 and the same stuff here. And, but it happens that there are particular choices of these coefficients, so they produce a nice, that they'll produce nice and easy combination, which has Q series expansion, actually. And this is what I will start talking about now. Any other questions? If you have, if you had these electric critical points, then you wouldn't, I mean, then those guys must be symmetric, but it's almost never, in the transmission series, it's almost never. Well, yeah, because I mean, once you go to this modified model space, where you don't, you question all your own base engagement information. For example, the irreducible and, like trivial for flight connections, they'll have different dimensions. So for example, irreducible, they will always, always be known as related in this approach. So this was a kind of proposal how to, well, some explicit conjectures, which propose how you can extend the definition, one possible way to extend the definition of this WRT environment to the, any complex K. But we want, we want something more. We don't want just some analytic behavior on K. We want some sort of, if you want to categorify the WRT environment, we want, as in John's point, we want to have some nice expansion in Q. These integral coefficients. And so for example, so if I consider, so suppose we found some analytic continuation, and so let me just mention, for example, a very simple case where there is actually no, I mean, the analytic continuation can be obtained very simply. It's not necessary to do all this machinery. And so you can, well, up to some simple, simple, very simple factor. It has the following form. Well, let me, well, if I want to write dependence on K, it'll be as follows. So this is a result, this result can be obtained by similarly to the exercise of our Lenspace L31, which was in the, I think yesterday's problem set. So you see, even here we obtained some analytic compression in a, so of course as usual Q is, it is a two pi I divided by K. So even if we obtain some analytic expression in K, it's not, it's hard to categorize it directly. There's some, there's some mess going on. And so one, but one can make a statement about that there is some universal way to reshuffle the WRT environment, so that we obtain some nice expressions which we can try to categorify. So let me consider to do this. So again, I will consider the modly space of a billion SU2 flat connections on M3. And so this can be realized as the Homes from pi one of M3. So now, so we require, a billion means we require, we require all the monomies to map to the, to the maximum torus of SU2. This will be U1 and the conjugated action then we'll just act as a Z2 while symmetric action on U1, on the maximum torus. And of course here we can replace pi one to its abilinization, just H1 of M3. And we actually will be interested in the set of connected components. And this can be identified with the Homes to the, of the just a torsion subgroup of H1 of M3 divided by Z2. Because the three, well, if you have some three components, they will just result in the kind of, in the enhance each point to just a product of some U1s essentially. So if you're interested in connected components, we can replace this H1 by its torsion subgroup. And then let me introduce following clinking pairing. So this is a billionaire pairing on the torsion part of the first homology of a three manifold valued in rational numbers, modular integers, and it's defined as follows. So let me consider certain cycle A and cycle B representing the torsion elements in the first homology. So I need to tell you where do I map it to. And to do this, I can take two, so since B in torsion, this means there exists some integer, let's say integer number P such that P times B is a boundary of some two-chain B prime. And then I want to map it, the stuff to the algebraic intersection number of B prime with A divided by N. And of course, this is only defined, well-defined modular one. So in particular, this link in pairing can be suit, it provides explicit, oh yes, thank you. Provides explicit isomorphism between the pentagonal dual of the torsion part of each one and the torsion of each one itself. So in particular, if I take some element here, I can construct a home by considering to pi exponential to pi linking number of A pairing by linking number of A with element here. So in particular, I can identify this home of torsion one to the torsion of each one itself. Questions? For example, let me give you some example. Example, so LP1, Lenspace LP1, so H1 is that P, so the whole sink is torsion and this linking of A and B is just A times B divided by P mod one. So now I can make the following conjecture. So this is, well, this appeared in the work by Gukov, myself and Balfa and also there was another work by Dupé. So let me state a bit of a simple reaction of this conjecture, which, so in this particular reaction it's actually fails in some cases, but there is some small modification which should do, but let me don't go into those details. So the conjecture is that the conjecture is that the conjecture is that the conjecture so the conjecture is that, let me write it in the form. So there exist some elements that had A where the label A belongs to the torsion of each one of MCA divided by Z2. So Z2 action is again induced by this, so first of all identify this space with torsion by itself and the Z2 action, which is induced by the Wildrop action is just the flip of sign here. And so the quantities which belong to the following space, one over two C, q delta A times series in q is integral coefficients where C is some positive integer and delta A is some rational number. So this is a set of indices such that this thing, the WRT variant is related to them with the following expression. So I sum over a pair of indices A and B belonging to this set, torsion divided by Z2 and there will be two pi i k times linking and pairing of A with itself and there will be some matrix SAB. This is not just one in the very beginning, this is not as this S matrix of the MTC. It's very different. And I want to take the limit of those whole thing to k goes to the, well, q. So this stuff is some series in q and some more of the statements as they are convergent in the unit disk and then I can take a limit. Of this expression, I don't have, I take a limit of this expression to q goes to exponential of two pi i divided by k, where SAB is the following. So SAB is independent, is k independent. So only dependence on k is here and the most important is the dependence which is very simple and also here. But what is the importance? So S is independent on k, but it's important that the whole other, like after this recombination, all other dependence on k is repackaged into some series in q. So this is some normalization coefficients, the belizer of element A with respect to Z2 action and the square root of the number of torsion elements. Any questions? Yes? Well, it's kind of this statement is kind of consistent with the statement which you mentioned. Well, okay, let me, I was just about to mention. So first of all, okay, so before mentioning this, let me mention the following remark. So this factor which appears here is just, so in the exponential, we have just the value of transseminence functional on the corresponding abelian-flat connection because of this relation which is one of the exercises today. And the other remark is that conjecture is consistent with the transseminence function which is consistent with those resurgence conjectures which is the stronger version of the asymptotic expansion conjecture. And so this is because, is that although it's naively it looks like there is the only sum of abelian-flat connection but it is the point that the non-abelian connections are packaged here in some way so that if you take, so if you take a particular term, a term is fixed A here, so you have the following thing. So this is, so this expression in terms of the, this quantity is ii which I did this before as the following expression and B. So it's the sum, so there is ii for abelian-flat connection contributes with coefficient one but then there are some sum of contributions from irreducible flat connections such that of course we want to, there are some self-consistency conditions that this sum over A is, should be the same as coefficients and B which appeared in the previous conjectures. But moreover, it's, I mean using this kind of wall crossing formula one can actually repackage by kind of choosing a particular contour. One can express of course all these guys just by starting from this B corresponding to abelian. These functions B alpha xi where alpha corresponds to just abelian connection, in choosing particular contour. So of course this, I mean suppose this conjecture is true this doesn't, in principle, doesn't uniquely determine those series in Q, so you can have a series in Q which gives something trivial when Q goes to a root of unit but from physics we expect, from physics expect that hat A to be a new independently defined invariance and moreover they have, from physics we expect them to be categorized in a way similar to how one of categorization of Jones polynomial. Moreover, expect that hat A is to be obtained as some graded early characteristics of certain homology spaces associated to a C manifold. We should also have carry this index A and so this is why actually the name kind of the word BPS appears. This lecture title, so the, those guys A are BPS states of 6D 2,0A1 type super conformal field theory on M3 times disk times R where R has a meaning of time along which, along which time direction along which we can, we can see the quantizer or series and consider this Hilbert space from which we just take a BPS part and A labels boundary condition on disk, on the boundary of this disk. Sorry. It's invertible even if you have a BPS sorry. It's invertible if, well to be precise to be careful I have to assume that M3 is D2 homology sphere then S is invertible. Well I don't want to do this, I want some, I want to, that's what I'm saying is there is no, even if S is invertible there could be none unique that hat. I can add some function which one is like pachamer symbol which one is just a total or something. I want to calculate that hat's some some independent proposal which one can do precise for some subclass of manifolds which I mentioned briefly now and I don't think this approach is, well I mean yes, you can try to do this. I mean yes, I mean this can help but not much I would say. And yes, I'm saying if like by itself it doesn't fix, there are a lot of ambiguity from just this, I just add something multiplied by pachamer symbol. So they can be independently defined for certain subclass of manifolds which are called plumbed manifolds with some extra condition on the, with some condition on linking matrix. And so plumbed manifolds means it's a, it's a, it's a, it's a, it's a, it's a, it's a surgery on a link of unnotes which can be encoded in what is called plumbed graph. So graph means just, so each vertex correspond to a knot and the edge means that the unnotes are linked in the, in the canonical way. And just, just write one formula and so let me denote by ai are framings. So for each unnot here I associate some integral number which denotes framing. And so it can be given in terms of some sort of integral of some generating function where I, I have a number of variables for which I integrate to be the same as number of vertices. And then I also have this, for each vertex I have this factor that minus one divided by that associated to the specter times to the power which is two minus valency of the vertex. And then I multiply it by, by a theta function corresponding to m which is a Lincoln matrix. So this has explicit expression as the following sum of our lattice. So l is the number of components as, as before I consider lattice with quadratic form given by the Lincoln matrix. And so one can check that it's particular, it's the invariant under Kirby moves which, kind of Kirby moves which relate the graph to another graph. And the graph and their statement that so any, any, any, any two plumb manifolds can be, can be related associated to, to different graphs can be related by, or by the special graph Kirby moves which, so there, there's some combination of, of this Kirby and Rob moves which I mentioned before which takes you one graph to another graph, in particular. Okay, so, and so also let me mention there was a relatively recent work by Gukov and Monolesko they also consider a kind of generalization of this story to kind of manifolds this to write boundaries. And okay, but yeah, I'm sorry, I'm out of time. Let me stop here.