 picture, I'm showing it to you at some fixed value of u. So again, I'm showing it to you at u equals 1. OK, so in this case, the sigma u is given by the equation lambda cubed plus phi 2 lambda plus phi 3 equals 0. So that has some branch points where the discriminant of this is 0. The discriminant is the cube of this minus the square of that. So it turns out there's four discriminant points. And for each one of them, we shoot out the trajectories. And what do we get? Well, we get some slightly more interesting picture. Right, first of all, the things can cross each other now. They couldn't before, because before, they all satisfied the same differential equation. They lived in a global foliation. But now, they satisfy different equations, so they can cross each other. And when they cross each other, I told you our rule, when they cross each other, we just shoot out another one. And so sure enough, here, when they cross each other, we shoot out another one. And we get some picture. OK, slightly more complicated looking picture, but fine. Once again, we can kind of vary the phase and see how the picture changes. There are some critical moments where something happens. Well, it's a picture in the patch of CP1 that's missing infinity. So it's a patch. Yeah, so we're taking our curve on which everything is happening to be CP1, but we're allowing everything to have a singularity at infinity. So all the pictures are drawn in the complex plane. No, no, no, they don't come back. Yeah, good. So I mean, that's a dynamical question you have to answer in each example. In this particular example, the singularity is strong enough that it attracts them and they never come back. That's actually what makes these examples much easier is the fact that they have the singularity that attracts all the lines. Had we not had that singularity, the lines would tend to kind of wind ergodically around the Riemann surface. That makes the pictures much harder to draw. OK, so this is our new notion of spectral network. It's introduced in a paper of Greta Moore and me. OK, fine. So now, as I've alluded to, there's a new phenomenon that occurs here that didn't occur in the sort of pure two-dimensional examples that we talked about before, which is that Sn of theta and u jumps at some critical phases. Theta equals theta critical. So let me go back to the sort of simplest example of that. Well, actually, here's the very simplest example. This is a case of a quadratic polynomial, but here this already shows you the basic feature. So now I've just got two zeros. As I change the phase, there is a critical moment, we're coming close to it, where the topology of the picture suddenly jumps. You see two lines going into the northwest and southeast, and then, bang, suddenly they're going the other way. And right at the moment when the jump occurs, you can see what's kind of special. What's special is the appearance of this trajectory that directly connects to the two branch points. So at theta critical, we have a picture like this. We have a trajectory that goes straight between these two branch points. So that kind of thing has a name. This is a classically studied thing in Teichmeler theory. So if you Google for, it's called a saddle connection. If you just Google for saddle connection, you get, there's also something in like welding that's called a saddle connection. But if you Google saddle connection, Teichmeler theory, then you get a lot of references to the right thing. So we're going to, these are going to be a new kind of particle for us. We're going to keep track also of the charge of this particle, and the charge of that particle is something simple and geometric here. Namely, let's remember we have this covering, sigma u, which is branched over these points. So associated to this saddle connection, there's a very natural homology class up here. Namely, I just take the saddle connection, and I lift it up to the cover in both possible ways with opposite orientation, there's some way of fixing the orientation. And exactly because these are branch points, this thing will be a closed cycle. And I'll call that cycle gamma, it lies in the lattice capital gamma sub u, which by definition is the first homology of the spectral curve. So we'll call this saddle connection with charge gamma. Well, that's not the only kind of topology change that you get in this business. So in some cases, the simplest kinds of topology change come from the saddle connections, but there's also other things that happen. So other, let's see, how can I say it? Other objects that cause topology jumps for the spectral network. So one that can happen already in this double cover situation is you can have trajectories that actually go, start at a branch point and end at the very same branch point. And when you get them, you actually get a ring of them. So you get something like this. So a whole family of closed trajectories. So let me call that a ring domain. Another thing that can happen in the higher rank situation, but not in the SL2 situation, is you can have something like this. You can have three leaves of different foliations that come together, a 1, 3, yeah, 3, 2, and 1, 2 like this. When this kind of object appears, it also causes a topology change. I'll call this a finite web. And maybe I'll show you a picture of that one, how that one happens. Here's the simplest example of that one. So this is with, this is an example where I have phi 2 is 4z dz squared and phi 3 is epsilon, whatever epsilon is. Epsilon is 1, okay. Phi 3 is dz cubed. So there, I'll just show you what the pictures look like for that example. So as we change the phase, here's the spectral network. And now as we change the phase, it starts to look like this. And now we're coming close to the critical moment. Okay, and you see what the picture looks like here. And now we go through the critical moment where there's that three string web. And on the other side it looks like this, so it goes. So, okay, so we have these kind of special kind of finite objects that can appear inside of the spectral network. And when they do, the topology of the thing suddenly jumps. Now the importance of these jumps for us, well they're gonna be important in our scheme for setting Higgs bundles, but first, they're physically important. So these jumps are associated to BPS particles. So again, BPS particles, but this is a kind of BPS particle that we haven't talked about until this moment. So so far we've been always talking about things that live on the little string. But now we're gonna talk about BPS particles of the full four dimensional theory in its vacuum U. So for example, for G equals SL2 I could define, let me define the invariant that counts these BPS particles, omega of gamma and U, so this is for any gamma in the lattice gamma which was remember was the first homology of sigma U. Omega of gamma and U is counting, so it counts the number of saddle connections with charge gamma minus two times the number of ring domains with charge gamma. So also for the ring domain you can define a charge, just lift this loop up to both sheets, take their difference and you get an element of gamma. So that's a kind of, so that's a mathematical definition of a number the way you're supposed to think of it physically, what do you mean by generic? They don't occur in these polynomial examples. They occur essentially generically in any example that's fancier than these polynomial examples. So they're kind of equally in sort of typical examples, they're sort of equally generic as the saddle connections. So there may be some domains of the Hitchin base where they don't exist, but there's plenty of domains where they do exist. So physically this quantity, omega of gamma of U is a BPS index counting BPS particles of charge gamma in the theory T4 of G in vacuum U. Ah, very good, very good. So we didn't talk about it in lecture, but in the notes I wrote down the kind of, the general definition of such an index in terms of the representation theory of the supersymmetry algebra in N equals two supersymmetry in four dimensions. And so just like the N equals two comma two situation you count things, but you have to count things with signs in general in order to get an invariant. So here unlike the two-dimensional case, in the four-dimensional case there's a bunch of different kinds of short representation. So in fact the saddle connection is giving you one kind of short representation which is called the massive hyper-multiplet. And the ring domain is giving you a different kind of short representation which is called the massive vector-multiplet. And so I wrote about what those representations actually are if you look back in the end of the notes from the first lecture. But anyway, I think all we need to know right now is that there's some BPS index that counts these things like the indices I defined for you before. And this number omega of gamma that we're computing in this way is one of those BPS indices. Okay, you sum over all theta. So you look at the theta where the saddle connection appears. Yeah, so for every charged gamma there's only one theta where the saddle connection could appear. In fact, this is the argument of Z gamma. Yeah, maybe that's one of the things I should say. For these particles, so just like before, they have a central charge, Z gamma, which is just the period of the holomorphic one form over the cycle gamma. That gives the central charges of these particles. So now in the end, I wanna try to sketch out what's one kind of geometric application of this stuff, of this construction. Yeah, yeah. In the examples from before it would be zero. Because this has to do with, well it would have been zero for two reasons, right? It would have been zero because this homology lattice would have been trivial. But also morally it should be zero exactly because this has to do with the four dimensional physics. This has to do with a phenomenon that happens in the four dimensional theory, T4 of G and C. In our previous examples, there was no four dimensional theory. Everything was happening in just two dimensional physics. So this is a new phenomenon that occurs here and didn't occur in those other examples. Let's talk about it later. For sure there can't be any BPS particles because all the periods will be zero. This lambda is exact in that situation. Okay, so let me try to sketch for you the application. So among the properties of this spectral network, SN of theta of u, so it controls in the same sense as before, it controls Stokes phenomena happening on the curve C. For families of connections, so just as before we had these two families of connections which are called NABLA H bar and NABLA Zeta. So here too, if I take in fact any Higgs bundle E over C whose spectral curve is given by sigma u. So yeah, so I take my curve, I've got my Riemann surface. I put a Higgs bundle on that Riemann surface. Its spectral curve is gonna be one of the spectral curves sigma u where u are the points of the Hitchin base. And now I wanna study, eventually I'd like to do non-Avillian-Hodge theory, so I'd like to solve Hitchin's equations, but I don't do it directly. Instead I study these one parameter families of connections first. So let me remind you one parameter family of connections. So this one parameter family of connections, NABLA Zeta was Phi over Zeta plus the Tern connection DH plus Phi dagger Zeta where H is the harmonic metric, the solution of Hitchin equations associated to our Higgs bundle, our Higgs bundle E Phi. Yeah, Higgs bundle I should write as E Phi over C. So you recall that whenever you have a Higgs bundle, by solving Hitchin equations, you can produce this one parameter family of flat connections. And now I wanna study this one parameter family of flat connections. The way I'm gonna do it as before is to study its Stokes phenomena at small Zeta. And for that, the key tool is exactly this spectral network. So let me say exactly what I mean. So the way it works is a little bit subtler than before. So before, in the previous story, I said, well, you look in the complement of spectral network. In every domain of the complement, you just get a distinguished basis of sections. And that distinguished basis, you know everything about their asymptotics. That was a good way of studying those connections. Here we don't get exactly that. We don't get exactly a distinguished basis of NABLA Zeta flat sections in each domain. Rather what we get is a distinguished line decomposition. So before we were kind of trivializing this connection completely in every domain of the complement of the spectral network. Now we're not trivializing it, what we're gonna do is we're gonna reduce it to something abelian. And so here's the precise conjecture. So the conjecture is like this. So there exists a family of flat GL1C connections. I'll call them NABLA Zeta ab over the spectral curve, sigma u, so remember the sigma u which projects to C, let pi be the projection. Such that the following things happen. So number one, these connections have to carry somehow the same information as the connection that I'm really interested in studying. So what I want to do is to study this connection. I'm gonna relate it to these connections over the spectral cover like this. So the statement is that away from the spectral network, the theta equals argument of h bar, there's an isomorphism, yota which identifies NABLA Zeta with the push forward of NABLA ab comma zeta. What that means in concrete terms, what that means in concrete terms is if I have some path that's avoiding the spectral network, let's say the spectral network is somewhere over here. If I have some path in the base curve and I wanna do the parallel transport of my flat connection along this path, to understand this flat connection, what I'd like is to understand this parallel transport. Yeah, they're both flat. Yeah, absolutely, these are flat. Sorry, that's very important. Hitching equations exactly means that this is one parameter family of flat C star connections and we're trying to describe in some efficient way that one parameter family of flat connections. If we can do that, we can, for example, find the hypercalametric. So we wanna understand these connections. The way we're gonna understand them is to describe their parallel transport in a concrete way. And the first thing I'm saying is, away from the, away from this spectral network, the parallel transport can be related just to parallel transport in a line bundle over a sigma. So like in this picture, we've got some rank three bundle. We're trying to do the parallel transport and we just relate it to a sum of three line bundles here and we do the parallel transport in those three line bundles. It's not as good as having a distinguished flat basis. If I had a flat basis, that would be even better, but this is almost as good. I just relate it to three a billion things. So that's what this number one says. Number two says, so away from the spectral network, it's as good as it could be. All of the spectral network of types say IJ, this isomorphism jumps, but it jumps in a very specific way. It jumps by kind of like the change of bases that we wrote before. It's a unipotent matrix, which looks like the identity plus some number times the elementary matrix EIJ. So just the kind of change of basis that we were writing before. That same kind of change of basis intervenes here. So that says if I wanna do the parallel transport along this path, it'll be almost the a billion parallel transport, except that when I get to here, I'll have to splice in some explicit unipotent matrix. That's still pretty good, we can control that if we know exactly where the spectral network is. Okay, now, the conjecture's not done yet. So far this part is, if you only wanted this, it wouldn't be that hard to get. So this conjecture is in GMN, by the way. If C was compact, I would formulate the same conjecture actually, but it would be much, when you press me about details, it would be very tricky to say it precisely, right? Because in that case, the spectral network is dense. So I have to be very careful in, yeah, so morally yes, but in practice it's a serious thing. Aaron Finneas has done some work in that direction. But for now, let me just stick to, I'd be very happy to do it just in any one of the examples I showed you. Okay, so I'll finish just by stating the rest of this conjecture. So one and two are kind of algebraic in nature, but the third thing is definitely not. So suppose I define now a function x gamma of zeta to be the holonomie of nabla, what's my notation, nabla ab zeta around the loop gamma. So here's sigma again, here's C. I've related my connection down here to an abelian connection up here, and now I just look at its holonomie around a loop. Those are in some sense the only real invariance of this connection. I think it's holonomie around the loop. The statement is that that function has nice controllable asymptotic behavior. It goes like exponential of z gamma over zeta plus something of order one as zeta goes to zero. So this is a function with a kind of