 OK, so in fact, I was listed as organizer of this conference. Actually, my role is just to humanitarian one to help Anton to find a nice building which available three days. Sure, of course. Yeah, and also Anton asked me to talk about samsikic. It's not actually my active memory, this subject now. But I'll just give a very short review of what is going on in string theory and how one can understand and generalize the structures. But first of all, I just want to say that there is a following simple question. One can calculate volume of unitary group with a natural with respect to a measure related associated with natural Romanian metric. My trace of square is killing metric. So it's some number. You can calculate it's actually a double gamma function. It's a product of volume of spheres. It's very easy to see. But then I can expand it as maybe a log of volume, get a volume. And now I can see the volume to power inverse, take a logarithm, and try to expand as n goes to infinity. Yeah, so it's simple exercise with sterling formula. But it's not a gamma function. It's a double gamma function. And you get something like some minus n square of log n minus square root of 2 pi log n. And then I write, I hope so. And then you get series with g greater than 2, earlier characteristic of modulus space of curves without markpoints to power n to power 2 minus 2g. Yeah, that's one knows from a horror in Zegir. It's Bernoulli number divided by something by 2g minus 2. This is also one can calculate. So it's one can really easy check. This is really not a big deal. But the question, can we make intrinsic proof of it? And at the moment, it's not the case. So I think it will be a really good goal to find an intrinsic explanation. So it's actually part of general patterns that have considered integral of space by n matrices take logarithm and expand to get some high genus expansion, which we know in some class of kind of integral situations. But in general, there should be some inner explanation. So I will speak about the following thing. There is a model, will be b model, and a categorical language. So what is a model? Suppose we get a compact, for a moment, simple electric manifold. And then there are a group of between invariants, which are virtual fundamental classes of model space of stable maps, which are in commulges of certain degree x to power n times mj n bar, regression coefficients. d here is a degree of a curve, which I interpreted as just a integral homologic class. And this dimension is calculated through some formula. It's half of real dimension of x minus 3 times genus of my curve minus 1, 1 minus genus, plus number of mark points, and plus integral first gen class of tangent bundle over homologic class d. So there are such classes. And it's well known that the whole thing can be organized to co-homological field theory. So you have vector space h equal to just commulge of x. And you get a map from h to power n to commulge of mj n bar. You take the sum of all the things with weight with powers q to power d. It's kind of depending on formal parameters, qi where i ranks from 1 to r, which is rank of second homology. You calculate, you can see the kind of qi to powers integral over corresponding basis of the representative of this commulge classes over divisor d. It takes some of all possible degrees. So you get a commulge field theory depending on some parameters. And now there is a description of what goes into Bay model if it exists. And Bay model's story is not so obvious what was originally the story if genus is equal to 0. All this reduced to a story about variation of host structures. Sorry, I left exponential host structures, at least part of this gen zero invariance. Namely, if mere object exists, it should be a complex algebraic variety of the same dimension as x with holomorphic algebraic volume element. And a potential, we just map to, which is again, polynomial map, a double should be proper map. And then again, this will be object dual to x in symplectic formats also will depend on r parameter where r is. And one can identify parameters. One with another, such that logarithm of qi, i equal to 1 to r, will be ratio of two integrals. You integrate exponent of w times volume over certain cycles and divide by integral of exponent of w of volume of gamma 0, gamma 1, gamma r, are some integer chains. Not necessarily close, such chains that can integrate exponent of potentials. They go to infinity at the direction of an exponent decay. Maybe I should put 2 by square root of minus 1 here. Yeah, so this coordinates will be kind of transcendental coordinates on model space of these varieties, these potentials. And one, if you call it different integrals, one can translate to structure about gen zero part of invariance. But for hygienists, it's kind of extremely painful. So it goes from hygienists. How to describe in terms of these B models? Not really clear at all. Let's consider some kind of most central case in the whole story. You just look on this equation, kind of central case and dimension of x is 6 or 3, or complex dimension 3. And first chain class is 0. Then in this case, what you get essentially for each of, then it means that curves of any degree of any genus without mark point should form 0 dimensional model space. And so it should count again, you just get these numbers. And all the equations that we get, homological field theory became completely vacuous in this case. The whole story is defined by just all information will be some series FG, where G greater than 0 will be functions in R variables, which are expanded to some form of power series and for small genus with the logarithm corrections. So F0, it's kind of understood through variational whole structure. But for hygienists, there was a model proposed by physicists, kind of Feynman integral, proposed by Bershatsky-Chikotya-Guryev-Affa. It's called Cadiro-Spencer theory. What is Cadiro-Spencer theory? One can make a very general class of Feynman integrals with full form. Suppose you get a differential gradedly algebra in degree, which sits in degrees 0, 1, 2, and 3. And that's for degrees. And you have a differential of degree plus 1. And also, suppose you get a pairing, which is kind of non-degenerate. Why I put comparatives is because larger will be infinite dimensional and compatible with a bracket. So A, B, C, maybe no causal rule signs. It's OK. And compatible with differential. Then one can write Chern-Siemann's functional on kind of G1, say. You integrate a gamma with d gamma over 2n plus gamma gamma gamma over 6i. You can see the Chern-Siemann's of gamma is this pairing. And then you can try to integrate exponent of Chern-Siemann's functional modulo gauge transformation by degree 0. So Cadiro-Spencer theory is a particular example for some Lie algebra. And what is Lie algebra? It is the following. You can see the d bar forms on collabial three-fold, which is will be merudul. Y potential will be 0, in this case. You can see the d bar form with values in divergence-free holomorphic vector fields. So locally, you can see the tensor product of functions forms in Z bar, depending on Z bar, and holomorphic vector fields in Z with 0 divergence. This slight problem with the pairing. Because if you have two forms, let's say A and B, there are two forms, which A belongs, let's say, 0i form. And this gives you 0 part of tangent bundle. And B belongs to 0, 3, minus i form. Uncontrollably multiply, because the product will be 0, 3 forms. And then it should do something with holomorphic vector fields. But because it's divergence-free, it should be one can substitute to form. It says it's d bar closed to form, to comma anything form. Holomorphic vector fields, you contract this holomorphic column element to get two form in holomorphic direction. Because d bar closed, you can write d minus 1 of this. You get one form in holomorphic direction. So what you do is essentially integrate your place by forms. And the pairing A, B will be integral of B inverse of A with B. Because we say if it's closed, it should be total derivative of something. There is some slight problem along some finite dimensional space with the definition. But you get some field theory. And this guy's Bishorovsky's category of opposite. If you start to calculate exponent of Chairman Simon's action for this theory divided by h bar and take logarithm expansion, you should get h to power 2 minus 2g minus 2 times fg. Yeah, unfortunately, I think it's a very ill-defined, this all Feynman integrals diverge. But also there's really no way to canonically regularize it. So it's extremely bad shape. But it gives some idea about a holomorphic anomaly equation. A holomorphic anomaly equation, it's kind of very close relative to what Maxim Kazaryan explained just before. Essentially, it says that he didn't actually explain what is a holomorphic anomaly equation. He said that you get some infinite dimensional space. You get some decomposition to Lagrangian subspace and functions. So it gives a Lagrangian submanifold. And then you try to quantize this Lagrangian submanifold to get a vector and a value representation. And this holomorphic anomaly equation means that if you kind of change parameters of your series, this vector stays the same. So it's more of the same as these equations which he explained. So now what are these categories? In a sense, it's a way to regularize this BCOV and improve that genetic series which come from symplectic geometry coincide with series coming from complex geometry. So this was a project which was started more than 10 years ago. And this was a project which is not yet completely satisfactory shape, which says the following. So how we encode this Gromov fitting invariance through some algebraic data? So for simply, the algebraic data will be the following. It will be finite collection of complex numbers, which can be thought here as a critical values of w in this example. It could be, in principle, this potential should be 0. It should be just 1 number 0. Then for each number, you get some algebraic gadget which is called a smooth compact infinity category with collaborative structure. And so the category CI, in this example, this potential is that, actually, let me explain, it's a critical values of w or it's the same as eigenvalues of operator of quantum product by first-gen class. So this one can make comparison between A and B model. And so you get this infinity category. The category will be the following. It will be, if I try to define Foucaille category for manifold, it consists of Lagrange manifold. Lagrange manifolds will be some kind of abstraction to existence of this object in the category. And one can shift by identity template by some eigenvalue, get something non-empty. So there are not one Foucaille category, but several Foucaille categories corresponding to eigenvalues. And here are the things called the category of matrix factorizations. And so it's some algebraic gadgets, smooth compact infinity category. Whatever it is, one can say it's some finite dimensional super vector space if I choose some generator plus a bunch of tensors, or some tensors, actually infinite bunch of tensors satisfying some equation. But modular gadget equivalents, you get finite number of parameters. So this thing is kind of algebraic number of parameters. So in principle, one should have again some kind of mathematical programs working effectively with this stuff. Like in Maxim Kazarin's talk, but it's completely different calculus. The Calabria structure, it's also one can formulate in this way. Calabria structure is the following gadget. For each category, we can associate something called negative cyclic homology. And this is actually a module over Taylor series in one variable, which in this subject is variable denoted by letter u. It's come from kind of non-commutative geometry and con operator and so on. And the whole thing seems to work when this model is actually free of finite rank. So you get vector bundle. And this Calabria structure is a section of line bundle with vanish at 0. And derivative should be invertible element more since the same sense as Maxim explained here. So it will be some invertibility property. It will be Calabria structure. And then one can make. But this thing, the category of Calabria structure gives not much. Each individual category gives a factor from its homology, or there are some Ho Chi homology of hn at u equal to 0 to power n, maps to, for each category, ci with fiber, it maps to homology of model space of curves, but without bar. Without bar and with arrow. With arrow it means I can see the stable curves with tangent direction at each point. This is pure algebraic game and how it roughly constructed. One can encode this infinity category seeking some cyclic tensors. And the cyclic tensors you put make from the invariance according to ribbon graphs. And it will be co-chains in the ribbon graphs. So that's basically, so it's pure algebraic construction. And the plan to prove this whole thing was that this pure algebraic construction, one can specify for Foucaille category identify with grammar between invariance, or specify with complex manifold in cities some kind of regularization of this Cadare Spencer theory. So there's a problem. There's an arrow and there's no bar. And to go from to replace mjn head by mjn bar, one need to make an additional choice. Choice is the following. Actually, I forgot to say they get a vector bundle with flat connection and with regular singularity for each category. And the choice is the following. You can see the direct sum of hn of this category C i multiplied by kind of one-dimensional bundle this connection which I denote by exponent of w i divided by u. So you can see the connection whose in one-dimensional function whose solution is exponent. You take direct sum. You get a bundle of this flat connection with second-order pole on a formal disk. And identify these things as formal neighborhood at infinity at 0 of a connection of the following type. This will be du du, kind of trivial bundle. And connection consists of given by two matrices. By hn you denote how should change or how should complex. What is hn? Nectarically homologin. Hn? Oh, negative hn. OK. And the other in the C i? C i of category. OK. Ah, OK. Thank you. Yeah. So you take direct sum, get some formal guy, and now you just want to make some gauge transformation to get just two matrices a and b. There will be, again, finally many parameters how to do it. And if you do it, then you extend your theory to mj and bar. If you do it differently, extend in different way. And these different choices are related by this given telegroup. So it was very nice. And there was some kind of account of this story by Kevin Costolo. But somehow we also communicate with him a lot. And there was some kind of final problem. We cannot really make it to the closed formula. And the reason is completely idiotic. This is a pure combinatorial equation how we go through, make more less space. This stabilization should give some tool to go to mj and bar. And eventually we construct some cell complex which homotopic will into mj and bar. And for example, we want to calculate the integral of fundamental class, like fg. And the problem is this complex is of infinite, it's infinite dimensionals, homotopic will into mj and bar. It's infinite dimensional. And I don't know any good, nice representative of the fundamental class. It will be represented, it will be closed formula, and then we can kind of state the physicists that the problem is solved. Yeah, what is, yeah, yeah, eventually, yeah. So it's really kind of some stupid, stumbling blocker. But that's why it's not really yet kind of formal, official kind of picture of the story. Yeah, yeah, so that was a picture about closed gromo-fitting invariants. Now I want to say a few words about, just before going on, just I want to mention some important thing. This was my proposal by Sergey Baranikov. And it's slightly different linear algebra. So instead of having a infinity algebra and all this data, he has a vector space, which is also infinity algebra, and a dimensional vector space. And a lot of elements in symmetric powers over cyclic powers of my vector space. Satisfying some, you get tensors in such things. Satisfying some, again, more carton equation. And then gives the classes homology of mg and bar. Now I should put prime. And prime is, it's a contraction of mg and bar. Namely, if you have on your stable curve, you get components which are free, which don't have marked points, you ignore the complex structure on such guys. Yeah, that's actually was a space which somehow appeared intermediate in my proof of this area function of written conjecture, some intermediate space. And the classes are here, not on mg and bar. So it's not terribly perfect story one should really remove this prime. And it's not clear how these parameters are related to these parameters. It is not to me. Why not to pull them back? You can just take it. Why not to pull them back to the... No, no, no, maybe we get... Because it's the elements in homology. And co-gonology to also. Also, okay, good, okay. But this will satisfy the hegematics. It's probably not, I'm afraid. Yeah, okay, so it was about close invariant. And now there's a pretty interesting story about open invariant, open closed gv invariant. And let's me concentrate on the case of Calabi-Yau III falls. Yeah, so you see that it's kind of case which is completely kind of perpendicular to this semi-simple case which makes an expense of... The freedom is huge. You get plenty of functions and no equation of... Can I make one there? Oh, just before going on, I want to say that it's really kind of color from all this picture. For Calabi-Yau III falls, one can... This gauge choices, one can make it's pretty explicit in geometric situation. If you get Y Calabi-Yau III falls and you have a choice of volume element. It's a variety, but I should choose volume element. And also, I choose a subspace. This was denote some sort of orthogonal space, a value of VF, I've forgotten you. Which you... Complimentary space, L, yeah. And you choose L in H3, YC, DRAM. Complimentary to terms of Hodge filtration, to H3-0 plus H2-1, which is actually called F2 of H3 is Hodge filtration. Then if you get such things, then the whole story, all this gauge choices will be automatically made. And then we get number FG for energy. And... Maybe at least two. And this number satisfies some question if you rotate L. But in particular, what you can see is that this FG are functions on some algebraic variety. Because so it's a G, it's kind of algebraic function on model spaces of this Y, volume, and L. This is algebraic variety. And it's correspond to really kind of algebraic parameters for categories, coherence shifts. And for example, if Y is mirror dual to quintics, then Y depends on one parameter, volume depends on one parameter, and this Lagrangian space in four-dimensional space depends on three parameters. You get five-dimensional algebraic variety. And FG will be thinking of this. And FG, which you can see in actual ground-fitting invariance, is obtained by restriction to some transcendental sub-manifold here. But so we see that it's algebraically dependent. So if you consider series coming from genetic function, there will be polynomials in first five series. And Yaw and Company checked it. And, yeah, so, yeah, so, but up to now, and also Albert Kliem calculated maybe first 50 polynomials. But because this technology is still not available, so he used some kind of boundary data. But for very large genes, they are not sufficient. You get some freedom. Gamow of canonical equation does really determine two things completely. So, so, so theoretical is answer exist, but in practical terms, it's not yet here. What is the, is there a definition of these FGs, except? Yeah. No, that's, it's only definition, yeah. Yeah, essentially, yeah, Castello has some paper when he have a definition of G without going to constructing logical field theory at all, yes, but it's very hard to extract any number. Now it's about open close ground between invariants. Suppose my Calabi of three-fold is a symplectic manifold, contains three-dimensional Lagrangian, the manifold which I'm afraid to, maybe I assume it will be a sphere, so, yes, suppose I get a sphere, Lagrangian sphere sitting in my three-dimensional Lagrangian manifold. Then one can try to count the following guys. One can count make a genetic series FGH, or number FGH, it's the sum of degrees, Q to power degree, and here I get a number of curves of genus G in H-holes and boundary belongs to this three-dimensional sphere. Yes, strictly speaking, it's not really well-defined problem in normative geometry, let's get boundary, model space with real boundary. But then there was suggestion by Vito Yakovina how to regularize a problem. One considers each curve and also draw something like roughly gradient lines on underlying sphere connecting points. Yes, so roughly speaking, one can consider maybe degenerate surfaces which consist of several components, and there will be some gradient, it's a bit long story, some gradient line connecting point on one circle to another circle in the three. And then the whole guy will have no boundary, so it's really well-defined numbers for homological spheres. Yes, so you get a parameter, it looks like you get new variable to raise to power H. Yes, and what is this new variable? And then there was a picture by thesis of a conifold transition. If you get a sphere in Kalevius three-fold, then I can construct a new family, one parameter family and one parameter will be responsible for this H of symplectic manifolds. Namely, do the following. You consider neighborhood of a sphere, neighborhood looks like a cotangent bundle to a sphere. And you can also identify this complex quadric in CT-C4. Then this complex quadric you can degenerate. You get kind of singular complex variety, so it means a contract S3 to a point. But now we can smooth in the complex geometry, make a kind of blow-up, but only get a kind of small blow-up. You glue inside CP1 instead. If it's a full blow-up, you glue CP1 cross CP1, but you just contract one. So you glue CP1. So in topological, you contract and reform, you get S2. And what you get? You get total space of two-dimensional bundle over CP1. Yeah, so you replace three-dimensional sphere by two-dimensional sphere. And now in this symplectic manifold, it has extra new homologic class, homologic class of the sphere. And then it means that it has more parameters as a symplectic manifold. You can change a symplectic structuring parameter corresponding to the sink. And the claim is that closed Gromok-Wittern invariance for kind of x, maybe tilde, for x tilde equal to open-closed for original for x and this dimensional sphere. And more or less, it's kind of clear, because if you consider this curve, then after this procedure, you get a curve and have some copies of CP1 added to this curve. Yeah. Yes. So before you get curve plus three-dimensional sphere, which is not really anything reasonable, but after this transition, you get a curve and copies of CP1. Yeah, the same one can do with several S-Series and these joint spheres. Then you get more parameters and get things depending on more parameters. Yeah. I want to say what is corresponding to B-model in the category language model. In B-model, it's actually opposite situation. Suppose you get Calabi-Yau three-fold and it contains what's called minus one, minus one, a rigid rational curve, which is rational curve is normal bundle O minus one plus O minus one. The typical rational curve should be this minus one, minus one. If you get this curve, you can contract, then you get a one-parameter family of new Calabi-Yau three-folds of Calabi-Yau, Y tilde, and how it's obtained, you contract a curve to a point and then try to smoothen it. So it's exactly this procedure, but in a sense, it goes opposite way. You don't blow up now. You contract a point, you get things, but now try to smoothen it. So go from singular guy to quadratic singularity. It's absolutely a parameter for deformation. So that's the story, and also in categories, it's also very nice, namely, you can see the category of coherent shifts on my Calabi-Yau right here, that's abstract category, which could be Foucaille category, and then this curve C, O of C is its object of this category and curve C, and it's a spherical object. So it means that X group from this object to itself, arc homology of three-dimensional sphere, then this contracting to curve to a point makes singular variety, and perfect plexus on a singular variety, it's a quotient category. Oh, it's not a quotient category, sorry, it's semi orthogonal complement to O. It's E such X from E of the spherical object is zero. Actually, it's perfect plexus on this category, it's a kind of full subcategory, it's kind of singular variety, and you can deform it using, as a category, a triangulated category, and you get this new family. This thing's actually not really yet done, so one can guess it's, the whole thing, behave nicely, so you get any three-dimensional Calabi-Yau category, you get spherical object, and you get new one-parent family. And now, and also this thing should be kind of a theorem, that this open-closed invariance are closed and don't fit invariance, so one should find some kind of formulation and watch as open-closed group invariance in two different frameworks, kind of, in B-model in more abstract categorical model. So meaning of open-closed invariance in B-model is the following, in B-model is integral of exponent of chance simons divided by h-bar of a watt, over, for the algebra, will be another, the algebra will be d-bar forms, actually, as well as there. In general, one get, this open-closed invariance will be defined for B-model and plus any, let's say, coherent shifts or coherent shifts or galmorphic bundle, which is spherical object, could be rigid galmorphic bundle, yeah, so consider d-bar forms, as well as endomorphisms of this bundle, and then this Calabi, that will be another quantum field theory, of halomorphic chance simons, yeah, there is actually much better situation here, this halomorphic chance simons, you also get some Feynman diagrams or expansion, and the whole thing is divergent as for Kaderi-Spencer theory, but there is canonical organization of all Feynman diagrams, so the whole thing is well defined, and in pure categoricals, I think it should be used to matrix integral, that's it, and categorically, you calculate the matrix integral, yeah, and here there's a really interesting point here, the whole thing exists, there's also kind of this additional choice, which I made, this gauging, this gauge choice, should kind of affect this matrix integral as well, because it's how we define what are stable curves, and so the, again, think which is not yet really done, so one can make a guess, to define the chance simons integral in algebraic situation, algebraic model, one needs a measure on space of objects, in the category of my bundles, yeah, and I need a way to define measure, and it's not clear at all, because this Calabi structure gives what's called shifted, minus one shifted, symplectic structure on model space of object, what does it mean that you've considered deformation of any, roughly, you've considered deformation of any holomorphic bundle, or any particular category, on the standard space will be x group shifted by one, and get this Calabi pairing gives you symplectic structure of degree minus one, on the super space, and then in this situation one get a, also chance simons function will be some function whose commentator is itself equal to zero, but for this odd symplectic structure there's no canonical volume element, unlike even case, and it's additional data, so we have a kind of classical master equation that things commit to the server, but there's no way to write what is Laplacian, makes, doesn't make sense without volume element, and volume element is something not one for free, and I think it's really coming from this additional trivialization of negative cyclic homology, which appear for, and this is not yet, from identification, negative cyclic homology should be, Hohschild homology is seriously new, and it should realize this line vector bundle. Yeah, that's actually interesting algebraic gadget to translate everything to algebra, and then we can see that we can translate infinite dimensional integrals as finite things to some game with tensors. Yeah, so that's what's going on, yeah, and in fact, if you have, you see I write it, if it's, if it'll be homomorphic bundle, get kind of six dimensional field theory, but if it's coherent shift supported on some in the curve, it looks really two dimensional theory, because this curve could be contracted to point it in zero dimensional series, so it should be reduced to matrix integrals on the nose stuff. Yeah, so in fact, this holomorphic chain of simons theory, depending on dimension of support of your shift, will be quantum field series in different dimensions, zero, two, four, and six. Yeah, so you get this holomorphic chain of simons, this kind of quantum field theory in dimensions zero, two, four, and six. Actually, I want to say that this one important thing, in this open closed things, I have two indices, genus and whole, and here I should get also two parameters, I get h bar, but also consider n copies of my wave vector bundle, yeah, so I should, so there are parameters h bar and n, and genus and h, they correspond to each, there are two parameters, can there, some kind of Laplace transform going here. Can we say QFT mean two dimensional topological point of field? No, no, no, quantum field, not topological, yeah. No, there will be some super symmetry here, so there are tools to calculate stuff, but it's not topological. Yeah, in principle, in some special cases, one can write things in a model. There was one point here, essentially, yeah, all this nice story, I have to say, but up to now, there's really no example of compact Calabria 340, we even know these numbers, yeah, so it's all, because of all things, it's not fully done in this algebraic transition, yeah, so it's, at least not in the sense that it can show numbers to physicists, yeah, so this, and almost all things which do physicists explicitly, it's some kind of limit case, say, you consider non-compact symplectic manifolds, so consider non-compact symplectic manifold, which can be, say, the formula, it's xv, it sits in some compact symplectic manifold, when we have divisive infinity should be semi-ample, so it means that intersection with any curve would be non-negative. Then, so, symplectic manifolds, from which the whole story can work, usually realizes complement to divisor, and then consider gv invariance for x bar with omega n plus, add some parameter, kind of one over epsilon times the class of divisor infinity, it should be positive, non-negative class, so it makes sense, and when epsilon goes to zero, you can go to the limit, and only curves which do not intersect the divisor infinity, so in a sense, open manifolds are kind of limits of close, guys from which there is a good picture, unfortunately I don't know good algebraic translation what is non-compact manifold, yeah, so it's things which is still missing, but here we have plenty of calculation, we've done, and in particular one can apply to the case when x is a cotangent bundle to some three-dimensional manifold times three, which is on a situation, and then this open grammar fitting invariance in the story, it means that you calculate channel Simon's theory, real channel Simon's theory, but for un bundles, the n goes to infinity, and it's spent in n, divided by h bar, so you got two parameters in h bar, and in particular this question which I think about the volume, it should be some kind of limiting case for a case of three-dimensional sphere, yeah, so this explanation should go through this thing, yeah, so one can get also three-dimensional theory in, is one of degenerate cases, yeah, okay, and finally we want to see what is the role of topological recursion from this perspective, yeah, and topological recursion when you make generating series, it will be serious in infinitely many variables, yeah, in principle the whole stuff also work in very general situation, I get gravitational descendants, but how what will be geometric meaning series in infinitely many variables, in variables like t1, t2, maybe p1, p2, infinitely many variables, and how are you going to try to see these variables, the picture is that Lagrangian manifold which you consider now is not one connected, but it has first homology, suppose it's Lagrangian manifolds across R2, so it's some kind of thing like this, and then if you get now you count your manifolds with boundary, each boundary components will wrap some number of times, and one can see that its number of times will be, if it's, and some situations could be kind of always positive, so this number of times will be a, a, a, i times wrap is supposed to weight Pi to this thing, yeah, so it's a way how to introduce infinitely many variables, and, and, and what does it mean, this infinitely many variables, yeah, so this, another way to arrange the whole thing was to have symmetric functions in, you can have functions, something like t, j, n of n, of n variables for each n, symmetric in these variables, and when you expand, you get some Taylor coefficients which will be wrapping, I think, and, so the way to do the following, this variable z1, zn can be sort as parameters of objects in 4k category, because when you have this s1 cross R2, one can put on this, it's like a manifold, but also it can be if I put rank one local system, which depends on one parameter, and my guess is it's, these functions which appear in functions of many, many variables which appear, this is the i runs through some algebraic varieties, some kind of algebraic curves, this algebraic curves parameterize objects in even 4k category, and that's, again, should be kind of translated to some pure algebraic game, but in principle one can see, one can go to more examples, for example one can see the things, these two generators of first homology, and it will be, instead of curves, we get surfaces and so on, yes, and pre-abstructly, of course, in real life, in principle, we can get Lagrange manifolds with arbitrary rank of h1, so we get a huge family of objects of categories, and my guess is it's, again, chance, so we get some object depending on zi, and my guess is it's what you calculate here, to calculate chance-samus, exponent of interval of chance-samus, oops, near easy one plus, plus easy n, yes, that's, that's maybe how one can really see this stuff, okay, thank you. For fukai category, there are no poles. Sorry? You're in for objects of fukai category, there are no poles. Poles? Yes, because the functions which we are studying, they have poles and diagonals. But homes, no, but here there's something singular, because homes between, if you have objects, depending on parameter, then homes really jump on diagonal, identity morphism disappears. Yes, homes are between different objects. No, no, but you can see the, if you can see the, under morphisms of this guy, this dimension of homology jumps when points start to coincide. No, just, yeah, that's it. Yeah. On that point, so if you take a carburetor, so called as a tangent boundary, and so your chance-samus is there, chance-samus, right? Yeah. My question is, what happens if you take a similar direction? No, no, if you make a general Calabius refold, and yeah, I've considered a general Lagrangian manifold sitting in Calabius refold, x6, say, yeah, then the channel-samus will be, in a sense, one can write formal stuff. This Lagrangian manifold will contain infinitely many loops, which will be boundaries of holomorphic disks, or maybe collection of loops with boundaries, of course. It will be isolated plenty of loops, and you kind of modify chance-samus section by adding, consider connection, which are not flat, but have some defect, and yeah, so it's pretty complicated story. Yeah, so it's going to be set, it's kind of local functional. It's, it has some information coming from ambient manifold here. One can write some action on this way.