 So this is joint work with Matijn Kohl and Thies-Laraca, and so I work over C. So I consider a smooth projective surface with geometric genus positive, so with a volume of two forms and first homology with that coefficient zero, that H been ample line bundle on S, then we can look at the modelized space, which I call M R C1 C2 of rank R H semi-stable cheese on S with standard classes C1 C2. We know I have called the definition of stable and semi-stable as it was in Thomas lecture, and for simplicity I usually assume that stable is equal to semi-stable and so then this modelized space is projective, it will be usually quite singular, but it has a certain expected dimension which is the, which I denote by V D or V D of R C1 C2 throughout the whole talk, so 2 R C2 minus R minus 1 C1 squared minus R squared minus 1 the holomorphic Euler characteristic of S. So now Waffa and Witten in 1994 in their famous paper, strong coupling test of S variety gave in some sense an explicit conjectural formula for the generating function of the Euler numbers of modelized spaces C in rank two in terms of modular forms. This is not quite true for one thing, it's maybe not precisely the Euler number and secondly it is more general, this generating function also computes other invariants and these invariants are now called the Waffa-Witten invariants and they have now recently been given a mathematical definition by Tanaka and Thomas in terms of modular spaces of Higgs pairs. So I think Richard hasn't yet talked about it but he maybe will tomorrow. So let me then briefly recall what these are. So let S H be a projective surface with an ample divisor, Higgs pair on S is a pair of a torsion free sheaf on S which is called E, E on S and a homomorphism from E to E tensor KS which is trace free and there's a stability condition which I don't tell you and then this N equal to N S H RC1 C2 is the modular space of these R stable Higgs pairs. So here it's written again, so this is just a pair of a sheaf and the homomorphism of the sheaf from the sheaf to the sheaf tensor KS. So this modular space admits what is called a symmetric obstruction theory. So we have tangent space and obstruction space which are dual to each other, you have the virtual tangent bundle and so this also means it has expected dimension zero and so you expect to get a number by integrating class one over its virtual fundamental class. Now this doesn't quite work because N is not compact so one cannot really say what that's supposed to mean but so it's compact but it has an action by star just by rescaling this fee from E to E tensor KS. So this fee is called the Higgs field. So I'm always moving this thing to where I want to show you, can you see it? Yeah, okay. And so it has this C star action and the fixed point locus is compact. And then they decide to just define the invariance using this, namely if N was compact then one could compute this integral of one over the virtual fundamental class by virtual localization and the formula would be this you sum over the you integrate over the fixed point locus one over the virtual normal bundle of the fixed point locus. So the virtual normal bundle of the is the part of the virtual tangent bundle where the action of this C star is not trivial. So if it was compact then this would be the formula how you compute this. Now as this is not compact as if or not defined so you use the right hand side to define the left hand side. And so that's what they do and so in this way they say what it means to what the number associated to this modelized space is namely formally integration of one over the modelized space given by this local virtual localization. So this will only give you a rational number because you know you have done this trick. Okay so now one can say a few more about these modelized spaces. So the fixed point locus has a decomposition into well unions of connected components and these unions of connected components are parametrized by partitions of the rank R we look at sheets of rank R and we look at this namely the piece corresponding to a partition lambda where lambda is equal to you know whatever lambda 1 to lambda k and parametrized space a phi where e decomposes as a direct sum of vector bundles where the rank of the vector bundle is lambda i the ith part of the partition and the phi only maps from e i to e i minus you know always maps e i to e i minus one. So in particular and so and then so the last one see e1 just maps is the zero map to zero okay and so so this gives us a decomposition what what so this gives us the decomposition of this fixed point locus and so we have two distinguished one one is one component is the one corresponding to the trivial partition which just consists of the number r so it means e we have e of rank r and phi just is the zero map and in this case if phi is the zero map it means we don't have a pair of coherent sheaf and such a homomorphism we just have the coherent sheaf so it turns out that this modular space this part of the modular space is just the modular space of stable sheets and we have also the vertical component where the sheaf decomposes into a direct sum of sheafs of rank one and then always one goes to the next okay so these are and then there are others corresponding to other partitions lot of your question does this mean partitions need not to be descending no i i mean yeah no i don't um no i don't think so i mean i think the partitions should be descending but you know i hope i'm not making a mistake and have it the wrong way around but i think that the partitions have to be descending but okay anyway this i think it should be okay as is i mean okay so now now we can make a partition function out of this this whole thing we sum up over all these parts how does one make this more yeah so um so we make a generating function of this so we sum q to the virtual dimension corresponding to rc1c2 for the normal modular space divided by 2 r some trivial pre-factor which serves to in the n ket modular forms and then we inter we have this thing that we had defined the integral of one over the modular space and we have this sign so we take this generating function like this in q which gives us all these numbers and this is the buffer between partition function now what now by definition this splits into a smaller partition function one corresponding to every partition lambda because for partition lambda we just look at the contribution of the parts of the c star fixed point locus where you know because after all this integral of one over this thing was defined in terms of doing something on the fixed point locus which is decomposed into into pieces parameterized by partitions so we can just sum up the terms corresponding to a certain partition this gives us the partition function for lambda and we so we can write it like this so this is just a sum of all lambda of the partition function corresponding to lambda and so um okay and so okay so now let's look in particular what is the reason to divide by r in this partition function well i think the point is i mean it's all just somehow aesthetic thing but um i mean it somehow is nicer if one divides by r it maybe also corresponds to somehow to the some kind of physics thing but by itself you know it doesn't have a big me but it somehow needs to slightly nicer formulas but i don't think it is of any real importance the powers of q are to do have to do with the fact that one finally wants to get something modular but there are i think not so important um now we can in particular again look back at the modular space of sheaves so this also carries a perfect obstruction theory but now not a symmetric obstruction theory this perfect obstruction was defined by much mucizuki and its virtual dimension is precisely virtual dimension against the beginning so it also has a virtual um tension bundle and a virtual fundamental class um which lies in the correct dimension uh chow chow group of the correct dimension of the modular space and then one can also define the virtual Euler number of this modular space by just integrating the correct um churn class of the virtual tension bundle over the virtual fundamental class and tanaka to tomas proved that indeed the virtual Euler number of the modular space is equal up to sign to the to the invariant that was defined by virtual localization for this the corresponding component of the modular space of the experts okay so everything is compatible um and so therefore if i take the part of the partition function maybe except that i forgot this r to the minus one the part of the partition function which corresponds to ah here this is obviously q to this power um which corresponds to uh to the partition tutorial partition r is just generating function of the Euler numbers of the modular spaces of sheeps so in particular the r for written partition function contains as part of it the generating functions of the virtual Euler numbers of the modular space of stable sheeps so which in case for instance the modular space is equal to the actual okay so there's also the the title so is other questions so um are there questions um so um the title of this yeah there is a question in the q and a is the left hand side then in the result of tp computed using the variant function no which one we are now the right hand side is uh no no they it doesn't work with the variant function no so that is i think something that Kanaka and Thomas found found out if you try instead to compute with the variant function you get the wrong result you will really have to do this this virtual local is this formal virtual localization this gives you a different result from what you would get with the variant function and somehow the what you get with the variant function is not is not good numbers in particular it doesn't match things you get from other sources so i mean that Thomas explained once in a talk which i listened to okay so is the interpretation of q like an intrinsic interpretation of q besides that you want to get something modular well it just a i don't really know what that would mean uh yeah i i don't i i mean at least i don't think i have such an interpretation i don't know precisely what how one could do that i mean like this it just counts the basically that the dimension of the modular spaces so you you kind of in some sense you basically have a generating function in c2 or why would you expect it on the modular what why would you expect it to be modular i wouldn't expect anything like that to be modular but the physicist expected to be modular this is called s2ality and it has something to do with this has something to do with gauge theory and you replace magnetic fluxes or magnetic things with the electric things and this gives you this but i don't know what i mean so i don't think there's a good mathematical relation actually well there are maybe some reasons but if there is no reason to expect it and i mean whatever maybe somebody else can say something better but you know there's you know you can relate it to something with the instanton partition function since you can see something is modular for some other reason but it's not so ah i can say one thing to this question so one you have two things here for one thing in the case of three surfaces you can compute all these things rather easily and you get always something which is related to the Dirichlet to the to the deliquent you know to the delta function to the delta model of form and you also have the blow-up formula which tells you what happens if you blow up your surface in a point and this is somehow can be expressed in terms of tether functions so at least here you have two instances where you have a modular forms and at least the tether functions you can see somehow directly by looking at the wall crossing that you would get that but i don't know that's not a deep reason it's just how it comes to pass in if one looks at it in a if way okay so now let me look at this so i wanted to briefly talk about yesterday first i will call a modular form so maybe people know what modular forms are so maybe just write it here so we have that's a holomorphic function from the complex upper half plane to the complex numbers which is has this transforms in this way under the action of sl2z so f of a tau plus d divided by c tau plus d is c tau plus d to the k times f of tau this is a modular form of k and you have two generators which is t like this which sends tau to tau plus one and s which is the more interesting one which gives sense tau minus one of the tau and furthermore you have this property that a modular form should always have a q development like this you can write it as a power series with an only positive and non-negative degrees a and q to the n where q is e to the 2 pi i tau a modular function will be a quotient of two modular forms of the same weight which also means that it's invariant and by definition okay and we can also consider modular forms and modular functions for finite index subgroups of sl2z for instance term of zero n which is this one so the subgroup of sl2z where the lower corner thing is congruent to zero mod n so now the s duality of physics predicts some behavior here that this generating function behaves in a nice way under modular transformations in fact one can look at the so-called angular dual partition function or the partition function for the Langmuir-Stuart group which is this so we take the buffered written partition function from before and we sum it up overall w in the second homology model of r times the second homology with this in the summand we have the power of the r through the unity w times c1 so the intersection product here so this is and obviously this only makes sense it's implicit in this that the r for written partition function only depends on w modulo r times the second homology so this is the Langmuir-Stuart and for this partition function the conjecture of r for written says that these two are related by this element s in the in the in the modular group sl2z so if I replace tau by minus one over tau in the w for written partition function I get up to some relatively elementary factors and this model of this transformation factor we get the one from the Langmuir-Stuart group and so that's a rather strong restriction for these things a rather strong property so if one looks at it carefully one will see that this operation exchanges the part of the partition function corresponding to the trivial partition r so which corresponds to the order number of modular space of sheaves and the one of the so-called vertical component where the partition is just one one one one one and in addition if for instance r as a prime number Thomas shows that this partition function for lambda will be zero unless lambda is the trivial partition r or one to the r or the other ones are zero so in particular by s duality one believes in it if r is the prime number if I only compute for instance this vertical w for written partition function so one to the r this determines the whole w for partition function in particular it gives me the order number of the normalized basis of sheaves okay and now I want to tell you what one gets for the vertical r for written index so I first recall result of larka who gives us a structure formula for the vertical r for written index so this goes so I first recall to you some model of forms and so on so you have the df the eta function which is this infinite product we have the discriminant it is the eta function to 24 this is the model of formal with 12 and here we consider the tether function some tether functions for the ar lattice so the ar lattice is just z to the r with the intersection form you know on the standard basis given by two's on the diagonal minus one on the things next to the diagonal on both side and otherwise zero and then we take if or if we put zero here we have for the l then we have just some over v to the in z to the r q to the one half dv and in addition we come we consider some kind of shifts of it where we shift the element of v by some rational multiple of a certain vector and and this gives us a number of tether functions here okay so for the moment we only use the one with zero but later we will see the others so in addition we need this notation so everybody knows what delta ab is but in this case it's slightly different so if a and b are two elements in the second homology then delta ab is sent to be one if a minus b is divisible by r in the homology and it's zero otherwise so it's kind of okay and now there's this theorem of larga for the virtual of a written invariance if you take the so if i fix the rank r then we have some universal power series c zero c i j my j from one to r minus one such that for all surfaces as a with z zero and h one as i said equal to zero we have that this vertical generating function is given in this form so first we have something with a delta function for the pi of pi of s then we have our tether function here to the minus ks squared divided by eta of q minus ks and then we have this unknown power series c zero to the ks squared then we take here this delta so we have the sum over all tuples beta which are beta one to beta r minus one in the second homology we this contribution is only non-zero if c one is congruent to the sum of the beta i times beta i in the modulo r times the second homology we have here we multiply by the product of the cyber quitten invariance of the beta i and we multiply by the product of the c i j to the power of the section numbers of these classes beta i so this is this general formula and this is proven but with these unknown powers and so just to remind to you this sw of beta i is a cyber quitten invariant of course so this is a cyber quitten invariant associates every homology class and the second homology an integer and these are in principle are some c infinity invariance of four many more folds but if s is an algebraic surface they are very easy to compute and for instance if s is a minimal is minimal surface of general type then we have only two classes for which the cyber quitten invariance are non-zero namely for the zero class in the second homology and for the canonical class which minus 1 to the power of s and just what i i reiterate what i said before this delta in the formulas comes from the fact that one can compute the invariance for k3 surface and for k3 surface only this term is there and all the other ones are zero because k is zero and there are no cyber quitten classes and or whatever i mean and for and this tether function for the this expression here with the tether function of the ar lattice comes because of the blow-up formula and so in this expression here in theorem of lager we know the terms in the upper line so we put this psi r sc1 q just the combination of the terms that we don't know and we want to know so if we know that c we know the whole vertical generating function and so let's try to understand what c is um so this is yeah um well obviously yes i know how one proves it i mean in substance if i have time which is kind of now maybe unlikely i would kind of say it later but i can briefly say so one can somehow you know by reducing the problem to you know you see here we are in the vertical part so the the sheaf is a direct sum of rank one sheaves so in some sense these are ideal sheaves of zero-dimensional sub schemes tensorized by a line bundle so you can imagine this has something to do with Hilbert's schemes of points so you can eventually reduce the whole thing to an intersection number on a product of Hilbert's schemes of points um and then one uses you know this somehow this auto is out of the links would lean in me um that intersection numbers on Hilbert's schemes of points in terms of sum total logical sheaves can be expressed as polynomials in the intersection numbers in the question and if they nicely fit together then you also get a product form and all this is satisfied here so one somehow reduces it to to some known facts on Hilbert's schemes of points but obviously it requires a lot of work but you know that's it follows eventually from this co-borderism argument on the Hilbert's scheme of points I mean if there's time I can show you um anyway now I wanted to show you the conjectural formulas that I have until rank five uh so for simplicity that you know just so that the formulas look simpler because in that case you know formulas depend on the Zabekwitten invariance and we assume that s is minimal minimal surface of genotype so that only we only have these two Zabekwitten classes for which the Zabekwitten invariance are non-zero so um so now I again remind you of some of the things we had I want to I had these tether functions for the ER lattice I look at the quotient of two of them so the one for zero where I don't shift it and the one for L this both of you know these two are both modular forms so the quotient is a certain modular function and here I can recall what they are and I also recall you what this delta AB is and then with this we can first look at the rank two case in the rank two case that's actually more or less what Waffawitten say if you compute this phi this is just it gives you one if c1 is congruent to zero mod two and it gives you minus one to the chi of four s times the tether function if c1 is congruent to s mod twice the commode so that's a rather simple that's the this formula in this case now in the rank and this this kind of statement follows essentially if one believes that there is a simple formula this would follow from the draw come and now sometime ago with the my time we computed the rank three case which is a bit more complicated so again if so it's again in terms of these tether functions now for the e2 lattice it's always in terms of the tether function of the r minus one lattice in rank r and so we have here we take this the quotient of these two tether functions for one and for zero to the ks squared times chi of s and so this we have if c1 is equal to ks or to minus ks modulo three times the second commode and something more interesting happens when c1 is congruent to zero because we get this new included this power series x plus and x minus and again to this power ks squared so in this case we have to solve a quadratic equation so as I told you modular functions are fields so I can look at field extensions and so I have this x plus and x minus are the solutions of this quadratic equations where the coefficients are again given by this quotient of tether functions so that's that's this the statement in rank three and now in rank four it gets so just to say again so these tether functions somehow just come from the blow off formulas but in already in the case of rank three the formula is more subtle by introducing these extra x plus and x minus now in rank four I the formula is somehow given in terms of Ramanujan's optic continued fraction so this is this u of q third of two q to one eight one plus q divided by one plus I mean you can see it this actually also can be expressed in a simple way in terms of as a quotient of products of eta functions in this expression you can see it's also modular function because an eta function is a modular form of weight one half and so it's weight three halves divided by three halves so this is a modular function but it has this nice continued fraction decomposition and now the conjecture is a little bit more complicated than in the rank three case but so you can anyway see we now as ingredients have these again these tether functions corresponding to the e3 lattice I mean the quotient of tether functions to the e3 lattice and then we have this u in these formulas and in addition we have this z where z and z to the minus one are the roots of this quadratic equation so we are in this extension of again and you can see it looks kind of in some sense has some nice symmetries also so if c1 is congruent to zero modulo the second the four times the second homology then you have this term z to the z to the minus one divided by this to the ks squared and you have then again the same term to the ks squared we replace z by z to the minus one if c1 is congruent to two ks then you essentially get the same where now maybe u is replaced by u to the minus one and something slightly different happens and if c1 is congruent to ks or minus ks the formula looks a bit simpler you don't need the z anymore but you only have this u of q squared or u of q squared to the minus one so these two are related by putting u of q to u of q squared to u of q squared to the minus one so this is this formula and now I got stuck I just want to get rid of this okay and now we finally I want to tell you about rank five so this is now in terms of this the famous word as a manui and continued fraction so we have q to the one fifth divided by one plus q divided by something and the something is of the form one plus q squared divided by something and that's of the form one plus q to the three divided by something and so on and this now is a rather non-trivial fact also has a product decomposition like this I think that's maybe the watcher the watcher's ramanui and identities and one can also prove it's also I think not totally obvious that if I take it out to the minus five minus eleven minus r to the five this is just this quotient of eta function eta of q divided by q to the five to the six okay and now in terms and I also introduce two auxiliary functions this beta one which is something with these again our theta function and then an expression something three times r to the minus five plus two minus eight times r to the five and then the better two is the same where I place here the theta function with one to that with two and replace r by r to the minus one and then I get this conjecture we have again this contributions when c one is congruent to zero to ks minus ks two ks or minus two ks and here you can see these functions ahead the beta i um the uh and but we also have some unknown ones with z x plus x plus minus y plus minus so these are we don't know what they are but in terms of these we have this expression um and now these are the solutions of three quadratic equations so x plus minus y plus minus so here we have the equation for x which is in terms of the beta one the theta function of theta functions for one and then like this the better for one and then we have this we have the y where we have somehow replace one with two and r with r to the minus one and then we have the last one for z which looks very similar to what we had in rank three and rank four for the x type equation so this is the final conjecture so for this rank four case okay so this is what one gets sorry sorry and there is a question in the q and a is there some intuition for this higher rank formula intuition behind uh yeah okay well not i mean so because of all this uh astrology stuff obviously we expect to have some modular functions and so uh what we it's actually there's not so much intuition so what we do and if i at the time i would explain how we do it is we compute these this generating function up to a certain order in q so basically up to uh up to the 12th order in q so that means we know all these functions up to order up to q to the 12 and then you know then we just play with them until we find some relations among them and some relations to uh known modular forms and functions and this is what we come up with so but it is you know by itself we kind of believe we have modular forms that we can see from the rank three case it's not as simple as one might hope so we are willing to solve some some equations and do kind of things but otherwise i mean i don't think there's you know you just look at these and you try to find relations between them until everything is determined and that's what we did here so um but i i don't have a deep reason why the formula has to look like that but you know a lot of effort went into it and also a lot of effort went into writing it as simple as possible regardless of how it looks you know i mean you can write the same formulas in a much you i think it's no difficulty to write the same thing on on three slides if you you know it's just this is the best way we could do it okay so we also so in all these cases the these unknown functions c0 and cij so are modular functions in a finite algebra extension of the field of modular functions for gamma zero r we also have some partial results of range for rank six and seven you know expressing things there again in terms of these data functions and so-called how mobile for so anyway this we have to see now i want to also now i want to briefly i mean how much time okay i briefly talk also about the horizontal or for written events which are just the the um just the virtual or the modular space of sheeps so we i started by this structure formula of larka for the vertical or for written invariance he actually proved it we are not able to prove the structure formula but we give a conjecture for it and then using this conjecture we can determine you know again until rank five the generating function for the virtual or the numbers with the modular spaces of sheeps on any surface with bg bigger than zero and h1 equal to zero so let's see so now somehow there's some kind of duality between the the one with one to the r so the vertical and the horizontal case so it's maybe not so surprising that instead of looking for the data function of the ar lattice we look at the data function for the dual lattice of the ar lattice so the dual lattice of the ar lattice is just z to the r with the bilinear bearing like this which we're now the intersection matrix is the inverse of the matrix for the ar lattice and then we write down a value so if l equal to zero we again write just some or v q of this one half intersection of the thing with itself but we also have some shift with l and z where now we don't shift the argument here but we put a sign or a unity depending on the multiple of l and then again we consider the quotient through such things and call it tl dual and again such a thing is always a modular form the quotient is always a modular function so we have have these and now we have this conjecture which gives a very similar structure to to that of laraca so for in any rank we have again as many universal power series d zero d ij such that for any surface you know with this sum cpg bigger than zero and so on the virtual order number of the modular space is the coefficient of q to the basically q to the one over two r times the virtual dimension of the modular space we have this term here which will make the whole thing modular of this expression here so we have a trivial pre-factor we have something with the delta function for the care of s we have something which looks very similar to what we had for the horizontal for the vertical invariance to the minus ks squared only now with the dual teta function and then we have the same expression as before with this d power series instead of with a c power series again summing over better two words better in the second homology with the cyroquiton classes but instead of having this delta of the of c1 with the sum of the i better i we take an rth root of unity to the power of this and sum them up okay so this is some kind of i mean maybe some kind of transform or whatever and then we multiply by d ij to the power better i better j so this is the conjecture and then again as in the other case we we take out the part which is which we don't know this just the lower line here so these are powers series which have explicitly given this is just the number and so this d0 this is kind of the part we need to know to know the whole generating function of all the order numbers of all the modular spaces okay and so now we have a so now we want to one wants to determine this via s2ality and so what we found is that the s2ality you know of that wafawit can say is equivalent under this previous conjecture to the statement that d0 is related to zero by a modular transformation and d ij is related to c ij by modular transformation and somehow in some sense in the simplest possible way namely the statement is if we put again q equal to two pi i tau tau in the complex upper half plane then d0 is just c0 at minus one over tau and d ij of tau is just c ij at minus one over tau okay so this is the conjecture fits with all the things we have because we have been in some low cases computed in some cases the other side and so given this we can we can now compute we can use the result we have for the vertical variable invariance to give us the horizontal one this is now because of this s2ality transformation there's a dictionary how you translate the formula for the vertical variable invariance and the formula for the order numbers of the modular spaces and this goes as follows so this is so c we call this generating function and then we have the following so we assume this conjecture holds then this generating function for corresponding virtual order numbers is obtained from the one for the vertical wafawit invariance by replacing first this delta c1 lks by the rth root of unity to the to the lks times c1 okay and this is really some kind of you know almost we transform that you do this then we this the the quotient of theta functions with respect to the r minus one lattice is replaced by the corresponding one for the dual lattice and then finally so this works like this it already gives us what we have in rank two and three because these are the only ingredients that one had in rank two and three in rank four we had this u and so we replace in all the formulas u of q squared by this expression in this rational this portion this product of theta functions which is again a model function and finally in case in the case of rank five we replace r of q so the rotor romanouian's continued fraction by this rational function in r of q namely we take one minus phi times r of q divided by phi plus r of q phi is the golden ratio one plus squared of five over two and so this gives us the formula for the Euler numbers of the modular spaces up to rank five for I mean the virtual on numbers for whenever pg is because in zero and h1 is equal to zero okay now obviously you know much of this was conjecture maybe now I should say how much time do I have I have 10 minutes so I can maybe say try to say a few words about how one goes about doing this I hinted at it before so this is about the computation of these virtual brands so we go back to the definition of this buffer written to the fixed point look with a modelized space so we have so so this parametrizes direct sums of rank one sheaves and the phi will always send the map from ei to ei minus one tensor ks and so we can take as I said ei to be an ideal sheaf of a zero-dimensional scheme tensor line bundle so s and i is the hybrid team of n i points on s li is a line bundle and obviously if this hicks sheaf if if this is supposed this is a non-zero map so if this exists you must have that that this tensor product is effective so there is a section okay so we can put this together to something so if n is an r minus one tool of positive integers better is a tuple of classes in the second homology all of which are effective we can take this product of Hilbert schemes denoted like this we take product of the linear systems corresponding to these classes so better is the the tuple of the better i call it better like this and we can look at the so-called nested Hilbert scheme which is a set of all r tuples of subs schemes and curved classes r minus one so that izi of minus c1 is equal to is contained in the ideal of ci minus one and this this incidence variety will be a a subscheme of the product of the Hilbert scheme and the product of the linear systems and by what i just said it is at least believable that the union of them of these fixed point loci corresponding to the partition one to the r is isomorphic to a union of certain of these nested Hilbert schemes so we can instead of we can work on these nested Hilbert schemes there's a way how one can decide which nested Hilbert schemes occur i mean we will not go into detail we have this quadratic form applying to the classes a i which are basically the classes of this better i's and there's a precise condition when this nested Hilbert scheme will be i so will be will correspond to the component so this is one to the r is one to the r not r corresponds to to this part of the of the r for written model I say I mean I don't go into detail here so only this n like this means the sum so n is a tuple of non-negative integers and n in brackets like is this sum of these integers and then we have this expression it doesn't maybe it's a I don't you anyway can't take it in in this detail there's just some explicit form and now Golan Poe and Thomas define a virtual fundamental class on this nested Hilbert scheme which coincides with the one one gets if one does virtual localization on that one gets on the fixed point loci if one does when one does this local virtual localization pretty fine the invariance the one for written events and I kind of try to very briefly say what this is it doesn't matter very much we have here the pullbacks of the ideal sheets of the universal sub schemes and then in terms of this so when one takes the virtual fundamental class of this nested Hilbert scheme one can look at its push forward to the product of the product of Hilbert schemes on the linear systems and this is given by product you know you have the cyber product of cyber written classes and you have a certain integral related to essentially ideal sheets of universal sheets or more business between them and and somehow protocol bundles on on these linear systems and you intersect it with a class of the Hilbert scheme and you know the point class so corresponding to line one line bundle each in the product of linear systems just an explicit format doesn't matter precisely what it is but you have an explicit formula like this which where you also see where the cyber grid name variants come from and so this also you know this formula also plays a role in in La Raca's proof of the no I'm close the door and so then okay they don't you know in La Raca's formula the word so this is also how La Raca proofs is forming and you can see how the cyberquitten invariance for instance come and then if better is effect so now I one has to so this gives us the fundamental class and now we have to see what to integrate or wait so if one does this virtual localization one has to integrate one over the Euler the equivalent Euler class of the virtual normal bond and so we have to say what is the virtual normal one and this one does in terms of somehow the universal sheet so for every better we consider some line bundles such that k minus beta i is equal to the difference of these line bundles and this is this and the sum of the first joint classes are given c1 and then we get some line bundles on the product of s and the Hilbert scheme and the product of the linear systems just taking the tensor part of these line bundles and then the tensor part of total logical or one bundles on this linear systems and we use all this data to make a sheaf so universal ideal sheaf tensor these line bundles tensor t to the minus one where t is the equivalent character of the action corresponding to the c star action and you can see if you look at this e the i corresponded to the ideal sheaf of zero dimensional subscheme the l corresponds to our line bundle and and so you can see where is it this corresponds precisely to the fact that that e is the sum of the i and the i look like this okay so this is how it comes to pass and now you know is because we have this or one here there is a total logical there will be a total logical map here and this gives us an equivalent x pair on the whole of of sn times s times sn times product of the beta and so one can define the one can compute the virtual tangent bundle in terms of this so the the contribution of the as i said the contribution of this how's it called the nested Hilbert scheme to the invariance is given by this integral over the virtual fundamental class of one over the Euler number of the virtual normal bundle and and the tangent bundle can explicitly be computed in terms of this universal sheaf like this by I mean it doesn't matter precisely what the formula is you have a relatively simple I mean you have a formula that you can evaluate it and so it doesn't matter precisely what it is but you can see that we have an explicit formula for the virtual fundamental class and we have an explicit formula for the tangent bundle and we are supposed to compute this virtual Euler number so everything is given to us and and if one looks at it this stuff is given in terms of universal ideal sheaves and total logical bundles on Hilbert schemes so these are all things which are you know have been studied which one can work with and so if one puts it all together one gets that this generating function for the vertical of written invariance is generating function like this where the strange quadratic form q that we had comes here and the sum of the n i comes here and otherwise it all looks like this this is our written class come from what we had in the in the formula for the virtual fundamental class and then we have some expression whatever epsilon of of the a the n and the t which kind of gives us the whole expression that we get if we want to compute the to some some the virtual fundamental class and the port word of the virtual fundamental class enter the class the term classes of the virtual tangent bundle and but one can explicitly say what this is but it's a very complicated expression because you get an explicit equivalent class on this product of Hilbert schemes of points we can compute what the constant term is and we can divide by it so we divide we take the generating function after we divide by the constant term this is some nice power series so as I said the thing that we integrate over the Hilbert scheme of points and is a universal expression in terms of turn classes of logical sheaves universal ideal sheaves and these classes ai you know the ai of which which are the components of this a and then as this general who borders an invariant story which which says which i mean we proved many years ago that whenever you have to integrate such a thing over the Hilbert scheme it can be expressed by a universal formula by the intersection numbers which are in question so the the intersection numbers which occur here are kaio for s the self-intersection of the ks squared intersection of the ai with ks and the product of the aij so this integral is a universal polynomial in this stuff or rather it's a powers it's a whatever power series in t whose coefficients are universal polynomials in this and so okay so that we know on the other hand we can also see if we take if our surface would be the disjoint union of two surfaces then if we take this product of Hilbert schemes of it this will actually be the product over all ways over all ways of writing our vector of numbers as a sum of two such vectors of the product of the corresponding product of Hilbert schemes over the factors and if one works this out what it means for this integral it means that actually this function g that we want to compute here is just you know if we have s is a disjoint union of surfaces and this a you know this were was a was a a tuple of line bundles or of homology classes so that a is just a one on s one and a two on s two then the whole expression will just be the product and now if i have these two statements for one thing the expression is given by polynomials in intersection numbers and second it's a product if we do like this then it follows formally from this by the way how such numbers can work that there is a product formula for this in terms of some universal power series there's a simple argument which proves this that you can write it like this and now our so this is for instance the proof of this gives the proof of la haka's result but it also allows us to compute these terms now because now we want to compute this in order to determine all these power series we need to be able to compute this gsa for sufficiently for the correct number of tuples of a surface and a tuple of classes in the second homology on it and we take the surface to be toric surfaces and these classes to be the classes of toric line bundles and we do it in such a way that these tuples of all these numbers are linearly independent so if we can determine gsa for all of these we know we know all these power series but now we have a we have a toric surface a toric line bundle so we can use equivalent localization to compute so these here was these two products of here with schemes of points have a c star action with finding many fixed points these fixed points are parametrized by tuples of partitions um no because a a subscheme constituted in a point which is invariant under the c star action is one which is generated by monomials in terms of local equivalent coordinates and these are parametrized by partitions so we get the fixed points are parametrized by tuples of partitions and everything is given in terms of some universal cheese on it so then the contribution at each fixed point if one does use the localization formula for this i mean the abort localization at these fixed points is given just by some combinatorial expression in terms of the data of the partitions and now if one is very smart one put try to understand this in terms of symmetric functions as we are not we instead write a program so i write a program to compute this up to q to the 12 to compute all these things up to degree 12 and with this we know all these power seeds for everything okay and then finally as i said in order to get uh our uh the formulas ahead before now we we kind of stare at this power series we have gotten and play with them and we try to find some relations between them until we find enough so that everything is determined and we get our formulas and so we have therefore these conjectures but we have also proven them uh you know up to a certain dimensions for all modular spaces up to a certain expected dimension this conjecture is you know we have come to the conjecture by proving it up to i mean for many examples okay or you know or just for in some sense for any surface up to a certain expected dimension of the modular space okay maybe that's all i wanted to say i'm sorry i went slightly over but um okay so what kind of structure do you expect in general so in the most optimistic well um you know in some sense well it's not so clear you know if we look at at these formulas here um i mean the first thing is i mean there is a certain i mean maybe the side is a bit bad but um there is a certain structure you know but it gets each time a bit more complicated i mean if one was very optimistic one would say okay we we do something with these with these theta functions we have to add some other modular form and then we have to solve some quadratic equation and modular functions but i don't know whether it's as simple as that or not um you know in the moment you know it's as one question said this is not based on some very much information no we we have found a way to you know compute up to a certain degree and match the data with some explicit model expressions but they are a little bit complicated and so i don't know precisely what more to expect but i mean what i can certainly say is that one would always have such an expression like here where you have um you know the sum over the delta so delta c1 and l times k for k from you know all the possibilities from whatever 0 to r off to r minus 1 and then you get something to the power ks squared and then you have this ks here and i mean i can say a little bit more but um yeah but you know what these functions are is not so clear i mean you should always get some modular functions and they should be in some relatively small extension of gamma zero n and you might always find a formula in terms of the help module or something but you know i'm not sure uh you know what precisely to expect maybe it was a long answer to see nothing but anyway okay but if you sum up all the partitions then you expect kind of this algebraic dependence to drop out or yeah but the function you sum up over all the partitions down though um in what sense which algebraic dependence on did you look you don't get a model of form you get an extension i'll try the extension of a model of function or um but um you know let me try to understand i mean so you know if you look for instance at this case you know you have these two contributions the one for um um uh for uh so you have the contribution for uh for for the for the trivial partition r and for the for one to the r these are kind of independent summons they don't really mix no we just sum both of them it's only that under uh s duality they are interchanged but i don't know and the whole expression is modular that's true so okay so so that is obviously you know this is actually in some sense you're right so you have all these terms are modular for some subgroup but when you sum them all up they get commuted in such a way that the whole expression is somehow modular for uh in some sense essentially well for maybe for actually i think for uh gamma zero ah so in that you're right but still if one can only compute one side then it's always one computes one and the other is determined by that so that the whole thing becomes modular i don't know whether that helps you know there you know the whole thing is modular but there's so many terms and you know it's not and you know the modular also involves this kind of free transform i mean so it's not completely clear to me how this gives us uh but maybe we can discuss at some point but i in the moment don't have a clear idea what i expect do you expect some uh fox space to be behind the story so to see this visor is about the expectations of some operators okay now that is um well in some i mean whether it's just a fox base that's a little bit but one would expect there is something like that obviously in the case of Hilbert schemes which is the rank uh one case that is precisely the case and so one could expect that there is some kind of uh the algebra like that but it must be you know a little bit more uh more complicated no it's not just a fox space and it you know it also maybe cannot you know it's not you know yeah i expect there will be something like this some vertex algebra whatever which is responsible for this but i don't see uh now how you know how this will be the case but there's something should be the case but i don't know how so sorry more questions so this is some hope for this two two partition i mean the rank for two two partition yeah i we haven't thought about it i once mentioned it so um to martin and he explained me why it was not so easy so uh it's a little bit complicated obviously you could say in first approximation you're looking at a product of two modellized spaces of sheets of rank two and then you do something similar to what we did with products of Hilbert schemes with that but there are many complications because stability doesn't quite match and so on so it should be somehow possible but i expect one would have to maybe do something with uh some stability conditions i mean maybe in some you know use some kind of maybe i don't know maybe some war costing in the right categories or whatever to you know because it it doesn't quite match on the nose that these are just you know you you cannot just say you take one modelized space other modelized space you integrate over it it's somehow different but somehow it's clear what one would like to do and then one would have to find out how to correct it into something which actually is right and in the moment i can't know how but certainly it should be possible in some way but okay and we haven't seen a question now we have a question in the chat is anything known or expected about the contribution of the two two component in rank four well this was just asked no i mean we haven't done the computation because we don't know how to do it in in particular we didn't produce a prediction i mean i think it is well let me see i don't think so i'm pretty sure it i mean i would have to check but i i think it doesn't follow from s to le t i mean there might be some restrictions coming from s to le t but otherwise it's it's not that you can't just deduce it from the vertical partition function and we don't know very well how to compute it because in first approximation as i said it's done by integrating over products of modelized spaces of rank two sheaves but it's not quite true and we don't really know how to how to make it work the other questions in the dm of lullacor so there was some exponent of k squared and also negative k square so if we just multiply the same thing under the under such exponents then it's true yeah i know so obviously this is actually i should have said it obviously that we have this term here with minus k squared is just our convention the point is you know obviously we could absorb the term with minus k squared into the term with the c zero we don't do this because this is this term is the one which comes from the blow up formula so kind of the interesting relevant you know so we somehow expect that we get the simpler formula for the c zero or you know something that we can understand better if we just pull this term out because this term we know that this has to be there in some form we could have absorbed it into this but it somehow kind of makes the formula less clean if one doesn't pull it out so we only want to keep the term that the part that we don't know but if you wanted we could multiply these two out and this would still you know it's it's not a theorem that you know obviously there's no problem multiplying these two and it's still something to the power k square that's true it's just so that we get a simpler formula for c zero. Thank you. Any more questions? All right, thank you.