 So you want to solve the topological string on the compact Kalabiao and so I will address a couple of questions which are different to the local case and so on. So we want to solve the topological string on compact Kalabiao 3-fold and I would 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 scalar 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 this is a generating function basically for all these numbers and these numbers are chromo-fitton invariants and we would like to know them. And the t, this t parameter is hidden here, so t is something like the volume of a curve. So you have b field and you have, so it's somehow the complex volume of the curve complexified with 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 this scalar volumes in, so these are the scalar parameters. 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-fibred kalabiao in a more or less closed form in that I expand in the base of the elliptic vibration. So for every fixed class in the base I give you a closed form and so that's the aim of the talk. So of course there's a certain motivation, so of course I said already this solves chroma-fitton theory, solves GV theory 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 theory because these invariants 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 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 depends here on the right spin and the left spin and the mass. And so 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. So for instance first I will sort of don't resolve the right spin, so I write something like just JR like this and then you write a parameter for the left spin and then you write something which measures the energy. But since there are BPS states, this energy is fixed by a charge lattice and this charge lattice turns out to be this Cormor-G lattice. So then this is where the Kappa class comes in. And so at the end you get something which you can write, and this was done by Gopakumar and Waffa. You can write it, well also by Necrosov independently somehow. So you can write it 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 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 really use in the following. And that formula, I mean again it goes back to Narain, Taylor and then was made more general by Gopakumar and Waffa. So this formula looks like this is basically involves this quantities here. So these are BPS indices and they can be in Z. So note that if you sort of resolve this right spin, so maybe resolve J right and then you get this minus one gets a U and then you get on the right hand side things which depend on J left and J right and not just on the genus and on Kappa and these are actually in N and that's sometimes interesting. So this is more like a real counting problem, this is like an index counting problem and 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 formulas that you will see. So then you get here 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 then you get this IG Kappa and then you get GSM over two and then two G minus two and then you get again this E to the MT 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 numbers of GS it has actually poles. That's pretty bizarre. So let me say note the poles. So these come from genus zero because then this is minus one and so note the poles at... So these poles of course will play a role. So if they are rational then we get the poles. And that's already, I mean, this is one of the starting point of this non-perturbative completion that Marcos and his friends are studying. So that is also true for the compact case and for the non-compact case. But now let me say generally that... Let me not arise this formula for a second. So I raised already a 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 two G minus and then write just FT like this. So this is better for this purpose. And you see these things are Kela line bundle. So they live in the Kela line bundle. I mean this comes from special geometry like we heard in the talk of beloved. So this is like a section of L2G minus two. 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 Kela transformation you get a problem. And I will come to this point later but you can remedy for this. You can say okay I take GS and I take a period of that. Now the period transforms in the inverse line bundle. So then this formula certainly makes sense. But then it has any good modular properties because the period is some complicated object and it transforms very complicated in a modern drawing. 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 saying in the local case this has beautiful application but in the global case this thing is very much spoiled this property. So this would be only in the local case. This is a huge difference for the global case. But let me now come to the geometries that I want to consider. And these geometries have very much to do with the talk that you hear by Martin Kohl. So except that I am not looking at elliptic surfaces but I am looking at elliptic three folds. So let me sort of make this small remark. So basically local Kalabiao they are always sort of say of the type you take the total space so M what is my Kalabiao 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 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 Kalabiao 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 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 outlandish when you look at the history of Kalabiao. So for instance if you take the degree 18 constraint in let's say 9 weighted projective space. So maybe we should just put it down. Okay, so then you have this Kalabiao. So this is one of the earliest things. This thing is a degree 18 hypersurface in this projective space, weighted projective space. So it's 6, 9, 6, 1, 1. 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 u and then of course it's already elliptical. 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 3 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 u prime for the second. And then you get a z to the 6. So the z to the 6 you don't see there because this has a singularity. You need to blow up parameter and this is the z to the 6. 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 Calabria, but you can replace this base. So in this case the base is actually p2. But you can certainly do this for any base. So that's very simple. Yeah, so this guy has exactly the same situation and in Martin Colt's talk it has a co-dimension 1 has only codiref fibers e1 and an entire co-dimension there are more singular fibers. But the important point in codimension 1 is only e1 codiref fibers. 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 grammar fitting theory on this base, 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 then I want to give you this algebraic form. This is basically like a b-model expression to see how it connects to the local case. Because if you take this pb0 and this depends on u and m and you set it to zero. 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 this is the spectral curve of a matrix model and it has many nice properties. And you can also see how you achieve this. How do you achieve this? So basically you say xe. This is the class of the fiber. So you take it 8b to the a to the 6. So the a is this, the b is this. And then you send this to zero. But you keep somehow b fixed. And then you get this... So I should say this depends on b. And then it... Well, I mean this is this m, a mass parameter, but then it also depends on u. And the u are basically the same u. Only you have a term up between them. So basically there is u to the 1, 6. So this is an term up. So that's the way you get this local geometry. But now let me change gears a little bit and say the same thing from the point of view of the a-model. So for the a-model, of course I've given you the geometry. It's basically an elliptic vibration over b. And then you can use... You can use Leris spectral sequence in order to get all the topology from the base if the fibers are not so singular. So the question was already asked. So today, similar like in the talk of Martin, we take one... We have one section and only Kodaira I1 fibers in Codim 1. And then everything is actually given by this spectral sequence. So now we use Leris spectral sequence to get the topological data, the topology of m just from b. So that's easy exercises and it works very nicely. So let me introduce some notation. So it doesn't pay to keep this formula. So basically, now we introduce some classes. So we introduce the class of the fiber which I call T and I call this exponential thing Q. And then... So this is the... Now we go to the homology. So what do 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 want to calculate the scalar parameter, you just take what I said before, the b and the omega, the symplectic form and integrate it over this curve class. So this is this volume. And then you have dual devices and these dual devices... I also need name for them. And then you have the class beta. These are classes in the base. So these are in H2BZ. 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 bases. So this just follows from the geometry of this. And here I have classes which I call dk. So maybe I call this also k. So that might be k of them. And that is just the... So here we have, of course, E. So we have a projection map from E to B. Where it's here. And then this I call P. And then this one is, so to say, pulled back from the base. So this is dk. And these are living on the base. So these are these classes. And now you can calculate all the intersection numbers just based on these classes. And the intersection numbers of the base. So that's an 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 if I say c, that ck 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 d... We have, let's say, dE times dk. So this is now intersected with this base classes once. And this is the c1 of the base times ddk. So this dk is this guy. So this is the divisor on the base. And this number I will call 12... Sorry, ak. So this is the definition of what ak is. And then you have intersection of dE, dj, dk. And these 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 base is, of course, zero, because it's a vibration. Okay, so then you can get more. So you can basically get a c2 of m. So now this is in the total space times dE. And you calculate it. And it's... So this is like a Hertz-Pochriemann-Roch formula. And it's like this. And you can also calculate this one on this other types of devices. And it gives 12 ak. That's what I introduced as ak. And then the Euler number is actually minus 60 times c1 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 to exhibit the modularity, I have to make a little change in the... So 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 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 were e to the 2 pi i tk, then this shifts these classes of the base by... So they were before tk, and now they get shifted like this. This tau. 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. I don't like this coupling 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 is that in the new base, if I use these guys, then these guys are actually zero. And by the way, these dk's don't change. So they are just the same. Okay, so now I can start... ...stating the main theorem, and then I try to prove the theorem. And this has very nice, so to say, insight in the wave function transformation of the compact topological string. 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 these base classes that are defined, and you have the string coupling. And now you decide, so to say, to expand it in terms of the base classes, as I said. And you pull out beta equals zero, which is special. It's not Jacobiform, but the other will be Jacobiform. That's why you pull it out. And then the rest is a very systematic expansion. So it's a B in HMZ, and then it's ZB tau GS, and then it's QB. So basically the whole thing that I want to explain to you today is why are these guys Jacobiforms and how you can calculate them and how you can eventually solve the problem. So that's the point. So these guys are the main characters in the rest of the talk. So what are the properties? So I give you a list of properties of this ZB TGS. 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 what is this? So the ZB tau GS are Jacobiforms. In fact, they are Meromorphic Jacobiforms, not weak. So these guys have poles, which the poles come basically from here. So they are Meromorphic Jacobiforms. And these Meromorphic Jacobiforms, 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 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 and the base minus one. So that's the index. And then this is not completely true. It's 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 if you make the S transformation on this Q beta, then you will find that this actually depends on the parity of this curve. So basically it is 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 this is a multiplier system, so the set Bs have to spill out these factors to make this expression invariant. Okay, so then the poles are, I will talk about the poles, but the poles are basically imposed from that. So they are basically over Q and that means they are the torsion point of the elliptic argument. The most important thing of course, which physically, is that you basically identify Z with G with Z, which is the elliptic argument of the weak Jacobiform. That's the most important point in this. So I don't know how much I should say about Jacobiforms, 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 Jacobiforms, 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 lives 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 string coupling. And so they can be holomorphic, meromorphic, 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 quasi-periodicity. And the modularity says, okay, the modularity says that if you say this is this normal transformation CT plus D, so it's the projective transformation, it says that if you 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 Z of gamma, then this thing is basically invariant up to 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 M and then the C comes up, this is the C really from here, and then it's C squared and then it's over C tau plus D and then as I said it's invariant otherwise. So it's just the phase. So this is the modularity and the quasi-periodicity says 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. So it's e to the minus 2 pi M and then lambda tau plus 2 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 comes in. So 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 Weierstrass 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. Well I mean the Jacobi forms have an expansion if you expand them in Z then you get the quasi-modular forms you get the Eisenstein series E2, E4, E6 and so on. So I guess that's all what I want to say about Jacobi forms but now at least at the general Jacobi forms so now let me sort of come to the property 2. So the property 2 is the following the property 2 is that the set B are not arbitrary Jacobi forms but they are Jacobi forms of a very particular type they have the Dedekind-Eder 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 this also is sort of in the title then finally and then the denominator is something which goes from L to 1 to B2 so this is the 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 class of the base and then we have tau Sz 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 so this theorem says basically the weak Jacobi forms are finally generated and freely from by the Eisenstein series E4 the Eisensteins and I call this actually Q and then E6 and then two other arguments so these arguments have just weighed no index and then there are things which have index and that is sometimes called gamma minus phi 1 so 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 1 and index M and then it's the and I said you know Jacobi forms of course the theta functions they have a multiplier system but there's one theta function which goes into itself and that's theta 1 and so this can be used to build a Jacobi form and you have to square it and then you take eta to the 6 and you see this has so to say weight this has weight 1, this has weight 3 then you have minus 2 and then to the question that it was already asked you can sort of have two interesting expansions so let me write this like this X1, so there is a product expansion which is sort of nice and useful actually 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 an expansion in Z and then this expansion coefficients will be actually out of the ring of quasi-modular form so this is spanned by E2, E4 and E6 so here is like this and then it goes with order Z squared sorry, this is order I'm off in the order so this is just Z to the 4 so it's only even so for instance if you want to do open strings of course you knifely would say you have to do 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 GF anyway so 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 torsion point because you see this guy goes with Gs with Z 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 put here this power of this A so it's basically to reproduce this torsion points and I said this is another generator so there's another generator so the other generator is called B and it has weight 0 and index 1 so B people write like 0, 1 and this is, well maybe I should just be brief so it's one half of the elliptic gingers of K3 that I suppose everybody knows that is actually so you take K1, 2, 3, 4 and then you take a ratio of theta functions so this is theta at the argument tau and Z equals 0 that's why you don't take the 1 and then so it's anyway has weight 0 because you have this nice ratio and then the weight comes from this theta so this is KT and then here is the full argument and you square both of them and that's the elliptic these are the other Jacobi theta functions so this is generated by these things 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 0 all 5 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 and the index and the weight of that Don Suggie has given you the index and the weight of this and of course this is classic and so then I just can build an object which has the right index and the weight and it will be let's say Z1 well, let me just so just Z1 is as I said phi1 over eta 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 one is very primitive so it's minus Q over 48 and then 31 Q cubed and then 113 P so this has a weight 12, 16 no index in this case because the formulas work out this gives already infinite number of predictions for higher genus 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's 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 invariance give you integer numbers and sort of this were the things that well you can also write it as Thomas and Thomas invariance and so on so let me come to the property 3 so the property 3 is also very nice so the property 3 says that once the index is negative or smaller than actually smaller equal than 0 so there's a property 3 property 3 and the claim is if if smaller equals 0 then the phi beta are completely fixed fixed by the vanishing by the known vanishing IG beta so these are the BPS indices and basically you know that if G is greater so this happens if G is basically too great it's too big so this happens if there's a little bit that depends on the base but it basically goes with the base degree so if B is bigger than this number then this thing will be 0 and you can show that if the index is negative then this grows very small this room of a weak Jacobi forms and then 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 observation because all these things precisely the things that Soto and these people classified they were rigid in the base so they rely on a negative self in the section curve then this is negative the index 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 non-trivial than the usual local series that we look at for a 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 are not doing this very good but that's why we now work with them so let me say some interesting aspects of the proof so 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 I mean first you have to establish modularity but I don't want to go into the details basically there's a formalism which comes from the derived categories so basically in this case there's a Fourier-Mokai kernel so in homological mirror symmetry if you have an elliptic vibration you have a certain property of the Fourier-Mokai 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 Fourier-Mokai kernel this is the only thing that you have to know so this is is theta so this is the douche-criminant e times m minus m times e and then and then you subtract also c1 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 the a-brains and then you know that the s-transformation on f and anything on f in the a-brains can be written out in terms of this Fourier-Mokai kernel so that's 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 Fourier-Mokai kernel like this and then you have p and for some reason we also tensor it or always with so this has to do with my shift of bases 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 this thing this works basically for 4 fold, 3 fold k3 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 1 minus 1 and then 0, 0, 0 so this is basically the s this is the block which is the s and then the other thing so to say here is a block which has a k and then there is a block which has c, k, j so basically I write this 0, x, k f, 0 sorry I introduce also a t x, t and then I have f, 0 f, t and f, k so this is basically 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 in the same and it's a bit ugly so x, k so this 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 1 and 0, 0, 0, 0 and it has a, j and then a that is a no, 0 then is c1 squared and 0, 0, 0 and then minus a, j and so there's all the section numbers 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 a system of let's say where you have 2 from the fiber 8 from the, 9 from the base then you could also do this by analytic continuation but the mathematicians give you a very nice answer and this answer is basically due to Bridgeland, he does this for case 3 and other people like Yao and company have done it for higher dimensional cases and we can I mean they did it sort of wrong so then we had to do it again but it's anyway standard S transformation is even simpler so the T transformation the T transformation you just tensor by you just tensor f0 just by this line bundle so then so T would be T of f0 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 1, 1, 0, 1 and then other classical intersection numbers 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 these are the these are the space time generators of the space time duality and if you have an object on the Kalaviau it should better be modular I mean this is what string theory tells you and now I want to make the one important point before I finish so the one important point is so if you look at Witten's wave function transformation property I already alluded to this a little bit then so you have so Witten says okay we have this to the G equals 0 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 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 so it's because so let me say a couple of things so basically he sets this like this H bar and then he has an H bar squared and then he has an alpha beta gamma this is originally 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 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 differential with respect to this complex structure okay and then you see and many people have looked at this including BCOV and you say 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 the first one 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 Kehler transformation otherwise this object makes no sense in the beginning I mean because it is not invariant 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 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 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 Gopakumawafa formula so you know already that Z goes to Z plus 1 is actually an 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 to have to shift by tau that's trivial just write it down this expansion you shift this by this amount and then you use that as 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 this is very trivial in some sense but you have to see the following I mean you don't really change the polarization because 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 invariance of it but here we got lucky because we have this monodromies that just change so to say the 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 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 quantity look at this at G equals 0 so 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 sort of absorb this I mean I sort of introduce a new parameter like this well this is the H so it's just a power counting parameter so now I take the derivative of respect B 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 which is not vanishing because this one is a holomorphic derivative so then everybody says I mean like Bershatsky and DiGraph and people who talked about the OSV conjecture they all say forget about the genus 0 part this thing has to be modified and they modify it simply by excluding all the consequences of it at genus 0 and genus 1 actually genus 1 is also a problem so precisely the cases where you have this killing field on the world cheat 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 Strominger there is an unholomorphic pre-potential and this has very much to do also with the talk of 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 so it's the covariant derivative let's call this unholomorphic pre-potential S and you see this one is more trivial but 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 is 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 point that I'm making is this thing works definition of strong engine for the 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 whether in BCOV series are propagators and what I call S is actually S so that's why I called propagator so I take the bar of the propagator of BCOV and then I take a factor of the scalar factor to give it the right scalar weight so this is the scalar factor so this has for instance minus two and then 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 the right weight and this thing is actually fulfills it seeing what is then this equation is it really fulfilled or not if you take the different definition and it turns out it 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 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 is 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 in Malin 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 complex conjugate is precisely what you need it's precisely if you take this S as the propagator you precisely fulfill this equation and why I'm saying this I'm saying this because from this equation if it's correct then you can get also the index so you just calculate it and use the properties of this in terms of topological numbers and then you get the index so the other properties I mean the property 3 is easy to establish the property 2 is not so easy to establish also part of the thing that the paper is not out and but yeah maybe I should finish