 Да, так что вчера Ян introduce a little bit of Chan-Simon theory and we'll have more Chan-Simon theory today and I want to make kind of some comment, some mathematical wisdom, which I learned from Alexander Kirill of concerning, which is very relevant. It's from, it's a small poem by some famous Russian poet Kazimaportkov middle of 19th century. It's about myosotis. Myosotis is an invasive plant, it's kind of small blue flowers. It's common name is Forget Me Not in Russian's Nizabutka. Nizabutka is the same name. And so it's about some poor guy Pachomich, who is behind the carriage, the place is not sitting, just staying and it's shaking very much. And he has a huge bunch of this Forget Me Nots and then he had blisters on his heels and then back at home he curled them with a comfort oil. And then reader in this fable cast off the Forget Me Nots. The two jokes here place it, but conclude this. If you have blisters to get rid of your pain, like our Pachomich, cure them with comfort oil. So what I will explain is three dimensional manifolds, Chan-Simon theory, it's kind of like Forget Me Nots. We shouldn't forget to forget them. Okay, so now I'll start with kind of point of view on Chan-Simon, we don't have any Chan-Simon section, nothing like this, no three manifolds, no groups. So I will start with certain sequence of polynomials. There are some nice polynomials, kind of invented by Euler. You see that E01, EEN01 is n factorial. It counts permutations with certain number of drops, some combinatorial properties. So you have the sequence of polynomials, which satisfies some recursive rule. And EEN plus 1 is equal to x1 minus x dxEN plus 1 plus nxEN. Define them by recursion of these polynomials. In fact, it's kind of a complicated way to see things. You define rational functions through these polynomials and have x is xEN of x divided by minus x to the power n plus 1. And to start, it's much simpler in terms of rational functions. You just take logarithmic derivatives. So I get these rational functions. A, now make new rational functions. R, getting some other sequence of functions. Look how I define them. I say that r0 plus r1 h bar plus r2 h bar square plus blah blah blah. It's equal to exponent. So I use only even indices of polynomials. The rest are just to define recursion. Okay, just a second. So I get these nice rational functions. And the main hero today will be falling guy. So it's not a function formal expression, because this series is never convergent. It's kind of asymptotic expansion. And this satisfies Q difference equation. So actually it's a unique solution of Q difference equations, which has some asymptotic and x goes to 0, goes to maybe 1. The next goes to where? When x goes to a small positive number. You mentioned 0 and 1. Sorry? When x goes to 0, maybe I think it's 1. But yeah, uproar if consider, try to solve this equation. Sorry? There is an h bar. No, I didn't get it. H bar is kind of very small formal parameter. And when you try to solve this equation, it's in this form times certain series you can multiply by freely, by exponent of any series in h bar. And this is normalization. Yeah, yeah, okay, yeah, okay. Let's do like this. Yeah, so get this function. And now the next thing. In all this series, whatever you can call, you study some new functions of several variables, which are given by the following thing. You consider kind of n-fold integral, and dimensional integral. But this function is not defined by one different situation. Sorry? The function is not defined uniquely by one different situation. By symmetric expansion it's uniquely defined. If you write a certain series in h bar. Oh, in this particular form. If you write in this form and put this condition, and with this initial condition it will be uniquely defined here. So it's not a function, it's kind of formal expression. You can see some kind of prefactor times the series, yeah. Now you can see the functions of n variables. T i belongs like in C star, independent on h bar. And then multiply by exponent. And here you do something like this. You take sum of a i j log x i log x j of h bar, plus sum of b i log x i, plus maybe some c h bar. And then also multiply by exponent of t i, divide by h bar, and that product d x i log x i. Okay. And here what you have, you get a, b and c, which you can form to n plus 1 matrix. It will be symmetric matrix with rational coefficients, with n plus 1 coefficients. And what we should do, we should consider various integrals like this. For generic values of parameter t, or maybe for special values of parameter t, the generic values are really nice. For generic values these things have kind of automatically more singularities. For generic it has automatically more singularity, and you get finitely many singular points. And you get finitely many series in h bar. So what's the line below that the h j cannot be? M i, M i. B i, again. It's symmetric matrix, yeah. J is equal to a g i. Okay. So what are critical points? Critical points for the integral is a solution of the system for equations. 1 over x i, t i, product over j, x j, a j. We get the system of equations. And for generic t it will have only finitely many solutions. For special values it could be some variety. And this story about vanishing cycles. But for generic t it has more critical points. All more for generic t. And this expression will have certain form. Of course it will be 2 p h bar 2 power n over 2. Then exponent to something, again some critical value divided by h bar. Critical value depends only on matrix a. Depends on b and this g. And then certain series in h bar, which will be factorially divergent. And what I claim is that all these guys are resurgents. Resurgent. And what people study in terms of some boundary condition, whatever, it's expressions like this. Nothing more goes on. So three dimension manifolds. I forgot what is it. Critical value. But it depends only on aij and variable t. Rational matrix aij, not on b and c. And some variable t. And some of critical values. Sorry, any critical? Some of critical values. But there are only finitely many solutions. So you get finitely many series. It's how many solutions this algebraic equation for generic t. And this is what I think with expansion. This integral in critical point will have more singularity. So there's really no question what to do. And of course when you have kind of bad value of t then you have degenerate manifolds. You get vanishing cycles, story. Right. Since you get rid of channel 7, do you justify that that's exactly what... No. Okay, so... No, no, no, I don't want to hear. I'll do that. I'll do that. Yeah, yeah, okay, yeah, okay, yeah. Yeah. Sorry, you say in synthetic expansion that it's not a function. It's already a series in h bar. Yeah, but since it should be independent. This is critical value independence on h bar. Depends on draw h bar to power minus 1 only in this exponent. And I consider all the rest. The resurgence is already present in each factor phi. In each factor phi is... Yes, yes, yeah, okay. Yes, yes, I will go to it in a second. Don't... Don't rush, okay. Yeah. So get... For this... You get plenty of various series. Maybe depending on parameters. The claims are all resurgence. And if it depends parameter, it will be kind of analytic. Depends of... But the claims that same series in h bar can appear in many different ways. In infinitely many ways. So change metrics, change number of variables and change critical points and get the same series. So plenty of such identities. How parameters depend on... No, no. This is my fable. Forget about... No, no, no. I have no idea. No, no, it's forget it. It's kind of... You have to spoil your brain. Yeah, okay, yeah. Yes, yes, yes, yeah. You know too much, yeah. Okay, yeah. No, no. This was my introduction for this, yeah. Yeah. So there are many identities. So the first things, for example, you can do... Just free transform in logarithmic coordinates, yeah. You can do free transform. I don't have exact formulas or not. But it's essentially the same function of t. It's kind of... Maybe multiply by exponent of log t squared. Let's do like this. Yeah, so if you make formula for free transform, you get the same function. Or you do something... Martin, can you write a little bit that t exponent raised to power of what? Exponent of log x or multiply by exponent of log t squared over h bar. I suppose something like this, yeah. Just look on critical point and get right formula. Yeah, yeah. So there are plenty of identities. Or there's a kind of... Again, you consider one-dimensional integral. Consider integrate. But t will be equal to 0 or to 1. So I don't have... It will be special value. If you want to calculate these things, then the critical values... Critical points will be x equal to... And... Of course, minus 1. Yeah, I choose first one. And the expansion near the first critical point is the following. Is equal to exponent minus 1 over 40 2 pi i squared over h bar. So what's the second critical value? Minus. Minus. It's conjugate number. Yeah. Then we get square root of some crazy algebraic number. Then we get h bar to... What is it? Square root of pi. Square root of h bar. Our rational powers of h bar also appear in this game. Sorry. No, no, not rational powers. It's an exponent minus h bar over 40. There's a pre-fact of h bar. Okay, but you have... Yeah, you get up to this square root of 2 pi h bar. You get these things. And this is also a legal expression in this game because you get exponent of rational number times h bar. Yeah, so it's one of identities. I don't know how to prove this, actually. I checked it on the computer. I discovered it on the computer, but... Yeah, so there are many identities. And again, if you want to remember about three manifolds... No, this one expression is equal to another. Yes, yes, I checked maybe 100 coefficients. No, I have an expansion about maybe 100 coefficients. So it's something like this. And kind of analogy is the following. What does mean the same thing appears in many ways? These three manifolds, you can make higar splitting... Make like a surgery of some note, whatever, in infinitely many ways. Infinitely many... Let's say, higar splitting of three manifolds. But it's definitely some kind of game independent of three manifolds and we should understand the source of all these identities between various expansions. Okay, yeah, yeah, so I just said... I didn't really show you any example of resurgence yet. Maybe... Just I want to draw some picture. I just want to keep. We have this equation for function phi, yeah? And let's call this operator y hat. And x hat will be f of x goes to x goes to f of x. Then satisfy quantum torus. You get quantum torus. And you get this equation. Go to classical limit. You get a curve. And this curve is equiblically 1 is equal to x plus y inverse. And when consider a logarithm of norma from c star squared to c, then this curve will be some kind of amoeba. A little bit log absolute value of x, log absolute value of y. And it will be certain curve like this. And now what we do? Okay. What will be geometric picture? Now I won't kind of tell you what is geometric picture for this integral. You have, let's say, maybe in one variable. You draw this Lagrangian variety in c star squared. And intersect with some Lagrangian variety, which is straight kind of affine subspace in logarithmic coordinates. Because even consider graph of differential of this quadratic expression, you get affine Lagrangian subspace. So intersect. This thing is kind of various Lagrangian subspace, which are straight cylinders with rational slope. And now we do something similar in many dimensions. We will take many copies of this red guy. And we choose some kind of Lagrangian green guy. And this form said that this Lagrangian is a graph of differential of quadratic form. So it's open cell in Lagrangian-Grasmanian. But one can go to this kind of limiting case. Let's consider something, which is not a graph of one form, kind of a delta function in x variable. So what we're interested in? We're interested in just if you, in this kind of limiting case, when you replace this a, j by like infinite matrix, what you get, and have just one variable, you get the value of f at a given point. So what we are... Sorry? Yeah, no, no, no. The claim that it's this way of writing, it's really complete, because we have to do the second part. Oh, you mean if you have delta function? Delta function, yeah, yeah. So you need two Lagrangian. I need two Lagrangian. One Lagrangian is product of f, graph of differential of product of fx, you see. Yeah, okay, yeah. Yeah, so we should have this expression r. Yeah, just by definition of fx, for given point x you just get its exponential pre-factor constant square root of minus x, and then series with r0 of x. So consider given x in c minus 01, yeah. We get this guys rn, and we get sequence of numbers, yeah. And let's denote something like zeta, it will be 2 pi i log x. I assume that x is not a negative real number. Zeta is different from zeta from the upper number. Ah, it's not Riemann zeta, yeah, sorry, yeah. It's just variable zeta. I'm not complaining that this was another... Okay, now we want to do kind of Borel summation. We want to study series like this. For Borel summation we divide by factorials and make the claim, you get analytic continuation outside of something cuts. But first I just want to write your asymptotics for this rn, for latch n. Claim that rn should multiply by various things. Maybe it's called zeta with index x. Some things, zeta will be variable here and zeta with index x will be specific number. It will be position of singularity. Singularity in Borel plane. And now you do the following. You consider zeta x to power n divided by n factorial. And then the asymptotic expansion will be the following. If n is even and n goes to plus infinity, the asymptotic will be... fn is even will be 2 zeta x r1 divided by n minus 1 plus... Can you raise the blackboard? I'll raise the other one. So r1 at x. At x, yeah. Plus 2 zeta x, maybe cube. n minus 1, n minus 2, n minus 3. r3 of x, etc. And if n is odd, we get 2 divided by n r0 of x. Yeah. And if you've been on my second lecture, they're exactly the same things which happens with stealing, coefficients of stealing formula. And even coefficients have asymptotic control by small odd coefficients and vice versa. Big odd coefficients are controlled by small even coefficients. And what is the natural explanation? Sorry, what is on the left-hand side? There is no r, no rn on the left-hand side. Oh, sorry, sorry, rn. Right, yeah. Okay, yeah. Maron of x, yeah. So I get the things kind of depending on parameter x. The same things called uniformly. What here really goes on? It's a nearest singularity, yeah. And what is going on? Maybe I can use it first with both. Is there a sum on the left-hand side? Sorry? Is there a sum on the left-hand side? I don't know. That's for single term, right? There is no sum on the left-hand side. It's expansion for coefficients for large n, yeah. And these things come because we have, maybe, we have two even and odd, because we have two closed singular points. I will just draw the picture. So what goes on in the Barel plane? I get 0.0, when we start to expand our things. Then I get 2 pi i log x. And start to make a cut here. Then I take 2 pi i of log x plus 2 pi i. Again make cut. And then it will be 2 pi i log x minus 2 pi i. Cut and so on. Yeah, so I just continue cuts here. And I have opposite cuts. So the singularities are all branches of 2 pi i times all branches of logarithm and negative numbers. There are two arithmetic progressions. Sorry? Ah, no, no, it's for large n. It means that this will be the leading term. If we remove this for larger n, it will be this guy. Of course, for finite n, it kind of makes, it doesn't make sense, yeah. So I get this series at this point. Analytically extend to these things. And behavior near all these points. 2 pi i log x plus any integer times 2 pi i. May plus minus. This point is the following. It's called, I don't know, it's kind of like zeta m. Maybe zeta xm, yeah. This n plus minus, yeah. And the behavior will be the following. It will be, it will be holomorphic function of zeta minus, whatever, plus minus zeta xm. Plus, again, plus minus 1. I think it's maybe plus 1 up, half plane, minus 1 in the log half plane, times log of zeta minus this singularity. And multiply by the same series. So the jump will be the same function. It's like for sterling formula, you get just one series, which is appearing everywhere. Here it's the same story, you get the same one, story appears everywhere. Now that's, what? Yeah, sure. No, no, I started this lecture exactly to avoid this question. Now I have no answer to this questions. But I don't think this is a treatment. I have no idea. No, it's a space of a sort of connection. I have no idea. I have no idea what I'm talking about, sorry. Sorry, sorry, I don't speak this language at all, yeah. Connections, I don't know what it's going to be. Yeah, okay, yeah. I just mean this specific example, I don't think comes from a treatment. It's the same example. It's not given that that results in a treatment. Okay, yeah. I don't understand. Let's understand the model first. Yeah, yeah, you have this. Sorry? It's very akin to examples from treatment. And they're covered by your general situation. Yeah, just, now what goes on in H plane? It was barrel plane, now what goes on H plane, we should now make cuts just again exactly the same points, but now we make cuts going from zero. And my kind of resummed function, we should have analytic function in sector, and if we go through this corners, we make some transformations. What are these rays? Rays are exactly the same rays that started from zero. It will be stock's rays, yeah, infinitely many. And for example, one can now have function f up and f down. Choose two kind of opposite domains. So this function on f up, it's defined on a sector, but it's analytically continued to some larger guy. You, if it's always a function defined on a sector, in this formalism it's always extend to a larger, I think you add 90 degrees on the both side. So f up actually defined somewhere here. And f down is defined somewhere here. So you can compare them in left and right things. It's really very similar to what we have with Steering formula. And in terms of analytic wall crossing structure, I have kind of infinitely many transformations and I compose all of them. And to say that it's analytic, I want to say that it will be kind of analytic transformation, but it will be one by one matrix. Yeah, so what I'll have, I'll have one by one matrix and I have some jump formula for a kind of X in, let's say positive real number line. If you make the analytic continuation, you have certain jump formula f up of h bar is equal to f down h bar. It times some interesting product and similar formula for negative, I will skip it. Similar if you do not invert. You invert, you just invert variables. Yeah, like this, yeah. Where Q1 is exponent minus 2 pi i log X divided by h bar and Q2 is exponent of minus 4 pi square divided by h bar and assumes that h bar is kind of positive. Ah, ah, it's written as positive. Yeah, so things are very, very small. They get convergent series. They get convergent. It's not a formal expression. It's actually analytic function in h bar in this domain. So everything is well defined. Here we see some kind of Jacobi forms and when we make this product it's some kind of Laurent series into variables and you can see what monomials appear here. You get just two arithmetic progressions. But these two arithmetic progressions are exactly what we see that means you have here by applying central chart. When I apply central chart to this picture you get exactly this position. I cannot read what is in this picture. Coordinated somewhat. It's a look on monomials which appear in this expression. Ah, okay. Yeah, so the claim is this guy actually belongs to series in Q1, say, and Q2 divided by Q1. Yeah, it's formal power series, yeah. And the monomials which appear you get just two guys. And if you apply central chart, this is my lattice kind of square. I apply central chart. You get exactly the center points on left-hand side and right-hand side. You get these functions. These are actual analytic functions. And very easy kind of serial claim. These functions essentially the same as up to very small modifications. Fadeev dialogographies is actual analytic functions. And what's central chart? Sir? You mentioned central chart. Yeah, these numbers. It's map from Z-square to... Ah, it's just a value on the elements of the lattice. Yeah, yeah. I shouldn't ask about geometry. No? Ah, this is... I don't know. It's very geometric, yeah? Sorry? Ah, it's some interval, yeah, some periods of commode of pair, yeah. Something like this, yeah. Yeah, I think it's actually commode of pair. You take union of this variety. You have union of two curves. And consider periods of commode. Yeah, a gamma s will be kind of like h2 of c star-square relative to kind of red... This curve 1 is equal to x. That's why my inverse and union x is equal to 1. And you get interval to form commode of pair. And the interval to form will be this central chart. Yeah, so it's a very, very basic example, yeah. But it's kind of an upgrade of resurgence for Stirling formula, which... to the quantum-dolography. And somehow it's kind of new, yeah, because it was paper for logarithm of quantum-dolography, of the Schaeff and Gropholides, but not for dialog from itself. Again, we claim this... It's up to simple corrections. It's just a spadeev dialog function. Do you want to write what is it? Or maybe Yon will write it, yeah. I don't want to go to the sexual thing. Okay, so it's... This is a very simple game. This correction is a number which does not detect anything. It's some kind of... some very little something here, some normalization for this. Yeah, in fact, what I wrote, you said that it's a general pattern. I was lying to you, I explained like a semi-simple part of the story. And there is something about extension, some nilpotent. I think which is really essential. And now I contradict to myself and go to three manifold stories. In three manifold stories, people have observed something very bizarre. You have a contribution for trivial connection and for non-trivial connections. And if you start a contribution for trivial connection, get expansion, then by this extended extension you see other non-trivial connection. But you never see trivial connection. And for trivial connection, you see all of them. So it's kind of... Stock synthesis is really asymmetric. Yeah, so it's... In a kind of simple example, if you get kind of gradient lines going one direction, it will be in another direction. Here it's asymmetric. You get certain... You see when you do... In a barrel plane. Yeah, yeah. So there is something... a bit funny here. Yeah, in all this quantum invariance, there's also... People write some kind of finite num sums. I don't want to go to this in general, but you do the folic. For example, you have n. You consider h bar is equal to exponent of 2 pi i over n. Oh, no, 2 pi i over n, yeah. And q... It will be root of 1. Close to 1. And then you do... Instead of integrals, you get kind of like finite sum. For example, you can see the function... Function on a finite set. 0, 1, n minus 1. Given by... At least your... What are values of this function? So value of this... q factorials. And then... Essentially, you get this a symmetric phi from this story. You should consider the limit when j over n goes to a constant. In integral 0, 1. You consider a very large n, consider a very large j. So this ratio... Say it's a number, then... You can see this story. And if you look on this function, it should satisfy some, again, q d from the equation. You can see the operator x hat. Here will be... You can see this matrix. y hat will be cyclic rotation. So x y hat is equal to q y hat. And it looks it will be solution of the same equation. But it turns out not. So if you apply this equation, this operator, and maybe called function phi n, I know again phi finite. Phi finite, you get vector 0, 0, 0, except the first term. You get some nonzero number, more or less one. So it means that it satisfies... It doesn't something nonzero, and it killed by 1 minus x. So the right equation, which satisfies this guy, is this guy, is this one. So one considers this q... Such q difference module, which will be reducible. It will be not simple module. It will be extension of... reducible q difference module. And... Okay, maybe I just draw the picture. What is the support of this module? If you go to classical limit, you add also vertical line x is equal to 1. So support will be this guy plus another vertical cylinder. If you consider classical limit, you get just union of two curves. Disjoint union of two curves. And now you want to solve this equation and gain wave function in this plane variable. And there will be one solution along one original curve. There will be the original dialog, but you get kind of another solution of this nonhomogeneous equation along vertical curve. But because it's vertical, it's better to use different coordinates not to project to y-coordinate. So now we should intersect with horizontal cylinder. y is equal to constant. And if you look what is going on, you start to solve the equation. If you consider Fourier transform, these things you get 1, 1, 1, function identically equal to 1. So it means that now you have the following equation. Minus 1 minus exponent of h bar y dy times y applied to some function psi is equal to 1. Because function 1 is killed by Fourier transform function 1 in y variable goes to delta function killed by this guy. And for this equation we need solution of the form psi is equal to psi 0 y h bar psi 1 of y that's all. psi i psi 0 psi 1 are polynomials in y. You substitute to get something like this, maybe make the following recursion. polynomial 1 and then for each n greater than 1 it says that psi n is equal minus sum over k equal to 1 to n 1 over k factorial y over dy to power k applied to y times psi n minus k y. If you substitute these things to this equation expansions in series in powers of derivative eventually get this recursion and you get again some things like this c0 equal to 1 then 1 is equal to minus 2 psi 2 is equal to minus y2 plus 2y2 and then maybe show a bit more these are really funny polynomials. You see here this kind of first coefficient is one of factorial and the last coefficient is factorial so things go up and down and again you get this polynomial and claim which is kind of not scientific yet is that for given y if you take sum over psi n y divided by n factorial z to the power n make this barrel transform it will be again analytically continued with singularities at what at very obvious things if you look things about commode of pair and periods dialogue of two plus integer times 2 pi i times dialogue of y plus integer 2 pi i squared so it will be shifted rank to letters by one amount and jumps will be certain integer will be certain integer stocks factors and you get your original guy yeah so it will be kind of like one extension of another and jump from this we know yeah and this is the whole picture but jumps so we are trying to calculate with Veronica on computer yeah it's quite funny but it's ah it's integer stocks factors and multiply by this guy okay yeah so you see that it's I think it's kind of really the simplest possible example when you have this extension behavior but we still don't understand it in whole totality yeah I think it's simpler than any whatever 3 manifold story you have it's really the most basic example and now I just want to finish kind of this very general remark about resurgence yeah you see rational functions, rational coefficients or polynomials with rational coefficients if you substitute rational number you get series with rational coefficients or and what is amazing that after dividing by 10 factorial you get to something whose singularity will be some interesting numbers and what I see we have examples in kind of 4 different kinds what's the original story when consider integral exponent of integral of let's say polynomial divided by h bar times some volume element and consider such guys then singularities will be difference between two critical values yeah critical value one minus critical value zero something like this it's difference between two critical values and if you start with kind of things of rational numbers you get algebraic numbers you get algebraic numbers or you get algebraic functions if you get things depending on parameters this case of dimension 1 I have now integral of have one form instead of function and here I get new things for example things I get singularities will be more general in periods of one dimensional integrals for some point to another point then here in this situation we see things like 2 pi squared dialogues 2 pi times logs yeah it's two dimensional integral and then 2 pi times logs by by by Cajay Iwaki and Marcus Marini we see that if you do something some grom of written invariance and some of some Galabios reforges compact or not compact then we get I think in local case we get 2 pi times two dimensional integrals and maybe three dimensional integrals for three forms of cycles in Galabios reforges in Marcus talk which is that even here we miss something should be maybe some periods of case resurfaces depending on parameters should be some kind of natural sequences of rational numbers of polynomials depending on some functions of rational functions of some parameters and said that singularities will be like periods of case resurfaces or maybe general two dimensional integral and definitely we have kind of next level complexity what comes next versus periods periods of high dimensional varieties appears singularities some barrel transforms, some natural gadgets defined to rational numbers ok thank you questions if a question about this last way there is an integrable system for this you vary your whatever what is integrable system here it's a bit shaky and here it's transcendental integrable system yeah about this kind of difference equation I had a comment 20 years ago we studied something which was very similar in fact your psi I guess is the exponential of the solution to a different equation and we were studying the inverse of the difference operator itself when you can write the inverse of the difference operator using decomposition but in reality in the background you just had decomposition of one of the hyperbolic tangent as a sum of simple poles so the consequence is that when you take the Borel transform of a simple pole you get an exponential and so we automatically get for the solution for the logarithm of your psi we basically get a series which proves that you can study the singularities and we get exactly that pattern and more or less it's meromorphic in the Borel plane but then when you take the exponential you can apply the rules of alien calculus and you understand this is one possible explanation for the resurgence phenomenon that you have pointed out here so I can show you the details but it's really strange to take lower pole to see it well maybe not exactly that because you have the dominant term with the i logarithm to remove first here just stop it exactly that can be we combine by generating series I mean there is a super trick at the moment I can show you more questions? no 0, 1, 2 its pair of complex programs somewhere 3 partially if your calabria is local but you still multiply by 2 pi i respect to integrals you add c star we have conic bundle and period of conic bundle okay more questions? thanks again for your questions