 Okay, so it seems clear that in this lecture I'll get through most of this stuff, but I won't get to the kind of concrete computation you do to get results in hyper-killer geometry from this. And so there'll be an extra lecture at two o'clock, which will be super informal, and no one should feel obligated to come, but if someone wants to hear that part of the story, that's where it'll be. But so I want to get as close to that as I can today to kind of set it up. So last time we were talking about a complex manifold C, which we thought of kind of abstractly as a modular space of some quantum field theories. Concretely, in all of our examples, what it actually was was a modular space of polynomials, but those polynomials were supposed to be producing for some quantum field theories. And then from those polynomials, I told you how to produce a bunch of other stuff. So first of all, a Higgs bundle on C, and this Higgs bundle was carrying two interesting one-parameter families of flat connections, which I labeled novela h-bar and novela zeta. There was, over C, there was a branch cover, which you could think of as being the spectral curve, if you're familiar with Higgs bundles, and you thought about Higgs bundles a little bit, the most obvious thing would be to take the spectral curve of that Higgs bundle, and that's exactly what this branch cover is. If you think about it in physics, you might rather think of this branch cover as being the set of classical vacua. So for each point of this parameter space, I've got a quantum field theory that's massive as a finite set of classical vacua. Those were the points of the spectral curve. Anyway, we had a branch cover, and then I described for you an algorithm for drawing something called spectral network on C. Where's the picture? This is not a picture of that. This is a picture of that. This is the one that we talked about at some length, coming from a family of quartic polynomials. And I guess the last thing to say about that spectral network is its purpose, as we introduced it, was to compute these BPS indices, mu of, so I wrote the mu of i, j, and z, and rather let's write it, mu of a and z. So here z is a point of our base, so a point of that picture there. A labeled a pair, i comma j, so i and j were sheets, oh well, let's say i and j are just points of sigma sub z. So the picture is, here's our base curve. Here's the spectral curve. So some branch covering, branched over the x's in that picture. And then the label i and j might label two of these. That's two of the two vacua of this quantum field theory, but it's also just two points of the spectral curve. And the meaning of this index was that it was counting BPS states. If I would draw a picture of what those states represent, the picture is we're in a two-dimensional quantum field theory. So space is just one-dimensional, so here's a picture of space. The theory has kind of asymptotic boundary conditions or asymptotic vacua provided by these points i and j. So let's set it up so that in one end it's in vacuum i and one end it's in vacuum j. And then what we're studying is the solitons, think of them as being just some kind of particle that lives on the line. And we're just counting the number of BPS states of this kind of particle. So this particle, I'll label it with a little a. To say a means all of the kind of charges carried by this particle. In this simple example, a doesn't carry any more information than just the vacua i and j actually. In the next example it will. Okay, so that's the sort of rough physical picture of what we were doing. This is the picture we drew. Last time we realized this very concretely in terms of just some classical differential equation defined on this line. And this mu was in fact counting solutions of that differential equation with signs. Okay, so that's the picture we had. Oh, one last thing was that we also had a number. So these BPS solitons, remember they carry this complex number called the central charge. And so we had a bunch of complex numbers z sub a, which in our examples were just the difference of critical values of the super potential. Anyway, that's part of the structure that we had. So now to do the story that I'm really interested in, I want to upgrade this whole thing. And what we want to do is now fix c to be a Riemann surface. So in my example so far, c was always a vector space. But now I'm going to take c to be a Riemann surface. I'm also going to fix, so now I'm just telling you some data that I'm going to fix. So I'm going to fix a Riemann surface, fix c a Riemann surface, a le algebra g, which I'll take to be always SLN, I think. And now, given this data, let me remind you of something that you heard about in Laura-Chapasnik's lectures and maybe also in some of the research program lectures, which is given these two data, there's a nice vector space called the Hitchin base. So by definition what it is is, so it's the base of Hitchin's interval system, but concretely what it is is it's just a vector space defined like this. So a point, so practically speaking, a point u in the Hitchin base is the same thing as the data of phi two up to phi n, capital N, where phi little n is a holomorphic n differential on c. So in other words, if you haven't seen that kind of thing before, you shouldn't get too scared by it. So locally it just means phi n would be represented by a holomorphic function f of z, let's call that holomorphic function tn of z, dz to the n. So it's essentially just a function on the curve, but then it transforms with this Jacobian under change of coordinates, yes. In the story, as I told it up until this moment, those two data are actually the same. So sigma doesn't have any non-trivial homology classes. So that's true, that's a defect of how I drew it. You're absolutely right, I'm so used to drawing it like this that it kind of looks like it has some non-trivial homology classes. Yeah, it's an important fact actually that there are no periods in this part of the story. So yeah, let me draw it better. Yeah, but indeed, in just a moment, we'll introduce that kind of fancier story in this context. But here I really did mean just a pair i and j, nothing else. We could have put in twisted masses and then that wouldn't have been true. We could have put in twisted masses, so it would be possible to kind of jack up the 2D story in such a way that it would be slightly more sophisticated than this, let's not do that. Okay, so I've got, my data's a Riemann surface, this Lie algebra, and then I told you there's the Hitchin base. So now there's a kind of physics fact. So the first physics fact is that given this data, there is a four dimensional N equals two supersymmetric quantum field theory. I'll write it just T4 of G and C. But I guess you should know it's usual name. So it's usual name is a theory of class S. And I think the reference for the construction of these things is in Witten, Gaiotto, Moore, Nitzki, and Gaiotto. Okay, anyway, there is some 40 N equals two supersymmetric quantum field theory, just depending on those data. And so the theory only depends on this, but it's effective physics depends on a little more. So if you actually wanna study what happens in this theory, it turns out that you need a little bit more data. You need to choose which vacuum you're in, and that data is exactly a point of the Hitchin base. So that's why I mentioned the Hitchin base in this context. I'm gonna introduce punctures later. Yeah, Greg is pointing out that although everything I said is true, somehow the easiest examples and the examples which we're gonna be able to deal with later, C will actually have to be equipped with some punctures. And so instead of doing holomorphic stuff, we'll do meromorphic stuff. But I wanna leave that at it first, so we'll get to that in due course. Okay, so when you ask a question in this quantum field theory, the answer often depends on a point of the Hitchin base, and that's fine. Now that doesn't get us all the way to what we want. Because before we were interested, our story before was about two-dimensional field theories, not four-dimensional field theories. And now I'm describing a sort of slight upgrade of that. So my upgrade should also have to do with something two-dimensional, and that's the following thing. So this quantum field theory, T4 of G and C, admits what's called a surface defect, which I'll call, I guess, S sub Z. And the reason for the label Z is that the surface defect depends on a parameter, Z and C. So what does a surface defect mean? So literally, let's think of what the space time looks like. Well, everything's gonna be time translation invariant, so let's just think about the space. The picture is gonna be, the picture in space is gonna be, so you've got your three-dimensional space where the theory T4 is kind of living. And then we're introducing into that space, oh, yeah, okay, let's make this space blue. And then here it'll also be blue. So we're introducing into our three-dimensional space, we're introducing this defect, which is in one-dimensional space, so it looks like a string. So we're taking our three-dimensional universe and putting a string in it. Okay. And so now, I mean, it's not, it's not literally a two-dimensional system. I mean, it's some kind of coupling between a two-dimensional and a four-dimensional system. On the other hand, from the point of view of like, what are its symmetries, you would say, this looks a lot like a two-dimensional system. Its Poincare invariance is exactly the Poincare invariance of a two-dimensional system. And if you work it out, its supersymmetry is also exactly that of the two-dimensional systems that we studied before. So it has kind of ISO 11 and N equals two, two supersymmetry. So this system looks in many ways just like the kind of two-dimensional systems we were studying before. That's right. So we have the symmetries of the four-dimensional universe. That would have been bigger. That would have been ISO 31 and N equals two supersymmetry, which is an algebra. It was in the notes. We didn't talk about it in lecture. An algebra with eight fermionic generators. And now I'm saying, when we put this thing in, obviously it destroys some of that symmetry, right? Like, we don't have translation invariance in this direction. You don't have some of the rotations. So, but inside of that, yeah. So this ISO 11, you think of as kind of sitting inside ISO 31. So it breaks the symmetry. It breaks it down to exactly the situation we were studying before. So this is gonna be in some way similar to the things we studied before. But now we've sort of realized our Riemann surface now as a modulized space of N equals two, two supersymmetric systems. And so now the dream is that it'll now carry all the structure that we were doing before. And we'll see that's more or less right. So the four dimensional spacetime is just R4, R3 comma one. Then inside of that R3 comma one, we're putting an R1 comma one, which is where the surface defect lives. Yeah, we're putting this R1 comma one, I should have written it in coordinates I guess. So it's like X1 equals zero and X2 equals zero. Inside of the R3 comma one, which is coordinateized by X0, X1, X2 and X3. Yeah, the smartest thing would have been to say that in the beginning rather than to just draw this picture of a line sitting inside of R3. But that's what we're doing. Yes, it does. Look at the notes to see that construction from point of view of six dimensional physics. Okay, so, but that's right. All this is supposed to come from six dimensions somehow. Okay, anyway, so given a point of the, given a point of the Coulomb branch, or a point of the Hitchin base, phi two up to phi n in B of C comma G. So we have, I'm just fixing notation and reminding you of these spectral curves. If the spectral curve, sigma sub u, which is given by the equation, lambda to the n plus phi two lambda to the n minus two, down to phi n equals zero. So that lives naturally inside of the cotangent bundle of the curve. Lambda is naturally a cotangent vector. And so the picture, so now the picture really is kind of more like the picture that I drew before. It really can have cycles in it. Lambda is like the differential of the central charge. So yeah, it's, I mean the slogan for it is often called the cyberwitten differential if you want to name for it. Okay, and so, so we'll label the sheets, label locally the sheets of sigma by, you know, the index i that goes from one up to n. And those sheets now correspond again to vacua of the surface defect. So before the sheets of this covering were the critical points of the super potential. So those were the kind of classical vacua. Now I don't have a super potential, they're not critical points of anything, but still I want to say these play the same role as they played before. Okay, now what we want to do is we want to study, again, VPS particles living now on the surface defect. So as before, now I've got my surface defect living inside of this three dimensional space. And I want to study particles living on that surface defect. So again, they'll be interpolating between two vacua, say i and j, but now the charge is a little subtler. So as Greg was alluding to. Now the charge A is not anymore just determined by giving two points on the spectral curve. That would be the i and j here. Now the charge is gonna be determined by giving a path from i to j. And there could be many different paths. So that means here there can be many different kinds of particle that interpolate between the same vacua i and j. Before we only had one kind of particle going between i and j. Now we'll have many different kinds. So the charge A is living in the homology classes of paths in say sigma sub z over some point z from the lift z i to z j. So coming back here, if I've got a point z, some particular kind of surface defect, a soliton is gonna correspond to a path between the lift z i and the lift z j. Those are the charges of the soliton. Okay, and so we want to compute some indices, mu of a and z, before I just wrote mu of i, j and z. Okay, and so as before, yeah, sorry, sorry, sorry, in sigma, thank you. Sorry, of course it can't be a path in sigma z. Sigma z is just a discrete set. Yeah, sorry, of course the path goes through the whole sigma, thank you. Yeah, so now the key tool as before is gonna be a spectral network, Sn of theta. And now it also depends on this point of the point of the hitch and base that we fixed on C. So let me tell you how to construct this spectral network. I make a definition. So yeah, so now I'm gonna tell you its construction. So I'll define an i, j trajectory. So now for a little while, this is just gonna be some kind of a complex geometry construction with no physics in it. I'll define an i, j trajectory of phase theta, i trajectory of phase theta on C to be a path along which lambda i minus lambda j multiplied by the phase e to the minus i theta is real. What are lambda i and lambda j? So on the spectral curve, we have a tautological one form just because it lives in the cotangent of C. So yeah, sigma carries tautological one form, holomorphic one form, lambda. And so just by pulling it down from here to here using a local trivialization of the covering you'd get n different one forms down here, lambda one, lambda two and so on. Which is the value of this one form on all the different sheets. So you have one one form on the cover that gives you locally n one forms downstairs. Those are the one forms I'm calling lambda i and lambda j. And so we've just got the surface C downstairs with a bunch of one forms and we just look at the trajectories along which lambda i minus lambda j multiplied by this phase is real. Okay, so if you think for a minute about what those trajectories look like you discover the following thing. So the i, j trajectories give a local, it's local because I'm using i and j to label two definite sheets. So it's only locally defined. Locally defined foliation of C which has singularities at the branch points where lambda i equals lambda j. So you have a foliation except that it has these kind of so let's look in a neighborhood of one of those branch points. The picture of this foliation looks like this. Okay, so now just knowing this now I can tell you the definition of the spectral network. It's the same kind of you should think of it as exactly the same kind of picture as before. It's not literally okay in a neighborhood of any branch point actually could rig up a super potential so that it would be literally an example of the thing before. But globally I cannot do that. So indeed locally these are exactly the same kind of pictures we were drawing before but globally they're not. Yeah, I mean morally you should think of this lambda you should think of lambda as being like dw the derivative of the super potential that we had before if you want to convert it directly. Okay, so then so the definition now is that the spectral network of theta and u is constructed by the following algorithm. So you start with you start by shooting three ij trajectories from each branch point of the covering sigma u over c. Well, I'll show you a picture of it in a second. So from each branch point we'll just shoot out three trajectories. There's solutions of a differential equation so I can just integrate that differential equation forever follow it around the surface. And then if they cross, so it may happen that I'll have a trajectory carrying the label ij which crosses a trajectory carrying the label jk and then I just declare that I'm gonna add to the network another trajectory of type ik. So if they cross if an ij and a jk cross like this shoot another trajectory from the intersection point from the crossing point and then you just keep going like that. So then you continuously integrate this differential equation. You make more and more trajectories every time they cross each other. If an ij crosses a jk it shoots out another one and so on. Now what kind of thing does this give you in practice? So yeah, so this was the old example and now we come to the new world. So okay, so here's an interesting one. So this will be kind of a running example for a while. So let me write it somewhere where I won't erase it hopefully. So for example, let's do the case where g is SL2. So that means we're talking about just a double cover. So in that case the data of u is just equivalent to the data of a quadratic differential and I'm gonna take my quadratic differential to be, oh sorry, right, and c is Cp1. Oh sorry, this is not great notation. So the data of u is equivalent to the data of a quadratic differential and concretely I'm gonna write a formula for the quadratic differential like this. I'm gonna write it as z cubed plus z plus u. U is a complex number. And so this is, so what I'm writing for you here is a kind of one parameter family of quadratic differentials. So in this case, I'm taking my hitch and base to be just the complex numbers. Now this is not literally an example of a situation I described before. Before I said I was gonna do only holomorphic quadratic differentials but the simplest examples are examples where you allow meromorphic lens. And so this is a kind of example of the meromorphic extension of what I was describing. So we're studying Cp1 and we're studying quadratic differentials which are allowed to be of exactly this form, a cubic polynomial and we vary only the constant coefficient. Yeah, that's right. That's right, in that case but the only example that I got of this kind where it just had two sheets, that would have fit exactly into this framework except that the quadratic differential phi two in that case was just z. So in that case, you just had a branch covered with one branch point and no period. Okay, and here I'm drawing the spectral network that you get from this procedure where I took u equals to one. So the game is, so what's exactly going on here? So the orange crosses are the branch points of sigma u to u, sigma u to, yeah, to c. Let's see. Which in this case are just the zeros of the quadratic differential phi two. But in this case, sigma u is just given by the equation lambda squared plus phi two equals zero. So the equation I wrote before in this case just becomes lambda squared plus phi two equals zero. In other words, this covering is just the covering by the two square roots of phi two. So it's branched exactly at the zeros of phi two. It's a cubic polynomial, so it has three zeros. There are the three zeros. And now the rules of the game are, we're supposed to start by taking the trajectories, the ij trajectories which are given by this equation we implement that equation here. So in this case, it just says the trajectories are just the trajectories where e to the minus i theta times the square root of phi two is real. So literally take the square root of this polynomial times dz and look at the directions where it's real. That's what we're doing here. And now we start by shooting them, we shoot them out of the branch points and we let them evolve. And it makes this picture. Okay, fine. Then we can vary the phase and as we vary the phase, the picture kind of twists around. There's some interesting moments where it changes topology which we'll come back to. Okay, fine. Let me show you one more interesting example. So this one more interesting example will be, so again, so this is gonna be the SL three situation. Again, I'm gonna take CP one, meromorphic things on CP one with a singularity just at infinity. And in this case, the family I'm gonna take is just phi two equals literally dz squared and the cubic differential phi three will be z squared plus u dz cubed. Ah, sorry, sorry. In the previous example, I should have shown you one other thing which is we had this parameter u. So what happens when you vary the parameter u? So the whole picture again just kind of deforms. So the zeroes move and the picture deforms. Okay, so here too, when I show you the.