 Okay, what's the sound? Yeah, there's some, okay, what? Try to what? Yeah, I'm talking. It is on at this recording in progress, it says. Okay, so it's a pleasure to be here and give a talk for the birthday of Tom Rofka, the star whom I know for many years and know for even longer. So you can remember when I was a postdoc hearing a talk about the structure formula for Donaldson invariance by Komama Mofka, and then later I had to talk in the Albert Star, who was still as a postdoc about the new cyber-quitment invariance. And I've met him occasionally over the years. And over the years since then, I always, you know, one always sees him with this engaging smile, so that I hope will... And so, yeah, I wish you happy birthday. So this also reminds me of the times when people in gauge theory were still talking to me because I was doing things related to gauge theory, but now I do algebraic geometry. And so let me talk about computation in written invariance. Okay, so we talk about... I want to talk about modular spaces. So in algebraic geometry, we look, for instance, at the modular space of semi-stable bundles on an algebraic surface with an ample class. So modular space would be a space parameterizing some interesting objects and mathematics. And in differential topology, everyone could talk about the modular space of a mid Einstein matrix on a fixed complex bundle with, you know, with an opinion study metric. And then the Kobayashi-Hitching correspondence says that there's a homomorphism between these two. Usually one considers complications between these two spaces. In algebraic geometry, the modular space of semi-stable coherent sheaves is the rank R, Schrodinger class C1, C2. And in differential topology, or gauge theory, the Ulmberg complication by ideal instantons. And these have been used, for instance, to... Yeah, you can turn off the sound. Also bothers me. What? Oh, it was mute. No, no, no. It was mute. And so this allows us to compute Donaldson variance in algebraic geometry of algebraic surfaces. And now it doesn't want to. Okay. So now let's look at, in algebraic geometry, we take a project of algebraic surface with a hyperprint class. And I will assume during the talk that the first vector number is zero, or maybe even that it's simply connected. So the B plus, B2 plus of S should be bigger equal to three, which is equivalent in algebraic geometry to Pg being bigger than zero. So that means there's an everywhere... There is a non-bendishing holomorphic two-form on S. Then we want to look at the modular space of rank R semi-stable sheaves on S with given Schrodinger class C1, C2. I've written down what semi-stable means in algebraic geometry. It means that the sub-sheaves are not too big. They don't have too many sections. So a sheave you can think of if you don't know what it is as a vector bundle with some singularities. And now this modular space in algebraic geometry will usually be singular. It has something which is called expected dimension, which is this number I've written down here. And this BD, we don't know in the moment precisely where this number comes from. It's the expected dimension. It's the dimension that the modular space should have. Now we will have to understand what that means. Where does it go? Okay, so just... If I write the number like C2, I mean obviously the evaluation of C2 on the fundamental class of S and so on. So now, what I've written in advance, I will talk a bit more about them later, but I first give a first approximation. So for instance, in the original paper where I've often written, there was a conjecture formula for the generating function of the Euler numbers of the modular spaces of the rank two sheaves with given first-journ class and running second-journ class in terms of modular forms. So let me state this briefly. So for simplicity we assume there exists a smooth connected curve in the canonical linear system. That means there is a zero set of holomorphic two form which is smooth and connected. This is just for simplicity. And then again we have these intersection numbers. There's ks squared, which is the same as c1 of s squared and we have chi of 4s which can be expressed like that. The holomorphic Euler characters which can be expressed like that in terms of turn numbers. And in this case, the virtual dimension of the modular space is given by this formula that I wrote there. And then the conjecture of Waffa and Britten or the formula of Waffa and Britten is the following. We take this eta function which is a standard model of form of weight one-half and we take the theta function for the lattice z. So sum over x to the n squared and then we have the generating function for the Euler numbers of the modular spaces just in terms of this. So we write down this expression 1 over 2 eta of x squared to the 12 to the chi of 4s and the other term to the ks squared and we take the coefficient of x to the vd. The vd is the virtual dimension of the modular space I'm talking about in terms of the general class of c1, c2 that I've written on the other side. So this is the formula. Now there's... Well, you know, so on the same surface, you know, it depends for one thing, but yeah, so if you fix the surface, which is given by ks squared and chi of 4s, then the modular space of the same virtual dimension have the same order, I guess. So that's our only way to precisely like that. Yeah. But you know, okay, so this is the formula. But one should maybe be a little bit careful because it's not quite clear whether this is actually true. I mean, this is the Waffa-Witten formula, so this is actually... We now want to see in what sense this might be true. So this modular space will usually be in algebraic geometry quite singular and its dimension might be different from this virtual dimension. So in differential topology or geometry, one then usually takes the equation which defines the modular space, one deforms it and one looks at what one is given. Algebraic geometry, one doesn't do that. One keeps the modular space as is and adds some virtual structure and remembers what would happen if one wanted to deform it to something smooth. And so this extra structure is given by an obstruction theory of dimension BD. So I want to briefly say what that roughly is. And this obstruction theory then allows us to define virtual analogs of all the invariants of smooth varieties. So what is this thing? So at every point in the modular space which would correspond to a sheaf, we have the modular space as a tangent space which in this case is this vector space x1, ff, trace 3. So if f is locally free, this is just h1 of the enomorphisms of f, trace 3. And the obstruction space is the x2. Miemann-Roch will compute that the difference of these dimensions of these two vector spaces is the number VD. And the Quranishi tells us that there will be an analytic map, the Quranishi map from a germ, from a neighborhood of the tangent space from 0 in the tangent space of the modular space at f to the obstruction space such that an analytic neighborhood of our point f in the modular space is isomorphic to the inverse in which of 0. And so for instance, if the obstruction is 0 or this Quranishi map is a submersion, the modular space will be non-singular of the dimension of the virtual dimension, but otherwise it could be also very singular. And so now we want to capture this in some virtual sense. So a so-called perfect obstruction theory is a way to capture all the tangent and obstruction spaces at all points in the modular space at the same time. So it's a complex of two vector bundles of finite rank, holomorphic vector bundles on the modular space, such that for every point in the modular space, the tangent space at that point is the kernel and the obstruction space is the cooker. So, you know, as I said, it means we have kind of tied together the abstractions together via two vector bundles of finite rank, holomorphic vector bundles. And then we can define things. So we have the so-called virtual tangent bundle, which is just a formal difference of these two vector bundles. The virtual dimension is the difference of the ranks of these bundles. And the difficult part is that one construct the so-called virtual fundamental class, so a class in the homology corresponding to the top homology of a smooth, complex manifold of dimension, the virtual dimension, which behaves like the fundamental class in some way of a smooth manifold. And so this is actually the difficult part, but anyway, one can do that. And then one can define a virtual version of the order number, which is just the integral or the evaluation on the virtual fundamental class of the top general class of the virtual tangent bundle. And then our conjecture, which has been confirmed in many cases, is that the rougher-witten formula will hold if we replace the actual order number of the modernized space by the virtual order. Okay, so we have this. So this is kind of the first approximation to the rougher-witten statement, but rougher-witten actually look at something more general, which I just gloss over very fast because I don't even know precisely what the symbols mean that are here now, because you are the experts, and I'm just the amateur. But anyway, we have, say we have a Riemann Romanian manifold, the 4-manifold with the SUR bundle on it. Then rougher-witten in the original paper asked us to count the solutions to a pair of PDEs for a connection on E and two fields B, a self-dual 2 form in the joint bundle of E and gamma 0 form in the joint bundle of E, which is given by this formula. So this bracket with the point is somehow the lead bracket together with the contraction. So if you look at this, it's what does it mean to count solutions to such a thing, this B and gamma both are completely unbounded, so it seems to be impossible to make sense of saying we want to count the solutions of this. Now Tanagan and Thomas have found a way to translate this. So if you have a scalar surface, this becomes, if you can reformulate this as a statement about a vector bundle and an endomorphism of the vector bundles with values in the holomorphic two forms on S, and so then they define an algebraic geometry, the analog, so let S H be a projective surface with an ample divisor. A Higgs pair on S is a pair, E phi of a torsion-free sheave on S and a homomorphism from E to E tensor KS. KS is the bundle of holomorphic two forms, which is trace-free. And there is a modular space of stable such sheave where the stability condition is similar to the one we had before. And then, so we have this modular space and this modular space also admits a perfect obstruction theory which is what is called symmetric which means that at every point the tangent space is dual to the obstruction space. So that means that the expected dimension of the modular space is zero. So kind of virtually this is a finite set of points. And so we should define the Waffa-Wittnien variant as the virtual count which is the integral over the virtual fundamental class of this thing of one. Now there's a problem with that obviously because we still have no compactness. Now we have this parameterizes pair of a vector bundle and this this Higgs field so homomorphism from E to E tensor KS which can be rescaled arbitrarily so it's not compact so there's no virtual fundamental class. So then one uses a trick. So if this thing was compact one could compute in a different way by the bot residue formula. So you just integrate over the so you have there is a C star action on this modelized space by just rescaling the Higgs field. And so then the bot residue formula would say that you can instead integrate over the fixed points of this action one over the Euler class. Now there's a virtual version of this where one does everything virtually. So one there's a virtual fundamental class induced on the fixed point locus and this has a virtual normal bundle which is the part of the tension the virtual tension bundle on which the action is of this C star is non-trivial and then we just formally write down the bot residue formula and we define this to be the integral over one on the modelized space. And so this is then the definition of the buffer written invariance. Now if I want to study this one has to see what this fixed point locus can look like. So I claim now that this fixed point locus has a decomposition parameterized by the partitions of the rank of the bundle. And this is as follows to a partition lambda we get the part where the E splits as a direct sum of sheaves of rank lambda i and the now equivalent map which you know C star equivalent map phi, if we have a fixed point has the will then be, has to be in such a way that it's just always maps E i to E i minus one, tens of K s. And obviously then the last map has to map, has to be the zero map. And so we have in particular two special components. One is a so-called horizontal component where the partition just consists of the trivial partitions where it's just E and the map phi is the zero map. And so that means this modelized space is nothing else than the modelized space of sheaves of rank r with Stern-class C1 C2. And we have the vertical component where this splits into sheaves of rank 1. Now, so there are all these wonderful numbers we have defined so we can put them in a generating function. So which we have done in this, so up to some stupid pre-factor and this normalization by dividing by 2r and the sine, we just have the generating function of the Waffen-Witten invariance where we queue to the virtual dimension of the modelized space of sheaves. And so that means this modelized space of the Waffen-Witten has virtual dimension zero. And so we look at this generating function and then according to what we said we can write it as a sum of such functions one for every partition of the rank r. No, it's just that lambda will parameterize the contribution of the fixed part corresponding to the partition lambda. Okay. So let's look a little bit at the so-called horizontal part so where the partition is trivial. So as I said, the modelized space of sheaves has a perfect obstruction theory of virtual dimension vd and we have this virtual Euler number and then Tanaka and Thomas show that this virtual Euler number is up to sine just equal to the contribution to the Waffen-Witten invariance corresponding to the to the horizontal part and so we see that this generating function this part of the partition function corresponding to the horizontal partitions is just the generating function for the Euler numbers of the modelized space. So in particular the Waffen-Witten partition function contains the generating function for the virtual Euler numbers as part of it. And so the formula that I mentioned in the beginning comes from that. Now there is something called S-duality which is the origin for the fact that there should be modular forms in this and so this predicts the behavior of this generating function under modular transformations. So you should have SL2z somehow acting on this generating function and something nice should happen. So that means we write Q as the I tau where tau is from the complex upper half plane and then you have the two operations which generate SL2z is tau goes to tau plus 1 which sends Q to itself so this doesn't change the generating function but the other one is tau goes to minus 1 over tau so the non-trivial element and something should happen there and kind of be the and so this has to do with looking at the so-called the partition function for the Langnitz-Duell of SUR so we sum over all cosets of H2 of SZ minus R times H2 of SZ with a sine ER as a R-thruit of unity we take the generating function the SUR partition function for C1 equal to this W so it only depends on C1 modulo R times H2 and so we take this generating function and the statement is of W for Witten also is that if we replace or I mean our version of it that if we replace tau by minus 1 over tau so the non-trivial element in SL2z so when Q is equal to E to the pi out tau then up to sine and the simple transformation factor we just get that the SUR partition so our partition W for Witten partition function from before translates into that for the Langnitz-Duell and so it is not completely evident but if one works out what this means in concrete cases one finds that this operation will interchange the contributions of the horizontal and the vertical path and it is known by Thomas that if the rank is a prime number then the contribution for partition will only be non-zero if the partition is either the trivial partition R or it is 1 to the R and so the whole partition function consists only of these two parts and they get interchanged by this S2 additive transformation so using if we believe in S2 additive as a prime number then if we know the vertical partition function we know all the Waffen Witten invariance so this is our motivation for looking at the vertical Waffen Witten invariance now so again write down a few model of forms there is the eta function there is delta which is the most standard model of form eta to the 24 and I also look at the teta function of the ER lattice so is that to the R intersection matrix given by the ER matrix so we have the 2's on the diagonal 1's on the diagonal things next to the diagonal and then we maybe minus or anyway okay and so and we also introduced this delta AB which is just saying if we have two numbers in the second comology we say delta AB is equal to 1 if A is congruent B model if A is H2 and 0 otherwise so we always and then there is this theorem kind of structure theorem for this vertical partition function so there are some universal power series which do not depend which only depend on the rank R so they are independent of anything of the surface or whatever we are interested in C0 and Cij for one smaller equal to i smaller j equal to r minus 1 such that for all polarized surface and all C1 we have this generating function is given like this we have something with the delta function to the holomorphic Euler characteristic we have this teta function so this with the 0 is the same as the one without the 0 divided by eta to the minus KS squared and then we have this universal power series C0 to the KS squared and then the sum over all r minus 1 triples of classes beta i and then the contribution is only non-zero if C1 is congruent to the sum of the i times beta i and then we have the product over the Ziber-Gritten invariance of beta i times Cij to the beta i times beta j so this is the formula and we have only these unknown power series that the Ziber-Gritten invariance I use the algebraic geometry convention so the Ziber-Gritten of Bi is what you would call the Ziber-Gritten invariant of 2 beta i minus KS so that it becomes a characteristic homology class and to simplify what I say later if the canonical linear system contains a smooth connected curve then there are only two non-trivial Ziber-Gritten invariance so which are not zero the Ziber-Gritten invariant of 0 is 1 and the Ziber-Gritten invariant of KS is minus 1 to the KiO4S all other ones are 0 so that then the formula would become a bit simpler but that's a method if it's called theorem it's a theorem if it's a physics theorem it's called conjecture no I mean that's yeah I mean okay but you know so now so we take the non-trivial part of this thing so everything in the lower line so everything which we don't know explicitly and we put it as a new generating function so if we know this phi we know all the the vertical we want to compute them so now we compute them we are able to compute them at least to get a concrete conjecture up to rank r equal 5 and again for simplicity to make the formulas look simpler as we turn out to be more than complicated enough we assume that KS contains a smooth connected curve so that we don't have to sum over Ziber-Gritten class and so so I put some notations so we take a quotient of two such theta functions for the AR lattice one is the one we had before and now one shifted by some class in the AR lattice and we call this TAR and so it's a quotient of two modular forms of weight one of a certain weight and so it of the same weight and so it's a modular function kind of invariant under the action of the modular group and then again I recall this definition of the delta AB and then with this which we have the following so we can first look at the rank 2 case so in that case the conjecture of afferent Witten that I said before if we take the s duality relating the Euler number to the vertical invariance and then put this together with this translate into this formula so that this phi is just delta so it's it's one if c1 is equal to zero the first part and then if c1 is equal to ks then it's minus one the chi of s of this quotient of theta functions so this is this and then this is the rank 2 case rank 3 case is a little bit more complicated but not much so we have contributions only if c1 is equal to zero ks or minus and we still have these theta functions now for the e2 lattice not for the a1 lattice but in addition we have this x plus and x minus which are the roots of a second order equation so we take kind of in so we know that this ta one is a modular function so modular functions are filled so these are again modular functions in some kind of extension field and so we have this and then now I will very quickly go to rank 4 and 5 because it gets more and more complicated and basically incomprehensible but you know just to see that there are explicit functions and that one can work them out and that the problem is not simple there's not a simple answer there's a complicated answer which we don't completely understand and in particular it's not so clear how one goes to get so much more complicated time so this is the conjecture for rank 4 so we see the structure the general structure is the same we have contributions when the first term class is zero order 2 ks or minus ks or plus ks and we have again these quotients of theta functions for the a3 lattice play a role but we also in addition have this continued fraction of Ramanujan which is the simpler which is a nice modular form which also has this product function in terms of theta functions and then finally we again cannot just take kind of the ready made modular forms but we have modular functions but we again have to solve this quadratic equation in modular forms to get this addition to Z so but anyway you can see it's a very complicated it looks very complicated but you know it is completely explicit so we know all the if all the of written invariance if we have this this formula yeah yeah yeah well I mean no we haven't really figured that out I mean it's somehow also not maybe enough cases I mean it's obviously you have you know it's some extension of gamma zero and but it's even bigger than that and so I mean all the group is even smaller so I mean we don't know precisely I mean obviously that would be very important to figure out but I mean in some sense as we can only do one at the time I mean we haven't even kind of tried to figure out where precisely these modular functions live maybe that would be a reasonable exercise you know I mean obviously you can work it out there but and then I very quickly flashed to you the rank 5 one so there this is done in terms of the Rajas or just Ramanujan continued fraction which is maybe one of the most well-known continued fractions and formulas of of Ramanujan so you have this continued fraction which can also be written as this infinite product and then we get this amazingly complicated formula but you know at least we can see the structure is the same so we have a contribution if C1 is 0 KS minus KS 2KS minus 2KS for all the the cosets of you know of multiples of KS mind modular 5 times KS and then we have some so we have this in addition I which are certain expressions in this Ramanujan functions and finally we have these theta functions now for the A4 lattice it's always AR minus one and then we now have to add to this thing the solutions of three quadratic equations in modular functions and so this is the most complicated formula that I want to present and but you know I just the point is there is a formula which is completely explicit unfortunately it's not the formula of the kind which one would think are I know these ones so okay then it's obvious for 6, 7 and 8 it must be this because there's a clear pattern at least in my mind there's no clear pattern okay and now we want to go to the whole Wafawit-Ninven it's not just a vertical one so we conjecture a similar structure formula to the one of La Raca for the horizontal Wafawit-Ninven now which is the same as the generating function for the order numbers of the modular spaces of sheaves and so in this case everything is kind of dual we replace the AR lattice by the dual of the AR lattice so the intersection form is replaced by the dual matrix and we take a similar quotient of theta function for this dual lattice and then the conjecture is very similar to the one before so we have some trivial pre-factor then we have a certain expression so eta 1 over q to the 1 over r to the 1 half to the square for s we have something to the ks squared and then we have again unknown power series d0 to the ks squared and di j to the product of cyberquitten classes and then we have some over r minus 1 tuples of cyberquitten classes the product of an rth root of unity to the i times beta i times the first churn class we are talking about so in this case the formula does not only depend on the rank and the surface but also on the first churn class in this form and then the product of the cyberquitten classes of these beta i's so this is this formula where these d0 and di j are universal power series which only depend on the rank what is di j well I don't know in general what I mean I haven't thought about if one puts q equal 1 I don't know whether I mean it's not clear to me why I mean so okay you have a model of formula so you could go to the cusps and see what happens there okay so that's maybe an interesting question but obviously you could go to other cusps in some sense you can see that we see in a moment that setting tau to minus 1 over tau replaces the di j by the c i j so one thing is that if you go to one other cusp you replace horizontal by vertical and vice versa so in that sense there is this thing I mean if you are near so not just go to the cusp but the neighborhood of the cusp so if that's maybe not your question but let me just say it so so S2LET translates into relation between the di j and the d0 and the c i j so this one can actually kind of see that this is reformulation and so and it's a very simple relation if I put q equal to 2 pi i tau tau is in the upper half place then d0 of tau is c0 of minus 1 over tau and di j of tau is c i j of minus 1 over tau so the simplest possible relation so the value of a certain cusp of the di j is equal to the one of c i j at another cusp and also the development in the neighborhood of a cusp of one is equal to the development of the neighborhood of another cusp so in that sense I mean don't know where that helps your question and so thus conjecturally we know all the rough of it in variance in ranks up to 5 so in particular we know the Euler numbers of the modular spaces and in fact one can easily work out what this means we have explicit expression in terms of modular forms or modular functions of the what d0 and di j are and so c0 and c i j are so we can also the things with explicit formulas for the d0 and di j up to rank 5 okay I don't know how much what is the time so I have still a few minutes no so then I can say a couple of words about why one should believe in all these conjectures you know what kind of work we actually did instead of just sitting and having some interesting dreams about numbers so and so I want to briefly say how one can check these conjectures and find them so so let me talk about the horizontal of a written no the vertical of a written invariance so this the part the vertical part of the modular space as we had seen parameterizes pairs e phi phi is the Higgs field where e is a direct sum of rank 1 sheaves the map phi always sends e i to e i minus 1 so in a chain like that so if I have a sheaf of rank 1 on a surface it is the ideal sheaf of a zero-dimensional scheme so basically finite set of points tensorized by a line bundle and so the zero-dimensional schemes are of length n so corresponding to second Schrodinger class equal to n are parameterized by the Hilbert scheme of points and the we had this phi is a homomorphism from e i to e i minus 1 tensor ks so that means that if e i is equal to i z tensor l i then l i minus 1 tensor l i dual tensor ks must be effective so there must be a holomorphic section of this okay so we are in this situation and so we can work so therefore our thing is given by the ideal sheaves of r sub schemes so points in the r-fold product of the Hilbert scheme of points on the surface and so if we fix these bi so these the first-journ class or whatever or just a line bundle l i minus 1 tensor l i tensor ks for i from 1 to r minus 1 then this fixes a component of this fixed-point locus corresponding to 11111 where these line bundles are these and this component by forgetting the line bundle projects down to the product of Hilbert schemes of points so if we take the generating function this is subjective so we get the whole we get all the Hilbert schemes of points like this if we do this and we have the restriction of the virtual class on them on this of a written fixed component and we have can restrict it to this thing corresponding to beta and we can push it forward to the product of Hilbert schemes and one can explicitly say what it is so it will be a product of these cyber-gritten invariance of these bi where these bi say are the first-journ class of these line bundles that are called bi multiplied by some class gamma which depends on this beta cap product to or whatever with the fundamental class of the product of these Hilbert schemes and so if you want to where this gamma of beta is an expression in certain classes of certain universal sheeps on the Hilbert scheme there are kind of two kinds of universal sheeps so which I have written down here one is the basically the ideal sheeps kind of globalized over the Hilbert scheme we can tenserize them by line bundles and the other one is structure sheeps of the sub-schemes I mean globalized okay so we have these universal sheeps and so we have this expression which is explicit in these things and then there is an old result with Elling student-learn that if you just have one just on the Hilbert scheme of points you take any integral or evaluate any expression then this can be universally expressed so independent of the surface just in terms of the holomorphic Euler characteristic as a polynomial in the holomorphic Euler characteristic of the surface ks squared ks times this line bundle to which you make the universal sheaf and see one of the line bundles squared now we globalize this over the product of the Hilbert schemes and then we get this expression that these universal power series c0 and cij of q depend only on the vector of these intersection numbers I wrote here on the surface namely ks squared bi of ks, bi times ks and bi bj so they only depend on this tuple of numbers and so we can compute what they are in general by looking at as many examples we just need to look at as many examples as we have as there are elements in this vector which are such that the corresponding vector of numbers is linearly vectors of numbers are linearly independent then what we compute for them will determine c0 of q and cij and therefore the invariance for all surfaces so now in this case we can just take linearly independent tuples we can take our surface just to be p2 or p1 times p1 and then vary the bi, these line bundles so on p2 or p1 times p1 we obviously have an action of c star times c star with finally many fixed points and we choose line bundles to which this action lifts so equivariant line bundles to this thing so so then one can show that in this case the action on the surface will lift to an action on the Hilbert's scheme of points which still has finally many fixed points and these fixed points are parametrized by tuples of partitions of n if we take the Hilbert's scheme of n points and so therefore we can now use localization so what residue formula what localization to compute c0 of q and ci of q in terms of the combinatorics of partitions now as we are not smart enough we don't find a nice closed formula in terms of whatever symmetric functions and so on but we just put it on the computer we compute up to a very high degree in q and we know the first whatever 20 coefficients of this power series and these are enough to to guess and check that the formulas or the formulas I told you are will be should be the right ones and then I should also say that we have this relation between the horizontal and the vertical invariance so the horizontal invariance can be treated in a similar way there is a so called Mochizuki formula which computes virtual intersection numbers on the on model space of sheaves to other more complicated intersection numbers of river schemes of points which however can again be related to intersection numbers of universal sheaves and so then we can use the same method to compute them and we see that we can also compute it's a bit more complicated so it takes also more effort to the computer but we can compute something and this gives us shows us that this s duality holds up to a high level and for instance that the relation between the d ij and the c ij is true up to the level we can compute and so we are confident it's true okay so this was all I wanted to say yeah they involved no so I mean the thing is the Waffen-Witt makes some assumptions like for instance they this is some I don't know whether it's precisely this assumption so for instance they might make the assumption that the canonical that there is a that the canonical contains a smooth connected curve or otherwise they have an expression when they have a more general formula they have an expression in terms of the connected component of the canonical curve and now one can translate that you know once one knows what cyber grid invariance are then this formula this expression in terms of the connected components of the canonical curve is precisely in terms of cyber grid invariance so the connected components of the I mean essentially the connected components of the of the canonical divisor are the cyber grid classes I mean in this algebraic formulation so in that sense it's just I mean the cyber grid invariance are in some sense in their formula only it's before they invented them yeah so you'll say one shouldn't expect new invariance from these formulas no I mean the the whole yeah so this whole thing somehow shows that I mean I mean everything that I've ever computed with these modular spaces always can be expressed in terms of of cyber grid invariance plus some universal power series so it appears that these modular spaces at least if one computes invariance like this do not seem to contain any more information than the cyber grid invariance that's true so if one is interested in finding invariance to classify for many folds it's kind of nonsense to to do this but I'm more interested in the structure of the formula