 to asymptotics of the double material. We want to make a relation to, in exact, to a value of, to a value at finite k. So, and in order to do this, so one has to talk about, so one way to do this is to talk about what is called analytically continued transimons theory. So what is the goal? The goal is some sort of define SQ for k, a complex number. So away from a positive integer. And so one of the motivation from this is that, so we know that for example John polynomial, the color of John's polynomial of any link is, so it's a Laurent polynomial in, or depends on the normalization, is Laurent polynomial in Q inverse, where Q, so if you want to relate it to the level in the transimons theory and the level is related by S to pi over k. So this defines, this is some analytic function in Q or analytic function k. So this is defined for any complex k. And here, so this kind of, the fact that this is a polynomial in Q is very important part of the definition of Havana homology, of course, the verification of this thing. So we want, so this is a part of the motivation of this. So how do we do this? So the basic idea is that, so again, if you have some sort of finite, if you have a finite dimensional integral, suppose over some just real line of some function of x, which is analytic function on x, yet actually can always understand it as a sort of integral over some contour, which is line here in this particular case of on the complex plane, where we can see that now f, we extend this function f to a complex plane, to the whole complex plane, and now we have integral over a line in the complex plane, now we are allowed to do some sort of deformation, contour deformation, as we want to simplify expression. And so we want kind of this general perspective to apply in a sense to some infinite dimension context. So this is kind of, so we want to define, so naively, I mean, suppose the naively definition if you want, so the of analytically continuous assignments will be as follows. So this will have some labels gamma, and it will depend on k, so k will be some also just complex number. And so we wish to define it as follows, and so gamma will be a contour in a certain space, so some sort of half-dimensional contour. So first of all, what is this, or what is this space? So naively we want, as here, we want some sort of complexified version of the space of SU2 connections, model gauge transformations. But what we actually need to do is a bit different thing. So again, so naively to do a quantification of the space of SU2 connections, we just replace SU2 connections to SL2C connections. But we want to quotient, when we do quotient, we have to be careful. So we want, first of all, we want now this to be defined for any complex k, and we also actually want this manifold in order to do some sort of statements. We want this, it would be fine to have this manifold, to be a smooth manifold. So what we want to do is we want to quotient our based gauge transformations connected to trivial. Gauge transformation. So based means that it's, so the gauge transformation is a map from, is determined by a map from a three manifold to now SL2C. And so we connected to trivial, that we can continuously deform it to our trivial map. And based means that we fix it, we fix a point in M3, and we require that G of this point is identity. So the first modification makes this manifold to be smooth. So if a quotient, the base gauge transformation act freely on the space, space of all connections. And the second modification, it leaves the requirements that k should be integer. So the requirements that k should be integer came from the fact that we wanted the whole expression to be invariant under all gauge transformations. But if your only quotient with respect to base gauge, sorry, gauge transformation connected to identity, then we can put any complex k here. And well, I mean by the smooths that this acts freely. Well, I mean, I start with some sort of affine space. I mean, so far this is some sort of naive heuristic, but from which I want to make some precise connection. I mean, I start with just the space of connections, it's just some, so I see all these bundles of materials. This is just from affine space. Yeah, just some from affine space and quotient by some group which acts freely. So I would, I think it's smooth. So the base motivation, so this would be smooth. And connected to one is that we want to, we want to this thing. Exponential of trans times actually to be well-defined. Because if I quotient over all guys, this is not gonna be well-defined on this. This is not gonna be well-defined functional on this space. Yeah? Yes? Yes, yes, because I mean, again, so this one can see so this is some affine space and now I quotient with respect to, so this gauge transformation, they all have one. So in general, I mean, kind of, if I start from some sort of affine space E and quotient with respect to some group, the fundamental group is by zero. Now it's three. So before the fundamental group was Z and now the fundamental group is three. Sorry, say it again. This is SL2C, can I? Yeah, well, I can understand. This space, so let me understand as a cover indeed of the usual space of SL2C with some projection. Where here I quotient over, make the usual quotient over all gauge transformations. Not based. Well, no, the trans times functional. So I can understand this some big space and so I have a value of trans times functional which is defined, which is now takes values in C. So before, if I want to define trans times functional here, it will take values in C mod Z. Now it takes values in C. So I can understand this kind of the space to be fibered over a complex plane. And so if this is kind of, if this is a real part, then this kind of, the extra quotient means a quotient with respect to translations by Z in this direction. No, no, but here I want K to be complex. Not, that's the point. I want K to be complex, not integer. I want this to be a certain map to complex. So complex numbers on the space over which I integrate. By the way, this is kind of this approach is due to Witten and also Konsevich. Although there is no actually published work by Konsevich, but he gave, you can find some talks. As usual, you can find some talks of Konsevich. Okay, so, good. So, and then the point is that, so gamma is a contour which represents, which should represent a cycle in a element in a certain homology group. So this would be a homology of, so half-dimensional mid-dimensional homology group of the space relative to a certain subspace. Let me call it B with integral coefficients where B is, so let me be, I can understand kind of B as a subspace of the boundary of the space where the value of transiments, the imaginary part of the transiments of, so let me write it like this. The real part of IK times transiments is much less than zero. Essentially, if I understand some, what happens here? So there is this big space, a SL2C, and there is, so it's non-compact space, but at a certain point when we go away, there will be certain regions where this function becomes exponentially small. And we only allow, of course, for this integral to be kind of, at least heuristically well-defined, we only allow gamma to go to those regions at the non-compact directions of the space. Okay, and now there is the following statement. So let me denote it by, I don't know, H. H has a basis given, well, represented by what is called left-shut symbols. So where alpha will be in the levels is labeled by the connected component of the modified space of SL2C flat connections. So this, so there is a subspace of critical points of transiments functional here, which I denote by MSL2C flat. Again, before it can be explicitly expressed using the HOMS from pi1 to M3 now to SL2C and times that. So you see the difference between the usual space of flat connections is that first I don't quotient with respect to conjugate action, and this corresponds to the fact that I quotient, here I quotient only sort of base-gear summation. And there also be the extra Z factor, which corresponds to lifting the flat connections in the true space of SL2C connections, model all-gear expansions to this universal, to the universal cover. So in particular, if I have some critical points here, so they will do this, or they don't like this, so maybe they may be non-azillated. And here, so they project it to some critical values, they will all come in towers labeled by Z. So suppose I explicitly choose, so when you define gamma alpha as the steepest, as a union of steepest descent pass starting from a compact mid-dimensional subspace of connected component of M tilde, flat SL2C libelate by alpha. So we label connected component by alphas. So how does it, you look like, roughly you can imagine, so they have the steepest descent flow, so you look like this, gamma alpha. So if this is, for example, gamma alpha, the connected component will be by alpha d, this will be gamma alpha d, and if this is alpha one, if this is connected component by alpha two, then this will be gamma alpha two. And what's important here, one can argue, easy to argue, that all the extra symbols with respect to this map, they all project it to raise, going in one direction, okay? And so, okay, so this contour, which we had before, then it can be, since this is a basis, this contour can be decomposed with respect to, with some integer numbers, with respect to the corresponding basis. And so in particular, so the integral, if I have some integral over the contour, gamma, this will be a direct sum with coefficients and alpha, where i alpha, where i alpha is integral over this basis of the corresponding left shift symbol, okay? And the border property here is that, so if you want to use, you can find a dimensional intuition, the important property here is that, it's uniquely, the value of such contour over left shift symbol is uniquely determined by asymptotic expansion around corresponding connected component. I make it precise. So one can, so this is intuition kind of, and now I want to make some precise, so suppose you have all coefficients of the asymptotic expansion, then there is two all-order, then you can write explicit formula for this guy in terms of these coefficients. I'm here, suppose my contour gamma has the composition, has the following decomposition with respect to this basis contour. Well, they can be, in principle we don't know, but I mean they can be determined from topology of the space. Well now I don't make any statement about what do you know by N alpha, but I will make some conjecture which tells you what are the constraints on those coefficients. No, this statement is independent. It has nothing to do with N alphas. This is just a statement about particular contour over left shift symbol. Yeah, this statement has nothing to do with those N alphas. I'm just saying if you know the coefficients N alphas, the question of this contour is reduced to the question of correlation over those contours. That's, well, okay let me, yeah I know I'm out of time, but well let me make kind of not, let me just start to make kind of first part of the conjecture which I want to make. So conjecture A, suppose this perturbative invariance which I continued before, well defined. So we are under those assumptions that those people consider, those people define those perturbative invariance, then consider the following series. And so we define those series and the statement that has finite radius of convergence and moreover can be analytically continued to a cover. So let me continue here to cover of complex plane minus a bunch of points removed where alpha are values, are critical values of transimmon's functional that is values on the corresponding connected components of the flat connections. So of course here we can see that the values plus z, so all possible values here, all possible. And the integral b alpha xi into the minus xi over contour which I, I take a contour, I start with xi alpha and so this is a critical value corresponding to this flat connections. So I consider any lift to the universal cover here doesn't matter and I go in the direction determined by the argument of K. So this is essentially in this plane, I can think xi as living in this plane and this is a contour, which I'm taking. And so in the statement is convergent if contour does not pass through other critical values. So this means if I continue this way, it doesn't counter the critical values, any other critical values. And so if, sorry, then if this can actually if true, then I can define I alpha by this integral. Okay, let me, sorry for going over time, let me stop here. Questions, yes? Well, in order to make this space over which I integrate to be smooth because the, I mean, the usual, this picard left, so we are kind of using some sort of picard left, the heuristic of picard left here in finite dimensional set, a case and there we want the manifold over which we integrate to be a complex smooth manifold. Yes? Well, the K determines this direction because the steepest distance, like these left symbols in the definition, they have, so the steepest distance pass with respect to, sorry, well, I wrote this kind of the condition so with respect to real part of IK transimons. This is just transimons, yeah, this is just transimons. So here the large, in this picture, the large gauge transformation make a shift by integer in this direction. Just, well, sorry, okay, sorry, I, so yeah, just, yeah, yeah, the large gauge transformation make a shift of transimons by integer. Yes, because I mean, the choice of these bases depend on K. So I mean, the homology says the definition of homology itself depends on K. Yeah, this is what I will, I mean, this was the first part of the conjecture, the second, there will be other part of the conjecture which discusses the behavior of those guys with respect to K. So here, again, so everything here is defined, is defined for particular K here. This definition depends on K. So once, and there is assumption that this contour doesn't go through other critical points, but I want to say, well, there's also some statement one can make what happens if I go to other critical points.