 Where's my eraser? Yeah. An interesting thing happens here. If you think about what that's doing to the curvature density, so if I take it, you know, think of S4, conformally S4 is just R4 with an extra point, a natural family of conformal transformations is just scaling of R4, and look at what that does to S4. So if I scale, maybe I've thrown out the south pole, scale towards the north pole, then if I think about the point-wise norm of the curvature of this connection, for the original connection, it's just uniformly distributed. The connection was invariant under isometries of, you know, full isometry groups, so this point-wise norm is invariant in the isometry group of S4, it's uniform distribution, but these conformal transformations, what they do is they take this guy and they start to pile it up. So the curvature starts to pile up at a point and in fact, you can show that in some trivialization, this connection is given by this formula. If I do this scaling, so if I pull back by scaling, then, so I scale the coordinates by lambda, and I rewrite this as like this. So as lambda goes to infinity, this, that's a lambda squared, so as lambda goes to infinity, this becomes the imaginary part of this guy. This is gauge transform of the trivial connection by a singular gauge transformation. So again, x is a quaternion. This is imaginary part of quaternion, this is quaternion multiplication. This is a unit quaternion defined away from the origin, and if I do think of the formula for how a gauge transformation acts on connections, you can check that if you act on a trivial connection, that's, by this gauge transformation, that's what you get. So this connection converges as kind of natural, it converges away from this point, the curvature is going to zero away from this point, it's converging to a flat connection, and then the limiting guy, when you look at it, you can fix it by a gauge transformation, which is not a global gauge transformation, that doesn't make sense at the origin, but it fixes the problem that the connection has at infinity. So, sorry, tau is this guy, tau lambda is these translations, so tau lambda of x is lambda x, thought of as acting on the force sphere, zeta. This is in a trivialization. So I've written this as an imaginary quaternionic valued one form, that's what a connection should be in this case, the algebra of sp1 is the imaginary quaternions. So it turns out that this connection that we constructed, the very symmetric one, is an expression for the connection one form in a certain trivialization. Yeah, so on a chart in S4, a chart in S4, and then you have to trivialize the bundle on that chart. So, very explicitly, we can see what happens in this family, and that's kind of a new phenomenon. I mean, we'll see how that plays out. Okay, so that's a new thing that can happen. We can have bubbling. So this is called bubbling, by the way. And it was discovered by Karen Ullenbeck, actually first in the context of harmonic maps, and then when she moved to study engage theory, she saw that it happened there as well. Well, actually, it's an interesting question. I mean, in fact, you can think of it that way. The question is, can I think of, like breaking of Morse trajectories? And in fact, yes, you can, because another conformal model for the force sphere is S3 times I. And this is just shooting things off the end of the cylinder, but, and it's a big but. I can do this in any direction, at any point on the force sphere, and I'm only allowed, if I'm going to think about it as Morse theory, I'm only allowed to think about it one time as a cylinder, but this can happen, you know, so if I think of, this shows you how, essentially the only way it turns out, Morse theory is going to, the Morse theory compactification is going to fail. So, if I look at R times S3, which is conformally S4 minus 2 points, well, I could view that particular situation as just, the energy is just sliding out that way. That's all it's doing. Okay, but I could have picked a different point and then the energy can also concentrate at that point. So then just the broken trajectory compactification isn't good enough. You're going to need to kind of keep track of where you might bubble as well. Okay. Okay, so now, we need to, yeah, it doesn't, but there's a natural generalization of it that does, which we will maybe state today, maybe not, we'll see. Let me see what else there was. It's over here. I'm going to say next. All right, so I want to indicate some of, some of what you need to do to start to make some progress in understanding these equations from the analytic point of view. So, step one, you need to understand a little better if we have P over X, which might be a four manifold with boundary, might be a compact four manifold, might be a cylinder. We need to understand how to deal with connections mod gauge. So I think in the first lecture series, you saw that a bit about how to do that in the Abelian case. So let me, I'm just going to sort of state some things, which you can read more about in the notes, but so remember, this is the way things act, or in a trivialization, so this is just a map from some ball to the group, and this is a one form on the ball with values in the Lie algebra. And this is a formula for the action. Now, we need to complete these things. So far we've been thinking about smooth things, and it turns out to be a good idea to complete these things as Hilbert spaces or Bonnock spaces. So we're going to look at space of LPK connections. So take background connection, which is smooth, and then take an LPK value, one form with values and add P. Now, if you look at this action, the way you have to differentiate the gauge transformation, so the natural group that acts is the gauge transformations of, well, you can look at the notes to see how to define this a little bit more carefully, LPK plus one gauge transformations. So if I have a gauge transformation like that in LPK, sorry, LPK plus one, then the covariant derivative of that gauge transformation with respect to the connection will be in LPK if you need a, well, actually, to be a tiny bit careful in the borderline case. So we need these to be continuous usually. So sub-left embedding theorem tells you, we need K plus one minus N over P to be positive. That's where the continuity comes from. If we're on an N manifold, so just in dimension four, the borderline for G is, well, either L41 or L22. Anything a little stronger than this will work. You can still work at the borderline, but it's a little trickier. You have to be, you know, that's like a bit radioactive. You have to make sure you wear your protective gear because it's easy to make mistakes, but it's still very useful to do things at that level. Okay, now, so you can check easily in this setting that, you know, you can give this the structure of a Banach-Lee group or Hilbert if P is two, and this is just an affine Banach or Hilbert space, and the action is smooth. Smoothly on APK, the quotient house dwarf. You should think that's a tiny bit surprising because this is not even, G's not locally compact, so it's a rather horrible thing. Nonetheless, the quotient space, as long as you have continuity, is a house dwarf space, and in fact, it's kind of a, it's a, quotient is a Banach, I'll say orbifold. So what I mean by that sort of, I'm not going to give a precise definition, but what I mean kind of colloquially by that is that it's a Banach space quotient, quotient by the action, smooth action of a finite dimensional Lee group. So it's really, you should think of it as a smooth manifold that has a tiny bit of singularity. And importantly, how do you prove any of these things? Here's A, there's an orbit, G.A. The key thing in understanding transformation group theories, understanding the slice to an orbit. And there's a nice notion of the slice in this context. So that's connections of the form A plus little a, where d star of little a is zero. So little computation that you can do that tangent space to the orbit is equal to dA of sigma, or sigma's section of the Li-algebra bundle, little adp. So in this direction is the image of dA, a natural complement is the L2 orthogonal complement, which is this, or if there's no boundary or it turns out I should take the hodge star and restrict it to the boundary if there's boundary. Okay, so that's the slice. What is the epsilon? I need to have A be small in LPK norm. So what you can show, rather by the inverse function theorem, that's why working in a Banach space is nice, that implies that if I look at SA epsilon, act on it by g, this, sorry, this is my picture of SA epsilon. Now I'm going to, the slice, now I'm going to act on it by g. So if the stabilizer of A is trivial, this takes A plus little a and g and just maps it to g acting on A plus little a. Okay, so you just take the slice and you translate it by the action of the group. If there's no stabilizer, this is a diffio onto its image. Okay, so now, why do I? Well, why do you pick that one usually? That's the boundary condition that means you're orthogonal to the orbit of the full gauge group on that compact manifold with boundary. You could take gauge transformations that are the identity at the boundary and that would give you a different boundary condition. And what we're doing, we really want to look at if it's unrestricted on the boundary. Yes, that's correct. I mean, so if it's a manifold with boundary, if it's a manifold with cylindrical end, we're going to do something else at infinity. Okay, so my erasers keep ending up here. More generally, the stabilizer of A is a finite-dimensional lead group, that's a kind of general fact that you have to prove along the way. And a slightly more general statement is that if you take, so the stabilizer acts on the slice, so you can take the quotient of this thing by the stabilizer if the stabilizer is non-trivial and then that, you've quotiented out a nice Banach space, Banach manifold by a finite-dimensional lead group and that's a model for the neighborhood of that thing. That's a free action by the way. Anyway, so, but it, you know, it's important to realize that this equation is the slice equation. That's making us move our connections in a way that we don't end up having gauge equivalent connections nearby. And what I want to do next, and construct the slices, let me say one tiny philosophical point, which people tend not to be very good about in literature. So if you, it's a really good idea. So the inverse function theorem easily gives you this result that where you control the LPK norm. Now, if you imagine, you know, as you change P and K to get bigger and bigger, well, that's making these balls smaller and smaller and smaller. But if you've proved that it, this map is a diffeomorphism onto its image for some P and K pretty small, with a little bit of care, you can prove that it's still a diffeomorphism on that kind of big-ish ball, but with respect to any stronger norms. That, anyway, that's a useful technical thing to keep in mind. And anyway, yes. Yeah. Well, no, I mean, okay, it depends what you want to do. I mean, you'll see in a second why I wanted to do this. So let me state the basic workhorse theorem, Olinbeck's fundamental lemma, which is the basis of where compactness, the compactness that you can prove comes from. So I'm going to state this for n less than or equal to 4 because it'll be convenient a bit later. And there exists c and epsilon such that for any L21 connection, and note L21 means the gauge group is L22 and you're in the slight danger zone, no matter for any L21 connection, there exists an L22 gauge transformation g. P is just the principal bundle over the n-ball. There's a gauge transformation, so a map from B to g so that if I apply the gauge transformation to the connection, I compare it to the trivial connection, then the following things hold. Well, first it's been gauge fixed with respect to the trivial connection, meaning that you have the boundary condition because the foreball has boundary and most importantly, you get control over the L21 norm of the connection. In terms of its L2 norm of its curvature, so that's not an epsilon, that's less than or equal to, sorry, for an L21 connection, I forgot the key hypothesis, if the L2 norm of the curvature is small, so in sum what this is saying is that L2 curvature is small on the ball, then I can first of all gauge fix it, and once I gauge fix it, it satisfies this estimate, the L21 norm of the connection is controlled by the L2 norm of the curvature. And here this would be false if we used a different boundary condition. And what that's saying is that this energy function on connections on the ball, it's kind of a gauge equivalence classes with no constraint on the boundary. On the ball, the energy functional has the trivial connection as a strongly non-degenerate minimum. So you really, as you move away from the trivial connection, you really have to pick up some curvature. I mean, you're reading it backwards. If you move away, this is getting bigger, so the curvature must be getting bigger. So it's strongly non-degenerate. And anyway, we'll see that, you know, so this estimate is already starting to, it starts to play really well with this bubbling phenomenon that we saw. Let me say a couple words about that. Because this is conformally invariant. So I can, you know, if I have a compact manifold, it's conformally invariant, so it doesn't matter what size the ball is. I get that estimate, no matter what size and dimension for it. You have to be a little bit careful with the L2 norm of A, but anyway. So, you know, if I have, but anyway, that control on the fact that you don't have to worry about the size of the ball, it eventually shows that, we'll see next time that you can, that you get convergence away from finitely many points. I'll sketch that next time. And this same thing, by the way, it holds for dimensions less than or equal to four. Now, if the dimension is less than four, then if I take a little tiny ball and scale it up to unit size, the L2 norm of the curvature is actually decreased. So in dimensions less than four, this same estimate proves a very strong compactness theorem. There's no bubbling in sequences. Life is quite good. Okay. I've gone slightly over time. Did I go fast? Okay, that's it. Any, yeah. What about...