glad and delighted to talk in honor of Maxime Koncevich. And so I will talk about interactions between two subjects, deformation quantization and the Hadelberg geometry. And for both of these subjects, of course, we all know the amazing influence of Maxime's work. So yeah, I mean,

so maybe to start with, let me mention that this is a joint work with several collaborators,

Tony Pontef, Michel Vacquier, and Gabriele Vizzozzi, and more recently with Damien Kalak. And the main goal of this talk is to give an overview of the construction of deformation quantization for what we call shifted symplectic or Poisson structures that will be explained in

the talk a little bit later. And maybe I should mention immediately the important corollary of this construction is the existence of a quantization or deformation quantization of G of X, where a bungee of X is a certain modularized space of G bundles on X, but considered

as a derived modularized space. So a big part of the talk will be concerned about defining this properly or at least giving some ideas about what is this object. And more importantly, the space X is going to be of arbitrary dimension. That's the purpose of introducing

this shift here is to take into account the fact that X can be of any dimension. So some references, or maybe I can skip this slide very quickly. There is a paper where

these shifted symplectic structures are defined and studied. And there are two preprints where most of the content of this talk here, we find some details about the content of this talk. And there is a work in progress about shifted Poisson structures that will contain the details of this talk, the full details. And here in progress, I must admit I have lower

standards than Denise about what in progress means. We didn't start writing yet. But it's in progress in the sense that we think the maths are done somehow. And we are very motivated to start writing, of course. OK, so I will start by some overview of deformation quantization

for manifolds that will be generalized later. So let me start with Fedorov quantization, or at least this is what people refer to Fedorov quantization. So I start with a smooth algebraic manifold. And omega is a symplectic form on X. Here everything is

going to be algebraic. So it's not even complex algebraic. It's algebraic over a field of characteristic 0. So I will never mention any transcendental or analytic argument. Then based on fundamental ideas of Fedorov, I think Bezukovnikov and Kaledin

constructed a canonical quantization of this pair X omega, where this is realized as a formal deformation of the category of coherent or quasi-coherent shifts on X. And the two

important properties of this construction, first, it's purely algebraic. So as I said, it doesn't refer to any analytical machinery or anything. It works over any field of characteristic 0. And it's also based on formal geometry, and in particular, the formal Darbulema.

We will see a little bit later that this is going to be an important piece of what I'm going to say today. And the strategy of the construction of this quantization, I can explain it very quickly. Maybe Dima can correct me if I say something wrong. So first, you start at the point X, and you look at the quantization of the formal completion.

So I take this X hat here is the formal completion of my variety at little x. And I can quantize this formal completion because the quantization here is realized as the category of modules over some viral algebra with respect to just the symplectic form on the tangent space.

And then the important part of the work is to glue these local constructions. And for this, there is a tool which is called the Grothendieck connection, or at least I've seen that this is called the Grothendieck connection. Maybe it has other names. And which is a tool to say how these families of formal completion varies when

x varies in x, and to glue these locally defined categories, which are modules over the viral algebras here. And maybe an important comment is that the viral algebras

themselves do not glue, but the categories of modules do so. So there is something that you cannot really expect the existence in the algebraic setting, at least the existence of a deformation quantization on the level of rings of functions, but only on the level of categories of modules. This will explain why later when I will try to generalize this, I'm always going

to consider quantization as deformed categories and not just deformed algebras. There is another deformation quantization, which is the famous Koncevich's deformation quantization of Poisson manifolds. So now I start with a

smooth algebraic variety, and I have an algebraic Poisson structure on x. And Koncevich proved that p can be used to deform or canonically deform the category of quasi-quotient chips on x. And again, this is a purely algebraic construction, and it's based on the so-called formality theorem, which has a deep statement,

which I don't claim to understand all the subtleties in this statement, and maybe I will avoid referring to it in this talk. I will mention some generalizations, but which are some kind of easy generalizations of this theorem here. Okay, so our goal is to extend these two

constructions to the case where x is no longer a smooth scheme. So we want more general objects, and possibly it's going to be a very stacky object. So it's going to be like an orbifold, but

where the points have big stabilizers, and these stabilizers can be non-locally constant, the dimension can jump, and so on. So they are going to be very stacky thing, and they are also going to be very derived. So this is something I will explain in a minute, meaning they are very far from being smooth variety, because they are very

singular, but they come with some extra structure that looks like a non-linear structure that makes them a little bit closer to be smooth schemes than what they look like. Another generalization is that this symplectic structure or Poisson structure can be shifted in a way that

is going to define later, but the idea is that instead of having an isomorphism between the tangent sheaf and the cotangent sheaf of the variety x, I will have an isomorphism between the tangent complex, and because of this stacky here and of this derived here, this is now going to be a complex of vector spaces, a complex of shifts of vector spaces, I will have an

identification between the tangent complex and the cotangent complex with a shift here, and n can be any integer. We will see that it's related to the dimension when we will apply this to g bundles on the space, it has some relation with the dimension of the space.

Okay, so what kind of examples we have in mind for x that motivates this work? First, the modular stack of local systems on spaces of arbitrary dimension. I think I already mentioned this at the very beginning. So of course the emblematic example is local systems on the

compact Riemann surfaces, but we want to extend this maybe to three-dimensional manifolds or dimension manifolds of higher dimensions. And already for compact Riemann surfaces where the shift we will see is zero, these spaces come already with some interesting non-trivial

and stacky structures that have to be taken into account. So even when the shift is trivial say, there is something interesting. Another kind of example is the modularized stack of compact objects in a Calabi-Odigi category. So this can be thought as a non-commutative version of

example one in some sense. So these Calabi-Odigi categories can be thought as a non-commutative manifolds, and this modularized stack of compact objects as the modularized space of points in these non-commutative manifolds in some sense.

And finally, there are also examples like BG. So G is a reductive group here, and BG is just the classifying space of G, say considered as an algebraic stack. There is this perf object, which is a modular for perfect complexes. So it's a modular of complexes of finite dimensional vector spaces up to quasi-morphisms.

And we aim for having a quantization of this guy also. And like higher examples, like higher Ellinger-McLean spaces for V coming with a quadratic form, I mean, can be symmetric or anti-symmetric depending of the parity of n. This also provides examples

where quantization can be done. So to explain what the main statement, I should start by telling you something about derived schemes. So everything is over a field of characteristic 0. And in algebraic geometry, the

schemes are locally modeled on spec A for A being a commutative ring, or say a K algebra, because everything is over K. And in derived algebraic geometry, it's almost the same thing, except that the local models for the derived schemes are given

by spec A for A, a commutative DG algebra. And this can be defined as follows. So first, my commutative DG algebras are not positively graded. That's a convention. So they look like this. Nothing in positive degrees. That's a convention.

And there is a notation that I will use often. I think I realize now that I never use this notation later. So it's completely, maybe pi 0. We will see. But that's a general convention you can find in literature. And then the spec A I was mentioning. So A is a DG algebra now. So A is of this form here. And spec A, by definition, it's a space, which is the spectrum of

the first cohomology group, the H0 of this thing, so A0 modulo the image of this guy. This is an algebra. I can take it. This is an affine scheme corresponding to this algebra. And on it, I have a sheaf. And now this sheaf is a sheaf of DG algebra itself. It's not

a sheaf of ring anymore, but this one is a sheaf of DG algebra. And it is defined as follows. So if I have U, an open in this spec here, it's given by, say, a basis for open is given by the locus where function f does not vanish. This function f, it lives really in pi 0,

but I can lift it to H0. And I can localize the DG algebra with respect to f. And I get a DG algebra. That's the value of OA over this open U. Now, maybe you're going to tell me that this f here, there are some choices. I can choose a different lift. This will be only

quasi-isomorphic. So this sheaf here is not actually a sheaf on the space, but it's a sheaf well-defined up to quasi-isomorphism. You can construct it in many different ways. But there are some issues about things not being, you have to glue things up to quasi-isomorphism to

make sense of this definition. And global derived schemes are defined as, well, naturally, it's the same way as we define our schemes, say. It's just there are pairs, xox, where x is a space. That's the underlying space of my derived scheme. And

ox is a sheaf of commutative DG algebra on x. And the condition is that, locally, it looks like a local model. So locally, it's equivalent to spec A for AODG algebra. So that tells you what are these objects called derived schemes.

Now, these derived schemes, unfortunately, form an infinity category. So that's one of the homotopical flavor of the subject, which is not something bad. It's something good. I mean, even if you don't like infinity categories, it's something good, believe me.

And but the nice thing is that if really you look at the theory from the point of view of infinity category theory, derived schemes behave very much like schemes. So the usual yoga of algebraic geometry of schemes extends to derived schemes as soon as this infinity

categorical nature here is taken into account correctly. So for instance, we can define fiber products. So in the same way that we have fiber product of schemes, we have fiber product of derived schemes. Derived schemes have derived categories. So if I have a derived scheme, I can talk about quasi-coherent sheafs on it, which are essentially sheafs of

DG modules over this sheaf of DG algebra and so on. I have a tile, smooth, flat maps. This is just a sample of examples. And one important thing, they have cotangent complexes, which is the correct analog in the setting of derived schemes of sheafs of one form or

cotangent sheaf of smooth varieties. We will use heavily this thing in a minute. So two important facts. I just mentioned them, but let me be a little bit more precise.

A derived scheme has a derived category. Again, it's an infinity category, but let's just use the world category here. So it's a derived category, DQQ of X. It's defined locally. So DQQ of spec A is just a derived category of A DG modules. And then globally, you have to glue these local

categories. So one way is to talk about sheafs of OX DG modules that locally satisfy some conditions. Or there is a gluing procedure by just integrating these local constructions over a fine open subset in X. And another important object is the cotangent complex. So

a derived scheme has a cotangent complex, which is an object in this derived category. And you have to think of it as the cotangent sheaf of X. But it's really a complex in the sense where cotangent complex is a bit misleading because it's a DG module. There is an underlying complex of sheafs behind this object here. And you can define it locally on one of these affine

models. This is just given by cotangent complex of the DG algebra A, which is an object here. It's an A DG module. So concretely, take a DG algebra, maybe make it quasi-free up

to quasi-isomorphism, then compute KL1 forms on this DG algebra. That's a DG module. This is a model for this LA. And work a little bit to prove that this can be done globally on X. Characteristic zero. I didn't say that. I did. Everything is characteristic zero.

OK. So derived schemes are nice. But derived schemes are not really enough for what I'm going to talk about. So I'm going to introduce derived stacks now. So it's a complicated definition that I will not give to you. But I will give an idea of what

they are. There is a concrete approximation of what is a derived stack. Oh, this concrete thing is attributed to Gabriele. OK. So derived algebraic stacks, by definition, or by approximation, if you want, are smooth groupoids in derived schemes. So what does

this mean? It means I have diagrams like this. So X0 here is a derived scheme. That's my underlying space, if you want. And X0 is also a derived scheme. And I have two maps, source and target. I have an identity map from X0 to X1. So you want to think that X0 is a derived scheme of objects. X1 is a derived scheme of maps. And then you have

a composition map from, if I get two maps that match on X0, then there is a third map, which is the composition. And et cetera, meaning a bunch of actions. I'm not going to write. If you want to make this correct, you have to talk about a single groupoid kind of object,

something like this. But there is a way to, as I said, it's a concrete approximation. Oh, and I forgot to say something. I said smooth groupoids. So there is a condition is that these two maps are smooth maps. It's not written on this board, but this is important.

And so, morally, derived algebraic stacks are quotients of derived schemes. So here, I take the quotient of X0 by an action of a smooth groupoid. This is what it means.

OK, so a large class of examples are given as follows. It's really a large class of examples, in the sense that many derived algebraic stacks might not be of this form, but have the stratifications that are of this form. So it generates a very large class of examples. These are quotient stacks of affines by a group G. So here, G is a reductive group

acting on a commutative DG algebra. So this action, you can make sense of it as saying that the DG algebra A lives inside the hop G. I think Sasha Goncharov already mentioned

some representations of a group inside the category of representations of another group. This is a little bit the same thing. I have a DG algebra inside the category of representations of G. And the corresponding groupoid, because I was telling you that you have stacks of groupoids, is the action groupoid. So this spec A is X0. The first map is the projection.

The second map is the action. And this form a groupoid because I can compose elements in G. And the map in the other way is the identity map, and so on and so forth.

And I will come back to this example to explain what's the derived category, cotangent complexes, and so on. This is a nice class of examples where everything is rather explicit. So the geometry of derived schemes. So I was telling you that the geometry of schemes extends to derived schemes. And the geometry of derived schemes extends to derived algebraic stacks in a non-surprisingly. It's not very surprising, but for instance,

we have derived categories. So if I have a derived algebraic stack, I do have a derived category of quasi-coherent sheaves on it, and I do have a cotangent complex. These are the two main ingredients in this talk that we need.

And I will also mention the tangent complex, which is a dual to the cotangent complex. It's the analog of the tangent sheaf. So what's happening when I have one of these quotient stacks? Here everything is nice, and it can be described purely algebraically.

First, the derived category is just the derived category of G-equivalent ADG modules. So A is a DG algebra inside of G, and I can take DG modules inside of G. And this, by definition, if I localize along quasi-isomorphism, defines me this equivalent derived category.

And if this is defined correctly, it's always equivalent to this purely algebraic description here. There is a definition of DQCO effects in general that gives you back this. This is not a definition. It's a theorem, if you want. That's the test, exactly.

And the cotangent complex is given by, I hope I didn't make a mistake here, because I always mix fibers and co-fibers. I think it's OK. So I have the cotangent complex of the algebra A. This is the dual of the action. So little g is the Lie algebra of G. It acts by derivation on A, so I have a map from forms on A to these things, the dual of the

Lie algebra tensor A. And I think the fiber in this category of equivariant DG module, mean really the cone, or the co-cone, the homotopy fiber of this map. And this is my object LX inside here. So it's a very natural formula. I'm just saying,

I take the tangent space of spec A, and I mode out by the Lie algebra of A. Now, just for applications, I will mention the higher stacks. So there is a notion of

derived algebraic higher stacks, which is a mixture about what I just said, and higher algebraic stacks in the sense of Simpson, for instance. So you can take quotients of quotients of quotients several times by taking actions of higher groupoids and create even the larger class of the higher stacks. And they are interesting because they

allow you to treat some examples. So for instance, they are needed to include the following important examples we have in mind. The first important example are shifted cotangent space. So suppose X, to start with suppose X is a smooth manifold. This T star

is just a total cotangent space of X. Now, I want to shift this. Then there is a formula. I can just take the symmetric algebra over the tangent complex with a shift by minus n. Functions

on the cotangent is the tangent. So this is like a shift of the G algebra over X. It's not concentrated in negative degree, as I said. So you have to be careful about what this spec means. But still there is a way to define this correctly. And that's by definition the

shifted cotangent. And there will be important examples for us because they carry canonical shifted symplectic structures in the same way that cotangent spaces are. They have a canonical symplectic form. So these are important basic examples to produce symplectic,

shifted symplectic DR stacks. And you really need higher stacks because as soon as n is positive, this is actually very stacky. It gets more and more stacky if X is a scheme, say, and n is 1. This is a 1 stack. But if n is 2, this is a 2 stack, and so on.

So it's a linear stack over X. And it's interesting already when X is a smooth variety. So out of a smooth variety, you have a family of higher derived algebraic stacks which are non-trivial and meaningful just by taking this formula. So out of smooth variety, you create interesting examples like this. And the second other example, so it's the example here. So this was the

first one, shifted cotangent. The other one is perf. That's the stack of perfect complexes. It's a classifying object for complexes with finite total cohomology. And you want to take

these complexes up to quasi-isomorphisms. And this is why it really is a higher algebraic stack and not only a 1 stack, say. Well, I won't say more about this, but we think it's a rather fundamental example to put in the game here. And you really need derived higher stacks to

make sense of this. OK, the derived stack of perfect complexes. Oh, surprise. Oh, and there is a slight generalization. This is the object in the derived category of k,

the base field, say. I can also take a moduli of objects, say, compact objects in a given nice enough digi category and produce higher derived algebraic stacks this way. OK, so now formal localization. So I'm going now to explain how these, so I mentioned this Basel-Kavnikov-Kaledin's

work where you start by doing something formally at a point. And then you move the point, and you have to glue all this formal completion together by means of this rotund connection. So I'm going to explain how it works in this derived setting here. This is something I call

formal localization, which is not maybe a very good terminology. But it means that we are going to look at the family of formal completion at each point and glue these things globally on the moduli space, I mean on x. But I want to do this in the setting of algebraic stacks I just mentioned. OK, so why do we do this? We do this because of the non-vanishing

of local Durham cohomology in the algebraic setting. This is already true for a smooth algebraic variety. I can localize Zariski or et al locally as much as I can. I will never make the Durham cohomology zero locally by looking at the Zariski or et al topology.

And this prevents for having a Darbu lemma, because if you have something in Darbu form, the corresponding symplectic form is exact. So if you have a symplectic form which locally is non-zero in H2, then you will never find a Darbu coordinate. And then you cannot apply the

strategy saying, I have Darbu coordinate, I take the Vail algebra, and so on. So this is why we really use this formal localization is because we have to localize more than just et al locally. And one way to do this is to localize formal locally for which the Durham cohomology vanishes. So that's the explanation why we do this.

So this is what I was just saying. It prevents any Darbu type of statement for Zariski or et al topology. Well, there is an exception here I should mention. It's a little bit too early to mention this. And in strictly negative, I didn't introduce shifted

symplectic structure. But when n is strictly negative and x is a scheme, maybe it's also true for stacks in some sense. Joyce and collaborators actually proved that the Zariski topology is enough to have Darbu coordinate locally. So there is something very specific to negative shifts where forms are always zero in Durham cohomology.

OK, so the solution is the formal localization. And it goes as follows. So we're going to work at the formal level. So x is going to be a Dirac algebraic stack, little x a point in it. And this is going to be the formal completion of this big X at little x.

I'm going to explain how it is defined concretely on the next slide. So why is this going to work? First, this formal completion satisfies that Durham cohomology vanishes. Durham cohomology here, well, I told you that there is a cotangent complex. You can continue the work and see that there is a Durham complex using this cotangent complex

and define Durham cohomology and just prove that in a formal Dirac stack there is no cohomology because Durham cohomology doesn't see the nilpotent structure. So it just reduces to the base point. And then the second thing is that I will have to explain how this family of formal

completion or individually each formal completion if I make the point x moves in x, they can be glued to reconstruct the space x itself, or at least some invariance of the space x itself. And this glue here that we use is something called the Grothendieck connection

I'm going to explain. Is the Grothendieck connection the same as the Fedorov connection? I don't know. I have the feeling that no. I have the feeling that that Fedorov has this thing that he chooses a connection for some... It's not this? Or maybe it's the other way? It's the same connection, and from that he gets the infinite connection.

Okay, Dima. Where is Dima? Is it the same thing as Fedorov's connection? Yes and no. Yes and no. Okay, so let's move on.

This is what you thought as Fedorov's connection. Okay. I apologize to call it Grothendieck connection. I don't know. I don't know. I mean, someone told me it's called the Grothendieck connection, but I've never seen any paper of Grothendieck. It's there. Yeah, I mean, it's there. We will see why it's there. Okay. So it's the connection on, if you want, it's the

connection on jets or something like this. Let me define it in a different language. So let me start with the Dirac algebraic, possibly higher, stack. And we define, I mean, I learned this definition from Simpson, at least as a functor. I think that's his

definition. I'm pretty sure. It was called the formal groupoid by maybe authors before him, but I think as a functor, at least I learned it from him. Okay. So it's the functor X Dirac that sends, okay, so Dirac algebraic stack is a functor on DG algebras, in the

same way that the scheme is a functor on algebras. And I define a new functor sending A to, I kill the non-Dirac part, so I just keep the underlying algebra, if you want. And I take the reduced part of it, and I take the points of X with coefficient in this algebra. So it's a way to identify in X all the points which are infinitesimally closed. Each time you

have two points in X which are very closed from the infinitesimal point of view, or behave to the same formal neighborhood, you just identify them. Is this an actual higher stack? It is an actual higher stack. It's no longer derived because of this formula.

Well, if you apply this to BG, what do I say? No, it's no longer derived because it's actually eta over the base. But it depends how you make sense of it. Okay.

I had a comment, and I forgot. Okay. If it comes back, I will let you know. So it's a new non-algebraic, so it's not an algebraic stack anymore, but it's a Dirac stack. It's a functor on commutative DG algebras. And it comes with a map from X to X Dorham, which is the quotient map. And you really want to, I mean, Tony Pantheff will give a

lecture later this afternoon. I strongly encourage you to go there, of course. And he's going to tell us that this X Dorham here is a space of leaves for a certain notion of foliation. And here there is only a unique leaf. But now the space of leaf, suppose you have a foliation,

you want to identify infinitesimally the points along the same leaves. And this is what you get if you do this. So it's like there is a general notion of foliations on derived stacks, out of which this X Dorham is the final or initial object. I mean, Tony will tell us about this later.

Okay. So what are the fibers? So P is my projection from X to X Dorham. What is interesting are the fibers of these things. So the fibers are exactly the formal completion. So if I pick a point in X Dorham, so it's just a K point, just a point of my underlying,

of my stack X, just a K point. I take the fiber, I find a formal stack, which is the formal completion of X at X. If you don't know what's the formal completion, take this as a definition. It works perfectly fine. And it's a little bit more general than this. I have a Cartesian square

where this is my projection two times, and this is the formal completion of the diagonal of X. So again, if you don't know what it is, take this as a definition. But what I mean here is that there is a general definition of formal completion of the diagonal, and that you can prove that it goes like this. It has a tautology to prove this, but just by definition,

almost. Okay, out of which you recover this thing, because if you take a point in X, take the fiber, you will get the fiber of the formal completion of the diagonal, which is just the formal completion of, at that point. So it's an interesting, this map here is interesting in the sense that it gives you the fibers of the form, it gives you a family

of formal completions of X at all the points. And it leaves over this X the ham, and the fact that it leaves over this X the ham exactly tells you that it has a connection. So there is this, you got here that objects like sheaves on X the ham are like sheaves with connection. So if I have an object over X the ham, you want to think of it as a connection

of the corresponding family. So this diagram is the quotently connection by definition. Just saying that I have, I have a vibration, I have a flat connection on it, because it leaves over X the ham. So this is my vibration. It has a flat connection because it comes from something over X the ham, and the fibers are these things.

Okay, what's the, what is it good for? It's good for because of the following formula. Suppose I have a sheaf on X, can be a quasi-coherent sheaf, but we will see it can be a sheaf of categories also. I can take global sections of X with coefficients in O, E, I should say.

I apologize. And because global sections are compatible with direct images, because they are themselves direct images, this is the same thing as global sections of X the ham with coefficients in this push forward here. And the push forward is interesting because

if I take the fiber of this push forward at a point there, I do get the value of the sheaf on the fiber, which is a formal completion. So this is something called, I mean, this is something like extremely classical, and which says essentially that if you have like a vector

bundles on X, sections of that vector bundles are the same thing as flat sections of the bundles of jets in it, something like this. Did I say it correctly? You don't care, okay. Okay, key observation. My sheaf E can be a sheaf of categories.

So I can look at U goes to DQCO of U as a sheaf of categories which are linear over X. So that's the sheaf of OX linear infinity categories over X. That's a tautological sheaf. It's like the category called structure sheaf.

And then if I apply what was on the last slide, I do get that global sections, of course, is just a derived category of the total space. And I can write it like this. Global sections on X the ham with values in this push forward.

In other terms, objects in the derived categories are families of objects in the derived category of the formal completions, which are flat for the Grothendieck connection. So to give an object in the derived category of X is the same thing as to give a family of objects over each of the formal completions in a way which is flat with respect to the

Grothendieck connection. And this is what we're going to use to obtain this quantization in a minute. We will actually deform each of these categories in a compatible way with a flat connection. So by this formula, I will deform this category here.

Let me state a theorem. I should have mentioned all the authors for this theorem, so I will do it orally. Let me state it first. X is again a derived algebraic stack. It can be

a higher derived algebraic stack. It has to be with some mild condition like locally of finite presentation. And the statement says that there is a sheaf of OXDG Lie algebra L, which lives on X, such that this formal completion of the diagonal, which is something that lives over X, is a formal spectrum of a certain sheaf of DG algebra.

And this sheaf of DG algebra is the Shevaly complex of the Lie algebra L. So why is it a completed Shevaly complex? Let me avoid mentioning completion here. So it's just a symmetric algebra over the dual sheaf of L,

shifted by minus 1. And the differential here is the sum of the co-mological differential and the Lie bracket that gives me a differential. And as such, this is sheaf of DG algebra or commutative DG algebra on X. And the formal spectrum of that thing gives me back the

completion of the diagonal. So one comment. This L is the tangent complex of X with a shift, and the bracket is the class of the object X. So I should mention here Misha Kapranov

for a proof of this statement when X is a smooth manifold. And this theorem here of Benjamin-Rényan is an extension of this to the case of general derived algebraic stacks or derived algebraic n-stacks. It's a way to control

completely this formal completion of the diagonal just by a sheaf of DG lie on X. So on a technical note, when you say DG lie, do you mean L infinity? No, I mean DG lie. I'm sorry. I actually mean DG lie.

Then, of course, you have to be careful about that. But the thing is, this L is going to be perfect, so the dual makes sense and so on. There is no strange thing happening. But if you want, I can also think of it as an L infinity Lie algebra. Of course, I can. So what you're saying is locally it's really nice.

It's glued in some infinity category. OK, the theorem lies a bit. Let me come back. So I said that the formal completion of the diagonal is a formal spectrum. So it's like I find at least as a formal stack.

I'm lying a little bit. I mean, it's not Benjamin who is lying. It's my way of stating his theorem. I'm doing some approximation here because I'm avoiding to talk about the stacking phenomenon. So this formula is not quite true. You have to do something a little bit more complicated. But still, it's perfectly a good enough approximation to perform the

quantization. So I will continue to think that this is true. Oh, that's not true. This is true. It is true. This statement is not true. What is true is that it's good enough to do the quantization. So what is true is that quantizing this shift of algebra is enough to quantizing the whole space.

OK, it was just to be completely honest, but maybe I should have just avoid saying anything. It's true, for instance, when x is a scheme or where x is a derived dell'indman fault stack, this is completely true.

OK, and another way to interpret the statement is to say that the Dirac category of the formal completion as a category over x is actually the category of modules over this shift of algebra.

And as we know that this leaves over x the Ram. I said that there is this Grothendie connection on the formal completion. These are functions on the formal completion. So the Grothendie connection extends. So this really leaves on x the Ram. So there is a shift. It's a completed shift of commutative DG algebra, AL, on x the Ram, whose category of modules

gives me back the Dirac category of x, because now if I take DQCO of x, these are flat AL modules, and these are just AL modules on x the Ram, if I see that LL leaves here. OK, so what I've constructed is a shift of commutative DG algebra, which is completed.

It leaves on x the Ram, meaning it has a flat connection, and flat modules gives me back objects in the Dirac category of x. There is a small thing here to say that the L itself does not leave on x the Ram, because the L is the tangent complex,

and it's not flat a priori. So L itself does not descend to x the Ram, but the Chevalier complex does. And this is related to the fact that we need to deform the category and not the shift of functions and so on. OK, so how do I perform deformation quantization now?

I'm going to deform this shift as a shift of algebra on x the Ram. So let x be a Dirac-Dirac stack, possibly higher, and then I fix an integer,

and I give this definition. So a deformation quantization of level n on x is a formal deformation of its Dirac category, which is considered as an e-n monoid or infinity category. I will say what this means now.

So it's not only a deformation of the category, as we saw for deformation quantization of Poisson manifolds or symplectic manifolds at the very beginning, for which this n was zero. But now I have these shifts, and then it means that I need to take into account the monoidal structure here, together with some certain kind of symmetries on it. OK, so what e-n infinity category means?

Well, it consists of an infinity category T with maps like this. So I have for any k, n can be negative. Don't leave before I say it. There is a trick that you won't like, but

OK. So it consists of maps where I have the space of configuration of k points in Rn. Infinity category is kind of linear. It's linear, and these are linear functions. It can take it to be a digi category to be more

careful. And this is the configuration space of k points in Rn. And I have a map with two infinity functions. And this is the tensile product of T by itself. So it means I have a family of multilinear operations parameterized by the configuration space, plus some associativity and equations and so on. So if n is negative, of course,

this doesn't make sense anymore. And I use this trick. I hope you won't get mad at me. I'm going to do this. So e minus n deformation makes sense. So I will say that a quantization

of level n is an e minus n deformation. But the parameter now has a degree, which is 2n. Sorry? Why n? Could be e, whatever you want. You know, I think the next talk I will give on the subject will ask the same question. I will give the same answer. I don't know. I mean, that's the natural thing to do.

What is behind here is that e n monoidal deformation over a parameter in degree zero or some degree is the same thing as e n plus one monoidal deformation, sorry, e n plus two monoidal deformation with a parameter shifted by two on the left.

Yeah, it means that you remember the parity. It's exactly the same notion. It's exactly the same notion, but you need a formality to make sense of this. There is something that's- It uses formality. It uses formality. I don't know how to make sense without it. OK, it's- OK, it makes sense, OK. So maybe it means that brutally you two-periodize the situation

and remember just odd and even shifts. It means a little bit more. It's like semi-periodic somehow. Characteristic zero, oh, come on. Characteristic zero. Can you make simple algebra system each algebra? I can make simple algebra if you want. Yes, but then these symplectic structures,

I don't know how they work. I'm sure that Dima knows how they work. OK, there is a small thing that this A L is not bounded, so I need really to take spectrum of simple show gadgets. Just to double check, negative here is the stackier situation, is that right?

No, no. Negative is the derived situation. On the shift of algebra, negative is the derived situation and positive is the stacky situation. So this A L goes a little bit on the positive side if you are in the stacky situation.

OK, so it is meaningful for all n. OK, I don't know. Yeah, I'm not comfortable with this, but this is the only thing I know. There are two exceptions when n is minus one and minus two. There are things you can try to do, but when n is very negative, that's the only notion I know about.

It's very special and it's probably the hardest. OK, of course. Now there is a reason why it is the hardest. OK, I will say it. So what do I want to say? Deformation quantization two. So let's suppose I have a

derived algebraic stacks, possibly higher, and I have this DGL on it and I have this A L, it lives on extra ham. Now there is this lemma, which is not very hard to prove, why you need to work a little bit maybe, that if I have a formal deformation of this shift of algebras as something over extra ham. As an E n plus one algebra, this defines me a

deformation quantization of level n on this guy. Yeah, I'm not deforming the Lie algebra, I'm deforming the Chevalier complex. The problem is that L doesn't live on extra ham. That's the problem. A L lives on extra ham, but not L. L is the tangent complex. OK,

so if I want to construct deformation quantization of x, I have to deform this L L as an E n plus one algebra. And again, with the same convention when n is negative with this shift thing.

Oh yeah, this is a comment that it sounds like a stupid statement. It's not because of this approximation I've made. This is not interesting. Oh, sorry. Why is A L an E n plus one algebra? A L is a commutative algebra, so it's E n for any n, in a canonical way.

Can you use the Poisson bracket yet? No, no, no, no. Deform this commutative. Yeah, exactly. I deformed the commutative bracket. Yes, but then I thought maybe you were going to use the Poisson bracket. Later. The x is just a stack, there is no Poisson structure on it. Of course. Oh, not yet. Not yet. I'm just saying that if I want to deform this guy, I just need to deform this algebra here.

OK, now I assume that x has a simplex structure of degree n. I'm not going to give the full definition, but let's say it's an identification of the tangent complex with the cotangent complex with a shift, as objects in the Dirac category, plus some relatively non-trivial

closeness structure. You have to say, this is the underlying form, and you have to say it satisfies Dirac equals zero. And that's a bit messy, and you need higher homotopies and so on. So there are some, well, that's the content of this paper I was mentioning

at the very beginning, where we give all the details about how to do this. So now I'm going fast here, but if you have such a simplex structure, this shift LAL carries a Poisson bracket of degree minus n, compatible with the algebra structure.

And the comment here is that this is a non-trivial statement. I mean, it's just a technical thing, but this is only an isomorphism in the Dirac category, which means that the form is non-degenerate only in cohomology, so it's only up to quasi-isomorphism. And it makes a complication when you want to dualize it to a Poisson structure.

So this thing is really, so that's the really in progress part of this very last preprint we are talking about. So we think now we have a full construction here, but it's complicated at the moment. And I've heard that Rosenblum and Getz-Gury and maybe Costello

announced also another way to go from symplectic to Poisson, which is more global than the proof we have at the moment. So OK, it's difficult, but it's somehow done, I think. But we also have a paper with a case in paper.

There's a comment about that. Yeah, it's how to do it in DigiSense or Digimantra. Ah, OK. And then one upshot is that it only gives a weak Poisson structure on this A L. And here I use the opportunity to mention this thing that this can actually be made into an actual Poisson structure of degree n. That is the work of Valerio Melani on this.

It helps actually seeing this as a Poisson algebra with a Poisson bracket of degree minus n. It's an actual sheaf of Poisson algebra on x-derham. OK, so now how do I do the deformation quantization? Suppose n is not 0. That's the

easiest case. A L is a P n plus 1 algebra. P n plus 1 algebra is just what I meant. It's commutative plus a Poisson bracket of degree minus n. Commutative algebra plus a Poisson bracket of degree minus n, which is compatible with the product. So it's an En plus 1 algebra because I can use formality here. So I can choose an equivalence

between the En plus 1 operand and the Pn plus 1 operand, which is a theorem by Maxime Koncevich. And I can see this Pn plus 1 algebra as an En plus 1 algebra, and I can look at it trivially as a deformation of the commutative underlying algebra.

And this gives me a deformation of A L, so it gives me a quantization. When n equals 0, things are getting complicated because, of course, E1 and P1 are not equivalent. But E1 has a filtration whose associated gradient is P1,

and there is this famous Koncevich's formality theorem that tells us that deformations of E1 algebras and deformations of P1 algebras can be identified. But at the moment, we don't know how to make this work on A L due to the fact that it's DG and it goes in the wrong direction, so it goes in the positive direction.

So I have the feeling that things should go through, but should be fine. OK, this is a feeling for us, so maybe it's a theorem. OK, still some technical problems to solve for us, at least, maybe. But it seems OK in the non-degenerate case, in the symplectic case, because you can write down explicitly the quantization by some

wild algebra, but this depends on the Darbou lemma that was passed to me by Kevin Costello several years ago. So there is a Darbou lemma in this setting that helps dealing with the non-degenerate case, and in general, it should be possible to do so,

and different from 0 or n equals 0 seems, you know, OK, so let me state a theorem. Any derived algebraic stack, possibly higher with a Poisson structure of degree n, where I didn't define Poisson structure of degree n. I did tell you what symplectic means. Admits a canonical quantization of level n, up to some universal choice of equivalence between

en plus 1 and pn plus 1. OK, so now how do you construct? I have two more slides, so it should get into the two more minutes. So let me come back to this theorem. Now I want to apply this theorem, so I need some statements to construct Poisson structures of degree n,

or symplectic structures of degree n. I need a machine to construct examples, and this is achieved by the following theorem, that if I have x and y two derived algebraic stacks, then there is a stack of maps, and if I assume that y, so let me see, y is symplectic of degree n,

and x is oriented of dimension d, so like it behaves like a topological manifold oriented of dimension d, then this map is stack as a symplectic structure of degree n minus d,

and you were mentioning AKSZ, but that's an algebraic version of this AKSZ formalism. And the other thing is that BG for g reductive perf, they both have natural symplectic structures of degree 2, for BG there are some slight choice to do, and if you put one and two together, it gives you zones of examples. So the following the algebraic stacks admit

a quantization of level n, g local systems on a manifold, where I take log g on k, where k say is an oriented compact topological manifold of dimension d, here n is 2 minus d. Bounded

coherent chips on carabials, like I can take the moduli space of bounded coherent chips on z, where z is a carabial variety of dimension d, again n is 2 minus d here. So in the first part, do you sort of use that it's a Poincare duality space?

Sure. You don't really need that it's a manifold? No, just that it satisfies Poincare duality at the chain level, and compatible with the multiplication in cohomology when you need some, I don't know if Poincare duality space is enough by definition. And flat bundles to state the last thing, so log g the ham of z,

so that's the moduli space of flat g bundles on the smooth and proper variety z of dimension d. So here n is 2 minus 2d, because the dualizing dimension of the ham common g is 2d here. Okay, final comments. I want to come back to Sasha's Goncharov's talk,

where he mentioned this quantum Hodge field theory. I think there are some not understood yet relations between his talk and the quantization of this log g the ham,

because this log g the ham of z has a non-ambient Hodge structure by the work of Simpson, plus some thing which I don't think no one really wrote correctly at the moment. You have to include the stacky and the derived structure of the moduli space into this non-ambient Hodge theory, which is not taught yet I think. And then this whole structure probably extends to

the quantization in a sense where it's not clear how to make this, but I think you have a suggestion that there is an action of an infinitesimal group on it, on the category, say. And the reason why it should extend is that because the symplectic structure is compatible with the whole structure.

This is pretty obvious that the semantic structure on this log G is compatible with the all the constructions, all the extra structure coming from non-linear Hodge theory. So that's closely related to what Koncharov told us about quantum Hodge theory, even though the relation is not clear.

To finish, I should say that there are many equations remaining, like we can ask for quantization of Lagrangian maps. There is a recent paper by Dima and co-authors, there is one about quantization of Lagrangian sub-manifolds using a gain for mild geometry. So this probably can be adapted or co-isotropic maps,

as well as we have these maps with boundary conditions. There is a work of Damian Kalak that says that this theorem saying that maps as a semantic structure extends to varieties with boundaries and so on. There are also some two special cases, n being minus one, n being minus two,

for which you can find a refined quantization. The case n being minus one seems extremely important for DT Donaldson Thomas invariance, and that's the content of the work of choice and co-author these days. Okay. Thanks. Let's thank the speaker again.