 when this critical point crosses the real line. So Fi, these solutions, Fi, are only piecewise, piecewise continuous. The jump on the other hand is of a pretty simple kind. For example, here, if you think about it from this contour C2, the integral over this contour C2 is actually equal to the sum of the integral here and the integral over that contour. So in this case, so in this explicit case here, F1 stayed constant, but F2 jumped by, plus or minus, I'm not sure, jumped by F1. So you've got a basis of solutions. It's a nice basis of solutions with nice uniform asymptotics. This is the good news. And what you could think of as bad news is that you don't get those asymptotics from everywhere-defined continuous solutions. Rather, to get those asymptotics, the solutions have to suffer some jump somewhere. So that's sometimes known as the Stokes phenomenon. If you think about how these things can be consistent, you could get confused because on the one hand, we're saying that these things have absolutely uniform asymptotics. This asymptotic is perfectly continuous. On the other hand, we're saying the solution itself is not continuous. How can that be? Well, if you think about what happens at the moment when the thing jumps, F2 shifts by a multiple of F1. You know the asymptotics of both of them, and if you work it out, what you'll see is that the asymptotic behavior of F1 is massively subleading compared to the behavior of F2. In other words, W at X2 is bigger than W at X1. Therefore, this contribution, at the moment when the jump happens, this contribution is exponentially small, and so it doesn't prevent the asymptotic behavior from being uniform. Yeah, you could analytically continue them. Absolutely, so what Greg is saying is there's another thing I could have done. I could have insisted, so in Greg's version of the world, I would have taken this function somewhere and then analytically continued it everywhere in the complex Z plane. So then I would be happy here. The thing would be absolutely, you know, would be entire, would have no jumps, but what I would not be happy about is that then, eventually, it would fail to have the right asymptotics. When I continue it far enough, the asymptotic behavior would be messed up. So I actually, for our purposes, it'll be more convenient to be in this world of things that jump but have good asymptotics everywhere rather than to have this entire thing, which is somehow non-canonical. You have to make some arbitrary choice of where to start. Okay, so now, if you think about where this jump happens, so what I just told you is that somewhere in the Z plane, there's these canonical solutions, totally canonical solutions, but somewhere in the Z plane, they have to jump. Yeah, yeah, yeah, you would just let the contour continuously deform. It might actually be branched, by the way. I'm not so sure that that's an entire function because when you move Z, so imagine I start with the, oh no, it is entire, no, of course, there is an entire function, yes, yes, yes, sorry. Yes, absolutely, there's an entire function of Z. The equation has no singularities. Yes, forgive me. Okay, so what's the story? Well, the jumping occurs exactly when the left-shed symbol from X i runs directly into the critical point X j. That's the moment when the jumping occurs. But that's equivalent, I'm not gonna prove this for you in real time, but you can work it out. That's equivalent to saying that there exists a BPS soliton in the sense that we were talking about before. A BPS soliton connecting X one to X two, sorry, I just called it a BPS ij soliton before. BPS ij soliton with phase, the phase of the central charge, theta equals argument of h bar. So finally, the picture is like this, is there's a canonical basis, a distinguished basis, canonical basis of nabla h bar flat sections, which jumps whenever there's a soliton. In other words, so I'll call these solutions psi i of Z. The basis jumps when Z crosses, well, when a BPS soliton appears, but that's what we call the spectral network. Sn of theta, where I take theta to be equal to the argument of h bar. And this canonical basis has kind of good asymptotics, good well understood asymptotics, like the asymptotics that wrote there, as h bar goes to zero on a ray. So you fix a ray, you fix the phase, and then you ask what happens to the asymptotically as theta goes as h bar goes to zero. There's a nice basis for which you control that asymptotics, but that basis exists that basis is single-valued only in the complement, only in the complement of this picture, so like, so this picture, so for that reason, this picture actually has another name, it's called the Stokes graph. It's not in any coordinates, it's on the bundle E. It's in the bundle, yeah, sorry, so this picture is happening over C, so nabla h bar. So remember, this is a family of connections labeled by h bar and C star in the bundle E that I described for you before over C, the parameter space of field theories. So the statement is that this parameter space of field theories is carrying this canonical kind of pencil of connections. Yes, yes, Z, yeah. Absolutely, this is in the Z coordinate. Right, that's an important point actually. We use the X coordinate mainly as a kind of crutch for getting a very sort of explicit understanding of why these Stokes jumps happen. But ultimately, the statement E is just that there is this canonical basis of solutions of the equation which jump exactly at this special network. Okay, the X's are not in this picture. Were you to try to write an explicit formula for the solution of the equation? In these particular class of examples, you could write them as integrals over X. But ultimately, the equation that we're solving is an equation in the Z variable that doesn't have any X's in it. Yeah, okay. So, okay, so let me come back to this phenomenon of the changes of basis. Let's say one more thing about it. So, when the parameter Z crosses one of these walls, one of the walls of that picture, carrying the label IJ, so it carries an IJ soliton, the basis of solutions of this equation is transformed. So there's a matrix giving the change of basis. And that matrix turns out to be pretty simple. It's the identity matrix plus the elementary matrix EIJ times the soliton number mu of I and J. So remember, each of these walls represents some bunch of solitons. It has a number associated to it. We defined those numbers last time, which is the soliton number. And that soliton number is what controls the explicit change of basis here. And now in particular, I guess I'll refer you to the notes for details of this. So in particular, now you can do the following thing. You imagine a situation where three walls come together, say a wall of type one, two, wall of type one, three, one, two, one, three, and two, three, sorry, one, two, two, three, one, three. So in fact, let's do even the simpler case. Let's do just this case. This is the case that we saw explicitly in some pictures last time. And I said it was associated with the phenomenon where of wall crossing, where a soliton in this range of parameters doesn't exist, and then in this range of parameters it does exist. And I told you there's a general formula that you code you off a wall crossing formula that tells you exactly what the jump is gonna be. Now, we can re-derive that just from the stuff we've been saying about canonical solutions of this equation, namely what you do is you consider the change of basis going this way, and the change of basis going this way, and you require that they be equal. They have to be equal because there's a distinguish basis here, there's a distinguish basis here. There's one change of basis matrix between them. You can compute it by going this way or compute it by going that way. Requiring that they're equal. In other words, just requiring that you have an actual flat connection here. Produces, tells you what this mu one three has to be. So if you consider the change of basis along two different paths, you get that you code you off a wall crossing formula. Well, maybe, but maybe, well, I mean, this is at a fixed value of, yeah, okay, so there's a wall of marginal stability that kinda looks like this. That's where the mu jumps. Okay, so there's a little calculation here. It's just an identity among three by three matrices, which you can find in the notes. Okay, but let's press on. Yeah, no, you're computing what the number mu one three is gonna be. Yeah, the fact that there's a wall here, that's just dictated by the, you know, these paths are trajectories of some differential equation that I wrote last time. And just knowing that is enough to work out the kind of local structure here that if you have a one two wall going this way and a two three wall going this way, there is some one three trajectory that emerges from here. And the only question is whether the soliton number here will be zero or not zero. You might first think it'll just be zero. There was no soliton here. There should be no soliton here. It turns out that that's wrong. And you can see that it's wrong just by considering these changes of basis. Roughly the change of basis matrix here doesn't commute with the change of basis matrix here. So multiplying them just in the opposite order won't give you the same, won't give you the same as the product here. To fix that, you have to stick in one more matrix and its coefficients are fixed by all the other coefficients. So it's just a little identity among a three by three matrices. Okay, yeah. I'll just, it is the same for, I'll just write the, I'll just write the thing. Yeah. So the two changes of basis on the two sides would look like this. On the left side, you would write this matrix. Let's see, yeah, I wrote it the same. Let's make this two three, this one two, this one two, and this two three. Now the two changes of basis matrices look like this. The one on the left is a product of two elementary matrices. So there are some, there's some mu here along this wall and some mu here along this wall. And those appear, I'm calling them mu one two and mu two three. That's the change of basis that you have on the left side. On the right side, you go through the one two wall first. And so the order of the product is backward. Oh, but the thing that's first is when you write last. So mu one two, mu two three. Now this by itself would not be an identity. These two products are just not equal. So what saves the day is to stick in a third matrix with a, I'll call it mu prime one three. So then this becomes an actual identity. And it's true exactly if mu prime one three equals the product mu one two, mu two three. So that's a special case of the wall crossing formula that I wrote last time. So that's how it comes out. So it's just some identity just multiplying three by three matrices. Okay. So now there's another aspect of this story which is gonna be important when we upgrade this thing to sort of more interesting Higgs bundles, which is the following. So there's something called TT star geometry discovered by Chikodi and Vafa. And what they said was that in addition to all the other structure, I told you that this Higgs bundle has, it carries one more thing. Well, I guess this will be the least surprising thing now. It carries a Hermitian metric. So far everything was kind of holomorphic, but now we'll do something that's not holomorphic. A Hermitian metric H, which obeys, well the TT star equation or the also called the Hitchin equation, the sort of higher dimensional Hitchin equation. And what that equation says is that the curvature of the churn connection plus the bracket of the Higgs field with Phi Dagger, Phi Dagger I mean is adjuvant with respect to this Hermitian metric is zero. So yeah, so by DH, just to write it, so DH is the churn connection in E, it's the unique connection in E that's compatible with the holomorphic structure and the Hermitian metric, so the churn connection. Phi is holomorphic, covariately holomorphic. That's implicit in, no, no, I don't have to write that separately. It was already a holomorphic section of the bundle, the churn connection is compatible with the holomorphic structure. So that says that Phi will be covariately constant. Now I want to reinterpret this equation in the following way. Well, sorry, let's say one thing first. So just to give you a concrete feeling for what kind of equation this is, in the cubic case, we could write this metric literally. In that particular case, this metric happens to be diagonal. It doesn't always happen. And in that case, the equation star, this might feel a little abstract. So here, the equation star becomes literally a scalar PDE, the Laplacian of U, del del bar U, minus E to the two U, minus E to the minus two U, absolute value of Z squared is zero. So it's just some concrete scalar PDE. Yeah, this one is actually, this is an interesting sort of model solution for studying Hitchin's equations. It's kind of the simplest example of this structure. Anyway, in general, we have some complicated PDE for our Hermitian metric in this bundle. What I want to say about it is, I would like to study that PDE in a slightly indirect way. So here's what I would like to do. So having a solution of Hitchin equation is absolutely equivalent to the following. It's equivalent to saying that the connection, the complex connection, nabla, I'll define nabla zeta, so zeta in C star, I'll define it to be the churn connection, which is unitary, plus zeta inverse times the Higgs field, plus zeta times the adjoint of the Higgs field. The statement is that that connection is flat for all zeta and C star. So having a solution of Hitchin's equations, this sort of difficult PDE, is just absolutely equivalent to having this one parameter family of flat connections that comes to you in this particular form. And now, this kind of family, we can study it in essentially the same way as I studied that family nabla H bar. If you look at the behavior at small value of zeta, it looks very similar to the small H bar behavior of the family I wrote before. And indeed, this is basically due to it's proven, I think. We can study its flat sections just as we discussed for nabla H bar. Now we won't have these kind of explicit formulas where you integrate over some contour, some explicit function. But nevertheless, they have the same Stokes graph, the spectral network, the same kind of formal asymptotics, same actual asymptotics, canonical basis with the same asymptotics and jumping behavior. And so if you had a good enough control over those, if you had a good enough control over these flat sections, which you can get just by studying this Stokes behavior, you can actually use it to reconstruct the solution of Hitchin equations. So the idea is to use this to reconstruct H. So that's the real goal why we're doing all this. We want to get a better handle on, that's one reason why we might do all this, is we want to get a better handle on the solutions of Hitchin equations. We want to understand them in a more explicit way. And the idea is that through this kind of indirect sounding business, first study this CSR family of flat connections, then control the behavior of the flat sections for these by knowing something almost just topological about their Stokes behavior. That'll be enough to let us, well, you still want to get an explicit formula, but you'll get information about H that you couldn't get easily in any other way. And so this is a program that was, this is still kind of an old story. It was basically explained by Jakodi Vafa and Dubrovn. Okay, so now in the next lecture, just the aim is gonna be to upgrade this whole story. So we'll replace these parameter spaces C by a general Riemann surface. And E by a general Higgs bundle. And that will involve somehow structures, which are very, very similar to what I've described here, but just with one new ingredient. And maybe I'll, so I had hoped to get much further, but maybe I'll show you a picture of the new ingredient, just so you get some idea what the deal is gonna be. So we're gonna draw spectral networks. Now there'll be spectral networks on the Riemann surface, whatever the Riemann surface is, but as usual, the simplest examples are the ones you can actually draw. So here by Riemann surface is CP1 with some punctured infinity. And we're gonna have, again, a kind of spectral network, which is counting some kind of particles. Now there'll be particles, as I'll explain in a, not just a two-dimensional theory, but a kind of coupled system, which is partly two-dimensional and partly four-dimensional. That'll be the interesting.