 Does somebody say go go? Okay, great Okay, so this is number two. I was gonna try and slow down, but then I noticed there was this on the desk Which says to the people far and back says go fast so I Don't know if that was for me, but I'll take it Okay, so So yesterday we saw that if you look at the churn simons function on Principal bundle on a three manifold then its critical points are flat connections and in doing some computations We saw that that there was something interesting going on If we just looked at flat connections on some special three manifold cyphert fibered spaces then the count Saw something interesting about four manifolds that they bound We also saw that if we looked at certain representation spaces of knots that the homology of those representation spaces was Looked like the Havana homology of the knots and so we we want to figure out a story That makes some Nice meaty story that those things fit into so the idea floors Fantastic idea was to try and do Morse homology for things like the churn simons functional And I just wanted to quickly throw up a few things from finite dimensional Morse theory, which we'll see In the what how they play out on that in the infinite dimensional setting a bit But you know here's everybody's favorite Morse function the height function on the Taurus We look at gradient flow for some choice of Ramanian metric Look at the energy of a path here. It is maybe want to put a half in there. Maybe not. I didn't today a Crucial thing is that never mind if it's a flow line the difference in the If you take a path Somewhere in here Then you can compute the difference by looking at the energy plus an extra term Which vanishes sorry wrong term It vanishes if If the this is a solution to the gradient flow equation so if you have a solution to the gradient flow equation that energy is controlled by the by the change in the in the value of the function and a key player in the story of Morse theory and some tellings of it is understanding the space of flow lines. So it Is m a b is the Is the set of gamma from r to m that solve the gradient flow equation Downward gradient flow equation and they limit As t goes to plus plus and minus infinity A gamma of t is a or B All right, that's so this plays a fundamental role and in fact it's Quotient by translation plays a plays a bigger role The way it plays a the way it's used What's important important property of it is understanding compact a compactification of it. It's called the broken flow line compactification I look at this space of trajectories mod translation and then It turns out that the only way that a sequence of Trajectories mod translation can fail to converge is if it converges to a broken trajectory. So here's an instance of a Thing in the compactification is a union over possible breaks. So there's a flow line from yeah, sorry That's equal to m check. Yes. Thanks Yeah, so this is this is so m check bar is a compactification of it And this is what it looks like it it counts trajectories. So here's a here's there's just c1 in this case and B So you compactify by adding There's two c1s in this situation You know so that that's a picture and just to give you a hint about how the Compactifications proved that's proved in the in some detail in the preamble to the notes, but if you take a flow line and look at it's the where look at the point-wise norm of the Gradient along the flow line. So that's what this picture is meant to represent well We know that the integral of that is finite because it's If it's a flow line between a pair of critical points the integral of this is finite So it's going to break up into pieces where You know, there's a be finally many pieces where it's big and then most of the time. It's small Now where it's small One important thing if you're going to find dimensional manifold It's easy easy to check and you need as a hypothesis in general if the Gradient is small then that tells you that you're in that need needs to tell you that you're near a critical point Now you can prove as is done in the notes that if you so here the energy small So pick some point you must be near a critical point in somewhere in here because they know the change in in the function small here so that Integral of this quantity is small over an interval of some fixed length So there must be some point in the interval where that the gradient itself is small so at somewhere it's close to a close to a critical point and then You can check that if you're close to a critical point then in fact You stay close to that critical point unless the energy changes a certain amount again. That's done in the notes. So What you find Eventually is that if you have a sequence of trajectories you can cut up the sequence up to translation into bits where things are hanging around near critical points and then there's some moment some finite length Intervals where you're moving between neighborhoods of the critical points and that eventually leads to this compactification Okay, so that's that's a finite dimensional Morse theory and Now we want to see so we want to do Morse theory for the turn simons function. We saw We saw that the gradient With some set of conventions the gradient of turn simons Sorry, well we saw the differential of turn simons Actually, let me say that We saw the differential at a connection on of turn simons was this Integral right, so we want to take we want to So here's this affine space of connections in P and We choose choose a Ramanian metric on on y then The tangent space at a connection Like the space of connections in affine space for one form, so it's tangent space is The vector space for which is an affine space one forms with values and in the algebra And so we can define given a metric on why we can define an inner product By taking the using the hodge star and The inner product on the Lie algebra to give us an inner product and then with respect to this the gradient of Turn simons is The hodge star of the curvature So the downward gradient flow equation Says This And we want to make a Key observation about that Namely let's Look at Four manifold, which is the cylinder times y with a product metric Then Yeah, well Let me Just sort of yeah product metric. So this is a four manifold So I think you saw in the first week You know that if you look at the hodge star on the four manifold It's square is one on two forms and So you can is it interesting equation I Want to call this connection be sorry about that if you're taking notes Now This connection be a solution To this equation is a one parameter family of connections on a three manifold I can view that as defining a connection on the four manifold. So view BT as giving us a Connection a on R times y and that the key observation is that Can't use that this equation dagger Dagger is equivalent to to a Being anti self-dual i.e. that the Hodge star of f a that's a four-dimensional connection. So star squared is one. So there's two eigenspaces and Sorry That's equal to minus f a now the two Two important miracles that Happened which don't have a analog in the finite dimensional Morse theory setting so the first of these is that This equation has a bigger symmetry group than this equation. So this equation gives us a one parameter family of connections on on R times y not every connection is As it has given a one parameter family you can always do a gauge transformation so that they are but this Equation is symmetric under gauge transformation solutions to this equation are preserved under gauge transformations on this four manifold Whereas solutions to this equation are only preserved by gauge transformations on the three manifold So that that's the first miracle The second miracle is that well this equation it makes sense on any four manifold not just on a cylinder. So this makes so symmetry group is 4d gauge transformations and Makes sense any 4d manifold there we go Lovely, okay now I want to explain that that These identities have a complete analog in that situation Okay, so let's define that Energy of a connection to be the L2 norm of the curvature which is written this way in this situation So an observation here so hodge stars is Conformally invariant Indimension for on two forms so that implies the energy is in conformally invariant so that's another another miracle and The analog of the equation that I just erased is that the integral one half the integral of the trace of F a wedge F a So it's a close. This is a closed-form manifold. This is a churn bay form This is equal to Okay, so what happens is this quantity which you should compare to F of gamma of t zero minus F of gamma of t one You'll see in a minute why that's true. This quantity is equal to the energy Minus the sorry that's plus the L2 norm of F plus square So again You just as before if you have a solution to the anti-self-dual equation F plus is zero then This topological quantity controls the energy right, so as a as a corollary State two corollaries sorry This this is just That analogy From the finite dimensional case that this quantity which we'll see in a minute Why that's analogous it is equal to this so you know This is the quantity that compares to the drop in the energy. I mean drop in the function Which are supposed to think of the churn-simon's function This is the energy and this is the norm of the equation that we're interested in so that there's a very precise analogy here and So so the first corollary if x is closed So that makes sense. It doesn't have to be a cylinder. It can just be a closed four manifold then Then what we have is that a is ASD so Connection in a principal bundle say print a principal s u2 bundle P Is ASD if and only if it's energy? equal to 4 pi squared C2 of P Evaluated on the fundamental class of x. So And if it's not ASD It's greater Right, so the there's a topological lower bound for the Yang-Mills functional in dimension 4 and Yeah, that's wonderful And yeah, and in particular If the second-churn class is negative this evaluation is negative. There's simply no ASD connections Corollary to is that If X is a manifold With boundary an empty boundary then a is ASD if and only if The churned Simons of the connection is equal to the energy Maybe there's a half in here, but so this Guy remember if we if we were on a cylinder This quantity just measures the Is equal by definition essentially to the difference in the terms value of the churned Simons at the two ends but More generally, it's just giving you a definition of the churned Simons when you have a manifold with boundary so in particular this the churned Simons integral churned Simons function doesn't This doesn't depend on the extension of the connection to the four manifold except up to a and You know up to four pi squared times an integer so again there's a characterization You know so let me say that this is that The formula that's exactly analogous to the story in Morse theory when you're on a cylinder Okay, so lovely That's that's where we get started now of course Morse theory is not so simple I mean setting it up. It's not so simple and The reason is because there I There are many ways to see that that there's going to be some kind of problems one of which is that there are already Interesting solutions on on the force sphere, so I want to give you An example of an interesting solution that's called the basic instanton and it's It's due to the 11 polyac of Schwartz and Tupkin something like 1975 or so They found an interesting solution to the ASD equation the many different descriptions of it One nice description is to think of So look look inside the quaternions at the sorry Two by two quaternity, sorry H2 look at the unit sphere then let Su2 which again is sp1 the three sphere inside the quaternions We're gonna let that act by Gonna make a right action this guy, okay, so this Is my eraser If I take the quotient that's HP1 that's the force sphere So we have this there's s7 so one of one of the beautiful hop vibrations You know so s7 is an s3 bundle over s4 and There's a there's a natural connection in this story Namely if I take some point and look at the orbit So here's the orbit of some Say v times sp1 I make a connection by Using the Ramanian metric standard round metric to find that define the horizontal space for the connection to be the orthogonal complement so it's the Orthogonal complement to the vertical tangent space there's projection pi and It turns out that this is a an anti-self-dual connection And Let me Let me quickly sketch why that's true, so you can again there are many different proofs You can do it by hand that's on the homework, but you can also do it because because this connection is very symmetric You notice that sp2 so So sp2 acts on s7 preserving a so sp2 is the Isometries of h2, which we are thinking of is here. We're acting This is a member Quaternions aren't commutative so we're acting on the right to define the principal bundle and There's a left action of two by two Quaternion matrices on this and we look at those quaternion matrices, which are also isometries. That's the group sp2 That acts and What I want to say about that so then Look at So that acts it preserves the connection. Oh, yeah, and by the way what the curvature of the connection That's a two-form with values in adp which when we can identify so We can identify Two forms on the base with values in adp Those are two forms on the total space of the bundle with values in the vertical space that's the so thinking of the connection sorry thinking of the curvature of a connection as a Two-form on the total space of the bundle which transforms when I do the right action by the adjoin action Well, that's what it is so it's a section of the sky and Now let's consider the stabilizer of Let's take Now V not to the be the point just one zero in h2 V not brackets that's it's Equivalence class in s4 and look at the stabilizer of V naught That's just That's just the diagonal matrices you can check easily diagonal matrices where they're each just unit quaternions now the And the connection is preserved by this sp2 action and so we can So it's curvature is an sp2 Equivariant to form on the total space and What you can check Is that Yeah, this is so p is s7 Okay, so I mean, let me just Once you think about this it's just an exercise and let me tell you what the exercise is so think of this At the point at the point v naught that's a map from Lambda 2 of the force sphere to the Lie algebra little sp1 now this splits as a direct sum of So this is this is lambda 2 of the force sphere split splits as Sorry, the eigenspaces of the hodge star This is an equivariant map So it has to you know and both of the this these three representations are each irreducible representations of sp2 and what you check What you check is that That you know That sp2 action is trivial here trivial here non-trivial here and Sorry, I'm not saying it So the thing I should say is the thing that preserves v naught is Just this subgroup where you have one here and q1 here and you can check that that That subgroup acts Self-dual it acts Trivially on both of these guys non-trivially here So the only option for the curvature is that it's a map from here to here So I mean just check by symmetry anyway, so You can check it many other ways Anyway, so there is an interesting solution is very nice symmetric connection on the seven sphere and We can generate other solutions from it. So we noted that the ASD equation is conformally invariant So we can act on a by the conformal group of S4 So 5 1 Now A is actually well strictly speaking if we're acting on a it's Acted on by spin 5 1 but never mind that a is a is fixed by The original one by sp2 which is spin 5 so we get Get an SO 5 1 mod SO 5 family of connections and that you can think of as Hyperbolic 5 space. So there's a 5 the ball so from this one connection. We've already constructed a 5-parameter family of Different solutions to the ASD equation