to be here and listen to all those really inspiring talks that really synthesize all of the wonderful things Maxime has been teaching us throughout the years. So I gotta

say an anecdote too, right? Everybody is telling one of those. So I met Maxime for the first time 20 years ago, and I tried to talk to him about the formation quantizations of K3 surfaces, and he wasn't interested in talking about that because he knew the answer. And he said,

okay, I'll explain it to you in a moment, but let me tell you this great thing I just saw. I was riding the bus in Cambridge, that was in Massachusetts, and I saw a billboard with a quote from Albert Einstein saying, imagination is more important than knowledge. So this is actually much greater than the formation quantizations of K3s. And I didn't believe him, but you know,

now I'm older and to a large extent because of Maxime Weiser, so I see how great that lesson was. But anyway. That's okay. I wasn't there, but then you gave a talk at Harvard. This was

your senior set. He was really excited about that billboard, and I really wanted to talk

about that. Anyway, so I want to talk about this story about variations and potentials

in derived geometry. And it's based on a couple of joint works with a subset of all of those people that Bertrand already mentioned with the addition of Lute Mill. So what this

story is, it's about understanding foliations in derived geometry and the leaf space of foliations in derived geometry. And the reason you're interested in that is because, in fact,

isotropic and Lagrangian foliations in shifted symplectic geometry turn out to be closely related to the existence of potentials. And so there is this old puzzle which we face

as mathematicians coming from mathematical physics, which is that moduli spaces of geometric objects or a vacua or for whatever the physicists are interested in, in physics are always given in some global way as the critical loci of potentials. Whereas for us, there are

some intrinsic objects that have nothing to do with any ambient spaces, any actions or anything like that. And in fact, very often, the spaces we describe may not even be describable as critical loci of potentials. And so in that sense, they're not physical. And so

there is this conceptual problem of trying to understand which moduli problems arise as critical loci of potentials. And this story with the foliations actually comes as a mechanism for providing an answer to that problem, especially isotropic foliations. And so yeah, so I'll talk

about this leaf spaces of foliations and their relationships to the De Butte theorems in derived geometry. And the most interesting part of the talk will be the examples and two general

constructions for producing these foliations and potentials, which are, I think, quite intriguing. So I got to apologize for about 20 minutes, it's going to be kind of dry because I need to set up the language and explain the concepts that we're going to be working with.

But once we have that under our belt, we'll start actually seeing some interesting geometry. So bear with me for a while. Don't fall asleep. So OK, so here's the setup. I'm going to talk

about foliations in the affine setting. So suppose that we have a non-positively graded commutative differential graded algebra over the complex numbers, the same way it appeared in Bertrand's talk. And we're looking at the derived scheme, which is the spectrum of that. And so the most economic way to define a foliation, which is not the way you really

think about them and work with them, but the logical reason is best to think about it this way. It's a mixed graded commutative differential graded algebra, such that its weight zero piece is isomorphic to our algebra functions on the derived scheme.

And there are two conditions. The weight one piece is a perfect module. So it's like a vector bundle on our derived scheme. And ignoring the mixed structure, ignoring the second differential, this algebra is free, freely generated by its weight one piece. And since

it's going to be relevant, let me just recall what the mixed graded commutative differential graded algebra is. So it's a bicomplex. It's a bi-graded vector space,

which is equipped with a graded product, which is associative, which is graded commutative with respect to either of the two gradings. So one of the gradings is written as a superscript as the homological grading. And one is written in parentheses. It's called the weight grading. And it's equipped also with two derivations for the product. One is of

homological degree one and weight degree zero. And the other is of homological degree minus one and weight degree one. And those are anti-commutes, so they give you a bi-complex.

So it's a bi-complex which has a homological differential and a homological differential, but also has an algebra structure. So that's what the mixed graded commutative differential graded algebra is. And oh, that's the problem. OK, thank you.

So this is really what the affiliation is, but let me try to explain that and reformulate it in a way that's more familiar and makes it easier to work with. So

since it's a differential graded algebra with a homological differential, but ignoring the mixed structure, the other differential, it's freely generated by its weight one piece. I can actually describe it in terms of that perfect complex, which is the weight one piece.

But before I do that, let me just give you the two standard examples, which are really the model examples of what's going on. One is the trivial affiliation, which is given by the weight zero piece is our algebra that was part of the definition, and there is no weight one piece. The weight one piece is zero.

And then there is the tautological affiliation, which is given by the derang complex. So we have the cotangent complex, which is the complex of forms on our derived scheme. And it has a homological differential because it's a complex, and it also has a mixed structure,

which is the derang differential. And so the only non-familiar thing here may be the fact that I've chosen a weird grading. I've shifted the form degrees so that all my forms sit in negative degrees. And that's so that we have compatibility with cyclic homologies and functions on the loop space. But it's not really essential. I mean, it's just a convention.

OK, so these are the two standard examples of variations, and they really model what's going on. So you can say what the variation is in this language in terms of the derang

complex. So you can think about it as a triple. L is a perfect module over our derived scheme. We have an anchor map from the cotangent complex to that perfect module. And we have a mixed structure on the free algebra generated by that perfect module,

shifted by one, so that the algebra map induced by the anchor map is a map compatible with the mixed structure. So it intertwines the derang differential

on the derang algebra with the one on the foliation. So why it's called foliation? So foliation is something common. I'm explaining. It's coming. Yeah, so there is a dual notion of that,

which is, so if you dualize this perfect complex to the dual perfect complex, then the mixed structure dualizes to a differential-graded Lie algebra structure. You still have the anchor map. And again, the condition is that it gives morphism of shifted

Poisson commutative differential-graded algebras. So this is more familiar. This is similar to the classical notion of foliation when you talk about an integrable distribution in the tangent sheath. The only difference with the classical differential geometric notion is that there is no injectivity condition because we are in the derived world.

So any morphism works. So you can, in the previous one, on the previous slide, we had it as quotients of the derang algebra. Here we have it as Poisson sub-algebras in the algebra vector fields or polyvector fields. And they're not really quotients or

subs. They're just things to which there are algebra maps or things that map to the vector fields. That's the only difference that's specific to the derived setting. So these are all equivalent data. The data of the mixed CDGA or the one coming from the

cotangent complex, it's actually better than the one coming from the tangent complex when you start doing descent. So the ones that's written in terms of forms, it's much better

adapted to writing gluing conditions. But it was also foliations of simple examples of Lie algebra. Yeah. So the previous slide, the thing that we have is really a Lie algebra in the derived setting. But Lie algebra is convinced to get some manifolds which is contained.

Yeah, exactly. The conventional affiliation may give a special example, but this is kind of unique. Say it again. Commutative, kind of. No, in the algebra, Lie algebra, it's a particular case. Yeah. Lie algebra, even in ordinary differential geometry, is a particular case of affiliation

in this sense, but may not be affiliation in the ordinary sense, depending on the behavior of the anchor map. OK. So this is what happens on affine guys. And then you can map foliations to each other by saying that you have maps between these vector bundles that generate the foliations,

and they intertwine the anchor maps and the mixed structures. So this is what happens on a given affine guy. And now the interesting thing is, because you want to glue these, you have to say how you base change them. So if you have a morphism between affine derived schemes, which comes from a morphism of CDGAs,

you can ask on how to base change. And you base change the same way you base change the same thing, basically, as we call it, actually, algebraic. So you pull back the...

So there is this notion of a base change affiliation, which is not the pullback of the vector bundle that generates the affiliation, but it's a pull-push construction. So you pull it back. So you base change by tensoring with algebra A prime. And then you take the push out under the co-differential, which is the map on the

cotangent complexes. And that's the base change of the foliation bundle. And once you define it that way, then the mixed structure descends along the right morphism. And that gives you a notion of affiliation. The anchor map is the one just tautologically

defined by this diagram. So that gives you a notion of base change affiliations. And again, let's look at some examples. If you take the trivial affiliation, then when you base change it by a morphism, you get the

foliation which is tangent along the fibers of the morphism. And I'll talk about it. I haven't defined it. I'll define it a couple of slides later, but it should be obvious, say, in the ordinary differential geometric setting what that one is, just vector fields tangent along the fibers. If you base change the tautological foliation,

you always get the tautological foliation. So there are complexes. They are functorial. And here is an amusing example. If you take a point, a foliation on the point in this derived setting is just a differential graded algebra.

And the base change along the morphism is a split foliation. It's the foliation that's this differential graded algebra extended by linearity on your space plus the vector fields with the standard Librac. So hopefully this gives you some sense of what these guys are.

And the good thing about this base change is that it allows you to descent. And you can define now foliations for derived spaces and derived stacks.

So you can look at the assignment that goes from derived affine schemes to the space, the mapping, the classifying space, the groupoid of foliations on the affine scheme. And this is a stack in that autophagy with this base change notion. And then you can define a global

foliation as an element in the mapping stack. If you have any stack, derived stack, in that autophagy on the stack of affine schemes, you can take the map from that to foliations. And that's a pattern in this derived stacky geometry. The stackiness is always

a problem because a lot of notions that we want to work with like forms and vector fields and differential operators, they have bad descent properties with respect to smooth morphisms. And you have to stackify every time you try to globalize them. They make perfect sense on affine locals, but once you start to globalize, you need to do something

like homological descent on hypercovers. And that's what this thing actually achieves. So there is a, I don't want to spend time, there is a way to actually write that in terms of local data on affine derived schemes that probe your stack. But I don't want to go

into that, so I'm going to skip that part. It's groupoid means an infinity groupoid for me. There are no truncated things. There are no truncated things in this talk. Everything is

infinity. Yeah, so the tangential foliations that I promised. Suppose you have a morphism of derived stacks, a locally finite presentation. The tangential foliation is given, so I didn't give you the description in terms of local data, but it is the vertical tangent complex,

the cotangent complex along the fibers of the morphism. If you want to write it in terms of an anchor map and mixed structure, you have to take the big cotangent complex on the big ital side, because the Ram differential doesn't exist on the quasi-coherent cotangent

complex globally. And the anchor map is just the restriction from global forms to vertical forms. And yeah, so this is the relative cotangent complex. You have the restriction map from

global forms to vertical forms that, of course, if you take the quasi-coherent acting on the big cotangent complex, you get the quasi-coherent cotangent complex, and this gives you the restriction map. And the only thing that doesn't exist on the quasi-coherent version is the Ram differential, which you have to keep there. So that's the tangential foliation.

This is an important remark, even though it's not going to, well, it's going to appear in one of the examples, but it's not going to appear essentially in this talk, but it's really important, so let me make it. You can define the tangential foliations for morphisms which are

not morphisms of algebraic derived stacks and are not locally finite presentations, and there are many important examples where you really need to do that. So they make sense when the target is a formal derived stack, and again, this is slightly delicate, so I'm going to skip the technical definition,

locally finite presentation. The formal derived stacks do not even need to be locally finite presentation. They can be almost locally a finite presentation, which means that they have cotangent complexes with coherent kukumologies, but maybe unbounded torah amplitude.

And they can also be informal derived stacks, almost a finite presentation locally. And this actually has been studied extensively by Nick Rosenblum, and there are many important examples coming from representation theory which are exactly of that type.

Yeah, so I said the classic operator will not be perfect, but in this case will be coherent kukumology and unbounded torah amplitude. And let me just give you some examples of these guys to convince you that they're important geometrically.

They arise really naturally. One example is a formal stack that Bertrand discussed last, in his lecture, is the DRAM stack, which will reappear very soon and play a very prominent role in our discussion. So it's a formal derived stack of locally finite presentation, but it's not

algebraic. The derived Hilbert and quals hems of Chiu Kang, Fontanin, and Kapranov, they are almost locally finite presentation. And RAN spaces are informal and almost locally finite presentation. So all of those are important, and you would like to work with them,

and they do appear when you start dealing with geometric settings all over the place. Okay, let me just, since I'm talking about tangential fullyations, let me just mention two

examples that the trivial and the tautological fullyations are tangential. The trivial fullyation is just a tangential fullyation for the identity map, and the structure morphism to a point gives you the tautological fullyation. So those are particular examples of tangential fullyations, and in fact, with the exception of one construction in this talk, all the

fullyations we'll be dealing with will be tangential. Okay, now the one thing that we really want to extract from this formalism is leaf spaces. So we want to be able to take a derived guy and quotient it by a fullyation. And as in usual geometry, the output of that

process is not going to be a space, but it's going to be, or a stack, and it's going to be a formal stack. And so you have to actually, depending on the generality of formal geometry you want to allow yourself, you have to work a little bit for that. So here is a

definition which is actually a theorem. So this is what Bertrand mentioned. So these quotients are defined by universal property, and they exist. So if you have a derived stack with a finite presentation and a fullyation on the derived stack in the strict sense, so given by a perfect guy, then there exists a formal derived stack, which is the leaf space

of that fullyation, and a quotient map, and it's uniquely characterized by universal property. The universal property is that if you map x to any other formal derived stack, so that the tangential fullyation to the map, to the test stack maps to the given fullyation,

then that map on fullyations induces a unique morphism on the, from the quotient to the test map. And this is unique. So that's the universal property, and you can construct this

leaf spaces, which are formal stacks, and we'll be using them very soon.

So again, the reason, yeah, maybe just let me just say a little bit about quotients. They behave exactly as you expect them to behave. First of all, these quotients are really sensitive to new potents or to derived structures. I mean, if you reduce everything, so if you truncate to something which is non-derived and pass to reductions,

the quotient by fullyation gives you an isomorphism as a higher undereived stack. So the quotient by the trivial fullyation doesn't do anything, just gives you x.

The quotient by the tautological fullyation gives you exactly the derived stack. Which Bertrand defined it, you attach to a commutative differential gradient algebra the points of x on the reduction of that, which means first you kill all the derived pieces,

and then you reduce by new potents. And this is Carlos's definition of the derived stack. And when you work out this universal property, this is exactly what you get. For the quotient by the tautological fullyation. If you take the cotangent complex for the leaf space, it is exactly what you expect it to be.

It's the cone of the map from the cotangent complexes from x to the fullyation bundle. You can, in fact, ignore the new potents in x. You need to keep the derived structure,

but you can ignore the new potents, so you can kill the derived structure, kill the new potents, take the reduced undereived space sitting inside as a closed sub-scheme in your derived stack. You can base change the fullyation. Now, this is not gonna be

fullyation in the strong sets, it's not gonna be perfect. But it will be in, locally, almost a finite presentation. And then the map on quotients is an isomorphism. So we can't prove it in this generality, we can only prove it for the strong fullyations, for the perfect ones.

So this thing exists in this particular case, but in general, we don't know how to prove it for general fullyations, which are almost locally finite presentations. Okay. Now, if you have a reduced algebra of finite type, nothing derived, non-DG, and you have a derived fullyation on it,

then the algebra functions on that leaf space is exactly what you expected. It is the completed, so the total complex, the negative cyclic complex

of the completed symmetric algebra, FL shifted by minus one. So it's just, you have this mixed complex and you take its total complex. So, again, if you understand formal spectra properly, which is what Bertrand was

discussing in his lecture, in the affine case, the quotient by the fullyation is just the formal spectrum of this total complex that there are multiple of the fullyation. So here is probably the most useful example. If you have a morphism of derived stacks, locally finite presentation,

and you look at the tangential fullyation, then the quotient is the relative formal completion of X along the fibers of F. So what you have to do is the derived stack, whose value on commutative differential gradient algebra is the fiber product of the points of

X. It's the fiber product of the points of the target over A with the reduced points of X over the reduced points of the target. So, you know, if the target was a point, you'll get exactly the DRAM stack. So this is like the family of the DRAM stacks along the fibers.

And there is a reason why we call it a formal completion because if the morphism is smooth morphism of schemes, then you really get the relative DRAM stack along the fibers. But if it's a closed immersion, you get the formal neighborhood. You get the functions

on the formal neighborhood of X inside Y. So you get the formal completion of Y along X. So there is a slight caveat in this isomorphism, but it's technical. I'm not going to discuss it right now. But they are isomorphic if you interpret what isomorphic means correctly.

I mean, they're extra structures which are not quite matched, but as algebra isomorphic. Okay. So we have these very nice properties of quotients, and we have this relative completion, which is a quotient by tangential affiliation.

And now I want to bring the symplectic geometry into this game. And see what can we say if we have not just foliations, but have foliations which are Lagrangian or isotropic. So if you have, say, a fine derived scheme and an n-shifted 2-form,

which doesn't have to be symplectic, just closed n-shifted 2-form. So in algebraic language, it's just a morphism from this mixed grade, commutative differential grade at algebra,

generated by a piece in one homological degree, 2 minus n, and one way degree, 2, to the theorem algebra. And so that's what n-shifted closed 2-form is.

And an isotropic structure on such foliation is just a homotopy between the induced morphism to the foliation from the symplectic form with the anchor map and the zero map. So that choice of a homotopy is a structure. You need to keep track of it, but

that's what it means for the foliation to be isotropic. And you can also define Lagrangian foliation. Yeah, I mean, I actually wrote down here explicitly how you write isotropic structures in this derived sense. You can write closed 2-forms as infinite sequence of forms, terms sitting in terms of the cotangent complex with

the corresponding degree, which are annihilated, which are co-cycles with respect to the total differential in the double complex. And then H is just a co-chain that bounds that form in the

double complex corresponding to the foliation. So in the affine case, it's something very explicit. I mean, it has a large amount of homological data on it, but it's something completely explicit. I mean, it becomes complicated when you start gluing things. And then you can play that game with gluing again. You can define a morphism of isotropic

foliations. You can define a base change of isotropic foliations. This is where you need the form not to be non-degenerate, because when you start pulling back,

non-degenerate forms can become degenerate, but we don't care. I mean, the notion of being isotropic doesn't care about non-degeneracy. So you can define base change of isotropic foliations, and again, the same trick works. The assignment that sends a shifted symplectic

or affine scheme with a shifted 2-form to the groupoid of isotropic foliations is stuck in that topology, and you can define a global isotropic foliation as an element in the mapping stack. Then you can put a non-degeneracy condition on the isotropic foliation. Once you

have one for a non-degenerate 2-form, then the non-degeneracy condition is a purely algebraic condition. It's just the condition that says that the foliation is isomorphic

as a perfect complex to the comormal bundle of the map. So you just write down the composition with the symplectic form to the quotient by the foliation, and you want this to be a quasi-isomorphism. So it's exactly, I mean, it's a complicated formula, but it's just

the most nifty thing you can do. It's the standard notion of being Lagrangian. So the reason we care about these guys is, as I said, because, so this is Lagrangian when it's a quasi-isomorphism, and the reason we care about them is because

we want to talk about potentials, and this is related to the Derboud theorem. So in classical symplectic geometry, the local structure of a symplectic manifold is given by the Derboud theorem, which says that, say locally in the C-infinity setting,

or formally in the algebraic setting, the symplectic manifold is isomorphic to the standard symplectic structure on a cotangent bundle. And you would like to have something like that in this derived symplectic geometry. The problem is that you have more cotangent

bundles, more than one cotangent bundle, so you have more potential local models than just a cotangent bundle. So the most nifty thing you can do is, what Bertrand described in his talk, is the shifted cotangent bundle. So you just take the tangent bundle shifted by

minus n, take the symmetric algebra, and take the spectrum of that, that carries a natural exact shifted symplectic form. So that's a perfectly nice linear shifted symplectic manifold, differential-graded manifold, or a stack. The problem is that not every shifted symplectic

derived stack is isomorphic locally to this. So there are deformations of these guys which are not locally isomorphic to them. So the derived critical locus of a function is equipped with an n-shifted symplectic form, and in general, it's not locally isomorphic

to a shifted cotangent bundle. So if you have a function mapping you to the m plus first shift of the line, so you can think about this as a degree m plus one cohomology class with coefficients in the structure shift of n,

this, it has a well-defined critical locus which is equipped with a symplectic structure which is shifted by n. So there is a drop in shift by one because this is a Lagrangian intersection picture. So if you have a shifted function, n plus one shifted function, its derived critical locus

is equipped with an n-shifted symplectic form. If that function is constant, then you get the shifted cotangent bundle, or locally constant, then you get the shifted cotangent bundle. But if it's not constant, then the derived critical locus is not necessarily isomorphic to a shifted cotangent bundle, and that's very easy to see in examples.

So you have a choice here whether you think about these guys as your local models or these guys, and since I said these are more general, they don't specialize to this even locally, it's better to use those as your local models. And so if you have a hope for a Debut theorem,

you want to say that a shifted symplectic manifold is locally isomorphic to the derived critical locus of a shifted function. And so that's what we want to do. And this leads to this remarkable Debut theorem that Dominik Joyce and his group proved.

If you have a derived Dylan-Manford stack with an n-shifted symplectic structure, where the shift is non-positive, then at all locally, it's isomorphic to the derived critical

locus of a shifted function. So this is Dominik Joyce and his post-docs, graduate students, and impacting on schemes is risky locally. So it's a very strong statement. And now the problem we were trying to solve, where these variations were needed,

was to try to understand what's the ambiguity in the choice of these local structures, and when can we hope for them to be global? So this tells you that locally you always have a potential, but maybe sometimes you have globally a potential. So when can you hope for the

variations help you answer? So the point is that potentials always exist if we have an isotropic variation. So here is the theorem. Suppose that you have a derived stack locally finite

presentation and an n-shifted symplectic form on it. And assume it's exact. Of course, if you want to have a potential, it better be exact. The ones that are shifted symplectic forms on derived critical loci are always exact. And suppose that it's equipped with an

isotropic variation, so a variation with an isotropic structure H. Then the claim is that on the leaf space of that variation, there is a shifted function, n plus one shifted function, and a map from X to the leaf space factors through the derived critical locus of that function.

And in fact, not only factors, but it's symplectic. It pulls back the shifted symplectic form on the derived critical locus to the original shifted symplectic form. And that's completely global. If the variation exists globally, then on the leaf space you get

this function, and you can identify the whole space as a derived critical locus. If in addition the variation is Lagrangian, then that map to the derived critical locus is also a tau. It's a very natural statement. And it's actually not very

hard to prove once the technology is set up. I mean, where is the function coming from? It's coming from the fact that the form omega is zero on the leafs of the

variation for two reasons. It's zero because it's exact, and it's zero because the variation is isotropic. So you can find the homotopy between these two zeros, and that's the function. It's just as simple as that. And it really connects with the result of Joyce and company,

where there was no exactness condition, but there was a condition that we were local with nonpositive shift. And the point is that if you work out through the Hodge theory, it turns out that if you have a closed P form, which is n shifted with a negative shift, it's always exact.

So that's something very peculiar for forms with negative shifts. And this is not a tautology. I mean, it's a statement with content because the corresponding Dirac homology is not zero. So this is really a property of this mixed complex of forms.

OK, so let me show you examples. And I want to really show you two constructions of these global potentials that come from these variations, which are cool. So of course, the basic example is a derived critical locus. If you have a smooth scheme and a regular

function, so this is just an undereived ordinary scheme, then you can take the derived scheme x to be the derived critical locus of w. It has a minus one shifted symplectic structure. Now, this better be coming from a potential. We know it's coming from potential. But in fact,

this potential is coming from a foliation. What's the foliation? You have the map from this derived scheme to z. And you can look at the tangential foliation for this map, the inclusion map from the derived scheme to z. And it has a Lagrangian structure,

which you can write explicitly in terms of the function w. And the quotient is exactly the formal completion of z along the undereived critical locus. And the potential is just the restriction of the function to that formal completion. And the map is just the map from

the derived critical locus to the formal completion. And the pullback of the symplectic forms is the fact that the pullback of the function is the pullback of the function. So this is a tautology, but it should be there as a check that what we are doing is correct. And you can do this with shifts. So this was for an unshifted function,

you can do it for a shifted function. And again, you get the same foliation gives you a shifted function. And again, you get the potential is a pullback from the formal completion.

Another example, if you take a cotangent bundle, so you take a smooth manifold, it's cotangent bundle with the standard symplectic structure, then you have a projection from the cotangent bundle to M with Lagrangian fibers. The tangential foliation is Lagrangian. The quotient by the tangential foliation is, of course, as a undereived space

or as the space underlying the stack is just the total space of the cotangent bundle, but the fibers are completed in the Dirac sense. So it's the Dirac stacks of the fibers,

the family of Dirac stacks of the fibers. The function is just zero, but viewed as a one-shifted function. And the derived critical locus is the shifting of the one-shift cotangent bundle, which is just the cotangent bundle. So again, in this case, you get exactly what you

expect. More interestingly, you can do twisted cotangent bundles. So suppose that you have a smooth manifold and you have a symplectic twisted cotangent bundle. So those are classified by elements in the hypercocomology of the stupidly truncated down complex in degree one.

So they are affine bundles over the cotangent bundle equipped with a symplectic form corresponding to that class. The tangential foliation mapping to the base is still Lagrangian. The problem is that the symplectic form is not necessarily exact anymore. Twisted cotangent bundles are not always exact, depending on the twisting.

If the symplectic form happens to be exact, then you should be able to find the twisted function, one-shifted function on the drum completion of the fibers. And it's actually not hard to identify it. If you see where this function should live, it should live in the degree one

cohomology of m with coefficients in the structure shift. And it is just the function, the class that bounds the class eta. So it turns out that eta will give you an exact twisted cotangent bundle exactly when it's in the image of the

Durham differential with respect to the Hodge filtration. And whatever bounds it is exactly the function. So and then you get identification of this with the derived critical locus. And let me just point out that this is, of course, interesting on m's which are not compact

because this function f is not going to be unique. It's going to be unique only up to a function which is locally constant or only up to something that's coming from H1 with coefficients in c. And of course, if we had something locally constant, then there is no shift because dff is zero.

So what you really get is a shifted cotangent bundle again or the original shifted, twisted, unshifted twisted cotangent bundle. So you cannot have m to be something for which Hodge theorem holds if you want to get interesting examples. But for affine m's,

you get interesting examples. Okay, one purely derived example, and then I'll give you the, how am I doing with time, 10 minutes? Okay, great. So a purely derived example which is neat. So suppose that you have a symplectic manifold which is a point,

with the zero symplectic structure, but you view it as a one-shifted symplectic structure. The zero symplectic structure you can put in any shift. So on the point, you can talk about shifted symplectic structures of any shift. So we can actually describe Lagrangian

affiliations. We have a finite dimensional vector space, and you look at the map from the point to the classifying stack of that space. So the stack classifying affine space is modeled on that vector space. The tangential affiliation is given just by the dual space with the zero anchor map and the zero mix structure. An isotropic structure is a self-homotopy

in this negative cyclic complex that has a zero differential. So it's actually an element in the second wedge power of the dual space. And it's Lagrangian if and only if this

two-form is a symplectic vector space. So a Lagrangian affiliation for this, so a situation in which this map from the point to the classifying spec of a vector space is a Lagrangian affiliation, if the tangential affiliation is Lagrangian,

it's just the choice of a symplectic structure on the vector space. Now, the quotient by this Lagrangian affiliation is just the classifying stack of the formal group, which is the completion of v at zero. And by the theorem, there should be a two-shifted function from that formal group, from that classifying stack

to a one two-shifted function, so that the point with the zero shifted symplectic structure is the derived critical locus. And so in order to find that function, you need to compute the second cohomology of Bv hat with coefficients in O.

And you can actually do that because we have the Breen calculation, which in characteristic zero was redone by Carlos. And it says that if you have a finite dimensional vector space and you look at the Eilbert-McLean stack kvn, then the eigth cohomology of kvn with coefficients in O is either a

exterior algebra or a symmetric algebra, depending on the parity of n, or it's zero if the degree over the homotopical degree is not an integer. So in particular, in this case, when we're doing something two-shifted, the answer is exactly wedge to a video,

and the function is just H. So it's a complete circle. OK, so I have another example about integrable systems, but I'm going to skip that,

even though it's interesting. I want to show you two constructions of isotropic foliations, one of an isotropic foliation and one of a Lagrangian foliation, which are interesting, can give you new realizations of modulized spaces of critical loss of potentials. So suppose that we have an oriented C-infinity manifold of odd dimension,

compact. Choose a Morse-Mell function, so it's a self-indexed Morse function, where the critical values are the indices. And choose a regular value between the two middle

critical values. So now, if you take the half of the manifold that corresponds to everything smaller than that regular value, you get a submanifold with boundary,

which when you include it in M, you get the homotopic equivalence on the k-dimensional skeleton. In fact, any such guy will do. I mean, this is a way to construct it. Now, if you have a complex reductive group and you look at the derived stack of local systems

on M, so these are maps from M to BG as a derived moduli, then this guy carries a 2-d-shifted symplectic structure, as Bertrand explained by this AK-Zik formalism. And if k is greater than

or equal to 1, this 2-d-shifted symplectic structure is negatively shifted. So by this theorem that I mentioned about the Hodge weights, it's exact. So you have the derived stack of local systems on an odd-dimensional compact-oriented manifold as soon as the dimension is bigger than

1, it is an exact shifted symplectic manifold or shifted symplectic stack. And the claim is that the tangential variation for the restriction map from local systems on M to local systems on this half, this Higgert half of it given by the Morse function

has a natural isotropic structure, depends only on the orientation data and the shifted symplectic form. And so you can find a shifted function, 2 minus 2k shifted function on the quotient by that variation, which is the relative Durham stack. And the moduli of

local systems with its shifted symplectic form is the critical locus of that. So for three manifolds, this just gives you an incarnation of the trans-Simons functional if you choose a generalization of the trans-Simons functional to higher dimensions. But unfortunately,

the off-shell space has no natural interpretation of a space of connections. It's really very derived. So that's one construction. The other has to do with non-abelian Hodge theory.

So it's very similar. Ah, maybe I didn't say that. So you can prove that this structure is isotropic. Unfortunately, we cannot prove it's Lagrangian. I believe it's Lagrangian so that this map is really a tall. And in fact, like, when M is simply connected, we can prove it's Lagrangian. When, in any case, I mean, there are some topological

conditions we can prove it's Lagrangian, but we cannot prove it in general. I think it's just stupidity. But it's certainly isotropic. So it is really a pullback of a derived critical

locus, symplectic form of a derived critical locus. You mean for shifted functions? Yeah, you can define shifts of vanishing cycles for shifted functions. But no, because these are formal stacks, you really have to be very careful with your,

with the, you know, the boundless properties of your complexes. So we can formally define it. I'm not sure that it actually exists because the cons may not be well defined

if it's unbounded complexes. So we haven't tried really. I mean, it just, you can try to mimic the standard definition, but that, you know, maybe the, if you don't get bounded below, by bounded above complexes, maybe you're not going to be

able to do it. So, okay, but let me do this example. So suppose that we have a smooth projective variety of dimension D, and you look at the moduli stack of rank and local systems, again, as a derived stack. So this is now equipped with 2-2D shifted symplectic structure.

And as Bertrand pointed out, the stack X, it has a tangent complex, and the tangent complex has a natural Hodge filtration from non-abelian Hodge debris, coming from the moduli space of lambda connections. So if you look at the relative moduli of lambda connections, in fact,

the shifted symplectic structure exists relatively in the twister space. And this, the C star action on the moduli of lambda connection gives you the Hodge filtration on TX, and the map given by the contraction with the symplectic form actually is a filtered

quasi-isomorphism for the Hodge filtrations. And as a consequence, if you look at the middle degree, the middle step of the Hodge filtration, the middle degree step of the Hodge filtration when D is odd-dimensional again, this middle step of the Hodge filtration has a canonical Lagrangian filtration structure for the shifted symplectic form.

It also works on the Higgs moduli, but here it's more interesting because on the Higgs moduli everything is split. And so you can take this filtration quotient by it and get the potential,

so you can write a moduli of local systems on an odd-dimensional projective manifold as a critical locus of a potential. And in fact, this filtration given by the middle step of the Hodge filtration is again a tangential filtration. So you have the local systems, the derived stack of local systems of rank N, and you can look at the derived stack of modules.

So this is like DG modules over the full-derang complex of N, which as all modules are of rank N. But you can look at DG modules over the truncated derang complex in the middle step of the Hodge filtration. And you have a restriction map from those, and the middle step

of the Hodge filtration is just the tangential filtration for that. So if K0, if we are doing one-dimensional curves, then this is just the ordinary map from the moduli of local systems to the stack of bundles. But the moment we go in dimension bigger than one, this map

actually on truncations, so this is just the truncation, right? But this map on truncations is an isomorphism. So you don't get a new space, but you get a new derived structure. And the full one truncated stack is recovered as a critical locus of a shifted

function on the relative derang stack. And you can actually identify that function explicitly there in terms of the symplectic structure. And I think I'll stop here.

Okay, so in this case, thank you very much.