 Okay, so we'll do it later and we're speaking about vertical, what are we going to do next? Okay, let's hope we actually get to the vertical, I mean, until now it was always, the announcement was always slightly different from what was said, it was always slower than predicted, but anyway, so we had defined the vector written there by formal localization of the space. And then we have defined 0, so it's not compact, so it is then actually defined as by localization, we see some of the components, connected components of the x-propos, but as to mind you, we have an element n, which is the x here, which is the only effect on this thing, and then we can localization formula, formally, this thing was compact, this is what localization tells us, and we set the definition for fixed point locals, here's the point that we should see. Now we want to describe the fixed point locals, we can assume, we can choose e to be equivalent, so that there is a c-star action on e, just means now we are on the fixed point, looks like we're just means that we're fiber-wise action. Remember that, that means we have a weight composition, direct term over some i, this is the action, so this is, we say, we set things up, we also have an economical class, we don't choose the typical action of an economic class, but we have an action of weight 1 on the, so the action, that's if we have an equivalent for morphism from e to e times of ks, then it will be invariant as a c-star, if not only the weight drops, it drops the weight by 1. An equivalent morphism of weight 0, so here the weight of this thing is i, the weight of this thing is j plus 1, so j plus 1 must be equal to i, so and so we can just therefore describe this thing like this, and so from this we can describe what the fixed point is, that's the fixed point, so I should say here just choose the weight to be from 0 to l, I can always rescale, I can always, I can always put myself in this situation because if I multiply the whole action by a power of t, nothing changed, so I choose it like this, so what are the fixed points, so e will be written as sum and it's 1, so the s, direct sum of all of them, some of them match the i part to this part, and like this obviously we'll send this one here, and this one has no other chance than going to 0, I assume we know this, and in addition to this notion of stability, one looks carefully what it means here, and it shows that all these maps, all the maps must be injected, except for the stability is some kind of condition that the kernel has to be in some sense smaller than sheet itself, and one looks out what this means that it gets this, so we have the, well okay, so this in particular, so this has subsheets of some sum and positive rank e, and so we can see therefore that as this maps an objective it means that the rank of e2, e2 must be smaller than what that will be what and so on, so the ranks of these sheets form a partition, obviously each such configuration, but when I fix such a partition, corresponding locals in the modernized cases are on different connected components, so in theory we don't know precisely whether the part corresponding to given position will be for itself or union of connected components, but we can kind of just say informally that these are connected, and nc stars, this shows you over a partition number, but n number c star is the local square, this we have this partition in the ranks, so there's components which interest us, because they are somehow the simplest, which correspond to the two most trivial partitions, so we have, the first we have horizontal partition where everything is just one lump, so just here e2 is zero, maps and the phi is zero, and now the stability condition, as you know, is that somehow things which are inverted under the map, so in particular things which lie in the kernel of phi have to be smaller than we have to satisfy the usual stability condition, so that means if the map e is not, is zero, then f e5 is stable if and only if e is stable as a sheet, so it follows just the modernized states, so this is called the horizontal one, so we have this somehow, this fixed point loci, one of the fixed point loci, it's just the modernized space of stable sheets, but then there are others, the other one, the little super good component is, you know, basically you can write a diagram for partition, you know, so it's like say you make a partition of nc, right, a diagram where we have the length of the individual pieces and then it can be horizontal, means it's just one piece, one or another, and we can have the vertical one, this one, this is the partitioned one, okay, so in this case all the sheets with EI have ranked one, and later when we study it, we want to relate it to the standard space of sheets of rank one, which is the point, so again, I mean, I just like, I think like generating function for the whole thing, there's a lot of which partition function comes to the three factor, which I can write one minus the general classes, as the virtual dimension of the modernized space corresponding to all absolute ones, I wish I could write minus two, if you think of this power dimension, we have this integral over, so this total integral, so this was defined, and so now as we have this connected components of the fixed point locus, and this was done by summing over the connected components, right, this sum of pieces corresponding to this connector corresponding to this lambda, corresponding to partitioned up, as I had described. It was somehow related to the virtual order number of the modernized space sheets that was supposed to be part of the virtual generated function, and in fact, the advantage here should be the part corresponding to the fixed point locus, corresponding to the stream of this is a sign, which is minus one, so that means the integral distribution, so in other words, the contribution, so this means the contribution of this part, so why is that, again, kind of sketch, we had seen, so f is fixed there, so then if one wants to first order the information of this pair, one can perform e, phi is just a homomorphism, and so it's just an element in the vector space, so the form means that, so the tangent space, we have a virtual setting, the expectation is zero, so the obstruction space may be the x1, obviously, and so therefore, the virtual normal lambda, which is the locus, precisely, hypervised, is this space minus the obstructions, so this is therefore just, so I also cheated it because it was trace free homomorphism, so it's trace free, so this is therefore home, on the other hand, we know what the virtual tangent lambda to the modular space of she is, this is x1 minus x2, without the so that means this is equal to, so this is, you know, here I say hypervised, but obviously to make it into a bundle, you place x1 by xip and so on, to make it into a family, but anyway, so then this is will be equal, so in case theory, the virtual tangent bundle are just with, just with hypervised, so m here, we have the dual, and so I showed you that hypervised a normal bundle, the virtual normal bundle is equal to the dual of the tangent bundle of this thing, and so therefore, this gives us that if I integrate this modular space, this thing is just a modular space, which I'm saying, and okay, so now it has to hit, so the way it works out, so you take this, then when you integrate it to, you have this equivalent parameter t, we'll take the coefficient of t to the zero of one over this, this is just, when you work out what this means is, this is just that one over the Euler class of this is the same as the top turn class of minus this element, so obviously the Euler class of minus, minus an element k to the, is one over the class of k theory, and not enough, the Euler class is very similar to the top turn class, and in this particular case, if you work it out, because one actually is interested in taking the coefficient of t to the zero, which is what one integrates is, it's precisely this, so this will be equal to modular space of the tangent bundle, so this will be super precise, but this is how it works out, so therefore, you see at least one can compute easily the contribution of one component in the sense that we know how this relates to things that we know from before. Now, sorry, sorry, I don't know why I got this, I obviously wanted to rise up, obviously it is the whole point of it, I hope it goes on for a few things, obviously I wanted to say that it says it, because that's the model of basic sheets, the basic filing that the other one, so now there's another theorem which maybe I was wrong, most of these, the contributions of most of these six point rules are actually zero, or many of them are zero, mainly this is the following, so statement, so if I look at this integral over the normal one, this will be zero, the partition has a very simple shape, namely, unless the partition, if you write the diagram for it, which is the direction the partition is equal to. In particular, so there is a, it's done by a four section of the partition, so now this implies, in particular, for these ones are, for example, at least the words I want to talk about, so now I want to, maybe I want to briefly introduce, there is a, this conjecture which somehow relates this to the form, which one can also formulate the terms of this, of the partition function, and so I can, let me, just for those who don't know, which one would not use the functions, but not too much, so more or less from the pk, that is the problem of the function, from the conjecture of the plane, the locus c, where the imaginary part is tau, bigger than zero, some of us, which satisfies some nice creation of the conjecture of the transformation, a vcp, and set to zero, so we have a, some sense, this says, essentially, sign two generators, the vt, which is this element, corresponds under section that we do here tau, s, other generators, which is like this, basically, we should have this very property under this term, and in addition, one wants some kind of, actually supposed to have a morphic, but one also has a notion of being holomorphic, infinity means, as it's invariant, so it says here, if we look at it, that it's invariant, and the tau goes to tau plus one, so it has it, we need a development, and the tau is tend to be written as sum for n, a n, n, some perfect number, and now if all of this could be all n for t in integers, but we allow all the positive processes for the morphic infinity, and the modular function is a quotient of two holomorphic, of two molar forms, okay, and so it's clear from the definition that modular functions form a t, which we will later use, and we can also consider modular forms for subtools that I said to z, for instance, we can look at gamma zero, and which would be set for, set to z, such that this c is converging to zero. We can take some kind of behavior under, so now here we have written everything in terms of q, so q is, if you want to write down something like that, the function would be invariant, that's how this tau goes to tau plus one, and now the modularity, there's something that would happen when tau goes to minus one for tau, and so we look at the following, we look at some other partition functions, the island rule partition functions, so that they're really right, so now this could be our times, some root of unity, so it's joint class, so section number, and usually one bit of w, right south as thing, so we just have any combination of things with some overall, so one can show that by definition it's more or less clear that it should depend only on the class modulo r times this, because you know at least with the modularity of sheaves, if you twist change c1 by r strength, it just means oh she is twisted by line one, but the modularity is the same, so we can write out this thing, and then the conjecture, the first way is that, you know, as I said, we write q, it's also on the other half thing, and then if you take z, take this usual one, replace tau by minus one over tau, then this gives us things to one, and so, but there's a tiny, you know, some tiny extra factor, so r minus one over s, and we have a modular factor, so it transforms like a formula for weight one half, as you are thinking, so this would be this conjecture, and it's not here now, but the doubt is that if you look how it's written here, this transformation tau goes to minus one over tau, will somehow exchange the horizontal part of the vertical part, and if I'm believing this conjecture, and the rank is prime number, but one only has to compute one of them, the other one is the theorem, but in addition, obviously one wants to check this conjecture, too. We want to start out by stating here we will see how the structure comes from the geometry of the model, so here you see that first the theorem and then afterwards say, for the first time, just write it, let's assume the theorem is in surface, with the e.g. e to the zero, and say h1, assume the rank is universe power series, c0, the rock series, such that the written in range can be written as follows, but there are some complicated terms, but anyway, minus one into the r minus one into the r, so right here there's some data, okay, so I will have to write down what this is, and then now we have our series, so first we have here, first just ask that is congruent model of r to zero, unless the sum of this class is y by i, I mean delta is equal to one, but zero in r, and so if it is the difference, whether these people or not, you can come on to one after, so we have this, and then statement, right, to take the product, one e to the zero written in range delta i, okay, so we have this very complicated formula, but then, so what we see is that, so if you remember the seven written in range, in any case here, for instance, we assume that there is a, and so remember that seven written in range, maybe at some point in proper definition, there's something written in range, how many quotes there for the in range, but by services, and so if, for instance, s has connected, the only classes for which the seven written in range are not zero, are only in range which associate the number, any class on the second homology, that this number is most of the time zero, for instance, in the case that, that the s has a connected column to third, then they're only these two, and in this case, the seven written in range of zero, c to one, so these are not as mysterious as they seem in algebraic geometry, they are very sensitive, and so this is the description here, so I have here more, let's give you have some fixed policies which I maybe don't explain, but you can see it's some kind of, apart from the fact that I, so you can see also here, you have something for the minus k square, something for the k square, so this is, therefore, there's no statement in this, okay, this is a non-trivial, this is kind of trivial, I can always, I could absorb this term into this, and if I don't know what c0 is, then I also don't know what this is, so this thing is actually not a statement, it's just a normalization, but the term for the power k for s is this one, is a statement, but otherwise, it's some kind of university, state in a very kind of similar to the one we had for the previous two more points, but certainly expressions can be written university in terms of the product form, now we don't have quite the product form, of course we also have this, I was written in variance, but we have kind of, it says that such a thing can be written in the universal way like this, then the only additional non-trivial statement is that this power, so I mean if you remember in the beginning, we had this kind of portability, there was a point where there was a bright sound, something reasonable generating function that once was built with two points, then this has always universal product form in terms of intersection numbers that are in the question, and now we have to still see what s are, delta is equal to delta, delta of q, which also is equal to, this is just p times product one minus q, which is also there, and then we have this theta function, this is the theta function, so theta is, I can write it down simple, so this is sum over all q, over all say vectors v and z to the r, I take q to the inner product of v itself, I have to say what this, I have to say what this inner product is, and so inner product is confined to the r matrix, so maybe when you know this is matrix, diagonal tools before and directly next to the diagonal on both sides, we have minus one, so we take the derivative z to the r, we take this and we get some numbers for this, okay so this is this formula, now I think, so I think I stopped here, I stated this thing, then we have to see what this projector means for this, and then we want to go into the detail, we want to maybe prove this and also see what one can say about this unknown power series c i j, see what is the simple case of this part of the formula, yeah yeah I mean that's one of the things I want to say, we know them until rank five, but they are okay, they will be some functions which we get kind of complicated, so in the in the two cases it's very simple that's the higher the rank, the more complicated, I don't have, we don't have any idea whether what there's a general formula for the arbitrary thing, only it's more, I don't know whether there's anything I haven't seen, but I actually have no idea, there was a prediction I saw which was actually not true, but otherwise I don't, I don't know, tell you that.