 OK, well then, good morning. So my plan for this first lecture is to introduce various equations which would have come to us via the way of physics. And it turns out that they're useful in mathematics. For some of them, we know this. For some of them, we expect this. And towards the end of the lecture, I want to explain how they all sort of fit in one framework, and in some sense, are all the same equation. And I want to start with something that hopefully you all know. Let's just do some review of anti-serve duality and Youngville's theory. So the setup for this is some sort of minimal setup that you always have to do for gauge theory. You take g to be a compact connected league group. And for convenience, I will take it to be semi-simple. So this thing is called the structure group of the theory. And I want semi-simple so that the Lie algebra has a canonical inner product that comes from the killing form, or the negative of the killing form. And so this is the group in this theory. Of course, this all happens on some space. The space I call x comma g. x is some manifold smooth. Of course, g is a Riemannian metric on this. I want this to be oriented, because you want to integrate. And then you need a g-principle bundle over this space to do some gauge theory on it. So this is some other manifold p, has some map down to x. It's the right action of g. Let's be pedantic and give this a name. And this quotient here, this group acts freely. And this is the quotient of this action. So this is a g-principle bundle. And the gauge theory sort of happens on this thing. And then just want to fix some notation in case you use different notation. I denote by script a of p. This is the set of connections on p, actions on g-principle connection. And this is a quite boring space. This is an affine space modeled on one forms on x of values in the adjoint bundle of p. And here adjoint bundle of p. This is the bundle that comes from the representation of g on its Lie algebra, the adjoint representation. It's mafthrag g is supposed to denote the Lie algebra of g. So this is the setup. And all of this is acted upon by the gauge group, g of p, gauge group, or the group of gauge transformations. This acts on everything that you can see over there. Well, it doesn't act on everything on this blackboard. The very left doesn't act. So this is the basic setup. And there's a map that associates to a connection that's curvature. So we give that a name as well. So if we have a connection, we can associate to it its curvature. And we denote the curvature by capital F of a. This is a two form also on x. And it takes values, the adjoint bundle. You should think of this as taking the derivative of a in some sense. Good. OK, so all of these are things that were naturally used in differential geometry. But we don't want to use them. We want to sort of study this space, but equivalently with respect to the gauge group. We need to have some more structure than just this. And it turns out that there's a natural functional on the space, which is just the L2 norm of this. And it's called the Young-Mills function. This is the, let me write this maybe more explicitly. Can I repeat the definition of at p, please? So when you have these principal bundles, they have a right action of g. So now if you have anything else that g acts on, you can make a bundle out of p and that space by making p times the space that you want. In our case, the space is the Lie algebra. And we can act on p with g in whatever way this is given. And we can act on g in the way given by this representation. And this quotient is the adjoint bundle. This is a vector bundle, because this is a linear. This is a representation, exactly. This is a vector bundle. Oh, this is omega 1. This is the one form that takes values in this vector bundle. The vector, these one forms are sections of the quotient bundle. We take sections of the quotient bundle, tensor at p. This is what this means. OK, good. So back to the Young-Mills functional. This is a natural energy functional on the space of connections. This norm here uses the norm that comes from the killing form, and I think the other stuff is reasonably sensible. OK, this is a natural energy functional. In the generality, I have defined that it could be infinite, because the space is not assumed to be compact. The second, I want a non-compact space. OK, so this is the Young-Mills functional. And you can formally, at least, compute the equations satisfied by a critical point, the so-called Euler-Lagrange equation. And this is the Young-Mills equation. If you compute this, you see it's the d star of fA, simple looking equation, but maybe not completely harmless. OK, and one can say some stuff that I will maybe not say in detail, but sort of Maxwell's equations can be formulated in this. This is an important equation in physics, but we don't really care about what the physics says. OK, so if you're an analyst, you would try to solve this equation, right? And then this is not easy to solve this. Maybe try direct minimization. This doesn't really work if the dimension is reasonably high. But if you're a geometer, maybe you observe that there's a special solution to this equation. Or maybe even if you're a topologist, you will observe this. So an easy way to solve this, let's give this a name. A is flat. A flat means that the curvature of this connection vanishes. I mean, then certainly it satisfies this equation, because if I apply something to 0, but I stay 0, this implies the Young-Mills equation. And this actually gives us an interesting class of solutions, and they're completely characterized by the fundamental group of the space. So if you want, maybe there are some exercises in what I've discussed so far already for you, if any of that doesn't really make sense to you. But here's maybe the first exercise that you should think about. You can look at all principal bundles of structure group G up to isomorphism. And then you can look at all flat connections on these bundles, modulo gauge. And I claim that this space has a natural bijection to space that you all know, hopefully. This is a bijection. And the bijection is with representations of the fundamental group in the group G, modulo conjugation. So this G acts on G by conjugation. So this is the first exercise. And the map just comes from doing parallel transport. This is the map comes from monodromy. It's a good exercise in covering theory, if you understand why this is true. So this is, in some sense, an extremely satisfactory answer if you're an analyst, because it becomes a topologist's problem. And you're done. You go to maybe one of you in the audience, and then they have to solve it. But it turns out that this is maybe not that easy to compute. And you can actually use this both ways. So let me not say more about this. But this is useful in both ways. This is not just the way to make your problem someone else's problem. But this is also not very interesting, because this really just depends on the topology. So yeah, there's a question. Yeah, DA star. OK. So this connection on the principle bundle, so the question was, what is DA, or DA star? The connection on the principle bundle associates covariant derivatives or connections on any bundle that you associate with it. So in particular, you have this adjoint bundle here. And this inherits a covariant derivative. And then in the usual way that you extend the derivative on functions to a derivative on differential forms, it extends to differential forms valued in your vector bundle. So this gives you a map from differential forms on x with values in adp, degree 1. And all of these spaces have inner products, so you can take the formal adjoint. And this is the formal adjoint. Hope that most of the gauge group related to G. So the gauge group is the group. This was the question in the chat, I think, maybe. How's the gauge group related to G? The gauge group is the group of automorphisms of this bundle, so these G-equivariant maps from P to P. And this is how this affects the gauge group. One can say something more, but maybe I will not say more about this. OK. Good. So we have these solutions that come from flat connections. Here's another interesting solution. So I don't know who first discovered this. Maybe this was an obvious observation immediately when people developed the theory of maybe Erismann already noticed this. But the next observation I think I can pin down to precisely who figured this out. So far there's no physics except for maybe the young who is functional, but very natural thing to discover. In 1975, four physicists, Berja Wien, Poljakow, Schwarz and Chupkin, they made the following observation. So I'll abbreviate their names as BPST because their names are quite long, and it's four of them. They observed the following. So there's something special that happens in dimension four. It doesn't happen in other dimensions quite in the same way. So the dimension of x is four. Then the Hodge star operator maps two forms to themselves, and there are two eigenvalues. So you can compute that the square is one. So the eigenvalues must be plus or minus one. And they observed the following. If your connection is anti-safe dual, so anti-safe dual means that when you write this over there, it means if you take the curvature and you apply the Hodge star to it, you get minus the Hodge star. You can also write this as saying that the safe dual part, one-half of FA plus, if this vanishes, then this connection must automatically satisfy the Young-Mills equation. Because the Hodge star in this dimension is star D star. The curvature connection is self-dual, anti-safe dual, so you just get this minus sign here. But this expression, the exterior derivative of the curvature, this always vanishes. This is the Bianchi entity, the basic structure property of a curvature, that satisfies this equation. So somehow, we have been able to solve this equation which differentiates F by just imposing an algebraic condition on F. This should be much easier to solve. So this means that in particular these solutions, these anti-safe dual connections, they satisfy the Young-Mills equation, but in fact, something much better is true than this. You can actually write down precisely what the Young-Mills energy is in terms of a topological invariant if you have an anti-safe dual connection. So in general, you have this energy identity, the Young-Mills energy of A is given by 1 fourth integral, while it's basically the norm squared of this thing, which let me write it in this form. It's this plus something else, and the something else is topological. I'm going to write this in some really obnoxious way, because the way that I'm writing it, there's going to be an expression that has a meaning later. So this is written as 8 pi squared. And then some number H check of G. You're not supposed to know what that means. It's some number that depends on G. Actually, we will sort of take this number out in a second times k of A. I mean, this is certainly true until I tell you what k of A is. This is the definition of k of A. This k of A is an important thing. It's called the instanton number of the connection. And it turns out on a closed manifold to be a topological thing. This holds with k of A. So I'm now taking this number back out again, because this has to be normalized for specific reasons that minus 1 over 2, then I'm taking this number back out. In general, it's the integral of fA which fA over x. This is a 4 form on x that takes values in a tensor product of the Lie algebra, a joint bundle with itself. But we can compose this with this contraction map that comes from the killing form. And this is an actual 4 form that we can integrate. And if you remember some Chen-Wei theory from a differential geometry class, then you identify this as, I'm sorry, I'm even taking the 8 pi squared out again. If you remember some Chen-Wei theory from your differential geometry class, then you will recognize this as 1 over 2 times this number. The Pontriagin class of the adjoint bundle paired with x, the fundamental class of x. So this is if x is closed. And in that case, this number really only depends on p. If it quite generally only depends on p, even if x is non-compact, but one has to say more technically precise things about this. Anyway, this thing has a special significance. This is called the instanton number. This is another exercise to prove this formula. You know that the formula doesn't actually contain this number, so I don't need to tell you to prove what this number is to do the exercise. I will tell you, however, that this is some number that comes from representation theory. If you do this for SUn, let's do it for SU2 instead of the most important case. For SU2, this number is just 2. For other ones, you can look at the paper of Atiyah Hitchin and Singer from 1978. And they will have a table with these numbers. Any questions? Am I stopping now? Yes, yes, yes, only depends on the principle. Yeah? Very good. OK, so I repeat that, because I would have said exactly the same thing next. So from this formula, we see that we can write, this was your question, so all of the credit goes to you. The Young Mills Functional, you can decompose into two parts. One of them is just the square of the positive part of the curvature, and then a topological number. That depends on the principle bundle and the structure group. In particular, this tells you that the Young Mills Functional is always bounded below by whatever this number is, because this is whatever it is. The connection has no influence on this. If the x is close, and this is non-negative. So you can make this as small as possible by making a anti-self dual. So the lesson you learn from this is that they don't just satisfy, ace the incidence, don't just satisfy the Young Mills equation, they're absolute minima for the equation. You haven't just found the critical point of this function, it's really the bottom. You can't go any deeper. So a fancy way to say is that they saturate the topological lower bound. This is what it is. OK. Very good. So they didn't just observe that this is true. They also found an explicit solution of this equation. Yes, that's right. So your question or your statement was that the number k a, in general, it depends on a. But under reasonable assumptions, like x being closed, it only depends on the bundle. Yeah, henceforth, I will sometimes write this if the space is closed. And one time more, I will write this, because we will be on a non-compact space. OK. Good. I mean, this is already good observation. But in hindsight, it's probably obvious. If you, even obvious observations have to be made somehow by someone at some point. OK, so but they also found the solution. This thing is called the BPST Instanton. And this thing is called Instanton number, because it tells you how many instantons can be in your bundle. What would be obvious? I mean, so if you think about connections and principle bundles and so on, and you know that you have this equation, and you know. So this is an equation that you actually have to solve. But there's another equation which looks very similar, which is always true. This is this Bianchi identity. So if you meditate over this for long enough, I guess you should be able to discover that if you can somehow relate star of FA to FA itself, that would be a good idea. And it would help you to solve this equation. But someone had to observe this first. And I guess it was these guys. I mean, in hindsight, everything becomes obvious. So if someone tells you something is true, once you digest it, it's hard to understand why it once was complicated. So they also found a solution. And the solution they found on R4. And to write the solution, it's very convenient to identify R4 with the quaternions. The quaternions, hopefully you all know what they are. But in case you don't, I would just briefly say that it's a normed division algebra. Similar to the complex numbers, but doesn't just have i. It also has j and k. And they satisfy the relations that i squared is equal to j squared is equal to k squared is equal to minus 1. And this is also the same as i times j times k. So these are the quaternions. And the quaternions naturally relate to a special league group, namely the league group SP1, the same as SU2. This is just the unit elements of the quaternions. The quaternions have a norm. This is the basis. And you take this basis to be orthonormal. And then the condition is just p is equal to 1. This is SUP. This is a group. And there's a very easy formula for them. So let me say, this is the league group. The league algebra, therefore, little SP1 is just the imaginary part of the quaternions. And the solution is given by following formula. It's the imaginary part of q. Sorry, I should say something before. So q is sort of the identity function on the quaternions. Every quaternion, if I'm somewhere on the quaternions, I can say the point where I'm at is a point in here. And it's just the identity. So you can take the function q. You can multiply it by dq conjugate. It's just q conjugate by flipping the signs on i, j, and k. This is some one form that takes values in the quaternions. It should take values in this. So we take the imaginary part. And then this is not an instanton, but it's an instanton if you divide it by q squared plus 1. This is a one form on h with values on SP1. Because this is not quite what I told you connection is. But this is defined on h. There's only the trivial bundle. The trivial bundle has a canonical connection. It's just product connection. And if you add this one form to the product connection, you get the BPST instanton. And on trivial bundles, one usually identifies the connections with just the one forms for this reason. So it's a good exercise to check that this thing actually is anti-self-duel. And q, it just maps, it's just a coordinate function. It's the identity. It's a name for the identity function. Does that make sense? It's like when you write the coordinate on the complex plane, you just write z. It's just on quaternions, people write q. So it's also a good exercise, actually, to try to find if you know that the connection must be of this form times a function to find this function. It's a good exercise to do that. Yeah? Yeah? It does, because, I mean, OK, yeah, it doesn't, yeah. Oh, OK, OK, OK, OK, OK, OK. But Paul asked, the Yang-Mills function looks like it doesn't depend on the orientation. But this seems to say that something depends on the orientation. But I mean, when you define, so this is actually differential geometry question, when you define the integral of a function, you have to say how to define the integral, and you have to say what the volume form is. And if you flip the sign on both, it flips the sign on both. So it does, let me be, I'm not quite sure. Does this answer your question? OK, good. So there's a similar identity with, you can also put a minus here. There's a similar identity that goes on. You flip the sign here and here. Good, so it's a good exercise to check that this is actually. So there are two exercises. The first exercise is to understand, if you actually mean understand what it means to compute the Yang-Mills function with respect to the killing norm of the inner product that comes from the killing form, is to compute that the instanton number of this thing is 1. This is not, I mean, this is an integral from 0 to infinity. You can do that. And the second exercise is to check, so this is 1, 2, to check that f plus of i actually is 0. And for this, the hint, compute dq, which dq bar, vary explicitly and see the expressions that show up there. And from that, I think you can do it. OK, good. OK, so this is some physics. This, they do this in a two and a half page paper. And I guess if a normal person would have just read that, they would have said, OK, it's an interesting thing. But at here, I really realized that this is a deep observation about the connection between geometry and physics. And he probably instructed all of his graduate students to work on problems related to this, connect this to algebraic geometry, and so on. They wrote this famous paper with Hitchin and Singer that computes the index formula of the modular space of this. They computed with Hitchin and separately Manning and Dreenfeld computed all solutions of this equation on R4 when the structure group is SUN. This is the ADHM correspondence. And then, of course, in 82, Simon Donaldson realized that all of the modular spaces of solutions of this equation and their compactification that was explained how to do by Uhlenbeck before, you get invariance of four manifolds and so on. So I hope you all understand that this is a significant equation, profound impact in mathematics that probably cannot be. I mean, the impact of this simple observation that it had on all of mathematics I think is very hard to underestimate. Overestimate, sorry. So then probably all of you who have ever actually tried to follow the proof of anything in Donaldson theory, you've seen that this is really complicated. Start by the analysis is quite complicated. And then, is there any question? If there are no questions, then we will come to the second equation that comes to us by the way of physics. So a few years after the introduction of the Donaldson and variance, the polynomial invariance, Witten realized he wanted to understand what this means physically. And he realized that from a physicist's perspective, you can somehow understand this as a topological from the certain physical limit of the n equal to supersymmetric Young-Mills theory, whatever that means. This is some physics theory. And then together with Cybeck, they studied this theory in a different limit. And they observed that you can describe the same physical theory by a different gauge theory. In particular, so this is I think a development from 1994. 1994 is that Cybeck Witten sort of observed by some physics magic that the SU2 Donaldson theory. By this, I mean the polynomial invariance that we extract out of the modular spaces. If you don't know what that means, it's not going to be important. But these are some numbers, basically, that come out of the modular space. So this theory, you should be able to compute them in terms of a U1 gauge theory. So this is a physics conjecture. Butable by U1 gauge theory. And this U1 gauge theory, I'll explain what this is in a second. This is the Cybeck Witten theory. This is the theory that gives us the Cybeck Witten equation. And you would expect that to be much simpler. This is an Abelian group. If someone says U1 gauge theory, expect it can't be much more complicated than Hodge theory. Maybe it is, in fact, much more complicated than Hodge theory, as you will see in a second. And it's much richer as well. I've heard that mathematicians have written down the same equations, but didn't realize the significance of that without this. But they've never actually seen someone's old notes where this was written down. Anyway, so this is the so-called Cybeck Witten theory. So in mathematics, this is called Cybeck Witten theory. In physics, they have a Cybeck Witten theory, which doesn't, where these equations don't show up. So what is this? OK. For the setup, this is a little different from the setup that we studied for the ASD instantons. I mean, it's somehow slightly more complicated. But at the end, it's simpler. So here's how the setup works. So if someone gives you Riemannian manifold, you can always construct the frame bundle. I'm sorry. Bear with me, OK? This is going to explain this to you in a way that maybe it's not the way that you usually see, because we will have to do something like this later on. Something called the frame bundle. So I denote this of the tangent bundle with the metric. I denote this by FR. So what this is, is simply at every point over x, we choose concrete, orthonormal basis, positive orthonormal basis, in fact, of the tangent space. If you have a different orthonormal basis, they're related by a transformation. In this case of SO4, so this is an SO4 principle bundle, simple example. I guess if you learn differential geometry from Czern's book, he tells you to work on this space throughout, never work on the manifold anyway. OK, so this is a natural thing that always comes with any Riemannian manifold. And then when you want to put structure on a manifold, it means changing this bundle to something different. So if you want to put a complex structure on it, it means that you want to change this group from SO4 to U2 and something. So you would like to modify this group to a different group. So if you have studied some spin geometry, then you know that the spin structure is basically changing this group from SO4 to spin 4. What you need to do for the cyber-written theory is to change this to spin C. What we need to do is we pick a spin C structure. So I will now finally tell you what a spin C structure is. I will also later tell you what the Dirac operator is. Maybe today, OK, pick a spin C structure. So what does this mean? I mean, this is something that's not much more complicated than this. This is spin C4. I will tell you what spin C4 is in a second if you don't know what this is. You pick a spin C4 principle bundle. This is supposed to be a Muffrack W. I'm sorry. Can't do much better than this over X. So this is not a spin C structure. This is just a spin C4 principle bundle together with. And it's important to distinguish this also for spin structures. Sometimes people get confused about this and then together the map from W to the frame bundle. And we demand that this W is equivariant under the action of spin C variant. We're not going to tell you what spin C is. The spin C4 has a very explicit description. So spin C in general is spin N. In our case, it's going to be spin 4 times U1. So spin N is the double cover of SON. N is sufficiently large, so 4 is fine. And it contains an element minus 1. So if you quotient by the element, you map to SO4. And of course, U1 also has an element called minus 1. So you just quotient by that. So and then this acts on SO4 because I can just forget this factor and map down to SO4. Call this. I don't think we need the name for this. So in general, whenever you have a homomorphism from one Lie group to another, there's a notion of a reduction of structure group of a set principle bundle along this homomorphism. Spin C structures are one example of this. Any geometric structure is basically an example of this that you usually use. OK, so you fix this data. And then you get a lot of structure out of this. Out of this comes the following. So out of this come two bundles, W plus and minus. They're the complex. So these are the, I denote this by W plus. I think it's what Francesco called S plus and S minus. These are the complex spinner bundles, they're positive or negative. There's a Clifford multiplication with some map from TX. You can use this either to make the elements of this are called spinners or sections of this generally are called spinners. You can make a positive spinner into a negative spinner by multiplying with a tangent vector. And you can do the same way the other way around. Thing is called the Clifford multiplication. This is sort of the globalized version of these matrices that Francesco introduced in his very first talk that he was writing over there on the little board. What else? The line this. So let me maybe say right away that sort of important thing that you have to do to write down the usual type of equation is you can see that you can extend this to the exterior algebra. So if you have two vectors, you just multiply them in order. So we only need the extension to wedge two. And then you observe that since they flip the chirality or whatever design you want to call, when you do this with a two form, it doesn't anymore. So you can, for example, make a positive spinner, take a positive spinner, make a new one out of this by multiplying it with a two form. And then I have not defined what these are, but I will tell you what they are maybe later or tomorrow. You can check that when you do this with a self dual two form, on these bundles you actually get zero. So this vanishes when you do it on wedge minus. Let's also make this on covectors. This is these bundles. They have metrics, so they're isomorphic to each other. If you do this on negative two forms, you get zero. And in fact, so this is an injection. You can prove this as well. And it identifies the subspace of self dual two forms with a special subspace in here. So you can check, in fact, you can check that induced map on self dual two forms. Doesn't just go to arbitrary endomorphisms, but it goes to skewer joint endomorphisms, which are trace-free. So I denote this by little su of w plus. So this means skewer joint, trace-free. This is an isomorphism. But this is going to be important because we will be able to write down an element of this and we want an element of this. So any questions? Yeah? Oh, yeah. Thank you. The question was how I go from gamma defined on tx to gamma defined on wedge two of tx star. OK, yeah. So when I have an element of this, I can decompose this into wedge w. I can make sure that they are orthogonal to each other. And it doesn't matter if I think of them as vectors or covectors because they have a metric that identifies them. So let's not bother with this. So then I make this thing act on the spinner. So the spinners are usually called phi or something fancy to make everyone scared. And then this is just defined as gamma of v first and then gamma of w second acting on phi. If you assume that they're already perpendicular to each other, if you don't do this, then you anti-symmetrize this. Oh, this is an element of this w plus. This phi is an element of w plus. And I do this. So I assume already that v and w are perpendicular. Otherwise, this is wrong. This is why Paul got scared. OK, good. OK, so we're now at page three of 15 pages I wrote for the first lecture. So OK, well, let's try to press on anyway. OK, so more structure comes out of this than this. OK, I told you what the spinners are. OK, so now we need to differentiate these guys. There was another question. Sorry, can you repeat the question? I don't think I understood. Ah, OK, OK, OK. So this is a good question. The question is, what is the structure of these bundles? OK, good. So these are complex vector bundles. They have a Hermitian metric, and they have rank two. Hermitian of rank two. That's right. That's right. OK, so the question is about the plus minus. There's some larger bundle called the full spinner bundle, and that has a natural operation on it with eigenvalues plus and minus one. And this is the plus eigenspace. I mean, one of the pluss, the other one is the minus eigenspace. But tomorrow, I will tell you more concretely what they are in terms of really elementary stuff, right? Just quaternions, multiplying, quaternions, and so on. You can also axiomatically just define a spin structure as something like this with some properties, but that doesn't generalize very well. Any more questions? The volume form gives the opposite splitting. So this confused me until the spring. Well, it didn't confuse me until the spring. I just didn't bother with it until the spring, when I had to explain why there's a difference. I will explain this to you tomorrow why there's a difference. Volume form gives you the opposite splitting. It's absolutely bizarre. Anyway, so it's better not to worry about these things and just, there are also choices that you think don't actually exist that you have to make. But if you are interested in this, I can tell you hours about this later, maybe not today in the lecture. Okay, so we have this. Okay, so now we also want to differentiate things because ultimately we want to define some Dirac operator. So how do you differentiate things? You need a connection, and you don't need an arbitrary connection on this. We want the one that induces Levi-Civita on this. They have a name that calls spin connections. So spin connections, W, so just connections. Let's give them a name A on W, which induce Levi-Civita on the frame bundle. Or if you want to think of this in terms of the induced connection on the spinner bundle, it just means that this operation here is parallel. It's the same if you have seen that before. Okay, so the space of these guys, I want to denote as connections on W that induce Levi-Civita on the frame bundle, so I put comma Levi-Civita. This is absolutely not the usual notation, but later we will need some notation of this nature. Okay, so now I can write the equation, so finally. So what's the Cyberg-Witton equation? This is an equation for connection, a spin connection, and one of those positive spinners, and it has two components. The first component says that you want the Dirac operator defined by A and gamma applied to phi to be zero. So now we'll tell you what this means. So you pick some point, choose your favorite orthonormal frame. Some that I call the frame E1 up to E4 and differentiate using the connection A and the direction of EA. Spinner, this gives you a new spinner, at least defined locally where this frame is defined, and then you just multiply one of those vectors. And then this comes out to be a negative chirality spinner field, and I want this to be zero. This is the definition of the Dirac operator. Should remind you of the formulas that Francesca wrote. You differentiate, you multiply it by these matrices. It's just this. So the second equation is a little more complicated. It's of the form, so. It's the save to a curvature of something, but it's not, so this connection here, the structure, if you take the curvature of this, the curvature takes values in some huge bundle whose fibers look like the Lie algebra of spin C4, but we don't want all of that, right? At the beginning I said it should be an abelian gauge theory. So a convenient way to say this is that you look at the connection of the determinant bundle, so this is the connection on the exterior, the second exterior power of W plus, for example. This is the same, by the way. I mean, now you cannot solve this exercise, but if you know what this is, you can solve this exercise. See that these actually are the same. So you want this to be equal to something that comes out of the spinner. Okay, so one thing you can do. So okay, here we see I can make a self dual two form if I have a skewer joint thing, right? Okay, I cannot really make a skewer joint thing, so well, I can, I can, so. But let's make first a self-adjoint thing. So you can take the spinner and you can form this quadratic form, then sort of you take the spinner, sort of takes inner product with the spinner, but with the spinner again. This is a self-adjoint endomorphism of W plus. It's not trace-free, you can make it trace-free by subtracting the trace identity. So now it's trace-free, it's not skewer joint, but I can make it skewer joint by multiplying it with I, but this actually takes values not just in wedge plus, it takes wedge plus times I because this is a connection on the unitary bundle. So I can just take gamma inverse of this directly and it's going to give me an element of this. And then for reasons, I want to put a one-half in front of this, okay. So this is the Cyberg-Witton equation and this looks complicated at first sight, but it turns out to be much easier to work with. Much easier to work with. And so now that I've defined this equation and maybe let me say that immediately after this was introduced in Witton's 1984, 1994 paper, people reproved basically everything they used to be able to prove. Donaldson's theory, the Cyberg-Witton theory, many new things were approved with this and it's technically just a much easier theory to work with. And there are two big open conjectures left from these early days that one of them is a simple type conjecture. We're not saying anything about that. And the other one is the conjecture by Witton that, so out of this you can construct invariance and you can put them in some complicated generating series and you can do the same thing with Donaldson variance and there's coordinate transformation as opposed to do, that makes them the same. This is Witton's conjecture and there have been approaches to, towards this and a lot of work, source of initial suggestion by Pitzley-Gadzian Turin and work by Fian and Linness but as far as I know it's not completely settled yet even though people have worked on this for 28 years. Anyway, so this is the Cyberg-Witton equation, okay. So far this is all supposed to be classical stuff and I think anyone who, yeah, there's a question. The definition of a spin connection, okay. So can I repeat the definition of a spin connection? I can repeat the definition of a spin connection. It's a connection on this principle bundle W with the property that the induced connection on the frame bundle agrees with Levi-Civita. There's another question? Yeah, it's, that's right. So what you, the question or statement was if I just think about this as pushing down the horizontal distributions, yes, that's one way of doing it. But there are like a million ways to think about this. This is one of them that I think geometrically is clear. Okay, so, so it turns out that there are many more equations that physicists have proposed all of which have some interesting, fascinating connections with gauge theory. I mean, they come from quantum gauge theories that are not as well explored as ASD and Cyberg-Witton and so one of them is the Waffa-Witton equation. That was actually discovered, I think, just a few months before the Cyberg-Witton equation. I think the paper just predates the Cyberg-Witton paper slightly, but who knows, I think Witton was just generally working on sort of ideas around this. And there's a second equation called the Kapustin-Witton equation which sort of has fascinating connections which we will hear more about on Friday to the Jones polynomial. It was originally discovered because Witton wanted to understand the Geometric Langlands program, sort of very, very complicated conjecture in algebraic geometry. Comes out of a more complicated conjecture in number theory and anyway. So I think I cannot reasonably say anything about the Kapustin-Witton in the remaining 22 minutes, but let me say something about Waffa-Witton. Okay, so in physics there's some conjecture that goes back to the work of Monton and Olive. It's called electromagnetic duality. It's some conjecture. So I think for Maxwell's equations, we have all seen that you can exchange the electric in the magnetic field if you flip the sign on one of them, the equation and the vacuum are still satisfied, some sort of simple symmetry. But they observe that this should be true in sort of a much more general framework. And this is really, even in physics, this is really a conjecture, right? This is not something that they know how to argue for rigorously, physically as far as I understand. And let me try to say what it says. So this is something that's supposed to happen for a large class of gauge theories. And apparently the best gauge theory, the best chance of this being true comes from N equal four, whatever this means, super symmetric youngest theory, right? This is supposed to have, this is a physical theory. And in this context, it's called S duality. So let me explain what this means. So this physical theory here, this depends on two parameters. The first parameter is what there's a general parameter, which is the structure group, right? This tells you what kind of a gauge theory this is. And then there are two constants. One of them is the coupling constant, which is called E, coupling constant. And then there's another parameter, which is called theta. This is supposed to be the angle. And it turns out that this coupling constant, this is always positive. They actually like to normalize this in some sort of bizarre way where the way that the coupling constant actually appears is four pi squared over E, sorry, four pi over E squared. Don't ask me why they do this, but that's what they do. So, and then the conjecture says, so S duality conjecture says that the theory is invariant under doing the following. So you don't do anything to the angle, but you change the coupling constant. You invert that thing. You invert this to one over four pi, okay? So this is easy. We can all invert this. There's a complicated thing that you also have to do. You have to replace this group by its dual group. This goes to the Langlands dual group. Let me not explain what this means in any detail. But this is an operation that you can do to a compact leak group. If you do it twice, you get the original group back. If you do it to SU2, you get SO3. In general, to understand this, you have to understand how to associate to a group. It's root datum. If you know what that means, you will have observed that there are two things that you can flip around. If you flip them around, you get a new group. This is the Langlands dual group. Yes? So the question is what this means. This is physics talk. I'm not sure what this means precisely. I have not spent an extraordinary amount to figure out what this means. No, no, here you don't square. As far, oh, sorry, no, no, no, sorry, yeah. Okay, no, no, you also square this. Okay, thank you. I don't know. I mean, we will combine this. Yeah, I'm sorry, sorry. Definitely the square has to be there. Otherwise, it's not true. Otherwise, it doesn't give the right thing. Okay, so, okay. So this is a very old conjecture in physics. It's a very important conjecture, sort of, I think this was one of the first sort of deep dualities they conjectured in physics, sort of remarkable. Actually, also cyber-written, sort of one part of cyber-written argument, also seems to use this electric magnetic duality. Okay, so in 1994, also, Waffa and Witten, they tried to test this conjecture. Okay, so this is extremely difficult to test because even physicists don't know how to compute what this means. So then physicists have this thing that they can do called a topological twist, and then you can do some magic, and then if you're lucky at the end of this, you get some PE, and then you can imitate what people usually do, right, when you do Donaldson theory or cyber-written theory, and the equation they arrive at is the following. I promise it's much more simpler than this, right? So I can write it down probably in two minutes. So this is the Waffa-Witten equation now. So this is, again, the setup is like for ASD. You have a principle bundle P, you take a connection on this. They try to engineer a particular structure of this equation, which you will see. It's like doubling up the ASD equation, or it's a little bit like taking the cotangent bundle of the ASD equation. So there's a second field called B. This is a two-form, sorry, this is a safe dual two-form on X with values in the I've got a bundle. And then there's another field, which is the most often is zero, but let me write it anyway. This is just a section of the adjoint bundle over X. So then you write down something that looks a little bit like this, okay? So if you wanted to guess this equation, it would not be too difficult to guess this. So a kind of Dirac equation that you can write in this setting, where we know how to differentiate this, gets us a one-form. We know how to differentiate this. We can get a three-form or one-form if we take the dual, so let's do that. Take dA star B plus dAC. If you just had to guess some reasonable equation, I would write this down. And then, well, FA plus, right, that worked well for ASD, so let's do that again. A little bit suspicious about the sign that I wrote here, I think that's wrong, I think it goes on the other side. Okay, so FA plus is supposed to be some quadratic form in B and C. So something easy we can do is we can take the commutator between C and B on the adjoint part that gives us a two-form, so let's do that. So, but this somehow, this doesn't seem like it's enough. So there's actually something you can do to these two-forms to make another two-form. We may be in the interest of time, not tell you what this is, but there's a natural bilinear form that you have on this that basically says, this basically comes about as follows. You think of these, of a self-dual two-form, you think of this as an endomorphism of the tangent bundle, two endomorphisms you can compose, and this is basically this composition. So, yeah, good. So this is the Waffa-Witton equation, and then you can be, you can sort of throw all reason away and say, well, this looks like the cyber-Witton or Donaldson variance, surely there must be four manifold invariants that come out of this, right? So then there's a wild guess. There's four manifold invariants, which are just defined by counting the modular space of these Waffa-Witton solutions on P. There's a modular space of solutions of these guys, you count them, and they're supposed to give you integers, and this is supposed to give you invariants that are called the Waffa-Witton invariants. I mean, none of this makes sense. The way I just said this, because we all know that to actually do this you need to do some actual work, you need to compactify the space for cyber-Witton, it's no problem because it's automatically compact, to do some transversality that's maybe not so hard, and then you can do this. But if this really is an invariant, it really depends on what your compactification is, it's not, I mean, some actual thought has to go into this. But let's sort of not worry about whether, yeah? No, so if the Waffa-Witton equations are not related to the geometric Langland's conjecture, there's a different one that is, so the appearance of this Langland's dual is very mysterious, and I think also Attia observed that, so there was a paper before in physics that sort of explained that the magnetic charge should correspond to roots, the electric charge to core roots, or the other way around, and you should swap them and that corresponds to Langland's duality for the groups, and Attia realized that this is, mathematicians already know what this is. Anyway, yeah, no, this equation apparently cannot explain geometric Langland's, or physically explain, but there's another one that, that, I mean, whether it can explain geometric Langland's, or not, this is a different question, but you can sort of phrase the concepts in the language of physics. Interesting mathematical problems come out of this. Okay, so, good. So you're supposed to be able to do this. So in algebraic geometry, people know how to do this for a project of surfaces. They have some technique of doing this. This is the work of Tanaka and Thomas from 2017. I guess I think you will hear more about this next week in Götzsch's talk. In differential geometry, we really don't know how to do this. I will say maybe more about this tomorrow, why we don't know how to do this, but let's just press on and pretend that these numbers exist. So, what the wild guess is that out of this four manifolds, the question is what is the wild guess? The answer is the wild guess is that out of this four manifold variance arise. But how precisely, it's not so clear. I mean, this is a very, very successful topic in algebraic geometry now. Okay, so let me tell you, this itself would not be interesting. So these invariants are supposed to not give you new invariants. What is interesting is the connection with this s-duality over there. So let me say, let's pretend we know what these are, and then Waffa Witten say you are supposed to write down this thing. So you're supposed to do your first combine, you combine the coupling constant and the angle into one number. So the real part we take to be the angle and the coupling constant we take to be the imaginary part B squared. You observe that this gives you something in the upper half space. Imaginary part of z is positive. This is the upper half space. And that might already make you a little bit suspicious about what's supposed to come out of this. Okay, so now you define a function in the upper half space. But the function depends on all sorts of things. It depends on the manifold, but it also depends on this parameter and the group. So then you're supposed to define this function as follows. Me, sure, I'm not messing this up. So you take the sum of these Waffa Witten invariants whatever they are, and you somehow want to put this in a generating series. Okay, so you do this in a sort of more complicated way than you would maybe naively guess. And I'm pretty sure when you read the Waffa Witten paper you see that they first made the naive guess and then sort of fiddled around with it until it works. Because the outcome is also not what they expected, I think at the very beginning of this. So natural thing would be, I mean, if you see this upper half plane here, the natural thing would be to say to write e to the two pi i tau times the instanton number. The instanton number, this is why we find the instanton number before. It would be a natural guess, but it turns out this is not quite right. You have to shift this down by some number that depends on x, which I will give you in a second. So here, S of x is supposed to be the rank of your group plus one divided by 24 times the Euler characteristic of x, why this is true? There's no physical argument. They computed some cases and then this one works. But that's fine. The sum over all isomorphism classes of P's, thank you. So this is some function, you have the half space. It's not obvious if this can purchase anything, but it's something you can write. Oh, this is the center of the league group and you just count how many elements it has. Thank you. Thank you. The question was what this is. It's the center of the league group. Okay, so then the prediction of S duality. This is not actually the prediction of S duality, I would say, but it's what they say. So if this is really true, S duality, if this is true on the nose, you would say, okay, inverting this number means doing this transformation. You do this merbius transformation, minus one over tau. You can see it preserves this, but it flips the sign on this. Sorry, it inverts this. And then you change this to the length of the dual group. And you would guess if, I mean, if you say the physics is invariant, you would say it should be the same, but it's not the same, there's some factor. The factor is very interesting. It's plus minus tau over i times w of x over two, and then times this function. If you have not seen anything like this, it might seem bizarre to you. If you have seen similar things, this might sort of seem obvious that something like this should appear. Let me tell you what these numbers are, plus or minus one are given by minus one to the rank of g over four, times the order characteristic of x plus the signature of x, and then maybe looks familiar to you, and w of x is just minus the order characteristic, times x, okay. So this is a complicated conjecture, right, okay. So if this is true, some magic happens, right? I mean, if this cannot be true, just by accident. Now, we know that on the upper half plane, there's a bigger group that acts the modular group. SL2Z acts on the upper half plane, and SL2Z has two generators. One generator does this, this generator is called, this is S of tau, and the other generator is T. It just shifts tau by one. So if this, for some reason, is an integer, certainly it's invariant under this, right? If not, then it's still the shift by tau plus one and it's a predictable thing. Okay, so this is the S-tuality conjecture. This is a very complicated thing, yeah? Yeah? Oh, sorry, sorry. This is plus minus one, plus minus. This is the sign. This is the sign that shows up here. Yeah, sorry, it's not upside, yeah. Yeah, it's a weird thing to write, just plus minus is equal to. As sum of all, I take, for every isomorphism class of principle, I take one representative, and I sum of all of them. So for example, if this is SU2, I sum over all principle bundles. I take, for every second-gen class, I take one principle bundle, yeah? Oh, this is the upper half plane. Yeah, it's the usual action by Mibius transformations. Okay, and then this is supposed to remind you of modular forms, right? So this is like a modular form, but the space is not the upper half place. The space is sort of two copies of the upper half space. One is labeled by G, the other one is labeled by L. There's some tricks that one can do to get, ultimately get only a sort of a modular form, but maybe of some higher level. Out of this equation, if you want, I can tell you more about this later. The question is, is the coincidence that this is called S duality and this is S over there? I don't know, but I suspect not. I mean, I seriously, I mean, I had the same question. The physicists, you never know. I suspect that this is why it's called S duality. I mean, they usually also call this electric magnetic duality. So I have only one minute left, so let me say. My plan was also to introduce a framework that generally explains how to think about these. I will do this tomorrow, and then I will tell you that this is a reasonable framework to study all of these equations and similar equations and maybe give you some other reasons why these equations are interesting. And then I will tell you sort of why doing things like this is hard, and I will tell you what we know about this. And even though it's hard, I will tell you some things that we can do and I will tell you what we can't do. And maybe on Friday, I will tell you why we can do the things we can do. Yeah, that's it. No, no, no, no, no, no, no. No, they should not be, they should not be, they're supposed to have the same information as even less information maybe than Donaldson variance. You can do this for arbitrary, the question was what kind of structure groups? You can do this in principle for arbitrary compact connected D groups. This is also important because otherwise it's a little bit boring if you only do it for one, right? I mean the really the fascinating thing is that the Langland's dual shows up here. So if you do it only for UN, the Langland's dual is itself. It's not so interesting when that's what algebraic geometry is to a care. Hi, this is a very good, do you want me to write this down? I'll show you later. So Waffa and Witten, okay, they write this down, right? And then they set out to actually compute this in some cases. And they compute it in particular for the K3 surface. I can show you the formula for the K3 surface later. And so they compute for the K3 surface for SU2 and SO3 and they check that this works. And it involves Dedekind eta functions and so on. It's, I mean, it's good evidence that something is going on. Yes, the question was if for that form of S duality G is simply less, and this is a good point. So in Waffa Witten original paper, they don't mention this, but it's right. If S, if G is not simply laced, apparently, you don't do this transformation, but you multiply with some number N that depends on the difference between the long and the short roots. Can be two or three. And apparently there's also some correction to this statement when the league group and the Langlands dual league group don't have the same league algebra. Two questions. The first question, if there's heuristic how to go between S cyber Britain and ASD, I mean, yes, there's this program by Pitstree Gutsch and Turin. There's some other equation that has some modular space, contains both of the modular spaces. You should get some co-bordisms. So this should give you some relation. The physical intuition, I cannot say anything about this. I mean, I have 15 pages of lecture notes for the first lecture. Maybe I cut it down to what I actually said and I put it on my web page. So maybe try to go to my web page later today. If you don't find it there, email me and I can send you something.