 going to the lectures themselves. Let me tell you that. So I do realize that I was very quick as the last lecture. And there was a lot of material, but that was a conscious choice. If you want to come to somewhere, you have to be quick at basics. So the good news is that for the next two lectures, after this one, so for the lectures three and four, this material will not be very important. So you still have some time to catch it up. So but if you are interested, then please do go to notes and read the details. OK. I will be still probably quite quick this lecture, but I will also slow down tomorrow. So take this as a good news. In any case, so I have finished the last lecture with the current classes. And first, we have defined an invariant polynomial for the Lie algebra UK into r. And this was psi is mapped to the determinant, essentially 1 plus i over 2 pi psi. We have written this as let me put lambda here, lambda to the power k plus c1 of psi, lambda to the power k minus 1 plus, and so on, ck of psi. And so what I've said the last time, if we apply those polynomials to the curvature, so if a is a connection on a principal bundle p, then we can compute p, so in this case, cj of f a. And we can think of this as a closed form on the base manifold, so we can take its cohomology class, and this gives us some class in the 2j's theorem cohomology of the base manifold m with values in real numbers. So here are some maybe before going to the property. So let me also define c, so the total current class to be the sum cj of p and j is here from 0 to k, where c0 of p is by definition just 1. In any case, so here are the most important properties. So the first property is c0 of p is 1. So if you wish, this is just a definition for any principal bundle, let me write. So if you have a principal UK bundle, we can always construct the associated vector bundle, where the action of UK on ck is the tautological one. Well, this is now a vector bundle, so let me denote this by p. Now, so I can think of, so I can talk about trend classes both for principal bundles and for vector bundles. And if I take the pullback, so consider the trend class of the pullback bundle, this is the trend, so this is a pullback of the trend class of the initial bundle. Now, the set property is, if I take the written sum of two vector bundles, say e1 and e2, take its trend class. This is just the trend class of e1, the trend class of e2. So you see why both principal bundles and vector bundles are sort of equivalent. It's easier to talk sometimes about vector bundles like here, while sometimes it is easier to talk about principal bundles. And the last property is that the total trend class of the tautological line bundle of p1 is 1 minus a. So let me define what is going on here. Now, as I said, this is just the tautological line bundle of p1. Now, what we know is that the second cohomology of p1, say with set coefficients, is canonically isomorphic to set. And the generator here is given just by the Poincare dual of the fundamental class of p1. So here, I will normalize so that a is the Poincare dual to the fundamental class of p1 up to a point, yes. Right. OK. So these are the most basic properties. So this is called the pullback. This is called the witness sum. And this is called normalization. Some properties which are sometimes also quite useful are the following. So if you take the j-strand class of the dual bundle, this is minus 1 to the power j Cj of e. And also, if you know that your bundle splits as, say, e1 plus some trivial line bundle of, say, rank r, then the trend class, so the j-strand class of e is 0, provided j is bigger than k minus r. So k here is the complex rank of e. OK. Now, this is something that you can easily prove from the definitions. And in fact, I think this is part of the exercises to do this. But what is not quite clear from the definition is the following property. So this is called the integrality of trend classes. Now, here is a theorem. If you have in our ambient manifold m, sub-manifold n of, so let it be closed oriented manifold, say, of dimension 2j, then the claim is that the integral over n of Cj e is an integer. Well, if you wish, what stays here is just the pairing of the homology class of n with the j-strand class of e. Now, this is quite striking properties if you start from the definitions that I have chosen for defining trend classes. In other words, what this tells us is that the j-strand class of e is in the image of the integral homology of m inside the real homology. Now, how can we see this property? So here is one way to see this. So for this, let me assume that k is 1, so that in fact, we have a u1 bundle. Or this is just the same as saying that we have a Hermitian line bundle, so let it be l. And the fact is that there exists an n, maybe quite large. And there exists a smooth map f from m to cpn. But the following property, so the property is that if you pull back the tautological line bundle of pn, this gives you l. And it's not very hard to construct this map, so essentially, this is just a very straightforward argument. But in any case, so Levas takes this as granted. But what we know is that opn minus 1 is a sub-bundle of the trivial bundle, cn plus 1, so that we have here a natural connection on this line bundle. And we can easily compute the coverage of this connection. And we can also integrate over p1 of this fA. Well, if you normalize this appropriately, so i over 2 pi, I think this is minus 1. So here, p1 is just a standardly embedded complex line in pn. In other words, what this tells you is that the first-gen class of the tautological line bundle of the protective space, so the term class of this lives in the integral cohomology group rather than the real cohomology group. And by the properties of the term class, we know that c1 of l is just a pullback of the first-gen class of o minus 1 of pn. And this is then in h2j m with the z coefficients. Well, to be more precise, in the image of the integral cohomology in the real cohomology groups. So for those of you who know definitions of the tautological definition of the cn classes, we could have taken this as a definition of cn classes so that if you have a map f, so that the pullback is our original line bundle l, then its first-gen class is a pullback of the generator of the second cohomology group on the protective space. And this would be probably even more desirable, a little bit. So this definition contains a little bit more information there because we have also torsion classes, but this actually won't bother us too much. OK, are there any questions to that? So isn't this approach obvious that you can only do this for complex? No, you don't need complex bundles here, so it's just easier to describe. You could do this also for real bundles, but then this is all very much related. And actually, you will see this in the exercises. So no, I mean, it's not really important, but it's just easier to describe. OK, so the next thing that I wanted to discuss is the transimons functional. Now, we specialize here even a little bit more, so what we have is a principle SU2 bundle. If you wish, this is just the same as saying that we have a complex rank 2 vector bundle, such that the second exterior power of E is a trivial line bundle. And if we take xi in SU2, then the first polynomial that we have defined, so C1 of xi, was i over 2 pi trace of xi. But the trace of xi is 0, just by definition. So the first trend class is trivial. And the second trend class, or the second polynomial, is i over 2 pi squared, the determinant of xi. And now you can compute easily that this is for an SU2 matrix, is just minus 1 half trace of xi squared. So all in all, what we have is 1 over 8 pi squared trace of xi squared. And so if m is an oriented closed for manifold, then I can identify the force cohomology group of m, say with real coefficients or with integers, with reals themselves. And the second trend class of phi is n, just the integral over m of the trace fA wedge fA, up to the coefficient 1 over 8 pi squared. OK. So now we will use this formula. So now I will assume that m is a 3-manifold, which is maybe let us choose a different notation. So let y be a 3-manifold, again closed oriented. And I will use the following fact. So the fact is that there is a 4-manifold x4 such that y is a boundary of this 4-manifold x. So in other words, what I'm saying is that each 3-manifold is a boundary of some 4-manifold. This is something specific to dimension 3, so this is not true in any dimension. But anyways, that's what we have. And this is fact 1. Second fact that I will use is that any SU2 bundle over y extends to x. If I have an SU2 bundle over the boundary, I can extend this to a bundle over the whole manifold x. And if I choose a connection a on p, then I will have a connection a. So I can choose an extension of this connection to a connection on the principal SU2 bundle over x. OK, very good. So what we can now consider is the tensimons functional of a. And I will define this just with this formula. So this is 1 over 8 pi squared the integral over x phase Fax, which Fax. Now, of course, here is a question. Why is this is well-defined? So in fact, it does depend in general on the extension. However, if I think of this as a function which takes values in real values mod z, this is now well-defined. And it's easy to understand why this is so. If I choose any other extension, so now I have two manifolds with the same boundary, I can stick them together to form a closed four manifold. And I know that this integral here will take values in integers. So the difference on the two bits is just an integer. And that's why if I mod out by integers, I will have a well-defined number here. All in all, I mean, this shows a very close relation between the transimons functional and the Chan-Wheel theory. We turns out that this functional can be computed explicitly. And I will give you a formula in a minute. So the outcome is that the transimons functional of A is just 1 over 8 pi squared. And the integral over y A veg dA plus 2 thirds A veg A veg A. So what I'm using here is that any SU2 bundle over three manifold, in fact, is trivial. So I can choose a trivialization. I can think of the connection just as a one form over y of those values in SU2. Now all these expressions make sense. Yes, of course. Cheers. If this integral is not long. Normally the term class is integral, right? So why is it not just a list here? I mean, right? This is an integral, but if you are on a closed manifold. And here we have a manifold with boundary. Right. It's not anymore. OK. So now we have an explicit form of this function so we can compute its derivative. So if I just compute at the point A on, say, A dot, it stands out to be 1 over the 4 pi squared integral over y of fA veg A dot. So from here, we can see immediately that this function has critical points. And these points are where the character actually vanishes. So what we have proved is the following proposition. So the critical points of the trans-simon's functional are those connections A, where the character is 0. And such connections are called flat connections. But that's just a variable. So this is a tangent factor to the space of all connections at the point A. So what we know is that we can apply gauge transformations to connections. And the character is essentially unchanged when we apply this. Well, up to what to do. But anyway, we have an action of the gauge group on this space, and we can take its quotient. So let me define m flat to be the quotient of all those A's. Such that fA is 0 by the gauge group of our bundle P. And this is called the moduli space of flat connections. So this is our very fast example of a gauge theoretic moduli space. And essentially, it's the only example where we have a very concrete description. Now it really turns out that this space is just the same as a space of homomorphisms from the fundamental group of y into SU2. So SU2 is here really not essential. So this works for any group divided by the conjugations. So it means that if I have two homomorphisms, I can conjugate by a fixed element of SU2. And such homomorphism is said to be equivalent. OK, in any case, what we can easily see from here is that m flat is compact. And this is really easy to see because the fundamental group of y is finally generated. So we have an embedding of this space into SU2 to some power, say n, modulo SU2. But this is a compact space. So the quotient is also compact. Therefore, this is a closed subspace in here. And therefore, also compact. Now what we also can prove is that what we can see is that m flat is not necessarily a manifold. m over the dimension of m flat is finite, but it can be arbitrarily large. Now why is it not really a pleasant news? Because you see, what we are studying, we are studying critical points of a function. And generically, we would expect to see a finite set of points, but this turns out not to be the case, at least in the full generality. In any case, we can show that, or it is possible to show, that we can count in a certain sense the number of points in this space. And so the number of points in m flat, at least in the case when this is really a finite set of points, is an invariant of our manifold y. And this stands out to be the Carson invariant, up to some normalization. In any case, I don't want to go into details here, but there will be an extra talk on Wednesday, I believe, where you will learn how to define the Carson invariant and how to bypass those difficulties, that m is not really neither a manifold, nor actually a finite set of points. All right, if there, yeah, I'll say it again. Yes, so what I mean here, I haven't actually defined the topology in this space, but I could have defined this as a smooth topology. But because of this identification, we know that this is a topological space that has topology induced from this space. Yeah, yeah. Okay, so one more topic that I wanted to discuss today are the Dirac operators. And this is the following thing. So we start first with a vector space. So v is a vector space, say the dimension of v is finite. And for this, we can construct an algebra in a canonical way, at least if v is in, say, Euclidean. And this is done as follows. So what we do, we take the tensor algebra of v, that is r plus v plus v tensor p and so on. And we divide this by an ideal generated by the relations v tensor v is minus norm of v squared. What you can easily see is that this is a finite dimensional algebra.