 topological string s duality and research. Okay, thank you. And that's why I thank the organizers for putting together this wonderful workshop and for giving me the opportunity to speak here. So it's wonderful to be here and I want to also see lots of you maybe for the first time after the pandemic. So what I want to talk about today is I'm going to spend a good amount of the motivation and telling you how these words relate to the main theme and the workshop they mean, perhaps theory and mirrored symmetry. And I love you all so much and I and their physics. So this is based on so the S-quality part which will come very late in the talk and maybe only in an incomplete way. So this is based on work and work in progress. But then most of the rest of the talk will be based on these papers in reverse chronological order. So this is with a paper with Bob Holland's from Harry's work in Edinburgh and Ivan Puddy from Hamburg from last year. And it was paper before with our friend Sahel who's now in Madrid. And your question on Ivan Puddy from Hamburg as well as papers with mine before then. So of course there is the very vast topic that ties and bids on works of many other people. So let me just mention for this talk. So my so these papers were heavily influenced by a program of the origin of the last take five, six years. And some of the aspects of these two papers were heavily building on work of Cairo Puddy. They're going to talk about different equations and part of it is also based on or very similar to work of Iwaki, Poitin and the king. And of course the resurgent part for what I'm going to talk about there was earlier work by Puddy and Siappa. And there's been a lot of work on non-perperperative logical symmetry by Puddy Hatsuda, Grancy and Mareño, but also earlier work of Hatsuda, Mareño and Riyama. And part of the story ties to all the work of Iwaki and Iwaki and Mareño and Vapa as well as some more later revisiting of the team that was laid out here by almost the same authors except that Puddy is here. But Cairo is then in paper, so Puddy is still like how Puddy is Mareño. And part of the story, especially of that paper also relates to recent work from last year of our Grancy, General Hau and Andy Neitzel. Okay, so this is just a quick overview. So as I mentioned, I'll be spending a lot of time on the motivation and introduction and always spending less and less time as we progress. But I hope to give you some idea of where this is going. Okay, so the motivation and let me also use the motivation in order to tie this up, the theme of the workshop. So we had already seen and took those off. So we had a quick review of mirror symmetry. So let me just require that sort of early indications of mirror symmetry is that there is a mirror symmetry on the hot diamonds of Calabria prefolds. So X and X and check would be a pair of Calabria prefolds, such that their watch numbers could change. So this was an early observation with scanning of Calabria prefolds and the work of the instructor. But it became very interesting when this became a statement not about isolated Calabria prefolds, but of families of Calabria so the picture would be that you really think of a family of people where you identify so the mirror pairs are really mirror fibers of these families. And you would have the same sort of underlying base states, which this is also as the logarithms. But these two families or these two vibrations are very different. So what is the thing? So of course, per fiber, you would still match the hot numbers, but there was something much more powerful to match, namely, there's a local system, or there is a bundle of complex vector bundle over an M together with a path connection that was matched on both sides. And so here, the underlying physics is, of course, a little bit like both McWarner and from a buffer. So there is really a meaning, physical meaning for what's going on in this deformation space and how you can translate this into differential equations. So eventually the math statement is that they have a complex vector bundle together with a spectrum called X and V with a path connection that is being matched on both sides where the meaning of what the complex vector bundles are is very different and also the meaning of the path connection. So on this side, as I'll refer to this side as the V side, so this will be a variational path structure problem together with a flat galvanine connection. And here there will be another flat connection which matches this one, which is the quantum homology. And the reason this is quantum homology is because the power of meritometry is that there is a distinguished trivialization of this vector bundle and then in the distinguished trivialization using distinguished coordinates, you can look at the connection one forms of this flat connection. And part of the data of the connection one forms is what is of fzero, a pre-potential depending on some distinguished local coordinates here, which eventually so there is some generalization of all of these, but let me just give you a flavor of what's going on. So in terms of the local coordinates, this thing is periodic and the expansion coefficients are rational numbers and these are the genomes they are from a written invariance with nothing to do with x checks, but rather with x. So this is normal written. So that was sort of the very convincing in the work of Candela's where this was worked out for the Mericlyntic and then all written invariance for the Quintic. Okay, so there is a higher genus generalization of this so that let me introduce a formal generating function with an additional follow parameter that are called lambda. So there will be a summation over a genera from zero to infinity and then there will be analogous objects to this f. And these will be in the same sense of this one, this formal generating function of genus g normal written invariant. Okay, so where does this come from? So this was part of the data of the flat connection. Where this comes from, you probably need some more construction and this is also what I want to motivate, namely you take the exponential of this. Okay, so what I should mention for the rest of the talk, here really lambda is a formal parameter and the series is as in public, namely you cannot, so it doesn't make sense in lambda it has zero radius of movement. So this object turns out to be a limit of an object that you obtain from some extension of this setup, namely there is a process of geometric quantization. And this is also, I will want to motivate this whole story sitting in much theory, which we have heard a lot about. So there is just a rigorous mathematical for there should be developed a rigorous mathematical framework for a process of geometric quantization. So these the fibers of both of these bundles are complex vector spaces but more over there some tactic and there's some real places inside there. And there's a formal process of geometric quantization that you can apply for these real sub vector spaces of the fiber that these. So in a nutshell what you get, so this is most clearly stated here. So you don't get a bundle of vector spaces but instead you have a bundle of Hilbert spaces and then you also have a flat connection, which identifies neighboring Hilbert spaces over the same logic. And so there is two objects that are called Z couple logical string. So I've had to both sides so one of them on the A side and that's the X which should also match an object on the B side. And both of these are flat sections of this, you know, so that's, that's quite sort of B2B and yes, yes, yes, but this is not, yeah, we have to talk about this later. So this will not be relevant. Okay, so this is one side of the story and then Sokto also told us that there's the homological mass between conjecture and savage. And now let me phrase this this way so you have in that context you're identifying categories associated with X and X check. So hopefully for him she's on X will be identified with Lagrangian's X check. But now we're again in the isolated case. So there is no family here. And it turns out that also from mathematical physics perspective matching the whole categories is too much. So what you need in order to both get back to a larger space and also to look at subcategories is a notion of stability. So there's a bunch of matches stable grant sheets using some distinguished stability condition and these should match special Lagrangian. And so again, there is a normative geometry that you can associate so stable Korean chiefs as well as special Lagrangian and these invariants will be called Donald and Thomas invariants. And merit symmetry will tell you, okay, if you have some generating function of the keyed invariants that you can associate looking at X, then this should match also the normative invariants that you'll get from special Lagrangian on X check, given that this ability condition here is really the same thing that would select the special Lagrangian. Okay, so the input from physics is that the subcategories that match here are attached to what are called DPS objects. So DPS should match DPS objects in many cool theories. So this is just the physical input that will tell you which objects to look for. Okay, and then of course, the question zone here, only one side had a normative interpretation. So this thing here gave rise to the moving theory, but there is no normative information on the D side. Here, there is really no more information on both sides. However, what was understood in particular in the last two decades, there is the stability it leads to while crossing phenomena. So some objects are only stable in certain regions of this M. And there is also a physical object which is supposed to give these partition functions. So there will be some formal partition function of DPS. Both of these should be obtained from the corresponding thing there. And then the question is of course about what is the relation between Z-Romov-Witton and Z-DT. And so there is a relation on this A side. So Z-Romov-Witton of X. It's conjectured to be the same thing as Z-DT of X on the same Calabiao manifold. So this is equal to Maolik, Negrosov-Ungod and Vandam. And there is a physics counterpart that will tell you that the Z-DT-S becomes for compact Calabiao geometries as the black hole is related to the black hole. So more recently, so this is only for some set of DT environments. So all of them more recently. So this is where it's nice to work out from Bridgend. So from Bridgend formulated really from the while crossing problem of DT environments. So NDT environments associated to some abstract setting of Calabiao-T category. So this led to rigorous formulation in terms of the Riemann-Hilbert problem, which would be formally solved. Yes. Not yet. So not for what I've done. So I'm only looking at the vector modelized space. But very, very good question. So it turns out that the dependence on lambda secretly sees the other. Okay, very good. So what I think is asking, so a priori part of the theme of this workshop is hot series, so variational hot structure on this level. Here and variational hot structure, a priori only knows about Dino-Zero, Gorma-Whithin brands. If you construct mathematically this quantization problem, then you also know about biogenes and Gorma-Whithin brands. But a priori variational hot structure does not know about DT environments. So it seems. But yeah, so from this and many other works, it seems that there is data of DT environments, which is really hidden in the dependence on this formal parameter. Okay, so originally, there was a wall crossing of DT and in some cases, the solution of the Riemann-Hilbert problem also gave all genus Gorma-Whithin theory on the same space. So the question is whether there's also the other way around. So is there a way to extract information about DT from Gorma-Whithin, although priori this has nothing obvious to do with Donaldson Thomas environments and that the answer is going to be yes. And it turns out that there is this formal generating function is bad as a series of lambda, but the badness in lambda and how to cure the badness in lambda encodes information about another set of environments in the same manner. So that was the end of motivation. And let me know for the rest of the talk be very specific. And just one example, because this is also the one example where we have all the glory details, but the example is already it's simple enough to write down all the glory details, but it's also generic enough to see what kind of general structure to expect. And as you may have already guessed, the example is going to be the result of 24, which has already appeared in many other books. Okay, so introduction means I'm going to introduce all the players. So as I said, we'll be very specific. So the pair of cannabia 3-folds on the A side, this is going to be the total states of the French rule, lambda over D1. So this is the small resolution of the French rule, I think. And the mirror to this will be given by the following space. So it will be given by an equation. So all these variables are in D2 variables and C, and then there's two variables in D star. Such that they satisfy the following equation. So this is one particular choice of framing of the glory map of there. So before we proceed, then we mentioned maybe two things, and what are some of the work for older work. So this space is a conic vibration, and there is a further thing that we obtain when this conic degenerates. I don't have these coordinates as ideal. There is another curve that you obtain. So that's the space x, y, and C star, where just this equation is. And I should say also fuel, capital fuel, is the exponential of t. And this will be thought of the t here, or the fuel will be considered here as a complex structure, the formation of the underlying gladiol. So this is called the mirror curve. And there is this work of Lagrange-Bankhoff, Demi-Morino, and Vafa back in the days. There is a formal process of thinking of, so again, here, C star, that's where this is detected space. And there is a natural subjective form. So you can also look at the formal quantization problem here. So this is what these guys did, which amounts to acting on a habit space, where these C star variables do not move anymore. And then you can associate to this an operator acting on states of a habit space, and there is a quantization problem. I think back in the day, it was thought that this quantization problem is related to the one leading to the logical swing, but it turns out that it's not, but it is related in other interesting, yet to be fully understood points. So maybe let's just decide everyone. So there's a quantization of a curve associated with this mirror. But for the moment, we'll just take the A side. So this is where we all have normal written, and out of normal written, we'll get amounts and commas of x. But there is interesting links to the other side as well, which I will only mention in time. Okay, so the mirror pair, now let's look at C and normal written, or let's look at point B. So log of C and normal written is for this space. So in general, for the result 24, in general, this will be a classical piece. So this is a polynomial of degree three. And then there's some contribution of constant paths. So this does not depend on the key at all. And this is universal for any polynomial threefold. This is the same expression of every genus just multiplied by the order factor. And then there is a thing which changes with every polynomial threefold, which are called f. So f-field that was determined from physics in early days by Rupakumar and Rafa, using a duality with John Simon's theory. And so Rupakumar, Rafa, Rafa engaged an expression, maybe nothing this form, but essentially that's the end of the question. Before I write down the expression, let me recall some special functions. So there is a fully logarithm with index s and s can probably be any complex number. One way to introduce this is in terms of a power series, where sometimes n is infinity and z is n. So this expression is smaller than one, but you can energy to continue this. So this is also interesting branches. So that's a fully logarithm song. And this features very prominently in this expression, namely you have that first term is a fully logarithm with index s-field. Then you have confirmed, which doesn't have any number. And then all the other terms starting in genus two have the same structure. These are fully logarithms, three minus two g. And Bernoulli numbers are obtained from the generating function. So that's sort of the old orders expression. And then if you want to spell this out, then you look at the rational coefficients. These will be the normal written invariance of this space. So everything is fully determined. And moreover, you see that the series is empathic. Look at the growth of the Bernoulli, sorry, missing here. Even with the vectorial downstairs, this series is still empathic. So it doesn't only make sense of the formal qualities. However, also due to work of Goporkmur and Bapa later, turn out that in general, there's a procedure of resumming or writing down instead of a series in lambda, just taking packaging the same information and writing it down as a series and exponential of i. Goporkmur and Bapa, resummation of this other thing. Quite the sense of dv, lambda and d of the same geometry. And for this geometry, it looks like the following. And the integer that's here, which is one, that's the integer invariant of the same geometry, which is the Goporkmur and Bapa. So now if you can look at this expression, and now this is not formal anymore in lambda, but you can actually think of this as a meromorphic function where lambda becomes complex, has infinitely many poles along the real line, but otherwise it's fine. So it's worth asking what happens in lambda anywhere else in the context that you can take this expression. A physics question would be, okay, is this expression a non-perfective completion of this thing here? And non-perfective completion mathematically, you should think of as an underlying analytic function of your asymptotic series expansion and the answer, which I'll give you in a few minutes is also yes, but then some precise sense. Okay, so let's move on. So one question that I think saw in the Gromov-Witton period at Geno zero is completely determined by the flatness of the variation of hot structure connection that you have. So one thing that I was long looking for is whether there is also some flat connection that extends in the lambda direction, whether there's any differential equation in lambda that this thing satisfies. And it turns out, so maybe there is, but what I was able to find that is in this very simple example, from the asymptotic expansion, there's a nice properties of these 40 logarithms under derivations that you can use. And you find that there is something closer differential equation to make a difference. And that's a very short paper of when theorem generalizing some instruction from a lot of recursion, mainly that this formal series satisfies the following finite difference equation. If you take this and shift feed by lambda by the right hand side, so all this lambda check in positive and negative directions and subtract from this voice. So this is some discrete one plus type of equation. Then what you get on the right hand side is something very simple and just log out one minus. So all of this is okay. So this is sort of the equation satisfied by this asymptotic experience. And now you can ask the game, but just sort of finding another question for an answer that you already have. But the other question, as you know also, as soon as you have the type connection at zero, allows you to continue also to look for some analytics behind this. So maybe okay, so I'll skip some connections of this and other things that maybe let me write and jump in. So that's the difference equation. So from this different equation, as I mentioned, it was a different question that you can ask to give the same solution, but it also gives more. So it turns out there's an actual analytic function in lambda, which also satisfies this difference equation, which is however analytic in lambda. So one way of writing this is in terms of some integral representation. So this is the number four, and the other one is zero. And this particular integral representation is only valid in some regime, but there's other integral that you can give. Let me just put in three units. Okay, so that is sort of the f and t. So that's something I will say for satisfies this difference equation star as well. And the question is, what's the content of this? So there's this particular if you try to write it in this form, is it the same as fgb or is it something different? And it turns out it's something different. So it is partly fgb, which you see people are simple, relatively complicated. But then there is another part which is given in terms of some other partition function that I will not mention further in this talk. But the relevant thing here is that in the other one, all the variables appear. So the lambda appears in terms of one. So this is a part that I skipped. So the important thing is now it's some expression in e to the i lambda, as well as some other expression in e to the i one over lambda, which appears here. And this also this particular form of putting it matches sort of the general lore of the perennial of what it's called the topological string spectral theory. Okay, so the question now is this is a ninth and I think function satisfying the same different equation. Is this now the longer the completion? Or is this really the nice and the object behind from over here? And the answer is again, yes, but only one of infinitely many other possible choices. And this will not possibly be one of the one of the Okay, so I should mention for here, this is also a little sign function. So this also already appeared in a lot of work on the fashion time in series. Also by then, there were some works using this function, also a lot of physics work. And you would think, but it's sort of the new insight was to see that it satisfied the same difference. And they could have gotten it as an analytic solution of the same problem underlying them as a public extension. So let me skip a few things, but at least mentioning one other thing that we have since then understood, which also applies to all work on funds. Namely, once you have identified this as another solution of this equation start, it turns out that there is another, so there is, so F and P also satisfies another kind of difference equation that I've already started. Namely, whereas in FGD everything in P appeared in the exponential, so it was manifested integer periodic, and P is no longer integer. But instead, there is some inhomogeneous term appearing, which you can write in the following way under the dialogue. And here, in the argument, there is again something surprised by one of our lambda, which will be quite important. And the punchline of the talk would work pretty well. So, okay, so you had a different situation, you found some analytic solution. Is this distinguished in any way? And it turns out yes, but there isn't too many others. And this is what you can systematically get from studying Borrell with summation of asymptotic series. So let me just give a two-minute overview. So if you had an asymptotic series in lambda with some positions a n, lambda to the n, starting in zero. So a typical source of these series being asymptotic is these positions go vectorially. So then you have zero radius convergence, which for example, we just put in here the n-tutorial, and it's immediately. From that, what you can do is look at Borrell's transform. More instead of the variable lambda, you introduce a new variable zeta. And just take the same coefficients and divide by n factorials of the shift here. This is the convention. Both providing my n factorial and my n-tutorial. And then you hope that this is a series or this is a power series that has some finite radius convergence. And when you have this, then you also can look at what is the analytic configuration. And then you hope to get some neomorphic function. So you can have this, then you can go back to something that is lambda dependent, which will be called the Borrell sum of i in terms of the original variable, which is then obtained by reversing from by term this factorial, which you obtain in terms of the n-tutorial. And the n-tutorial is naturally done along the real line. But sometimes what you get here is a neomorphic function that has seniorities along the real line. So you can allow yourself any infinite rate in the complex zeta plane. And you'll call this the Borrell sum in the direction of zeta. Okay, so this is something that we did with, so this is n-tutorial. So I'm building on work of Karo-Rolibis and LaShine. So you can take the asymptotic series, which was the normal written generating function, and then explicitly look at what the borrell transform is. And there is lots of work you have to do there, but you get an explicit expression for what the borrell transform is. So let me just, maybe not write all the details, but this is the one that is and there's some other term reversed in that side. Okay, so why I wanted to write this down. So there's a sum and the integers. And here you have manifested something that is neomorphic and zeta. And the folds are at the following values of zeta. So it's for all of the values of y i and t plus i k times m. And here both, so m is any one zero integer and k equals two. Okay, so you have something neomorphic borrell transform. So now you can ask how to get the borrell sum and what is the put energy function behind all this story. If you look at where the singularities are, then you'll find that there's infinitely many singularities along some rays. And these accumulate from both sides to the positive and negative imaginary axes. So what you can do now is you have to avoid to allow all of these infinitely many rays, there's infinitely many singularities. So what you need to do is to find this laplace transform by avoiding these rays. So each one of these is labeled by some integer k. So there's this integer k and the integer m will give you the infinitely many singularities along the way. So what you can do is just define some half line row anywhere in between these. For example, row k will be lying between k and k minus one. And now the question that you can ask is if I do this borrell sum in two different ways, avoiding the singularities on the ray k. So let's say before and after the ray, what is the difference between the two analytic functions that I get. So first of all, what you get if you can write down this is something that is analytically in a whole half thing. And this is what I will end the talk with. So I'm writing delta lk. So this means the difference of the borrell sums of f row lambda k. The difference of the borrell sums along two different rays before and after this singular line. And this is something that we computed and is given by the following. So here what you should recognize that this, and these will be called Stokes factors because they connect the analytic functions which have an overlapping domain of analyticity. But if you want to extend this, then you have overlapping functions. And you can interpret the whole thing that the exponentials of these are really not functions but sections of the line bundle. And these are the transition functions of the line bundle. What I wanted to point out, so this is also the inhomogeneous term of double star. And there is a dual equation. So if you have these analytic functions, you can, this is the s duality part. First of all, if you have these analytic functions, you can expand in one over lambda the whole thing. You get a different asymptotic series which again has its own borrell trend form. And it has its own singularity structure. And it has its own Stokes factors. And the Stokes factors of the dual, so that's obtained asymptotic series in one over lambda and the dependence on k becomes k over lambda. The Stokes factors are related to the inhomogeneous terms of the first difference. It seems that the two different situations that you get to know about the Stokes factors of the corresponding other asymptotic expansion. But this is not yet s duality. So s duality is something that is still in progress, namely for duality or some full modular structure. So here you only have the inversion. For full modular structure, you also want some object that is in brand under integer shifts. And this is something that you can obtain by taking products of these analytic functions over the whole half thing. And maybe let's just mention this. So then what you get from all of these products of z is related to what is called an elliptic gamma function. It was studied by Federer and Varshanko and many other people. And this in particular is that the logical spring coupling is not only a formal parameter and not only a complex parameter, but also has modular structure or some modality that has yet to be understood. Questions anyone? This whole thing. Yeah. I see what you're saying. I think the Stokes automorphism, if you is, if you look at the exponentials of this, so we interpreted this as a different anomaly. We got then the difference or the ratio of z rho k plus one divided by z rho k, then given by the exponential of this. So we interpret this as transition functions in this line boundary. But I think you should also be possible to write this down as a morphism acting on these objects. So it's much clearer if you don't work with the logical spring, but then if you do the resurgence program, what we did with Lotta and Ivan, that relates to the quantization of the miracle, because then out of the F, what you're saying is what is called the world's symbols of the exact of your TV of the miracle. And there on the world symbol, you know, so again, combination of these factors will be the Stokes automorphism acting on the words. Yes, yes, I think so. So what what you're pointing out, so there is, so I mentioned that the beginning, okay, there's a complex vector bundle that's over m, and there's a connection, but what has been so sorry, the connection here is not a single one, but there's a whole family. So there's a family depending on the complex parameter safer. So there's a back to the star connection, where part is data, whether it was one, but later it was understood that this data can be extended to one, and what you want to then look at the twister, how many way extend this, and then this connection, this extended connection for where you lift the vector bundle over m times t one, this has Stokes phenomena at zero and so zero and infinity are irregular similar points of this. And sort of this is the underlying object, which ties in nicely with a lot of things. And at the end of the day, I think that's what phenomena and this connection should be related to the word. But this is only what I should be as far as I know, those. So I think that this particular one is, so I think there's work of our graphene at all, where there's with entrance times theory, where some of these can be contained with limits of some of the thick blanket, but it's a very good question. So I think, yes, but not that I know. So again, so what I discussed is only these, sorry, so here, the normative interpretation of this really from things that we're asking for the one that was here was also the Linnerton Thomas and Brian corresponding to a particular sheet, who was sent to a charge is this, I think, so this interpretive as the center of charge corresponding to some methods and amendment written by Linnerton Thomas and Brian so all the jumps have a normative geometry and stages and but if I miss it correctly, so what you're asking is all the other things if you look at the asymptotic functions for all the infinity many here and the ethnic public dimension in Namba have the same, the same thanks also for asking there is a limiting whereas some along the imaginary axis, which gives you on the nose just a little bit more about without and so all of these additional corrections are just some of these but there is another interesting thing and if you look at the analytic thing and expanded in one of a number and it looks like it wants to be an interpretive so it's one piece is again from with the theory of X, but there's additional pieces which took on their structure look like a normative geometry of open I don't know so then the question is what is the S rule of logical things being on the same level okay okay yes yes yes yes great question so that's that's what we looked at with a lot of column so what I mentioned earlier is from the mere construction for and this is only for long-compact and I cannot be able to get the notion of America and the quantization of America leads to a difference operator however the difference operators in the variables of the curve the difference operator than I have is on the modular space the difference operator that I wrote can be again if you're ready to be exponential can be written as the following so acting on something so there is some object that you can obtain from the topological string and then the equation becomes this and this again looks like a quantization of a c star an equation even c star and square and what I think would be the general story is that you have a very different operator and on the curve and part of this difference operator structure and then we translate it under the logic or in other words the curve that finds the quantum mechanical problem where the moduli are fixed or the moduli are the analogs of energy levels and complex energy levels and then I think what's going on but yeah so there's lots of indication levels is that the difference operator is then a quantization of the same problem if you work in the right variables so if you get rid of the curve dependence as you will do for the harmonic constant and by going for sine and cosine and then you only have variables which are what are these called action angle variables so then the quantization of the infected space becomes the quantization of another infected space which is better than that particular problem in other words also the object that you get here you can think of these as the analog the higher genus analogs of periods so then the object from the curve which satisfies the difference which are analogs of arbitrary maps and this is the analog of translating what the arbitrary maps satisfy or what the period or what the higher genus deformation of the period