 about gauge theory, but it will be more an introduction to gauge theory. So more advanced topics will be covered later the next weeks, but in this lectures, I wanted to cover basics of gauge theory, so that you have something to rely on for the next weeks. In any case, so let me start with a basic idea behind essentially any gauge theory as long as it is concerned, it was manifold invariance. Okay, I see. So what we have is, we have an infinite dimensional manifold. So the dimension of C is infinity will make this more precise later in the course. And the Lie group G, which acts on this infinite dimensional manifold, and this Lie group will be typically also infinite dimensional. Now a second piece of data is a map F from C into V, where this is a G presentation. And so what we do is, this is required to be a G invariant. So F is G, F is G F invariant, which means that F at M point G is G inverse F of M. So the standard convention is that the Lie group G acts on this manifold on the right and acts on the left on the target manifold. That's why you have G inverse over here. And if you take a fixed point, so say the origin in V, and consider the pre-image of the origin, you still have an action of G on this sub-manifold. And if you quotient this, you will obtain something which is known as a modular space. So this is Y that's very G. Now, this may be a bit ill-behaved because it may fail to be, for instance, a manifold, it may fail to be an isotopological space. But let me assume that if I remove points from here, which have non-trivial stabilizers, I will get a manifold. So we'll define M star with those points in the pre-image so that the stabilizer of M is trivial. And I quotient this by G. So I will assume that this is now a smooth, smooth, compact, oriented, manifold dimension D, a finite dimension D. As such, this has a fundamental class. So we have M star as a well-defined homology class in the HD of a space. So let me write B star set, where B star is simply the space that we have started with. So if you remove points with non-trivial stabilizers and divide by the gauge group, we have some topological space, and we have a well-defined homology class in this space. Okay, very good. But we want to have numbers out of that. So this homology class sort of is an interesting object on its own, but we want to extract numbers in some way. And this is one way to do this. So assume we have a short exact sequence. So in other, well, it's too short. So there is another group G. In other words, we have a normal subgroup G zero in G and the quotient is the straight G, which is assumed to be a finite dimensionally group. Now we can form also the so-called framed modulate space. So this is M hat star. So again, I will take the preimage of the origin, but I will divide by the subgroup G zero. And what we will get in fact, we will get M hat star as a principal bundle over M with the structure group G. So this is something that I will explain today in the lecture in more detail. But the point is that once we have a principal bundle, we can associate to this characteristic classes. So say eta and assume this is in HD of star set, right? Once we have a cohomology class in degree D and we have a D dimensional manifold, we can integrate this class over M and what we get is an integral like this and this is the invariant we are after. Right, so what we want to understand in this course, why is this well defined? How the details work and why is this interesting? So the rough plan for the lectures is like this. So today I will tell you a few details about bundles, connections, characteristic classes and so on. In the next two lectures, I will tell you some tools from analysis. So the keywords here are Fridholm maps, elliptic operators and so on. And finally in the last two lectures, I will talk about an example of a gauge theory which is particularly well behaved with respect to this scheme and this will be the cyberquit and gauge theory. If there are no questions to that, let us start with basics. So if there anything to erase the blackboards, yes? So essentially what you sort of want to define is that the C star is the set of all those points, M in C is a trivial stabilizer and quotient this by G. Maybe you want to put a little bit more restrictions so what you want to have is that this is an isopological space, something like a manifold, this would be an ideal case. Any other question? Thank you very much. So I assume that most of you know most of what I'm going to say today. The purpose of two lectures today is to give you essentially a bunch of keywords so that you have a chance to pick up something that you have perhaps missed in your lectures or having a chance to look up yet. Any case, so let us talk with vector bundles. So vector bundle is essentially just a family of vector spaces parametrized by points in a say manifold M or more generally of a topological space. And this is assumed to be locally trivial which just means that you know whenever you pick a point M in your base manifold there is a neighborhood U and a map psi say from the, so we have a natural projection map into M from the pre-image of U into U cross RK and we have the diagram which, so here is psi in U cross RK and this diagram is U cross RK commutes. Now if you have a vector bundle a natural thing to consider is the section so what is a section? This is just a map from M to E such that so this is a section of the composition of pi and S is a identity on M. In other words, when I have a pick a vector in each fiber and this depends mostly on M. So I will always assume that I work in smooth category but you can of course work in topological category or any other categories that you like. Now if you have a vector bundle you can make constructions with this vector bundle. So essentially any constructions that you know from a linear algebra which doesn't require choice of base goes through to the category of vector bundles. So in particular you can construct say lambda P E so E dual and the morphisms of E and all these are vector bundles and so on. Now since the topic of this school is to make connections so let us make a connection here you go. So from a basic analysis it is useful to differentiate functions and we want also to differentiate sections and here is how it goes. So if you have a comma of E you know the space of all sections then the covariant derivative is a connection is a R linear map which satisfies the Leibniz rule which means that NABLA of F times S is D F tensor S plus F NABLA S and this holds for any smooth function on the base manifold M. Now where you can easily prove that any vector bundle has a connection but if you have a connection it's never unique so if you have one connection NABLA you can easily construct another connection say NABLA prime which is just NABLA plus A where A is a one form on M those values in the endomorphisms of E. On the other hand you can also show if you have two connections and you take their difference then the difference is a one form those values in the endomorphism of E and what you establish in this way is our basic fact is that the space of all connections this is denoted by A of E is an affine space modeled on omega one and E. Okay, now once you have a connection you can construct a sequence very similar to the Duran complex so what you start with you start with zero forms whose values in E so these are just sections of E and you can consider one forms with values in E and so on. So here you have the covariant derivative which ends exactly in the space of one forms you can extend this to an operator DNABLA which I will define in a second. So once you have one form or maybe even any K form so that the alpha be a K form whose values in E so assume you can write alpha locally as omega tensor S where omega is a K form on M and S is a section of E so we'll define DNABLA of alpha to be D omega tensor S plus minus one to the degree of omega wedge NABLA S so in that way you have an operator DNABLA here and DNABLA answer one. Now the basic fact is that unlike for the Duran complex the composition is not zero anymore so this will be DNABLA as well. Instead if you compute DNABLA DNABLA the DNABLA squared where you can establish that this is an algebraic operator which is FNABLA this is a two form as values in the endomorphism part bundle of E and this is called the curvature of NABLA. Now if you pick local coordinates say X1 and so on XN, yes, thank you. Right, if you choose local coordinates on M then what you can do, you can ask yourself what is NABLA D over DXI, NABLA D over DXJ applied to a section S. You can compare this to NABLA D over DXJ, NABLA D over DXI again applied to S and the difference is exactly the curvature form of NABLA applied to D over DXI D over DXJ. So in other words the curvature form is a measure of the non-commutativity of partial covariant derivatives. So unlike in the classical analysis partial covariant derivatives do not commute. Okay, now locally which means if you pick a local trivialization of your vector bundle you can write your connection NABLA as D plus say A where A is a one form on M with values in say GLKR so just in matrices and it easily computes that the curvature form is given again locally as GA plus A wedge A and so A wedge A here means simultaneous multiplication as matrices and taking wedge product of the elements. And now if you have one connection you can construct many, many connections in a very standard way. So if you have a GE which is a section of the endomorphism bundle of E, maybe let me denote by GE the set of all those endomorphisms of E such set at each point G, this is indigable so we have GLEM, now this is called the gauge group and if you have one connection you can construct and you pick some G in the gauge group you can construct another connection which acts on the section S just by, so what you do you multiply G with S you have another connection you apply your initial connection NABLA to Z and you apply the inverse of G to Z. And so in that case NABLA and NABLA G are called gauge equivalent. And so what you can prove quite easily is that the curvature essentially is tensorial with respect to this gauge equivalent section of the gauge group. So here is an exercise for you. So compute the curvature of the connection NABLA G. So one way to deal with vector bundles is the following so you can define your objects for instance your connection in a local trivialization and then show that this doesn't change when you change your local trivialization. And with this, so this is a way which is very, very much used in physics. But mathematically, good move at this point is to consider all possible local frames at the same time. And so what this yields is a notion of the principle bundle. So what is a principle bundle? This is the following thing. So you have a manifold P and again a projection say pi the base manifold M was the following properties. So first of all, part of the data is also a league group G which acts. So G acts on P and pi is a gene variance that is pi of P. Secondly, for any point M in M, the preimage of M has an induced action of G and this is free and transitive. And this is also locally trivial in the same way as a vector bundle is a locally trivial structure. So in other words what we have here is not a family of vector bundles but a family of league groups parameterized by a point of your base manifold. Now, if you have a vector bundle, there is so if economical way to construct the principle bundle, so here is the definition. So E is a vector bundle. Then the frame bundle of E is just the set of all isomorphisms from say R, say M and this is a set of all isomorphisms from RK into EM. So it's easy to see that there is an action of GLK and this makes the frame bundle into the principle GLK bundle. Now, it turns out that you can invert this construction. So if you have a principle G bundle, you can always construct a vector bundle and this is called the associated bundle. So I assume that P is a principle G bundle and you have representation rho, so G into GLK, say R. Then what you do, you can construct the space P times RK and you wipe an action of the group G. But the action is, so you take P and say X, multiply this by G, say on the right and this is just pi G rho G inverse X. And what you can easily establish is that the quotient is a vector bundle with a natural projection to M. Right, what this is known as the associated vector bundle. Now, what I'm trying to say essentially is that there is a set of equivalence between the notions of vector bundle and principle bundle. However, it is sometimes much more convenient to work with principle bundles rather than vector bundles. And so find the advantage which you can get here if you vary your representation rho, you will get many vector bundles which are associated to the same principle bundle. Okay, now there is also a notion of a connection for the principle bundle and this is as follows. So connection on P is just a one form, A was values, so on the total space of P was values in the l-algebra of G. So G is just the l-algebra of G with the following properties. So first of all, A is G active variant which means the following. So I have an action by an element of the l-group G on the principle bundle P, this is denoted as RG. So I can pull back my one form omega and this is required to be the joint action of G minus one on omega. So omega takes values in the l-algebra, this is naturally a representation of the l-group and so we can act with this element on the element of the l-algebra. And the second condition is that A, let me just write, so A applied to K psi is psi for any psi in G. Now what is K psi? K psi is just the infinitesimal action of G on P. In other words, you take one parameter subgroup generated by psi, this gives you a curve on P and you take its tangent factor at the point zero. Now,