Introduction to wall-crossing, in physics and differential equations
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Neitzke, Andrew
It has been known since the early 1990s that wall-crossing phenomena are crucial for understanding the spectrum of particles in supersymmetric quantum field theories. More recently, the rich algebraic and geometric structures underlying these phenomena are beginning to be understood. I will give an introduction to this story as it is known today, via examples, focusing mainly on cases related to integrable systems introduced by Hitchin. In this context, the story turns out to be related to the WKB analysis of families of differential equations.
 All right, so it's a great pleasure to have the opportunity to be here to talk about the subject. So thank you. I thank very much the organizers. 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 Calabi-Aa 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 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 neurosymmetry, in hypercalor geometry, in renewable 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. OK, 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, to the extent that I say anything original in this talk, it's part of a long-running joint work with David DeGato 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 at it 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 quark 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. OK. So in general, we might be interested in a. No, this is no function. It's not a real value function. Of course, yes, no function everywhere that I cross it. And say yes or no. Yeah. The point is that it's not an example. A real value function, I will have a function, a purely real value function, and it jumps. It functions a year's note, all of it is like that. Well, take the function to be one if the. No, but this is cheating. I call it pi e. No, it's because the actual motion. It's because the object is like that. It's because it's a function. I mean, just language, our language has no language. It's cheating. It's just. It's cheating. Your language has wall crossing, what the physical system? You call it yes or no, you call it one zero. The physical system is not actually a physical system. No, no, no, no, it's just a method of notation. It's a mathematical matrix. And you call this number, yes you call one, or no, call zero. But this notation is just language. But there are coming some questions where the answer changes. Let me show you an example. Yesterday, in Greece, what you call black or white, what you call, yes? I don't know the part of it. Let's use the example. Let's use it. Let's use it. 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, imagine I look at the set of all 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. But can you give a really convincing example? And compared to this, it's like what you call any man or what is an ape, you know? And the reason people say this is not... It's not biology. It's here a notion of limit. It's kind of a question about large volume limit. Nobody sees us now. So if you do not know. No, no, no, it's symmetrical approximation. Go to infinity. No, 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. So you can make a number of systems more complex, but you can make a number of systems more complex, 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, and in our attention to those, we get a problem that we have a chance of actually solving. So a little more precisely, you know, 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 of 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 change? Okay, so that's the kind of problem that arises in physics that we would like to be able to solve. So what exactly staples the region in the mathematical language there? You'll see. Now I think maybe you won't be satisfied until I show you the actual mathematical example. So I'll come on to that. So the first cases where this problem was solved were systems in just two space-time dimensions. They were studied in the early 1990s by Shikodi and Bafa. And in that case, they gave 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, Seinberg and Witten solved this problem in the case of one very specific gauge theory, this so-called pure N equals 2 super young mill 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. 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 inspiration for all the new progress in this subject is that mathematicians came to the rescue here. So they formulated a theory of generalized Donaldson-Thomas invariance. They're defined in a totally geometric way. And in many particular physical systems, you can formulate their generalized Donaldson-Thomas invariance and they seem to be exactly these omega of 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 Soibelman and Joyce Sonsang. 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 Soibelman wrote down. So let's fix some data. So let's fix a Lie algebra of A to E type and let's fix a compact Riemann surface, let me draw there, with a bunch of mark points on it. So for every such choice, for every choice of an A to E 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, 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. And 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 absorb 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 over theta over 2, I mean, because theta plus, or theta to theta plus pi, it's... I think I want really theta. So indeed, so, oh, very good. So indeed, the foliation... I guess the way that I've written it here... 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, we have... 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, you know, 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 the... 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 to... roughly when it comes close enough to one of these poles, it'll be attracted and fallen. 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 okay, 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 zeros 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 2 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 2 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 where 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 2 fixed and just varying the parameter theta, as you vary that parameter, this picture, this critical graph of the 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, 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 which occurs just at this critical value of the phase. So, okay, 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 it 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 counter-bly many phases. Well, why is that? It's because, as I'll explain in a second, they only have counter-bly 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 the 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. And 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 0's of phi 2. Those are the places where the two square roots come together. 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 0's. 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 0's, 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 is measuring the topology of my special trajectory. So to every one of these things, I have an associated homology class or what I'll call a charge. And now, okay, 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 can define, you know, 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 leval 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 being, 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 is theta. Okay, 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, you know, 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, for a special trajectory with any particular topology. Okay. So okay, 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 two, and we fix a homology class in the spectral cover called 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? Ah. 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, will count that, with this funny number, that as minus two. Okay. 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 for 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? 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 zero or one. And in every example I've ever actually looked at, you always get either zero or one. 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, so if there are really two of them that occur simultaneously, then I would count that as two. But in general, you know, in every example I'll show you, the number will always be just zero or one, so you're free to think it's zero if there's none and one if there is one. Okay, great, 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 one, 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 zero to pi, we found just exactly two saddle connections. So I could call the lift of this cycle, say gamma one, and the lift of this other cycle, say gamma two, so those are two little loops on the spectral cover. And so our invariants are omega of gamma one equals one, omega of gamma two equals one. And in fact, if I vary theta from pi to two pi, I'll see the same picture again repeated, but then if you keep track of the orientations carefully, you get minus gamma one and minus gamma two. So according to the definition of these invariants, we have omega plus or minus gamma one equals one, omega plus or minus gamma two equals one, and all the other invariants are just equal to zero. Okay, that's simple enough. Okay, 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 one. So we've moved a little bit in the parameter space of quadratic differentials, and now we see, instead of just two, we see three of these saddle connections. So, and 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, now we have, so this cycle here is the cycle that I used to call gamma two. This one is the new one which didn't occur before. That one has charged gamma one plus gamma two, and then finally there's this one which has charged gamma one. So after we've changed our parameters a little bit, now we have three of these saddle connections instead of two. So concretely, one of these invariants has jumped. The invariant which has jumped is this third one, omega gamma one plus gamma two, which used to be zero and now it's one. 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 zero to one. So this is the most basic example of this wall crossing phenomenon. So our parameter space, it was our parameter space, the space of Meromorphic quadratic differentials with some fixed singular behavior. On one side, we had this saddle connection, on the other side we don't have it. Okay. 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 zero and at infinity. A singularity of order three in both places. And so I represented as a cylinder, so zero and infinity play symmetrical roles. So here in this picture, once again, there are two saddle connections as theta varies from zero to pi. Unfortunately, they occur at the very beginning and very end of this animation. So here are the two branch points, and here at this moment you see two trajectories coming together. Bang, 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 one, all the rest are equal to zero. 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, we have these closed trajectories. And so correspondingly here, the invariance have become seemingly much, much more complicated. So here we have this infinite collection of non-zero invariance. So for any value of n, this is equal to one corresponding to a saddle connection. And then we have also this one critical moment where we had the closed trajectories. And so there's one omega which actually equals minus two, according to this definition that I wrote down. And again, all the others are equal to zero. So here we had just two non-zero invariance. Here we had this infinite collection of non-zero invariance. OK. 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 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 n, but I'll write it as the Ham from the lattice gamma in the C star. So this is a torus who's sort of... 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 pi 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 n, 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, can Savitch and Soilbhulman 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... Is it the insert period of the group structure in the torus? No, no, we're not using the group structure of the torus. No, that's right. This is a very nonlinear automorphism. So it's an automorphism of the torus just considered as a complex... It's a very directional transformation. Yeah. 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's the sigma of gamma as some tricky sign. And this pairing here is just the intersection pairing on the homology of the torus. Okay, so, can Savich and Soibelman 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 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, you know, 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. And the phases at which 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 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 one and gamma two. And on the other side of the wall I have gamma one, gamma two, and also a new one has been born with charge, say, gamma one plus gamma two. So this picture typically looks something like this. Okay, so Kutayvich and Soybelman tell us 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 walls, in other words, every time you meet one 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 my rational transformation. So what we're building up, yeah, what we're building up is some very complicated looking automorphism, some very complicated looking 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. It's on exactly what is the order. That's right. Well, okay. For a minute just imagine that the path begins somewhere and ends somewhere. It depends where you start. Well as it turns out, it'll turn out that it doesn't depend. The equation is one of the four. It doesn't depend on the basis. The basis doesn't depend on the basis. It's the equation. It's the equation. No, no, no, it's the equation. It doesn't depend on the basis. Right, so for a minute let's just suppose that the path begins somewhere and ends somewhere. And so we go around this path and 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. Contractable up stairs and down stairs. This is down stairs. There's no up stairs here. So this is contractable down stairs. Contractable down stairs, 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 crossed with a circle. But you aren't allowed to hit the point where the... I'm not allowed to go right through this point. Well, I mean, in a sense the formula will say that you are allowed to do that. But A priori, let's not allow that. Because exactly at that moment these omega of gammas maybe are 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 is this to which all contractable curve, this thing will depend on what's epic class? Yes, then there's... But so this is better for me, just from the fundamental group somewhere, right? Yeah, I mean, let me formulate it for now only for contractable. So, okay, so... It's a problem by the problem of the topi, right? If you want it low, you know it can't be low for you probably. Yeah, it's a bit divergent. The typical theory is it's going to be divergent. 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. It's not a condition, it's a fact, it follows from the beginning to the end. 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 actually a wall crossing formula. So, it's a wall 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 two. Now I go around this path and here I cross some lines corresponding to the non-zero invariance at the quadratic differential phi two prime, different from phi two. 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, if I keep track of the orientation, 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 one to theta two, 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 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 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 one and this is gamma two, here I would be taking gamma one before gamma two, here I'm taking gamma two first, then gamma one, 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. Isn't it some kind of zero curvature condition? 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 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 one. That was what happened in the first example that- Why don't you want to answer that question? Is not this a flat connection that you study in a region? No, this is not a connection over the curve exactly. This is a connection which lives over the parameter space of quadratic differentials. So, no, it's something a little bit trickier. But formula is connection with the structure of the group of the two. That's right. So, there is such a connection in the story. But formula is there. Yeah. Okay, so, but I'm trying to say something much simpler. So, suppose for example that I have just two charges whose intersection product is one. 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 on one side of a wall. And then I look at a wall where the central charge is z gamma 1 becomes a line 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 is exactly that if I have two saddle connections on one side of a wall, whose inner product is one, 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. I'm not saying intuitively why it has to be in the middle, not one of the sides. Oh, because so you have a linear map. So, you have a, sorry. So, you know, the ordering is controlled by the phases of these z gammas, right? 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, then because it's 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 going on here. And that's what controls the ordering of all of these products. The phrase is circle, it means between and the circle. Sorry? Phrase is circle, it is not between there. Well, there's no, there's no between, but here when I drew this path, you know, that path kind of breaks the symmetry of the circle. That's a circle, no. You know, it's small interval. Yeah, I mean, you're just looking at a little interval where you're taking this contractible 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 one, k gamma two to the power one. 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, indexed 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 one. And then in the middle, you have to put one special thing which occurs with the exponent of minus two. 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 minus two. So, okay. And the claim is that, in fact, 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. So, then in that definition of k, there has to be some deep thought that we can understand and we can tell us. Right. The definition is given. Suppose I know nothing but the definition. I come and 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 two, gamma one plus gamma two and gamma one. So, 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? You mean, in the second example, you have gamma one plus gamma two equals two. Right. You already know that right-hand side will be like that. Well, okay. So, how does one know... How does one derive this formula? Just as a purely algebraic question, how do you derive this formula? Feel it somehow. I feel it. How do you feel it? Yeah, 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 a purely algebraic question, how to find the thing on the right side from the thing on the left side. But this thing can inform you, just like that. But this formula looks a little bit like a composition depending on which borax of algebra you choose. Like normal ordering, if you mean gamma two is positive... He's talking some feeling here, so he feels it somehow. Gamma two is positive... No, it's not that they're positive and negative. They're all positive. They're all positive. I mean, it's more like the following. No, but if I think... No, he means upper triangular lower triangle. No, but that's not what it is. They're all upper triangular. That's exactly what I'm saying. So, I mean, 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, no, no, no. But I mean, here's the sort of... They're more prism... Sorry, just the group of... They're automorphisms of the torus, but they accidentally... They also preserve some symplectic structure, which I haven't mentioned. But say, yes, but I think the information in the source is a cream-orange group, right? Yeah, no, no, no, but here's the completion, because I think... You see, there are things in formal power series. Yeah, she's... There's some completion with any sort of old Baratian... No, no, no, no, no....perimorphisms of the torus. Yeah, it's... So, what's the speciality? They're complex. They're symplectic. They're complex. Yeah. That's right. Yeah, 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. And 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 kind of finite dimensional analog of what we're doing here. Yeah, you can order positive rules. Yeah. Excuse me. You said that on the right hand side, you have k gamma 2 and k gamma 1 in the opposite order. That's right. But in the last line, I don't see them at all. Oh, sorry. Excuse me. This should be... The product should be from 0 to infinity, not from 1 to infinity. Thank you. Sorry. Could 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 braid groups? Let's say I'm not aware of a connection with braid groups. So, so, okay, so, so, so now you can ask, you know, why is it right? Why do these... 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, 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 at 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. So, we're just going to do something else with quadratic differentials. Okay. So, so first of all, actually, we're not going to quite just use the quadratic differentials. We're going to rather use Higgs bundles. So, it's just a slightly fancier object. So, 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, 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, you know, locally, what does it concretely mean? It just means I'm, we're making a matrix, a matrix of one form. A matrix 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, that, so a pair like that we call a Higgs bundle. So, okay, 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, 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. Okay, so instead of the quadratic differential, we'll think about these Higgs, these Higgs bundles. And now the spectral curve which we used before, so before I was looking at the, this double cover that just came by taking the two square roots of the quadratic differential. So now another way of, 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, of my Higgs field. So, okay, so the point of the spectral curve are the eigenvalues of the Higgs field. Okay, so okay, so the, you know, 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, the, the space of Higgs bundles is not just equal to the space of quadratic differential, but rather it's a vibration over the space of quadratic differential. So by passing from phi to, 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 exact, form a line bundle over the spectral curve. And so what we get here is really a complex inter-rubble system. Roughly what you have is the Jacobians of the spectral curves, which are some, some tori. But today we're basically only interested in this phenomenon that takes place just over the base, just over the space of quadratic differentials. And so I won't care much about exactly where I am on the, on the torus fibers. I'll just take any Higgs, any Higgs field to correspond to that quadratic differential. Okay. Okay, so, so 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. So, okay, maybe it's unnecessary for this audience, but I remind you what is a, what is a flat connection? So again, locally, so a flat, a connection is something that allows you to differentiate sections, to differentiate sections of your, of your bundle. So locally, again, if I've trivialized, if I've locally trivialized my bundle, then I could write the, so then its sections are just some little two by two, some little two by two vectors of functions. And to differentiate one of them according to this section, well, 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, you know, if, if, if locally I can find a basis for my bundle e, which is annihilated by the connection. So, you know, a flat connection, in other words, 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. Okay. And so, you know, if I'm given a flat SL2C connection, you know, one of the things I can do is I can take its parallel transport, by which I mean, you know, sorry, I said parallel transport, but I should really say the monodromy operation. So, you know, I started in some neighborhood, I have the local solutions of this equation, nabla s equals zero, then I, 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, but they, they differ from the original solutions just by an element of SL2. So, what we get is a, 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, right? The power gauge to summation makes no sense. Up to isomorphism. Yeah, up to equivalence of connections. So, but on the other hand, the map, you know, concretely, if I'm, 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. Okay, so I was, I was talking on the one hand about 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, in the cases with singularities, which we'll come to in a moment, it's Simpson and Descorps 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. Suppose someone 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, it's, this is going to be a c star family. And what that family looks like, well, it looks like this. It's of, it's of this simple kind of a three term form. So, here we have 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. So, 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, you know, roughly the, to be flat, says the bracket of Nabila with Nabila 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, 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, so, given the 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, where the quadratic differential had some singularities. And so, correspondingly, the, the, the Higgs field that we consider will have some, will have some singularities. And so, corresponding to that, you know, we don't get connections on the whole curve, but we get connections with singularities at the same points. So, you know, if the, 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. 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. So, there's 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, 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, the Gage invariant information. So, it's replaced by some, something a little more complicated, the so-called Stokes data. I don't want to say, 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 zeta goes to zero. But independent of that, there's also this, this other Stokes phenomenon that occurs here. So, this kind of, this kind of example sounds harder, but actually it turns out that the most tractable examples are, are this kind of example. Because here you can get something non-trivial, even if you take the curve to be Cp1. Right, 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. Okay. So, 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 with zeta, 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, you know, because it's, 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 as zeta goes to infinity, but there's a symmetry exchanging zero and infinity. So, doing zero is essentially enough. So, understanding this is like, well okay, we want to understand this. Okay, so what's the idea? Well, the idea is, so this family of connections comes to you in this explicit form. And so in particular, you know, 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, so very naively, you would think that to find a solution, it might be enough just to deal with this, just to deal with this first term. So, what does it suggest? Whoops. Oh yeah, sorry. What it suggests is that, you know, 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, no, 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, 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, the solution will not exactly be given by that. But what you might try is to make a series in zeta whose leading term is given by the thing that I wrote. So, let's try, so suppose you try that. So, 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. Okay, 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. I mean, 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, you know, all the dependence on a base point is only here, only in this integral. And so, it would say that, you know, in order to evolve the equation, in order to work out its monodromy, all you have to do is evaluate some integrals of these forms. All right, 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 where you can 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 one and lambda two, 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. They can't exist because of this obstruction from the monodromy. 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 say, okay, well this is 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 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 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, I mean the following claim, so this is part of the joint work with Gato and Moore. The claim is that if I, 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 zeta 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't 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 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 you can come back to the original question. The question that we started with was, well, sorry, not yet 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 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. And now you can 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 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. And 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 sort of 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, but 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. So, 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 whole NOMI function, and you find that the sort of good way of studying the asymptotics is indeed to do this WKB procedure. You just relate it 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. Okay. 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. Okay. So, there's another way of thinking about this construction. 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 you have an SL2 connection over a curve, there's a sort of useful way of organizing its invariance. So, organizing the whole NOMIs around some path, namely the trace of the whole NOMI around some path gets expanded in terms of some simpler objects. So, these things here are the whole NOMIs of some 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 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 spectral cover. So, this is the kind of the slogan for this is that 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 walls. And this cutting and gluing is exactly taking care of the Stokes phenomenon. The fact that 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. 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 data of the SL2C connection over the curve is just equivalent up to, the data of this connection up to equivalence is equivalent to the data of the C star connection up to equivalence over the covering. And so, what that says is that there's some coordinate system on the moduli of, on the moduli of flat SL2C connections just identified with 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 fucking Gondroff. Okay, so this abelianization 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 abelianization. Okay, 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, find you in theta just a little bit, then the spectral network just changes by an isotope. And a variation like that 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. All right, fine. Yes, I'm... But on the other hand, so as I vary these parameters further, I might cross one of these critical phases where a special trajectory appears. So 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 of expanding the holonomie 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, you know, if I change theta, I don't change this connection at all. If I move in the hitching 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, oh, 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 1 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, if you like, 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 of this k gamma now rates to the power of minus 2. So in general, as we move around in parameter space, the spectral network deforms, and the jump of these function... these x gamma, which before they were formal coordinates on some torus, now they have a concrete meaning as the whole of the 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. So in particular, now if I come back to the... so 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, my... these x gammas parameterizing this abelian connection have to jump, and they jump by exactly this k gamma. On the other hand, 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 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 Kansai-Schabelman-Syntlektomorphism formula. Well, and the explanation is that if you lift it to this connection, it is just a statement that nothing changes. That's right. So the statement is that if you study a single non-abelian connection, or a non-abelian, an SL2C connection which varies continuously, that 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 WKBSyntotics. But even if you don't study WKBSyntotics, there is this procedure. Now, that procedure depends on the choice of spectral network. 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. Yeah, that's the statement. So, okay. So what have I said? So we were considering some counting problem that had to do with these special trajectories of quadratic differentials. And what we found is that, first of all, they obeyed this Kansai-Schabelman, this kind of universal wall-crossing formula. And moreover, that if you want to explain that formula, you have to see it arise in some natural way. 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. So that suggests a kind of natural problem, which is what if I wasn't just interested in SL2, but what I was interested in SLK, or more generally, I mean, it could be any league group, but let's just start with SLK for K bigger than 2. And so here, following the exact same kind of strategy, you meet a new kind of counting problem, which so far hasn't been much explored. So these objects here, these special trajectories, are a kind of classically studied thing. People in Teichmeler theory have studied them a lot. But the higher-ranked analog, which I'm now going to briefly tell you about, has been relatively little explored. I think it's an interesting problem. Okay, so let's just start again. So now, I'm going to start with the Higgs field. 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 Higgs field. So, you know, locally, one way of thinking about it is that, well, the covering is the same as, giving the covering is the same as giving K just holomorphic one-forms, or more concretely, the Higgs field. Just take the eigenvalues of the Higgs field, that gives you K 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. And now, so it seems, so in this work with Gaiauto and Moore, we've tried to generalize the whole WKB story that I just told you to this situation. And it seems that it works, 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 case K equals 2. 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-ranked version of those trajectories. So as before, we're going to fix a phase, and now we'll define a network. 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. Now I'll describe them by differential equation, and they're not anymore leaves of a global foliation. Now they're going to be carrying some label, 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, well, it says this. So you take lambda i minus lambda j. That gives you some one form, and you have to do that one form when evaluated on your tangent vector, tangent vectors to your lines should have phase theta. So that equation, that's a first order equation. It defines for you, again, locally a foliation. You're taking the leaves of this foliation. Now globally, the different kinds of line can mix. And then, so now, the local structure around the branch points is going to be much the same as it was before. At each branch point, now the branch points are labeled by two sheets of the covering. So again, let me assume that I'm in the situation of simple ramification. So we only have places where two sheets collide. Sorry, you have a foliation on the spectral curve? No, there's no foliation on the spectral curve because the leaves of this foliation depend on a choice of two sheets. So if you like, it could be on some symmetric product. But I prefer to think of it just directly on this curve itself. So each of the branch points is a branch point labeled by two sheets. And the foliation with label ij around this branch point is going to be much the same as it was before. So we have three prong singularity. And in particular, it has three distinguished trajectories which are going to be among our stokes lines. So 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 branch point. And now, of course, globally, I would have a problem of trivializing this cover. So I'll write explicitly, enjoying this picture, I'll put an actual branch cut. This one is labeled by ij. This one is ji. And this one is ij again. So they alternate. And if you go through here, you cross a cut. 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. But sometimes, there are not any more leaves of a global foliation. So it may happen that they would actually cross each other. And if they cross each other, if I have an ij one and a jk one to intersect, 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. And in addition, we're going to generate a new line, a line of type ik. So this is, in a sense, this is kind of an echo of something that occurred in the story of two-dimensional wall crossing a long time ago. It is essentially, by that formula, exactly. So, yeah, somehow in the abelianization procedure, it's exactly this formula which is forcing you to give birth to these lines. But recently, we learned that this had actually been studied before. So people in WKB had noticed this phenomenon. OK. 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, so you let them evolve according to this differential equation, but when they cross, you have to give birth to new ones. And now, so here's an example of what this procedure might look like. So here I took the case of just a three-fold cover. So this is a Higgs field on a rank three bundle. And I took the curve to be, again, the only things I can plot really are, take CP1. So here it's stereographic coordinates. 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 covering has a total of six branch points. Those are the six X's that you see here. And what you see is that the thing, at least in this particular example, the behavior is defined by a priori, a totally crazy procedure. But at least here, the behavior is actually reasonably well under control. So the trajectories travel around. When they collide, they can give birth to new trajectories. But eventually, all the trajectories get attracted down to these singularities. This would be a both-patching system. The sphere with three mark points. Yeah. SL3. SL3. SL3. Right, exactly. It's SL3, a sphere with three mark points, and with singularities of the most generic kind at those three points. So first-order poles with generic eigenvalues at the three points. So, OK. So this, at least in this particular example, the behavior is a lot like what we had for the trajectories of quadratic differentials. Now, it's an unfortunate fact that not every example is so well behaved. But at least there are some, so in general, it may be that this procedure defines for you actually a dense set of trajectories. And in that case, we just have to live with that. But at least there are many good examples in which it's really like what we had for the quadratic differentials. So this is the object that you study. This is the object that gives rise to the stokes phenomenon in these higher-rank examples. OK. So is it true that any trajectory is going to a pole? I think it's, I believe it is true. So as I say, there are really no theorems about this. Experimentally, I believe that it's true that if you have the poles, 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. We first thought that this can't happen, but it now seems that it actually does happen. 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. And there are some examples in which it never happens. There are some parameter spaces where it doesn't happen anywhere. OK. So now as before, just as before, we could study the solutions of Hitchin equations and the corresponding families of flat connections. And just as before, the asymptotics are going to be controlled by a kind of a bilionization. So you express the monodromes of an SLK connection downstairs in terms of some simpler quantities, which are holonomes of an abelian connection upstairs over the spectral cover. The problem is we're solving now the same equation of the nabla S equal 0. Exactly. The connection is SL3. Yes. SL3 or SLK, yeah. SL3. Right. We look in WKB. We're not solving exactly. We want to look in WKB. Well, we are solving it exactly. The statements that we derive from this are exact statements. I mean, this formula that expresses the monodromy in terms of the monodromes of an abelian connection, that's an exact formula. I mean, you derive that formula. What leads you to consider that particular construction, the motivation is that you're interested in studying the WKB asymptotics. But no, the point is that there really exist actual solutions that have those asymptotics. So this relation between the monodromy downstairs and the monodromy upstairs is an exact relation. And then again, we have the same kind of story that we had in the Rank II case. So you have this network that gives you the network of stokes lines, controlling the Zeta Go to zero asymptotics. And when this network jumps, so now again, we can ask, suppose I vary parameters, how does the network change? And the network's topology may jump, and at that moment these X gammas also jump. But as before, the cool thing is you can study very generally what will be the form of that, what will be the form of the jump. And you find just as in the Rank II case, that they're always controlled by these kind of transformations that appeared in the Kinsavich-Soylman formula. It is the same K. It is the same K. Because I mean, after all, this K, as it was defined by Kinsavich and Soymelman, was just an automorphism of a torus, a binaural transformation of a torus. 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. And so no, it's really exactly the same formal structure as there was in the case of Rank II connections. Is there a way to understand this procedure? I mean, I believe the addition is a very rough procedure. I would say that we would go from SLN to SLN minus 1, SLN minus 10. Yeah, so I'm sorry that I didn't include more about exactly how you do this. I've said it in a slightly rough way, but it's an absolutely precise thing. I mean, is there a way just to simplify first? I mean, you have your SLK connection, so is there a way to construct a cover where you have SLK minus 1 connection? I mean, all the levies of all the parabolic? OK, so there's something like that that will happen if you consider a point of the Hitchin base, which is very special. So imagine that you consider the case of GLN1 and 2. Then you can reduce the thing to GLN1. What you have to do is consider some very special points of the Hitchin base where the spectral cover has N1 sheets together, where it's a non-reduced thing that's like N1 copies. So then there is a kind of stepwise version of the thing. So now the formal story is just exactly the same as it was before, except so now 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. Now in this higher rank situation, the classification of possible kind of topologies is somehow much, much more complicated. So let me show you the first new thing. So here's the situation. This is again in rank k equals 3. And I'm just studying this little cluster of three branch points. And here we see the following thing happen. So right now the topology jumps. And so the new object that appeared at that critical moment was an object that looks like this. So we've got these three branch points. 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. At the next one, it's the pair 1, 2, 3. At the last one, it's the pair 3, 1. And there's some critical moment. The critical moment is the moment when these three trajectories come together at a common point and they make this kind of three string network. And that three string network is the object that's kind of responsible for this jump in the topology. So now we're finding a new kind of invariance, the invariance that count these jumps. This isn't the kind of object that contributes. This is a new kind of object that contributes to these invariance. But that's not the only new kind of object. So once you allow that topology, you have to allow much more general, well, sorry. You just study this thing and you see what you get. But it doesn't include just that. It includes, certainly, you could have a more general kind of tree. You could have something like this. You could have something with five vertices. You could have something where you attach a loop to it. 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. There's sort of arbitrarily complicated structure in this problem. 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. And so in particular, let me show you one particularly sort of crazy thing. Oh, yeah. So this is the picture that I just drew. So this is the sort of before and after picture of this spectral network. So this one, if you work it out, by the way, this one contributes just one. And very only equals one. So it's really just like the sandal connections. But in general, you get 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 looks like this. So this is the locus along which the trajectories are kind of hitting each other head to head. And in this case, you go away and you calculate the procedure for calculating the jump of the abelianization map. And you just calculate it in this case. And what you find is that it is indeed given in this same formula, 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, which are exponentially growing. So the story in this higher-rank case seems to be somehow much more intricate and much less explored. Nevertheless, 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 invariants. 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 just showed you a slide ago also could have been derived, in principle, just from the wall-crossing formula. So here, it's wall-crossing between two things that have inner product three. So you have two things, called gamma 1 and gamma 2. You consider this product of this simple product of transformations, and now you try to rewrite it as a product in the opposite order. 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, you find this product of transformations with exactly these coefficients. So there's some new counting problem here, which is still part of this whole Donaldson-Thomas invariant structure, which is able to give you fairly complicated stuff. So let me just sum up. So I told you in the beginning what this wall-crossing phenomenon is, where it came from in physics. I didn't tell you much. And then I described some particular examples. For a particular physical theory, the whole story becomes very geometric. And so we studied wall-crossing for the special trajectories of quadratic differentials, and then more generally for the degenerations of these new objects, which we're calling spectral networks. So they both come sort of equally naturally from quantum field theory. And so in each case, one way to think about the objects which you're trying to count is they have something to do with the asymptotics of the monodromy for this system of linear differential equations with a small parameter. And from that point of view, the wall-crossing formula, once you sort of absorb this way of 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. So as I said, the special trajectories of quadratic differentials are kind of well-studied subjects, although I think the statement that they obeyed the wall-crossing formula hadn't been noticed before. So now that story is somehow connected to the theory of Donaldson-Thomas invariance. And then there's this new theory for 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 had studied this higher K problem quite a lot. Okay, great. So I don't understand completely what they get, but there exists voluminous literature. Great, so then correct this. So I so far don't know very much, and by connecting with these guys perhaps I'll learn something. Yeah, thank you. Okay, I think more about other questions. Any clue what is this omega in the question of quantization of the fictitious system? So far I don't understand the relation between this story. It's very similar what you're doing. You're almost quantizing the fictitious system. Yeah, yeah. So I mean, so yeah, no doubt these things have some role to play in the story of the quantization, but so far I don't know what it is. Okay, I have another question then. In WKB theory you have a small parameter, but it's not clear you have a symmetry between zero and infinity in this data, 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 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. So I don't know whether, I mean, here I was describing a pretty detailed thing about, you know, precise formulas for the asymptotics of the monodromy. I don't know, so the proof that, so we gave an argument for why these local solutions exist. And in that argument, we use the fact that the thing has this symmetry. So it's apriori possible that the results that I told you really only hold for the special class of things that come from Hitchin systems. Or it might be that it works for every family, I just don't know. Does the abilityization map depend on the fact that it has, is this special formula? So the abilityization thing is a totally general thing which you can do as soon as someone gives you a spectral network, and as soon as someone gives you a... Would that be the same, would you get the same thing if you use any complex WKB problem with the same... Yeah, I think, I mean, you could formulate that whole procedure, you could indeed formulate that whole procedure without the structure in infinity. But would you get the same, the abilityization map, would it be the same map? It would be the same map, yeah. But you have the sine sigma, which was minus the male case, and in general it is not, you know. The sine, no, we've worked out that sine, so that, I mean, that sine ultimately has to do with a tricky thing in WKB, so I told you that when you have a flat connection, you abillionize it by a flat connection over the spectral cover. 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 whole nomenus minus one around the branch points. So, 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 sine. But the process of getting rid of that sine introduces this sigma gamma somewhere else. So, you can either work with these twisted, these sort of twisted flat connections. Flat connections would have this minus one everywhere, and then everything is totally canonical. Yeah, exactly. So, in WKB, this is a well-known thing, but it occurs generally in this abillionization process, even if you don't think about it as WKB. That sine is just in there. If you want to get rid of that sine and just deal with ordinary flat connections, you can do it, but the price you pay is this little sigma gamma. 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 completely agree with you that for second-order differential equations, indeed, it can be bypassed. So, there's a more efficient way of completely avoiding this whole scheme. So, I wonder perhaps in your problem, it might also help. Yeah, I don't know. I mean, our interest in these structures was, of course, partly motivated by WKB, but partly not. So, yeah, if it turns out that this stuff is not necessary for WKB, I mean, that's okay with me. If you do the hydrogenous scheme, instead of starting CP1, let's take the torus with one markpoint. People know that there are relations. I mean, in some other quantization equations, there is a relation between torus with one markpoint and a sphere with four forms. Do you see those things here? Or you don't study the hydrogenous case? The hydrogenous case, we've done a little bit. I mean, we certainly did study that particular case of a torus with one puncture for SL2. Is it somehow the problem is same as the sphere with four markpoints? 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 singularity. Yeah, you take regular singularities for both, but then I think you have to impose some further relation before they become the same. There's one thing that I know which is that the moduli of SL2C flat connections on the four punctured sphere is, if you write it concretely, it's like an affine cubic surface. But more precisely, if you fix the eigenvalues of monodromy around the four points, you can… You have to fix the resultant friction on there. Right. Recalculate. So this is a relation we're equal to star angle equal to four. Yes. Right. So I put that if you put some restriction on the… So then the affine cubic surface, I mean, for the… for the… Taurus 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. Sometimes it is possible to refine WKB by some non-perturbative corrections. Can you comment? I mean, I think in a sense… In a sense, the stokes lines that we're drawing are exactly the kind of non-perturbative corrections that you need. I mean, I don't have a direct comment, but… 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. So I'm writing here… We're writing here a formula for the trace of the holonomy around some loop. Now, 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 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 subleading 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 Gertrude series. No, no, even in this case. I mean, well, the WKB series 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… No, no, 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. Thank you.
 All right, so it's a great pleasure to have the opportunity to be here to talk about the  subject.  So thank you.  I thank very much the organizers.  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 Calabi-Aa 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 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 neurosymmetry, in hypercalor geometry,  in renewable 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.  OK, 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, to the extent that I say anything original in this talk, it's part  of a long-running joint work with David DeGato 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 at it 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 quark 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.  OK.  So in general, we might be interested in a.  No, this is no function.  It's not a real value function.  Of course, yes, no function everywhere that I cross it.  And say yes or no.  Yeah.  The point is that it's not an example.  A real value function, I will have a function, a purely real value function, and it jumps.  It functions a year's note, all of it is like that.  Well, take the function to be one if the.  No, but this is cheating.  I call it pi e.  No, it's because the actual motion.  It's because the object is like that.  It's because it's a function.  I mean, just language, our language has no language.  It's cheating.  It's just.  It's cheating.  Your language has wall crossing, what the physical system?  You call it yes or no, you call it one zero.  The physical system is not actually a physical system.  No, no, no, no, it's just a method of notation.  It's a mathematical matrix.  And you call this number, yes you call one, or no, call zero.  But this notation is just language.  But there are coming some questions where the answer changes.  Let me show you an example.  Yesterday, in Greece, what you call black or white, what you call, yes?  I don't know the part of it.  Let's use the example.  Let's use it.  Let's use it.  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, imagine I look at the set of all 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.  But can you give a really convincing example?  And compared to this, it's like what you call any man or what is an ape, you know?  And the reason people say this is not...  It's not biology.  It's here a notion of limit.  It's kind of a question about large volume limit.  Nobody sees us now.  So if you do not know.  No, no, no, it's symmetrical approximation.  Go to infinity.  No, 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.  So you can make a number of systems more complex, but you can make a number of systems more complex,  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,  and in our attention to those, we get a problem that we have a chance of actually solving.  So a little more precisely, you know, 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 of 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 change?  Okay, so that's the kind of problem that arises in physics that we would like to be able to solve.  So what exactly staples the region in the mathematical language there?  You'll see.  Now I think maybe you won't be satisfied until I show you the actual mathematical example.  So I'll come on to that.  So the first cases where this problem was solved were systems in just two space-time dimensions.  They were studied in the early 1990s by Shikodi and Bafa.  And in that case, they gave 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, Seinberg and Witten solved this problem in the case of one very specific gauge theory,  this so-called pure N equals 2 super young mill 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.  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 inspiration for all the new progress in this subject  is that mathematicians came to the rescue here.  So they formulated a theory of generalized Donaldson-Thomas invariance.  They're defined in a totally geometric way.  And in many particular physical systems, you can formulate their generalized Donaldson-Thomas invariance  and they seem to be exactly these omega of 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 Soibelman and Joyce Sonsang.  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 Soibelman wrote down.  So let's fix some data.  So let's fix a Lie algebra of A to E type and let's fix a compact Riemann surface,  let me draw there, with a bunch of mark points on it.  So for every such choice, for every choice of an A to E 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, 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.  And 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 absorb 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 over theta over 2, I mean, because theta plus, or theta to theta plus pi, it's...  I think I want really theta.  So indeed, so, oh, very good.  So indeed, the foliation...  I guess the way that I've written it here...  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, we have...  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, you know, 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 the...  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 to... roughly when it comes close enough to one of these poles, it'll be attracted and fallen.  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 okay, 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 zeros 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 2 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 2 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  where 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 2 fixed and just varying the parameter theta,  as you vary that parameter, this picture, this critical graph  of the 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,  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 which occurs just at this critical value of the phase.  So, okay, 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 it 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 counter-bly many phases.  Well, why is that?  It's because, as I'll explain in a second, they only have counter-bly 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 the 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.  And 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 0's of phi 2.  Those are the places where the two square roots come together.  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 0's.  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 0's, 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 is measuring the topology of my special trajectory.  So to every one of these things, I have an associated homology class or what I'll call a charge.  And now, okay, 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 can define, you know, 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 leval 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 being, 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 is theta.  Okay, 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, you know, 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, for a special trajectory with any particular topology.  Okay.  So okay, 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 two,  and we fix a homology class in the spectral cover called 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?  Ah.  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, will count that, with this funny number,  that as minus two.  Okay.  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 for 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?  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 zero or one.  And in every example I've ever actually looked at, you always get either zero or one.  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, so if there are really two of them that occur simultaneously,  then I would count that as two.  But in general, you know, in every example I'll show you,  the number will always be just zero or one,  so you're free to think it's zero if there's none and one if there is one.  Okay, great, 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 one, 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 zero to pi,  we found just exactly two saddle connections.  So I could call the lift of this cycle, say gamma one,  and the lift of this other cycle, say gamma two,  so those are two little loops on the spectral cover.  And so our invariants are omega of gamma one equals one,  omega of gamma two equals one.  And in fact, if I vary theta from pi to two pi,  I'll see the same picture again repeated,  but then if you keep track of the orientations carefully,  you get minus gamma one and minus gamma two.  So according to the definition of these invariants,  we have omega plus or minus gamma one equals one,  omega plus or minus gamma two equals one,  and all the other invariants are just equal to zero.  Okay, that's simple enough.  Okay, 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 one.  So we've moved a little bit in the parameter space of quadratic differentials,  and now we see, instead of just two, we see three of these saddle connections.  So, and 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,  now we have, so this cycle here is the cycle that I used to call gamma two.  This one is the new one which didn't occur before.  That one has charged gamma one plus gamma two,  and then finally there's this one which has charged gamma one.  So after we've changed our parameters a little bit,  now we have three of these saddle connections instead of two.  So concretely, one of these invariants has jumped.  The invariant which has jumped is this third one,  omega gamma one plus gamma two, which used to be zero and now it's one.  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 zero to one.  So this is the most basic example of this wall crossing phenomenon.  So our parameter space, it was our parameter space,  the space of Meromorphic quadratic differentials with some fixed singular behavior.  On one side, we had this saddle connection,  on the other side we don't have it.  Okay. 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 zero and at infinity.  A singularity of order three in both places.  And so I represented as a cylinder,  so zero and infinity play symmetrical roles.  So here in this picture, once again,  there are two saddle connections as theta varies from zero to pi.  Unfortunately, they occur at the very beginning and very end of this animation.  So here are the two branch points,  and here at this moment you see two trajectories coming together.  Bang, 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 one, all the rest are equal to zero.  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, we have these closed trajectories.  And so correspondingly here, the invariance have become  seemingly much, much more complicated.  So here we have this infinite collection of non-zero invariance.  So for any value of n, this is equal to one corresponding to a saddle connection.  And then we have also this one critical moment where we had the closed trajectories.  And so there's one omega which actually equals minus two,  according to this definition that I wrote down.  And again, all the others are equal to zero.  So here we had just two non-zero invariance.  Here we had this infinite collection of non-zero invariance.  OK.  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 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 n, but I'll write it as the Ham from the lattice gamma in the C star.  So this is a torus who's sort of...  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 pi 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 n, 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, can Savitch and Soilbhulman 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...  Is it the insert period of the group structure in the torus?  No, no, we're not using the group structure of the torus.  No, that's right. This is a very nonlinear automorphism.  So it's an automorphism of the torus just considered as a complex...  It's a very directional transformation.  Yeah.  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's the sigma of gamma as some tricky sign.  And this pairing here is just the intersection pairing on the homology of the torus.  Okay, so, can Savich and Soibelman 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 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, you know, 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.  And the phases at which 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 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 one and gamma two.  And on the other side of the wall I have gamma one, gamma two,  and also a new one has been born with charge, say, gamma one plus gamma two.  So this picture typically looks something like this.  Okay, so Kutayvich and Soybelman tell us 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 walls,  in other words, every time you meet one 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 my rational transformation.  So what we're building up, yeah, what we're building up is some very complicated looking automorphism,  some very complicated looking 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.  It's on exactly what is the order.  That's right.  Well, okay.  For a minute just imagine that the path begins somewhere and ends somewhere.  It depends where you start.  Well as it turns out, it'll turn out that it doesn't depend.  The equation is one of the four.  It doesn't depend on the basis.  The basis doesn't depend on the basis.  It's the equation.  It's the equation.  No, no, no, it's the equation.  It doesn't depend on the basis.  Right, so for a minute let's just suppose that the path begins somewhere and ends somewhere.  And so we go around this path and 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.  Contractable up stairs and down stairs.  This is down stairs.  There's no up stairs here.  So this is contractable down stairs.  Contractable down stairs, 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 crossed with a circle.  But you aren't allowed to hit the point where the...  I'm not allowed to go right through this point.  Well, I mean, in a sense the formula will say that you are allowed to do that.  But A priori, let's not allow that.  Because exactly at that moment these omega of gammas maybe are 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 is this to which all contractable curve, this thing will depend on what's epic class?  Yes, then there's...  But so this is better for me, just from the fundamental group somewhere, right?  Yeah, I mean, let me formulate it for now only for contractable.  So, okay, so...  It's a problem by the problem of the topi, right?  If you want it low, you know it can't be low for you probably.  Yeah, it's a bit divergent.  The typical theory is it's going to be divergent.  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.  It's not a condition, it's a fact, it follows from the beginning to the end.  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 actually a wall crossing formula.  So, it's a wall 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 two.  Now I go around this path and here I cross some lines corresponding to the non-zero invariance  at the quadratic differential phi two prime, different from phi two.  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,  if I keep track of the orientation, 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 one to theta two,  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 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 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 one and this is gamma two,  here I would be taking gamma one before gamma two,  here I'm taking gamma two first, then gamma one,  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.  Isn't it some kind of zero curvature condition?  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 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 one.  That was what happened in the first example that-  Why don't you want to answer that question?  Is not this a flat connection that you study in a region?  No, this is not a connection over the curve exactly.  This is a connection which lives over the parameter space of quadratic differentials.  So, no, it's something a little bit trickier.  But formula is connection with the structure of the group of the two.  That's right.  So, there is such a connection in the story.  But formula is there.  Yeah. Okay, so, but I'm trying to say something much simpler.  So, suppose for example that I have just two charges whose intersection product is one.  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 on one side of a wall.  And then I look at a wall where the central charge is z gamma 1 becomes a line 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 is exactly that if I have two saddle connections on one side of a wall,  whose inner product is one, 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.  I'm not saying intuitively why it has to be in the middle, not one of the sides.  Oh, because so you have a linear map.  So, you have a, sorry.  So, you know, the ordering is controlled by the phases of these z gammas, right?  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, then because it's 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 going on here.  And that's what controls the ordering of all of these products.  The phrase is circle, it means between and the circle.  Sorry?  Phrase is circle, it is not between there.  Well, there's no, there's no between, but here when I drew this path, you know,  that path kind of breaks the symmetry of the circle.  That's a circle, no.  You know, it's small interval.  Yeah, I mean, you're just looking at a little interval where you're taking this contractible 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 one, k gamma two to the power one.  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, indexed 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 one.  And then in the middle, you have to put one special thing which occurs with the exponent of minus two.  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 minus two.  So, okay. And the claim is that, in fact, 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.  So, then in that definition of k, there has to be some deep thought that we can understand and we can tell us.  Right.  The definition is given.  Suppose I know nothing but the definition.  I come and 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 two, gamma one plus gamma two and gamma one.  So, 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?  You mean, in the second example, you have gamma one plus gamma two equals two.  Right.  You already know that right-hand side will be like that.  Well, okay.  So, how does one know...  How does one derive this formula?  Just as a purely algebraic question, how do you derive this formula?  Feel it somehow.  I feel it.  How do you feel it?  Yeah, 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 a purely algebraic question,  how to find the thing on the right side from the thing on the left side.  But this thing can inform you, just like that.  But this formula looks a little bit like a composition depending on which borax of algebra you choose.  Like normal ordering, if you mean gamma two is positive...  He's talking some feeling here, so he feels it somehow.  Gamma two is positive...  No, it's not that they're positive and negative.  They're all positive.  They're all positive.  I mean, it's more like the following.  No, but if I think...  No, he means upper triangular lower triangle.  No, but that's not what it is.  They're all upper triangular.  That's exactly what I'm saying.  So, I mean, 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, no, no, no.  But I mean, here's the sort of...  They're more prism...  Sorry, just the group of...  They're automorphisms of the torus, but they accidentally...  They also preserve some symplectic structure, which I haven't mentioned.  But say, yes, but I think the information in the source is a cream-orange group, right?  Yeah, no, no, no, but here's the completion, because I think...  You see, there are things in formal power series.  Yeah, she's...  There's some completion with any sort of old Baratian...  No, no, no, no, no. ...perimorphisms of the torus.  Yeah, it's...  So, what's the speciality?  They're complex.  They're symplectic.  They're complex.  Yeah.  That's right.  Yeah, 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.  And 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 kind of finite dimensional analog of what we're doing here.  Yeah, you can order positive rules.  Yeah.  Excuse me.  You said that on the right hand side, you have k gamma 2 and k gamma 1 in the opposite order.  That's right.  But in the last line, I don't see them at all.  Oh, sorry.  Excuse me.  This should be...  The product should be from 0 to infinity, not from 1 to infinity.  Thank you.  Sorry.  Could 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 braid groups?  Let's say I'm not aware of a connection with braid groups.  So, so, okay, so, so, so now you can ask, you know, why is it right?  Why do these...  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, 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 at 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.  So, we're just going to do something else with quadratic differentials.  Okay.  So, so first of all, actually, we're not going to quite just use the quadratic differentials.  We're going to rather use Higgs bundles.  So, it's just a slightly fancier object.  So, 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, 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, you know, locally, what does it concretely mean?  It just means I'm, we're making a matrix, a matrix of one form.  A matrix 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, that, so a pair like that we call a Higgs bundle.  So, okay, 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, 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.  Okay, so instead of the quadratic differential, we'll think about these Higgs, these Higgs bundles.  And now the spectral curve which we used before, so before I was looking at the, this double cover that just came by taking the two square roots of the quadratic differential.  So now another way of, 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, of my Higgs field.  So, okay, so the point of the spectral curve are the eigenvalues of the Higgs field.  Okay, so okay, so the, you know, 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, the, the space of Higgs bundles is not just equal to the space of quadratic differential, but rather it's a vibration over the space of quadratic differential.  So by passing from phi to, 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 exact, form a line bundle over the spectral curve.  And so what we get here is really a complex inter-rubble system.  Roughly what you have is the Jacobians of the spectral curves, which are some, some tori.  But today we're basically only interested in this phenomenon that takes place just over the base, just over the space of quadratic differentials.  And so I won't care much about exactly where I am on the, on the torus fibers.  I'll just take any Higgs, any Higgs field to correspond to that quadratic differential.  Okay.  Okay, so, so 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.  So, okay, maybe it's unnecessary for this audience, but I remind you what is a, what is a flat connection?  So again, locally, so a flat, a connection is something that allows you to differentiate sections, to differentiate sections of your, of your bundle.  So locally, again, if I've trivialized, if I've locally trivialized my bundle, then I could write the, so then its sections are just some little two by two, some little two by two vectors of functions.  And to differentiate one of them according to this section, well, 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, you know, if, if, if locally I can find a basis for my bundle e, which is annihilated by the connection.  So, you know, a flat connection, in other words, 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.  Okay.  And so, you know, if I'm given a flat SL2C connection, you know, one of the things I can do is I can take its parallel transport, by which I mean, you know, sorry, I said parallel transport, but I should really say the monodromy operation.  So, you know, I started in some neighborhood, I have the local solutions of this equation, nabla s equals zero, then I, 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, but they, they differ from the original solutions just by an element of SL2.  So, what we get is a, 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, right?  The power gauge to summation makes no sense.  Up to isomorphism.  Yeah, up to equivalence of connections.  So, but on the other hand, the map, you know, concretely, if I'm, 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.  Okay, so I was, I was talking on the one hand about 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, in the cases with singularities, which we'll come to in a moment, it's Simpson and Descorps 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.  Suppose someone 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, it's, this is going to be a c star family.  And what that family looks like, well, it looks like this.  It's of, it's of this simple kind of a three term form.  So, here we have 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.  So, 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, you know, roughly the, to be flat, says the bracket of Nabila with Nabila 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, 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, so, given the 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,  where the quadratic differential had some singularities.  And so, correspondingly, the, the, the Higgs field that we consider will have some, will have some singularities.  And so, corresponding to that, you know, we don't get connections on the whole curve, but we get connections with singularities at the same points.  So, you know, if the, 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.  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.  So, there's 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, 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, the Gage invariant information.  So, it's replaced by some, something a little more complicated, the so-called Stokes data.  I don't want to say, 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 zeta goes to zero.  But independent of that, there's also this, this other Stokes phenomenon that occurs here.  So, this kind of, this kind of example sounds harder, but actually it turns out that the most tractable examples are, are this kind of example.  Because here you can get something non-trivial, even if you take the curve to be Cp1.  Right, 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.  Okay.  So, 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 with zeta, 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, you know, because it's, 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 as zeta goes to infinity, but there's a symmetry exchanging zero and infinity.  So, doing zero is essentially enough.  So, understanding this is like, well okay, we want to understand this.  Okay, so what's the idea?  Well, the idea is, so this family of connections comes to you in this explicit form.  And so in particular, you know, 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, so very naively, you would think that to find a solution, it might be enough just to deal with this, just to deal with this first term.  So, what does it suggest? Whoops.  Oh yeah, sorry.  What it suggests is that, you know, 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,  no, 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, 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, the solution will not exactly be given by that.  But what you might try is to make a series in zeta whose leading term is given by the thing that I wrote.  So, let's try, so suppose you try that.  So, 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.  Okay, 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.  I mean, 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, you know,  all the dependence on a base point is only here, only in this integral.  And so, it would say that, you know, in order to evolve the equation,  in order to work out its monodromy, all you have to do is evaluate some integrals of these forms.  All right, 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 where you can 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 one and lambda two,  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.  They can't exist because of this obstruction from the monodromy.  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 say,  okay, well this is 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 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 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, I mean the following claim,  so this is part of the joint work with Gato and Moore.  The claim is that if I, 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 zeta 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't 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 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 you can come back to the original question.  The question that we started with was, well, sorry,  not yet 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 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.  And now you can 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 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.  And 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 sort of 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,  but 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.  So, 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 whole NOMI function,  and you find that the sort of good way of studying the asymptotics  is indeed to do this WKB procedure.  You just relate it 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.  Okay.  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.  Okay.  So, there's another way of thinking about this construction.  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 you have an SL2 connection over a curve,  there's a sort of useful way of organizing its invariance.  So, organizing the whole NOMIs around some path,  namely the trace of the whole NOMI around some path gets expanded in terms of some simpler objects.  So, these things here are the whole NOMIs of some 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 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 spectral cover.  So, this is the kind of the slogan for this is that 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 walls.  And this cutting and gluing is exactly taking care of the Stokes phenomenon.  The fact that 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.  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 data of the SL2C connection over the curve is just equivalent up to,  the data of this connection up to equivalence is equivalent to the data of the C star connection  up to equivalence over the covering.  And so, what that says is that there's some coordinate system on the moduli of,  on the moduli of flat SL2C connections just identified with 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 fucking Gondroff.  Okay, so this abelianization 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 abelianization.  Okay, 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, find you in theta just a little bit,  then the spectral network just changes by an isotope.  And a variation like that 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.  All right, fine.  Yes, I'm...  But on the other hand, so as I vary these parameters further,  I might cross one of these critical phases where a special trajectory appears.  So 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 of expanding the holonomie  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, you know, if I change theta,  I don't change this connection at all.  If I move in the hitching 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,  oh, 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 1 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,  if you like, 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  of this k gamma now rates to the power of minus 2.  So in general, as we move around in parameter space,  the spectral network deforms,  and the jump of these function...  these x gamma, which before they were formal coordinates on some torus,  now they have a concrete meaning as the whole of the 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.  So in particular, now if I come back to the...  so 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,  my... these x gammas parameterizing this abelian connection have to jump,  and they jump by exactly this k gamma.  On the other hand, 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 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  Kansai-Schabelman-Syntlektomorphism formula.  Well, and the explanation is that if you lift it to this connection,  it is just a statement that nothing changes.  That's right.  So the statement is that if you study a single non-abelian connection,  or a non-abelian, an SL2C connection which varies continuously,  that 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 WKBSyntotics.  But even if you don't study WKBSyntotics, there is this procedure.  Now, that procedure depends on the choice of spectral network.  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.  Yeah, that's the statement.  So, okay.  So what have I said?  So we were considering some counting problem that had to do with  these special trajectories of quadratic differentials.  And what we found is that, first of all,  they obeyed this Kansai-Schabelman,  this kind of universal wall-crossing formula.  And moreover, that if you want to explain that formula,  you have to see it arise in some natural way.  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.  So that suggests a kind of natural problem,  which is what if I wasn't just interested in SL2,  but what I was interested in SLK,  or more generally, I mean, it could be any league group,  but let's just start with SLK for K bigger than 2.  And so here, following the exact same kind of strategy,  you meet a new kind of counting problem,  which so far hasn't been much explored.  So these objects here, these special trajectories,  are a kind of classically studied thing.  People in Teichmeler theory have studied them a lot.  But the higher-ranked analog,  which I'm now going to briefly tell you about,  has been relatively little explored.  I think it's an interesting problem.  Okay, so let's just start again.  So now, I'm going to start with the Higgs field.  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 Higgs field.  So, you know, locally, one way of thinking about it is that,  well, the covering is the same as,  giving the covering is the same as giving K  just holomorphic one-forms,  or more concretely, the Higgs field.  Just take the eigenvalues of the Higgs field,  that gives you K 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.  And now, so it seems,  so in this work with Gaiauto and Moore,  we've tried to generalize the whole WKB story  that I just told you to this situation.  And it seems that it works,  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 case K equals 2.  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-ranked version  of those trajectories.  So as before, we're going to fix a phase,  and now we'll define a network.  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.  Now I'll describe them by differential equation,  and they're not anymore leaves of a global foliation.  Now they're going to be carrying some label,  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, well, it says this.  So you take lambda i minus lambda j.  That gives you some one form, and you have to do that one form  when evaluated on your tangent vector,  tangent vectors to your lines should have phase theta.  So that equation, that's a first order equation.  It defines for you, again, locally a foliation.  You're taking the leaves of this foliation.  Now globally, the different kinds of line can mix.  And then, so now, the local structure around the branch points  is going to be much the same as it was before.  At each branch point, now the branch points are labeled  by two sheets of the covering.  So again, let me assume that I'm in the situation  of simple ramification.  So we only have places where two sheets collide.  Sorry, you have a foliation on the spectral curve?  No, there's no foliation on the spectral curve  because the leaves of this foliation  depend on a choice of two sheets.  So if you like, it could be on some symmetric product.  But I prefer to think of it just directly on this curve itself.  So each of the branch points is a branch point  labeled by two sheets.  And the foliation with label ij around this branch point  is going to be much the same as it was before.  So we have three prong singularity.  And in particular, it has three distinguished trajectories  which are going to be among our stokes lines.  So 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 branch point.  And now, of course, globally, I would have a problem  of trivializing this cover.  So I'll write explicitly, enjoying this picture,  I'll put an actual branch cut.  This one is labeled by ij.  This one is ji.  And this one is ij again.  So they alternate.  And if you go through here, you cross a cut.  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.  But sometimes, there are not any more leaves of a global foliation.  So it may happen that they would actually cross each other.  And if they cross each other, if I have an ij one and a jk one  to intersect, 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.  And in addition, we're going to generate a new line, a line of type ik.  So this is, in a sense, this is kind of an echo of something  that occurred in the story of two-dimensional wall crossing  a long time ago.  It is essentially, by that formula, exactly.  So, yeah, somehow in the abelianization procedure,  it's exactly this formula which is forcing you to give birth to these lines.  But recently, we learned that this had actually been studied before.  So people in WKB had noticed this phenomenon.  OK.  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, so you let them evolve according to this differential  equation, but when they cross, you have to give birth to new ones.  And now, so here's an example of what this procedure might look like.  So here I took the case of just a three-fold cover.  So this is a Higgs field on a rank three bundle.  And I took the curve to be, again, the only things I can plot really are,  take CP1.  So here it's stereographic coordinates.  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 covering has a total of six branch points.  Those are the six X's that you see here.  And what you see is that the thing, at least in this particular example,  the behavior is defined by a priori, a totally crazy procedure.  But at least here, the behavior is actually reasonably well under control.  So the trajectories travel around.  When they collide, they can give birth to new trajectories.  But eventually, all the trajectories get attracted down to these singularities.  This would be a both-patching system.  The sphere with three mark points.  Yeah.  SL3.  SL3.  SL3.  Right, exactly.  It's SL3, a sphere with three mark points, and with singularities of the most generic kind  at those three points.  So first-order poles with generic eigenvalues at the three points.  So, OK.  So this, at least in this particular example, the behavior is a lot like what we had  for the trajectories of quadratic differentials.  Now, it's an unfortunate fact that not every example is so well behaved.  But at least there are some, so in general, it may be that this procedure defines for you  actually a dense set of trajectories.  And in that case, we just have to live with that.  But at least there are many good examples in which it's really like what we had  for the quadratic differentials.  So this is the object that you study.  This is the object that gives rise to the stokes phenomenon in these higher-rank examples.  OK.  So is it true that any trajectory is going to a pole?  I think it's, I believe it is true.  So as I say, there are really no theorems about this.  Experimentally, I believe that it's true that if you have the poles, 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.  We first thought that this can't happen, but it now seems that it actually does happen.  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.  And there are some examples in which it never happens.  There are some parameter spaces where it doesn't happen anywhere.  OK.  So now as before, just as before, we could study the solutions of Hitchin equations  and the corresponding families of flat connections.  And just as before, the asymptotics are going to be controlled by a kind of a  bilionization.  So you express the monodromes of an SLK connection downstairs in terms of some  simpler quantities, which are holonomes of an abelian connection upstairs over the  spectral cover.  The problem is we're solving now the same equation of the nabla S equal 0.  Exactly.  The connection is SL3.  Yes.  SL3 or SLK, yeah.  SL3.  Right.  We look in WKB.  We're not solving exactly.  We want to look in WKB.  Well, we are solving it exactly.  The statements that we derive from this are exact statements.  I mean, this formula that expresses the monodromy in terms of the monodromes of an  abelian connection, that's an exact formula.  I mean, you derive that formula.  What leads you to consider that particular construction, the motivation is that you're  interested in studying the WKB asymptotics.  But no, the point is that there really exist actual solutions that have those asymptotics.  So this relation between the monodromy downstairs and the monodromy upstairs is an  exact relation.  And then again, we have the same kind of story that we had in the Rank II case.  So you have this network that gives you the network of stokes lines, controlling the Zeta  Go to zero asymptotics.  And when this network jumps, so now again, we can ask, suppose I vary parameters, how  does the network change?  And the network's topology may jump, and at that moment these X gammas also jump.  But as before, the cool thing is you can study very generally what will be the form of  that, what will be the form of the jump.  And you find just as in the Rank II case, that they're always controlled by these kind  of transformations that appeared in the Kinsavich-Soylman formula.  It is the same K. It is the same K.  Because I mean, after all, this K, as it was defined by Kinsavich and Soymelman, was just  an automorphism of a torus, a binaural transformation of a torus.  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.  And so no, it's really exactly the same formal structure as there was in the case of Rank II  connections.  Is there a way to understand this procedure?  I mean, I believe the addition is a very rough procedure.  I would say that we would go from SLN to SLN minus 1, SLN minus 10.  Yeah, so I'm sorry that I didn't include more about exactly how you do this.  I've said it in a slightly rough way, but it's an absolutely precise thing.  I mean, is there a way just to simplify first?  I mean, you have your SLK connection, so is there a way to construct a cover where you  have SLK minus 1 connection?  I mean, all the levies of all the parabolic?  OK, so there's something like that that will happen if you consider a point of the Hitchin  base, which is very special.  So imagine that you consider the case of GLN1 and 2.  Then you can reduce the thing to GLN1.  What you have to do is consider some very special points of the Hitchin base where the  spectral cover has N1 sheets together, where it's a non-reduced thing that's like N1 copies.  So then there is a kind of stepwise version of the thing.  So now the formal story is just exactly the same as it was before, except so now 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.  Now in this higher rank situation, the classification of possible kind of topologies is somehow  much, much more complicated.  So let me show you the first new thing.  So here's the situation.  This is again in rank k equals 3.  And I'm just studying this little cluster of three branch points.  And here we see the following thing happen.  So right now the topology jumps.  And so the new object that appeared at that critical moment was an object that looks like  this.  So we've got these three branch points.  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.  At the next one, it's the pair 1, 2, 3.  At the last one, it's the pair 3, 1.  And there's some critical moment.  The critical moment is the moment when these three trajectories come together at a common  point and they make this kind of three string network.  And that three string network is the object that's kind of responsible for this jump in  the topology.  So now we're finding a new kind of invariance, the invariance that count these jumps.  This isn't the kind of object that contributes.  This is a new kind of object that contributes to these invariance.  But that's not the only new kind of object.  So once you allow that topology, you have to allow much more general, well, sorry.  You just study this thing and you see what you get.  But it doesn't include just that.  It includes, certainly, you could have a more general kind of tree.  You could have something like this.  You could have something with five vertices.  You could have something where you attach a loop to it.  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.  There's sort of arbitrarily complicated structure in this problem.  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.  And so in particular, let me show you one particularly sort of crazy thing.  Oh, yeah.  So this is the picture that I just drew.  So this is the sort of before and after picture of this spectral network.  So this one, if you work it out, by the way, this one contributes just one.  And very only equals one.  So it's really just like the sandal connections.  But in general, you get 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 looks like this.  So this is the locus along which the trajectories are kind of hitting each other head to head.  And in this case, you go away and you calculate the procedure for calculating the jump of the  abelianization map.  And you just calculate it in this case.  And what you find is that it is indeed given in this same formula, 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,  which are exponentially growing.  So the story in this higher-rank case seems to be somehow much more intricate and much less explored.  Nevertheless, 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 invariants.  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 just showed you a slide ago also could have been  derived, in principle, just from the wall-crossing formula.  So here, it's wall-crossing between two things that have inner product three.  So you have two things, called gamma 1 and gamma 2.  You consider this product of this simple product of transformations, and now you try to rewrite it as a product in the opposite order.  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, you find this product of transformations with exactly these coefficients.  So there's some new counting problem here, which is still part of this whole Donaldson-Thomas invariant structure,  which is able to give you fairly complicated stuff.  So let me just sum up.  So I told you in the beginning what this wall-crossing phenomenon is, where it came from in physics.  I didn't tell you much.  And then I described some particular examples.  For a particular physical theory, the whole story becomes very geometric.  And so we studied wall-crossing for the special trajectories of quadratic differentials,  and then more generally for the degenerations of these new objects, which we're calling spectral networks.  So they both come sort of equally naturally from quantum field theory.  And so in each case, one way to think about the objects which you're trying to count is they have something to do with the asymptotics of the monodromy  for this system of linear differential equations with a small parameter.  And from that point of view, the wall-crossing formula, once you sort of absorb this way of 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.  So as I said, the special trajectories of quadratic differentials are kind of well-studied subjects,  although I think the statement that they obeyed the wall-crossing formula hadn't been noticed before.  So now that story is somehow connected to the theory of Donaldson-Thomas invariance.  And then there's this new theory for 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 had studied this higher K problem quite a lot.  Okay, great. So I don't understand completely what they get, but there exists voluminous literature.  Great, so then correct this.  So I so far don't know very much, and by connecting with these guys perhaps I'll learn something.  Yeah, thank you.  Okay, I think more about other questions.  Any clue what is this omega in the question of quantization of the fictitious system?  So far I don't understand the relation between this story.  It's very similar what you're doing. You're almost quantizing the fictitious system.  Yeah, yeah. So I mean, so yeah, no doubt these things have some role to play in the story of the quantization,  but so far I don't know what it is.  Okay, I have another question then. In WKB theory you have a small parameter,  but it's not clear you have a symmetry between zero and infinity in this data,  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 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.  So I don't know whether, I mean, here I was describing a pretty detailed thing about, you know,  precise formulas for the asymptotics of the monodromy.  I don't know, so the proof that, so we gave an argument for why these local solutions exist.  And in that argument, we use the fact that the thing has this symmetry.  So it's apriori possible that the results that I told you really only hold for the special class of things that come from Hitchin systems.  Or it might be that it works for every family, I just don't know.  Does the abilityization map depend on the fact that it has, is this special formula?  So the abilityization thing is a totally general thing which you can do as soon as someone gives you a spectral network,  and as soon as someone gives you a...  Would that be the same, would you get the same thing if you use any complex WKB problem with the same...  Yeah, I think, I mean, you could formulate that whole procedure,  you could indeed formulate that whole procedure without the structure in infinity.  But would you get the same, the abilityization map, would it be the same map?  It would be the same map, yeah.  But you have the sine sigma, which was minus the male case, and in general it is not, you know.  The sine, no, we've worked out that sine, so that, I mean, that sine ultimately has to do with a tricky thing in WKB,  so I told you that when you have a flat connection, you abillionize it by a flat connection over the spectral cover.  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 whole nomenus minus one around the branch points.  So, 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 sine.  But the process of getting rid of that sine introduces this sigma gamma somewhere else.  So, you can either work with these twisted, these sort of twisted flat connections.  Flat connections would have this minus one everywhere, and then everything is totally canonical.  Yeah, exactly.  So, in WKB, this is a well-known thing, but it occurs generally in this abillionization process,  even if you don't think about it as WKB.  That sine is just in there.  If you want to get rid of that sine and just deal with ordinary flat connections, you can do it,  but the price you pay is this little sigma gamma.  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 completely agree with you that for second-order differential equations,  indeed, it can be bypassed.  So, there's a more efficient way of completely avoiding this whole scheme.  So, I wonder perhaps in your problem, it might also help.  Yeah, I don't know.  I mean, our interest in these structures was, of course, partly motivated by WKB, but partly not.  So, yeah, if it turns out that this stuff is not necessary for WKB, I mean, that's okay with me.  If you do the hydrogenous scheme, instead of starting CP1, let's take the torus with one markpoint.  People know that there are relations.  I mean, in some other quantization equations, there is a relation between torus with one markpoint  and a sphere with four forms.  Do you see those things here?  Or you don't study the hydrogenous case?  The hydrogenous case, we've done a little bit.  I mean, we certainly did study that particular case of a torus with one puncture for SL2.  Is it somehow the problem is same as the sphere with four markpoints?  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 singularity.  Yeah, you take regular singularities for both, but then I think you have to impose some further relation before they become the same.  There's one thing that I know which is that the moduli of SL2C flat connections on the four punctured sphere is,  if you write it concretely, it's like an affine cubic surface.  But more precisely, if you fix the eigenvalues of monodromy around the four points, you can…  You have to fix the resultant friction on there.  Right.  Recalculate.  So this is a relation we're equal to star angle equal to four.  Yes.  Right.  So I put that if you put some restriction on the…  So then the affine cubic surface, I mean, for the… for the…  Taurus 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.  Sometimes it is possible to refine WKB by some non-perturbative corrections.  Can you comment?  I mean, I think in a sense…  In a sense, the stokes lines that we're drawing are exactly the kind of non-perturbative corrections that you need.  I mean, I don't have a direct comment, but…  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.  So I'm writing here…  We're writing here a formula for the trace of the holonomy around some loop.  Now, 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 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 subleading 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 Gertrude series.  No, no, even in this case.  I mean, well, the WKB series 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…  No, no, 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.  Thank you.
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