 So there's one thing I wanted to say before we get back to the compactness. So we saw that there were slices for the action of the gauge group, which gives the quotient space a nice structure. And then we can restrict that picture to the space of anti-self-dual connections, which are the zero set of the mat that sends A to F A plus. There's zero. Here's lambda plus. So inside this slice, where's my? Maybe the slice has acquired some dimension now. Inside the slice, there's a local model for what actually the anti-self-dual connection looks like anti-self-dual connections look like modulo gauge. And it's a fundamental fact, which you saw a version of in the lectures on the first week, that if you kind of linearize this picture, what you get, linearizing the gauge group action gives you the map that sends adp value of 0 forms to 1 forms dA. And then linearizing the curvature equation gives you d plus. You notice that A is d. That implies that d plus, this composition is F plus, is 0. So this is a complex. It's an elliptic complex, as you saw. And it's index. Well, I'm going to write it's negative index. Index is the alternating sum of the dimensions. We're really interested in the dimension of h1, so minus the index. And it's 8k minus 3 halves, the Euler characteristic of x. So this is in the case of a closed manifold, or 8k minus 3 times 1 minus b1 plus b plus. So this picture kind of gives you, so this is an elliptic complex. In particular, this tells you that if I restrict to the slice, then the curvature map is a nonlinear Fredholm map. And that gives you a nice package of stuff that tells you that the moduli space is locally, at least finite dimensional, if it's a differential surjective. Then of course, it's a smooth manifold of some dimension, dimension given by this formula, at least if the connection you're looking at is irreducible, has no stabilizer. OK, so that's something I maybe should have said a little bit earlier in yesterday's lecture. I'm going to go back to Ulinbeck's fundamental lemma. I'm not going to prove it. And I stated a slight generalization of it. So the statement I said at the end of the last lecture was the case where I just looked at one connection that has small curvature. But it's easy to prove using the same techniques. I'll say a little bit about that. If you take a pair of connections, each of which has small curvature, so there's some bound that Ulinbeck's argument gives you, epsilon, then, well, you can put both of them into this good gauge, Coulomb gauge with this boundary condition. But the interesting thing about looking at a pair of connections is that you can prove that the difference of the L21 norm of the difference of the connections is controlled by the difference in the L2 norms of the curvature. So the statement I had last time was, say, A2 is 0. So we're just saying that the L2 norm of the curvature controls the L21 norm of the connection in the good gauge. Here, we're saying something slightly sharper, and we'll see that that has a nice consequence. But let's, again, I'm not going to prove that in detail, but just go through one part of it. So a lemma on the way to the proof to kind of get you the feeling for what special in dimension 4, et cetera, is that, well, Ulinbeck's lemma holds, if we assume, actually, let's call this 1 and 2. So we're going to assume 1 that we've already got it in a good gauge. And instead of, maybe, 0, instead of 0, we're not going to assume that. We're going to assume something stronger. We're assumed that the L4 norm on the ball of both connections is less than some epsilon prime. Then that implies that 2 holds. So just to get you some feeling for why this lemma is useful in proving Ulinbeck's theorem, well, if you have control over the L21 norm, so in dimension 4 or less, that L21 embeds into L4. So this assumption holds if you know this. So if the curvature actually is controlling the L21 norm and is sufficiently small, then the L4 norm is sufficiently small. And then the idea, once you know this lemma of proving its theorem, is that you compare the set of connections that satisfy this conclusion of the lemma with a given hypothesis. You compare that with all connections that have small curvature. And you prove that the set that satisfies this hypothesis once the curvature is sufficiently small is closed and open inside the space of connections that have small curvature. And you prove that space of connections that have small curvature are connected. So all of them satisfy it. So somehow there's a thoughtful argument that goes from this lemma to proving both statement. And we'll see in this lemma where we use this. So this is just a straightforward computation exploiting the fact that we need to know that if we integrate over the foreball d plus d star of A, or this doesn't matter what the dimension is, then this is equal to the covariant derivative squared plus the integral of b squared on the boundary if the boundary conditions are satisfied. In particular, this is positive so I can get rid of it. So if I know this, so if the boundary conditions have to be satisfied and then I have this estimate. So on the other hand, if I look at, so again I'm assuming these guys are in good gauge, I look at the difference in the curvatures. In good gauge, this is one. That's what everybody has called it forever. It's a good gauge, it works. So you have to ask Greg. I would call it a gauge fixing condition. Maybe Greg would call it that. But somehow physicists are much less careful about language than we are, sorry. And it's true and we've adopted that language. I mean this is a gauge fixing condition, it's just meaning, just means that we're taking a slice. That's all it means. Anyway, this one. So there's another term, I mean if you write, this is sort of the Weissenbach formula in a manifold with boundary, there's another boundary term in fact. Well, the ball is flat so there's no curvature term. There's another boundary term which vanishes under the boundary condition assumption. So it's particularly simple. Actually let me, I don't want to do the calculation here because I'll run out of room. So we look at FA1 minus FA2 squared. That's equal to DA1 plus that minus DA2 minus D. And well it's in good gauge. So this is, let me say, yeah. Well first of all, this is up to some constant. I can write this as DA1 minus DA2 squared and then minus A1 minus A2 times A1 plus A2. Integrated over the ball. So I collect these terms together here. I collect these terms together. This is sort of a difference of squares. Sorry, I wanted to write this. A1 plus A2 and well this controls the covariant derivative by the Weizenbach formula over there. What about this term? Well, we use Cauchy-Schwarz inequality and we get that A1 minus A2 L4 squared times A1 L4 squared plus A2 L4 squared. And here you see the L4 norm coming in to play. But when we apply the Cauchy-Schwarz inequality, Holder inequality I guess. Anyway, so we're assuming these guys are small, right? And this is controlled by the L2 1 norm up to some constant, right? So what we learn is that this stuff we can control by the L2 1 norm times some epsilon that sucks a little bit away from this guy. The covariant derivative of the difference controls the L2 norm of the difference as well, as you can check easily. And that means that this is greater than or equal to some constant times A1 minus A2 squared plus A1 minus A2 squared. That's it. So it's a simple computation. The key is just this kind of rearrangement in here using the fact that the L4 norm is small. And one nice corollary of this, what? I could have used stronger norms. I couldn't have used weaker norms in dimension 4. The key thing is that I was using this bound, right? Which is true up until dimension 4 and then past dimension 4. Well, it just doesn't work, right? OK, so now a nice corollary of this is that if I look at the space of small curvature connections in a slice, so look at a set of gamma plus A so that gamma plus A is in the slice for some small epsilon. And well, that's sort of redundant. But anyway, then the curvature map, mapping to two forms on the ball, so I think of this with this as this is in the L21 topology. So let me state this. This is L21 topology, L2 topology on the ball values in SU2. This is a proper map, right? Well, that follows immediately from this estimate if I have small curvature, so in the set of connections with small curvature, I get control of the L21 norm. And then if the curvatures are Cauchy and L2, this tells us that the connections are Cauchy and L21. So it's a proper map. And that's kind of the precursor of compactness. So this is, we want to know something about sequences of anti-self-dual connections. And what this shows is that what I need to do to prove compactness, at least locally, is that I need to prove that for a sequence of anti-self-dual connections, I can pass to a subsequence where the curvatures are converging strongly in L2. That's all I need to prove. I only know that half of the curvature, F plus is 0. But anyway, I'm not going to, well, let me just state a version of compactness that's good enough. I won't prove it. Again, I think that simple calculation starts to give you a feeling of what you need to know. And one nice way to kind of set up compactness is to note that if you look at the small energy moduli space on the ball, it's a set of gauge equivalence classes of connections on the ball so that F plus equals 0. And the curvature is less than this magical epsilon. And what you can prove is that there's a restriction map to sub-balls that this is actually a compact map. And several different ways to prove this. I mean, you can think of this in the sort of maybe most standard way to prove this is to just show that once you put connections, anti-self-dual connections in a good gauge, then on sub-balls you have control over higher norms. And then this would follow from, let's say, less goalie theorem. There's kind of, in some sense, a more elementary proof that just uses this properness eventually. But so what this is telling us is that once we have a sequence of connections and we can find a ball where all those connections have small curvature, then on a slightly smaller ball we can assume that they converge after passing to sub-sequences. And so the interesting thing that happens in dimension 4, let's first think about a compact manifold. So let's take a sequence of connections, so ASD connections in a principal bundle P. So first of all, the energy of the AI is always, it's this with our conventions for pi squared k. So it's bounded uniformly. And then what we do is we cover, did I say what my manifold was called, maybe x by balls of some fixed size. So here's x. And we want to do it carefully, so fixed size, but so that we want to make sure we choose the cover carefully so that no point, there's a uniform bound for the number of balls that meet any point. All right, so then if we have that, then the number of balls that violate this, the size of the balls doesn't matter because we're in dimension 4. We'll index statement in dimension 4 scale invariant. So the number of balls that violate the requirement for any given connection is finite. And so we can pass to a subsequence where all the connections satisfy the hypothesis of Ulland-Beck's theorem on all the balls, but finitely many of the same ones. Yeah, but the key is that no matter what scale, it'll be the case that there are only a given bounded number of balls for which that fails. Because the total energy is bounded uniformly always. So there are at most 14 balls no matter what scale, say, which violate the inequality. Only 14 balls for any given connection, so you pass to a subsequence where the failure is only on a fixed 14 balls. At some scale, you pass to the next scale just looking, observing those balls. Again, there are only 14 that are bad. And you keep going, and eventually you get convergence. So it's just like Gromov compactness. Is it Gromov compactness? Yeah. What's the first one I'm saying? Well, it's this epsilon. That epsilon is given to you. We prove that for a four ball. For the flat four ball, there is a constant. It doesn't matter what the radius because of scale invariance is. It's exactly the same as in Gromov compactness. I mean, because the energy scale invariant, you prove an estimate on the unit ball once. Then you know it for balls of any scale. And as long as you're below that energy scale, no matter how small a ball you have, you're in good shape. So that's exactly the same. And they're only 14. No, no. I mean, you could prove a sharper statement where epsilon is the energy of the instanton on the four ball, on the four sphere. This is epsilon's, you know, the epsilon that you get out of the argument is much smaller. And it's a technical story, which, you know, first you go by proving that for the given epsilon that you get out of this mess, which is just, you know. I mean, this epsilon depends on, you know, first eigenvalue on the ball and Sobolev embedding constant, that sort of thing. That's all it notices. But then you could prove a sharper statement that says that, well, actually, you don't have any problem if your energy is below the energy of an ASD connection on the four sphere. So anyway, yeah. OK, so we happy with that? So we get eventually out of this, we get convergence away from, finally, many points. Yeah. That's just, it fails, well, because we don't, you know, we don't have, you know, if we have a uniform bound on the L2 norm, then we kind of lose on, you know, the scale and variance as I shrink the ball in dimension five or higher, the energy increases, right? So you don't really, no, no, no, I mean, this, well, this is solving this equation. And this is kind of general nonsense. And that's where, you know, like I said, it follows. What? Which, it goes wrong here. There's no, because we don't, that epsilon doesn't, you know, it's not scale and variance, the size of the ball matters, right? So it just doesn't start. Yeah, yeah, yeah. To localize where the, where the, yeah. So, you know, so, you know, maybe this ball was one of the 14 bad balls. So I get convergence everywhere else. Now I cover this ball by smaller balls. There are, again, only fixed number of balls, there may be two, but no more than 14, inside here. And then I do it again. Okay? Okay, okay, great. Yeah, it just gets worse, life gets worse and worse. I mean, like I said, there are statements you can prove in that context. And the statements are trickier. You get convergence away from a subset of Hausdorff co-dimension four, stuff like that. And again, trying to analyze the structure of what that subset actually looks like is an interesting topic of current research. Okay? Okay, I'm worried that if Vivek and Greg don't understand, then oh dear, I don't know how many other people understand, but my students do, right? Right? Okay, okay. So we've kind of sketched why Lundbeck and Packness holds. And then there's one more key step, which is, let me just put it here. We get convergence away from finally many points. And then we need, as we saw in the four sphere, when we looked at that sequence of connections that bubbled, we observed that we could actually fix what happened at the limit. So there's a removable singularities theorem. So if A is ASD on the punctured four ball and has finite energy on the punctured four ball, then there exists a gauge transformation for G from the punctured four ball to G so that G star A extends to an ASD connection. So G need not extend. As we saw in that example, the gauge transformation that does this job may not extend over the four ball. So what this means is we have a sequence of connections that are converging away from a finite set of points. We get a limit, which is defined away from that finite set of points. This is saying we can fix it, but in fixing it, we're gonna change the topology of the bundle. So P changes. All right. So now let's get back to the cylinder and I need to explain one more important thing before we can kind of sketch the construction of floor homology. So now let's think of R times Y. And I'm gonna say it. Yeah, so the behavior of the moduli space on a closed-form manifold was kind of controlled by local behaviors controlled by these two operators, D plus and D star. D plus gives us the linearization of the curvature equation. D star gives us the gauge fixing. And what I wanna do is, so it's nice to just package them into one package. We're gonna look at this operator on the cylinder and let me tell you what it looks like. So I'm gonna write my connection as some pullback connection from Y. So gamma lives here, it's not necessarily a trivial connection. Plus a one-form time-dependent one-form on Y plus a time-dependent zero-form times DT. So I can write any one-form as a bit with it, out of DT component and a bit with a DT component that I choose to do it that way. And then this operator, in terms of these guys, becomes this guy. So BC goes to, okay. So on the cylinder, I'm gonna write it in this particular form. What's important about this form is, first of all, it looks like D by DT plus a self-adjoint operator. This is self-adjoint. You looked at, one of the homework problems was to look at this operator on the three torus and you observed, I hope, that its spectrum was infinite in both positive and negative directions. Now, if you look at, sorry.