 So it's a great pleasure to have the opportunity to be here to talk about the subject, so thank you very much the organizers. Okay, so my title, it should have said introduction to, so wall crossing in physics and some application to differential equations. So in the last few years, maybe the last five years or so, there's been a big resurgence of interest in this problem of wall crossing. I'll tell you what it is in just a moment, but the funny thing is when you first hear about it, this wall crossing sounds like something that you would really rather not think about. It's kind of an annoying detail that's messing up whatever you were trying to do. You had some quantity which you were saying you were trying to count something, you were trying to count volume minimizing cycles in Calabria three-folds, or in physics you were trying to count some supersymmetric particles, and you thought it was going to be invariant. You thought it wasn't going to depend on parameters at all. And then when you look closely, when you try to get all the details right, you discover that there's a little problem. The thing is not quite invariant. When you vary the parameters of your system, the thing is kind of piecewise constant, but there's some critical moments where it might jump. So rather than having a thing that's just globally invariant, you wind up with a thing that has this kind of piecewise constant dependence on parameters. And so it sounds like something that you would rather ignore. You hope that someone else will come up with a better invariant that doesn't have this problem, but it seems that that's kind of the wrong point of view. So this phenomenon, this phenomenon of the jumping of invariants, which shows up in a lot of contexts, it turned out to have a kind of rich sort of inner structure to it, which turned out to be connected to a lot of subjects. So it showed up in quantum field theory, but on the other hand it somehow shows up also in neuro-symmetry, in hyper-killer geometry, in interval systems, cluster algebras, and important for us today, the theory of differential equations. In Donaldson theory, there's also the general phenomenon of wall-crossing. But here I'm talking about a particular kind of wall-crossing. So here I'm talking about this particular BPS wall-crossing, which is somehow not the same as the wall-crossing in Donaldson theory. At least I haven't heard of any connection between the two, so that's why I don't put that here. Okay, so my main goal today is to talk about some relatively simple examples of the wall-crossing phenomenon, the ones that are somehow connected to the Hitchin system, and to explain how they're related in particular to differential equations. So this is part of, you know, to the extent that I say anything original in this talk, it's part of a long-running joint work with David Egato and Greg Moore. So that'll be ultimately a pure mathematics story, but let me start by describing what the problem of wall-crossing is actually from the point of view of physics. So suppose we have some physical system, and suppose that it's something which if you look at it from a long way away, it would look like just a particle. So maybe it's like an atomic nucleus, but when you look a long way away, you don't see the inner structure, you just see a particle. So one of the basic questions you might ask about that thing is, you know, is it stable or unstable? Will it live forever, or will it eventually decay into some more basic constituents? Well, the answer might depend on the parameters of the system. So for example, you could imagine taking this particle and putting it in a box full of other particles, and then you could change the density of other particles in that box. Or if you're more ambitious, you know, if you're a high-energy theorist, you might imagine that you can even vary the kind of fundamental constants of the universe. You could change the cork masses or the coupling of all the interactions, and maybe the question of whether some particular atomic nucleus is stable will depend on the answer to that question. So as you vary the parameters, there might be some locus in the parameter space where the particle changes. It changes from being stable to unstable or the other way. And so those loci, evidently, they're a real co-dimension one in the whole parameter space, and so let me call them the walls. So on one side of the wall, the thing's stable. Well, no, there are some things that are really absolutely stable. For example, if you have the lightest particle that's carrying a given charge, that particle really cannot decay even in infinite time. Yeah, for example, an electron. But that's actually a good example. So the electron is stable, but if we would change the parameters so that some other particle carrying the same charge as the electron became lighter than the electron, then it would become possible for the electron to decay into that other particle. So that's a quite good example of this kind of phenomenon. And we'll be studying some very special field theories for which there really is a notion of absolutely stable particle. So in general, we might be interested in a... Is this yes or no function? It's not a real value function. Of course, yes, no function, everywhere there are crosses. And say yes or no. The point is that it's not a real value function. This is not an example. The real wall crossing, I will have a function, a purely real value function, and this jumps. If it functions a yes, no, it's always like that. Well, take the function to be one if the... No, but no, but this is cheating. I thought it would be fine. No, it's not because of actual worship, it's because of this object. It's not real value function, I mean just... Our language has no language. It's cheating, yeah, it's just... Your language has wall crossing, what's the physical system? You call it yes or no, you call it one, zero. Physical system, mathematical physical system, not actual physical system, so... No, no, no, no, this is notation. It's just method of notation. We call this number yes, you call it one, or no, call it zero. But this notation is just language. But there are coming some questions where the answer changes. Well, they gave a short example. That was one short example. Yesterday in Greece, what do you call... Black or white, what do you call it? I don't know what the problem is. Let's see this example. Let's see this example. Yeah, maybe the sort of mathematical instantiation of this you'll like better. But so in general, we may have a single physical system. We might have many different species of particle, and we'd like to study all of them simultaneously. And so correspondingly, there might be many different walls in our parameter space across which something happens. So here one particle decays, here another particle decays, and so on. And so the kind of question you'd like to understand is, first of all, where in your parameter space are the walls? And second, you know, imagine I look at the set of all sort of absolutely stable particles, and you could ask how that set changes when you move from one side of the wall to the other side. So concretely, imagine that you knew all the stable particles on one side of a wall, and you tried to determine what are going to be all the stable particles on the other side. That's the question we'd like to understand. Now today, this problem is really too hard to solve exactly. Well, can you give us a really convincing example? Because look at this. It's like what you call any man or what is an ape, you know? And the reason why people say there's no... No, it's not biology. It's here a notion of limit. It's kind of a question about large volume limit. No, but it's yes or no. It's not number. No, no, no. It's symmetrical approximation. Go to infinity. No, no, it's a number. You can say it's an invariant. I'm going to make a number momentarily. I'll make a number momentarily and then tell me if that number is satisfactory. So the problem that, as I formulated it so far, is actually... it's too hard to solve exactly for most sort of physically realistic systems. On the other hand, there are some kind of toy examples where it has been possible to do much better. And those are systems where the underlying laws of nature are somehow very special. They're so-called supersymmetric. And these supersymmetric models, superficially you first think they're more complicated. You take your particles and add, roughly speaking, one new particle for every old particle. They're sort of superficially more complicated but actually they turn out to be much, much more nicely behaved than a sort of random arbitrary system. So you should think of the differences somehow like the difference between real analysis and complex analysis. So in complex analysis, you throw in... you sort of double everything, you make all your numbers complex, but that actually makes the story much, much simpler. And in these kind of systems, there's a kind of privileged class of absolutely stable particles, the so-called BPS particles. And by restricting our attention to those, we get a problem that we have a chance of actually solving. So a little more precisely, suppose I have any quantum system, say a quantum field theory that describes the physics in our universe or some other fictional universe. So then this quantum system should have a Hilbert space. And in particular, we could focus on the Hilbert space just the subspace inside that Hilbert space consisting of just the one particle states. And so it's graded by some lattice, the lattice of global charges in the theory. So you can think of, for example, electromagnetism, then this gamma will be literally a rank two lattice just parameterizing the electric and magnetic charge of any given particle. So we have a Hilbert space and it's divided up into these pieces. And so for a sort of random physical system, each of these things would be a representation of just the Poincare algebra, just the isometries of spacetime. If it's super symmetric, it's a representation of something a little bigger, what we call super Poincare algebra. And then what one defines, one defines an integer number. So for every one of these Hilbert spaces, one defines an integer number, which you can think of roughly as being just a graded version of the dimension. But a little more precisely, it's not exactly the dimension, but rather you have this H gamma, you decompose it under the super Poincare algebra and you keep track of the number of times that some special small representations occur. Roughly it's the dimension, but it's not precisely the dimension, it's something a little fancier. So then we can formulate a question. So now the question becomes, so all the particles which sit in these small representations, they have in particular the property of being absolutely stable. And so now you can ask, suppose now that I vary the parameters of my super symmetric system, how does this collection of integers, how does this collection of integers change? Okay. So that's the kind of problem that arises in physics that we would like to be able to solve. So with exactly the same percentage of mathematical language there, you will see. Yeah, I think maybe you won't be satisfied until I show you the actual mathematical example. So I'll come on to that. So this problem, so the first cases where this problem was solved were systems in just two space-time dimensions. They were studied in like the early 1990s by Shikodi and Vafa. And in that case they gave sort of the most perfect possible solution of the problem. They had a formula which tells you if you know all the particles on one side, what will be the particles on the other side? In four dimensions, so which is somehow the more physically interesting case, the problem also is much, much harder. So it was solved in a few isolated examples. In particular, in 1995, Simon and Witten solved this problem in the case of one very specific gauge theory, this so-called pure N equals 2 Super Young Mills theory. And what they found was that in this theory, so in this theory you have a parameter space which is just a one-dimensional, one complex dimensional space. There was one wall. Inside of the wall there were exactly two distinct stable particles. Outside of the wall there was this infinite family of them. And so after that further progress was made just in some specific examples. But to solve the problem in complete generality seemed just hopelessly out of reach. So fortunately, the sort of inspiration for all the new progress in this subject is that mathematicians kind of came to the rescue here. So they formulated a theory of generalized Donaldson-Thomas invariants. They're defined in a totally geometric way. And in many particular physical systems you can formulate their generalized Donaldson-Thomas invariants and they seem to be exactly these omega gamma. So exactly the quantities which we wanted to study in physics, the same quantities were studied by mathematicians, particularly in the work of Kinsavich-Soybleman and Joyce and Song, they found a totally general solution to this problem of wall crossing. Okay, so now I want to switch essentially from physics to mathematics. So I want to show you a few actual concrete examples of this wall crossing phenomenon and of the wall crossing formula that Kinsavich and Soylbleman wrote down. So let's fix some data. So let's fix a Lie algebra of ADE type and let's fix a compact Riemann surface that we draw there with a bunch of mark points on it. So for every such choice, for every choice of an ADE Lie algebra and a curve with mark points, there is actually a corresponding four-dimensional quantum field theory and in that particular quantum field theory, the questions which I so far described to you in a kind of airy physical way, the questions, particularly questions about these quantities omega of gamma become now completely concrete and geometric questions. So okay, and let me specialize a little further. Let me specialize to the case where the Lie algebra G is A1, in other words, SU2. So we're going to do a construction and the construction is going to involve meromorphic quadratic differentials on the curve. So what's a meromorphic quadratic differential? It's just a thing which, so thing phi 2 of z, which locally in terms of some local coordinate z on the curve is represented just as f of z d z squared. So some meromorphic function times d z squared. So if I fix the singularities, so they're meromorphic, they're going to have singularities exactly at the points z i that I marked. And if I fix the order of the singularity at each of those marked points, then there's some finite-dimensional space of these meromorphic quadratic differentials. And that's going to be my parameter space. Okay, so what are we going to do with these quadratic differentials? Well, okay, so given one of these quadratic differentials and given also a phase, there's an induced foliation of the curve. So what is that foliation? So I take, it's easiest to describe it just to describe what it looks like in local coordinates. So we had our thing which was f of z d z squared. Locally, at least away from the zeros of this quadratic differential, sorry, away from the zeros and the poles, I could find a local coordinate w where I just absorbed this f of z into the definition of w. So w is like the integral of the square root of f of z. So find a local coordinate in which your quadratic differential just looks like d w squared. And then in that local coordinate, it's easy to say what this foliation is. You just take the foliation by straight lines. Well, with straight lines, there's a choice of what angle to take the straight lines. So that's this parameter theta. So the leaves will be just straight lines whose inclination is theta in the w coordinate. Okay. The theta or theta over 2, because theta plus, or if you take theta to theta plus pi, it's... I think I want really, I think I want really theta. So indeed, so very good. So indeed, the foliation, I guess the way that I've written it here yeah, sorry, the way that I've written it here is slightly ambiguous because I could always change w to w, w to minus w without changing this. So yeah, we could write it a little more precisely to deal even with that ambiguity. But for now, let me just say this. All right. So what does this foliation look like? Well, as I said, away from the zeros and the poles, it's just a regular foliation. At the zeros and the poles of this quadratic differential, the foliation has some singularities. So let me describe, for example, what happens at a double pole, which is somehow the most basic case to understand. At a double pole, if you just work out the local behavior of the trajectories around that pole, what you find is that the trajectories tend to spiral in. And the exact way that they spiral in depends on the relationship between this parameter m, which is the residue, and the parameter theta, which was defining the foliation. But anyway, except in some exceptional case, the trajectories are always kind of spiraling into these poles. And for a sort of technical convenience from now on, I'm always going to assume that my quadratic differentials have at least one pole, which is double or maybe higher order. If it's higher order, it still attracts the trajectories just in a slightly different way. So what's useful about that is that it kind of controls the global behavior of this foliation. So if you think of the way I define this foliation, in general, you might think that the trajectories behave in a very crazy way, that they move like ergodically around the surface. And in the case where we have these poles, that won't happen. As long as this parameter theta is generic, what will happen is that every trajectory has at least one end on a pole, so it's eventually attracted roughly when it comes close enough to one of these poles, it'll be attracted and fall in. And so for that, we use some fortunately, Strebel studied in great detail the trajectories of quadratic differentials. And from his results, you get this nice fact. So that's what happens at the poles. On the other hand, you also have singularities at the zeros of the quadratic differential. And at the simple zeros, if you work out the local behavior around a simple zero, you find a singularity which is just a three pronged singularity. So I'll mark the simple zero as always with this cross. And so let me assume for now, in fact actually, I think throughout this talk, let me assume that the zeros are only simple zeros. So for the generic quadratic differential, you only have simple zeros. And so then we have, for every simple zero, we have these three distinguished trajectories, the ones which go right into the zero. And so I'm going to focus in the construction, I'm going to tell you, we actually essentially don't use the generic trajectories at all. We only use the trajectories that are coming out of the zeros. And so for every zero, we have these three things coming out. Let's take the union of all those trajectories. I'll call that the spectral network. In this case, it also has another name. It's called the critical graph of the quadratic differential. So I'll write it w of phi two and theta. Okay. And so for a generic value of theta, what happens? So we have these zeros. And each zero is emitting three trajectories. And each of those trajectories travels around the surface in some complicated way and eventually ends up on one of the poles. So the picture you get is something like this. The surface gets divided up into these cells. So at least this is what happens for a generic value of the phase. Okay. So now what we're going to want to understand is how does this thing vary as we vary the parameters, as we vary the quadratic differential and as we vary this phase parameter, which we're calling theta. So, well, from the way it's constructed, it's reasonably clear that for a small variation of phi two and theta, this network is just going to change by an isotope. But as we make a larger variation, we may reach some critical place but the topology of the network actually suddenly jumps. So let's see a picture of that. So this is kind of the simplest example of such a jump. So here I'm taking the curve to be Cp1 and I'm taking my quadratic differential. So here's a quadratic differential which has a singularity at infinity but nowhere else. And it has two zeros, one at plus one and one at minus one. And what you see is that as we vary this parameter theta, so now I'm showing you holding phi two fixed and just varying the parameter theta. As you vary that parameter, as you vary that parameter, this picture, this critical graph or spectral network just varies by isotope except at one critical moment. So here there's one critical moment where, right there, that topology suddenly changes. So we had two trajectories that were going out here, they suddenly jumped and then we had two trajectories here instead. So and if you look at the moment where the jump happens, if you look at the moment where the jump happens, what you see is that the jump happens exactly at the moment when something non-generic happens is when you have a trajectory which comes out of one zero and goes just straight into the other zero. So that's a kind of special trajectory that we must at this critical value of the phase. So what are these special trajectories? So I told you that for a generic value of the phase, every trajectory has at least one end on a pole. It travels, it might start at a zero, but it travels around the surface and eventually ends up on a pole. But that's only the generic statement. So for some special value of the phase, it might happen that you have a trajectory which has both ends on a zero. So I'm calling that a special trajectory and those things come in two flavors. You could have either, we call a saddle connection which is what we saw in the last picture. Just one which travels from one, begins at one zero and ends on another one. Or you could have a closed trajectory. So that's a trajectory which begins at a zero, travels around and comes back to the same zero. Okay. So the moments when this critical graph suddenly jumps are exactly the moments when one of these special trajectories appears. Okay, so now we can formulate a question. So let's consider the following question. Suppose I now fix my quadratic differential and I allow this phase parameter to vary. Then you could ask, well, the sort of course question you could ask is, how many special trajectories will we see? So I vary theta just from zero to pi. How many times will this picture jump? How many times will I see a special trajectory appear? Okay. Well, okay, so let's start thinking about this question. So a first very preliminary comment about it is that these special trajectories, they can't occur too much. They can only occur at most at counterably many phases. Well, why is that? It's because, so as I'll explain in a second, they only have counterably many possible topologies. And the parameter theta where the thing occurs is actually determined by its topology. Well, in the following sense. I'm about to say what topology means. But yeah, roughly it means exactly the position on the surface. But let me measure that a little more precisely. So we have this quadratic differential. Given a quadratic differential, in particular, there's a sort of canonical double cover of the curve attached to it. Namely, you just take the two square roots of the quadratic differential. So as you move around on the curve, phi 2 of z, you look at the square roots of phi 2 of z. Because the thing was a quadratic differential, its square roots are naturally valued in the cotangent bundle. And so this spectral cover is living inside the cotangent bundle of the curve. So there's a sort of obvious 2 to 1 projection that just forgets the square root. And that covering is, of course, branched at the zeroes of phi 2. Those are the places where the two square roots come together. So OK, so we have this branched covering. Well, really over the complement of the poles. Never mind the poles for a second. The important thing is that this thing is branched at the zeroes. So now let's define our lattice gamma to be the first homology of the spectral curve. So before, in the abstract physics part of the talk, I defined some lattice, which was going to be the lattice of charges. So the thing which kept track of the charges of all the particles in the theory. In this particular theory, the thing that keeps track of the charges is just the first homology of this spectral curve. And now in particular, each one of these special trajectories, canonically, can be lifted to a one cycle on the spectral cover. So here's the picture of it for a saddle connection. So if I have a saddle connection here, it's something which exactly runs between two zeroes, which are two branch points. So if I just take the union of its two lifts up to the cover, that becomes actually a closed path on the cover. And there's a way of even determining the orientation. So we get a corresponding homology class. So that homology class is what I'm going to say during the topology of my special trajectory. OK. So then to every one of these things, I have an associated homology class or what I'll call a charge. And now, OK. So now I come back to this question of at what phase will the thing appear? Well, once we have a homology class on the cover, we define the most obvious thing to consider. We have a cover which is sitting inside of the cotangent bundle. The cotangent bundle has this tautological or levil one form. And you could just take the integral of that one form around the one cycle. Well, what is this one form? Concretely, when you evaluate it on the spectral cover, that one form literally is the square root of the quadratic differential. So for paths which lie on this spectral cover, you would think of this integral as you're integrating the square root of the quadratic differential. Or said yet another way, you're integrating, you're looking at the variation of this coordinate, which I called w. And so in particular, if this gamma is not a random homology class, but if it comes as the charge of one of these special trajectories, when I lift that spectral trajectory up and compute this integral, what I get is guaranteed to be it's not a random number, but rather it's valued in its phase as theta. So the periods for a special trajectory to occur at a particular phase, its period has to have exactly that phase. And now since there's only countably many homology classes, this equation can be solved only for countably many phases. And also, if you know what charge you're looking for, then the corresponding phase is just determined. So we know sort of where to look for the special trajectory with any particular topology. So now I can formulate finally what exactly are the sort of Donaldson-Thomas invariants or the BPS counting invariants in this particular problem. Namely, we're going to do the following thing. So we fix a particular quadratic differential, phi 2, and we fix a homology class in the spectral cover called a gamma. And then we ask, well, we form what first looks like a kind of funny combination. So we take the number of saddle connections which appear with that particular homology class. And then we subtract two times the number of closed loops which occur. Actually, I wrote closed-loop pairs here. That's because as it turns out, these closed loops... Where's the chalk? These closed loops, when they occur, they generally don't occur in isolation. So rather, if you have one closed loop in your foliation, then if you ask, what are the neighboring leaves of the foliation like, it turns out that they also have to be closed loops and so on until you reach a second branch point. So you get a picture like this. In other words, they sweep out just an annulus. So in particular, you have these two closed loops on the boundary of the annulus. And I'm calling that a closed-loop pair. So this whole structure, when it occurs, we'll count that with this funny number. We'll count that as minus 2. So we're counting these saddle connections in closed trajectories, but we're counting them in this very particular way. And these coefficients are really crucial. So the whole wall-crossing story that I'm going to tell, it's essential that you count the things in exactly this way. Okay. Excuse me, what do you call the number of saddle connections? The number of saddle connections, so literally, right at this critical phase. So we've got this gamma here, and so we take theta to be the argument of z-gamma, and then you look at the foliation at exactly that moment. And so, yeah, what you would, of course, usually expect is that you get either 0 or 1. And in every example I've ever actually looked at, you always get either 0 or 1. But in principle, so I asked the quadratic differential people whether it's possible to have more than one, you know, if you have a complicated enough surface, and they told me, yeah, they think there are examples. So where you have two things that occur simultaneously and they're actually in the same homology class. So in that case, you know, if there are really two of them that occur simultaneously, then I would count that as 2. But in general, you know, in every example I'll show you, the number will always be just 0 or 1, so you're free to think it's 0 if there's none and 1 if there is one. Okay. Okay, great. So that's, so now these are our invariants. So let's see an example. So let's take again the curve to be just Cp1. And now we're studying quadratic differential with a slightly worse singularity infinity. Before I took Z squared minus 1, now I'm taking Z cubed minus Z. So it has three zeros in here. Okay, here's one, bang, one saddle connection, which connecting the leftmost two zeros. And then in a moment we're going to see a second saddle connection appear. Bang, there's the second one. So in this case, as we vary this parameter theta from 0 to pi, we found just exactly two saddle connections. So I could call the lift of this cycle, say gamma 1, and the lift of this other cycle, say gamma 2. So those are two little loops on the spectral cover. And so our invariants are omega of gamma 1 equals 1, omega of gamma 2 equals 1. And in fact, if I vary theta from pi to 2 pi, I'll see the same picture again repeated, but then if you keep track of the orientations carefully, you get minus gamma 1 and minus gamma 2. So according to the definition of these invariants, we have plus or minus gamma 1 equals 1, omega of plus or minus gamma 2 equals 1, and all the other invariants are just equal to 0. OK, that's simple enough. OK, so now this is a very similar picture where I've changed my parameters just a little bit. So before I had z cubed minus z, now we've changed the quadratic differential to be just z cubed minus 1. So we've moved a little bit in the parameter space of quadratic differentials. And now we see, instead of just 2, we see three of these saddle connections. So there's a corresponding family of spectral curves. We vary the quadratic differential correspondingly. We vary the spectral curve, moves in a family. And if you compare the homologies, so this cycle here is the cycle that I used to call gamma 2. This one is the new one, which didn't occur before. That one has charged gamma 1 plus gamma 2, and then finally there's this one, which has charged gamma 1. So after we've changed our parameters a little bit, now we have three of these saddle connections instead of two. And so concretely, one of these invariants has jumped. The invariant which has jumped is this third one, omega of gamma 1 plus gamma 2, which used to be 0, and now it's 1. So that's the picture. So what did we just see? So as we deform the quadratic differential from this one into this one, this is what I just said, the invariant changed from 0 to 1. And so this is sort of the most basic example of this wall crossing phenomenon. So our parameter space, and it was our parameter space is the space of meromorphic quadratic differentials with some fixed singular behavior. And on one side we had this saddle connection, on the other side we don't have it. OK. OK, now here's a slightly fancier example. So here again I'm taking the curve to be Cp1, but now we're studying quadratic differentials which have singularities both at 0 and at infinity, a singularity of order 3 in both places. And so I represented it as a cylinder, so 0 and infinity play kind of symmetrical roles. So here in this picture, once again there are two saddle connections as theta varies from 0 to pi. Unfortunately they occur at the very beginning and very end of this animation. So here are the two branch points, here at this moment you see two trajectories coming together, back right there. The second one will be at the end. So just trajectories which are going around the cylinder in both directions, one that goes around this way and one that goes around the other way. And so correspondingly here we have just two invariants, two nonzero invariants, they're equal to 1, all the rest are equal to 0. But now suppose I deform that one, so now I'm again on the same parameter space I've just deformed this quadratic differential just by adding a constant term. And after making that deformation, well maybe you already saw it, so the picture has become much, much more complicated. So now as I vary the phase I encounter actually infinitely many saddle connections, infinitely many saddle connections which are winding more and more times around this cylinder. So let's see it again. So here's the first one, the second one, the third one, and so on, there's an infinite tower of them winding more and more times, and then right at the very middle, right at the very middle of that picture there's a critical moment where you actually have closed trajectories, where there's a trajectory that just begins at say this zero, winds around the cylinder and comes back to the same zero, and similarly down here. So right at that moment, right at that moment we have these closed trajectories and so correspondingly here the invariants have become seemingly much, much more complicated. So here we have this infinite collection of nonzero invariants, so for any value of n this is equal to 1 corresponding to a saddle connection. And then we have also this one critical moment where we had the closed trajectories and so there's 1 omega which actually equals minus 2 according to this definition that I wrote down. And again all the others are equal to zero. So here we had just two nonzero invariants, here we had this infinite collection of nonzero invariants. So now the sort of amazing thing, for us it was amazing, at least when we first learned about it, is that these wall crossing phenomena that we just observed are completely determined by this single kind of universal formula which was written down, I mean in the form that I'm going to describe it it was written down by Kinsavich Soibelman, I think an analogous thing was written down also by Joyce and Song. So the claim is that just knowing the spectrum of trajectories on one side, in other words just knowing about these two trajectories here using this wall crossing formula we can actually determine what will be this complicated infinite spectrum of trajectories on the other side. So let me now tell you how this formula works. And to formulate this formula you need an ingredient which at first looks kind of strange. So far I've been talking about quadratic differentials and now we're going to introduce something which at first seems to have nothing to do with the quadratic differentials. So let's consider just a torus. So this is just C star to the end but I'll write it as the hum from the lattice gamma in the C star. So this is a torus who's sort of... What is your gamma here? Oh gamma was this lattice, it was the first homology of the spectral curve. So another way of thinking of this is that it's representations of... well okay I'll come to it later. Gamma is the first homology of the spectral curve. So this torus, it's C star to the end but the way we presented here it's sort of canonical coordinate functions are labeled by choices of gamma. So for every gamma we get a kind of Fourier mode along this torus. A C star valued function. You just get by evaluation. Okay so we're supposed to think about this torus and now Kinsavich and Soebelman invite you to consider a certain automorphism of this torus. So to give you the automorphism I just have to tell you what happens to the coordinates of the torus. So for each gamma we'll define this automorphism called K gamma and the way this automorphism acts it takes a coordinate X, well let me call it X gamma prime. It maps the coordinate X gamma prime to again X gamma prime but multiplied by some factor. The sigma of gamma is a kind of tricky sign that enters into this story so it would be better if we ignore it for now. Just imagine that... In the initial period of the group structure no no we're not using the group structure no that's right this is a very non-linear automorphism so it's an automorphism of the torus just considered as a complex... It's a straight bi-rational transformation Oh automorphism, no no no not an automorphism of the group no no that's right a bi-rational map of the torus to itself just given by this formula so here the sigma of gamma is some tricky sign and this pairing here is just the intersection pairing on the homology of the torus. Okay so... Can Savich and Soilman say consider that automorphism? Okay and now what do we do? Okay so now we're going to draw a picture so each curve on this picture corresponds to one of the saddle connections or closed trajectories that that I showed you in the pictures before so concretely so here is the axis where we're varying the quadratic differential so let me use b for the space of these quadratic differentials, the parameter space and theta, well theta is the parameter theta that we had before and then for some quadratic differential we have a saddle connection that appears at some particular theta then I put a dot here so that's what this picture is the saddle connections or closed trajectories occur so I just plot the ones for which omega is not equal to zero so just the ones for which you actually have one of these things and now the fact that these lines are kind of branching like this reflects exactly this phenomenon of wall crossing so the moment when you the moment when you reach the wall is actually well it's a consequence of this formula or you can also see it directly the places where you reach the walls are exactly the moments when the phases of two objects become aligned so here we have two objects say these could be the guys with charge gamma 1 and gamma 2 and on the other side of the wall I have gamma 1 gamma 2 and also a new one has been born with charge say gamma 1 plus gamma 2 so this picture typically looks something like this something like this okay so to draw this picture and now the wall crossing formula says now to do the following thing so look at this picture and consider just any closed path on the picture any contractable closed path and then as you go around this path and every time you cross one of these one of these walls in other words every time you meet some of the saddle connections or closed trajectories you take a factor of this automorphism k gamma raised to the power omega of gamma or more exactly plus omega of gamma if you're going up and minus omega of gamma if you're going down so for example on this path here I would have a k gamma so k gamma was the automorphism that we considered just a moment ago sorry the myrational transformation yeah what we're building up is some very complicated looking automorphism transformation of the torus into itself these are non-commuting and so we have to take the product in a particular order indeed and so the order we take is just the order going around the path so start somewhere go around the path here you have an automorph here you have a transformation that's right it's a well okay for a minute just imagine that the path begins somewhere and ends somewhere well as it turns out it'll turn out that it doesn't depend right so okay so for a minute let's just suppose that the path begins somewhere and ends somewhere and so we go around this path we multiply together these transformations of the torus and now what the wall crossing formula says is that whenever you do this for any contractable path whatsoever when you come around back to the start this is downstairs there's no upstairs here so this is contractable downstairs contractable downstairs but this is not a path on the curve this is not a path on the curve this is a path in the parameter space of quadratic differentials and cross with a circle but you aren't allowed to hit the point where the I'm not allowed to go right through this point and in a sense the formula will say that you are allowed to do that but April or I let's not allow that because exactly at that moment these omega of gammas maybe you're not quite well defined so let's avoid that singular situation I won't go straight through here so then what the wall crossing formula says is that whenever you do this you multiply together all these transformations of the torus and what you get when you come around the loop is you should actually get the trivial but this is not contractable curve this thing will not depend on what the pick was yes then there's but so this is better formula just from the fundamental group somewhere right yeah I mean let me formulate it for now only for only for contractable so okay so yeah yeah it's a bit divergent yeah so so okay so here's at least a formula it's not clear yet why this is a wall crossing formula but here's a condition which you could put this yes here's a formula which indeed follows from the previous definitions but in a very non-obvious way so okay so let's see why is this the same formula so it's a one crossing formula because suppose I take my path p to be actually a kind of rectangular path like this so here I'm focusing on one particular sort of wall crossing event where I have some non-zero invariance on one side and some other non-zero invariance on the other side so suppose I go around this path I don't cross anything here here I cross some non-zero invariance at the quadratic differential phi 2 and here I cross some some lines corresponding to the non-zero invariance at the quadratic differential phi 2 prime different from phi 2 so what the wall crossing formula says in this case is that the transformation that I meet the composite transformation that I meet by crossing these guys should be just equal to the composite transformation that I meet by crossing these guys because in going around this path I'm going to meet the transformation here times the inverse of the transformation that I got here so in other words if I define a product if I define the product just going from theta 1 to theta 2 sorry here I called them theta and theta prime if I take the product of the transformations that I meet here it should be equal to the product of the transformations that I meet on this side so the wall crossing formula just says that these two are literally equal and now the the miracle about this is that this formula this condition is actually strong enough to determine all of the omegas on this side in terms of in terms of the omegas on this side so in that sense this thing is a wall crossing formula by the way you might ask why does it not simply determine them to be equal why doesn't this formula just say that the omegas on this side are equal to the omegas on that side and the reason is exactly that the product that I take on this side is occurring in a different order from the product that I'm taking on this side so here for example if this is gamma 1 and this is gamma 2 here I would be taking gamma 1 before gamma 2 here I'm taking gamma 2 first then gamma 1 and to compensate the fact that they don't commute I have to insert some new things in the middle to make those two products be equal yeah it is that's right it is a kind of zero curvature condition for some connection let me not try to formulate it now okay so so now let me say so these two wall crossing phenomena that we already observed are really just governed by instances of exactly this formula so first suppose I have two charges whose intersection product is just 1 that was what happened in the first example that no this is not a connection over the curve exactly this is a connection which lives over the parameter space of over the parameter space of quadratic differentials so no it's something a little bit it's a little bit trickier so that's right that's right so there is such a connection in the story but okay so but I'm trying to say something sort of much simpler so so suppose for example that I have just two charges whose intersection product is 1 so then if I look at these two automorphisms I look at their composition just k gamma 1 k gamma 2 so that's what I'll have that's what I'll have on one side of a wall and then I look at a wall where the the central charge of z gamma 1 becomes aligned with z gamma 2 on the other side of the wall I'll meet these things in the opposite order I'll meet k gamma 2 first k gamma 1 last and then k gamma 1 and k gamma 2 don't commute but in order to fix up that on commutation you just have to insert one more transformation k gamma 1 plus gamma 2 so this is an identity once you know it you can easily check it by hand just with the definitions of these transformations and so what it says what it says is exactly that if I have two saddle connections on one side of a wall whose inner product is 1 then on the other side of the wall I'll have exactly three saddle connections the two original ones and one more and that's exactly what we saw in this example that I showed you because to be in the middle they're not one of the sides oh because so you have a linear map so the ordering is controlled by the phases of these z gammas so you have a linear map z from gamma to c and so I'm just the ordering just comes from the fact that if z gamma 1 is say here and z gamma 2 is say here this linear z gamma 1 plus gamma 2 will be sitting say here and so the ordering is controlled by the phase of z gamma so the phases so you see the phases come in this particular order gamma 1 plus gamma 2 always comes between gamma 1 and gamma 2 so that's all that's that's all that's going on here and that's what controls the ordering of all of these products phrase is circular we mean between or the circle phrase is circle there is no between well there's no between but here when I drew this path you know that path kind of breaks the symmetry of the circle yeah I mean you're just looking at a little interval where you're taking this contractable path so on that little interval there's a definite ordering and the ordering is as I said yeah if you don't know in advance that you have to find this on the next page please if you don't know what to have in the right hand side can you deduce it right so that's the amazing thing is that yes you can deduce it yeah so it's indeed uniquely determined by this formula and here's a more complicated example of the same thing suppose I have again two invariants which are equal to one so here this is k gamma to the power 1 k gamma 2 to the power 1 and the only thing you change is that the intersection between the two charges should be two instead of one and in that case you try to do the same thing you take this product and you try to rewrite it as a product of things in the opposite order you find again that it can be done but the formula that you have to write is somehow much more complicated it now has infinitely many of these transformations an infinite tower of them in x by n another infinite tower multiplied in the opposite order and then in the middle you have to put one so all these occurred with the exponent 1 and then in the middle you have to put one special thing which occurs with the exponent of minus 2 so this formula is exactly governing what happened in our second example where we had on one side just two saddle connections on the other side we had this infinite tower of saddle connections plus one annulus of closed trajectories which we counted with this factor of minus 2 so okay and the claim is that in fact this is this is completely general it doesn't just happen in these two examples but it happens in every example so everything was hidden in that formula how what is k right in that definition of k there has to be some deep thought that we can understand and we can tell us the definition is given then suppose I know nothing but the definition I come I multiply left and side and I have to know why I will have this thing on the right hand side that transformation includes itself so how do you know you mean how did we know to make this particular definition of I mean there was a question before for example on the right hand side you have gamma 2 gamma 1 plus gamma 2 and gamma 1 on the right hand side with that homology you already know that right hand side will be like that that's right it follows from the definition so how do I feel it why and on the second example you have gamma 2 gamma 2 equals 2 you already know that right hand side will be like that well okay oh so how how does one know how does one derive this formula just as a purely algebraic question how do you derive this formula I feel it somehow I don't know so these formulas I mean maybe I should say you can check these formulas on a computer so if you want to determine this formula up to any finite order just as a purely algebraic question how to find the thing on the right side from the thing on the left side but this is something informal but this formula looks a little bit like composition depending on which you choose like normal ordering gamma 2 is positive I keep talking some feeling here gamma 2 is positive no it's not that they're positive it's not that they're positive and negative they're all positive I mean it's more like the following no but if I think no but that's not what it is so what I'm saying the sort of finite dimensional analog of this so this is taking place in some infinite dimensional group which is the group of symplectomorphisms but if you wanted the kind of finite dimensional analog of what it is no here's the sort of sorry just the group of there are automorphisms of the Taurus but they also preserve some symplectic structure which I haven't mentioned but actually the Taurus is a Cremoron group but here it has a completion because I think it's important to consider things informal of our series there's some completion with any surface all but actually so what is the speciality they conform with the symplectic the complex symplectic that's right I mean if you want the sort of finite dimensional analog of this it's really the following kind of thing so suppose that I consider this product where here I took the 1, 2 thing before the 2, 3 thing if I try to write them in the opposite order it'll have the following shape and then in the middle so these two things don't quite commute but they sorry I guess minus here these two things don't quite commute and to correct the failure to commute you have to insert this extra guy that's given by the sum of these two roots so that's the finite dimensional analog of what we're doing here you can order positive rules order in different ways excuse me you said that on the right hand side you have K gamma 2 and K gamma 1 in the opposite order but in the last line I don't see them at all oh sorry the product should be from 0 to infinity not from 1 to infinity, thank you can you explain why K is defined by the formula you mean why K is given by this particular formula so that's okay so that's what I'm going to describe in the second part of the talk so so far I've just told you this formula and I've told you that what looks like a miracle that this formula, this totally algebraic game actually correctly captures the wall crossing behavior of these trajectories of quadratic differentials is there some connection with red groups let's say I'm not aware of a connection with red groups so okay so now you can ask why is it why does this totally algebraic thing, just moving transformations of a torus, why does that have anything to do with the special trajectories of quadratic differentials okay so there's by now a few different ways of understanding this but let me choose one particular way of describing it here so we'll take what it first seems like a big detour so we're going to use the quadratic differentials now in a totally different way so we're going to think of them as having something to do with the asymptotics of solutions of one parameter families of differential equations so it'll take us a little while to get there so for a while you won't see these k gammas up here, we're just going to do something else with quadratic differentials so first of all actually we're not going to quite just use the quadratic differentials, we're going to rather use Higgs bundles just a slightly fancier object so suppose I have, again I have this curve C and I'm going to fix some rank two holomorphic vector bundle let me call it E and then let's consider a meromorphic section of the endomorphisms of E tensor with the canonical bundle the cotangent bundle of C and we'll take it to be traceless so if you like it's not just a vector bundle it's a vector bundle with a volume form and we're taking the thing to be valued in SL2 so locally what does it concretely mean it just means we're making a matrix a matrix of one form of meromorphic functions like this so that's what it looks like in every local patch where I choose a trivialization of this vector bundle so a pair like that we call a Higgs bundle so the pair of a holomorphic vector bundle plus this one form valued endomorphism that pair is called a Higgs bundle now so given one of these Higgs bundles we can recover a quadratic differential how do we do it, we just take the trace of the square of this of this field that we called phi so phi was a one form valued thing and so its square will be a quadratic differential we take the trace of that then that's a globally defined thing that doesn't depend on any trivializations so instead of the quadratic differential we take these Higgs bundles and now the spectral curve which we used before so before I was looking at this double cover that just came by taking the two square roots of the quadratic differential so now another way of thinking about that is that those two sheets, those two square roots of the quadratic differential are just the two eigenvalues of my Higgs field so the point of the spectral curve are the eigenvalues of the Higgs field okay so okay so the Higgs bundle has a little bit more information than the quadratic differential I mean, morally speaking the quadratic differential only knew about the eigenvalues and now we're also introducing we're introducing this phi so it not only has eigenvalues but it has eigenspaces and so the space of Higgs bundles is not just equal to the space of quadratic differentials but rather it's a vibration over the space of quadratic differentials by passing from phi to its trace squared I forget a little bit of information one way of saying the information that I forget is I forget the eigenspaces the eigenspaces form a line bundle over the spectral curve and so what we get here is really a complex integrable system, roughly what you have is the Jacobians of the spectral curves which are some tori but today we're basically only interested in this phenomenon that takes place just over the space of quadratic differentials and so I won't care much about exactly where I am on the torus fibers I'll just take any Higgs field to correspond to that quadratic differential I just reminded you what's a Higgs field we're going to be interested not just in Higgs fields but also in flat connections over the curve maybe it's unnecessary for this audience to remind you what is a flat connection so again locally a connection is something that allows you to differentiate sections to differentiate sections of your bundle so locally again if I've trivialized if I've locally trivialized my bundle so then its sections are just some little two by two vectors of functions and to differentiate one of them according to this section where I take its derivative its partial derivative in the usual sense and I add to that some traceless matrix acting on my section so locally the connection is just represented by this pair of traceless matrices az and az bar and then we say that the connection is flat exactly if there exists if locally I can find a basis for my bundle E which is annihilated by the connection so a flat connection giving a flat connection is equivalent to giving some system of linear differential equations which are just these equations some compatible system of linear differential equations and so if I'm given a flat SL2C connection one of the things I can do is I can take its parallel transport by which I mean sorry I said parallel transport but I should really say the monodromy operation so I started in some neighborhood I have the local solutions of this equation nabla s equals zero then I continue them using this equation nabla s equals zero continue them along some path on the curve come back to the original point and I'll come back to potentially some different some different solutions they differ from the original solutions just by an element of SL2 so what we get is a representation of the fundamental group of the curve into SL2C up to equivalence and conversely you can recover the connection from this representation up to gauge transformation up to isomorphism yeah up to equivalence of connections so but on the other hand the map concretely if I'm given these matrices representing the connection locally if I'm given these matrices and I want to construct the corresponding representation it's in some sense a sort of complicated map to go from the to go from the explicit matrices to the to the corresponding monodromy representation okay so I was talking on the one hand the Higgs bundles on the other hand about flat connections now there's this miraculous fact which is I guess is due to Hitchin, Donaldson, Corlett and then in the in the cases with singularities which we'll come to in a moment it's Simpson and Decor and Bulge so the remarkable fact is the following so suppose someone gives you one of these Higgs bundles so we've got a quadratic differential we've chosen a Higgs bundle which gives you one of those then there's a corresponding totally canonical one parameter family of flat connections in that bundle so it's parameterized by a parameter zeta so this is going to be a c star family and what that family looks like well it looks like this it's of this simple kind of three term form so the original Higgs field here we have some connection which is actually unitary relative to a metric in the bundle and then here we have the adjoint of the Higgs field this is not a simple thing because if you look at this family of connections to require that that connection is flat and to require that it's flat for every value of the parameter zeta that requires some equations roughly to be flat says the bracket of nobler with nobler is zero and so that imposes some equations on this Phi, Phi dagger, and D and those equations are some complicated partial differential equations they're Hitchens equations so the theorem that this deformation exists is kind of a hard is a hard PDE theorem but nevertheless it's true so given a Higgs bundle there's this associated one parameter family of flat connections okay and to understand this wall crossing phenomenon we're going to understand it as some property of this family of flat connections okay well actually here we're considering the Higgs field that we consider is not going to be holomorphic but rather meromorphic because the quadratic differential had some singularities and so correspondingly the Higgs fields that we consider will have some singularities and so corresponding to that we don't get connections on the whole curve but we get connections with singularities at the same points so if the Higgs bundle that we consider if the Higgs field has just simple poles then this family of connections each member of that family will be a connection with regular singularities and so in that case we just encounter a representation not of Pi one of the original curve but rather Pi one of the curve with these points deleted, this monodromy around those singularities and if the if the Higgs field has higher order poles then the connection has some more complicated singularities so-called irregular singularity and in that case the representation well you still have a monodromy representation you still have a monodromy representation but the monodromy representation doesn't capture all the information about the connection, not even all the Gage invariant information so it's replaced by something a little more complicated the so-called Stokes data I don't want to try to say anything in detail about this Stokes phenomenon we're going to talk a lot about a different Stokes phenomenon which occurs in the asymptotics of this family of connections as Z goes to zero but independent of that there's also this other Stokes phenomenon that occurs here so this kind of example sounds harder but actually it turns out that the most tractable examples are this kind of example because here you can get something non-trivial even if you take the curve to be Cp1 the monodromy represent- well in fact you can take Cp1, even Cp1 which is a single singularity so then the monodromy representation would be absolutely trivial but nevertheless you can get something interesting here so okay so now we can formulate a question so so we have a Higgs bundle, we're given a Higgs bundle and we have a corresponding family of connections according to this according to the this non-Avillian Hodge correspondence and for each one of them we have a corresponding representation of the fundamental group so we have a family of representations of the fundamental group and they're varying holomorphically for Zeta and C star and we can ask the question how does this family behave as the parameter Zeta goes to zero or infinity in some sense if we want to understand this family completely because it's holomorphic for Zeta and C star understanding it completely boils down to just understanding what happens as Zeta goes to zero or Zeta goes to infinity but there's a symmetry exchanging zero and infinity so doing zero is essentially enough so understanding this is like we want to understand this okay so what's the idea well the idea is this family of connections comes to you in this explicit form and so in particular in the limit as Zeta goes to zero what would you think so you look at this connection and what it looks like is it's dominated by the first term so very naively you would think that to find a solution it might be enough just to deal with this first term so what does it suggest so what it suggests is in order to try to build a solution of this equation what should you do the key thing would be to just diagonalize this Higgs field because if you want to solve the equation is there an eraser so if you want to solve the equation you know the derivative of S equals some function times S that's essentially easy you just get that S is the integral of this function dZ so you know if we forget the difference between that D there and the partial derivative operator for just a second and if we diagonalize the leading term so if I have F over Zeta here then correspondingly I'll have the integral of F over Zeta here so so if we if we forget that D is not the partial derivative and if we diagonalize the leading term then you might think well you might look at least for a solution for a basis of solutions of this form so I just take the various eigenvalues and just integrate each eigenvalue so this is what you might try now well okay so the first thing you say is well let's find a solution which looks at least asymptotically like this let's try to do that well a little more precisely you know if you take a solution like that whose leading term is what I wrote and you plug it back into the equation of course you can't the solution will not exactly be given by that but what you might try is to make a series a series in Zeta whose leading term is given by the thing by the thing that I wrote so let's try it so suppose you try that so we're going to have a we're going to have a basis of kind of candidate solutions parameterized by the eigenvalues of the Higgs field and each solution is going to look like just a series like this well okay so so indeed you know you write this series you substitute this series back into the equation and you get equations which sort of iteratively determine all the higher terms so that's great so it looks like you've managed to solve the equation at least locally right you diagonalize the Higgs field locally you get this basis of solutions and now you've completely solved it so in particular what it would mean is that all the dependence on a base point is only here only in this integral and so it would say that in order to evolve the equation and work out its monodromy all you have to do is evaluate some integrals of the integrals of these forms alright so that would be pretty good so it would be great it would be great if this were true but actually it cannot be true why cannot be true well okay so suppose that there existed actual solutions of this form so we have them locally on every domain to diagonalize the Higgs field so then you could study the parallel transport indeed just by patching these solutions together and the little overlaps here would just be given by some integrals of this kind okay so that would be nice now the local solutions as we just said they're labeled by the eigenvalues of this Higgs field and so imagine that we move so we move around on the curve we move around one of the branch points of this covering so concretely these are the zeros of the quadratic differential so as you move around one of these points the two eigenvalues what I'm calling here lambda 1 and lambda 2 those two eigenvalues are being exchanged that's true for the leading term of this series but it's also true for all the sub-leading terms because they're determined algebraically by the leading term so if you had literal actual solutions of this form then the parallel transport around a branch point would exchange them well okay so that says that the parallel transport operator here exchanges these two solutions but that contradicts the fact that this connection was supposed to be a flat connection not only on the region where we can diagonalize the thing but a flat connection on the whole curve so here we're finding in this approximation which by the way is usually called the WKB approximation in this approximation we find a monodromy but that monodromy cannot be there cannot be there for the actual solutions so what went wrong? we wrote these formal series solutions and we've just seen that they can't be real they can't actually exist because of this obstruction from the monodromy so but they're defined by a totally reasonable procedure the only thing that can go wrong is that this series that we wrote is really only a formal series so the series has zero radius of convergence so formally it's fine but actually it is not fine, it has this problem okay well so then you might you might say it's only a formal thing and it's useless in the actual world but on the other hand this WKB approximation this is something that people use all the time and it does seem to have some use so how do we make sense of it? well we can try to make sense of these series not as an actual formula for an actual solution but only as an asymptotic expansion of a solution so this is a series which doesn't have to converge it doesn't converge for any fixed value of zeta but it gives the asymptotics of some actual solution well okay so as it turns out you can make sense of it as an asymptotic expansion but as usual with asymptotic expansions this asymptotic expansion is not going to be valid globally so we're not going to find solutions that have these asymptotics kind of uniformly as zeta goes to zero in any direction but suppose we ask for something a little bit less suppose we ask for solutions which have these asymptotics only in some sector in fact it seems that the right thing to do is to take a half space so let's fix fix some angle theta which is the same theta that appeared before as we'll see fix some angle theta and let's ask for solutions which have these asymptotics as zeta goes to zero just in the half space h theta so then the following claim so this is part of the joint work with Gato and more the claim is that around the generic point of this curve there are genuine actual solutions so now again depending on an eigenvalue of the Higgs field and depending on a phase theta there are genuine actual solutions which do have this kind of asymptotic expansion as it goes to zero but only in the half plane only in some half plane okay so let me explore a little bit the consequences of this claim so I said we have these local solutions but these local solutions can patch together smoothly if they did we would have the same monodromy problem that we had before saying that the asymptotics are only in some half space doesn't get rid of that problem so rather these local solutions are going to jump and they jump at some kind of co-dimension one loci in the curve which are the so-called stokes lines so here are the stokes lines for the zeta goes to zero asymptotics and the stokes lines for these local solutions of the equation we've actually seen already in this talk so if we go back again to look at this quadratic differential the stokes lines for these local solutions are exactly the lines of what I called before the spectral network so the claim is now that we're studying this family of differential equations we're looking for solutions with good asymptotics in a half space and the claim is that you can find such solutions but you can find them only in the complement of the walls of this spectral network and then there's some jumping so there will be some matrix that tells you when you cross any one of these walls there will be some matrix that tells you how the solutions in this domain are related to the solutions in this domain and so on and those are the so-called stokes factors okay so okay so now so now you can come back to the original question the question that we started with was well sorry the original question of wall crossing but the question of studying the asymptotics of the monodromy so suppose now that we have any closed loop suppose now that we have any closed loop on our curve and we look at the monodromy around that loop so now we're studying a family of connections so we study that monodromy in a family so we're getting some element of SL2C for every choice of this parameter zeta now you concretely try to compute that monodromy by the same strategy I was doing before you just patch together you have these local solutions and you patch them together but now you have to take account of the fact that you have jumps along the walls of this spectrum network so if you do that you find the following picture you find that the you find that the trace the trace that you're interested in so it by itself its asymptotics are somehow well you find that the trace can be expanded in this form so you have a sum over pieces which are labeled by the homology classes by homology classes of this spectral cover so for each piece we have some function of zeta standing here and they appear with some coefficients actually integer coefficients so the picture is that as zeta goes to zero in this half space the asymptotics of these functions so these are the functions which you would get by just naively doing the WKB procedure that I told you at the beginning so you just take the one form the tautological one form on the spectral curve and just integrate it around some cycles and what you're getting here is that the actual trace is given not just by a single thing like that not just by a single integral around some cycle and not even the sum of two which is what you might think if you have two solutions corresponding to the two eigenvalues but rather it's given by some sum of pieces each of which has each of the pieces has the kind of naive asymptotics that you would expect from WKB and they come to you in some complicated combination and these coefficients the coefficients that appear here are determined just by taking your path and looking at how it intersects with the spectral network so you start out trying to study the asymptotics of this holonomy function and you find that the sort of good way of studying the asymptotics is indeed to do this WKB procedure to get to integrals over the spectral curve but those integrals are a little bit more complicated than you might have thought because you have to keep track of how your path intersects with this spectral network so in particular now once you've got this far now you can recover the Zeta goes to zero asymptotics that you were interested in they're just dominated by whichever term has the largest coefficient of this exponential so there's another way of thinking about this thinking about this construction so so for a moment let me not think about asymptotics and just look at this formula as it stands so what we're saying is that if if you have a if you have an SL2 connection over a curve there's a sort of useful way of organizing its invariance so organizing the holonomies around some path namely the trace of the holonomy around some path gets expanded in terms of some simpler objects so these things here are the holonomies of some flat a flat connection which lives over the spectral cover so that's exactly what the WKB naively said that you could do it said that you could relate the holonomy parallel transport of your connection to just some integrals that you do over the spectral cover and here we have a kind of we have a sharp version of that the only trick is that given any path here you may have to expand it in terms of a bunch of different paths over the over the spectral cover so this is a kind of the slogan for this it's kind of a millionization so you're interested in studying SL2C connections over the curve and you relate them to something simpler you relate them to C star connections over this covering now of course it's not literally like push forward it's not like you have a diagonal connection and you push it forward that would give you something diagonal downstairs but the statement is that you take the push forward and then you do some cutting and gluing along the along the walls and this cutting and gluing is exactly taking care of the Stokes phenomenon the fact that the the local solutions couldn't be extended globally so this cutting and gluing just concretely speaking what it involves is just sticking in some unipotent matrices for the parallel transport every time you cross a wall you have to stick in some unipotent matrix and everything else is diagonal so okay so in particular what this says is that if you have an SL2C connection and this is supposed to be actually a one-to-one map so the datum of the SL2C connection over the curve is just equivalent up to equivalence is equivalent to the data of the C star connection up to equivalence over the over the covering and so what that says is that there's some coordinate system on the moduli on the moduli of flat SL2C connections just identified with the the moduli of C star connections over this covering and all I want to say is that in this way what you recover are some known interesting coordinate systems they're the so-called cluster coordinate systems on the moduli here it's just for SL2C connections and so those are also called complexified shear coordinates but for the higher rank which I'll hopefully get to at the end for the higher rank you get in this way some really new and exotic looking coordinate systems some of which have been studied by by okay so this is a thing which you can talk about without talking about WKB without talking about this asymptotics but what I've said here is that it arises very naturally if you think about the asymptotics you're sort of forced to think about this emulationization okay okay so now so now we can come back to the question we're studying these asymptotics and what happens now if I try to vary the parameters so I move a little bit in the hitch and base or I move my phase theta so we're studying asymptotics just in some half space and theta is controlling which half space we're looking at so let me remind you of what happens so if we vary the parameters phi 2 and theta just a little bit then the spectral network just changes by an isotope and a variation like that it doesn't affect anything about how if I consider a closed path say this closed path closed path that goes around just two of these things this WKB analysis is all about how the path intersects this spectral network and if I just change the network by a little isotope I won't change this WKB analysis so in particular these coefficients don't vary if I just vary the spectral network a little bit alright fine but on the other hand as I vary these parameters further I might cross one of these critical phases where a special trajectory appears and these were exactly the things that we wanted to count and at that moment the WKB analysis of these closed paths changes or in other words the abelianization I have kind of two different ways of abelianizing a connection one involving the spectral network before this crossing and one involving the spectral network after this crossing and those two structures are a little bit different and so correspondingly I get two different ways in terms of abelian holonomies so two different expansions so these coefficients omega are changing but on the other hand the original SL2C connection the original SL2C connection that you were studying doesn't jump it just varies if I change theta I don't change this connection at all if I move in the hitch and base then I move this connection but I move it in a continuous way so if these coefficients are jumping in order for the whole story to be consistent these abelian holonomies the holonomies on the cover have to also jump and we can calculate that jump so the way that these omegas change is just determined by the topology of how the spectral network degenerates and so you can really calculate that jump you can calculate it directly from the topology of the degeneration and so now what you get I'm sorry and so what do you get well so first suppose that the spectral network degenerates in a way that just includes one of these saddle connections so the picture is that the picture is that we have something like this and then after the degeneration we have something like this so here's two different spectral networks with two different corresponding ways of abelianizing your connection you can ask how do those two abelianizations differ so here I'm getting some functions x gamma here I'm getting some functions x prime gamma which are the holonomies of this abelian connection upstairs and the statement is that those two are different but they're different in a computable way and the way that they differ they differ exactly by this map which appeared in the wall crossing formula so this transformation so in this case x prime gamma equals x gamma times one minus x gamma naught where gamma naught is the homology class of the thing that appeared in the middle so the jump of the abelianized connection is exactly given by this k gamma so here's the moment where here's the moment where we're explaining or re-encountering the same formula that appeared there so that's a relatively easy computation what's a little bit harder is what happens if you have this annulus of closed trajectories but there so there's a scheme for working it out and when you work it out you find that the jump there is exactly this k gamma now raised to the power of minus two so in general as we move around in parameter space the spectral network deforms and the jump of these functions is k gamma which before they were formal coordinates on some torus now they have a concrete meaning as the whole of an abelian connection the statement is that that torus should be thought of as the torus of abelian connections and these transformations k gamma are just the jumps that occur when you try to abelianize a single fixed SL2C connection using different spectral networks okay in particular now if I come back to the so you know in each domain of this picture now I have a particular spectral network up to isotope in a particular way of abelianizing the connection and so in each one of these domains you know if I let my non-abelian connection very continuously these x gammas parameterizing this abelian connection have to jump and they jump by exactly this k gamma and if we go around a closed loop if we travel around a closed loop and come back to the original point the x gammas are defined by just a univalent procedure so we have to return to the original functions x gamma and that's exactly the wall crossing formula which we wanted to explain so the way we explained it is to find some concrete objects which actually transform by this funny looking transformation x gamma and then because those objects are are univalently defined when I go around a loop I have to come back to the original objects that's it okay so so the whole point was to explain this simplectomorphism formula and the explanation is that if you lift it to this connection it is just a statement that the elastic changes that's right so the statement is that if you study if you study a single non-abelian connection an SL2C connection which varies continuously there's a procedure for relating it to an abelian connection and that procedure is kind of forced on you if you try to study the WKB asymptotics but even if you don't study WKB asymptotics there is this procedure now that procedure depends on the choice of and as you move around in the parameter space the corresponding spectral network jumps and the jumps of the spectral network are exactly these K gammas that appeared in the wall crossing formula um yeah that's the statement um so okay so what have I said so we were considering some counting problem that had to do with the special trajectories of quadratic differentials uh and what what we found is that first of all they obeyed this conceivage soebelman this kind of universal wall crossing formula um and moreover that if you want to explain that formula or if you want to see it arise in some natural way um one way that it arises naturally is when you try to study the asymptotics of these families of SL2C connections that come from Hitchin equations um so that suggests a kind of natural problem which is you know what if I wasn't just interested in SL2 um or more generally I mean it could be any lead group but let's just start with SLK um for K bigger than 2 um and so here following the exact same kind of strategy um you meet a new kind of counting problem which so far hasn't been much explored so these these objects here these special trajectories are a kind of classically studied thing people in teichmeler theory have studied them a lot um but the higher rank analog which I'm now going to briefly tell you about has been relatively little explored I think it's an interesting problem um okay so let's just start again so now I'm going to start with the Higgs field um so and now make it SLK valued so just a rank K matrix instead of rank 2 so now I'll define the spectral curve to be again just parameterized by the eigenvalues it's just parameterizing the eigenvalues of this of this Higgs field um so you know locally one way of thinking about it is that uh well the covering is the same as giving the covering is the same as giving K just holomorphic one forms or more concrete the Higgs field just take the eigenvalues of the Higgs field that gives you K uh one forms which vary holomorphically now globally they have monodromes so we don't get K global things but locally you have K one forms um and now so so um so it seems so in this work with the Gayato and more we've tried to generalize the um the whole WKB story that I just told you to this situation um and it seems that it works um but the structure of the Stokes lines so the structure of the analog of this spectral network will be more complicated than it was in the in the case K equals 2 um so let me tell you let me give you now the definition a definition analogous to this definition of the trajectories of quadratic differentials I'm going to give you the higher rank version of those trajectories um so as before we're going to fix a phase um and now we'll define a network um so the network is going to be made up of Stokes lines and let me first tell you what's the local structure of those Stokes lines so before they were the leaves of this foliation um now I'll describe them by differential equation and they're not anymore leaves of a global foliation um now they're going to be carrying some label um labeled by two sheets i and j so in other words two eigenvalues of this Higgs field so you choose out any two eigenvalues and then you can write this differential equation so it says that um uh well it says this so you take lambda i minus lambda j um that gives you some one form and you have that one form when evaluated on your tangent vector tangent vectors to your lines um should be should have phase theta um so that equation that's you know first order equation defines for you again locally locally a foliation you're taking the leaves of this foliation but now globally um the the different uh kinds of line can mix um and then so so now um um the local structure around the branch points is going to be much the same as it was before um at each branch point uh now the branch points are labeled by two sheets of the covering so I'm I'm again let me assume that I'm in the situation of simple ramification so we only have places where two sheets collide no there's no foliation on the spectral curve because because the here the the leaves of this foliation depend on a choice of two sheets so if you like it could be on some product symmetric product but I prefer to think of it just directly on the on this curve itself um so each each of the branch points is a branch point with um labeled by two sheets and the foliation the foliation is labeled IJ around this branch point um is going to be much the same as it was before it has a kind of three prong singularity and in particular it has three distinguished trajectories um which are going to be among our our stokes lines um so there these stokes lines are going to be solutions of this equation where I and J are the particular I and J that met at this cover at this branch point and now of course globally I would have a problem of um trivializing this cover so right explicitly I'll you know in drawing this picture I'll put an actual branch cut then the stokes lines this one is labeled by IJ this one is J I this one is IJ again so they alternate and if you go through here you cross a cut um so okay so that's going to be the local picture each branch point is going to emit three of these stokes lines and then there's one further thing the further thing which is at first surprising is so now you emit these lines and just like before you let them just travel around the curve do whatever they want to do um but sometimes they're not anymore leaves of a global foliation so it may happen that they would actually cross each other um and if they cross each other if I if I have an IJ one and a JK one to intersect um then the rule is going to be that they should give birth to a new one so here I have IJ here I have JK and they cross each other in addition we're going to generate a new line um a line of type uh IK um so this is in a sense this is kind of an echo of something that occurred in the story of the conventional wall crossing a long time ago it is essentially by that formula exactly so so yeah somehow in the in the abelianization procedure it's exactly this formula which which is forcing you to give birth to these lines um but this is actually recently we learned that this had actually been studied before so people in WKB had noticed this phenomenon um okay uh so so okay so sorry so so so that's the definition of the spectrum network the spectrum network is you just shoot these three lines out of each branch point and then you let them do whatever they want to do but when they cross um so you let them evolve according to this differential equation but when they cross you have to give birth to new ones um and now so here's an example of what this procedure might look like so here I took um I took the case of just a three-fold cover so this is a Higgs field on a rank three bundle um and I took the curve to be again the only things I can plot really are take cp1 so here it's a stereographic coordinates but all this and we put all the singularities so it's a Higgs field with singularities and the singularities are at those three blue dots and I chose it so that the the covering has a total of six branch points those are these six uh the six x's that you see here um and what you see is that the thing at least in this particular example um the behavior is uh it's defined by a priori a total a totally crazy procedure but at least here the behavior is actually reasonably well under control so the trajectories travel around um when they collide they can give birth to new trajectories um but eventually uh uh all the trajectories get attracted down to these uh uh singularities this would be what switching system the sphere with three mark points yeah uh SL3 right exactly it's SL3 a sphere with three mark points and with singularities of the most generic generic kind at those three points so first-order poles um with generic eigenvalues at the three points um uh so um uh so okay so um so this at least in this particular example the behavior is a lot like what we had for the trajectories of quadratic differentials um now it's an unfortunate fact that not every example is uh um is so well behaved but at least there are some in so in general it may be that this procedure defines for you actually a dense set of trajectories um and in that case we just have to we just have to live with that um but at least there are there are many good examples in which it's really like what we had for the quadratic differentials um so this is the object that we this is the object that you study this is the object that gives rise to the um to the to the stokes phenomenon um in this higher rank um in these higher rank examples um okay um so is it true that any trajectory is going to a pole uh I think it's I believe it is true so as I say there are really no theorems about this um experimentally I believe that it's true that if you have these if you have the poles um that every trajectory will wind up at a pole the problem is that they may collide many many times with other guys on the way they may give birth to new things and you have sort of an infinite regress of these new things being created but this can't happen but it now seems that it actually does happen um it does happen but still there's sort of open sets in parameter space where it doesn't and so you may just consider those open sets as a way of getting started um and there are some examples in which it never happens there are some parameter spaces where it doesn't happen anywhere um okay so now as before just as before um we could study the solutions of Hitchin equations and the corresponding families of flat connections and just as before the asymptotics um can be controlled by a kind of abelianization so uh you express the monodromes of an SLK connection uh downstairs um in terms of some simpler quantities which are whole nomes of an abelian connection uh upstairs uh over the spectral cover in practice we're solving now same equation of the nabla s equals zero exactly the connection is SL3 SL3 or SLK yeah SL3 right we're we look in WKB we're not only exactly we want to look in WKB well we are solving an exact I mean the statements that we derive from this are exact statements um I mean the statement this formula that expresses the monodromes in terms of the monodromes of an abelian connection that's an exact formula um I mean it's it's you derive that formula you know what leads you to consider that that particular construction the motivation is that you're interested in studying the WKB asymptotics um but uh but no the point is that there really exist actual solutions that have those asymptotics so the formula is an exact the this relation between the monodromes downstairs and the monodromes upstairs is an exact relation um so and then again we have the same kind of story that we had in the um in the rank 2 case um uh when this network so you have this network that gives you the this network of stokes lines um controlling the zeta goes to zero asymptotics um and when this network jumps so now again we can we can ask suppose I vary parameters how does the network change um and the networks topology may jump and at that moment these these x gammas also jump um but as before the cool thing is you can study very generally what will be the form of that uh what will be the form of the jump um and you find just as in the rank 2 case that they're always controlled by this these kind of transformations that appeared in the consavage settlement uh formula um um it is the same k it is the same k because I mean after all this k as it was defined by consavage settlement was just an automorphism of a torus uh so you know a binational transformation of a torus um uh right now we have a k-fold cover but again we're just studying c-star connections over that k-fold cover so that's still a torus just a torus of larger dimension um and so no it's really exactly the same formal structure as there was in the case of rank 2 connections um uh is there a way to understand this procedure I mean I believe the addition is a very uh rough procedure I mean I would say that you would go from SLN to SLN-1 and SLN-2 no I'm yeah so I'm sorry that I didn't include sort of more about exactly how you uh how you do this I mean I've said it in a slightly rough way um but it's an absolutely precise thing I mean it's I mean is there a way just to simplify first I mean you have you SLK connections so is there a way to construct the cover where you have SLK-1 connection I mean all the levis that's what I'm talking about of all the parabolic okay so there's something like that that'll happen if you consider a point of the Hitchin base which is very special so like imagine that you consider like the case of uh GL N-1 and 2 then you can reduce the thing to GLN-1 what you have to do is consider some very special uh uh points of the Hitchin base where the spectral cover has kind of N-1 sheets together where it's a where it's a non-reduced thing that's like N-1 copies so then there is a kind of stepwise version of the thing um okay so so uh so now the formal story is just exactly the same as it was before um except so now the the jumps that you get before I can totally classify what happened the classification was you could have these saddle connections or these closed loops and nothing else um now in this higher rank situation um the classification of possible kind of topologies is somehow much much more complicated um so let me show you the first new thing so here's a situation this is again in rank uh k equals 3 and I'm just studying this little cluster of 3 branch points um and here we see the following thing happen so right now the topology jumps um and so the the new object that appeared at that critical moment uh was an object that looks like this so we've got these 3 branch points um at each of these branch points some pair of sheets is coming together so here it's the pair 1,2 and it's emitting a 1,2 trajectory um at the next one it's the pair 1 2,3 at the last one it's the pair 3,1 and in critical moment the critical moment is the moment when these 3 trajectories come together in a common point and they make this kind of 3 string network and that 3 string network is the object that's kind of responsible for this jump in the topology so now we're you know we're finding a new kind of invariance the invariance that count these jumps um this isn't the kind of object that contributes uh this is a new kind of object that contributes to these invariance um but that's not the only new kind of object so once you once you allow that topology um you have to allow much more general well sorry you just study this thing and you see what you get um but it doesn't include just that um it includes you know certainly you could have you could have a more general kind of tree you could have something like this you could have something with 5 vertices you could have something where you um uh you could have something where you attach a loop to it um at this moment I can't remember how to oh yeah so you could have something where you attach a loop to it like this um there's there's sort of arbitrarily complicated structure in this problem um but it seems to be just forced on you I mean if you want to study the SLK version of this story that's what the story is um uh and so in particular let me show you one particularly sort of crazy thing um oh yeah so this is the sorry this is the picture that I just drew so here's the this is the sort of before and after picture of this spectral network um and so this one if you work it out by the way this one contributes just one uh and very only equals one so it's really just like the sandal connections um but in so in general you get uh sort of much much more crazy things so here's a sort of exotic thing which happens already for k equals in the SL3 story so this picture is drawn on the cylinder so you should identify the left and right sides you can have a combination of trajectories which um which looks like this so this is the locus along which the trajectories are kind of hitting each other head to head um and in this case you go away and you calculate the um you know there's a procedure for calculating the jump of the abelianization map um and you you just calculate it in this case and what you find is that it is indeed given in this same formula you know k gamma to the power omega of gamma but now it contributes not just to omega of gamma so here corresponding to this picture there's some homology class which you get again by lifting these trajectories to the spectral curve and what you get here is invariance not just for that homology class but also for all the multiples of that homology class um uh which are exponentially growing so so the story in this higher rank uh uh case um seems to be uh seems to be somehow much more intricate and much less straightforward um nevertheless uh you know the same sort of formal story that I told you for proving the wall crossing formula works equally well in this situation and so this thing in fact that's how we extract these invariance um uh and so the wall crossing behavior of these new numbers is still completely captured by this wall crossing formula so in particular this sort of complicated sequence of invariance that I that I just showed you a slide ago um also could have been derived um in principle just from the from the wall crossing formula so here um it's wall crossing between two things that have uh inner product three um uh so you have two things called gamma one and gamma two you consider this product of this simple product of transformations and now you try to rewrite it as a product in the opposite order um and you find an incredibly complicated thing but in particular in the middle of that thing so there's a bunch of dots here for things that I'm not writing but in the middle of that thing um you find this product um of transformations with exactly these coefficients um so there's some there's some new counting problem here um which is still part of this whole you know Donaldson Thomas invariant structure um uh but which gives you know which is able to give you a sort of fairly complicated stuff um okay um so okay so let me just sum up so I told you in the beginning what this wall crossing phenomenon is sort of where it came from in physics um I didn't tell you much and and then uh and then I described some particular examples you know for a particular physical theory um the whole story becomes very geometric um and so we studied wall crossing for the special trajectories of quadratic differentials um and then more generally for uh generations of these new objects which we're calling spectral networks um so they both come from they both come sort of equally naturally from quantum field theory um and so in each case you know one way to think about the objects which you're trying to count is they have something to do with the the asymptotics of the monadromi uh for this system of linear differential equations with a small parameter um and from that point of view the wall crossing formula you know once you sort of absorb this this way when you're thinking about the story the wall crossing formula becomes something which is actually just automatic it's sort of a consistency condition for the whole story to work um so the so this as I said the special trajectories of quadratic differentials are kind of well studied subjects although I think the statement that they have this wall crossing phenomena um the statement that they obeyed the wall crossing formula I hadn't been noticed uh before um so now that story is somehow connected to this theory of Donaldson thomas invariance um and then there's this new higher rank about which essentially almost nothing is known so far which I sort of hope to advertise a little bit as an interesting thing okay that's it so thank you I would have many comments but let me just begin with the last one that in fact the Japanese school of exact wkb theory have studied this uh higher k problem quite a lot okay great so so uh I knew that they so great so so I don't understand completely what they get but there exists voluminous literature great so then correct this so so I so far don't know very much um and by connecting with these guys perhaps I'll learn something uh yeah thank you okay I after I think more about other questions any clue what is this omega in the question of quantization of future um so far I don't understand the relation between this story it's very similar what you are doing you are almost quantizing the future yeah yeah so I mean um so yeah no doubt these things have some role to play in the story of the quantization um but so far I don't know what it is okay I have another question then in wkb theory you have a small parameter right but it's not clear you have a symmetry between zero and infinity and it's not clear how it connects to general wkb right okay that is a great question so I mean the things that arose for us in physics um kind of automatically came in this very particular form these very particular families that have this symmetry between zero and infinity now of course the structures that we find look a lot like the structures that people in wkb have studied before and there they study it without that restriction um so so I don't know whether I mean here I was describing a pretty detailed thing about you know precise formulas for the robotics of the of the monodrome um I don't know so the proof that the so we gave an argument for why these local solutions exist um and in that argument we use the fact that the thing has this symmetry so it's a priori possible that the results that I told you really only hold for this special class of things that come from Hitchin systems um or it might be that it works for every family I just don't know does the map depend on the fact that it has is this special form well no the abelidization thing is a totally general thing which you can do as soon as someone gives you a spectral network um and assume as soon as someone gives you a would that be the same would you get the same thing if you use any uh any complex wkb problem with the same um yeah I think I mean you could formulate that you could formulate that whole procedure without the structure in infinity um but would you get the same uh the new addition map would it be the same map? it would be the same it would be the same map yeah um that you have the sign sigma which was minus in your case yeah and in general it is not the sign no we've worked out that sign so that I mean that sign ultimately has to do with a tricky thing in wkb so I I told you that um when you have a flat connection you abelianize it by a flat connection over the over the spectral cover but actually yeah I should have said so the most canonical thing to get is not exactly a flat connection over the spectral cover what it rather is is a flat connection which has holonomy minus one around the uh the branch points so it I mean it's the set of those things is a torsor over the set of flat connections but it has this little minus one now in many examples you're able to avoid that you're able to get rid of that sign um but the process of getting rid of that sign introduces this sigma gamma somewhere else um so you can either work with these twisted these sort of twisted flat connections the flat connections which have this minus one everywhere and then everything is totally canonical yeah exactly so in wkb this is known this is a well known thing but it occurs generally in this abelianization process even if you don't think about this wkb that sign is just in there um if you want to get rid of that sign and just deal with ordinary flat connections you can do it but the price you pay is this little sigma gamma oh yeah one more comment please you started by saying that all of this was a sort of annoying detail oh yes in a way I can I completely agree with you that for um second-order differential equations indeed it can be bypassed so there's more more efficient way of completely avoiding this all this this whole scheme so I wonder perhaps in your problem it might also help yeah I don't know our interest in these structures was of course you know partly motivated by wkb but partly not so yeah if it turns out that this stuff is not necessary for wkb I mean that that's okay with me yeah if you choose a higher genus instead of setting cp1 let's take the torus with one marked point people know that there are relations I mean in some other quantization of equations there is a relation between torus with one marked point do you see those things here or you don't study the higher genus case um with a higher genus case we've done a little bit I mean we certainly did study that particular case of a torus with one puncture um for sl2 somehow the problem is same as the sphere with four marked points um well there's some relation I mean in the limit where you take the residues to be some specific residues for the sphere with four punctures just regular regular singularity um yeah you take regular singularities for both but then I think you have to impose some further relation before they become the same I mean there's one thing that I know which is that the moduli of flat connections of sl2c flat connections on the four punctured sphere is if you write it concretely it's like an affine cubic surface um but more precisely if you fix the eigenvalues of monodromy around the four points you have to fix there is some restriction on there but they calculate so this is a relation between equal to star n and equal 4 yes if you put some restriction on the um so then the affine cubic surface I mean for the torus with one puncture there's also an affine cubic surface with some special parameter so in that sense the moduli spaces are the same I haven't thought about whether the abelanization procedure is also the same although I guess it probably is um sometimes it is possible to refine WKB by some non-perturbative corrections uh can you comment um I mean I think in a sense the the stokes lines that we're drawing are exactly the kind of non-perturbative corrections that you need um I mean I don't have a direct comment but um I mean the formulas that we write ultimately are sort of non-perturbative formulas and for example so maybe one thing to say is that for example here's an example of a kind of non-perturbative correction um so so I'm writing here a we're writing here a formula for the trace of the holonomy around some loop now this formula um this formula is actually an exact formula so it's an expansion of the trace in terms of these simpler objects which are the holonomies of this abelian connection now if you like you might say the leading term in that so there's for any particular quadratic differential or for any particular point of the hitch and base there's going to be the leading term which is the one where the real part of this is the biggest and then if you like you might say that the other terms the sub leading terms which are coming from other loops on the spectral curve are kind of non-perturbative corrections to that one so maybe that's sort of morally the yeah in this case you don't have the convergent series uh no no even in this case I mean the I mean well the WKB series for the for the sections or even for these functions X gamma of zeta I mean those functions X gamma of zeta if you expand them around zero they have they're not given by convergent series this is really asymptotic so it's not a I think this is just an unfortunate point of history that WKB theory started as asymptotics with divergent series in fact it's an exact theory okay so I fully sympathize with that point of view I mean here these are certainly supposed to be exact formulas but I mean it's true that if you expand them around zeta equals zero which is an interesting thing to do you do find this divergent series