 That's gamma. If you look at this bit, that's the Hessian of the Chern-Simon's function. Chern-Simon's function, its derivative term was something like this. So its Hessian is basically this operator. You have to put in a couple of hodge stars in here to make it into an inner product. And then the Hessian becomes this. So the operator that controls the deformations of the anti-self-dual equation looks like the Hessian. I mean, it looks like d by e t plus the Hessian, as in finite dimensional Morse theory. But because of the gauge group, there's some extra terms involved. And I want to explain an important point here. So this bit is, well, we'll call the extended Hessian. So the extended Hessian is its first order. It's elliptic, it's self-adjoint, and has a spectrum extending to plus or minus infinity. So here's the r that contains the spectrum. Spectrum's real. It's spectrum's discrete. Looks something like this. And here's 0. So if we're just trying to do finite dimensional Morse theory, I mean, the first movement, finite dimensional Morse theory is to understand the stable and unstable manifolds. What this picture is telling you, so dimension of stable and unstable manifolds are the number of positive and negative eigenvalues. Those are both infinite. Now, there's two interesting things to say about that in this context. So that looks like bad news, but if I so, yeah. So this is the space of self-adjoint FEDHOME operators. So let's think about the bounded self-adjoint operators. These aren't bounded, but just for concreteness. Look at the space of bounded self-adjoint operators on a Hilbert space. This breaks apart into three pieces. So because it's self-adjoint and FEDHOME, that means that 0 is an isolated, the spectrum near 0 is discrete. In this case, it's actually discrete everywhere, but anyway, spectrum near 0 is discrete. And then this breaks up into those operators that have only finitely many negative eigenvalues. That's this guy. They're essentially positive. There's this bit where it has finitely many positive eigenvalues. And there's another component, which has both infinitely many positive and negative eigenvalues. Or a little more precisely, the dimension of spectral projection is infinite for both positive and negative. Anyway, what's interesting is that this is contractable, these two. And this guy, it's topologically very interesting. Its loop space is, well, if it's real operators, it's classifying space for real k theories, z times bo. And in the complex case, it's z times bu, if this is for mission operators. Anyway, these are real FEDHOME operators, so this is the bit that we're interested in. OK, so we were quietly minding on our own business, and we ran into some interesting and serious algebraic topology as we're trying to do Morse theory. Now, in Morse theory, actually, so remember, I didn't say this yet, officially, but if I have two critical points and I look at the Morse function, I want to compute the differential of a, then say with my two coefficients, what I do is I look at, I sum over b so that the difference in the Morse indices is 1, number of points in MAB check. So this means that the dimension of the intersection of stable and unstable manifolds is 1, but I act by translation, so it's a finite, I mean, compactness tells you it's a finite set of points. And sorry, this is times b. That's the definition of the differential. And this difference is the thing that still makes sense in the infinite dimensional context. So if I have critical point a, critical point b, I look at the eigenvalues of a, look at the eigenvalues of b, well, then they move along. Picture like this happens. And in particular, some number of these paths cross 0. And if you think about it, the number of times it crosses 0 in a finite dimensional case is computing this difference. But we can always, I mean, in the infinite dimensional case, we replace defining these things individually by only worrying about their difference. Shouldn't I grade? Well, the eta invariant, I mean, in the finite dimensional case, there's no eta invariant. It's 0, right? And what? In my floor case, well, I mean, it depends whether I, I mean, if I want a canonical grading, then there are many ways to do it, eta invariant being one of them. So yeah, sure. And I mean, the eta invariant minus transimans is what's natural, because the eta invariant depends on the choice of metric, et cetera. And you don't want your grading to depend on those choices. Yeah, but I don't. I just want it to, I mean, look. You should do the best thing you can do. Having a detourcer when you can just have an honest canonical grading, not a good idea. If you need to have it, take it. Yeah, I mean, anyway, let's not get into that interesting bit of weeds for a minute. I mean, it's important to get a canonical grading. Now, yeah, what do I want to say? Yeah, right. So there's a spectral flow, which maps the loop space of this component to z, loop space or paths between two points, path space between a and b, which is picking up this z. I said that the loop space of the space of self-adjoint operators is z times b o. And the interesting thing that picks up the z is the spectral flow. Now, there's a really, well, slightly disturbing thing that happens. That plus is actually a zero. Yes, it's the interesting component. It's true for plus. It just doesn't have a very interesting image. I mean, there's spectral flow in all components. Anyway, OK, so in dimension, if I look at this quotient space, kind of think of the picture downstairs, well, this gp is basically maps of our three-manifold into s3, which has pi 0 equal to z. So that means this has pi 1 is equal to z, pi 1 of the space of connections mod gauge. And so a better picture of the quotient space, some kind of schematic picture, well, there might be some singularities. But it has fundamental group. And now, if I take my a and b, if I compute the spectral flow along one path and I compute it along another path, they're not homotopic. They might not be the same. And in fact, the spectral flow gives you a map from pi 1 of this bp to z, which in the case of SU2, its image is 8z inside z. Spectral flow around a sort of generating closed loop is plus or minus 8, OK? And yes, it is 4 times the dual-coxedron. The dual-coxedron number, by the way, is just telling you if I look at the inclusion of a basic SU2 in G, what multiple of the generator it hits. And that's, I mean, sorry, what did you say exactly? Four times the dual-coxedron. Yeah, so I think, yeah, the dual-coxedron number is twice what I said. So that was, anyway, OK, great. Anyway, so there's this spectral flow. Now, if I were doing finite-dimensional Morse theory, Chern-Simon's function is really a circle-valued function here, what I would think I have to do is Novikov-Morse theory, because I no longer have energy bounds. I mean, if I take the change, let's draw this picture again, if I take a path, a flow line, then the change in energy depends on the homotopy class of the path. And I don't get a uniform energy bound. I need to get Morse theory going, we need uniform energy bound connections in MAB. But yeah, there's some little way to win here, which is quite remarkable. This is sort of one of Floor's basic observations that, so in this case, the fundamental group is z. And yeah, what do I say? Oh, I still have it. See here, I have this differential. Now, let's take that. And the way we should modify this is, well, I need, here's my gamma, and I look at its homotopy class in the path space from A to B in BP. For the differential, I need to only look at flow lines that are in the given homotopy class. And then I need that the spectral flow along that homotopy class is 1. I want to only look at one-dimensional moduli spaces. Well, in this case, there may be a one-dimensional moduli space if I go along this homotopy class. But maybe along this homotopy class, it's nine-dimensional. They differ by multiples of 8. So I only want to sum over those where the spectral flow for the given homotopy class is exactly 1. But now you notice that at the spectral flow mapping, pi 1 of B to z. And I also have the sort of change in churned simons around loops that matched pi 1 of B. Well, let's say churned simons over 4 pi squared. That was the thing that was an integer. That maps to z. And these are proportional. In fact, the constant of proportionality is 8. So what that tells you is that in this homotopy class, that energy is fixed. Because there are many homotopy classes, but I pick one homotopy class, the one where the spectral flow is 1. And churned simons has a unique value on that homotopy class. It doesn't depend which curve in that homotopy class I pick. So these are proportional by necessity. So that implies energy bounds in a given, well, maybe, to say it more simply, given a homotopy class, there are energy bounds. And then the spectral flow pins down the homotopy class that I need to use in the definition of the differential. OK. So a version of the compactness theorem holds if you keep track of the homotopy class. And the only way the compactness theorem differs, let me see if I can try to state it. The only way it differs is because there can be bubbling. So let me say that. So here, the finite dimensional picture, we looked at the energy profile of a given connection, or a given Morse trajectory, or eventually sequence of Morse trajectories. And if I took a sequence of Morse trajectories, I could pass to a subsequence where the only interesting behavior was that there are finitely many chunks of finite many chunks of finite length, uniformly bounded length, where the energy is kind of big. And then these chunks might be moving apart, but in between these chunks, the energy is small. So that was the picture in the finite dimensional case. What can happen here is that the energy profile you should think of now can also have some bubbles happening. So as I have a sequence of connections, next guy in the sequence might have moved apart, and the bubbles have gotten stronger. That one's disappeared off the face of the Earth over there, and the sort of background energy profile looks like this. So what we should do, remember before when we tried to compactify what we did is we looked at spaces of broken trajectories. So maybe just one break for a moment. That would be something that appears in the compactification of MAB. But now we need to add to this picture, we need to add bubbles. And what bubbles do, I should refine the picture by looking at, first I define MAB plus some number of bubbles. So a point in the symmetric product of this guy. So I'm going to take as another thing that I'm going to add to my compactification. It's built out of things like this where, what do I say? So these are finitely many points with multiplicity in r times y. And then I have some instant on A. There pairs A and x1 up to xk in here up to the action of the symmetric group. And I have to tell you what I think, I should pick a homotopy class and what I should tell you what the spectral flow for such a pair is. I shouldn't call this L. I mean, well, it doesn't matter. The spectral flow is the spectral flow of the connection plus 8L. And then the compactification of this, let me call, then I want to take this and mod out by translation. So if I want to compactify MAB, the things that it's this thing. But then, well, first thing I'm going to add is the kind of things that we saw before, MC1 check times MC1B. But now, I need the gamma in there, the homotopy class. So these homotopy classes should, because these both end, this one ends at the point that this one starts and makes sense to concatenate these homotopy classes and then they should add up to this guy. So similar things with more breaks. But then I also need to include things like M check gamma. I'm going to call it minus 1 AB times, sorry, M check gamma 1 R times Y divided by R and M. So let me just try and say it in words. The formulas just kind of get too hairy. Before the building blocks are just, I want to look at the space of broken trajectories. For broken trajectories, it's important to realize that there is a homotopy class by concatenation. So the compactification before was if I wanted to keep track of the homotopy class was just by broken trajectories. Now, what I want to do is still look at broken trajectories, but I want to allow broken trajectories that have some number of points at which bubbling happens. So I take a point in such a modular space together with a bunch of points, mod out by R, and then I throw that into the pile of stuff from which I have to build up my compactification. The thing that's important, when you prove that D squared is 0, the thing that's important is to see this picture. You go from A to some C, where the Morse index is 2, and you want to see that the only thing that appears in the compactifications are things where there's one break. Now, what happens here is that the bubbling sucks off the dimension of the moduli space after you bubble, just the raw dimension is 8 less. But then you have to remember the point at which you bubbled, which adds 4. So the total drop is 4. And that means that here we're only looking at two-dimensional moduli spaces. When you, in the compactification of this thing, you don't see any of the bubble contributions. So there are no two. So here spectral flow is 2. So if spectral flow is less than 4, there's no contribution. So the compactification is still just as it was. This picture, as infinite-dimensional Morse 3, is what you need to see to prove that D squared is 0, at least mod 2. And then you can keep track of signs. And yeah, so that's sort of, well, one more. I mean, yeah. So one thing that doesn't happen on a cylinder, but for a cylinder, things are not so bad. Well, one thing I haven't talked about is reducible connections, which cause trouble. Now, all the floor homologies that I'm going to talk about next time, there will be no reducible connections. There will be no trouble at all. If I want to prove independence of choices, so there are a huge number of choices that you need to make to make the story work just as in the finite-dimensional case. There's a choice of metric. Well, the churn-Simon's function may not be a Morse function, may not have non-degenerate critical points, so I need to figure out how to perturb it. The moduli spaces might not be smooth. I need further perturbations possibly to make moduli spaces smooth, et cetera. That's all fine. And it works quite well if there are no reducibles. If there are reducibles, so improving independence as long as there are no reducibles, all you need is you just need to be able to do this picture in one, pictures like this in one parameter families. You just need to, you want to see that two chain, two, I make two different choices, the chain maps, the differentials. There's a chain homotopy between those guys, and that involves just moduli spaces of dimension one bigger, if you think about it. So that's still fine. The interesting problem happens when you try and define co-bordism maps when there are reducibles around. Then there can be technical problems, and it's more interesting. OK. I think that was the question. Any other questions? Yes, Josh. Well, no, it's really less than eight. Sorry, I shouldn't say eight. It depends what you want to keep track of. Sorry, the moduli space has to be at least eight dimensional for a bubble to happen. That's what I should have said. Sorry, thanks. So we're quite far from that. Anything else? OK. Bon appetit.