik cases, radius can be strictly less than one. And it means that you cannot make, if you try to make something called mirror map, it means that if a class of symplectic form is too small,

then you cannot make unambiguous meaning what is Foucault category. You get ramified map to modulus space of categories from killer modulus space. And I think it's also this problem with small area will affect this book of Witten program.

They say that for any symplectic manifold, blah, blah, blah, but maybe for some symplectic form is too small, the whole thing will just break down. Yeah, so the conjecture is that under some conditions, which we don't know yet,

and for good examples, the whole thing M n will be in kind of analytic part of the

sums, c i t a i, such that sum over there exists a constant

epsilon, such that sum over c i epsilon to power a i is less than infinity. Yeah, you can see the kind of like convergent series with real exponents.

And now the last issue about more objects. Yeah, the proposal is the following, which is not yet materialized.

First we choose some lambda in X, which will be compact singular Lagrangian subset.

So it's closed subset, and what is singular Lagrangian? For example, it could be maybe sub-analytic, if it's X is real analytic closed subset,

and its most locus will be Lagrangian sub-variety. Yeah, so it could have pretty horrible singularities, or maybe not so horrible,

and the story is the following. This Lagrangian set plus this orientation, z gradient, z gradient choice, z gradient data,

which we have on our symplectic manifold. So you don't do anything like spin structure on this guy at all. So you have this realization of twice canonical class on symplectic manifold. ...should produce a certain

dg algebra, comes like a lambda, over integers, and algebra will be of some the following, roughly of the following nature. We'll get a

maybe finite quiver, then edges, you'll have a set of edges, you'll have a map to integers, so edges will be kind of z-graded, and we can see the past algebra, we get

gradient algebra is z-graded. And now we introduce some differential of the following form. Differential of each edge will be finite expression,

finite non-commutative polynomial, non-commutative expressions with integer coefficients in previous edges. So it means that a set of edges is totally ordered. So you add them step by step,

you add one expression, differential of this guy, so you get some finite collection of some nice integer non-commutative polynomials, t squared to zero, you get this algebra, and then

it depends only on neighborhood of lambda. And then what your manifold, big symplectic manifold will do, x omega will give, in a totally clear way,

solution of Mauro-Cartan equation for a homological Hochschild complex for this algebra.

You deform in a homological Hochschild complex over a Novikov thing, for example, and then the object...

You have infinitely many positive and negative degrees, a lambda, it's great, it cannot have infinitely many... Of course, yeah, yes, everything, yeah, no problem, yeah. But now one can, if you get this

Mauro-Cartan equation, one can speak about finite dimensional modules, finite dimensional representation of this algebra, depending on each bar, so

algebra deform as well, and consider modules over finite dimensional over Novikov ring modules over deformed algebra should be also interpreted as object of this Foucault category, x omega b, objects of

enlarged Foucault category. In the case of smooth manifold, this thing is something very close to

group ring of the fundamental group, we consider trivial representation, yeah, that will be like original object which we have. I'm sorry, there you mean in the Lie algebra of Hochschild, is that what you mean? Yeah, it's D.G., yeah, D.G. Hochschild is D.G. Lie algebra.

Okay, so basically it's an infinite algebra, right? It's an infinite structure, but also with m0, maybe some things, okay, yeah, but that's...

So, what is the rough idea of this algebra? Well, this algebra, I think, this solution is more or less stabilized now. Yeah, I suppose, I guess, it's like graph manifold, like on plane I get a graph, you know, just an example, not true. Yeah, then I take a small, sorry, a lambda,

a lambda, rough idea of a lambda. Ah, provided that lambda exists. Lambda is subset, yeah. Lambda is given, I know what is this, empty set.

Yeah, if you get... Modules over A lambda should contain inside the compact Lagrangian, this blah, blah, blah. It's large, which means that it's bigger than fucai.

Yeah, no, in the original fucai it's a smooth Lagrangian, but it will be singular Lagrangian in some kind of shift data. Could be empty, no, for any lambda, for any lambda, and then we take inductive limit for all possible lambda. Yeah, yeah, one can, if one sits in another, one category sits,

is a full subcategory in another. Yeah, so, the story is a funk. This Lagrangian manifolds, this single Lagrangian manifolds are more or less the same as kind of what's called Weinstein domains, because they appear in

the following query. You have open neighborhood of L and vector field psi,

said that, let's say, L xi of omega now will be minus omega, opposite to the things which we have at infinity. And suppose this xi at boundary looks inward, is directed inwards. So you really need the domain on which restriction class of omega is

zero in second commode class, and you choose a representative which is directed inwards, and then apply the flow. So these things start to contract, and generally it's contract to something singular Lagrangian guy, and converse to singular Lagrangian guy is coming from this

neighborhood. Yeah, so it's, yeah, I don't know, there's a many skeleton in this, yeah. So, okay, but, yeah, so you get this flow which contracts in collective form, and

you get the singular Lagrangian, but now you do the following flow. On the same U, you consider something like epsilon neighborhood of L with some metric.

You can see the Hamiltonian flow, which is lambda, lambda, sorry, it's L lambda. You can see the gradient of distance function to lambda. You can see the gradient of the

boson positive for negative direction, so it's,

because orbits stay in a compact set, get the flow. And what will be this, very roughly, it's not exactly this quiver, you can see the...

I'll describe actually not some algebra, which will be quasi-isomorphic to A-lambda. Namely, what I do, I have this neighborhood, and now I choose points p-alpha in a connected component.

You can see the smooth locus of lambda. It has several connected components, like here. This interval, this interval, this interval. In each connected point, choose at least one point of lambda's smooth.

Then you can see the transversal Lagrangian disks. And in door with z-grading, in the sense which I explained, z-grading is the lift of some argument function.

Then we get something from which we can hope to make homes.

And now, consider the following algebra. Sum over all alpha, beta. And consider the inductive limit when time goes to plus infinity of c. And now you would do the following, if you get alpha and beta, you start to apply the flow, and then the disk starts to move.

The alpha what squared? Two-dimensional disk. What are you? Two denotes nothing. The alpha. Two, it's one-dimensional in your picture. It's a graph, you take... Ah, sorry. N-dimensional, yeah. It's n-dimensional disk, if I n-dimension to n.

In this case it's one-dimensional disk. And consider intersection points of your play exponent p times the flow, which I explained before, this Hamiltonian flow,

with respect to distance function to d-alpha and intersect with d-beta. You get infinitely many points. Yeah, so this, I think the simplest example if you take lambda is a circle. And if you get one guy and, let's see, and you start to apply this flow many, many times, you get things like this.

Intersection points will be larger and larger set, and eventually you get basis of monomials in Laurent polynomials.

All these intersection points come from this category. This disk itself, this inductive limit will be exactly get all around polynomials.

Yeah, that's a typical example. And what is the state of art?

What is expected is that alpha lambda is a section of Cauchy of algebras and gamma in homotopy sense.

And on smooth locus, this category is equivalent to finite dimensional complexes of a billion group.

And globally it's twisted a little bit by this second Stiefel-Whitney class. No, one can shift, twist this shift of categories by any class in H2 in ZMOT2.

Of lambda, lambda smooth, lambda smooth, yeah. And, for example, in this one dimensional example, if you consider triple point, then what will be locally this category?

Locally this category is a representation of quiver A2. You have three complexes, E1 and E2, and they form exact triangle. And it will be the same as a representation of quiver A2, this exact triangle.

So more general, if you get such guy, kind of K spikes, you get representation of quiver Ak-1. Yeah, so that's a very beautiful story, which I will not talk to here.

If lambda is smooth, what is A lambda? A lambda, if you forget about this ZMOT2, its chains, you can see them, suppose it's connected manifold, and choose the base point. Then you have a topological monoid, loops from the base point.

It's a monoid. And you can see the chains, singular chains of this monoid to get digital algebra negatively graded. This will be A lambda. If it's lambda is KP1 space, it's a group ring of the fundamental group. It's a generalization of group ring. Yeah, it's a nice object.

Yeah, and what I want to say is that there was a kind of breakthrough work by David Nadler. He proposed a finite list of possible singularities in any given dimension.

For example, in real dimension 2, it will be at most 3-valent graph. It will be 1-val... Maybe not just... Maybe... So it's just a question, it's about complexes and some exact triangles, the whole category.

And it's called arboreal singularities in higher dimensions. And what is not yet done, how to... So he deformed any kind of lambda without changing homotopy type to something with nice singularities.

Yeah, for a couple of graphs, you can deform to some trees. That picture doesn't change the category. And what is not done is how to construct this more katan class.

How to count holomorphic disks with a boundary on a single Lagrangian and get some contributions. Yeah, I think some people working on it, it will be done in some finite future. At least for this Nadler's class of singularities.

Yeah, so it will be... So it's almost a good science. And now I think I'll make a 5-minute break. I'll try to formulate certain conjecture.

Unfortunately, because of troubles on both sides, it's not really yet mathematical conjecture. And the conjecture of X-omega is, let's say, algebraic symplectic manifold.

And we choose, for definiteness, Poisson compactification with some Poisson structure.

And such that X bar minus X is, let's say, normal Poisson divisor.

And suppose also you get some class. Class B0 in H2X bar R-map to pi i, z.

I had some kind of seeds from which we can make deformation quantization. And I assume that I extend it to an analytic germ of germ in H bar of quantum categories.

I do not understand. The previous was given. What do you expect? I mean, all this is given. What do you extend? No, suppose now I have a family of...

And suppose I have a formal path to the Mario Kartan space of the following digital algebra. Consider R-gamma of X bar and consider a logarithmic polyvector field.

Like a divisor.

I have a formal solution that says first derivative coefficients of H inverse will be omega inverse.

Then Poisson structure gamma of H to the power one will be gamma. And some kind of formal path. Then I get a family of categories. I can twist them by also by B0. And what I want these things to be?

To be analytic germ in H bar.

Then I kind of want to speak about object also analytically, analytically depending on H bar.

Then for H bar less than sufficiently small we should get a category of objects in quantized...

When the conjecture ends? Oh, it's maybe, yeah, it will be very non-precise and very long. Yeah, that's a problem. Yeah, it's a formulation of kind of left hand side of conjecture.

Then we get category of object in quantized category. Such that the restriction to the boundary is zero.

I recall you that by this logarithmic formality theorem one gets not only one category, but category for each divisor, for each intersection. Then one has infunctors. And then one can ask about object whose restriction to the boundary is zero.

It will be analog of holonomic modules to get some category. And what will be another category? When you get this thing, in particular you can look what goes on inside of this story. You forget, you can see the map from this Markatan to the open part.

And inside you get a class in H2 of X.

You can see the restriction, you can see the solution of Markatan in... You restrict to X, a gamma of X, and it holds the story. And then the symplectic case. There was a story in the first lecture which you missed. Yeah, that if you formally deform open symplectic manifold, just in the case you get a formal pass and second cohomology.

No, you be filled with B0 is not trivial. B0, it's kind of separate, it's just camp as a free addition.

You forget about B0. Ah, you get a class which is omega over h-bar. And maybe you should add B0 later on. Yeah, because modulo 2 pi IZ. This will be B0, this class.

And now, what you do, you make fucai category of X, this real part of omega is symplectic form.

And as B field will be B0 plus correction term. Real part of omega plus real part of correction term. Plus imaginary part of correction term.

So we make this abstract fucai story. And the conjecture is that this is naturally equivalent. And it will be a kind of Riemann Hilbert correspondence in a very general setup.

I don't like something.

I mean, you should probably scale on the real part by page. Yes, yes. Yeah, exactly. Yes, that's it. And here you get imaginary part of omega divided by h. H is real? No, you should put h inside the real part.

Ah, yes, yes, you're right. So you're taking h real? No, no, no, small complex numbers. That will be the main point to put. Not real, but arbitrary. Yeah, yeah, yeah.

So to say these things, I need to say something about z-grading. It's kind of automatic in this case. And let me explain why. We're going to get a complex symplectic manifold.

The structure group of the tangent space is sp2nc. Sir? And this is homotopic equivalent to unitary quaternion, UN quaternionic groups.

And this is preserved. So it means that each tangent space has hyperkali geometry situation. And then it gives you certain... If you choose a structure, it gives you some almost complex structure.

But also it gives you kind of like three symplectic forms. So your original symplectic form is omega1 plus items omega2 on my manifold. So the real part will be omega1.

And then you get also three complex structure. And for complex structure compatible with omega1, you will have holomorphic 2-form. And take exterior part of holomorphic 2-form. You get volume form at each point.

And this volume form gives this grading procedure as I explained. So the grading is completely automatic in this case. And the space of choices is contractible.

So that's a big conjecture. It's not mirror symmetry. Also it's one category below looks like A-model. You do some Foucault category with holomorphic disks. And another category, it's like B-model.

You do maybe deformation of perfect coherent shifts. But it's on the same manifold and there's no duality. And it's kind of a new statement and not mirror symmetry.

And in fact, I really don't understand deep reasons why it should be true. It's complex and like it's own hyperteller. Yes, yes, but still, yeah. And there are really, yeah, it looks like almost pseudoscience

because we don't have a Foucault category. We don't really know convergence and so on, yeah. So first let me explain why it really has something to do with Riemann hybrid correspondence.

That's why it's a smooth algebraic variety of complex numbers.

And in fact, I want to stress here that it's what we choose only. Suppose we have not one variety, but variety depending on h-bar. Kind of holomorphically depending. So why it sits, it's maybe some simplification.

And all holomorphically depends on h-bar. Like you have not one curve, but family of curves depending on h-bar. Then as a category, we take a variety which is cotangent bundle,

which is compactified fiber-wise by Poisson manifold.

And I claim if we have this family of variety, we can consider a particular family of these categories.

And categories which we will consider, category holomorphically depending on h-bar, will be de-modules over y h-bar. Kind of h-de-modules.

Vector bundles on y h-bar with h-bar connection. And all the story is ingredients for h-bar not equal to zero.

Here we get a limit at h-bar equal to zero. Consider what's called Higgs bundles. Consider coherent shifts with complex support on cotangent bundle,

and support and direct image should be vector bundle on y. So we get these things, and this Foucault category in this case will be... For cotangent bundle consider local object of this Foucault category is representation of fundamental group.

No, this is, people proved it. And how is the constructive functor? If you get cotangent bundle, let's simply assume that y is compact.

It's true also for non-compact case. If y is compact, people proved it.

And the proof is a functor. You get cotangent bundle. And it took Foucault category in whatever version it is. Lagrangian manifolds with local system, or the singular Lagrangian...

No? No, it's not representation of P1, it's kind of... The category is called constructible shifts with locally of y, with locally constant cohomology.

Yeah, that's the right notion. Yeah, it's not representation of P1 here. And what is kind of the reason? If you have an object of your Foucault category, L, maybe with something,

then consider different guys, consider cotangent fiber. It's another Lagrangian guy, a little bit non-compact, but it's not really a problem here, because the intersection will be compact.

We'll get compact space of disk, we'll get homes bit from one to another. Home from I to L is well-defined. And we get some finite complex of vector spaces, finite dimensional complex of vector spaces.

But this object, if you shift y, it will be Hamiltonian deformation. And it shows that its objects are isomorphic and it identifies homes. This gives you roughly this... Sorry? It's part of other congenials, no.

Arnold conjecture eventually follows from all this stuff. And then that's how you map object Foucault category to a local system, and eventually people can handle to prove that it's equivalent. Yeah, but it's kind of very simple case.

It's local system, but we, in the last lecture, explained some singularities, irregular singularities, and so on. This can be done, I think, in the following way. At least for the case of contingent bundle of curve.

Suppose its y is a curve. Then you can see the contingent bundle. Again, compactify.

But it should be only first compactification. I will do something else. So, I remind you what I get. I get a kind of contingent bundle.

And at divisor 10 to infinity, which projects to my curve y, and this divisor 2 has a second order pole. Let's say, for example, y is a compact curve.

Then if I'm interested in some demodules with some singularities, what I do is start to make blowups. I start to make Poisson blowups.

And if I make blowups, I get different order of poles, or different order of zeros of Poisson structures, and eventually there will be certain divisors when I have first order pole. And these first order poles correspond to irregular terms in formal classification of demodules.

So I get this surface, first order poles, and a kind of semi-classical picture, which we considered spectral curves, or something which intersects this divisor infinity only, this smooth point of this first order pole. But now I can do something new.

I can take a point of this divisor when it gets first order pole, and make again a blowup. When I make a blowup, exceptional divisor will have zero order of poles, so it will be part of larger symplectic leaf. So I make a new symplectic manifold. And I can do it several times. And eventually, if I have any curve in my Cartangent bundle,

by these blowups I can achieve the situation that infinity meets only first order poles, but after many, many blowups, I can make it compact completely.

So this procedure gives some kind of modification of categories of demodules. If you look at what you are doing here, you essentially should do blowup once.

You do the following things. Recall that divisors when order of pole of omega is equal to one, they're canonically identified with C. They have canonical coordinate.

They're canonically identified with C, and coordinates are better denoted by something called lambda. And just in the case of regular singularity, if you make a blowup once,

it will be this coordinate lambda here. And so on each divisor, I fix some collection of, finite collection of complex numbers for each logarithmic divisor,

this lambda i, and then I make blowup, I get a new point of the symplectic leaf. So what is corresponding in terms of demodules? Just let's speak about regular singularity.

Suppose I get on a formal disk, I get a bundle with connection with logarithmic poles, and all this data will say the following. I identify, so I get a bundle,

it is not just a meromorphic bundles, but bundles over CZ with a set of delta will be, delta zero dz has no pole at zero,

so it means that connection has first order pole. And then if we consider residue of the connection, it will be endomorphism of the fiber at zero, the data will be the following. I will identify E, will identify with direct sum over i,

E0, E0i, and residue delta preserves on E0i, will be lambda i divided by h-bar times identity operator.

So this will be, all eigenvalues will be no Jordan blocks, and eigenvalues are fixed. And actually it's quite a good formulation,

because exponent of these numbers can coincide for certain h-bar, but still separated logarithms. And if I make additional blowups, I can also handle Jordan blocks. I don't know what to do in higher dimensions, but it's this kind of way to go from d-modules

to something which in classical limit has complex support inside the symplectic manifold. And this is the modification of category of d-modules, in this case. I consider this formal classification, I also choose some sub-lattices and such sums.

And there is a similar story in Foucault, in Foucault category.

In fact, here is a kind of alternative theory, I think the compact one is better.

The alternative theory is, consider not object with compact support, but such a restriction will be shifts with finite support on this union of logarithmic poles.

Do not make blowups at smooth points of log divisors.

So we get another boundary condition, so restriction of zero, that will be something. And on Foucault side, you can see the partially wrapped Foucault category. But if we do blowups, we consider just compact.

It will be a slightly different category. And this partially wrapped is not Calabi-Yau. It's called Recalabi-Yau.

And partially wrapped roughly means the following. This is my manifold text. And now my Lagrangian lambda, for example, a singular Lagrangian, can go to infinity in a certain direction. And then when I define homes I start to move a little bit

to things at infinity. So it's some story, maybe I'll return to it next time. And in this case, the relation with the stock's filtration

will be the following. It's kind of a rough picture of data. I recall you that the stock's data should be interpreted geometrically in case of curves. Not stock this, it's irregular terms.

We get a compact curve, we get some points, and then at points we get certain stock diagrams. And these stock diagrams we can now extend to a conical real Lagrangian guys.

So it will be some kind of non-compact. We get lambda sitting in cotangent bundle to surface, non-compact, singular Lagrangian. And the nice property here in this case,

there's no correction, in this case there's no correction from holomorphic disks, it will be exact Lagrangian, no deformation of A lambda. And then this A lambda, in this case modules, this will be exactly a better description

of Riemann-Hilbert correspondence. And it's related to this complex story that the cylinders go to this logarithmic device's infinity.

And the picture how to map from Foucault category to identify the stock's data is essentially the same. They can see the homes with vertical fibers, and when we approach a singular point, we deform them a little bit, and eventually we see these filtrations.

So how do we see the filtrations in this? I think because the intersection points of this, when we consider this vertical fibers approach singular point, then we'll kind of intersect these things in several pieces.

And yeah, that's roughly the story. Yeah, you can try to think that object of Foucault category is something with support close to this guy, and then the intersections with this vertical fibers will be.

Similar thing also explains this Riemann-Hilbert correspondence for quantum modules.

Like this. I put this in symplectic form, and I have algebra.

Algebra will be C of, depending on h-bar, z1-hat plus minus z2-hat plus minus modulus of relation, d2-hat z1-hat is point of h-bar. I have this algebra.

And again, the complication is, let's say, some toric surface.

And so we get toric surface, so the boundary gets a bunch of C-star. Think of C-stars. If you have such Poisson surface, then consider modules such that, let's say, in classical limit,

the curves which do not go to intersection points, zero-dimensional, they all go to other devices, so in tropical limit you get some curve

which goes tropically on several directions. And eventually one can make blow-ups at these points and make the C compact, if you want to go to one language or go to this wrapped foci,

if you go to another language. Actually wrapped foci is not a very bad case, at least for absolute value of Q less than one. Now for, if it means that h is not in IR,

actually,

how to map foci category, how to map to coherent shifts on elliptic curve

plus two anti-hardener single filtration, which I explained in the last lecture. Yeah, let's assume this picture in this diagram I don't have vertical directions.

You can change, apply this to the transformations, you don't have vertical direction. Then I get a vector bundle with two anti-hard single filtration and the picture is the following. You can see that, like in previous story, you can see the home from object, which will be the following.

It will be for each ZW, you can see the C*, Z1 equal to Z and W arbitrary. It will be my C*, cross C*, I consider some fiber,

which will be Lagrangian sub-manifold and local system will be of rank one and monodromy is multiplication by W. And I can see that this is unobstructed object of foci category, then I can make a home from this thing to your object.

And what you get? You get something holomorphically depending on bundle on C cross is coordinate Z times C cross is coordinate W. It's not locally constant,

because the deformations are not exact. But there is some, again, piece of wisdom in this foci category. If you deform Lagrangian manifold by non-lagrangian deformation, it's the first cohomology class of it, the speed of deformation.

It's equivalent to make local system in this first cohomology class. And then it's easy to see that this holomorphic bundle has flat connection, a long folation, a kind of holomorphic, one folation given by vector field

Z dO dz minus 2 pi I over h bar W dO dW. So on this star square you get this holomorphic, this homo don't change this direction, and the quotient space will be elliptic curve.

Exactly this elliptic curve. And similarly, if you approach now Z goes to 0 to infinity, then this intersection points will separate in several groups and you get filtrations. So it's automatically,

if Z goes to 0 or infinity, you will get two filtrations.

It all looks like some kind of pseudoscience, because I cannot get precise definition of OK category, I don't know convergence, blah blah blah. But there are some corollaries which are completely rigorous, formulously formulated conjectures. The first, I think it's,

this Riemann-Hilbert correspondence for quantum torus hypothetically generalizes to high dimensions. I can see the C star cross C star standard to power 2N.

The algebra will be Ah times N. I consider this product of arbitrary number of copies. What I can formulate, I will not give complete details now, but I can formulate the hypothetical equivalence,

N here, you're right. And let's, and let's, less than one, and assume that it's this.

Then holonomic, Ah bar to power N modules, are equivalent to certain explicit category, which can describe completely rigorous,

using all this machinery, and the category is the following. It will be inductive limit, of some full subcategories, under what set, I consider,

Lagrangian fans in R to N. What is it? Maybe I denote them again lambda. Lambda is finite union,

or some finite union of some pieces, and lambda alpha is, is a convex, full dimensional,

cone in a certain Lagrangian subspace, and alpha is defined over rational numbers. And cone with, kind of rational, it will be also rational cone,

and with the properties that, intersection of lambda alpha is lambda beta, interior of lambda alpha is lambda beta, is empty for alpha and omega beta.

Yeah, so good. I don't know how to draw four dimensions, you get some kind of arbitrary correction of this convex cone meeting each other. And then for this correction,

one can make certain dg, yeah, this is a typical singular Lagrangian guy, and then we get this small neighborhood, and then we get this whole algebra lambda,

and in particular we get this transversal disks and homes between them. We get certain dg category over integer subject objects, are set of alphas,

using, and this in fact can be calculated completely explicitly, because in this conical situation, it's easy to reduce the conical in cotangent directions, case and reduce to a series of shifts with some micro-local support, so it will be totally elementary object.

Yeah? In the one-dimensional example, or two-dimensional example, you have the Tate curve, so to speak, so in the higher dimensions are you getting some toroidal degenerations of abelian varieties

related to those fans? No, no, no, there's no abelian, no, it will be power of curve at the end of the day, it's nothing. Just one each. I have one each, yeah. But what is the story? We have this algebra,

we have this category, maybe called c lambda, and now consider another category, consider the perfect complexes of the nth power of elliptic curve, e is elliptic curve, yeah?

You get another category, called b, and for each Lagrangian, then this category, there was something nice here, sp to nz, maybe universal cover acts on b,

by Fourier-Mouquet transforms, then b contains certain category b0, b may be horizontal, I'll explain in a second way, it will be perfect complexes with finite support, or zero-dimensional support

on e to the power n. Just bunches of points and some commutingly important operators. This category is invariant under j, l and z, it's preserved by the action of j, l and z, dissecting by automorphism of e, n,

and then it implies that by equivalence, if you consider quotient space, it will be kind of like z-covering, such that it's grading, of the set of all rational Lagrangian subspaces,

j, l and z stabilizers of one subspace and from one subspace and go to another, is a covering of space of rational Lagrangian subspaces. That means that for each subspace you get, if you choose this z-grading, you get a well-defined category,

and now this kind of right-hand side of Riemann-Hilbert's correspondence will be the following. It will be functors from C lambda to B, category of functors, such that for each object, object alpha,

maps the object in this subcategory, corresponding to Lagrangian subspace l alpha. In the case of an n-pool-1,

you get exactly this picture with these filtrations, because you just get a bunch of rational rays. This subcategory will be semi-stable, think of the slope, and if you analyze what it thinks, it means you get two filtrations.

So that's one kind of non-empty and precise statement, and it's amazing, it works in precise conjecture, and it's amazing that it works in high dimensions, so it's much cleaner than d-modules when you have this irregular singularity, so it's really much simpler.

And another statement which is almost precise, when you get this in big correspondence, you get some families of quantum spaces and then Foucault categories, but then different families can give the same symplectic, same classes in H2,

like, for example, consider d-modules, and you change the curve. So you get extra parameters which you cannot see on Foucault side. And then it says that, forget about Foucault categories, it says that this object with compact support in the quantum cases, they're isomorphic under some kind of isomanodromic deformation.

Yeah, that's purely, maybe it makes sense even though form of power series, yeah, so it's kind of interesting statement by itself without referring to Foucault categories, because you get two more parameters on complex side. Can you give me an example?

No, for example, a family of curves depending on h-bar, and consider h-bar connections, the claims that you get equivalent categories for any two families. Constant family, non-constant family, yeah. Okay, thank you.