 So, you want to solve the topological string on the compact kalabiao and so I will address a couple of questions which are different and to the local case and so on. So we want to solve the topological string on compact kalabiao three-fold and I would also call this M and what I mean by that so it's now maybe more specific is the string partition function so it's the closed string partition function. So this will depend on certain kalab parameters and it will depend on the string coupling and we all know it's a sort of perturbative expansion and then there will be a class kappa let's say an h2mz and there will be a sum over the genus g equals 0 to infinity and then there are so this is a generating function like 2g so this is gs 2g minus 2 and then there are numbers that we want to know so let's say r so they are labeled by this club kappa let's put it up and then g and then there is a parameter which counts let's call this q kappa so this counts the class kappa so it's a generating function basically for all these numbers and these numbers are chromofitten invariance and we would like to know them and the the t this t parameter is hidden here so t is something like the volume of a curve so you have you have b field and you have so it's some some of the complex volume of the the volume of the curve complexified is the b field and you write it in this combination and then the q kappa is something like e to the t kappa and so in this way you get you get this scalar volumes in so this other calor parameter now in some sense this is a perturbative expansion you have to think whether it makes sense or whether there are better forms of it and so on but what I can give you today is basically is an attempt to give you this function for elliptically fibered Calabi out in a more or less close form in that I expand in the base of the elliptic vibration so for every fixed class in the base I give you a close form and and so that's the aim of the talk so of course there's a certain motivation so of course I said already the soft chromofitten series soft GV series or at least the counting problem of it and it would also solve this was already in Martin's talks it solves maybe DT on PT series because this invariance are related and then what is more interesting for me and where I get some structure out it also solves a BPS counting problem in five dimensions 5d and this BPS counting problem basically is you look at the 5d Poincare group and then you you see that there's the little group of the 5d Lawrence group so there's the little group and this is SU2 left times SU2 right and now there is certainly a mass to get this Poincare representation there's a mass and you sort of want to calculate something which which depends here on the right spin and the left spin and the mass and so what you what you want to do you want to take a trace of the Hilbert space of this BPS states and then you have various choices of instance first I will sort of don't resolve the right spin so I write something like just J R like this and then you write a parameter for the left spin and then you write something which measures the I mean some measures the energy but since they are BPS states this energy is fixed by a charge lattice and this charge lattice turns out to be this common so then this is where the kappa class comes in and so at the end you get something which which you can write and this was done by Gopal Kumar and Wafa you can write it while also by necrossof independently somehow and so you can write it as as a sum over let's say the genus which is positive and then there is an e kappa g and then this parameter y appears here so it's something like something that we also saw in Martin's talk already and then there's a 2g and then you have this e to the t kappa that measures basically this q so this this measures the mass and then they are fixed by this kappa class and this kappa lies in the lattice and this is because they are BPS states but then there is actually a very interesting formula that is implied by that which I'm really use in the following and that formula I mean again it goes back to Narain Taylor and then was made more general by Gopal Kumar and Wafa so this formula looks like this is basically involves this quantities here so these are BPS indices and they are they can be be in Z so note that if you sort of resolve this this right spin so maybe resolve J right and then you get this this minus one gets a U and then you get on the right hand side things which have depend on J left and J right and not just on the genus and on be on kappa and these are actually in N and that's something sometimes interesting so this is more this is more like a real counting problem this is like a index counting problem and for you can actually do both in the cases that I will discuss I only restrict myself to the index counting problem because the formulas are simpler but you can certainly do both with very similar formalism that you will see so so then you get here well M from one to infinity and then you get a sum over G from zero to infinity and then you get a sum over this kappa that we already had and and then you get this I G kappa and then you get G SM over 2 and then 2 G minus 2 and then you get again this e to the M T kappa okay so this formula is sort of slightly bizarre and it has been used the fact that it's slightly bizarre because if it is a perturbative expansion then at all rational and numbers of GS it has actually poles that's that's pretty bizarre so so let me say note the poles so this come from genus zero because then this is minus one and so note the poles at so this poles of course will play a role so if they are rational then we get the poles and that's that's already I mean this is one of the starting point of this non-perturbative completion that Marcus and his friends are studying so there is also true for the compact case and for the non-compact case but now let me say generally that so let me not arise this formula for a second so so I raised already formula which maybe you would have protested so why should you have protested so let me write this formula again so let's write it like GS 2 G minus and then write just FT like this so this is better for this propose and you see this things are Kayla line bundle so they live in the Kayla line bundle I mean this comes from special geometry like we heard in the talk of beloved so this is like a section of L 2 G minus 2 so then if you think this is a good object you have a problem because this comes in all powers of the exponential if you make a killer transformation you get you get a problem and I will come to this point later but you can you can remedy this you can say okay I take GS and I take a I take a period of that now the period works the spirit transforms in the inverse line bundle so then this formula certainly makes sense but then it's not anymore has any good modeler properties because the period is some complicated object that can transform very complicated and in some sense the reason that in the local case you have this is one you have a global section here makes this formula work if you apply it I mean for instance this wave function thing in the local case this has a beautiful application but in the global case this thing is very much spoiled this property so so this would be only in the local case and this is a huge difference for this for the things for the global case but let me now come to the to the geometries that I want to consider and this geometry is have very much to do with the talk that you hear by Martin Cole so so it except that I'm not looking at elliptic surfaces but I'm looking at elliptic three-folds so so let me sort of make this small remark so basically local Calabiao they are always sort of say of the type you take a total space so M what is my Calabiao is basically the total space of the anti-canonical line bundle over some base so what I want to do is so this is this and then and then this can be solved in many ways so what I want to do today is basically to change the setting slightly but not so much so we make a Calabiao three-fold which is compact so this is clearly non-compact because you have this non-compact line bundle but the only thing that I want to do is I want to exchange this compact this non-compact line bundle by an elliptic fiber so then you have this situation and M is compact and that is not at all very out outlandish when you look at the history of Calabiao so for instance if you take the degree 18 constraint in let's say nine weighted projective space so maybe we should just put it down okay so then you have this Calabiao so this is one of the earliest thing is a degree 18 hypersurface in this projective space weighted projective space so it's six nine six one one and now if you write this down you can immediately see that you get an equation which has this elliptic structure because you call this coordinate y you call this coordinate x and this coordinate you call you and then of course it's already elliptic so it goes y squared and then you have I don't know I don't care about the coefficient at the moment you have x to the three because it's degree 18 and then you get something like a x y z and then you have a product of this use let me call them you you prime for the second and then you get a z to the six so the c to the six you don't see that because this has a singularity you need to blow up parameter and this is the c to the six and then we have a b phi i from 1 to r u i 6 and then you get a polynomial pb u of m and this I write down because now you can see so this is the constraint so this is basically the constraint for this particular Calabiao but you can replace this base so if this is in this case the base is actually p2 but you can certainly do this for any base so that's a very simple yeah so this can this guy has exactly the same situation and then Martin Colts talk it has it has a co-dimension one has only codera fibers e1 and then at higher co-dimension there are more singular fibers but the important point that in co-dimension one is only e1 codera fibers and and then you can calculate the discriminant it has some cusps and so on so it's a complicated thing but I will basically make a prediction for the gram of fitness theory on this space which I think Martin also tried to solve it it's not so easy and then he resorted to the surfaces again and okay anyway so so then but I wanted to give you this algebraic form this is basically like a b-model expression to see what how it connects to the local case because if you take this PB 0 and use this depends on you and M and you set it to 0 well it's actually like UV this is actually the mirror curve so it sort of has a so this is the mirror curve and it's a spectral curve of a matrix model and it has many many nice properties and and you can also see how you achieve this you how do you achieve this so basically you say x e this is the class of the fiber so you take it 8b to the 8 to the 6 so the a is this the b is this and then you you send this to 0 but you keep somehow b fix and then you get you get this so I should say this depends on b and then it well I mean this is this this m a mass parameters but then it also depends on you and you are basically the same you only you have an Italian map between them so basically there's a you is you to the 1 6 so this is an Italian map so that's the way you get you get this local geometries but now let me change gears a little bit and say the same thing from the from the point of view of the a model so for the a model of course I given you the geometry it's basically it's an elliptic vibration over b and then you can use you can use service spectral series spectral sequence in order to get all the topology from the base if the fibers are not so singular so so the question was already asked so today similar like in the talk of Martin we take once we have one section and only Kodaira I1 fibers in Kodim one and then everything is actually given by this spectral sequence so so now we use Larry's spectral sequence to get the top the topological data the topology of M just from B so that's a easy exercises and it works very nicely so so let me introduce some notations so it doesn't pay to keep this formula but so so basically now we introduce some classes so we introduce the class of the fiber which I call T and I call this exponential saying Q and then so this is this is the now we go to the homology so what what do we have so we have we have a curve class I call this curve class e for elliptic fiber so this is the class of the fiber and if you if you want to calculate the scalar parameter you just take what I said before the B and the and the omega the symplectic form and integrated over this curve class so this is this volume and then you have dual devices and this dual devices I also need name for them and then you have the class beta these are classes in the base so these are these are in H2 BZ and then I have here I have well the dual devices go a little bit higher so here the dual devices I should already have a DE and I put a tilde here so this is actually the class of e which is the class of the section and then it's shifted by the canonical class of the basis so this is just follows from the geometry of this and here I have classes which I call DK so maybe I call this also K so there might be K of them and that is just the so here we have of course we have of course e so we have a projection map from e to B here and then this I call B and then this one also to say pull back from the base so this is DK and this these are living on the base so these are these classes and and now you can calculate all the intersection numbers just based on these classes and and and the intersection numbers of the base so that's a easy thing but I need it for some later work so let me just say that okay so you now calculate DE squared so this was this class here and this one just gives this just gives the integration of the second churn class so so if I say C that C K is always the churn class of the base and if it's not the churn class of the base I write the manifold okay so this is the churn class of the base and then we have then we have D we have let's say D E times D K so this is now intersected with this base classes once and this is the C one of the base times D DK so this DK is is this guy so this is a divisor on the base and this number I will call 12 8 sorry 8 K so this is the definition of what AK is and then you have intersection of DE DJ DK and this ones are just given by intersection of the base so this is intersection of the base so this is just over the base and you take the DJ DI and you integrate it over the base so this gives all the intersection numbers and the one which is three times things of the basis of course 0 because it's a vibration okay so then you can get more so you can basically get a C 2 of M so now this is in the total space times DE and you calculate it and it's so this is like a Hirz-Pochriemann rock formula and it's like this and you can also calculate this one on this other types of devices and it gives 12 AK this what I introduced this AK and then the Euler number is actually minus 60 times C 1 squared of the base so for instance in this case this is this famous Euler number 450 minus 450 is the second example that Candela's ever looked at this is this case okay so then for a reason I mean for to see the exhibit the model the modularity I have to make a little change in the so this is the this is the basis of the classes and the curves for the Morricone so this was the Morricone and I make a little change from that and and define DE so this was also why they are tilted before DE I shift again with this canonical pullback of the canonical class of the base and then that will lead to so if the QK where e to the 2 pi I TK and this shifts this classes of the base by so they were before TK and now they are get shifted like this this tau and and then why do I make this change of the base I basically make this change of the base because I didn't like this coupling so I like this coupling I don't like this coupling sort of which which which mixes the fiber and the base in a way that I don't like so basically now in this new base so the key property key property is that in a new base so if I use this guys then these guys are actually zero and by the way this case don't change so they are just the same okay so now I can start start stating the main a serum and then I try to prove the serum and and this has very nice so to say inside in the wave function transformation of the top compact topological string and and I also need this formula so let me make the claim so what is the claim so now so we of course our goal is to expand the string partition function and now I expand it in a particular way so you have the fiber class you have this base classes that are defined and you have the string coupling and now you decide so to say to to expand it in terms of the base classes as I said so and you you pull out beta equals zero which is special it's not Jacobi form but the other will be Jacobi form that's why you pull it out and then you have the rest is a very systematic expansion so it's a b in h m z and then it's z b tau g s and then it's q b so and basically the whole thing that I want to explain you today is why are these guys Jacobi forms and how you can calculate them and how you can eventually solve the problem so that that's the point so this these guys are the main characters in the rest of the talk so what are the what are the properties so I give you a list of properties of this of this z b t g s and so let's start with property one so they have basically three properties and I'm able to prove some of these properties so let's start with property one so so what is this so um so the set b tau g s are Jacobi forms in fact they are meromorphic Jacobi forms not weak so these guys have poles which the poles come basically from here so they are meromorphic Jacobi forms and these meromorphic Jacobi forms they have a weight and they have an index and the weight of this guy I call in general k and it's zero that's very good and the index is actually I call it m and it depends on the base class of course and it's one half times um times the class b times b minus the canonical class so one cb and if you know a little bit of algebraic geometry especially the genus of the curve in the base minus one so that's the index and and then this is not completely true it's up to a up to a multiplier system which is however very trivial so the multiplier system comes because this quantity is not completely sl2z invariant so the multiplier system comes because um if you make the s transformation on this q beta then you will find that this is actually depends on the parity of this curve so basically it is a minus one to the c1 times beta times q beta and if you make the t transformation on it then it has a similar factor so it's minus one c1b times q beta so so this this is a multiplier system which so so the so the set bs have to spill out these factors to make this expression invariant okay so then um the poles are yeah i will talk about the poles but the poles are basically imposed from that so they are they are basically over q and that means they are the torsion point of the elliptic argument so one of the most important thing of course which physically is that you basically identify z with z g with z which is the elliptic argument of the weak jacobi form that's the most important point in in this so so i i don't know how much i should say about jacobi forms so it's a pleasure to talk about them but many people know them or many people don't get too much out of it but let me say one very trivial very trivial thing so so jacobi forms so basically what is it it's a map from the upper half plane times c into c and well that's here in the upper half plane lifts what we call the elliptic argument sorry the modular argument so it's the modular argument and here lives what we call the elliptic argument and this is the one which is going to identify this the string coupling and so they are they can be holomorphic maromorphic whatever and now they have of course transformation properties so you say gamma is a matrix a b c d in sl2z and then you can make the following you have the following transformation properties so basically that's the defining transformation property is modularity and modularity and and quasi periodicity and the modularity says okay the modularity says that if you uh say this is this normal transformation ct plus d so it's the projective transformation and it says that if you uh now plug in here and you take the t of gamma and then there's also a z of gamma which is just z of c tau plus d and you put here the set of gamma then this thing is basically invariant up to uh a phase and this phase is related to the index of this guy so that is the m so it's related to the index and then it is um it is um m and then the c comes up this is the c really from here and then is c squared and then it's over c tau plus d and and then as i said it's invariant otherwise so it's just the phase and then the so this is the modularity and the quasi periodicity says if you if you it's it's sort of say a model it gives a reason to call this the elliptic argument because elliptic curve of course you can shift by one and you can shift by tau and it should not be invariant again up to a phase so that makes this an elliptic argument so if you take c plus mu plus lambda tau then again it's basically the same up to a phase um so it's e to the minus two pi m and then lambda tau plus a two lambda z and by the way here i forgot something important which nobody seemed to have realized so here is a factor of ct plus d to the k and that gives the weight and that is the other thing that that comes in okay so they so what i the index comes in here so the index is only coming as a phase okay and then of course i should say that you know jacobi forms anyway so many things are the wire stress forms the jacobi form the tether functions are jacobi forms with vector value jacobi forms they are jacobi forms so there are lots of jacobi forms that you know but the well i mean the jacobi forms have an expansion if you expand them in z then you get then you get the quasi modular forms you get the eisenstein series e2 e4 e6 and so on so um i guess that's all what i want to say about jacobi forms um but now well at least at the general jacobi form so now let me sort of uh come to the property two so the property two is the following the property two is that the set b are not arbitrary jacobi forms but they are jacobi forms of a very particular type they have the dedicated eta function raised to 12 times c1 times b and then they have a numerator phi b tau of z and this one is a weak jacobi form so this is a very strong statement as it turns out jacobi form and there's also is sort of in the title then finally and then the denominator is um something which goes from l to 1 to b2 so this is the uh numbers of the classes in the base and then it's an s from 1 to bl so bl is the l's component of this vector b that gives the uh a class of the base and then we have tau s z so i haven't told you what this guy is but in order to tell you this i have to go say a little bit more about the structural theorem of jacobi forms so there is a very important theorem of eichle it's okay so this uh theorem says basically the weak jacobi forms are finitely generated and freely from by the eisenstein series e4 the eisensteins and i call this actually q and then e6 and then um and then two other arguments so this this arguments have just wait no index and then there are then there are things which have uh index and that is uh sometimes called gamma minus phi one so this is this is the notation this is k and this is m here this is always like this k and m so this has k minus one and index m and um and then it's the and i said you know jacobi forms of course the jacobi the the teta functions they have a multiply system but there's one teta function which goes into itself and there's teta one and so this this can be used to build a jacobi form and you have to square it and then you take eta to the six and you see this has sort of say weight um this has weight one this has weight three then you have minus minus two and then to the question that it was already asked you can sort of um have two interesting expansions so let me write this like this um x one so there is a product expansion which is sort of nice um and useful actually um so this of course just comes from the jacobi triple product identity and is given like this and now to this other question so you can write this in terms of um expansion in z and then this expansion coefficients will be actually um out of the ring of um of quasi-modular form so this is spanned by e2 e4 and e6 so here is a is like this and then it goes with order z squared uh sorry this is order this is i'm i'm i'm off in the order so this is um this is just set to the four and so it's only it's only even so for instance if you want to do uh open strings of course you knifely would say you have to do uh odd jacobi forms they're even jacobi forms they're odd jacobi forms but you certainly get here only powers which are even in z and that means even in gs anyway so this is the this is the story and now you see why i picked this denominator and that's why i left this formula basically this denominator is precisely to reproduce this poles at the portion point because you see this guy goes with gs with that so it has poles and i sort of multiply here by s to get in this multi covering that generates this torsion points at the rational i i put this i put here this power of this a so it's basically to reproduce this torsion points and this is this i set this uh this is another generator so there's another generator so the other generator is called b by and it has weight zero and index one so b people write like zero one and this is well maybe i just be brief and don't well so it's one half of the elliptic jingles of k three that i suppose everybody knows that is actually so you take k one two two three four and then you take a ratio of theta functions so this is theta at the argument tau and z equals zero that's why you don't take the one and then um so it's anyway has weight zero because you have this nice ratio and then the weight comes from this tether so this is k t and then here is the full argument and you square both of them and that's that's the elliptic so this is these are the other jacobi tether functions so this is generated by these things and um and this is already extremely powerful the statement in the sense that so let's go back to b is equal to p2 well now i tell you for degree zero all five are degrees the only thing is that i have to fix a ring whose dimension you can easily determine because i have told you the dimension of the z at the index and the weight of that don't sucky has given you the index and the weight of this and of course this is classic and so then i i just can build an object which has the right index and the right and and the weight and it will be let's say z1 well let me just so just z1 is as i said phi one over error to the 18 and then it will be just a tau z and this is the only thing that i need to give you and this this one is very primitive so it's minus q over 48 and then 31 q cubed and then 113 p so this has has a weight 12 16 no index in this case because the formulas work out and this gives already infinite number of predictions for higher genes and so you can iterate this process and then since the ring is not so small at some point you would still run out of boundary condition but for instance it is easy to do it for b equals 6 in this case that's okay so you get a lot of lot of predictions and i can sort of convert them in bps invariants give you integer numbers and sort of this were the things that that well you can also write it as thomas and thomas invariants and so on so let me come to the property three so the property three is also very nice so the property three says that once the index is negative or smaller than actually smaller equal than zero so there's a property three property three and the claim is if is smaller equals zero then um then um the phi beta are completely fixed fixed by the vanishing by the known vanishing i g beta so these are the bps indices and basically you know that that if g is greater so this this happens if g is basically too great is too big so this happens if there's a little bit depends on the on the base but it basically goes with the base degree so if b is bigger than this number then then this thing will be zero and you can show that if the index is negative then this grows very small a small this room of a weak jacobi forms and then this this conditions solves it completely so this particular solves the m string the e string the refined m string the refined e string things which have singular fibers and so on so this this observation because all these things precisely the things that soto and these people classified they were rigid in the base so they they they rely on a negative self in the section curve then this is negative the index because of the formula i gave you and then you can solve it so this solves a lot of series but these are local series so this is not so it's the old thing for some reason the global series are really really much harder but they are more trivial more non-trivial than the usual local series that we look at for reason that i now explain so let me sort of in the last minutes try to say you some interesting aspects of the proof of it yes yeah yeah that's what i'm saying so they they are not doing this very good but that's why we now work with them so um so let me let me say some interesting aspects of the proof so i mean the i mean property one is certainly the most important one so let me try to make you understand property one because it's very simple so property one so you know that this um i mean first you have to establish modularity but i don't want to go into the details basically there is a there's a formalism which is comes from the from the from the derived categories so basically in this case there's a four-year mochi kernel so in homological mirror symmetry if you have an elliptic vibration you have a you have a certain property of the four-year mochi kernel so basically what people always well let me just sort of draw you one diagram so that you get that if you want to look it up you can see it so there's two projections you take m times m and you take two projections to the base and then there is this four-year mochi kernel this is the only thing that you have to know so this is uh is tether so this is the ruse-criminant and then it's e times m minus m times e and then um and then you subtract on the also c one of the base and then when you have this object then you can say that in the so this is basically homological mirror symmetry you have your you have the the a-brains but uh and then you know that the s transformation on f and anything in f in the a-brains can be written out in terms of this four-year mochi kernel so that's i mean that's uh so you take some pullback of this and then you take some left action on this map and then you take f and you take this this four-year mochi kernel sorry i should throw it like this and then you you have p and for for some reason we also tensor it also always with um so this has to do with my shift of basis um wow so the upshot is when you have this formula and you know these classes then you can write down an s transformation matrix entirely in terms of the of the intersection of the intersection of this thing this works basically for four-fold three-fold uh k three whatever so everything that has a mirror symmetry so basically then the s matrix i just give the s matrix so the s matrix will be something like one minus one uh and then um 0 0 0 0 0 so this is basically the s this is the the block which is the s and then the other thing so to say here's a block which have a k and then there is a block which has a c k j so basically i i write this like x not x k uh f not um sorry i introduce also a t x t and then i have f uh not f t and f k so this is basically is the the series period then this is the one which talks to the tau this is the one which talks to the base classes and these are the dual periods uh in the same and um and it's a bit ugly so x k so this um yeah maybe you're writing a matrix so then this one is a k as i said and then it has here 0 0 and then it has this intersection matrix on the base i k and then it's minus one and 0 0 0 0 and it has a j and that is um a that is a no 0 then is c 1 squared and 0 0 0 and then minus a j and so there's all the section number so the point is you can establish and you can also do this by just looking at the analytic continuation of the periods this is just a homological mirror symmetry argument if you are powerful and want to analyze the system of um let's say where you have 10 model i two from the fiber eight from nine from the base then you could also do this by analytic continuation but but the but the mathematicians give you a very nice answer and this answer is basically due to bridgeland he does this for case reason and other people like yaw and company have done it for higher dimensional cases and we can i mean they did it sort of wrong so then we have to do it again but the but it's anyway standard and then there's the s transformation the s transformation is even simpler so the s so the t transformation the t transformation you just tensor by you just tensor f not just by this line bundle so then so t would be t of f not you just tensor this by a certain line bundle and that's what we know as the b-field shift so that's and again we do a little bit shift by this class so anyway this matrix will be look similar all in the intersection numbers the only thing that now you have one one zero one and then other other classical intersection so why i'm saying this so i'm saying this because to show you that if you have a partition function and you have the symmetry then you would expect it to be modular because after all this these are the these are the space-time generators of the space-time duality and if you have an object on the calabria it should better be modular i mean this is what string theory tells you and and now i want to make the one important point before i finish so the one important point is so if you look at witton's wave function transformation property i already alluded to this a little bit then so you have so witton says okay we have this e to the g equals zero to infinity and then we have this what i wrote already before the only difference is he now calls this the wave function and then he goes into geometric quantization of h2r so then you do geometric quantization of h3mr and you want that this so to say doesn't depend on the polarization and since the polarization is given by the change of the complex structure you get a differential equation in this complex structure and this looks like a heat equation and it looks like this there's so it's because so let me say a couple of things so basically he said he sets this like this h bar and then he has an h bar squared and then he has an alpha beta gamma so this is originally how how he writes it beta so this is a bar index and this is a gamma and then this guy annihilates this wave function so this is the is the infinitesimal statement for the fact that this wave function should not depend on the polarization and since the polarization is induced on which complex structure you use there is a is a differential with respect to this complex structure this is a complex structure okay and then you see and many people have looked at this including bcuv and you say this this cannot be right well it cannot be right for two reasons and let me sort of give you the trivial reason so first of all well it is right but people thought it was not right and i should tell you why it is right after all so basically let me give you first one make a make a small remark so i said already that this is bizarre because here you have no invariance right you shouldn't take this rather you should take x naught some period you can take any period but this period should be so that it cancels the scalar transformation otherwise this object makes no sense in the beginning i mean because it's not invariant under scalar transformation so and if you do that and you look at this transformation you will see that x naught actually goes under st transformation because this is x naught it goes to x naught c tau plus d so if this goes transforms like this then this guys have to compensate that and if they compensate that then this guys must must be modular forms of a certain weight so it follows it follows that the fg are as a function of tau so this holds all in the limit of the base expansion are are forms modular forms of a certain weight and the weight you can it's just the weight is just given by this index 2g minus 2 and now comes the second thing that is very important but now this guy is almost quasi periodic because of the gopa comma waffa formula so you know already that z goes to z plus one is actually in okay transformation of this is because of the bps formula so if this is a correct transformation and you have this property you can see easily that you also have also have to have to shift by tau that's trivial you just write it down this expansion you shift this by by this amount and then you use that as an invariant and you have to see that this is actually not just a shift but it has to also be shifted by tau must be also possible so this implies this implies quasi periodicity so so this is very trivial in some sense so but you have to see the following i mean you don't really change the polarization because you you you you just rotate the a matrices changing the polarization means changing the b periods and the a periods you don't do this so if you would change the b periods then you would actually change the polarization and the whole thing becomes very intriguing so then it's not so easy to say what is the invariant of it but here we got lucky because because we have this monodromes that just change so to say the a period a periods not the b periods so the full power of this whole thing that i may be saying would be for the quintic when you have this transformation but you also change the polarization then it will be very very powerful even for this case i believe and then there was a second point that i should really finish quickly as i said i mean this this quantity look at this at g equals zero so in so at the at the lowest order in h so a lowest order of h bar this thing is simply wrong because if you pull this down then you get something so now i i sort of absorb this i mean i sort of introduce a new parameter like this uh well this was is the h so just a power counting parameter so now i take the derivative of respect be with h and look in the lowest order in h bar then of course this gives an anti-holomorphic derivative of f naught which is the pre-potential and it gives something here from which is not vanishing because this one is a holomorphic derivative so then everybody says i mean like like bachatsky and digraph and people who talked about the osv conjecture they all say forget about the genus zero part this thing has to be modified and and they modify it simply by say by by excluding all the consequences of it at genus zero and genus one actually genus one is also a problem so precisely the cases where you have this killing field on the world sheet are a problem so i'm saying this is not what you should do so what you actually should do there is a even in the old literature on special dramedy by stromenger there is a unholomorphic pre-potential and this has very much to do also with the talk of of bill uh so there is an unholomorphic pre-potential and this unholomorphic pre-potential has the property that the triple couplings are not given by this but they are given by the covariant derivatives of that um so it's the covariant derivative let's call this unholomorphic pre-potential s and you see since this contained uh i mean this one is is is more trivial but they they start containing the crystal symbols of the connection so they are not holomorphic so if this going to be holomorphic which of course is true in conformal field theory because you fix three points on the sphere there's no degeneration that is holomorphic but this one will be not holomorphic i mean this would not work if this is holomorphic so take this non-holomorphic so the so the point that i'm making is this thing works if you take this old definition of stromenger put a non-holomorphic pre- potential but then you can ask do you know that object and it turns out you know it it is precisely the propagator of bcov so um whether in bcov series are propagators and what i'm what i call s is actually s so that's why i'm called propagator so i i take the bar of the of the propagator of bcov and then i uh take a factor of the caler factor to make it to give it the right caler weight so this is the caler factor so this has well this had for instance as minus two and then uh i wanted to have i know it has two so i wanted to have it minus two one and this has clearly two two this factor so there i make it to a right weight and this thing is actually fulfills it and now you can think what is then this equation is it really fulfilled or not if you take the different uh definition and it turns out is fulfilled and it's very beautiful it's basically fulfilled by this ramanujan identity for modular form so you take well i mean it's so to say convenient to write this as a matrix of propagators which is um well you change the coordinate but it's basically two s s i s i so there are three types of propagators in bcov and so there's uh this is a change of matrix and then you have immediately a property of this big propagators which actually we have found in the paper with marcos and grim and some people uh in malin vise i guess and this is like c k m n and then it's the propagator again m i propagator n j and you see this is precisely when you when you complex conjugate is precisely what you need is precisely if you take this s as the propagator you're precisely fulfill this equation and and why i'm saying this i'm saying this because from this equation if it's correct then you can uh then you can get also the index so you just calculate it and use the properties of this in terms of the topological numbers and then you get the index so the other properties i mean the property three is easy to establish the property two is not so easy to establish also part of the the thing that the paper is not out and uh but um yeah maybe here should finish