 And the reason why I was vague is that it's actually not possible to do it in a globally consistent way on this picture. So there's some sort of further little covering I would have to make, consisting of the choices that you would have to make some further choice in order to land on a space where the sign is literally well defined. But you're right, it's a little worse than I'm making it sound, exactly because this is an important point. There's monodrome in this picture, right? When I go once around this picture, the two-vacuate interchange, I exchanges with J. So if I say that this one is plus one and that one's minus one, I got a problem when they interchange, something discontinuous had to happen. So yes, there's a problem, I mean, purposely ignoring that problem. C, Gaiotto, Moran, Witten for a discussion of this issue. Unfortunately, their solution involves replacing these numbers by, you know, phases, so I don't like it so much. But it's somehow the correct thing to do. Okay. But you're right, so I'm going to do everything kind of up to signs. Anyway, so let's come back to this and let's, yeah, let's ask what the picture's going to be here. So here, over W of X1, there's a critical point X1 and the local picture of the covering looks something like this. Now here, over X3, the local picture of the covering looks something like this. And now, now you can't tell by pure thought whether there is or is not a soliton connecting X1 to X3. It depends on how the sheets are kind of mixing with each other and going from this place to this place. And so correspondingly, the actual picture turns out to be more interesting. So, sorry, and I should say, I guess the key thing is that the answer is going to depend, so as I said, the answer somehow depends on the topology of this picture. But as the topology change, the topology will change exactly when this branch point comes along and crosses this line. That's the moment when the answer could possibly change. So in general, what's going to happen is mu of ij can change when the three vacua become collinear, when the three critical values W of Xi become collinear. So let me draw the picture of what happens here. Let me draw the picture of what happens here. So now I'm going to draw a picture in the parameter space of these polynomials, but the parameter space is two-dimensional, so I'm going to have to take a slice through it in order to draw it. So let me fix the value of Z1, say, and I'll fix it to be minus one. And then let's draw the picture in the Z2 plane. So here's the picture, Z2. And now there's two special points instead of one. So these are places where two of the three critical points coincide. So for a generic polynomial, it has three distinct critical points. These are the two values where that doesn't happen, the discriminant locus. And then there's a picture like this. So in this outside domain, all mu of ij are equal to plus or minus one. In other words, if you choose any two of the vacua, there's a soliton connecting them. If you chose your polynomial in this region, there's an interior region. And in the interior region, with some choice of labeling of the vacua, there's a soliton from one to two, a soliton from two to three, but there's no soliton from one to three. And so this is the first example of the wall crossing phenomenon. This mu of ij changes as we move in the space of polynomials. And so this place where it changes is called the wall of marginal stability. That's right, that's right. Those are exactly the places, right. If you look at any of the polynomials on here, the picture of the critical values, you'll see three collinear critical values. That's right. And everywhere else in the picture, the critical values are not collinear. Fantastic. So how did I get these numbers? That's the next question. How did I actually get this picture? Let me say two more things and then I'll get to that, which is actually the main topic. So first, just a few more little comments about this wall crossing phenomenon. So the first thing you could ask is, you kind of see it, but you don't believe it, right? What happened to the soliton? How could the soliton suddenly disappear? There was just some solution of some differential equation. How does it suddenly disappear? So what happens is, so let's imagine as Z2 approaches the wall, wall of marginal stability from the outside, what happens to the 1,3 soliton? And what happens to it is it starts to look like this. So let me draw the picture of it. So now I'm drawing again the picture as a function of s. So the picture is, out here it's in vacuum x1, and it has to end up in vacuum x3 somehow. And so at a generic point in this parameter space, it just looks like something that goes from x1 to x3. As we get closer and closer to the wall, what happens to it is it starts to look like this. It first goes from x1 to x2, then spends a lot of time near x2, very, very close to x2, and then sometime later it goes from x2 to x3. That's the picture of the soliton if you're very close to this wall in parameter space, but on the outside of it, if you're like there. And there's some long distance here, call it delta s, and what happens is that as Z goes to the wall, delta s goes to infinity. And so in that way this configuration kind of disappears from the spectrum. And so the physical interpretation of this is literally you have one particle which decays into two particles. It literally sort of reveals itself to be made of two different particles. Okay, so that's the wall crossing. Now, there's a formula that actually captures these jumps. It won't be a surprising formula, but let's say what the formula is. Oh, thanks, did I not say it? Yeah, I sort of have... Yeah, yeah, I've kind of gone around it, but I haven't exactly said it. Yeah, thank you. So physics fact, mu of i, j is BPS index in the sense that we defined last time and reviewed at the beginning of this talk in the Landau-Ginsberg model. So to keep track of the labelings, what I was calling the charge is this pair i, j. And the additive relation is i, j plus j, k is i, k. Sorry, this is why I couldn't instantly answer this question of sort of what kind of structure they live in. I forget what this is called. And the central charge, the central charge, Za, which is Zij, in this case is just W of xi minus W of xj. So this phenomenon of the three things three critical values becoming collinear is exactly the phenomenon I mentioned before of the two Zs becoming collinear. Okay, great. So, yeah, I guess we could call this next thing another physics fact. At a wall where Zij becomes parallel to Zjk, the indices jump and they jump in a specific way. The index mu of i, k jumps by, let's write, yeah, let's write it like this, mu of i, k jumps to mu of i, k plus or minus mu of i, j mu of j, k. This is a very important formula, which is why I've sort of thoughtfully crammed it into the, crammed it in here. But I'm putting it in, surrounding it in a big box. So this is called the Jakodi-Vafa wall-crossing formula. And so it's supposed to hold quite generally. So even though here we're talking about one very specific example of these Landau-Ginsberg models, the idea is that this is something universal that happens in any n equals 2, come into supersymmetric field theory. Maybe we'll talk a little next time about what are other examples. Okay. So we're going to come back to this formula. But now let me, at least briefly, try to talk about the question of, how did I get this picture? I mean, the Picard and Lefchitz didn't know about n equals 2, come into supersymmetric quantum field theories. So just for that reason, it can't be called the Picard-Lefchitz formula. I agree that in this specific case, it kind of boils down to something Picard-Lefchitz. As they said in their paper, the Jakodian Vapha were certainly aware of that. Okay. So yeah, how do you actually determine this picture in practice? So let me make a definition. It'll at first seem a little ad hoc. But, okay. So for any phase theta, I'll define Sn of theta, and Sn stands for spectral network. Sn of theta will be, so it'll be a subset of the parameter space. So the same space where this picture is drawn. In that parameter space, I'm going to draw a subset. And what it is, is it's the set of all z's, such that there exists a soliton. So there exists some i and j, such that mu of i and j at z is not zero. But not just any soliton. If I just said this, this subset would be the whole plane. Because there's at least one mu, not zero everywhere. I put the initial condition that zij, which I'll remind you is w of xi minus w of xj, has argument, the argument of zij is theta. So I ask not just that there's a soliton, but that there's a soliton whose central charge has phase exactly theta. And then I draw a picture of that. Okay, well let's draw what some of those pictures look like. I'm going to keep those examples. So in the cubic example, okay, so there I just have two vacuos, so i and j are just one and two. The only thing I need to keep track of is the difference of the critical values, w of xi minus w of xj. Okay, well the critical values are like z to the half, plus or minus z to the half. Now we're cubing them, so it's going to be proportional to z to the three halves. And what we're asking is that that should have some definite phase, say real. Let's do the case of it where it's real. So then the picture of this spectral network will be like this. Well, when is z to the three halves real? Okay, here's a place. Here's a place. So this is in the z plane. I could label them again by which kind of soliton has central charge real and, I'm sorry, not only real, real and positive. So here it's a soliton of type one, two. Here it's a soliton of type two, one. Oh, now I got a problem. This one is again a soliton of type two, one. To make this all consistent, you have to remember that there's a branch cut between those two vacuum exchange, vacuum one and two. So the kind of decorated picture looks like this. Okay, so that one I was able to draw just because I knew the soliton degeneracies already, right? Everywhere there's one soliton, so all I have to do is keep track of when does this guy have some definite phase? So this is the spectral network at theta equals zero. Okay, now how about the Quartic example? So this is good, so I'm going to draw it to this picture, because it's drawn in the same space. Oh, sorry, I didn't tell you what the arrow is. The arrow keeps track of the direction in which the soliton mass is increasing. So you just calculate the absolute value of that difference, wi minus wj, it's getting bigger in this direction. So it actually brings up an important point. So as I go to here, the mass of that particle is getting smaller and smaller until here it becomes zero. That's another sign that something, you know, kind of degenerate happens here. So, yeah, so for the Quartic one, so let me take the same slices before, so I'll fix z2 equals minus one, and I'll just draw it in the z2 plane. So then again, I have these two special points, which are the branch points. Now the branch points, as we just said, that's the place where some soliton mass becomes zero. And actually, from that, you can pretty easily convince yourself, well, okay, let me draw the picture first and then talk about it. So here's what the picture looks like. Okay, and now I'm even going to label these one, two, two, three, one, three, one, two, three, two, two, one, three, one, and three, two. So this I claim is the picture in the Quartic example. I'm not telling you yet how I got it. Let's just look at a few kind of structural features of this picture. So one thing is that around a branch point, the branch point is where two vacua are coming very, very close together. So if you just look at those two vacua that are coming very, very close together, you can analyze in a kind of universal way the solitons that connect those two vacua. And the answer is going to be exactly the same as the answer was in this case where we only had two vacua. So that shows you right away that this thing is this kind of three prong structure near the branch points. So near this branch point we have a three prong thing, near this branch point we have a three prong thing. The only other interesting feature in this picture is these points, right? So this point is the moment where if I'm exactly here, I have two different solitons, namely the soliton associated to this line, which is 2, 1, and the soliton associated to this line, which is 3, 2, two different solitons that both have phase zero. Remember this is the picture only of the solitons that have phase zero. At this moment I have two guys that have phase zero. That's exactly the moment where the wall crossing is supposed to happen, where a new soliton is supposed to either be born or decay. And sure enough, here we see a new line that exists on this side and doesn't exist on that side. That's the soliton that was born as a bound state of these two. This kind of structure where when two lines cross, another line is born is just a reflection of that wall crossing formula that I told you over there. Yeah. Why don't they go away? Oh, right, right, right, right. Yeah, so, right, because you figured they're kind of making this new one, right? Yeah, so the picture is... Yeah, how do you say it? I mean, you still have... Let's just think of it as... Let's think of it in terms of solutions of the equation. So we're saying you have a soliton that goes from one to two. That solution of the equations continues to exist. It's not that that solution of the equations suddenly goes away. Indeed, there's no obvious way that it could suddenly go away. Similarly, we have the... So that was the two-one. We have the three-two guy. That one also, nothing really special happens with that one as we go across this point. What happens at this point is all about the kind of longer one. The soliton that goes from one, starts at one, then spends a bunch of intermediate time at two, then ends up at three. It's that solution which exists only on one side of the wall and not on the other side. So that solution looks, this part of it looks very much like this solution, and this part of it looks very much like that solution. But there's nothing discontinuous or special that happens to those solutions at the wall. All that happens is that this kind of composite one appears or disappears. Okay, I'm not sure if that was a helpful answer. That's probably the best I can do at this moment. Other questions? Yeah. Yes, so indeed. So if I were to now change the phase theta, so this was the picture at theta equals zero. If I were to change the phase theta, I would kind of continuously deform this picture in such a way that this point moves and it exactly traces out this wall of marginal stability, because then we're looking at all the places with all different phases where you have two solitons at the same phase. But we're not constraining that phase if we sweep over all phases. In the problem set for today, for those of you who have or have access to Mathematica, I've provided a tool for how to see exactly that, to draw this family and to see that sweep out the wall of marginal stability. Do the walls of marginal stability always go through the critical points? Does every wall of marginal stability go through a critical point? Yeah, you'd think I would know that. I mean, it very often happens. Let me get you an answer offline. Okay. In the last three minutes, let me tell you how this picture was obtained. So, the idea is the following. The paths in the spectral network, the paths in SN of theta, they're pretty easy to describe. They obey a differential equation. The equation is just dZij by dt is, I mean, you can parameterize them so that dZij is literally e to the i theta. They're characterized by the condition that Zij stays the same phase. It stays the same phase, but its magnitude changes. You can easily, if you know the function Zij, which is perfectly explicit in these cases, it's just the difference of critical values. Well, that's not explicit. It's the difference of critical values. You can integrate that differential equation. So, what you do is you just start from the branch points. From each branch point, you make a little local computation to see which way the trajectories are coming out. And if you're like, this is the picture of all the salatons in the theory with mass less than epsilon for some small epsilon. And then you gradually increase your cutoff. You say, okay, now I want to look at the ones with a little larger mass. Well, it's not too hard to convince yourself that the ones you'll get are the ones you get by just extending the trajectories a little bit. Then what about the ones with a little larger mass? Well, you just keep extending the trajectories. And the only surprise that you have to worry about, the only surprise that you have to worry about is that, well, sometimes they're going to cross. And when they cross, there's a new salaton that can be born at the crossing. And so that means you're going to have to draw another line. And then you just integrate, then that line is on the same footing as all the rest. You just add it to your list of things and you integrate that one, follow it as long as it goes. Maybe it meets someone else and then they generate new ones, generate new ones, and so on. But that's the process by which you can draw this picture, a totally systematic process. And once you have that picture, then you can easily read out from it what the muses actually are. So that's the strategy for calculating these muses without ever actually having to solve this salaton equation. Okay. What I did not tell you is what all this has to do with Higgs bundles. So let's just say, as a kind of teaser for the next time, that this space C where I'm drawing these pictures also carries, in exactly this setup of the polynomial in Wendell-Ginsberg model, this space C carries a canonical Higgs bundle. And these trajectories have to do with studying families of connections that are defined on that Higgs bundle. Okay, so that'll have to be for next time, so thank you. Right. Yeah, the index actually is the correct index. I mean, when they don't lift, they kind of annihilate each other in pairs. So you make a complex, a complex has some differential that sort of annihilates two things at once, one with a plus and one with a minus. So the index is the correct index. Yeah. Yes. Yeah, good. Right, so usually we do it on a Riemann surface. Now we're doing it on a higher-dimensional thing. It's a co-dimension one. Okay, the 100% honest answer is that up until like a couple weeks ago when I was preparing these lectures, I never thought seriously about the higher-dimensional spectral networks. And therefore, all of my sort of thinking is geared toward this situation of taking a slice. And then you just draw an arrow. In the higher-dimensional thing, it's not exactly an arrow, yeah. But I don't think it's necessary to have that arrow, really. Yeah, we could talk about it more.