 Okay, is it working? So there was the following space SL, ASL2C with tilde associated with three manifold. So this was understood as a space of all SL2C connections on some closed-ended three manifold M3 where we do a quotient only with respect to based gauge transformations connected to identity. And of course, this can be understood as a, so there is a projection to the usual space of SL2C connections, model of all gauge transformations and the quotient is essentially done with respect to that times just a single copy of SL2C group. So up to this quotient by SL2C, this space can be understood as a universal cover. And so there was, so from this space, so we have a trans-simons functional maps to see. Contrary, so here as a trans-simons functional was only well defined model of that. And then there was the subspace of critical points which we denoted by MSL2C flat. So of course it's all depends on three manifold, sometimes they will omit it. Also the tilde, which is of course also projected to the usual model space of SL2C flat connections. And okay, then there were, so they have a notion of perturbative environs so they can be encoded in some kind of formal power series expansion which can be the generating function for them. So this is a formal power series in trans-simons level. So it starts with some power d alpha plus one and goes from zero. And then we denoted them my a and alpha. So alpha, so let me kind of, so alpha in principle should label the connected components here, but so I will abuse notation, I will take, I will label, I will choose alpha to be element of pi zero of this biggest space. But of course it's all these coefficients and the shift depend only on its projection here. Doesn't depend on which of the copies we take upstairs. Okay, and so in principle you can relate this shift to a dimension of the corresponding connected component and the formula is like this. So let me, so this is the corresponding connected component in m tilde flat. So, okay, well, yes, well here, this is supposed to be smooth because you don't, well, yes, this is a generic point. Yeah, well, this is a generic point. Well, this should be some sort of maximum or minimum, some stuff like this. I mean, well, I guess the minimum. Yeah, I think it should be minimum. Okay, so, but in principle we, yeah, okay. So now, and then what was the, so I started to formulate some conjectures. So first, so let me mention that the conjectures kind of, well, they can, well, not very explicitly, but they appeared in the work by Gukov, Marino, myself, where we also explicitly checked them for various examples. So first conjecture, A, but of course they are motivated by this construction, by this kind of, by this relation of transsemination theory to some sort of generalized Picar-Levich's theory proposed by Konsevich and Witton. So conjecture A is that, so if you consider the following power series, so let psi alpha be the value of transsemination functional on any representative from this, from the corresponding connected component, labeled by alpha, then, so we can, you want to construct the following series related to the series by some sort of rescaling of the coefficients by the gamma function. Let me make sure that I write this correctly. X alpha, d alpha plus n. And then the statement that it's, it can be, so first of all the statement that this has a, this series will have a actually finite radius of conversion, then it can be analytically continued to a cover of complex plane minus, minus, with critical values of transsemination action removed. And so again you can, kind of with the motivation of this is that, so you kind of understand this big space to be fibered over C by transsemination action and then there are, so this the connected, so there are some connected components of critical points and alpha. And so they can be kind of isolated, but they can be non-isolated and they always come in z-works of copies. And so they project, but they all project just points in this complex plane. And this is a plane where I can think of where my variable xi lives here. And so the idea is that this B of xi is some sort of, so the value of this is some sort of integral over some half-dimensional cycle, this thing, which we had in the past integral. So this is kind of heuristics behind this. And of course when we want to analytically continue on this plane, we can encounter singularities of this function where some of the, so when, well this part is not really important. Well I was just saying dimensions that, so yeah the value B alpha xi can be understood as the integral, and heuristically of integral over some half-dimensional compact, half-dimensional cycle in the fiber. So this is the cycle, so we start with some point, critical points are alpha, we take a cycle which shrinks at this point because this is a critical value, and then we want to kind of analytically continue, there can be some singularities when some other cycles shrink. And that's, integral of dA, well the same thing which we had before. So the integral of this exponent at two pi k times transimus. No, this is correct. Now I alpha, alpha will be integral over the whole left shift symbol. So this is, so you can imagine, so here I can, so just the value of B alpha of xi itself is just integral over the half-dimensional cycle in the fiber. But if I first integrate, which I will mention tomorrow, if I first integrate this function over a ray here, in a particular direction, this will introduce additional integration over this, along this direction, and this will produce the integral over the whole left shift symbol. So here I'm talking about not the integral of B alpha, but just the value of B alpha. Well, left shift symbol half-dimensional is the whole space, but here we're taking half-dimensional cycle in the fiber. And moreover, so the second part of this conjecture is that indeed the integral, this thing over the contour which starts at xi alpha, it goes along a ray. So this is a contour, let me denote it by gamma alpha along the ray with the direction determined by the argument of K, is convergent. So this is conjecture A. And so if it's true, then let me define pi alpha to be the symbol. And then indeed, so kind of intuitively, I want you to understand, so we first, we integrated over half-dimensional cycle in the fiber, and then we first integrate over the ray, and this produces me integral over left shift symbol, and this is what we want pi alpha to be. But now we want, so we want to kind of avoid, we don't really want to define it as integration. I mean, we cannot define it as integration over this infinite dimensional space. We don't know the reason, but if you know, if you know the perturbative environments, and they have in some particular, under some particular conditions, they have a precise definition, and one also can, there's some algorithm to calculate them. Then one can just define alpha to be this thing. Assuming they have, they indeed have these properties, so this conjecture is satisfied. Questions? So this here, of course, we want to assume that this ray doesn't pass, I mean, the ray lies in this, in this, we lie in this cover, so it doesn't pass through any critical values, any other critical values, and so the second conjecture will be about what happens, what should happen if, when it does pass through this singularity, and so of course, this, the connection B will assume that the conjecture A is true, and so the same is that there exists integers B alpha and alpha, and such that, so the first part, such that I alpha, so of course the definition of, the definition of this guy depends on K, well to be precise, it just depends on the argument of K. So if the argument of K is the same as, so here I might be not very careful with science, so if I want to, so if the argument of K becomes this, is equal to the critical value, so its current size, is such that it's, the ray will pass so this can be understood as, for example, so this is my XI plane, and so this will be XI alpha, this will be somewhere XI beta, and so I want to start with kind of this configuration, so I take argument of K to be a little bit less than this critical argument, and the statement that should be related to I alpha's, whereas the argument of K is a bit, is a critical value plus epsilon by the following, by some linear combination involving integral coefficients in which I have this form. Well the, well you have to, I mean, well you, sorry, but I mean your integral will pass through the singularity, or you have to, you'll have to regularize some of your integral, there's some ambiguity. I mean the integral is all defined only outside, and so you can imagine, so when you pass through this thing, this kind of, this contour, you can imagine this contour will go to a linear combination of the same contour plus something new, and so now the second part of conjecture, so to formulate the second part of this conjecture, let me introduce the following terminology. So I can separate my flat connections into two groups by the following properties. So our first property is that like first group of the flat connections or correspondingly they're connected components, I will call a billion, which means they can be conjugated into, so the holonomies, all holonomies can be conjugated into C star subgroup of SL2C, and the other, all other I will call a reducible. No, no, but I mean there are many, I mean they're actually exactly the same number of half dimensional cycles as left system, as connected components to critical points, but that half dimensional cycle is, so the half dimensional cycles, the complex half dimensional cycles in the fiber, they need one to one correspondence is connected components of critical subspace. So here I put, like to define this B alpha, I mean the B alpha I pick a particular critical point, and this is a half dimensional cycle, which corresponds. XI alpha is a critical value, and alpha is a connected component of subspace of critical point. So the second part of the conjecture tells, is the following, that there are very special, the coefficients are quite restricted. Well, sir, what do you mean? I think two. But okay, I mean there, yeah. I mean, I can imagine some special case when I mean kind of I have some degenerate here. I mean, but let me not go into detail. Okay, so the statement that this thing is zero whenever if alpha is irreducible and beta is a billion. But the other way around, so again, if alpha is irreducible and beta billion is non-zero for some, a billion where alpha is irreducible. So if this holds, then it actually means that using this wall crossing phenomenon, wall crossing formula, we can always, I mean suppose we know somehow this coefficients M, which more or less has a meaning of the monodromy of the half dimensional cycles in this formulation, and then we can express alpha, which is irreducible from our beta, where this is. So if we wanna assume that this, yeah, this is the moral. No, I mean here, oh, what happened? I mean, you can, in principle, you can have, you can have something. Yeah, you can have, well, you can have some, well, you'll have infinite sum, right? Sorry, say it again? Yes, so for, in this irreducible case, I think. Yeah, yeah. Well, it's some sort of, well, I mean, it's some complicated, it's some function of K, which obtained from those environments by this procedure, which I mentioned. Like you do, you consider some generating function I would say kind of all of them, essentially. For example, even, for example, to determine, to determine, for example, just like the position, the position of the singularity. So for example, one is like the weak statement of this whole thing, is that you can, like starting from the, starting from the perturbative environments at one point, at one, at one, at one flight connection, you can determine a value of, for example, a certain value of transseminence functional at the other flight connection, potentially. And this is actually determined by the asymptotics of this finite type of environment at one point. So kind of the, each diagram here is determined by some asymptotics of the environments there. So it's not just, it's not simple, but still it's quite, I mean, it's quite amazing that it works in some examples. Okay, and now the conjecture C is that, so let me take numbers. Well, let me assume that they are rational. So take some numbers for each, rational numbers for each alpha, such that, so I want to require if I take a sum alpha or the lift, some of the all lifts of an SU2 flight connection. So inside SL2C flight connections there, of course SU2 flight connections. And so if I pick a particular component in the model space of SU2 flight connection and the sum of all lifts, I want this to be one. And for any SU2 flight connection. And if I take the sum of all lifts of non-SU2 flight connections. So the SL3C flight connection, which cannot be conjugated to SU2 with imaginary part of transimons less or greater than zero, then I want to require this to be zero. And so let me write it here. If I take the sum over an alpha, so the same thing as before, but with imaginary part of transimons greater than zero, then it should be equal to some, it should be equal to some integer. So if I take, well, let me write it in this way, of some integer Nc, which levels this SL2C flight connection with imaginary part of transimons greater than zero. And so the statement that there exists those numbers Nc, such that if I pick an alpha satisfying this sum rows, so kind of, I fix the sum over all these towers in the universal cover, then my double ERT invariant, so this guy was defined for positive integral z, will be equal to this sum for alpha when I restrict K to the plane. So this can be ensured as a stronger reaction of much stronger reaction of the asymptotic expansion connection. It's not just you relate asymptotics of the double ERT invariant with this perturbative invariant, but you relate exact value. Exact value with those guys alpha, which are constructed from perturbative invariant. Questions? Yes? Well, because if this would be true, then the asymptotic expansion conjecture would be false. I mean, those guys, I mean, these conditions are fixed by that you want asymptotic expansion conjecture to be true, at least. So for example, all SU2, so in the stability assumption connection, only SU2 appears, and they appear with kind of, with coefficient one, essentially, with some proper normalization. But those guys with positive value of trans diamonds of imaginary part, they can contribute with some multiplicities, which we don't know, but one can try to fix them by looking at this, at the topology with half dimensional cycles in this space, in principle. Yes, it's just, it just shifts by this, by like two pi i k times integer. That's why when you stick k to z, to z plus, you only care about those sums. So of course, you can always choose this, like you can always choose this sum to be, if you want this sum to be fine, and just, in each of these sums, take only a single non-zero value. So this would be a simple choice. But sometimes you get a nice, so of course, the different choices of an alpha will give you different analytic continuation, which has the same restriction to integer ks. But the point is that, so one can of course take them just to be, again, once, in each of the sum, in each of the sum, only single is one, the other, or other is zero.