 I would like to talk to you about the case functions of the maximally symmetric Young-Mills theory in four dimensions. So we would like to make it very sketchy, a half integral. Actually, it's not. Is this better? Ok, is that better? Ok, thank you. The basic path integral over the Young-Mills field. So we have 50 action as of A is as usual one over G squared, over the four manifold sigma which I take to be an algebraic surface, compact algebraic surface efforts, star F plus a topological term i theta over 8 pi squared trace FHF. And I should say this. And this tau on the left hand side is the complexified coupling constant G squared where it includes the theta. So tau is theta divided by 2 pi plus G squared. Ok, so you see this is always lying in the upper half plane just because G squared is always positive. Now written this way, this n indicates the gauge group U n. Now written this way, this partition function is very hard to evaluate. This has no supersymmetry, but we include the supersymmetry in order that basically the partition function then just localizes on classical solutions or the instanton solutions. I will say a little bit more later. So schematically then the path integral becomes something like this we had fermions and scalars. I won't go into the details of writing out the action completely for this period. And one of the motivations you may have to study this to evaluate this path integral are physical dualities. And the famous one is the electric magnetic duality which goes back to Montounen and Olaf from 1977. And then Wafa Witte made the connections in the 90s that basically this z n of tau should transform into a model of form under S L 2 z in S L 2 z. Ok, so I won't go into the details of the weight. There are weights multiplying this function, but I mean for the purpose of this talk I will skip it. Was there a question? It was about the weight. Ok, you can look them up in the papers to get everything right. Ok, are there any other questions? Ok, so to answer this question what we will do is to attempt to evaluate this partition function explicitly and see to what extent this is a modeler form. Ok, so one important result of the other result of the paper of Wafa and Witte in 96 is that this path integral localizes on the Hermitian Young Mills equations. Equations which basically state that if he integrate the covariant derivative of A with J over the surface this should be proportional to the identity matrix of the adjoint representation of F. And that the 0, 2 and 2, 0 components of the field strength are equal to 0. If you take your connection to be an SUN connection then this is equal to 0 and this statement just says that the star of F is equal to minus F on a complex algebraic surface. These are some vanishing theorems. There were some vanishing theorems which I am also skipping over since it is a one hour talk. There were some vanishing theorems to get to this statement. Thank you. Oh yeah, I should say that J is the scalar form of your algebraic surface. Any other questions? Ok, so this is more a physical language. Now we, maybe before explaining the first geometric correspondence, then I will also say that this path integral was also argued by Wafa and Witte and that this path integral should make it also a function of J since you see that the Hermitian Young Mills equations depend on this scalar form J which is roughly something as the sum over instanton numbers times the Euler number of instanton modelized spaces and then weighted by the classical action evaluation, evaluated at the instanton solution. So these are the, this is the question. At the moment I am just assuming it is a complex algebraic surface such that it is in particular scalar and it has a scalar form. I am not assuming at the moment anything that it is sort of that. All of it is algebraic surface or… Yeah, I thought there were some issues with v2 plus equal to 1. So what I will do actually, I will kind of redefine this partition, redefine the right hand side in a more mathematical precise fashion by introducing some mathematical invariance here and then just working with that partition function. So this is, it is a bit more motivation. You might want to argue more vigorously that this path integral really localizes to this and you have to analyze these vanishing conditions. But I will basically take it from here that I need to, I want to determine the generating function of these Euler numbers of instanton modelized spaces. Yeah, this is evaluated at the, I will explain this in detail. If you evaluate this for a given instanton number or given turn class, you will get a specific exponent. It will be the next part of the discussion. So the Donaldson-Ulembach-Yau theorem connects these Hermitian-Young Mills connections to geometry in particular to holomorphic vector bundles. These holomorphic vector bundles are classified by their turn classes and let me just recall the formulas of the turn classes in terms of these field strengths f. I guess the most basic one, the rank of the vector bundle corresponds to the end of the gates group. The first turn class is up to some constant equal to the trace of the field strength. And then this combination of first and second turn classes which is essentially the second turn character or instanton number is essentially the second term in the action, trace f, fxf. So this is basically the dictionary of your, between your solution, instanton solution and the turn classes of the corresponding vector bundle. The vector bundle determined is the same state? Yeah, indeed. The first, the other thing. So often I will abbreviate these numbers just by gamma. Then indeed the Donaldson-Ulembach-Yau theorem says that these vector bundles are semi-stable in those equations. Vector bundles, so what does this semi-stable mean? To this end we define a so-called slope phi of j of gamma equal to the first turn class dot the scalar form divided by the rank. I view this now as a number leaving out the integration but kind of implicitly assuming the integration over the forms. Vector bundle g is semi-stable if and only if for each sub-bundle g prime in g that combination phi j gamma prime where gamma prime is the turn character of g prime is smaller or equal than phi j of gamma. So this is how the dependence on this scalar form appears in the context of the vector bundles. First we saw it appearing in the differential equations for the insult zone or for the Hermitian Young-Mills equations and here it comes as a stability condition on the vector bundles. The modellized spaces of the semi-stable vector bundles are quite well studied in the literature and one of the already quite old results is that we let this be the modellized space of the semi-stable vector bundles with turn character gamma and with respect to the scalar form j. Then this space is smooth compact if two conditions are satisfied at least if the rank and c1 are primitive. This is a primitive vector and then we get a condition on the surfaces that j dot the canonical class of the surfaces is negative. So with the formal requirement it says that the second extension group is equal to zero and therefore it's smooth. But then if this is a smooth and compact space we can define topological invariance of this space and in particular I will consider these omega, gamma, w's, j's where on the right hand side they make a generating function of the Betty numbers or the dimensions of the comology groups of the modellized space and w is the generating variable. This is i is zero to two times mc. On the right hand side I have w to the power i which is the complex dimension of m so that if there is Poincare inequality and there is Poincare inequality for these modellized spaces and this is a polynomial which is symmetric under exchanging w and w inverse. Questions about this? Relatively basic invariant. So we would like to be a bit more general we would like to relax this condition that r and c1 are relatively prime where that's a primitive vector. However on the mathematical side we need to become a lot more complicated because you have these strictly semi-stable objects and I will just say basically in words what we are looking at but for the purpose of this talk I cannot really go in too much detail. So what we are, we'll look at this so-called virtual Poincare function that you consider and noted by this curly i, j which goes on the name virtual. Head back to this story if r, c1 so in this case where r and c1 are relatively prime this polynomial would be related to this object simply by multiplying a vector w minus w inverse. The moral of the story is this object exists for any gamma and if r and c1 are relatively prime you get a Laurent polynomial just by multiplying with w minus w inverse. You can also get polynomial invariants from these guys using a map called the the statistic logarithm mapping it to Don St Thomas invariants but in that case there is a less clear link to precisely what, yeah then these numbers are conjecturally so-called the dimensions of the intersection comology so everything becomes a bit more abstract and difficult and I would like to not go into that direction. It's not a virtual Poincare polynomial it's a virtual in draft rank. It's a virtual Poincare function I didn't call it polynomial because it has these rational terms in the denominator and if you have a larger increasing rank you get increasing it's a stacking invariant, yeah. Well it's not really a it's a function it's a rational function Okay so this is all basically to say that we can feed into a more precise mathematical definition of these generating functions with or somehow related to this topological twisted and it goes for supersymmetric young mill speed. So this is all related to the coefficients of this exponent of the e to the power e to the action. Let me say now a few words of how this action how we can evaluate this exponential how one can show that this exponential evaluated on the Hermitian Young Mills equations takes the following form this q to the power 1 divided by 2r c1 plus squared times the complex conjugate of q times r delta minus 1 divided by 2r c1 minus squared with a q as usual e to the 2 pi i tau and then c1 plus squared is equal to c1 dot j squared and we divide and we normalize it so we divide by j squared and this is equal to then c1 squared minus c1 minus squared minus so I should say this this lattice of h2 sigma sigma z for the I'm considering these surfaces sigma which have b2 plus equal to 2 1 so we have an indefinite general indefinite lattice with one positive direction and rest negative directions and this scalar form is by definition essentially positive and we can use it to project a general factor and in particular the first turn-class of the bundle in order to the positive definite subspace and then to the negative definite subspace and this is what we see appearing here in the exponent and delta it was the other unknown it's simply a combination of the turn-classes it's c2 minus r minus 1 divided by 2r c1 squared okay so then a more mathematical precise definition of what we started with from the supersymmetric case theory point of view is this is a generating function where we sum over all first and second turn-classes and we keep the rank of the bundles fixed and we sum over all these gamma w j and then we have this exponent here in the minus such that the left-hand side equals the right-hand side because you gave the definition of the left-hand side before of sd basically well, obviously it will take a couple of steps to work it out but it is there always exist in A at least such that the right-hand side such that it's equal to the right-hand side no, it should, for any A with the same, with turn-character corresponding to r, c1 and c2 the action evaluates to this just using that trace fhf is proportional to the second turn-character and then the trace of f is essentially the first one and that starf you can relate then starf to f using j because j is self-dual on our questions and now one way we can sorry, I should connect these w and z so this w is e to the 2 pi i z and now one way, the first point now where we will see modular forms appearing is that you can make so-called theta a decomposition from this function due to the fact that these modulized spaces of instantons isomorphic if you tensor all the bundles with a line bundle so this will change your second and first turn-character not the rank but only the second and first turn-character but the modulized spaces of these turn-characters are isomorphic and essentially due to this relation these invariance are isomorphic and we can write this generating function as a sum over now a finite coset mu and lambda slash r lambda it's complex conjugate r mu of tau times a theta function r mu but these are, I will give the definition theta functions are sum overall and the sum overalls are lambda times q a minus squared minus k minus squared and this lambda, I guess I should say is and then the h's are of mu so our sum overall, the second turn classes are the sum of first turn classes is taken into account by the theta functions sum over the second turn classes that's where the non-trivial invariance are just weighted by r times delta okay so now I will specialize for the rest of the talk to the Hitchaboo services where one can explicitly determine these functions 8 where all the difficulty is c1 is equal to mu that's where the c1 is information of the c1 is so this gamma is other questions the geometric underlying is this tensoring of the bundles by a line bundle but it's not very deep in some sense but except that these are now honest theta functions which transform as modular forms of a given allomorphic and non-allomorphic weight so we see some modularity appearing but now in order to go ahead we should determine these guys and see whether they are modular forms restrict to the Hitchaboo services these are a fiber these are p1 fibers over p1 it's also p1 so the second homology is is two-dimensional and we have two generators the base one is c and the fiber one is f and then the intersection numbers just to be complete our c squared is minus l c and f are equal to 1 and f squared is to 0 so we have the following 0 you have the following intersection matrix and you see it's of indefinite has an indefinite form now let us parametrize this space of these scalar forms since this is a two-dimensional lattice the space of j's is also two-dimensional even although of course it's for the stability only only as projectivation is important so I don't know that we have two integers m and m but it has c plus times and then so basically what the structure of the calculations is that the partition of these h's are known if you go here at the boundary for j 0, 0, 1 sorry that is the and then you can do use walkers and formulas to go to any j inside the inside the scalar group so let me briefly give these generating functions for here if you go to sit at the boundary of the scalar constant space we'll use a color this is the space of j m and n are both positive and we know that we have formulas for elegant formulas for choosing j at the boundary of the scalar cone and then using walkers we can go inside so h's h's of r c1, j 0, 1 is in fact equal to 0 if the intersection of the fiber and the first-term class is not equal to 0 modulo r and it is equal to a function r of z tau if it is modulo r and these functions take a rather elegant form in terms of Jacobi theta functions and elegant eta functions i minus 1 to the r minus 1 eta to r minus 3 theta 1 to r z the product of j is 1 r minus 1 theta 1 that's correct eta is the elegant eta function and these are the antisymmetric Jacobi theta functions if you set r equal to 2, 1 you'll find a back-gritches formula if you set r is equal to 2 then you find the result of Yoshioca from 1994 and for all the other rs it is proven by Moskovoy in 2013 and we also these formulas are also known if you would replace this curve by a higher genus curve and then for genus 1 it has been explained in string theory by Babaka Grigat how all these functions appear for arbitrary rank using a elliptic genus calculation so basically this whole calculation also continues to hold for the world surface just for this talk I'll explain it for genus 0 base curve base higher genus curve if you have a higher genus curve here you still have an upper carter which corresponds to the scalar form and again all these functions are known at the boundary of the scalar form have the similar structure as this and also all the right-hand sides are also known and proven and they are understood in string theory then for genus 1 because basically you can do this on the torus it will be interesting to see whether you can derive for other genera also these functions from string theory for arbitrary rank if you have rank rank 1 is also understood in string theory so let me now say a little bit about this wall crossing so if we restrict for now to the case of rank 2 we can really do it for arbitrary rank for simplicity restrict to rank 2 then the wall crossing occurs just due to walls where the slope of two constituents gamma 1 and gamma 2 with rank 1 are equal and then I don't have the time to go in full detail but then you can express this generating function is that j for an arbitrary j in terms of in terms of these functions and so called indefinite theta functions so what are the indefinite theta functions they are a sum over lattice points in an indefinite lattice let me call this one twiddle just to distinguish it from the lambda we had earlier for the second cosmology and what is special about these generating functions is that they are holomorphic in Q and minus 1 is what I use as convention so they are mostly a negative and now they are they are holomorphic in Q and in order to get a convergent generating function you need to insert some kernel which falls off fast enough for the terms which are positive in order to get a convergent generating function and then a classic choice of kernel which appears precisely in this context of model i of vector bundles is as follows the difference of two signs the sign of j dot k minus the sign of j prime for this j prime you can think of as being this point j 0 1 and this is the reference point of the point where you are interested in to move to so for example let me make a diagram do this let us minus 1 again this is your positive definite subspace then here we get a negative definite cone here there is a negative definite cone and then this is a positive this is again positive definite so in order to get a convergent series if we say j prime we take here to be sitting on the boundaries and this is j and this then if you then look where this is non-vanishing this is some cone in the negative definite subspace we are summing over a lattice points which are a negative definite subspace of the lattice in such a way that we get the convergent series and if we change the stability parameter we start changing the subset of lattice points we are summing over and this is how the partition function in this case takes to account the wall crossing of the vector bundles and this this picture was developed by Goetze and Saquille for Donaldson invariance and also for the vector bundles of rank 2 now it's important that these we get a holomorphic holomorphic theta function but because we are not summing over the total of the total lattice we cannot do this summation and this is not this theta tau phi does not transform as a model of form as we are used to with the theta functions over summing over indefinite lattices or the ones we had where we had these projections to this negative and positive definite subspace in the meantime this work was about 96 of Sveggers in his thesis of 2002 which basically told us by hand how to change this function to get a function that is transformed as a model of form and the changes is really is sub leading so we are not going to change the leading behavior of the function it's a sub leading but not a holomorphic change and the prescription is to replace essentially replace the sign of X over there following integral 2 times the 0 square root of 2 tau 2 X with a minus by U squared dU which is equal to the error function square root of 2 by tau 2 X and tau 2 is the imaginary part of that so couple things to note are this is now a C infinity function so here we had this continuous change across the walls but this is now in some sense smoothed out by the error function but if we send this tau 2 to infinity then we recover the sign of X let me make this slightly more precise by a formula which is relevant later the error function is equal to the sign of X minus the sign of X times the complementary error function of the absolute value of X and this one goes exponentially quick to 0 if X goes to infinity and another important property actually of the error function also is that the following operator the X squared plus 2 pi X the X acting on the error function that this is equal to 0 basically this identity implies that if we define a different kernel pi hat of k 1 half and now we choose these functions here tau 2 j naught k minus the same thing for j prime then theta hat is basically the same theta function as before but now we have the kernel pi hat is a model of form or transforms as a model of form so this is a result by finera it goes back to a 77 but it becomes increasingly more known that Svager's in his thesis used for some reason to explain that this kernel would do the job but this is finera has a very general statement about let's say with arbitrary signatures and the only thing you need to check is that there is a differential equation that needs to act on your kernel if the right hand side is equal to 0 and your kernel falls off fast enough then you have a theta function which has modular properties in some sense this is really put in by hand and if you say well now I just changed these indefinite theta functions which appear in my partition functions to include these arrow functions now I have a model of form and this now explains the electric magnetic duality is not very convincing or satisfactory so therefore in the last 10 minutes or so of the talk I would like to make another correspondence again more towards the physics where these BPS partition functions appear also in the context of hyper-multiple modellized places and from that context we this is joint work with Sergei Alexandrov, Boris P. Olien and Sebastian Barney from this spring we can derive these kind of functions and the way progress to higher rank was earlier not known so these modellized places now have nothing to do with the modellized spaces of these vector bundles these are modellized spaces of the scalars in hyper-multiple which appear in string theory compactifications and we choose our compactification manifold now to be a local hitchable surface on the canonical bundle over the hitchable surface and then these n equals 4 young mills appears as the theory living on the free brain instantons which correct the hyper-multiple modellized space so the link is n equals 4 young mills and the free instanton corrections to this hyper-multiple modellized space for the hyper-multiple modellized spaces they receive both alpha prime and gs corrections and among others they get corrections from all the d-brain instantons and the type 2b also of the d-free brain instantons and the special property of type 2b is what we will use is that type 2b as a S2LT group which is SL2Z in the way the d-free brains are self-dual so in particular the instanton corrections due to the free brain instantons have to be mapped to itself and therefore they have to be in some sense modeller invariant. In order to approach this question we will consider a specific function on the hyper-multiple on this space M8 it's known as the contact potential or the four-dimensional dilaton classically it goes as the volume of the labyow times tau2 squared and then it has all kinds of one loop and instanton corrections and you can see from this this first term this is weight plus 3 over 2 comma plus 3 over 2 and then this changes it to minus 1 half minus 1 half this is the modeller so the classical part transforms as a form and this is expected to be the case for all the corrections to this to this quantity type to be as the value acts just on tau as the usual modeller it also acts on the volume of the labyow there's quite some technology developed in order to include these corrections of d-brain instantons in particular due to koyotomor and peitske and also the alexandrov van doren peolien van doren and saueressig so let me just write down the most famous equation of this field I think quite a few people of you have seen this equation this is the equation for the so-called Darbuk coordinates which are Darbuk coordinates on the twister space over this hypermultiple model space they take the following form times in the first part is semi flat part and then we get an exponent there we get a sum over these bps invariance gamma prime t and an integral over prescribed integral over p1 zeta plus zeta prime zeta minus zeta prime k, gamma sorry I'm a bit sketchy here but so we have an integral we have an integral equation for these Darbuk coordinates which you can solve order by order in the instantons if you're interested and it is in terms of this integral over the twister space this is what I wanted to point out and the main thing is that basically this gave a proof of the Konseven-Scholmann-Rachlossing formula because these Darbuk coordinates are smooth across walls because discontinuities and these invariance were precisely cancelled due to discontinuities in these integrals so what we did in these projects of the spring and also going back to 2012 was that we took a large volume limit of these kind of twister integrals which also appear in this quantity for the of interest and we found that in the large volume limit of these twister integrals and the two instanton approximation we found integrals of the following form i over pi integral r minus d z over z e to the minus pi z squared minus 2 pi i z so this basically came out directly out of taking an approximation of an integral of a twister integral like that and with a bit of staring at this equation we found that this is equal to the sign of times the complementary error function of the absolute value of u times the square root of of pi and every by a thing combined together in precisely such a way that they appeared they were added should be added to the just the generating functions of the BPS invariance and we got a modular function which transforms as a model of form for that would be for the case r is equal to 2 well it would be if you do if you would start from 2a you would compactify your d4 brain you make time and as one you do a t-duality on that as one and you expand out to get 2b and then so all these these corrections of instantons are captured by identical equations to the hypermultiple model space yeah we can it's r3 times s1 times glavial but they compact no no no I am I'm compactifying type-to-be string theory on this on this space type-to-be as among its d-brains it has the d3 brains which I wrap on a four-cycle in the glavial then the world-form theory of those d3 brains is n equals 4nm those d3 brains correct the hypermultiple model space and therefore the generating functions first I was discussing them from the context of just n equals 4nm correct the hypermultiple metric of the string theory compactification yeah so we found that this is corrected the size correction is some type-to times some model or derivative of the n equals 4 partition function then so it wasn't an open question this is for for higher rank but this picture of multiple modelized spaces basically showed us how to find these generalizations of this integral which would also which would help us to complete some holomorphic sums over indefinite theta functions with more general signatures in particular n comma this was published in a more mathematical paper and has been followed up now by by a group of Katherine Brinkman and Larry Rowland the application to Gromov-Hitten theory also and by Kutla embedding it in the Kutla-Millson approach to integrals over theta functions if you start if you want to consider corrections of higher rank more d3-brains or higher rank gaze groups then somehow the most indefinite letters you will get as signature r r minus 1, r minus 1 so for rank 2 we had 1 comma 1 and it goes as r minus 1, r minus 1 and then from the from these expanding out these twist integrals we found maybe I'll write this formula it's most m2 looks like something minus 1 over i squared we get 2 integrals r minus i u1 dZ1 another integral over second variable r minus i u2 there you get an exponent sum over i is 1 and 2 i z i squared minus denominator is z1 z2 minus alpha z1 so this is it's a bit messy but this is the generalization of this one necessary for letters of signature 2 comma 2 and then we were able to passing it together to functions we call the generalized error functions which you have to put into your kernel and then you get a modeler function for rank 2 in principle higher can be extended to higher rank so I would like to really like to understand these corrections purely from the context of n equals to 4 young males or from the boundary of vector boundary model spaces but yeah I'm very happy that we already have some geometrical explanation coming from hypermultiple modeler spaces thank you for your attention