 Сегодня будет абсолютно спекулятивная лекция. Я попробую использовать идею из моих предыдущих лекций, в инфинитном дименциальном состоянии. Тогда будет немного introduction для tomorrow's Chernzymons Day в конце. Но я начну с 5-минутных интегралов с халоморфикой грандией, с баундром. Итак, L0 и L1 Let's consider the simplest possible. There will be a dramatic change of notation, so the coordinates will be p and q, not x and y, and the plant constant will be h bar today, not t. All right. So then these are complex Lagrangeans. And so then we can define the infinite dimensional manifold of passes. So these are c infinity passes with boundary conditions with L0 and L1. So actually this is a complex infinite dimensional manifold and it carries a closed one form. So it makes kind of a link to the previous talk. Now we can also choose, actually something interesting can be said even at this setting, but we can also choose Hamiltonian in addition to that and define the action functional. So Hamiltonian, 0, 1. So it will be a halomorphic in coordinates q and p and say c infinity in the variable t, which today will be time. And so then we can define, we can define the action functional, which, well, it's, you know. Yan, can you raise the blackboard? Yeah, it's a standard, yeah, I see. Thank you, Sasha. Please keep reminding. Okay. t plus integral. You have two boundary terms, because pdq should be closed for formula 0, 1, which is differential of some function. Yeah. You should remove this two boundary terms. So far, yes, I should remove this boundary, but then I will perform the path integral. It will be, phi is just an arbitrary, well, it's a smooth map. Yeah, no, but action will be, the right action is, if you want, you should add two boundary terms as well. Yeah? Yeah. If you have some function on a 0, 1, whose derivative is false pdq. Yeah, that's correct. All right, so let me say, plus boundary terms, depending on a 0 and a 1. All right. Then, the critical points, critical points should be solutions to the trajectories to the Hamiltonian system with Hamiltonian h. In my main example, h will be equal to 0. with Hamiltonian h. Well, and so it defines the flow, the one parameter group of simplectomorphism, maybe not everywhere defined, but it's not even a group, because it's time dependent. Semi group, I think, yes. It's just a family of simplectomorphisms and actually probably they are not well defined globally. Partially defined, simplectomorphisms, depending on t, gt, and so we can, just forgetting that it's not globally defined. So, we can apply this family to one of the Lagrangians. So, this is L0, this is L1, and so we have a family. All right, so it's really, it's very bad. But what can I do? Just put t upstairs. Put t in the bracket. Where do you want me to put t in the bracket? Is it an index? Ah, it's an index. Here, all right, good. So, anyway, what I want to say is that I can apply, I do not need this intersection thing, so I can consider the intersection of these two things. For example, if h is equal to zero, well, and this is sort of a main object, which corresponds to to the set of critical points and zeros in all previous examples in all our lectures. For example, if h is equal to zero, so this is just this and this is the same as critical points. All right, so then what we can do without this? So, we can look now for the path integral, you'll define, but we can kind of think of it as an infinite dimensional analog of the exponential integral, which we considered previously and ask about the wall crossing structure in some sense. Or we can try to give a meaning of it in the way which Markus mentioned yesterday, that we can consider a formal expansion just using Feynman rule, consider formal expansion critical points, so we get a bunch of series and then we can study the interaction of these critical points, which is not a quantum problem, like, for example, let's kind of assume that unreal, well, it's not very realistic example, that the set of critical points is just, in some sense, in infinite dimensional space there are more. Then we can think that after the choice of an appropriate scalar metric, again, an infinite dimensional sense, we can shoot a symbol out of the critical point and then we can try to integrate, to give a meaning of this integral perturbatively as an integral or the local symbol, which corresponded to the local isomorphism between the Raman-Bettica homology in my yesterday lecture. And so modular summary normalization, because you remember that we considered some modified integrals, we'll get the formal power series, which is expected to be resurgent, but also this is something local, but for different critical points we can look for, say, gradient lines, we join them and this is reduced to some problem in, say, partial differential equations, but anyway, it's some problem, which is not quantum, and having this data, this is purely formal, so it's kind of a Feynman rule, and this studies something like gradient lines, so then in the end we can get the Valkrosian structure, which, if it's analytic, then we can hope that these series are resurgent. All right. So this is, if you'd like, sort of a Betty approach to the Valkrosian structure, one can try to develop the Raman approach, which I'm not going to discuss today. And actually it's also somehow related to what Markus said yesterday. He said that we either have integral and then we do something like what I'm doing here, or we have some differential equations, which we can study by some methods like study asymptotics using WKB, or topological recursion, whatever you have. And this means that, for example, one can forget about infinite dimensionality of the problem and look for a complex symplectic manifold, C2N, or more general complex symplectic manifold with the two holomorphic defaults, maybe intersecting, actually not necessarily transversely in general. And try to work within the category of holonomic DQ modules, say global, which I defined yesterday, and assign some holonomic DQ modules to this pair of Lagrangians. So then the purpose, if you do it properly, then this will be cyclic modules over the quantized algebra functions. In this case the quantized algebra function is just a whale algebra, but in general something more complicated, and so if there are cyclic, and if we can moreover find the cyclic vectors, quantum wave functions, so then we can kind of define, if you'd like, well, if we can define perturbatively the answer as a pairing, well, in the case when h equal to 0 it should be exactly the pairing of these two things, where of course we should explain what this pairing means. Roughly speaking it should correspond to the product of these two things, integrated or the symbols and expanded perturbatively in h bar. If h is not equal to 0, we should quantize if you'd like this parallel transport and apply some evolution operator and then pair in the end. All right, so this is something which I'm not going to discuss today, but we do have a theory of quantum wave functions and something can be indeed said in this direction. All right, let's see. Well, it's a generator of a cyclic module of this algebra. No, in this generality it lives over formal power series and the power is pairing. Again, it's kind of belongs something like this. So, but you can save some further words and kind of to be more restricted, but here the theory is formal. All right. So, now what's the relation of that story to this Betty de Ram and what's the wall crossing structure. So, again, as I said it's a very speculative talk, which means that no theorems will be proved. The point is that even if we have a function in the complex infinite dimensional manifold under some restrictions on the critical locus of this function, we can define a shift of vanishing cycles in pure finite dimensional terms. So, if, for example I don't know we have a function on say, maybe infinite dimensional C manifold which is holomorphic in some sense and assume that this is, for example finite union of too many z's. Okay. It's kind of a connected component. It's better to be compact but certainly it has to be finite dimensional. So, then imposing some conditions which actually people impose in nonlinear functional analysis. You can imagine that this function factorizes rather the neighborhood of each connected component factorizes into the product of two things. One is finite dimensional another is infinite dimensional but your function is of a quadratic form in the infinite dimensional direction. So, it's sum of f is kind of a sum of something. Finite dimensional and quadratic form in infinitely many variables. And so, then if you if you want to define what is the shift of vanishing cycles of that, so you can by analogy with finite dimensional case, you can essentially throw away this quadratic part because it does not affect the topology. So, shift of vanishing cycles kind of intuitively it's a relative, I mean if you compute cahomology it will be a relative cahomology of the neighborhood with respect to the boundary. And this quadratic part doesn't affect the boundary. Also, there is some subtlety here because you nevertheless you should choose some orientation shift in order to have a well defined even infinite dimensional dimensions you have to choose some orientation shift but let me put it under the carpet. So, what I would like to say that if if we do it with this s, then we can define if s is such that it satisfies some properties like this then we can define a collection of vector spaces one for each component which will be well some cahomology of this component with the coefficients well s has some critical value on this component so probably I should I should subtract it also we have h s is f s is f but f is kind of a general story s is f so we apply it to f or other another way f is equal to s and so then we have something like I don't know, shift z and we have we can define this this word was a local vector cahomology in my yesterday talk in the finite dimensional setting and then for two for two critical for two components like z, i and z, j points so we need to define some linear maps tij which should be defined in terms of trajectories between these two components alright so this is some sort of a general very generalized general considerations and I would like to discuss what it can give us what structures it will give us in the case of complexified Sharon Simons story so also although Sharon Simons doesn't look as a path integral okay you can anyway this is more general of course than just a path integral because you see these considerations can be applied to many other functional integrals so your your action functional doesn't have to be a function in the space of path but this is sort of a main example so now let's try to apply these informal considerations to the case of complexified Sharon Simons theory so for that we need dimensional manifold I don't know compact oriented without boundary closed compact oriented I need a gauge group now normally typically people fix a compact group as a gauge group in Sharon Simons theory but I would like since it's complexified so I will complexify it so like SU2 SL2C so this is GC this is G alright and so then there are kind of two possibilities we can consider so I will work with G connections principal G connections on M3 bundles with principal G connections I can consider this stack or I can so version maybe it's more convenient fix some point and consider consider G connections trivialized at this point so G connections trivialized at this point G bundles G bundles so I use just G connections bundle with G connections bundle with G connections if you want bundle with principal well listen let me just trivialize G connections trivialized at M0 alright so then this gives an infinite dimensional complex manifold it's frame connection FR states for framed alright and the group G acts on it now there is a closed one form it's Sharon Simons form on this space but people I mean typically in physics people prefer to go to the universal abelian cover and consider if you'd like let me call it multivalued functional multivalued functional which is defined on the connections by trace is a killing form on on the Lie algebra of G should I divide by 2 I would alright I think this is better actually it doesn't matter because I'm not going to use it to use this explicit form and again kind of more rigorously one should work with this with a closed one form this is better because the set of zeros is satisfied that property finite union component alright so then the critical set I guess on the space connections there is a single value no space of connections a single value if you divide by the case group it's no longer a single value we divide by a case group we divide by a case group principle bundles on three many forms provided G is simple and simply connected we divide by a case group because we trivialized this point slightly different but the way you wrote it is that the space of connections in a huge infinite dimensional space there it's totally single value it's only after division by the case group no it's another question it's always divided by a case group if they realize both of them then we agree right? yeah but that's what I was afraid of I said that kind of people like physicists I don't know they like to write it in this way you have some problems I would say there's no problem it's just a question great York are you happy? or should I think that almost everyone in the room knows what I'm talking about what is the definition? you think all kinds of things but you haven't said it so there is a closed one form this is anti-derivative this space of connections on bundles in bundles traversed base point quotiented by quotiented by the group G based on that point so whatever anyway I want to so maybe that's something like this yeah space of representative flat connections critical points are flat G bundles so then if we would like to decompose something so that's right place I don't know maybe I should use this blackboard as well so if we assume that 3D manifold is obtained by a surgery alone then we can define two complex Lagrangian submanifolds say L in and L out again in the standard way which is described in this famous ITL lectures on topology and physics and something like maybe 30-40 anyway many years ago so you consider tubular neighborhood for example if you not consider flat G bundles on the boundary and these are those which can be extended so G connections flat G connections extended inside inside and those extended outside so then the intersection is this it's better to write 0 so this form but it's the same as critical alright so this is the final okay good so then we are in this framework speculative if you like framework which I described before and we can kind of apply it blindly and to see what kind of structures will we get so for that of course I need to assume that this shift of vanishing cycles is well defined we can lift indeed Chairman Simons functional on the abelian cover of this space of frame connections or alternatively actually you can think of taking neighborhood of each zero component and think that Chairman Simons is defined in the neighborhood of this component alright so then we need something which is in general in Chairman Simons theory I think it's a black box it's a formal expansion so I should take this cahomology in fact it should be zero cahomology with coefficients in the shift of vanishing cycles this my local state space so this should be thought of as a generalized perturbative expansion corresponding to this component and in general of course if you think in terms of kind of Feynman perturbative expansion it should be something like this but this probably is true only when our component is just something like rigid like more critical point rigid reducible flat connection so in general we have no control what can it be so it can be serious well in fact with with rational coefficients rational okay rational exponents and also they can be a monodrama so it's some serious which belongs well it's still kind of no I cannot that's a black box not V I am talking about it so what I am saying so that we assume we are maps okay now but then what we can talk about that if ZG is a georbit of something rigid like rigid irreducible and flat connection critical point like flat yeah so then the RG are known so they are given by indeed by just Feynman expansion RG is a sum for trivial and graph but only in this case yeah this is Feynman expansion yeah yeah it depends on some element it depends on the propagator actually it's maybe zero homology maybe not to about homogenous complex yeah yeah I prefer to work with cahomology but I don't understand what you mean so you want to compute the exponential integral e to the is that what you are talking about? no the general philosophy is that to each critical component I would like to assign formal series and then to study the interaction between critical components and define the wall crossing structure in some terms formal series is a perturbative expansion at this critical component but critical component it can be component something finite dimensional then there should be some rule which is not known in general in physics as well which assigns to yeah it's not a question to me what's a physical meaning I am proposing some framework for which the physical it's like generalized Feynman rule but I think people like Mignov yeah in BV formalism yeah yeah is it physical meaning what? are you sure? mathematical physical meaning I don't know I mean you're talking about perturbative transience there irreducible flat connections they've not been worked out before yeah are the integrals convergent yes yes yes yes yes yes where's the table no no but again this is a framework and then you can try to give me I mean if you do not understand what's a perturbative expansion I don't think you can go very far so then it should be it should be these are isolated if you factorize by the action of G they're isolated so you do basically it's like free field type I agree in principle this is completely free no no no how to define them you see it's asium you can define it free field type connection what defines free field type connection this is kind of a black box you know there is this machine learning so you should you know we can talk about things which you don't understand at this level at this level so if you ask me how to define or if maybe say once again what is the source it's H0 it's compact well it's it's compact it's compactly supported it's H0 H0 it's complex of shift it's a shift of vanishing cycles we can give an example but it's not true I already wrote I mean Maxim told me you will see this perturbative expansion which is a sum of or maybe you wrote I mean it's a kind of a 90's no where the source is not where it's H0 which is used on trivial trivial connection sorry for trivial connections on trivial and so it appears in this axle load single paper it's secret number of course they do not speak in these terms not in these terms because it appears in some other terms 2008 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8