Symmetry in Geometry and Mechanics

Home , Differential geometry, Ehresmann connection, Lie derivative, Linear connection

Treball ﬁnal de grau

GRAU DE MATEMÀTIQUES

Facultat de Matemàtiques i Informàtica Universitat de Barcelona

Autor: Pablo Ruiz

Director: Dr. Ignasi Mundet Realitzat a: Departament de Matemàtiques i Informàtica

Barcelona, June 29, 2017 ABSTRACT

In this work we have tried to give a theoretical description, as well as a historical review, of the differential geometry behind different areas of physics (Classical Mechanics, Electromagnetism, Yang-Mills...) trying to capture the deep geometric nature of modern theoretical physics. Firstly we start by introducing classical differential-geometric notions from the modern viewpoint (emphasizing the role played by Bundles and Sections). Then we will devote to the study of Hamiltonian Systems both in the Symplectic and in the Pois- son formalisms, in here we will see two results that we believe are specially important, geodesic ﬂow from the dynamics point of view (where we will analyze the notion of geodesics through the lenses of physical theories, and a proof of Clairaut’s formula in classical differential geometry by the use of a fundamental concept in modern geometric mechanics, the moment map. Contents

0.1 Review of classical differential geometry ...... 1 0.1.1 Connections in TM tensor bundles and abstract vector bundles ...... 10 0.1.2 The importance of the Levi-Civita connection in classical differential geometry ...... 15 0.2 Élie Cartan and the revolutionarization of classical differential geometry ...... 18 0.2.1 Differential forms and exterior derivative ...... 18 0.2.2 Lie derivative in smooth manifolds ...... 25 0.3 Hamiltonian mechanics and Symplectic geometry ...... 27 0.3.1 Cotangent Bundle of a smooth manifold ...... 28 0.3.2 Symplectic and Hamiltonian vector fields ...... 29 0.3.3 Poincare’s lemma ...... 32 0.3.4 Darboux’s theorem ...... 32 0.3.5 Poisson Brackets on Symplectic manifolds ...... 35 0.4 Poisson Manifolds ...... 37 0.4.1 Hamiltonian vector fields in Poisson Manifolds . . . . . 38 0.4.2 Poisson maps/canonical transformations ...... 41 0.5 Moment Maps ...... 47 0.5.1 Theory ...... 47 0.5.2 Computation and Properties of Momentum Maps . . . . 53 0.6 More differential geometry, Principal and Associated Bundles (Yang-Mills theory)...... 55 0.6.1 Geometrization of the Kaluza-Klein theory of electromagnetism ...... 57 0.6.2 Principal Fiber Bundles and their associated Ehresmann connections ...... 61 0.6.3 The notion of Ehresmann connection ...... 64 0.6.4 Linear connections in Associated Vector Bundles comming from Ehresmann connections in Principal Bundles 64 0.6.5 The electromagnetic field as a U(1)-connection . . . . . 66 0.6.6 Line Bundle defined by a closed 2-form on a manifold . 68 0.6.7 Quantization using Complex Linear Connections in Com- plex Vector Bundles ...... 71 0.6.8 Hermitian Vector Bundles and Connections ...... 72

1 .1 Tensors, a brief description ...... 74

Bibliography 83 During the work we will be mainly using Einstein’s summation convention when we operate on tensorial objects, this just means that we omit the ∑ where we sum, and we know that we have to sum those indices that are repeated (usually one up/one down).

0.1 Review of classical differential geometry

This section pretends to be just a brief discussion of classical differential geometry, so that the reader knows and recalls some of the concepts that will be essential later, and to develop the language that we will be using in the rest of the work. We will recast classical differential geometry via the fundamental concept of fiber bundles, since they are the abstract framework that allows us to better understand how differential geometry works. We start with the standard definition of a manifold, that is mainly a result of Riemman’s revolutionarization of geometry two centuries ago that deeply changed the notion of geometry and physics. The abstract definition of manifold just tries to capture a simple idea, the fact that one needs n magnitudes to determine position in an n-dimensional manifold. How this quantities get represented numer- ically, or what do they represent is just a matter of choice, what we would call a choice of local patch in modern language. What we also require is that the multiple ways to look at this magnitudes coincide (that we have the same geometry/physics). Thus we will now give the formal definition that can be found in any standard book on differential geometry as a starting point. Definition 0.1.0.1. A manifold of dimension n is a Haussdorff second-countable topological space M together with certain structure ,that we usually call atlas (in the case of a smooth manifold we usually call it a differentiable structure):

1. An open cover {Ui}i∈I of M. n 2. A collection of continuous injective maps Φi : Ui → R called coordinate charts/local coordinates such that Φi(Ui) is an open set. We also require the existence of smooth transition maps, what this means is that if Ui ∩ Uj 6= φ, then the following map transition map is smooth: −1 n n Φj ◦ Φi : Φi(Ui ∩ Uj) ⊂ R → Φj(Ui ∩ Uj) ⊂ R , we say the charts are (smoothly) compatible. The beauty of this definition is that it can be restricted to the class of functions we are specially interested (smooth functions usually in differential geometry), but we could consider analytical functions, meromorphic functions, class Ck functions... Riemann himself used this to revolutionize classical complex analysis with the introduction of Riemann surfaces. 1 n We can interpret each chart Φi as a collection of n functions (x , ..., x ) on Ui, and we also interpret transition maps as a coordinate transformation in the sense that they can be interpreted as a family of maps: (x1, ..., xn) 7→ (y1(x1, ..., xm), ..., yn(x1, ..., xm)) One important remark to make here is that one can also define the notion of manifold in the infinite-dimensional case, and there is a whole theory built on them (by modeling them on Banach Spaces, rather than on the finite- dimensional Rn. Since the major focus of this work is on finite-dimensional manifolds we will stick to the definition we gave and restrict our attention to the finite-dimensional case. In sections of the work (mainly the last ones) some of the spaces that will pop up in the description will be infinite-dimensional, and thus we will have little to say about them, what we can get is an intuitive idea on the object, rather than a deep comprehension, that would obviously require us to develop the highly non-trivial theory of infinite-dimensional manifolds. Now that we have the fundamental notion of a manifold, that is essential to geometry and physics, we connect with standard calculus on euclidean space just by noting that trivially Rn is a smooth manifold with a natural smooth n n structure that has a unique chart, the identity mapping Idn : R → R . The formal notion of a manifold as we know it today was the result of a deep historical development, this definition is not essentially fundamental, what we want to emphasize is that this definition is nothing more than the formal definition that comes in hand with Riemman’s ideas. Since it is something that in mathematics usually gets forgotten, but definitions are not the start of anything, but a guide we allow ourselves to be clear about the ideas we are talking about, and definitions get superseded by others that are more advanced and that represent better the nature of the object (the case of connections, the notion of derivative, even the very nature of what does geometry mean). We could argue, in a similar fashion to what the german Philosopher Hegel had in mind with his criticism of Kant’s phylosophy, that the concrete definition/result has to be understood as a realization of the whole Historical Devel- opment. (In our case this just means that to better understand mathematical concepts one is almost obligated to analyze the History that led to them). The main contribution to this revolutionarization of geometry comes from Riemann, in his doctoral dissertation on the nature of geometry [11]. And this same notion turned out to be indispensable for Einstein while seeking for a geometrical description of the physical universe. In fact, due to the fact that differential geometry was not as well understood at the beginning of last century, Einstein had to develop a lot of intuition on how it worked, and needed a lot of mathematically-minded people to develop the mathematics needed to describe the universe at a cosmological level and to elaborate the best theory of gravity we still have. To continue we now introduce the notion of a smooth map between smooth manifolds, that is just a natural generalization of the notion on euclidean space (we just require that it works locally and for every chart).

Deﬁnition 0.1.0.2. We say that a map between two smooth manifolds f : M → N (of dimensions m and n respectively) is smooth if for every local charts φ on M and ψ on N the composition ψ ◦ f ◦ ϕ−1 : Rm → Rn is a smooth map (in the standard calculus sense). And further than that, we introduce the concept of "equality" inside the category of smooth manifolds, the diffeomorphism.

Deﬁnition 0.1.0.3. Let f : M → N be a smooth map, we say f is a diffeomorphism if it is an invertible function and the inverse function is also smooth. If we have a diffeomorphism between to manifolds M, N we say that the manifolds are diffeomorphic.

In a certain smooth manifold M we will denote the space of all smooth functions (what physicists call observables) in the following way:

C∞(M) = { f : M → R, f smooth}.

Hence we have a good abstract understanding of what a manifold is, but for the moment it is difﬁcult to connect it to the intuitive notion, this is done via a proposition that actually shows why the concept of manifold is so important, since it can describe a lot of geometrical situations one has already encountered in other contexts. One can interpret manifolds this way thus as zeros of functions in a certain manifold (usually some Rn). The proposition is the following:

Proposition 1. (Proposition 1.2.5 in [1]). Consider M a smooth manifold of dimen- ∞ sion n and some smooth functions f1, ..., fk ∈ C (M). Now we consider the set:

Z = Z( f1, ..., fk) = {p ∈ M| f1(p) = ... = fk(p) = 0},

We assume that the functions fi are functionally independent along Z, meaning that we have local coordinates (x1, ..., xn) deﬁned in a neighbourhood of p ∈ M such that xi(p) = 0 for every i, and that the matrix:   ∂ f1 ∂ f1 ··· ∂ f1 ∂x1 ∂x2 ∂xn  ∂ f2 ∂ f2 ∂ f2  ∂ f f  ···  1, ..., k  ∂x1 ∂x2 ∂xn  =  . . . .  . ∂x  . . .. .    ∂ fm ∂ fm ∂ fm ∂x ∂x ··· ∂x 1 2 n x1=...=xm=0 hast rank k (this matrix is just the differential matrix). If this happens then Z has the natural structure of a smooth manifold of dimension m − k. This proposition is usually called the regular value theorem, and is very used to show that certain sets (for example submanifolds of Rn) are smooth manifolds.

And this is also how manifolds entered mathematics if one looks at the historical development of geometry, since they were a necessity to express the notion of geometry needed in the classical study of Gauss on surfaces, but that needed to be superseeded by a more intrinsic notion of geometry. For Gauss, surfaces were always objects one looked to be immersed in euclidean space, the beauty of Riemann’s idea is to connect directly with one of the ideas of Gauss, namely its Theorema Egregium (one of the most striking results of Gauss). This theorem just says that curvature (that Gauss understood to be a property of the surface inside euclidean space) is in fact a property of the surface as an abstract object! It is an invariant that only depends on the intrinsic structure of the surface! This observation by Gauss shows us that we have to be very interested in the intrinsic properties of a manifold, and for physics this indispensable, since we are inside a certain manifold, we can not just picture an immersion into higher-dimensional euclidean space (although we could do that due to Whitney’s theorem in [?], it is better not to usually, since it doesn’t allows us a better comprehension of the properties of the manifold). Now we will brieﬂy review some differential-geometric objects that we can introduce in a manifold that allow us to understand the physics and the geometry. We will give abstract ideas of what the classical concepts used in differential geometry represent if one looks at them from the right perspective, we encourage the reader to relate this abstract notions to intuition, since they are in fact nothing more than the adequate formal expression of this intuition. One of the fundamental concepts of modern geometry is the notion of a Bundle. This will be one of the fundamental objects of this work, that we will use in a variety of different ways and constructions, thus seeing the power of this notion to describe geometrical situations. One can picture a bundle intuitively as the following: π : P → X where P and X represent two smooth manifolds and one thinks of P as being "over" X and as the map π as a kind of projection. The notion of Bundle tries to generalize the notion of product space P × X. In the sense that locally it works as a product (and the map works as a projection), but it may have a very different global topological structure! So we now deﬁne this object that will be essential for almost all what follows (in many different situations).

Definition 0.1.0.4. A fiber bundle is a structure (π : P → X,F) where P, X, F are smooth manifolds and π is a smooth surjective map. We require a fundamental property for this structure, we want it to look locally like a product space. Thus we impose that for every point p ∈ P we require that there exists an open neighborhood U ⊂ X of π(p) ∈ X and a diffeomorphism ϕ : π−1(U) → U × F in such a way that the map π (restricted to that domain) agrees with the projection U × F → U onto the first factor. We call this property local triviality, and the collection {(Ui, ϕi)} a local trivialization of the Bundle. We usually call X the base manifold, F the fiber and P the total space.

With this notion introduced we have the perspective of modern differential geometry, and this will be very useful to look at classical differential geometry through the glasses of the modern notions. We will try to use the modern notions whenever they are more clear in order to describe geometric/physical notions. We follow with another fundamental notion in differential geometry,

Deﬁnition 0.1.0.5. A vector ﬁeld X on our manifold M is an R-linear map X : C∞(M) → C∞(M), that acts as a derivation of the algebra of functions:

X( f1 f2) = X( f1) f2 + f1X( f2),

∞ for every pair of functions f1, f2 ∈ C (M). We denote by Γ(TM) the space of all vector ﬁelds, this notation just comes from seeing vector ﬁelds as sections of the tangent bundle, notation that we will heavily use later.

Now we see what kind of abstract structure do vector fields have, first observe that we can multiply vector fields by functions and we still get a vector field (hence we can look at Γ(TM) as a C∞(M)-module). Now, before we introduce further structure on vector-fields, we will describe a fundamental object in modern differential geometry and many other areas of mathematics, the notion of a Lie Algebra. This notion will also be essential during the work, since as we will be seeing during the work, they are a very good tool to encode the symmetries of a system. Thus we will enter the domain of abstract algebra for a second, the notion of a Lie Algebra is motivated purely in geometric terms (and many of the examples where they arise are purely geometric). Although that is the case, a general notion of what it represents is fundamental, since we will see Lie Algebras pop up in many different places during the work, and having a deep understanding of what they represent is fundamental to understanding modern geometry. We begin recalling what the definition of an abstract Lie Algebra is, since it is a concept that permeates both geometry and physics, and as well as now it will be needed later when we talk about Lie Groups (another fundamental object in differential geometry, we will define them later). Since we will be mainly interested in differential geometry and continuous transformations we will deal with Lie Algebras over R or C.

Deﬁnition 0.1.0.6. Let g be a vector space over R. We say g is a Lie Algebra if there is a binary operation [·, ·] : g × g → g that satisﬁes:

1. It is bilinear, for every x, y, z ∈ g and every scalars a, b

[ax + by, z] = a[x, z] + b[y, z], [z, ax + by] = a[z, x] + b[z, y]

2. Anti-commutativity, for every x, y ∈ g:

[x, y] = −[y, x]

3. What is usually known as the Jacobi Identity, for every x, y, z ∈ g:

[x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0

Remark. One of the first things to be noted in this definition is that Anti- Commutativity implies directly that for every x ∈ g, [x, x] = 0 With the abstract notion of a Lie Algebra, we now introduce a product in the space of vector fields, it is usually known as the Lie/Jacobi bracket: [·, ·] : Γ(TM) × Γ(TM) → Γ(TM) (X, Y) 7→ [X, Y] = XY − YX The details on why can we do this and still obtain a vector field can be found in (Lemma 3.1.10 in [1]) here we want to emphasize the idea, the formal result just follows from direct computation and it does not offer any deep idea apart from practice in dealing with vector-fields. What is important about this structure is that it turns the space of vector fields into a real Lie Algebra (it is needed to check that it gives a vector field and that it satisfies the Jacobi Iden- tity, the other properties are clear). This point of view will be essential in later sections, where we will deeply connect different Lie Algebras in a manifold and see their mutual relationship. Now we come to a fundamental object when describing smooth manifolds, here as in many areas of mathematics, we are interested in linearizing the structure we have, since we will then be able to apply all the machinery of linear algebra to our concrete problem. How we linearize in a smooth manifold is by introducing the notion of the tangent space at a point p ∈ M, that is usually denoted by Tp M, which is just a real vector space of the same dimension of the manifold. The intuitive interpretation of the tangent space one has in the theory of surfaces is indispensable, since the tangent space is nothing more than that naive picture we all have in mind, but in order to define it abstractly and intrinsically, one has to be a bit clever. There are two main perspectives on what the tangent space is, a more classical description of it (and you could say that it is more physical) is just as an equivalence class of trajectories that pass through the point, and that get identified if they have the same tangent vector, we interpret tangent vectors thus as tangent vectors to a trajectory, as concrete and visual things. Although this description is very good for maintaining intuition, it is better to have an abstract understanding that relates to the whole formalism, and that is the notion of tangent space introduced by Chevalley, that just relates to a fundamental observation, that tangent vectors are nothing more than directional derivatives! Thus we can identify (or actually define) the tangent space Tp M with the set of linear maps ϕ : C∞(M) → R that satisfy the following property at the point p: ϕ( f1 f2) = ϕ( f1) f2(p) + f1(p)ϕ( f2), note that this is nothing more than the condition of being a directional derivative! And we have thus a formal understanding of what do tangent vectors represent, and if the reader has never looked at this definition profoundly it is recommended, since it just describes in a beautiful and concise way tangent vectors from an intrinsic perspective, and it allows us to relate better to the notion of vector fields as derivations of C∞(M), since we now that at a point (tangent) vectors are also derivations. Now that we have introduced the linearization of the manifold at every point we can do a fundamental observation (that we commented before), we now have at our disposal the whole machinery of Linear Algebra at every point p in the manifold, and this perspective has been fundamental for the modern development of differential geometry. We could say that in a sense differential geometry is what one gets when one adapts Linear Algebra to work in the tangent space at every point and to vary smoothly from point to point. The importance that Linear Algebra has in modern mathematics should never be underestimated! With the tangent space at each point in hand we will now introduce the notion of the Tangent Bundle. The notion of Bundle is essential for modern differential geometry, since it has the advantage of putting different concepts under the same theoretical framework. Bundles are objects are indispensable for our description since they will be fundamental for the description of the physical universe in Electro-Magnetism and Yang-Mills theories (that can be viewed as generalizations of Electro-Magnetism) where Principal Bundles play an essential role. But now the reader can get an intuitive understanding of what the concept of Bundle tries to generalize when looking at the fundamental example of the Tangent Bundle.

Deﬁnition 0.1.0.7. The tangent bundle to a manifold M, denoted as TM, is just the union of tangent spaces at every point, G TM = Tp M. p∈M

It comes associated with a projection π : TM → M that just sends a vector v ∈ Tp M to the point of the tangent space where it lives, namely p. It inherits the differentiable structure of the manifold M by using the basis of tangent ∂ vectors associated to a coordinate patch (∂µ = ∂xµ ). Hence we have a Bundle, and a special type of bundle, what are called Vector Bundles, that is just a Bundle with the structure of a vector space at each point (in this case one clearly sees why), we will later further develop the notion of Bundle and elaborate further in this intuitive description. But have in mind that a Bundle will be a similar thing to what we have here (but not necessarily the tangent one).

One can also deﬁne the Cotangent Bundle in a similar fashion, just by taking every dual tangent space in the following way:

∗ G ∗ T M = Tp M, p∈M and giving it the structure of a differentiable bundle by using the dual basis to the one we used previously, which we usually denote by dxµ. This will be further explained in the section where we need to use it to describe mechanical systems that have that geometry (in the section devoted to classical mechanics and symplectic geometry). The interesting thing one can do with this Bundles, to really see the importance of the concept of bundle, is to consider what are called sections of a Bundle. These sections vary depending on the bundle, and for every bundle they carry different information about the bundle, we can view vector ﬁelds in the following way:

Deﬁnition 0.1.0.8. A vector ﬁeld X is can be understood as a smooth map

X : M → TM, such that X ◦ π = IdM, this is just a fancy (but very useful) way of looking at vector ﬁelds since what we are saying is we consider a map that associates a tangent vector at every point on the manifold M in a smooth way.

But the idea that we can understand vector ﬁelds like this is essential, since now we can really consider other types of objects like tensor ﬁelds just by changing the tangent space for an appropriate tensor product space (if the reader is not used to tensor products we refer to the appendix for a description of their theoretical and practical uses), and just apply the same reasoning we did here with the tangent bundle. The importance of tensor bundles are their properties with respect to changes of coordinates, this motivates physically that they are appropriate quantities to measure geometrical/physical quantities that really exist and are not an artifact of our choice of coordinates. And this is just how physicists usually understand (the local representation of) this kind of objects, as collections of numbers with certain transformation laws. The power of this reasoning both in geometry and in physics is that tensorial quantities/equations are the objects that ought to describe nature since, although computed in local coordinates, they represent information that exists independent of them. We can thus choose freely our coordinates to treat concrete geometrical problems if we are careful with the objects we use, and deduce from local computations in a coordinate chart intrinsic properties of the geometry of certain objects, and allowing to investigate the global behavior of certain objects. But remember one thing, physics always takes place locally, so physicists usually don’t care (although not always, when topological properties of the space become relevant) about the global aspect of their objects. In the introduction of the work by Robert Hermann on Yang-Mills theory [?], the author criticizes this view and he argues that it is indispensable to understand the geometrical structure of the physical objects to better understand physics. We quote two fragments of [?] (some of the concepts in the quote will come later in the section, the critic attitude is what we want to show now, and it is a very harsh criticism):

It was considered then –and mathematicians still largely think this way– that Riemannian geometry was the only true geometric faith. Thus, writing down the laws of physics in terms of Riemannian geometry (particularly the "differential invariants" of the Riemannian metric, the curvature) was considered the highest form of poetry. Physicists have long played their own version of the Oedipal game and have rejected Einstein’s aesthetics, but I am beginning to think that the old boy might have been right after all. One of the main objections to Kaluza-Klein in the 1920’s was the extra fifth dimension that it involved. After fifty years of science fiction (and a lot more mathematics) we are a good deal more broad minded about such things. (The physicists call it "internal symmetry degrees of freedom", while we mathematicians can call it "principal fibre bundles over R4with compact Lie group as a structure group" and make it seem more familiar.)

Although elementary particle physics is the field of science with the greatest intellectual appeal –and the closest relation to geometric mathematics– I believe that the main thrust of the field is completely distorted by the physicist’s attitude towards mathematics. Particularly since the rise of "gauge theories" as a crude phenomenological guide to the interpretation of the experiment (and they are so far no more than that) the subject needs a theory that puts together a suitable combination of intuitive physical insight and the "right" mathematical structure (and "right" includes all those classical elements of elegance, beauty,etc. [...]) I believe the mathematical and physical ideas we need are lying at our feet. "Modern" differential geometry, meaning fiber bundles, connections, etc., is obviously the raw material.

The rest of the Introduction in that book is highly recommended for a more complete view on the opinion argued by R. Hermann. With all the things we introduced previously we have, in a sense, destroyed the subjectivist attitude towards preferred coordinate systems as being more "real" than others. In principle one can choose coordinates the way one wants if one respects the rules of tensor calculus, the importance (for the physics) lies on the quantities that are purely geometrical and that do not depend on the coordinate system/observer. Thus we have just arrived at the central idea in relativity, the physical world is independent of the choice of coordinates one uses to describe it. There is an important thing to notice, this was not exclusively Einstein’s idea (he took it very seriously, and it was not at all an easy task) but it comes from Riemann, it is the kind of geometry Riemann was referring to in his Doctoral Dissertation! Thus the abstract framework of differential geometry is indispensable in modern theoretical physics, and one of the main difficulties of Einstein is that differential geometry was not as well understood as it is today by the start of last century. Now let’s do an example of a tensor bundle and sections on it that is interesting to be able to see how all this machinery is used. For example suppose that we want to study the symmetric covariant 2- tensor fields in our manifold, then we just take the bundle S2T∗ M defined in the following way (where S2 refers to the symmetric tensor product):

2 ∗ G 2 ∗ S T M = S (Tp M), p∈M and we consider the obvious projection and here we have it, we now just have to consider sections on this bundle and we will be able to study the tensor ﬁelds we were interested in! Thus, arbitrary tensor ﬁelds on our manifold (very important objects for a geometrical description of the physical universe) are now just described in a purely geometric way, as sections of a certain tensor bundle. All the examples we have just seen are examples of what are called vector bundles, since all the spaces we considered also have the vector space structure! And this vector space structure is essential for the notion of differentiation! In latter sections we will further elaborate on the importance of Bundles in geometry and physics.

0.1.1 Connections in TM tensor bundles and abstract vector bundles

Now we will brieﬂy recall the notion of a connection in differential geometry, just to get some intuition on what a connection means here as a tool for differentiating appropriately inside a manifold, this concept will be superseded in later sections, where we will introduce the views of Cartan and Ehresmann on connections, and we will have a more abstract understanding of the different concepts one has ﬂoating around in classical differential geometry, the following triad:

1. Connection

2. Parallel Transport

3. Covariant Derivative

This are very different concepts, and we must correctly interpret them as different things geometrically, the problem here is that we are not in the adequate theoretical framework to deal with the difference, so we will just for now comment brieﬂy on how classical differential geometry deals with connections.

Deﬁnition 0.1.1.1. A connection (usually denoted by ∇) is an operation on vector ﬁelds

∇ : Γ(TM) × Γ(TM) → Γ(TM)

(X, Y) 7→ ∇XY that satisfies the algebraic properties of the ordinary derivation in euclidean space (we will give now a generalization that allows us to understand the necessity of this algebraic properties). This definition can be generalized to arbitrary vector bundles (think of tensor bundles to get an intuition), if we consider a smooth vector bundle π : E → M, we define a connection on E as a map

∇ : Γ(E) → Γ(E ⊗ T∗ M), that satisﬁes the Leibniz rule for every smooth function f ∈ C∞(M) and every smooth section τ ∈ Γ(E), namely

∇(τ f ) = (∇τ) f + τ ⊗ d f .

So for every vector ﬁeld X ∈ Γ(TM) we can think of a covariant derivative

∇X : Γ(E) → Γ(E) if we contract the vector ﬁeld with the result of the connection applied to a section.

This definition is due to Jean-Luis Koszul, who gave the appropriate algebraic framework for connections on vector bundles. The covariant derivative we defined in general vector bundles satisfies the properties we wanted, namely:

∇X(τ1 + τ2) = ∇Xτ1 + ∇Xτ2 ∇ = ∇ + ∇ X1+X2 τ X1 τ X2 τ ∇X( f τ) = f ∇Xτ + X( f )τ

∇ f Xτ = f ∇Xτ.

∞ for every vector ﬁelds X, X1, X2 ∈ Γ(TM), every function f ∈ C (M), and every sections τ1, τ2 ∈ Γ(E). Thus we have the essence of a connection in a vector bundle understood as a covariant derivative, which is what one does in classical differential geometry (usually one works in the tangent bundle as a vector bundle).

What one has to learn now is how to compute with this object, since at first it seems highly non-trivial to do that. But as everything in differential geometry, as a global object is one thing, to do computations with it we need its expression in local coordinates. So we take a coordinate patch U ⊂ M that has certain coordinates (xµ). This coordinates induce vector fields (derivations) ∂ for every coordinate (∂µ = ∂xµ ). And with this vector fields we can write any other vector field locally. Now that we are in the local situation, we can consider the following vectors (that will contain all the local information about the connection):

Γµν = ∇∂µ ∂ν, this are usually called the Christoffel symbols of a connection, and usually understood as a collection of numbers in the following way:

λ λ Γµν = dx (∇∂µ ∂ν).

This coefficients have all the local information about the connection and allow us to differentiate (in a certain sense). Notice they work for a general connection, we have not imposed any special restriction on the connection we are taking (this is a very powerful idea) and allow us to do the following process: Consider two vector fields X, Y ∈ Γ(TM) written in local coordinates in the following way, µ ν X = X ∂µ, Y = Y ∂ν, now we want to use the christoffel symbols to compute ∇XY, and if one just applies the properties of the connection and recalls the definition of the symbols one thus gets the expression for the components of ∇XY in the following way

k k ν k µ ν µ ∂Y k σ (∇ Y) = (∇ µ (Y ∂ )) = X (∇ Y ∂ ) = X ( + Γ Y ). X X ∂µ ν ∂µ ν ∂xµ µσ Thus we see now what do really this numbers allow us to know, how the geometry of the space modifies the dynamics inside it!. And we see this since if we λ suppose for a second that the Symbols are all zero (Γµν = 0, ∀λ, µ, ν) we just see that we are computing the derivative from ordinary calculus. So if for a certain connection we have this (we know this is the case in flat space-time, later we will talk about the relationship between the connection and another object the metric tensor), then for those coordinates we will just be considering the usual derivative. But notice a very important thing, that doesn’t need to happen if we choose other coordinate systems! In another coordinate system on the same space those symbols may vary (since they are associated to another local patch), but we will be describing the same physical reality. Thus here we see one of the powers of differential geometry, the ability to choose the coordinates better adapted to the description of a certain system, more on that later. All what we did was adapted to the case of interest in classical differential geometry, but we could have considered an arbitrary vector bundle and a basis of sections (adapted to the problem in hand) and express the symbols, that would represent derivatives in arbitrary vector bundles in computations, but one needs to see how is all this happening in a familiar case. And now comes the fundamental idea of Cartan that is fundamental, interpreting connections as differential 1forms with values on endomorphisms. What does this even mean? We will later introduce the notion of differential forms in a manifold (you can understand a 1-form for now as something that "eats" vectors and gives us scalars), and talk again about all this notions in later sections, but notice for a second the following fundamental intuitive idea. Take a connection ∇ on a certain vector bundle, then for a certain vector field X ∈ Γ(TM) we have that

∇X : Γ(E) → Gamma(E), thus the connection takes sections to sections, if we interpret this pointwise (in each fiber) we get that ∇X gives us a map E → E at each fiber. And this is was the fundamental observation made by Cartan! Connections are thus nothing more than the assigning at each point in the base manifold of an Endomorphism of the associated fiber, this obviously done in a smooth way. Thus we could consider an appropriate bundle (that has as fiber the space of linear maps End(E) = E ⊗ E∗) and understand connections as well as sections of certain fiber bundles!. The appropriate bundle in this case would be what’s usually denoted, making an abuse of notation, by

π : E ⊗ E∗ → M.

Before we start talking about another fundamental object in differential geometry (the metric tensor), notice for a second that working with connections via this symbols is something that works in the case we are treating, that of finite-dimensional manifolds. If we want to treat infinite-dimensional ones, we have to be very careful with what connections represent there, during the work we will give some information here and there about the infinite- dimensionality of a manifold and how to work there, but bear in mind that it is highly non trivial. Sometimes, in the things we will explain, it is better to get the idea/picture on how the same ideas that arise in finite-dimensional systems can be applied to infinite-dimensional ones rather than working out the concrete analytical details for a precise problem (that usually can be done, although not always, and one should be careful). One can further generalize the notion of a connection to more abstract types of Bundles, namely Principal Bundles (which will be introduced later) but that are basically Bundles with a Lie group acting on the total manifold respecting the bundle structure (i.e. the fibers) and some more requirements on this action depending on the context, and we will do this in later sections. Now we define another very important object,

Definition 0.1.1.2. Let M be a smooth manifold, we call a tensor field g ∈ Γ(S2T∗ M) that is non-degenerate a metric tensor (notice that we do not impose positive-definiteness, that is important). In this definition we use modern differential geometric notions, namely that of sections on a tensor bundle introduced before. What all this is saying is that at each point p ∈ M we have a map

gp : Tp M × Tp M → R that satisﬁes:

1. It is linear in each component.

2. It is symmetric, this means that for every pair of vector ﬁelds X, Y ∈ Γ(TM) we have that g(X, Y) = g(Y, X).

3. It is non-degenerate in the sense that if for every vector ﬁeld Y ∈ Γ(TM), g(X, Y) = 0 then X = 0. In ﬁnite dimensions we can just think that, as a matrix we can invert it (detg 6= 0).

This is the same as saying that at each point the metric tensor is just a symmet- 2 ∗ ric co-variant 2-tensor, namely that gp ∈ S Tp M, and that it varies smoothly from point to point! Thus we see again that vector bundles and sections on them are essential in differential geometry since they give the correct abstract framework that generalizes the classical objects. Now we have to note a thing, usually mathematicians just restrict their study to what are usually called Riemannian metrics, where one requires that the metric is positive-definite (to automatically associate to it a norm of vectors as usual) and to define a notion of distance in the smooth manifold. But physicists are not really interested usually in that positive-definiteness (although they are in some cases) for a simple reason, the metric used in relativity comes from generalizing the metric one has for the flat-space of Special Relativity, Minkowski Space-Time. Minkowski Space-time, introduced by the german mathematician Hermann Minkowski to better study Electro-Magnetism turned out to be the natural setting for Special Relativity. In mathematical language, this space is nothing more than R4 (to describe space and time) equipped with a metric η = diag(−1, 1, 1, 1) or the signs flipped (that is not very important) if we take appropriate units and normalize the speed of light to c = 1. The group of transformations that preserve this structure is very important and is called the Poincaré group, studied by the french mathematician Henri Poincaré, it is a non-abelian "Lie" group of dimension 10 of fundamental importance in physics. In special relativity the bilinear product allows us to distinguish between three kinds of vectors in R4, if we consider a vector v = (x0, ..., x3) = (t, x1, ..., x3) ∈ R4 we thus get:

1. η(v, v) = −t2 + (x1)2 + (x2)2 + (x3)2 < 0

2. η(v, v) = −t2 + (x1)2 + (x2)2 + (x3)2 = 0

3. η(v, v) = −t2 + (x1)2 + (x2)2 + (x3)2 > 0

And it is important to distinguish (in General Relativity too) between the distinct trajectories that these give and their physical difference, where the 3-dimensional cones deﬁned by the equation

(x1)2 + (x2)2 + (x3)2 = t2, allow us to introduce the speed of light in Special Relativity (in this description normalized to c = 1) as a cosmic limit for speed by clearly separating the different types of trajectories of the physical system, noting that the fundamental idea of Einstein taken from electromagnetism is that the speed of light is a cosmic limit for the speed of particles and thus we restrict our attention to those elements that represent slower speeds (namely v ∈ R4 such that η(v, v) < 0). So to be able to describe Special Relativity we need a bilinear product that is obviously not positive definite and hence in physics manifolds are not at all restricted to have a positive-definite metric, other kinds of metrics are also needed for a description of the physical universe, and hence we need to study what are called pseudo-Riemannian manifolds that is what we defined before. 0.1.2 The importance of the Levi-Civita connection in classical differential geometry Now we are going to explore a deep relationship that we can find between the two concepts we just introduced, the connection and the metric tensor. This idea of associating a connection (on the tangent bundle) directly to a metric is an idea introduced by the italian mathematician Tullio Levi-Civita, together with Ricci-Curbastro they published their theory of differential geometry and tensor calculus in [7] and other books/articles. Einstein used heavily this sources (comming from the Italian school of differential geometry) as a way to learn differential geometry. Levi-Civita was heavily involved in the physics of his time, in the development of General Relativity in correspondence with Einstein and on Dirac’s equations for Quantum Mechanics. And we could say that in a certain sense, this two mathematicians, as well as Rie- mann, allowed Einstein to start describing the universe geometrically. So suppose we want to use the notion of a connection introduced previously in a manifold that already has a metric tensor. We can impose thus that the connection satisfies some properties that seem relevant: 1. We want it to preserve the metric in the sense that ∇g = 0, this just means that it satisfies the Leibniz product rule

Xg(Y, Z) = g(∇XY, Z) + g(Y, ∇X Z).

2. We also want it to be a torsion-free connection, in the sense that the torsion tensor associated to the connection to be zero,

Tor(∇) = ∇XY − ∇Y X − [X, Y] = 0.

If we look at this condition in the Christoffel symbols of the connection, and using that the Lie bracket of two coordinate tangent basis vectors is zero (by the symmetry of the second derivatives in standard calculus) we just get that the symbols are symmetric in the lower two indices:

λ λ Γµν = Γνµ

The classical result by Levi-Civita is that there is a unique connection on the manifold that satisﬁes this (Proposition 4.1.9 in [1]). And the strategy of Levi-Civita, followed by Einstein was to emphasize the fundamental role played by this connection associated to the metric of the space. This is due to the fact that given the metric, one can actually compute the christoffel symbols with the following equation: ∂g ∂g ∂g Γi = 1 gim mk + ml − kl , kl 2 ∂xl ∂xk ∂xm that using a shorthand notation for derivatives that we use a lot in this work looks like this: l 1 lr n o Γ jk = 2 g ∂kgrj + ∂jgrk − ∂r gjk , recall that we are all the time using the Einstein summation convention, the actual letter of the index doesn’t matter at all, just his relationship with others. We have thus determined explicitly the connection symbols in terms of the metric tensor (its inverse) and its derivatives. Thus a connection that just uses information about the metric tensor, that is the fundamental object we suppose we have in classical General Relativity, thus Einstein accepted without a doubt this connection as the one to describe reality with. The fundamental idea of general relativity is to use tensor quantities in a 4-dimensional smooth manifold associated to a metric to describe gravity, one of the fundamental ones being the curvature associated to a connection, and in the description we are now, the curvature of the Levi-Civita connection. We will later see how this works for concrete examples in later sections where we will use more differential geometric machinery that for the moment we haven’t introduced yet. Now we introduce the concept of geodesics of a pseudo-Riemannian manifold, they are a special class of curves on the manifold that are fundamental for both geometry and physics. We can understand the notion of a geodesic from different viewpoints, the classical perspective is that they are the curves that "generalize" the notion of straight line in euclidean space. Looking at geodesics from this point of view, the classical deﬁnition of geodesics is as curves α : I → M that satisfy

∇α˙ α˙ = 0. And this is the equation for geodesics if one uses ∇ as the Levi-Civita connection. Computation in local coordinates of covariant derivatives is done, as we saw before, by using the Christoffel Symbols associated to the connection, in this case we are in the Levi-Civita connection and we can compute this symbols in terms of derivatives of the metric. And thus if one chooses local coordinates (xµ) for the curve the geodesic equation is just

k k µ ν x¨ + Γµνx˙ x˙ = 0. And thus we "just" have to solve this differential equation for finding the geodesics. This is the usual definition of geodesics, but we have to keep in mind that it is intimately related to the Levi-Civita connection. To see another view on geodesics we briefly recall the Lagrangian formalism for understanding dynamics. This formalism starts with a given smooth function L : TM → R, and this is the function that captures the dynamics of the system. The dynamics of the associated are generated by imposing that they are extremals of a certain functional, thus imposing a calculus of variations problem. The associated functional is

Z t1 I[x] = L (x, x˙)dt. t0 We thus impose the Hamilton Principle, namely that δI = 0. This gives then rise to a description of the dynamics of the system, and it is equivalent to the well known Euler-Lagrange equations. This equations are imposed for every coordinate (xi): d ∂L ∂L = . dt ∂x˙i ∂xi Thus giving rise to the differential equations that govern the dynamics of the Lagrangian system. The fundamental observation here now is that

Geodesics can be understood under the Lagrangian formalism, they are the dynamics of a concrete Lagrangian function on the manifold.

The concrete Lagrangian that can be seen to generate the dynamics of geodesics in a pseudo-Riemannian manifold is

1 L (q, v) = g (v, v). 2 q One thus deﬁnes the Energy functional for every curve α : I → M as follows: Z E[α] = L (α, α˙ )dt.

Computing the variations for the action we get thus the dynamics of a certain "physical" system, one can see that they are in fact geodesics of the manifold (we will show this later in the Hamiltonian version of this fact, since it will further allow us to see the deep geometric structure behind all this). The physical interpretation one gets from this is that geodesics are "free falling particles" for the physical system deﬁned by the metric, since they have a pure kinetic term, there is no potential in the Lagrangian. And this connects heavily with mechanics, since now we can understand geodesics as those objects that are freely falling, and that is the information that the metric gives about the physics, we don’t need the Levi-Civita connection for understanding those trajectories now. We can thus use the connection to carry other relevant physical/geometrical information (the electromagnetic ﬁeld as we will see later). And with all this in mind, we want to emphasize what Élie Cartan noted to Einstein in General Relativity:

There is no special reason to choose the Levi-Civita connection as the preferred one.

Cartan’s idea was to mantain one of the conditions of the metric, namely that ∇g = 0, but to relax the condition on Tor(∇). The power of this idea is that it allows the torsion of the connection to carry the information about the intrinsic angular momentum (spin) of matter. The spin of matter in curved space-time requires that torsion is not restricted to zero but rather variable. Cartan’s view was essentially to treat as separate objects the metric tensor g and the connection ∇, since there is no reason or argument to why it should be zero apart from the fact that it guarantees the uniqueness of it. But to understand the physics of the universe we don’t need this uniqueness, in fact we may need non-uniqueness to actually view it as a quantity that is varying in the space of connections/fields (we will talk about that in the last section of the work). Einstein appreciated after a while Cartan’s deep geometrical appreciation, and became a proponent of the theory. But there is an important thing to mention here, this geometrically beautiful idea proposed by Cartan is not usually taken into account for practical computations done by physicists, although it is indispensable for theoretical understanding. The reason for this is that torsion seems to add little predictive benefits but at the same time produces less tractable equations, and hence finding the solutions becomes harder. But nevertheless it is the general theory of which classical torsion-free General Relativity is just a special case. We will end this section with just a definition, a definition that at first sight seems not to imply a lot, but that has in itself a very deep structure and that is essential to understand the role of symmetry in modern geometry and physics, the definition just tries to make two very known structures in mathematics play together.

Deﬁnition 0.1.2.1. A Lie Group G is a smooth manifold together with a group structure (such that the operations of product and inverse in the group are smooth).

So here we have it, a group that is a smooth manifold. In this short definition we have the basis for the comprehension of symmetry in differential geometry and the physical universe. We will not develop the theory of Lie Groups in this work, the only thing that we will mention is that there is a relationship between the notion of a Lie Group and the notion of a Lie Alge- bra. The relationship is that one can introduce a Lie-Algebra structure on the tangent space at the identity of the group (TeG) inherited from the Lie-bracket in a natural way, we refer the reader to any reference on differential geometry or Lie Groups for a detailed development of the theory. We now devote the following sections to further introduce necessary notions coming from differential geometry that we will indispensable for the later discussion.

0.2 Élie Cartan and the revolutionarization of classical differential geometry

0.2.1 Differential forms and exterior derivative

Now we turn to the concrete study of a fundamental object of differential geometry and needed for the following exposition. These objects are called Differential forms. They will be used in a variety of settings since they are useful to carry certain kinds of information about the geometry/dynamics/- physics of the system, and they encode deep topological information about the manifold. In the exposition we will try to give a practical idea of why they are useful, and show some of the ideas that led Élie Cartan to new perspectives in Geometry and Physics in later sections. We will try to give an overview and many technical lemmas will not be proved, the reader can check any book on differential geometry as reference. We will proof just those that are relevant for the exposition (if the proof introduces interesting concepts or ideas), and will give some examples of how they work. The fundamental object of study in this section, differential forms, can just be interpreted in the Vector Bundle framework as sections of the bundle POSAR WEDGE I TAL. We will later give a more intuitive way to think about them, but they are just, for now, antisymmetric co-vector ﬁelds of arbitrary rank. The exterior derivative is a fundamental notion of differentiation we en- counter in manifolds, but not the only one (as we have already seen in the ﬁrst section). In differential geometry one has various notions of what does it mean to take a derivative, and one has to be very careful with their difference and mutual relationship (since they are different objects usually), the problem is that in standard calculus on euclidean space the difference is very subtle and one usually doesn’t elaborate on this since it is usually unnecessary. In cartesian coordinates in euclidean space one can just do the computations correctly if one is careful with the rules, but for a better theoretical understanding and to be able to generalize those ideas to arbitrary manifolds it is indispensable to know the precise and concrete structures and objects we are dealing with.We have the following main notions:

1. Exterior derivative on differential forms (the one we introduce in this section). The idea of using this operation and exploiting it in concrete computations was mainly due to Élie Cartan. This notion applies to differential forms and is very algebraic, it doesn’t need further structure on the manifold, and allows a generalization of the classical calculus on R3 via differential forms. As a simple visual image, one can picture the exterior derivative (of a function) as nothing more than the gradient, in a certain sense (since we need to be very careful with the fact that vectors and co-vectors are different beasts and can not be treated in an equal footing, which is what is usually done in classical R3 calculus.

2. The Lie derivative (named after Sophus Lie), that we will introduce in the next section, is the "natural" generalization of the concept of derivative to arbitrary manifolds. In the definition of this derivative is one place where one starts to see that to be able to generalize classical calculus concepts to calculus on manifolds one has to be very careful and precise, since for instance one has to take care of the different tangent spaces at points on the manifold (something that doesn’s happen in Rn since there we just identify canonically all tangent spaces). This notion of derivative applies not only to functions and vector fields, it can be applied to every tensor field, to see how that varies along the flow of a certain vector field. We will elaborate further on this notion in the following section.

3. The introduction of a connection. This notion of differentiation requires that we introduce more structure into our manifold, an object called a connection (and how one chooses this is not trivial and can help us in our concrete problem). The classical approach to this is to get the Levi- Civita connection from the metric on the manifold, and start working from there. The important thing to realize is that it is not mandatory that we do that, and this ideas led Élie Cartan and Charles Ehresmann to a deeper understanding of what a connection means and in what settings do we actually get a notion of derivative out of it (more on this later).

We now turn to the study of the ﬁrst, differential forms. Marsden in [?] gives the following motto for differential forms:

The main idea of differential forms is to provide a generalization of the basic operations of vector calculus (div,grad,curl), and the integral theorems of Green, Gauss, and Stokes to manifolds of arbitrary dimension

And further than that, the important geometrical idea here is that they are coordinate independent. Their properties will not depend on our coordinate system and patch, since they are tensorial objects (the reader should check the Appendix on tensors for further explanation on this idea if the importance of this in physics and geometry is not already clear). We take a fundamental idea from one of the fundamental references in General Relativity [?].

All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics.

Now we go to talk about what is a differential form? (again) and very importantly why do we need them?. We will try to go from a very naive description (usually using local coordinates needed for concrete computations) and develop it into a more formal description involving exterior powers of vector bundles (the deﬁnition already given). We start with the deﬁnition given by Marsden in [?]:

Deﬁnition 0.2.1.1. A k-form on a manifold M, α ∈ Ωk(M) (we denote by k( ) ( ) k → R Ω M the space of k-forms) is a function α m : ×i=1 Tm M , assigning to each point m ∈ M a skew-symmetric k-multilinear (linear in each factor) map on the tangent space Tm M to M at m (in a smooth way).

We now will work in local coordinates to better understand what do these objects represent and what they are. The price that we will play is that all along this description will be dependent on the finite dimensionality of the manifold, but we will give some ideas on how can we adapt some ideas to generalize the concepts to infinite-dimensional situations. Bear in mind that in the majority of cases we will not deal with the (analytical) complications arising from infinite-dimensional systems, and that sometimes just the formal derivation of intuitions for infinite-dimensional manifolds is preferred. Let (xµ) be local coordinates on a certain patch U ⊂ M. We know that ∂ this induces basis vectors on each tangent space Tm M, { ∂xµ }, and that it also ∗ µ induces a basis for the cotangent space on each Tm M, {dx }, which is the dual basis of the previous one. We will make an abuse of notation that happens a lot in the mathematical literature and even more in the physical literature, we ∂ will denote the basis of tangent vectors by ∂µ = ∂xµ . Now we can locally expand in coordinates our geometrical objects (differential forms in this section) using this basis, this is a very fundamental idea and usually it is not emphasized a lot, there is a big difference between the global object and the local description in coordinates). And one has to cope with both perspectives at the same time, the abstraction of the deep structure of the object and his concrete representation. As an example we can write in local coordinates a 2-form ω ∈ Ω2(M) as:

i j i j ωm(v, w) = ωm(v ∂i, w ∂j) = v w ωm(∂i, ∂j) just by using the bilinearity and the local basis. Now we consider a family of functions: ωij(m) = ωm(∂i, ∂j) and conclude that locally (in the coordinate patch) we can represent our 2 global 2-form ω ∈ Ω (M) by a family of functions ωij : U → R that depend on the coordinate patch (and that one has to know how to transform in between coordinate systems, more on that later). More generally, following a similar procedure, a k-form α ∈ Ω2(M) can be written locally as:

( ) = ( ) i1 ··· ik αm v1,..., vk αi1...ik m v1 vk , recall that we are using the summation convention, so the sum is on i1,..., ik One of the fundamental operations one can do on differential forms is just the wedge product. We talk about this notion for antisymmetric tensors in the appendix (and the reader not familiarized with the concept should look at that part of the appendix). With this operation in hand we just call the wedge product on differential forms the operation of applying the wedge product (of antisymmetric tensors) at every point. Pull-back and Push-forward: These are operations one defines to be able to compare differential forms (and arbitrary tensor fields, but one has to be careful) and to transport them to other spaces. It works in the following way, consider a smooth function f : M → N, we thus define the pullback of a differential form ω ∈ Ωn(N) as a differential n-form on M, that we denote by f ∗ω ∈ Ωn(M)

∗ n f ω ∈ Ω (M)(v1, ..., vn : = ω(d f (v1, ..., d f (vn), this operation thus deﬁnes a map

f ∗ : Ωn(N) → Ωn(M).

Note that we cal also pullback functions by using f ∗φ = φ ◦ f . This map on differential forms satisﬁes some essential properties, that can be found in any standard reference on differential geometry (there is another fundamental property relating to the exterior derivative, that we will later comment on):

∗ ∗ ∗ f (ω1 ∧ ω2) = f (ω1) ∧ f (ω2)

Thus we are able of transporting differential forms (co-vector fields in general) from one manifold to another (or to the same to see if it has changed, as we will later do with the notion of the Lie derivative). Note that we could have done the same definition for arbitrary co-tensor fields, and it would work in a similar fashion. The neat fact comes if the function f is a diffeomorphism, since then we can invert it locally (the differential map is invertible) and hence transport tensor fields of all type by cleverly using the differential map and its inverse, the operation that transports tensor fields from M to N is usually called the push-forward. One important remark to look at what happens with the structure of a Bundle. In a bundle we have a pair of manifolds P and M and a map π : P → M, the essential observation here is that we can use the pull-back operation (π∗) to transport the co-vector tensor fields that we have in the base manifold to the total space! And this applies to metrics, differential forms, and other objects that are essential in both geometry and physics. To further investigate the notion of differential forms, we have to introduce the notion of the exterior derivative, the second notion of "derivation" that we see in this work, and it works in a rather different way compared to a connection. This exterior derivative is an operation on differential forms (usually called d) in the following way:

d : Ωn(M) → Ωn+1(M).

So it takes n-forms to n + 1-forms, in any reference in differential geometry that there is a unique object (the exterior derivative) that satisﬁes the following properties:

• If f is a smooth function, then d f ∈ Ω1(M) is just the differential of f .

• d is a linear operation.

• It satisﬁes the product rule, that is just:

d(α ∧ β) = dα ∧ β + (−1)kα ∧ dβ,

for every β ∈ Ω1(M) and every α ∈ Ωk(M). • d2 = 0, meaning that for every α ∈ Ωk(M), d(dα) = 0.

• d is a local operator, this means a fundamental thing, that dα(m) depends only on α restricted to any local neighborhood of m (we use only inﬁnitesimal information for the exterior derivative, although it is a global algebraic object). If U ⊂ M is an open set we require that

d(α|U) = (dα)|U,

where α|U just means the restriction of the form to this open set.

Knowing this, we call d the unique operation that satisﬁes this properties, the exterior derivative. And although it seems like a rather abstract object actually its computation in local coordinates is fairly simple and just involves derivatives from ordinary calculus. If α ∈ Ωk(M) is written in local coordinates as

= i1 ∧ · · · ∧ ik α αi1...ik dx dx , then the exterior derivative in this coordinates is just

= j ∧ i1 ∧ · · · ∧ ik dα ∂jαi1...ik dx dx dx . The fundamental property that relates this exterior derivative operation with the pull-back operation previously deﬁned is that if we consider α ∈ Ωk(N) and a smooth map f : M → N, we get the following equality:

d( f ∗α) = f ∗(dα).

VECTOR CALCULUS AND DIFFERENTIAL FORMS HODGE STAR Another fundamental operation one can define is the operator known as the Hodge star ?. To define this concept we firt need the notion of a volume form. First note that, as we said before, via pullback we can take for example the metric and extend it to forms. Suppose that X is an oriented manifold with dimX = n, and consider thus that metric extended to arbitrary forms, there is a unique n-form, called dx or usually dnx (in the physics literature) that satisfies the following to properties:

1. dx is positively oriented.

2. < dx, dx >= +1 or < dx, dx >= −1. this form is usually called the Riemman volume element form. With this concept in our hands we can deﬁne the notion of the Hodge star,

Deﬁnition 0.2.1.2. The Hodge Operator is the linear map ? : Ωk(X) → Ωn−k(X) that satisﬁes θ ∧ θ0 =< θ, θ0 > dx, for every pair of forms θ, θ0 ∈ Ωk(X). This operator allows us to view classical euclidean R3 with the glasses of modern differential geometry. First we will look at what does this operator do on the standard forms dx, dy, dzΩ1(R3) that one has in euclidean space:

1. ?1 = dx ∧ dy ∧ dz

2. ?dx = dy ∧ dz, ?dy = −dx ∧ dz, ?dz = dz ∧ dy, ?(dy ∧ dz) = dx, ?(dx ∧ dz) = −dy, ?(dx ∧ dy) = dz

3. ?(dx ∧ dy ∧ dz) = 1

And one can put into this modern framework the whole of euclidean calculus (we will not do that since we are not specially interested in those identities in this work, one can see a further development of this in [14]). One can deﬁne a theory of integration on differential forms, since this is done in almost every book on differential geometry, we will skip this process, just remind that locally we are just doing standard integrals from calculus. The most remarkable result one gets here is Stoke’s theorem, which says that if we consider an orientable manifold S we get the following equality of integrals for any form ω: Z Z ω = dω. ∂S S This properties of generalizing classical ideas from a modern framework is one of the main reasons why differential forms are so important in modern differential geometry. To end this section we will brieﬂy introduce the notion of the de Rahm Cohomology, that shows us that differential forms carry deep topological information about the manifold. The whole notion is based on the fact that d is a linear operation and that d2 = 0. This generates the following complex:

0 → Ω0(M) → Ω1(M) → Ω2(M) → Ω3(M) → · · ·

From this and using that d2 = 0 one knows that exact forms are closed. The important fact here is that the converse may not be the case, and this is what the de-Rahm cohomology tries to capture. The basic idea is to try to classify the different types of closed forms in a manifold. One says that two closed forms α, β ∈ Ωk(M) are cohomologous if they differ by an exact form, this meaning that α − β is an exact form. This induces an equivalence relation on the space of closed forms, and one can thus make a quotient that is called k the k-th de-Rahm cohomology group, and denoted by HdR(M). The fundamental result here is what’s called de Rahm’s theorem, which says that this are exactly the same groups as one would get if one considered another concept of cohomology (based on the topology of the manifold) called singular cohomology. What this means is that the relation between exact and closed differential forms depends on the topology of the manifold!. 0.2.2 Lie derivative in smooth manifolds

In a certain smooth manifold M we have a notion of differentiation that just uses the structure given, called the Lie derivative. This notion of taking a derivative in a smooth manifold is just the generalization of the classical definition on Rn. The biggest problem one faces when trying to generalize is that in Rn we have a way of canonically identifying all tangent spaces (via a translation, since Rn is an affine space). This fact, that is usually not talked about when one studies this things for the first time, needs to be thought of again when one is trying to adapt the concept to an arbitrary manifold. We will now do this process step by step to actually understand that the definition of the Lie derivative is not arbitrary and that comes from modifying adequately the Rn definition to deal with different tangent spaces. One thus defines the notion of Lie derivative of a form (or in general a tensor field) as follows (using the pull-back to appropiately treat different tangent spaces)

1 ∗ d ∗ L α = lim → [φ α − alpha] = φ α| = X t 0 t t dt t t 0 . This operation has a lot of properties that we encourage the interested reader to review in any book on differential geometry, we will pick a few of them:

d ∗ ∗ 1. dt ϕt α = ϕt LXα- 2. If f is a smooth real-valued function on the manifold we get:

LX f = X( f ) = d f · X.

3. For two vector ﬁelds X, Y, the Lie derivative is the same as the Lie bracket, in the sense that

LXY = [X, Y].

Killing vector/tensor ﬁelds

Now that we have introduced the notion of the Lie derivative we have the right structure to define the notion of Killing vector fields, named after Wilhelm Killing, a German mathematician. This concept has a deep relationship with symmetry, they are a possible language in which to talk about symmetry in a manifold (that is used a lot in General Relativity). What Killing vector fields try to study are certain vector fields X ∈ Γ(TM) that preserve a certain structure, usually a tensorial object, since tensorial objects are the ones one is usually interested in differential geometry and physics since they are the objects that allow us to describe systems in a coordinate- free fashion, indispensable for a correct comprehension of the physical world. Depending on the concrete objects one has, vector fields will be of a certain kind and have special properties. Knowledge of the existence as well as a description of these vector fields for certain geometrical/physical model is indispensable for a comprehension of the dynamics in those spaces. The main interest that physics has in these objects comes from General Relativity, since these objects allow us to study better certain solutions of the Einstein equations. And sometimes imposing a priori symmetries (or finding them) allows one to solve the equations in certain cases, and many solutions of the Einstein equations come from studying symmetries like this. The main interest in Killing vector fields in classical differential geometry is in associating them to the concept of isometries in a Riemannian or pseudo- Riemannian manifold (M, g). So what we want to study in this concrete case are vector fields that preserve the metric tensor. That the vector field preserves the metric tensor just means that along the flow of the field (time evolution of the system), the metric tensor doesn’t vary. So we will generally use the following definition:

Definition 0.2.2.1. Let X ∈ Γ(TM) be a smooth vector field, we say it is a Killing vector field for the metric tensor g if it satisfies:

LX g = 0.

Note that this condition on the vector field has direct consequences on the flow of the Killing vector field, that we denote by ϕt. We will look now at this flow as a collection of diffeomorphisms for every possible time. The Killing condition just means that the flow satisfies:

∗ ϕt g = g, recall that this is nothing more than saying that the vector field generates a flow of isometries. Now we will see that in fact Killing vector fields form a Lie subalgebra of the algebra of vector fields. We see this by just using a property of the Lie derivative and using the Killing condition. Recall a property satisfied by the Lie derivative, namely that

L[X,Y] = [LX, LY] = LXLY − LYLX

(Corollary 3.1.13 in [1]) With this property in hand now we just use the property to obtain:

L[X,Y]g = LXLY g − LYLX g = 0, the last equality obtained due to the Killing condition. We have thus found that Killing vector fields form a Lie-subalgebra of the Lie Algebra of vector fields Γ(TM). And this is one of the first places where we start to appreciate the deep relationship between symmetries and Lie Algebras, that we will further explore in following sections. Those Lie Algebras are essential for understanding symmetry in physics! Cartan’s magic formula

In general computing Lie derivatives of arbitrary tensor ﬁelds is not an easy task, the purpose of this section is to show that for differential forms we have an easier way of computing it that reduces the computation to the previous notion of exterior differentiation. The purpose is to show a way of computing (which will be very useful later) LXω, for a general n-form ω ∈ Ωn(M) and an arbitrary vector ﬁeld X ∈ Γ(TM) due to Élie Cartan (and in fact it is called Cartan’s magic formula due to its discoverer).

Proposition 2. Consider an arbitrary vector ﬁeld X ∈ Γ(TM) and a differential form ω ∈ Ωn(M), we then have the following equality:

LXω = iXdω + d(iXω), interpreting them as operators the formula can also be written as:

LX = iX ◦ d + d ◦ iX

Proof of this fact can be found in [1] and in any standard reference for differential geometry, we will use it mainly as a computational tool and to show results in the next section.

0.3 Hamiltonian mechanics and Symplectic geometry

We have already seen how Lagrangian mechanics work (briefly in the introduction), we will now describe how Hamiltonian mechanics work, emphasizing the role that the different geometric structures plays in this description of classical mechanics. We will start with the abstract geometric framework where Hamiltonian mechanics takes place (being the fundamental definition of a whole area in mathematics called symplectic geometry). With this explained the next step will be to see how this geometric idea is, in a way, a generalization of the structure one has in cotangent bundles over a manifold (that we will later explain how they work, in contrast to tangent bundles explained before). We will put special emphasis in the fact that this cotangent bundle will come equipped with a natural geometric structure (namely a 2-form ω ∈ Ω2(M)). So let’s start with the definition that will be fundamental in this work and start seeing some of its properties.

Deﬁnition 0.3.0.1. A symplectic structure on a smooth manifold M is given by a 2-form ω ∈ Ω2(M) satisfying the following properties:

1. ω is a closed 2-form (i.e. dω = 0).

2. ω is non-degenerate (i.e. for every p ∈ M, ωp : Tp M × Tp M → R is non-degenerate in the sense that if ∀Y ∈ Tp M, ω(X, Y) = 0 ⇒ X = 0). The pair (M, ω) is called a symplectic manifold.

Associated to this deﬁnition we naturally give, as with every structure in mathematics, the morphism that preserves it, thus we deﬁne:

Deﬁnition 0.3.0.2. If we have two symplectic manifolds (M, ω), (N, ρ), then a smooth map φ : M → N satisfying φ∗(ρ) = ω is called a symplectic map. If, in addition we have that φ is a diffeomorphism, we call φ a symplectomorphism. We will denote by Symp(M) = {φ : M → M|φ symplectomorphism }, the set of all symplectomorphisms of the manifold.

One of the fundamental examples of a symplectic manifold is the cotangent bundle (T∗Q) of a manifold Q (and in fact it is the example that motivates the abstract deﬁnition). So we will start by analyzing how this example works to get some intuition on our object of study in this section, symplectic manifolds.

0.3.1 Cotangent Bundle of a smooth manifold In this section we will recall from before the notion of the cotangent bundle, and after that we will see that it comes equipped with a natural symplectic structure. In this section we usually think of the base manifold as the position variables while the other variables will be thought of as momentum variables, and we will try to follow the standard notation (coming from physics) of call- ing position variables q and momentum variables p.

Deﬁnition 0.3.1.1. Given a smooth manifold Q we deﬁned its cotangent bun- ∗ F ∗ dle as the set T Q = q∈Q Tq Q (the disjoint union of all cotangent spaces at every point in the manifold).

It is a well known fact that this is also a smooth manifold (in fact a vector bundle over Q) of dimension 2n (if we have that dimQ = n), and hence we will not reproduce the proof (what we need to do is use all the structure in Q and lift it to T∗Q). We also have a natural projection π : T∗Q → Q deﬁned on the bundle, ∈ ∗ ( ) ∈ ∈ ∗ assigning to a point z T Q the point π z Q such that z Tπ(z)Q (in an analogous way to the projection on the tangent bundle). It is useful to note that the coordinates of this new manifold are nothing more than the coordinates we already have in Q, (q1, ..., qn) together with the associated basis of 1-forms (dq1, ..., dqn). In this respective coordinates we label our points using the classical notation coming from physics (position and ∗ 1 n i momentum), namely for a point z ∈ T Q, z = (q , ..., q , p1, ..., pn) = (q , pj) in a certain chart in a neighborhood U of z. This bundle comes equipped with a natural geometric structure (a 1-form θ ∈ Ω1(T∗Q)) that has multiple names: Liouville one-form, Poincaré one-form, Canonical one-form, or symplectic potential. We will thus give a construction of this form (but from a global perspective, using the bundle structure, just later we will see how it works in local coordinates). We thus start by considering the projection map from the bundle structure,

π : T∗Q → Q, this induces a tangent map

dπ : T(T∗Q) → TQ.

Now consider a point z ∈ T∗Q, since it is a point in the tangent bundle, we can understand this point as a linear map

z : Tπ(z)Q → R.

We now deﬁne the 1-form we wanted at the point z in the following way

θz = z ◦ dπ.

Notice how all this structures just link together in a natural way! Hence we can view θz as a linear map in the following way:

∗ θz : Tz(T Q) → R, and hence we have constructed a 1-form in the cotangent bundle in a natural way from a global perspective. Having this form one can naturally consider

ω = dθ.

And we naturally obtain a 2-form ω in the cotangent bundle. If one considers them in local coordinates with respect to the standard i coordinates (q , pj) one just gets the usual expressions

i θ = ∑ pidq , i and i ω = dθ = ∑ dpi ∧ dq . i This construction in the cotangent bundle is essential for a pair of reasons, since it is the natural geometric setting of many mechanical problems (in Hamiltonian form) and because it allows us to generalize this geometrical setting to an arbitrary manifold (not necessarily the cotangent bundle).

0.3.2 Symplectic and Hamiltonian vector fields The purpose of the section is to introduce the fundamental concepts of Symplectic vector fields, Hamiltonian vector fields, their structure (they both form a Lie subalgebra of (Γ(TM), [·, ·]), the Lie algebra of vector fields with the Lie Bracket), and how their difference relates to the first de Rahm cohomology 1 group of the manifold (i.e. HdR(M)). In all what follows we will consider (M, ω) a symplectic manifold. Definition 0.3.2.1. A vector field X ∈ Γ(TM) is called a symplectic vector field if it preserves the symplectic structure (any of the following two equivalent conditions):

1. LXω = 0. ∗ 2. ϕt ω = ω for all t (where ϕ is the flow associated to X). We will denote by Symp(M) the set of all symplectic vector fields. What we want to see now is that in fact Symp(M) (with the standard lie bracket operation on vector fields) forms a Lie subalgebra of the Lie algebra of vector fields. To see this we just need to see that the Lie bracket of two symplectic vector fields is again symplectic. Proposition 3. Let X, Y ∈ Symp(M),then[X, Y] = XY − YX ∈ Symp(M). Proof. First we start by recalling a property satisfied by the Lie derivative, namely that L[X,Y] = [LX, LY] = LXLY − LYLX (Corollary 3.1.13 in [1]) And now we just compute the Lie derivative using the property above: L[X,Y]ω = LXLYω − LYLXω = 0, since both X and Y are symplectic vector fields (LXω = LYω = 0). Hence we have just seen that Symp(M) is a Lie algebra in itself. Now we define the other fundamental concept of the section, hamiltonian vector fields. Definition 0.3.2.2. A vector field X ∈ Γ(TM) is called a hamiltonian if there exists a function f ∈ C∞(M) such that:

d f = iXω We will denote by Ham(M) the set of all hamiltonian vector fields. And ∞ denote by X f the vector field associated to the function f ∈ C (M). Just as before, Ham(M) also forms a Lie algebra. The best way to see this is via the Poisson Bracket structure that we will introduce later, just to mention briefly, this will be a Lie algebra operation on the space of smooth functions denoted by {·, ·}. This bracket will turn out to satisfy a certain property in relation to the Lie bracket, for two Hamiltonian vector fields XF, XG we will have that [XF, XG] = −X{{F, G}}. Although we still don’t know what this bracket operation exactly is, what this tells us is that there will be a certain function (computed with the bracket) that will make the Lie bracket of two Hamiltonian vector fields be Hamiltonian, and hence Ham(M) is a Lie subalgebra of the algebra of vector fields. Seeing this via this Poisson Bracket is in our view the more natural, since we start from the very start seeing the importance that this structure will have in our description of Hamiltonian Systems. Now we proof a standard fact on their mutual relationship, namely that: Proposition 4. If X is a hamiltonian vector field then it preserves the symplectic structure (i.e. Ham(M) ⊂ Symp(M)).

Proof. Since our vector ﬁeld X is hamiltonian, we have a function f ∈ C∞(M) such that d f = iXω. Now we just compute the Lie derivative of the symplectic form using Cartan’s magic formula:

2 LXω = iXdω + diXω = diXω = d f = 0 where we have used that dω = 0 and that d2 = 0.

Now we start to know some things about the relation between both concepts, let’s explore that further, viewing the property of being symplectic as a property of a certain differential form (similar to the hamiltonian one). After that we will start to see that in fact it is no more than a topological (in the sense of de-Rahm) condition.

Proposition 5. A vector ﬁeld X is symplectic if and only if its associated 1-form (iXω) is closed. X ∈ Symp(M) ⇔ d(iXω) = 0

Proof. Just use, as before, Cartan’s magic formula to obtain:

LXω = iXdω + diXω = diXω, so we clearly see that both things are equivalent.

So we see actually that symplectic vector fields induce closed forms, and we saw before that hamiltonian vector fields induce exact forms, this reminds us heavily of the de-Rahm cohomology. So naively we see that there is a strong relationship between the difference of hamiltonian and symplectic vector fields and a certain cohomology group, we have in fact seen the following:

Proposition 6. (Proposition 2.3.3 in [3]) For a manifold M, and recalling that in our work ω is always non-degenerate, we get the following exact sequence:

1 0 −→ Ham(M) −→ Symp(M) −→ HdR(M) −→ 0

Remark. We made emphasis in the fact that ω is non-degenerate since if that doesn’t happen the proposition doesn’t hold in general. Although to our object of study (symplectic manifolds) that is not important since we suppose we are already working with a non-degenerate form, one has to be careful if one wants to study the general theory of what are usually called Dirac manifolds. The general theory for that kind of spaces can be found in [2]. Thus we have an explicit characterization of the relationship between Ham(M) and Symp(M) relating to the topology of our manifold, note that what we have said implies that locally every symplectic vector ﬁeld is hamiltonian 1 (and globally if HdR(M) = 0. 0.3.3 Poincare’s lemma Now we will try to give some of the deep ideas that are behind the relationship between exact and closed differential forms (we have already seen in the case of 1-forms how this works in the previous section) by proving Poincare’s lemma, which basically says that every closed n+1-form can, at least locally, be computed as the differential of a certain n-form. In a sense we see with this section a bit why topology is so important in modern theoretical physics. What we learn (just as before) is that it is the de-Rahm cohomology of the manifold (and hence its topology since it is a topological invariant) what determines to what extent can we derive a closed n+1-form from a n-form. And this can seem like a very theoretical observation, but is really important and useful, and once we proof the lemma we will try to emphasize the importance of cohomology in physics, for instance in Electro-Magnetism.

Theorem 0.3.3.1. (Poincaré’s lemma). A closed form is locally exact, meaning that if a differential form α ∈ Ωn+1(M) such that dα = 0, then there is a neighborhood about each point where we can ﬁnd (locally) a form β such that α = dβ.

Proof. Since this is a local result, we will ﬁrst study what happens in euclidean space. Let U be an open ball around the origin in Rn and α ∈ Ωk(U) such that dα = 0. In local coordinates we write

= j1 ∧ · · · ∧ jk α αj1...jk dx dx . Now we will consider the following differential form

Z 1 ( 1 n) = ( k−1 ( 1 n) j) i1 ∧ · · · ∧ ik ) β x , ..., x t alphaji1...ik tx , ..., tx x dx dx , 0 where the sum is over i1 < ... < ik. This is just, in a sense, the inverse operation of the exterior derivative, and thus by direct computation on sees that α = dβ. From this local result and using the fact that pullback commutes with exterior differentiation, we get the local result in an arbitrary manifold.

With this in hand we see that we can always derive a differential n + 1- form from a differential n-form, at least locally. Globally we have to know the topology of the manifold to know if this is true or not, since the information is encoded in the notion of the de-Rahm Cohomology that we introduced before. But recall a thing, that physics always takes place locally, and hence for physicists it is very important and natural to impose the condition that we can ﬁnd a potential that describes the whole ﬁeld (this is done for example in the description of electromagnetism). And the idea of deriving the fundamental object of a theory from a Potential is essential in a lot of physical theories.

0.3.4 Darboux’s theorem In this section we will show one of the fundamental theorems of symplectic geometry, due to Darboux in [4]. Although the result can be generalized to inﬁnite-dimensional cases and correctly treated there, we will be concerned with the ﬁnite-dimensional case (although we will give a proof that can be adapted easily, due to Moser in [6]). The result naively says that "symplectic manifolds are locally the same", namely that two symplectic manifolds of the same dimension are locally symplectomorphic. Another way to look at this is noting that every symplectic manifold (M2n, ω) is locally symplectomorphic to the "trivial" model of a symplectic 2n n i manifold, (R , ω0), with ω0 = ∑i=1 dq ∧ dpi (where we take the usual coor- i 2n dinates of position/momentum (q , pj) ∈ R ). What this actually means is that for every point p ∈ M, we take an open neighborhood p ∈ U ⊂ M and a diffeomorphism

( ) → (R2n ) ψ : U, ω|U , ω0 ∗ = such that ψ ω0 ω|U (i.e. ψ is what we call a symplectomorphism). This proof is partially taken from [?] where he reproduces the proof by Moser.

Theorem 0.3.4.1. (Darboux’s theorem.)Let (M2n, ω) be a symplectic manifold of dimension 2n. For any point p ∈ M, there is a local coordinate chart in which ω is constant.

Proof. We assume that our manifold M = E and z = 0 ∈ E (if not we just do a translation), where E is a Banach space (in our case, in ﬁnite dimensions we just consider E a ﬁnite dimensional vector space, just isomorphic to R2n. Take ω1 the constant form such that ω1 = ω(z) = ω(0). Now we consider a continuous deformation of the symplectic form in the following way:

0 0 ω = ω1 − ω, ωt = ω + tω , 0 ≤ t ≤ 1

Note that the starting point of the deformation ω0 is just ω (our original 2- 0 form in our symplectic manifold) and the endpoint is ω1 = ω + ω = ω + (ω(0) − ω) = ω1 (hence this is why we defined ω1 = ω(0)). Now we have that for each 0 ≤ t ≤ 1, the bilinear form ωt(0) = ω(0) is non-degenerate. So, by openness of the set of linear isomorphisms E → E∗ (since they come from the condition det 6= 0, which is an open condition), and compactness of [0, 1], there is a neighborhood of 0 ∈ E where ωt is non- degenerate for every 0 ≤ t ≤ 1. We assume that this neighborhood is a ball. By using the Poincaré lemma, ω0 = dα for a certain α ∈ Ω1(M). We can further suppose that α(0) = 0 (replacing α by α − α(0) if necessary). We thus define a time-dependent vector field Xt by

iXt ωt = −α, which we can do since we have seen that ωt is non-degenerate. Since α(0) = 0, we automatically get that Xt(0) = 0, and from the local existence theory of ODE’s we know there is a ball on which the integral curves of Xt are deﬁned for a time at least one (technical theorem can be found in [?]). Consider Ft to be the ﬂow of Xt starting at F0 = Id. Now we just compute the Lie derivative: d d (F∗ω ) = F∗(L ω ) + F∗ ω dt t t t Xt t t dt t ∗ ∗ 0 ∗ 0 Ft diXt ωt + Ft ω = Ft (d(−α) + ω ) = 0 From this computation we deduce that ∗ ∗ F1 ω1 = F0 ω0 = ω, and hence F1 provides us with a chart transforming ω to the constant form ω1. And thus is we see this in coordinates, we have proven the existence of 1 n local coordinates (q , ..., q , p1, ..., pn) such that ω = ∑ dpi ∧ dqi, i using the fact that any constant symplectic form can be transformed into this canonical form.

Associated to the notion of a symplectic form ω we have the notion of the Liuville volume. This volume form Λ gets deﬁned as

(−1)n(n−1)/2 Λ = ω ∧ · · · ∧ ω n! . Applying Darboux’s theorem this volume form can always be written locally 1 n in the canonical coordinates (q , ..., q , p1, ..., pn) in the following way: 1 n Λ = dq ∧ · · · ∧ dq ∧ dp1 ∧ · · · ∧ dpn. The fundamental observation is that this quantity is purely defined in terms of the symplectic form ω. And hence (due to the property of the pullback with respect to the wedge product) this volume form Λ will always be preserved whenever ω is preserved (thus for all symplectic vector fields!). Now we will apply this ideas on Hamiltonian systems to geodesics: In the introduction we mentioned that it is possible to give an interpretation on geodesics that can be independent of the fact that one chooses or not the Levi-Civita connection, and that understanding geodesics from the physics viewpoint can enrich this perspective and allow us to encode information on the geometry/physics on the connection we decide to use. Now we will interpret geodesics in the Hamiltonian formalism. We have that geodesics, via de Levi-Civita connection are defined by the equation

k k µ ν x¨ + Γµνx˙ x˙ . Now what we will see is that this geodesics can also be understood as the dynamics generated by the Hamiltonian function 1 H = gij p p . 2 i j Thus consider we are in a manifold M that has a metric tensor g, recall that the notion of geodesic is purely local, and hence we also need to show it ∗ i in the local setting. Consider the cotangent bundle T M coordinates (q , pj) such that the 2-form ω has the canonical form (we know we can do this from previous results). Consider now a curve in the cotangent bundle

γ(q(t), p(t)), and consider the previous Hamiltonian as the function determining the dynamics, hence since we are in coordinates where the symplectic form takes the canonical form we have the standard Hamilton equations:

k ki q˙ = ∂pk H = g pi 1 ij p˙ = −∂ k = − ∂ k (g )p p k q 2 q i j i If we use this equations together with the fact that pk = gikq˙ (this viewed physically is just the usual interpretation of momentum as mass × velocity), we get the following:

j i 1 ij µ ν p˙ = ∂ i (g )q˙ q˙ = − − ∂ k (g )g g x˙ x˙ k q jk 2 q iµ jν ij Now we use the fact that (index manipulation) −∂qk (g )giµgjν = ∂qk gµν, and thus the previous equation turns into:

j i i 1 i j ∂ i (g )q˙ q˙ + g q¨ = ∂ k (g )q˙ q˙ , q jk ik 2 q ij and here it is (although we don’t yet fully see it clearly) what we wanted to show, it is a matter of re-arranging the equation and recalling the formula for the Christoffel symbols in terms of the metris, one thus gets the geodesic equation: 1 q¨k + Γk q˙µq˙ν. 2 µν And as we commented in the introduction, this physical/dynamical picture of geodesics is a very important point of view for us!

0.3.5 Poisson Brackets on Symplectic manifolds In this section we introduce one of the fundamental objects one can obtain from the 2-form ω we have on our symplectic manifold. We will see the importance this concept has in understanding the role that symmetry plays in Hamiltonian Mechanics/Symplectic Geometry. Now, using the fundamental concept of a Lie Algebra that we defined in the first section we will explore the concept of a Poisson Bracket. For the moment, this structure will be inherited from a 2-form ω in a symplectic manifold M, later we will see that this can be further generalized to consider arbitrary Poisson Manifolds. Definition 0.3.5.1. We define the Poisson Bracket of a pair of smooth functions F, G ∈ C∞(M) by:

{F, G}(z) = ω(XF(z), XG(z))), so clearly this bracket takes two functions and produces another function {F, G} ∈ C∞(M).

This bracket in itself is very important for geometry and for physics since it allows us to investigate the deep structure that permeates the subject, we start by noting that the following are equivalent expressions:

{F, G}(z) = dF(z) · XG(z) = −dG(z) · XF(z). We have a special name for functions F, G ∈ C∞(M) such that {F, G} = 0, the functions are said to be in involution or to Poisson commute. Recall that apart from the Lie Algebra structure that we have with the Poisson bracket, we also have a Lie Algebra structure on the space of vector ﬁelds Γ(TM). The good news is that they are not independent structures, they are intimately linked together as the following proposition shows:

Proposition 7. If [X, Y] = XY − YX denotes the Lie bracket of vector ﬁelds (i.e. X, Y ∈ Γ(TM)), and we have two smooth functions F, G ∈ C∞(M). Then,

X{F,G} = −[XF, XG]

Proof. The proof follows by the following computation:

ω(X{F,G}(z), u) = d{F, G}· u = d(ω(XF(z), XG(z))) · u

= ω(DXF(z) · (u),

Now we will start seeing some of the deep structure that this Poisson bracket has and to what extent it does incorporate the information from the 2-form of the manifold (we will in fact see that it has all the information relating to the dynamics!).

Proposition 8. A diffeomorphism ψ : P1 → P2 is symplectic iff for every pair of ∞ functions F, G ∈ C (U), where U is an arbitrary open subset of P2.

{F, G} ◦ ψ = {F ◦ ϕ, G ◦ ϕ}.

Proposition 9. If we have a certain hamiltonian vector field XH ∈ Ham(M) (or a locally Hamiltonian vector field), then for every F, G ∈ C∞(M) (or a locally Hamil- tonian vector field) we have the following:

∗ ∗ ∗ ϕt {F, G} = {ϕt F, ϕt G} The fundamental observation now is that this bracket induces a deep structure on the space of smooth functions. All this ideas lead naturally to the following insight in the nature of the geometry behind symplectic geometry and mechanics.

Proposition 10. The space of all smooth functions C∞(M) form a Lie Algebra under the Poisson bracket.

Proof. With the properties inherited from ω, the bracket {·, ·} : C∞(M) × C∞(M) → C∞(M) is clearly (real) bilinear and skew-symmetric, so what we actually have to check is Jacobi’s identity. We start by noting the following:

{F, G} = iXF ω(XG) = dF(XG) = XG[F], from this it follows that

{{F, G}, H} = XH[{F, G}] and by the proposition proven before we get

{{F, G}, H} = {XH[F], G} + {F, XH[G]} = {{F, H}, G} + {{F, G}, H} which is just Jacobi’s identity.

This proof gives us further insight on the nature of what does Jacobi’s identity represent in the picture of a Lie Algebra (since at ﬁrst it seems rather arbitrary). As Marsden says in [?]:

Jacobi’s identity is just the inﬁnitessimal statement of ϕt being canonical.

Now that we have seen that this new structure, deeply connected with the symplectic form, and that contains a lot of information about the manifold, its dynamics, symmetries and physics. We want to study this object in itself, to see what kind of information does it carry and how can we use it to understand dynamics, conserved quantities, and almost everything, just with the Poisson bracket. This motivates the following study on what are usually known as Poisson Manifolds, to put Hamiltonian dynamics in an even more abstract formalism that allows us to understand the rich structure of the geometry underlying the physical universe.

0.4 Poisson Manifolds

In this section we will treat a generalization of the notion of symplectic manifold by keeping just the properties of Poisson brackets needed to describe Hamiltonian systems. It has been a subject of study for many years, but in many different contexts (as we will later see). Deﬁnition 0.4.0.1. A Poisson bracket (or structure) on a manifold M is a bilinear on C∞(M) operation:

{·, ·} : C∞(M) × C∞(M) → C∞(M) that satisﬁes the following properties:

1. (C∞(M), {, }) is a Lie Algebra

2. {, } is a derivation in each factor, that is, for every smooth functions F, G, H ∈ C∞(M),

{FG, H} = {F, H}G + F{G, H}

A manifold M equipped with this structure on its space of smooth functions is called a Poisson manifold. Note one trivial thing, any manifold can be equipped with the trivial Pois- son structure ({F, G} = 0) for every pair of smooth functions.

0.4.1 Hamiltonian vector ﬁelds in Poisson Manifolds Our task in this section will be to generalize many of the concepts we had in Symplectic Geometry to the more general context of Poisson manifolds. We will begin by extending the notion of a Hamiltonian vector ﬁeld. To see the beauty of the abstract framework we begin with a proposition that consist a fundamental observation.

Proposition 11. Consider M a Poisson Manifold. If we consider H ∈ C∞(M), then there is a unique vector ﬁeld on M (XH ∈ Γ(TM)) that satisﬁes the following equality for every G ∈ C∞(M):

LH[G] = {G, H},

XH is called the Hamiltonian vector field of H. Proof. We start with the following fundamental observation, what happens when we fix a function in our Poisson bracket, what kind of object do we obtain? What we see is that in fact it is a very known object, we are just talking about vector fields! Recall that vector fields can be interpreted abstractly as the derivations of the space of smooth functions, and this is exactly what we get. To see this we just fix a component in our bracket for a certain function H ∈ C∞(M), we consider the following:

∞ ∞ XH : C (M) → C (M) Q 7→ {Q, H} in a certain sense we can interpret ﬁxing a component in the following way, reduced to a simple formula:

XH = {·, H}. We now return to our previous observation and using that the Poisson bracket acts as a derivation in each factor we know that {·, H} is nothing more than a vector field. Now, since we know that a derivation is represented by a vector field and that it determines it uniquely if we require that the flow ϕt of the vector field XH preserves the Poisson structure (this is essential), we have thus seen the property we wanted. Recall that preserving the Poisson structure of the manifold just means:

∗ ∗ ∗ ϕt {F, G} = {ϕt F, ϕt G}, which can also be written in the following form, the form one usually uses in practical computations: {F, G} ◦ ϕt = {F ◦ ϕt, G ◦ ϕt}.

Remark. Note that this definition agrees with the definition we gave previously in the case of the symplectic space. So if the Poisson manifold we are considering is symplectic we get exactly the same definition of the Hamiltonian vector field associated with a smooth function H ∈ C∞(M). The argument is the following, in the case of a symplectic space (M, ω), we defined the Hamiltonian vector field of H ∈ C∞(M) to be the one that satisfies:

iXH ω = dH, recall the property relating the Poisson bracket and the usual derivative of a function with respect to a vector ﬁeld (Lie derivative), for a pair of functions F, G ∈ C∞(M): {F, G} = XG F, and it just follows by carefully manipulating the operations in the correct way, for a pair of functions F, H ∈ C∞(M) (we will later just ignore F):

{F, H} = iXF ω(XH) = dF(XH) = XH F = LXH F, so if we ignore the F we just get that the derivations are the same and hence, they define the same concept. So we have seen that really the Poisson Bracket is the structure that takes care of the Hamiltonian formalism. We would like to elaborate further now on the last concept we mentioned on the proof of last proposition. Maps and flows (diffeomorphisms on the manifold) preserving the Poisson Bracket are fundamental. They are what lie at the heart of classical mechanics, the abstract description of the transformations needed for finding symmetries in spaces, what kind of symmetries can one find and much more. The abstract description of the physical world reduced to just the necessary structure for the description of the object, to see its fundamental properties. But we should never forget that this was done his- torically in a more practical and computationally oriented way, by just trying what seems correct. And in fact when one has to actually find concrete symmetries for a certain system (be it geometrical, physical ...) one has to usually appeal to intuition to guess the correct one (recall the classical problems faced by astronomers and mathematicians to reduce the dimensionality of problems arising in celestial mechanics by exploiting symmetry in every possible way and usually with physical arguments to deduce symmetries of the system). It is essential here to learn from the historical development of mathematics and physics, and from there we extract the knowledge that an abstract understanding of how symmetries work in geometric systems is fundamental for the understanding of their geometry and the physics of our universe. We will now follow with an observation, that will later lead to a neat proposition revealing further the structure and the deep relationship of the Poisson structure with the manifold. What we did before was associating a vector field to every smooth function on the manifold via the Poisson bracket, now we just take seriously this idea, let’s make a function out of it. Consider then the following map: C∞(M) → Γ(TM)

H 7→ XH = {·, H} This reformulation allows us to understand within an abstract differential- geometric formalism the heart of classical mechanics. Since what we do in classical mechanics is just playing with the Poisson Bracket in a certain differentiable manifold (and usually a symplectic space, and more concretely a Cotangent Space!). We could say that in order to understand mechanics, we should tackle the difficult problem, unveiling its geometric basis. T The idea that will be fundamental during the whole work is that we will try to seek this geometric heart of physics, a geometric hart that is not only part of classical mechanics, it permeates the whole of physics, the ideas of Einstein heavily influenced this way to think about physics. The fundamental idea of Ein- stein is exactly this, that physics is just geometry (maybe a difficult differential- geometric object) but geometry nevertheless, and it has to be understood as such, unveiling its geometric heart just as Einstein did with gravitation. Now that we know the importance of the abstract formalism in the understanding of classical mechanics, we follow with a proposition that shows us, at the level of Poisson Manifolds, the deep relationship between vector fields in a manifold and the Poisson bracket operation (the relationship between dynamics and equations!): ∞ Proposition 12. The map H 7→ XH, from C (M) to Γ(TM), is a Lie algebra anti- homomorphism. This just means that the following equality is satisfied for every pair of functions F, G ∈ C∞(M):

[XF, XG] = −X{F,G}, Proof. We just follow a computation manipulating in the correct way our objects and using Jacobi’s identity:

[XF, XG]Q = XFXGQ − XGXFQ = {{Q, G}, F} − {{Q, F}, G} = −{Q, {F, G}}

= −X{F,G}Q And hence we deduce that [XF, XG] = −X{F,G}.

Remark. We note to the reader that if he checks other sources, or some of the bibliography, depending on several choices we have been doing about signs and orders, in almost every formula there will be a different sign, so we have tried to use the same criteria all the time. But it is important to check in every formula one seeks about symplectic geometry how the author deﬁnes things. Now we turn to investigate how dynamics work under this abstract formalism, how do we describe them in this abstract formalism? We start by proving a proposition in this abstract framework to see how dynamics get written, we will see that they get down to very concise equations that clearly express all the aspects of classical mechanics.

Proposition 13. We consider M to be a Poisson manifold and a smooth function H ∈ C∞(M), then the following is satisﬁed:

1. for every open set U ⊂ M and every function F ∈ C∞(U),

d (F ◦ ϕ ) = {F, H} ◦ ϕ = {F ◦ ϕ , H}, dt t t t

if and only if ϕt is the ﬂow of XH. We usually write this in a very concise notation: F˙ = {F, H}

2. This property is the standard property that the Hamiltonian function gets conserved under time evolution of the system (conservation of energy). If ϕt is the ﬂow of XH then, ∗ ϕt H = H ◦ ϕt = H.

Proof. We will proof the ﬁrst fact and the second will be a consequence of it. Let z ∈ M, then we have the following:

d d F(ϕ (z)) = dF(ϕ (z)) · ϕ (z), dt t t dt t and also

{F, H}(ϕt(z)) = dF(ϕt(z)) · XH(ϕt(z)).

Note that the equations are exactly equal if and only if ϕt is the ﬂow of XH. For the proof of the second fact just take H = F in the previous one.

0.4.2 Poisson maps/canonical transformations

Now we understand the deep structure we have contained in the Poisson bracket. What we will do now is to introduce the morphisms that respect the bracket and see some of its properties and uses. Deﬁnition 0.4.2.1. We say that a smooth map f : (M, {, }M) → (N, {, }N) between two Poisson manifolds is canonical or Poisson if for every pair of smooth functions F, G ∈ C∞(N),

∗ ∗ ∗ f {F, G}N = { f F, f G}M Recall that we already saw that in the case of a symplectic manifold, we have an equivalence between symplectomorphisms and canonical (transformations). Now we will reformulate a classical result with this formalism in the following proposition:

Proposition 14. If ϕt is the ﬂow of XH, then ϕt is a canonical transformation that preserves the Poisson structure.

Proof. This is even further also true for time-dependent Hamiltonian systems but we will prove it now in the time-independent case. Let F, G ∈ C∞(M) be a pair of smooth functions and ϕt the ﬂow of XH. We will consider the following quantity

u = {F ◦ ϕt, G ◦ ϕt} − {F, G}ϕt, that is kind of a measure of the "no-canonicality" of the ﬂow ϕt, to what extend is this ﬂow canonical is what u tries to measure. Using the bilinearity of the Bracket we thus get

d d d d u = { F ◦ ϕ , G ◦ ϕ } − {F, G}ϕ . dt dt t dt t dt t Now by using previous propositions we get that

d u = {{F ◦ ϕ , H}, G ◦ ϕ } + {F ◦ ϕ , {G ◦ ϕ , H}} − {{F, G} ◦ ϕ , H}, dt t t t t t and by Jacobi’s identity we thus get

d u = {u, H} = X u. dt H

This equation has a unique solution ut = u0 ◦ ϕt. Since we know that ϕ0 is just the identity, this implies u0 = 0, and since it is the ﬂow of the Hamiltonian and by looking at the deﬁnition of u, we thus get that u = 0. And with this we have proven the fact, since u just measured how much "non-canonical" was ϕt, and we have seen that u = 0, thus ϕt is canonical.

This property is essential since the Poisson structure doesn’t change in the dynamics of the system. But there’s even more, and this is very related to exploiting the symmetry of the system, Poisson maps push Hamiltonian flows to Hamiltonian flows! What this means is that in order to change a "Hamiltonian system" from one coordinate system to another, one just needs to change the function defining the dynamics (the Hamiltonian). Here we start to see the power of the abstract formalism in order to understand the symmetries of the system in an easy way that usually gets distorted if all the mathematical machinery is not well understood. Proposition 15. We consider a Poisson (canonical) transformation between Poisson ∞ manifolds f : M → N, and a smooth function H ∈ C (N). If ϕt is the flow of XH and ψt is the flow of XH◦ f , then

• ϕt ◦ f = f ◦ ψt.

• d f ◦ XH◦ f = XH ◦ f . The converse result is also true, if f : M → N satisﬁes this properties then it is canonical.

Proof. Take any smooth function G ∈ C∞(N) and a point z ∈ M. From the previous proposition and by using the fact that f is canonical, d d G(( f ◦ ψ )(z)) = (G ◦ f )(ψ (z)) = {G ◦ f , H ◦ f }(ψ (z)) = {G, H}( f ◦ ψ )(z), dt t dt t t t

this shows us that ( f ◦ ψt)(z) is an integral curve of the vector ﬁeld XHΓ(TN) through the point f (z). Note that the curve (ϕt ◦ f )(z) is also an integral curve of the same vector ﬁeld through the same point, by the uniqueness theorems in the theory of ODE’s, we conclude that

( f ◦ ψt)(z) = (ϕt ◦ f )(z), and if we take the time derivative we get the relation

d f ◦ XH◦ f = XH ◦ f .

To prove the converse, we assume that for every function H ∈ C∞(N) we have d f ◦ XH◦ f = XH ◦ f . By using the chain rule we get

XH◦ f [F ◦ f ](z) = XH[F]( f (z)), ∗ ∗ this means that XH◦ f [ f F](z) = f (XH[F]). Thus, if we take a function G ∈ C∞(N),

∗ ∗ {G, H} ◦ f = f (XH(G)) = XH◦ f ( f G) = {G ◦ f , G ◦ f }, and hence f is canonical.

We have thus explored the basic properties where we have seen the intimate connection that the Poisson bracket has with the dynamics on our manifold, now we will turn to a more concrete problem and introduce some classical terminology. First we start by recalling a fact that we mentioned earlier, namely that we interpret the Poisson bracket of a function F ∈ C∞(M) with the Hamiltonian of the system as the time derivative along its ﬂow:

F˙ = {F, H}. This has amazing consequences since we obtain an equivalence between the algebraic formalism and the dynamics, we relate the derivative along a flow with a differential-geometric operation on the algebra of functions of the manifold. A first observation one makes here is to ask for what happens if a certain function F ∈ C∞(M) satisfies that {F, H} = 0. Recall that evaluating the Pois- son bracket with the Hamiltonian just gives time evolution, so this means that the value of F is a constant of motion, it is an invariant. Now we have seen thus that invariants are just certain subsets of the Lie Algebra of smooth functions, and this provides a deep insight on how symmetry works in classical mechanics. Notice this is a very similar observation to the one we did when we were talking about Killing vector fields and the fundamental importance of this algebras for understanding symmetries in geometry. But before we start dealing with those functions that are what’s called in involution (recall that F, G ∈ C∞(M) are in involution if {F, G} = 0). We will note a thing, that hamiltonian dynamics can be written in a beautiful way only in terms of the Poisson bracket using what we just commented about the time derivative along a solution:

Q˙ = {Q, H} P˙ = {P, H}

And now we clearly see how is the hamiltonian function generating all the dynamics in a concise way, since how much a quantity varies is just contained in the hamiltonian function we consider and on the Poisson structure of the manifold. Now we can understand a classical result in mechanics, known as Noether’s theorem. This theorem relates symmetries with conserved quantities and it is usually stated in the form of Lagrangian mechanics with some computations. What we want to emphasize is the geometrical meaning of this theorem, why it is just there "naturally" and not as a mere extern artifact. Using the formalism of Poisson manifolds and understanding classical mechanics there, the theorem just appears in the following way:

Theorem 0.4.2.1. (Noether’s theorem) If we have a symmetry (generated by a vector ﬁeld XQ) of a certain Hamiltonian system in a Poisson manifold M and Hamiltonian H ∈ C∞(M), then Q is a conserved quantity. And the converse also holds, this is the beauty of the theorem in this formalism, it just makes a natural relation between conserved quantities and symmetry.

Proof. The beauty of the formalism is that the theorem is just a direct consequence of the geometric structure of our space, and every quantity related to the dynamics relates directly to the bracket, we just use the skew-symmetry of the bracket. Since we have a symmetry, we have that

XQ H = 0 ⇔ {H, Q} = 0, using the skew-symmetry of the bracket we obtain directly that

0 = {H, Q} = −{Q, H} = −XH Q = 0.

Now we will talk a bit about what we have done, the necessity of abstraction in the theoretical understanding of the world (be it mathematical, physical or in any other theoretical investigation) and why we chose to explain the things this way. We start by seeing a very powerful advantage of this abstraction, that let’s us arrive at the core of the object, we are able to exploit its properties from inside the object, rather than from its outside via external manipulations that we already now where are heading (such as in the classical derivation of Noether’s Theorem). With the previous proposition we are thus at the geometric heart of Noether’s theorem, conserved quantities and symmetries are concepts with a direct geometric relationship in classical mechanics. It is not a mathematical artifact (as the physicist mentality usually thinks about it) that produces what we want, it is an essential property of our description of the physical world that is just revealed if one tackles the problem from the right theoretical perspective. After all this theoretical description we now turn to a concrete analysis on how all this machinery works in certain classical cases, why and how Siméon Denis Poisson introduced this bracket to analyze concrete mechanical problems is what is going to concern us now. To see that this bracket, although very abstractly deﬁned and talked about (but completely necessary for the correct understanding of it) in the modern formulation we have given before, is just a natural operation to consider. Thus let’s see how the Poisson bracket just appears naively in a classical Hamiltonian system, that we will consider for now to be the classical model 2n n i M = R with the canonical 2-form ω = ∑i=1 dq ∧ dpi. We will see now clearly what does the bracket represent in this simple case, it is just a clever way to organize the derivative of a certain quantity along the trajectory of a mechanical system. To see this we just consider an arbitrary observable (note that physicists call functions on C∞(M) observables) on the manifold R2n, and as a function it i depends on position, momentum and time, so we have a certain f (q , pj, t). Now, we just compute the derivative of this quantity along a solution of the Hamilton equations (the standard ones) for a given hamiltonian H ∈ C∞(R2n), and see what happens:

i d ∂ dq ∂ dpj ∂ ( i ) = ( i ) + ( i ) + ( i ) f q , pj, t i f q , pj, t f q , pj, t f q , pj, t , dt ∂q dt ∂pj dt ∂t we have obtained this just by applying the properties of the derivative, now we introduce into this equation Hamilton’s equations of motion and we thus get:

d ∂ ∂H ∂ ∂H ∂ ( i ) = ( i ) − ( i ) + ( i ) f q , pj, t i f q , pj, t f q , pj, t j f q , pj, t , dt ∂q ∂pi ∂pj ∂q ∂t the quantity we observe in the right (except from the time dependence) is what Poisson observed as essential for the description of the mechanics of the system, if we write it in the notation we have been using we get the expression:

d ∂ f = { f , H} + f . dt ∂t We have thus seen the importance of the Poisson bracket in the description of dynamics and what motivated Poisson to deeply study it for understanding mechanical problems. In a certain sense what this shows us is that the bracket is nothing more than derivation (correctly adapted to general manifolds) along the trajectories of a certain Hamiltonian (ant the bracket takes care of the whole dynamics of the system, independent of the choice of Hamiltonian!). So in this explanation we have said that the Poisson bracket, written in coordinates (position/momentum) in the canonical symplectic manifold R2n takes the following form:

∂F ∂G ∂G ∂F { } = − F, G i i , ∂q ∂pi ∂q ∂pi for every pair of smooth functions F, G. But how does this relate to our definition of the Poisson bracket abstractly? We just apply the formalism we gave to see that we will actually get this bracket in this context. Recall the definition of the bracket: {F, G} = ω(XF, XG), now just see that in the case of R2n computations are very easy to do, first notice 0 Id ω−1 = n , −Idn 0 so to compute the associated vector field of a function reduces to:

∂F ∂F ∂F ∂F = −1 = −1( ) = ( − ) XF ω dF ω i , , i ∂q ∂pj ∂pj ∂q And hence, as we deﬁned the Poisson bracket before, we just compute in this case: ∂F ∂F ∂G ∂G ∂F ∂F ∂G ∂G { } = ( ) = ( − ) ( − ) = ( − )(− ) = F, G ω XF, XG , i ω , i , i i , ∂pj ∂q ∂pj ∂q ∂pi ∂q ∂q ∂pi

∂F ∂G ∂G ∂F = − i i ∂q ∂pi ∂q ∂pi 0.5 Moment Maps

Now that we have already talked about Poisson manifolds in the previous section, in this section we will introduce a fundamental concept that allows us to exploit and analyze the symmetries of a mechanical system in a new way. In this section we will see how to obtain conserved quantities for geometrical/mechanical systems that have some kind of symmetry. The fundamental concept we will have to introduce is the moment map, that will allow us to explore the symmetry of our system- It is not just a mere reformulation of Noether’s theorem, but rather a whole new approach to symmetry, and a fundamental idea in the ﬁeld of geometric mechanics. We will ﬁrst introduce the theory and the abstract description and properties of the object , and later we will see how to use it adequately in concrete problems.

0.5.1 Theory

We will ﬁrst introduce the theoretical setting for our understanding of the moment map. What we emphasize is that Poisson manifolds are the appropriate setting for developing the theory, as we said in the previous section. Since we also want to analyze symmetry in a certain system, we will suppose we have symmetry, namely that we have an action of a certain Lie Group G acting on our Poisson manifold (in a canonical way).

Deﬁnition 0.5.1.1. Let P be a Poisson manifold and G be a Lie Group. The Lie Group acts on the Poisson manifold by a smooth left action by canonical transformations, this means that we have:

. : G × P → P (g, p) 7→ g . p.

We impose that these actions are canonical (that is what one usually ﬁnds in mechanics). We denote the action by a concrete element of the Lie group g ∈ G as g. : P → P. Recall that being canonical just means that the Poisson structure gets preserved by the action of the Lie group:

(g.)∗{F, G} = {(g.)∗F, (g.)∗G}, needs to be satisﬁed for every pair of functions F, G ∈ C∞(P) and every element of the Lie group g ∈ G. Note that if we are in the case of a symplectic manifold with symplectic form ω, then the action is canonical (preserves the Poisson structure) if and only if it is symplectic, wich just means that for every g ∈ G, (g.)∗ω = ω.

Recall that in this setting (the action of a Lie Group on a manifold) we get what are usually called infinitessimal generators of the action. To a Lie Algebra element ξ ∈ g we associate a vector field ξP on P that we obtain differentiating the action at the identity in the direction of the Lie Algebra element via the exponential map: d ξ (z) = exp(tξ) . z| = . P dt t 0 We will need two identities concerning Lie Groups and Lie Algebras that are proven in [?]. First, the flow of the vector field ξP is

ϕt = exp(tξ) . .

Second, we have the following:

−1 ∗ (g .) ξP = (Adgξ)P, that comes with a differentiated companion that we mentioned previously:

[ξP, ηP] = −[ξ, η]P.

We will now see how all this abstraction is present even in the case of the standard rotation group on R3. The Rotation Group: To show the reader this identities in a familiar case, consider the action of SO(3) on R3. The Lie algebra of the group is denoted by so(3) and is identified with R3. The Lie bracket is identified with the standard cross-product. For the action of SO(3) on R3 by rotations, the infinitesimal generator η ∈ R3 is just ηR3 (x) = η × x, then the general properties just become

(Aη × x) = A(η × A−1x) for every A ∈ SO(3), while the second identity just becomes the Jacobi identity for the vector product. Poisson Automorphisms: If we return to the general case, recall we had a canonical action that satisﬁes

(g.)∗{F, G} = {(g.)∗F, (g.)∗G}, if we differentiate this condition with respect to the element g in the Lie group in the direction ξ we get

ξP({F, G}) = {ξPF, G} + {F, ξPG}.

If we are in the symplectic case, by differentiating (g.)∗ω = ω we obtain

L = ξP ω 0, that is, ξP is locally Hamiltonian. If we are in a Poisson manifold, we call inﬁnitesimal Poisson automorphisms those vector ﬁelds X ∈ Γ(TM) that satisfy the previous mentioned condition:

X({F, G}) = {X(F), G} + {F, X(G)}, It is important to note that this vector fields need not be locally Hamiltonian (that is, locally of the form XH). And Marsden in [?] gives an example to show that this may be the case in certain Poisson manifolds. Thus we will restrict the scope of our description to certain kind of actions with special properties adapted to the kind of systems one usually finds in mechanics. We will be mainly interested in the case in which ξP is globally Hamilto- nian (and this need not be the case a priori). So we assume that there is a ∞ global Hamiltonian J(ξ) ∈ C (P) for the vector field ξP:

XJ(ξ) = ξP.

This equation does not determine J(ξ) though, since we could choose for example any other hamiltonian of the form J(ξ) plus any constant, that would yield the same vector field (a general Casimir function depending on the space). Note that ξP is linear in ξ, if we suppose we are in a finite dimensional manifold we can modify any J(ξ) to make it also linear in ξ and retain the equation that we want to be satisfied, namely XJ(ξ) = ξP. If e1, ..., er is a basis a of g, we can let the map be defined by Jˆ(ξ) = ξ J(ea) and make it linear. In the definition we gave before of the momentum map we could also just supposed we had a Lie algebra action on the manifold, in order to define the next, so Marsden in [?] uses this notion of a Lie algebra action to continue. Now we are ready to define the fundamental object of the section,

Deﬁnition 0.5.1.2. Consider a Lie Algebra g acting canonically on a Poisson Manifold P. We suppose, as we said before, there is a linear map J : g → C∞(P) that the previous mentioned condition for every ξ ∈ g:

XJ(ξ) = ξP.

Now the moment map (of the action) is the map µ : P → g∗ deﬁned by the following property satisﬁed for every ξ ∈ g and z ∈ P:

< µ(z), ξ >= J(ξ)(z)

Now that we gave the abstract definition we stop for a moment and recall what have we just defined and what is this object doing. What we have defined takes into consideration symmetries that are given by hamiltonian vector fields on the Poisson manifold. The function J associates to every Lie Algebra element the function that generates the vector field on the Manifold that the action is generating (a rotation by an axis, for example). The momentum map is the dual concept of this and thus a way of gathering information about how this happens, and to extract conserved quantities from the system, as we will later see. What we will do now is elaborate on how this concepts arise when one considers the familiar case of mechanics in euclidean 3-dimensional space. Let’s review the notion of angular momentum and how it fits under the picture we gave as "momentum", does it have anything to do? Of course, in fact the prototype that motivates the abstraction given before is in fact this example. Angular Momentum: We will consider now that we are in ordinary euclidean space and consider position and momentum (q, p). The angular momentum is given by the formula J(q, p) = q × p (standard cross-product in R3). Now take a vector ξ ∈ R3 and consider the component of J in that axis: < J(q, p), ξ >= ξ · (q × p). If we considers the Hamilton equations determined by this function of q and p we just get infinitessimal rotations about the ξ axis. This is the example that motivates the abstract definition, which is just a generalization of this essential geometric property satisfied by angular momentum. Relationship with Poisson brackets: Now we are going to see if there is any relationship between the moment map and the Poisson bracket we have on our manifold. First recall that we understand the Poisson bracket as the structure that is able to generate vector fields from smooth functions in the sense that: {F, H} = XH F. Now, the defining condition of the moment map can be written in terms of the Poisson bracket structure. We require that for every smooth function F ∈ C∞(P) and every ξ ∈ g,

{F, J(ξ)} = ξPF, which is equivalent to the deﬁnition we gave before that

XJ(ξ) = ξP, just written in Poisson bracket form. Note that the defining condition for the moment map µ : P → g∗, that it satisfies < µ(z), ξ >= J(ξ)(z) just defines a natural identification (isomorphism) between the space of smooth moment maps µ : P → g∗ and the space of linear maps J : g → C∞(P). The correct interpretation is to think about the collection of functions J(ξ) as ξ varies in the lie algebra g as the components of the moment map µ. This is why we are interested in the moment map as a generalization of the concept of momentum one finds in classical hamiltonian dynamics, since it carries the conserved quantities of a system with symmetries! Recall we denoted by Ham(P) the algebra of hamiltonian vector fields on the manifold. We will denote by

P(P) = {X ∈ Γ(TM)|X({F, G}) = {X(F), G} + {F, X(G)}, the algebra of Poisson vector ﬁelds. By applying a property of Poisson automorphisms mentioned before (Propo- sition 11.1.8 in [?]), we have that for any ξ ∈ g we have an associated Poisson vector ﬁeld ξP ∈ P. With this in our knowledge, giving a moment map µ is equivalent to specifying a linear map J : g → C∞(P) such that the following diagram commutes:

g ξ→ξP J F→X C∞(P) F P(P)

Notice that both ξ → ξP and H → XH are Lie algebra antihomomorphisms. For ξ, η ∈ g, using the bracket [, ] in the lie algebra (careful that we also denote like that the Lie Bracket on vector ﬁelds) and the Poisson bracket {, } on the manifold, we obtain the following:

XJ([ξ,η]) = {ξ, η}P = −[ξP, ηP] = −[XJ(ξ), XJ(η)] = X{J(ξ),J(η)}, thus obtaining the identity

XJ([ξ,η]) = X{J(ξ),J(η)}. We have seen thus a theoretical description of the moment map and we see that it is a very useful concept for the description of symmetries in a Poisson manifold since it plays nicely with the bracket structure. In the next section we will give some ideas on how to actually compute them, but for now we need to further study the theoretical properties of moment maps, but first we will give a brief historical overview to appreciate the importance of this object. Some History on the Moment map: The concept can be traced back to the second volume of Lie [?], where it appears in the context of homogeneous canonical transformations. And it was heavily used by Lie in geometrical problems. The modern history of the concept, as Marsden says referring to information from B.Kostant and J.M. Souriau, goes as follows. In some lectures given in 1965, Kostant introduced the moment map to generalize a theorem and classify all homogeneous symplectic manifolds; this is known as "Kostant’s coadjoint orbit covering theorem". These lectures also contain the essential ideas for developing the notion of geometric quantization. Souriau introduced the momentum map in some lectures in 1965 and put it in print in 1966 [?]. The momentum map finally got its formal definition and name in a paper by Souriau [?]. Souriau also studied its properties of equivariance and formulated the coadjoint orbit theorem. In Kostant’s quantization lectures [?], the moment map appeared as a key tool for quantization. Independently, A.Kirilliov did work on the momentum map in [?]. The modern formulation of the momentum map was developed in the context of classical mechanics, in the work of Smale [?]. He applied it in his topological program for the planar n-body problem. Marsden and Weinstein [?] and other authors have heavily used this concept too. Now that we know the history of the development of the concept and the importance it had (and still has) in the geometrization of mechanics, we will study abstract properties of the momentum map allow us to better understand what it describes. Conservation of momentum maps: One of the fundamental reason that momentum maps are important in mechanics is that they are conserved quantities, we thus get yet another view (now even more geometrical) of the classical Noether’s Theorem that exploits all the symmetries of the system. Theorem 0.5.1.1. If we have a Lie algebra g acting canonically on a Poisson manifold P which admits a momentum map µ : P → g∗. And if we have a hamiltonian function ∞ generating the dynamics H ∈ C (P) that is g-invariant, in the sense that ξP(H) = 0 for every ξ ∈ g. Then, the moment map µ is a constant of the motion generated by H, namely that µ ◦ ϕt = µ, where ϕt is the flow associated to the hamiltonian vector field XH. If the Lie algebra action comes from a canonical left Lie group action ., then the invariance hypothesis on H that we require in this formulation of the theorem is implied by the invariance condition (g.)∗ H = H, that just means that for every element g ∈ G of the Lie group, H ◦ (g.) = H.

Proof. By the properties we saw before, the condition that ξP H = 0 implies that {J(ξ), H} = 0. This clearly implies, for the description we have been giving, that for each Lie algebra element ξ, J(ξ) is a conserved quantity along the dynamics generated by the ﬂow of the Hamiltonian. This means that the values that correspond to the g∗-valued moment map µ : P → g∗ are also conserved. The last part of the theorem, the one referring to the case when we have a Lie group acting on the system just follows by differentiating the condition H ◦ (g.) = H with respect to g at the identity element of the group (recall the Lie algebra associated to a group is just the tangent space at the identity) in the direction ξ ∈ g to obtain the corresponding condition on the Lie Algebra:

ξP(H) = 0.

We have thus seen yet another version of the Noether’s theorem, this one relating to the notion of moment map in a deep geometric way. This version is essential since it allows both to have a better geometrical interpretation of Noether’s theorem, as well as a very useful tool for computing conserved quantities in mechanical systems and in geometrical problems. It is useful since it allows us to use the geometrical structure of the phase space to deduce relevant information on the symmetry of the system that is just not visible from an action (that is what usually describes physical systems in the physics literature). 0.5.2 Computation and Properties of Momentum Maps We will now see how one can take advantage of using momentum maps for understanding the dynamics of a system. We shall see that in the case of a cotangent bundle, the geometrical systems one usually ﬁnds in Hamiltonian mechanics, we will have much more information on the momentum map (we can give an explicit formula!).

Momentum maps on cotangent bundles

The fundamental result one can obtain in this section is the following, the proof can be found in [14]. Theorem 0.5.2.1. (Momentum Maps for Lifted Actions). Suppose that the lie algebra g acts on the left on the manifold Q, so that we can lift the action to the cotangent bundle P = T∗ M. This g-action on P is Hamiltonian with moment map µ : P → g∗ given by < µ(αq), ξ >=< αq, ξQ(q) >= P(ξQ)(αq), ∞ ∗ where P(X)(αq) =< αq, X(q) > is a map P : Γ(TQ) → C (T Q), in coordinates this map is just i j i P(X)(q , pi) = X (q )pj.

Concrete Examples

We will center our atention in only one concrete example, the proof of Clairaut’s Theorem using all this machinery we have introduced. Clairaut’s Theorem: We will now apply all this formalism to show a classic result in the study of surfaces and deduce it from the formalism that we use to describe mechanics (since this theorem is about conserved quantities). Let M be a surface of revolution in R3 obtained by imposing the equation q x2 + y2 = r = f (z), that just revolves the graph of the function f about the z-axis, where we require that f is a smooth positive function. In this surface of revolution we can consider the chart given by the following parametrization:

Φ(θ, z) = ( f (z) cos(θ), f (z) sin(θ), z).

3 We consider the usual metric Id3 in R given by the identity matrix at every point and we consider the pullback metric in our surface M, namely we will consider ∗ T T g = Φ Id3 = (dΦ) Id3(dΦ) = (dΦ) (dΦ), If one computes the differential of the parametrization one obtains:

− sin(θ) f (z) f 0(z) cos(θ) dΦ(θ, z) =  f (z) cos(θ) f 0(z) sin(θ) 0 1 thus the pullback metric turns out to be: − sin(θ) f (z) f 0(z) cos(θ) − sin(θ) f (z) f (z) cos(θ) 0 g = (dΦ)T(dΦ) =  f (z) cos(θ) f 0(z) sin(θ) f 0(z) cos(θ) f 0(z) sin(θ) 1 0 1 and hence, f 2(z) 0 g = 0 f 02(z) + 1 Thus we see that the metric g in our surface is independent of the coordinate θ, thus just means that the metric is invariant under rotations about the z- axis. Now we will see how our abstract ideas enter the game here. Start by considering the geodesic ﬂow on M. Recall that θ is an angle-variable that ranges from 0 to 2π, and hence what we have in our system is a S1 symmetry, recall that the associated Lie Algebra is just R. The momentum map associated with this symmetry is µ : TM → R given by the following expression: < µ(q, v), ξ >=< (q, v), ξM(q) >, as usual, where we just denoted both coordinates by q and tangent vectors 3 by v. Here, ξM is the vector ﬁeld on R that is associated to a rotation with angular velocity ξ about the z-axis. Hence we have that

ξM(q) = ξ(0, 0, 1) × q, or how the physicist would write it:

ξM(q) = ξk × q, where k usually denotes in the physics literature the z coordinate basis vectors. But the we have a better view of this object in our coordinates, it is nothing more than ξ∂θ. We realize the following by noting that both views are just related by the differential of the parametrization (by a push-forward): − sin(θ) f (z) f 0(z) cos(θ) − sin(θ) f (z) ξ dΦ(ξ∂ ) =  f (z) cos(θ) f 0(z) sin(θ) = ξ  cos(θ) f (z)  = ξk × Φ(θ, z) θ 0 0 1 0 and this last expression clearly equals the other description. Thus we get that

< µ(q, v), ξ >=< (q, v), ξM(q) >=< (q, v), ξk × q >, And we conclude from this computation the expression we wanted to ﬁnd:

< µ(q, v), ξ >= ξr||v|| cos(τ), where r = f (z) is the distance to the z-axis and τ is the angle between the tangent vector v at each point and the horizontal plane. Since we are considering geodesic ﬂow, we know that ||v|| is conserved, by conservation of energy. Thus what we can conclude is that r cos(τ) is conserved along any geodesic on a surface of revolution, the classical result named as Clairaut’s Theorem. 0.6 More differential geometry, Principal and As- sociated Bundles (Yang-Mills theory).

Now we return to differential geometry in order to study some geometrical spaces to better understand their abstract properties, and also see how they generalize the classical view of differential geometry. The geometrical objects we are about to introduce are the necessary tools to be able to talk about physics geometrically, and to correctly understand the notion of a connection (and this is very important in physics, since it is the geometric formalism appropriate to describe fundamental interactions). The correct understanding of these geometrical objects and their physical signiﬁcance is essential for an understanding of the physics of our universe. Modern Gauge Theories (in this geometric interpretation) try to describe physical systems with symmetries taking as a model the geometrization of Maxwell’s Equations of Electro-Magnetism. This theories are called Yang-Mills theories, that generalize the U(1) = {z ∈ C||z| = 1} symmetry one has in classical electrodynamics to larger groups of symmetries that in the physics literature are usually SU(n) (the special unitary group of n × n matrices with determinant 1), more generally we can consider arbitrary compact semi-simple Lie Group. The fundamental idea behind Yang-Mills theory is to try to describe the behavior of elementary particles using this non-Abelian Lie Groups. They intend to do this by using appropriate bundles and connections on them to correctly generalize classical electromagnetism. Yang-Mills theory takes connections in principal and associated bundles as the fundamental geometrical object to describe reality. These kind of objects were deeply studied by last century’s mathematicians (specially Cartan, Ehresmann and Koszul). An important thing to note is the following:

Note there is an important difference between general Yang-Mills theory and classical Maxwell electrodynamics, the group in classical electromagnetism is the Lie Group U(1), which is Abelian! In the general Yang-Mills situation we lose this property since SU(N) is not abelian for N ≥ 2!

We will not enter further for the moment on the technical details of these Lie Groups, we just recall the reader that SU(N) is a compact Lie group of real dimension N2 − 1 that is simply connected, it is also what is called a simple Lie Group. Also note that its Lie algebra su(N) can be identiﬁed with the traceless anti-hermitian n × n complex matrices with the usual commutator on matrices as the bracket. In particle physics one usually needs a concrete study of this groups for little values of N, in order to actually compute with these objects. We will not devote much time to that concrete analysis, since we believe that the geometrical description we are about to give is essential before thinking about concrete computations, and specially in this areas of physics! One should know what geometrical object one is handling and try to manipulate it accordingly, if not all this theory becomes a great mess incapable of understanding the geometrical nature of the physical universe. Concrete groups used in physics: The concrete Lie Groups of interest for particle physics are the following:

• Electro-weak interaction is described by the group SU(2) × U(1).

• Quantum Chromodynamics, the theory of the strong force, is described by the group SU(3).

And physicist have deeply studies this concrete groups and their mutual relationship (they form the basis for the Standard Model). They have developed a lot of computationally useful tools to extract results about this that relate to the experimental data one can extract from a particle accelerator, for example the one at CERN. Those techniques should be further studied to get a good knowledge of what kind of intuition do physicist use to deal with this objects. This view on Yang-Mills theories will not be the one we take here, but it is fundamental if we want to develop a theory that adapts the problem to the correct theoretical framework. A theoretical perspective that allows the intuition physicists have on this objects to be useful for integrating both views (theoretical/practical) into a unified theoretical and intuitive understanding of the universe. Nowadays, both views are very far apart from each other, and we emphasize the necessity that modern mathematical ideas enter theoretical physics as well as getting into mathematics the physical intuition of objects. We believe that this is a process that benefits both mathematics and physics, as one can clearly see in the geometrization of Yang-Mills theories we will describe in this section, as well as in mathematics in the study of Donaldson of the properties of 4-dimensional manifolds in [?], study that has in itself deep relationship to physics and that uses many ideas coming from physics. One can also see a deep relationship to physics in the ideas that have been floating around about the Poincaré Conjecture, which was proved by Perelman using some physical ideas. Other authors have been trying to take those physical ideas in the Poincaré Conjecture even further to better understand the intuition behind the result proved by Perelman, for example the paper by A. Kholodenko [?].

The description we will try to give tries to show the geometrical formalism as the natural apparatus to describe the physics. And we will turn over and over again on the same ideas, in deeper and deeper levels of abstraction to try to arrive at a theoretical description that combines both a natural (intuitive) description as well as a deep theoretical description that incorporates heavy machinery coming from modern differential geometry. We will try to introduce the concepts that become a necessity for the theoretical description in the right moment, to connect with intuition. We start thus by introducing the reader to the first idea, due to Kaluza- Klein, to try to develop a "unified field theory". We will recast this classical idea in terms of "modern" differential geometry. Then we will be in a good situation to try and generalize to Yang-Mills.

0.6.1 Geometrization of the Kaluza-Klein theory of electromagnetism We start by considering the standard lorentzian manifold of special relativity, M = R4. We denote our coordinates on M by xµ with 0 ≤ µ ≤ 3. Then we interpret this coordinates physically in the following way: • x0 = t is just the physical "time" coordinate.

• x = (x1, x2, x3) are just the "space" coordinates. We will now consider the following object,

θ ∈ Ω1(M), that in the coordinates we have can be written as

µ θ = Aµdx , these Aµ that appear as coefficients is what the physicist calls the electromagnetic field potentials, any classical physics books describing this will suffice to the reader if he hasn’t seen this description before, for example the Lan- dau/Lifshitz book devoted to classical electromagnetism in the famous series of lectures [?]. Now we consider the tangent bundle TM and we consider the associated coordinates in the tangent bundle associated to the ones we have, that we will denote as we did in previous sections by

µ (x , ∂µ), so vector ﬁelds on TM are denoted by:

µ V = x˙ ∂µ, Now we consider a Lagrangian deﬁned in this bundle (physicists, although not always, tend to prefer the Lagrangian formalism)

L : TM → R, and this function in the theory gets deﬁned as: q e L (xµ, x˙µ) = g x˙µx˙ν + A x˙µ, µν mc2 µ recall that the lorentzian metric tensor in ﬂat space is given by the matrix(here we are not normalizing the speed of light and thus pops out as a constant c):

c2 0 0 0   0 −1 0 0  g =   ,  0 0 −1 0  0 0 0 −1 the criteria for the signs is flipped in contrast with the previous sections, it is not important which one chooses if one is consistent during the computations with the one chosen for concrete calculations. There is a whole Holy war in the physics community about the signs for this metrics. As mathematicians we just understand them as different correct choices for the description of the system, and one is free to choice the preferred one for doing some concrete computations, thus it is a good idea to tell during every computation exactly what metric tensor are we considering (of what signature). The Lagrangian of the system is very similar to the Lagrangian for geodesics (not exactly since we have taken a square-root of it, but the idea is similar, and it will be important later) on the base manifold M to which we add a term coming from the electromagnetic field, that information of which we have inside θ ∈ Ω1(M). The quantities e, m that we introduced in the Lagrangian are just charge and mass respectively. Note one fundamental observation, that if the particle we try to describe doesn’t have charge (i.e. e = 0), then we just get the equations for the standard movement of particles in Lorentzian space, inertial motion. But when we vary the charge we introduce a perturbation of this dynamics, and the electromagnetic field is reflected in the dynamics. Now we want to know the dynamics of the system, as we know this is done by imposing a variational condition on the functional defined by the Lagrangian on the space of possible trajectories. Hence we have to compute the extremals of the following calculus of variations problem defined by Z d L (xµ, xµ)ds. dt We know then that the solutions satisfy the Euler-Lagrange equations: d ∂L ∂L ( ) = , ds ∂x˙µ ∂xµ so we do a pair of previous necessary computations:

ν ∂L gµνx˙ e = p + Aµ ∂x˙µ g(x˙, x˙) mc2 ∂L e ∂A = ν x˙nu ∂xµ mc2 ∂xµ Where we obtain the ﬁrst identity just by differentiating and the second one by noting that the Lorentzian metric we have is independent of position (translation- invariant) and hence we just differentiate the part of the Lagrangian relating to the electromagnetic ﬁeld. Now note a thing, the Lagrangian is homogeneous of degree 1 in x˙ since it can be written in the following form, if we expand the metric as (be careful with the difference between indices and powers) q e L = c2(x˙0)2 − (x˙1)2 − (x˙2)2 − (x˙3)2 + A x˙µ, mc2 µ this means that without loss of generality we may assume that ||x˙||2 = g(x˙, x˙) = µ ν gµνx˙ x˙ is just constant. Thus taking all this together and applying the Euler-Lagrange equation we obtain the following extremal equations:

1 d e e (g x˙ν) + ∂ (A )x˙ν = ∂ L = ∂ A x˙ν, ||x˙|| ds µν mc2 ν µ µ mc2 µ ν

that are equivalently written as

1 d e (g x˙ν) = F x˙ν, ||x˙|| ds µν mc2 µν if we deﬁne the following quantity: Fµν = ∂µ Aν − ∂ν Aµ, this object is known as the electromagnetic ﬁeld in the physics literature (it is also known as the electromagnetic tensor). We can interpret this object in the following sense:

µ ν ν Fµνdx ∧ dx = d(Aνdx ) = dθ, the electromagnetic ﬁeld is thus nothing more than the exterior derivative ν of the potential 1-form θ = Aνdx . And this are just Maxwell equations of motion! This is an essential observation! Later we will start to see why this view is so important. Now let’s follow through the ideas of Kaluza-Klein, consider Y to be a 5-dimensional manifold given by the following coordinates:

(xµ, φ), and consider the map π : Y → M that satisﬁes

π∗(xµ) = xµ, what this just means is that π is (locally) just the natural projection R5 → R4. We are for now just considering the local behavior if this structure, when we introduce the appropriate concepts we will say this is just a fiber bundle with SO(2, R) = U(1) as structure group and compact 1-dimensional fibers). Now we consider the following Lagrangian on the tangent bundle L 0 : TY → R given by: 1 L 0 = g x˙µx˙ν + (φ˙ − A x˙µ). 2 µν µ This Lagrangian, in a similar fashion as before, defines a calculus-of-variations problem that has Y as configuration space. And this may seem like the same situation as we did before but in a more complicated setting, but it is this more geometrical description that allows us to understand better the deep structure behind the ideas of Kaluza-Klein. This comes from the fundamental observation that:

The extremals are geodesics of a Riemannian metric on Y.

And what we want to emphasize is that this view should be regarded as more "natural" in a geometric sense. And this view has intimate relationship with the ideas of Einstein of deriving physical "laws" from geometric structures. Now we will compute the extremals using the Euler-Lagrange equations as before, ﬁrst some needed computations (Recall we are still using the same metric on the base manifold as before :

∂L 0 = g x˙ν − 2(φ˙ − Ax˙)A ∂x˙µ µν µ ∂L 0 = 2(φ˙ − Ax˙) ∂φ˙ 0 ∂µL = 2(φ˙ − Ax˙)∂µ Aν 0 ∂φL = 0

now we consider the Euler-Lagrange equations (recall that there is an equation for every coordinate, so now we have an aditional equation). Due to the 0 fact that ∂φL = 0, we automatically get that

d ∂L 0 ( ) = 0, ds ∂φ˙ we have thus obtained a conserved quantity along the dynamics given by the Lagrangian so

∂L 0 = 2(φ˙ − Ax˙) = ais a constant of motion. ∂φ˙

This is just a pedestrian way to use Noether’s theorem to compute conserved quantities, but that is very intuitive and allows the reader to get better idea of the natural relationship between symmetries and conserved quantities in a very easy case where the symmetry is clear and is just a translation. We compute the rest of the Euler-Lagrange equation and obtain:

d (g x˙ν) − 2a∂ (A )x˙ν = 2a∂ (A )x˙µ, ds µν ν µ µ ν using the tensor Fµν we deﬁned, we get the expression:

d (g x˙ν) = 2aF x˙ν. ds µν µν

We can get all this information together in the form of the following theorem,

Theorem 0.6.1.1. (Theorem 3.1 in [9]) The Lagrangian L 0 : TY → R for which the constant of motion a takes a certain value satisfies that dynamics on Y project onto the dynamics we had in M with the Lagrangian L . Thus the possible trajectories of all charged particles –with varying values of e/m – is in a sense "parametrized" by the set of all extremals of L0, by the geodesics of a Riemannian metric on Y. We call this metric the Kaluza-Klein metric, and it is a natural geometric object attached to the electromagnetic field. We can picture "charge" e as if it was parametrized by the coordinate φ, the fiber space of the bundle Y when considered as a fiber bundle over M = R4. The fundamental observation here is that this generalizes naturally to Yang- Mills fields. And this is the idea we want to show and that Hermann in [9] emphasizes, the deep relationship between Kaluza-Klein and Yang-Mills.

0.6.2 Principal Fiber Bundles and their associated Ehresmann connections Now we will give a view on what do the notions of Principal Bundles and Ehresmann connections mean. We are ready now to generalize the Kaluza-Klein construction to Yang-Mills situations. In the Yang-Mills situation we will be dealing with principal ﬁber bundles

π : Y → R4, with R4 as base space and whose "ﬁber" (the preimage of a point in the base manifold) is a compact Lie Group G. We further require in this concrete investigation that we have a metric on Y such that π is a Riemmanian submersion map and such that parallel transport is translation on the Lie Group. Be- fore we do all these construction we will review some of the concepts needed before, Consider Y, X to be arbitrary smooth manifolds and a map π : Y → X. Let G denote a Lie group that acts as a transformation group on Y. Now we introduce a fundamental deﬁnition:

Definition 0.6.2.1. We call all this structure (Y, π, X) and the transformation group G a principal fiber bundle with G as structure group if the following conditions are satisfied:

1. The orbits of G are precisely the ﬁbers of π, this means that for every g ∈ G and y ∈ Y we have

π(gy) = π(y),

and if π(y) = π(y1) then we can find a group element g ∈ G such that y1 = gy. 2. G acts freely, this means that if an element g ∈ G leaves a point fixed then it leaves all points of Y fixed.

3. π is a submersion map.

For most purposes (and certainly for ours) it is very useful to introduce an aditional condition. We suppose too that G acts effectively in the sense that every nonzero element of the group always acts non-trivially. With this conditions and the above, we have the following: each orbit of G on Y (i.e. each fiber of π) is identified with the group G itself. Note that in general the identification depends on the point of the fiber one considers, and is in general not a natural identification. We refer to Hermann in [9] for the details on this and the importance this ambiguity may have. This motivates another definition that is usually given for Principal Bundles, and the important fact that we want to emphasize is that they are just different views on a certain geometrical object. Later in his book Hermann defines this kinds of bundles like the bundles π : Y → M as with structure group G as the ones that satisfy this essential observation, that fibers just identify with the group, since it is usually better addapted to the geometrical understanding of physics that Hermann develops. This notion of bundles allows as to define a very related notion that is also essential, that of Associated Bundles. Definition 0.6.2.2. Consider that we have a Principal G-bundle π : Y → M. Let F be another space together with a transformation group action of G on F. Now consider the action of G on the product Y × F via,

g(y, f ) = (gy, g f ).

Now the orbit space YF = G (Y × F) is called the associated fiber bundle with canonical fiber F. For it to have a bundle structure we need a "projection" defined, consider

πF : YF → M (y, f ) 7→ π.(y)

One can see that the fibers of the bundle πF : YF → M are just identified with F itself (Theorem 6.1 in [9]). But that the isomorphism one gets is not "uniquely determined". Recall the important idea we remarked in the first sections, Bundles are important because they allow us to talk about sections. Remember that the concept was a priori not hard to grasp, if π : E → M is a fiber space over a manifold, the space of cross-section maps is just defined to be

Γ(E) = {ψ : M → E|πψ = identity}.

This is a basic object of interest both to physicists and mathematicians. And we make now a fundamental observation on the nature of this objects, if we are in the case were the fiber space is constructed using the "associated bundle", sections can be understood in a different and useful way. Thus we consider all the structure we introduced before, that we have an associated bundle πF : YF → M to a certain principal G-bundle π : Y → M in the sense we introduced them before, and recall that YF is a bundle with typical fibre F. Now consider a certain map ψ : Y → F that satisfies the following for every g ∈ G and y ∈ Y: ψ(gy) = gψ(y), that is to say that is invariant under the action of the structure group. Now Hermann introduces the fundamental observation in Theorem 7.1 in [9] , the following fundamental equivalence that allows for a very rich interpretation (more Gauge-Theoretic in the sense of a physicist)

Theorem 0.6.2.1. The space of cross-sections of the associated bundle Γ(YF) can be identiﬁed with the space of maps ψ : Y → F that commute with the action, in the sense given before: ψ(gy) = gψ(y), for every element g ∈ G and y ∈ Y.

We are now going to continue our discussion of Bundles by recalling the notion commented on previous sections, that of Vector Bundles. We will be for now considering the case where the action of G on F is a linear action. This means that we are considering that F has a vector space structure, we will call it E for the time being to remember that we are dealing with a vector bundle. We could choose any ﬁeld of scalars for this vector space, but for our geometric purposes it is enough to consider it as being either R or C. Denote by End(E) the space of linear maps E → E. And further suppose that we are given a linear representation of G on E, this just means we have a homomorphism σ : G → End(E). Now we deﬁne the transformation group action of the group G on E as follows: (g, v) → gv = σ(g)v, thus the construction of the associated bundle

Yσ = G (Y × E) provides a fiber space over M with each fiber isomorphic to E. Now note that although this isomorphisms are not in general unique, each of them preserves the linear structures (i.e. the fibers of the bundle can be given vector structures and hence Yσ → M is a vector bundle. Now let’s see what can we say about the space of sections Γ(Yσ) in this case. Recall that the space of sections can be identified with the space of maps

ψ : Y → E that satisfy the condition

ψ(gy) = σ(g−1)ψ(y) for every g ∈ G and every y ∈ Y. Notice that two maps of that kind can be added (due to the fact that E is a vector space), and this just shows us that there is a "natural" vector space structure for the space of sections Γ(Yσ). Thus what we have seen in this section is the essential importance of the notion of Principal and Associated Bundles. The "associated bundle" construction given above is essential for a deep understanding of differential geometry and physics, since it is the basis for their description.

0.6.3 The notion of Ehresmann connection In this section we will introduce an object of fundamental importance that generalizes the classical notion of connection in differential geometry, Ehres- mann connections, that further generalize the notion of Cartan/Koszul we already introduced, there are three levels of abstraction here!. This notion of connection comes from a fundamental observation that we can split a Principal Bundle in two parts, but that this spliting is not unique an thus one is making a choice. This idea of spliting the bundle comes from considering the differential of the bundle projection map π. We notice that by taking the Kernel (this being the vertical bundle) of this map we induce a splitting of TP into two parts, but we can still make a choice.

Deﬁnition 0.6.3.1. A Ehresmann connection on a principal bundle P is a choice of splitting TP = V ⊕ H, where we require that H (horizontal) is a G-invariant subbundle of TP com- plementary to the vertical bundle V.

This is one of the most abstract notions of a connection one can introduce in differential geometry, but it is fundamental to see the deep structure behind bundles and to correctly generalize previous notions, we will deal in later sections with the question of how to compute with this object, since at ﬁrst it doesn’t seem like an easy thing to do. With these notions introduced we are in the right path to start understanding the deep geometric structure of Yang-Mills theories. We will now introduce in the next section thus the fundamental abstract geometrical concepts needed for the description we are still not fully grasping.

0.6.4 Linear connections in Associated Vector Bundles comming from Ehresmann connections in Principal Bundles In this section we will come back to our previous description of connections and study some specific cases of physical interest. The immediate point for physics of all the formalism that we have introduced is that it gives a way of describing "instrinsically" the basic operation in Dirac’s monopole paper [?]. This operation consists on passing from an electromagnetic field as the curvature of an Ehresmann connection in a principal U(1)-bundle to the "Schrödinger equation" for a "quantum" charged particle that interacts with a classical electromagnetic field. In fact is is remarkable, as Hermann notes, that the global existence of a global U(1)-bundle together with a U(1)-invariant connection is what determines Dirac’s "quantization" condition!, we will not talk much more about this and further details on this question can be found in [9]. We thus need to develop further now our theoretical understanding of the concrete geometrical objects that are in play here, in this section we are considering a vector bundle π : E → X (with real vector spaces as fibers) over a manifold X. As usual denote by Γ(E) the space of sections, Γ(E) is here a C∞(X)-module. Recall that a (linear) connection ∇ for E is an R-linear map

∇ : Γ(TM) × Γ(E) → Γ(E)

(X, α) 7→ ∇Xα that satisﬁes for every smooth function f ∈ C∞(X), every vector ﬁeld V and evey α ∈ Γ(E)

∇ f V α = f ∇V α

∇V ( f α) = V( f )α + f ∇V α

What we want to see now an Ehresmann connection for the bundle E → X can deﬁne a linear connection in this sense. So suppose we are given an Ehresmann connection, namely a ﬁeld

e 7→ He of horizontal tangent subspaces. For every vector field V ∈ Γ(TX), there is a horizontal lifting, which is a vector field VH on E that satisfies:

VH(e) ∈ He ∗ ∗ π (V( f )) = VH(π f ) for every e ∈ E and every smooth function f ∈ C∞(X). There is an important thing to note here, namely that the second property may be written also as dπ(VH) = V, and with this we see that in fact the property just means the fundamental property that π maps an orbit curve of VH onto a orbit curve of V. Now denote by

t → exp(tVH)t → exp(tV) the ﬂow of each respective vector ﬁeld, the previous conditions just imply that the dynamics satisfy the following:

πexp(tVH)e = exp(tV)πe.

Definition 0.6.4.1. We say that the Ehresmann connection H is linear if each exp(tVH) (thought of as a diffeomorphism for each t) maps fibers to fibers in a linear way. What we will now do is use that linear Ehresmann connection to define a covariant derivative operation. Given σΓ(E), t ∈ R and a vector field VΓ(TX), we consider the following object:

σt = exp(tVH)σexp(−tV), which is again a cross-section of the bundle E. We thus deﬁne the associated covariant derivative to this connection as follows: d ∇ σ = σ | = V dt t t 0 With the adequate geometric framework in hand, we jump to the next section and view the power of this ideas in their main physical application.

0.6.5 The electromagnetic field as a U(1)-connection In this section we will apply all the heavy machinery we have developed to the concrete problem of the electromagnetic field. So now we will specialize to the case where we have a bundle π : Y → X with the specific spaces

X = R4, Minkowski space Y = X × C

Where we consider the natural projection for this bundle. In this case, the space of cross-sections Γ(Y) consists of maps of the form

α : x 7→ (x, ψ(x)), with ψ : X → C. And we will during this section why this is actually the appropiate notion to consider physically. We consider the standar coordinates for relativistic physics, (xµ) that we have used before and we will denote by z the complex coordinate. Assume for this theoretical investigation that all the adjustable constants of electromagnetism are equal to one. Thus the structure we have here is known from previous section, the bundle π : Y → X is just a vector bundle with a one-dimensional complex vector space as ﬁber (this is also called in the literature a Line Bundle). We know that a linear Ehresmann connection may be deﬁned by a one-form

µ 1 θ = dz − iAµzdx ∈ Ω (Y, C) (Here the tangent vectors v ∈ TY such that θ(v) = 0 are the horizontal vectors). If one considers the Aµ to be real-valued functions, then the parallel transport associated to the connection preserves the norm |z|2 = zz¯, this just means that the structure group is actually U(1) = SO(2, R), Harmann sees in [?] how one can associate this connection with an invariant connection in a certain principal bundle with U(1) as structure group. We will give a short explanation on how this is done in this section. µ Consider now a vector ﬁeld V = V (x)∂µ on the base manifold X. Now we consider its horizontal lifting VH,

µ µ VH(z) = iAµ(x)V(z )z = iAµ(x)V z,

we thus get that

µ µ VH = V ∂µ + iAµV z∂z. Suppose then that σ ∈ Γ(Y) is a cross section given in the form we said previously (recall that it is given by a function ψ : X → C. Now we want to see how do we compute its covariant derivative. We do this in a computation, start by writing ψ = σ∗(z), now recall that σt = exp(tVH)σexp(−tV), this gives rise to ∗ ∗ ∗ ∗ σt = exp(−tV) σ exp(tVH) . We now compute it’s derivative

∂ ∗ ∗ ∗ µ µ σ (z)| = = −Vσ (z) + σ V (z) = −V ∂ (ψ) + iA (x)V ψ. ∂t t t 0 H µ µ µ We have thus seen that ∇V σ = V(ψ) − iAµV ψ, and this can be written in a short form (but being careful with the "abuse of language") as:

∇V ψ = V(ψ) − A(V)ψ

µ A = Aµdx , what we see from this is essential! This has shown us how the electromagnetic field potential 1-form A operates as a connection (i.e. covariant derivative) on what are usually called in the physics literature "Schrödinger wave functions" x → ψ(x). Consider, as an example of the formalism, a "free" Schrödinger equation for a particle of mass m. The classical equation for energy in this case is just 1 E = p · p, 2m what this means Quantum Mechanically is that 1 (i∂ − δij∂ ∂ )ψ = 0, 0 2m i j where both i, j run from 1 to 3 and we have set Planck’s constant equal to one. If in this equation one replaces partial derivatives with covariant derivatives using the previous formula, to incorporate the action of a electromagnetic field, one gets 1 (i(∂ − iA ) − δij(∂ − iA )(∂ − iA ))ψ = 0, 0 0 2m i i j j be careful not to mix the imaginary unit i with the summation index! This equation we obtained is thus a differential equation for ψ, considered as a map ψ : X → C. Appart from that it has great physical importance, since it is the equation for a charged particle in an electromagnetic field. Notice that in this example we started with a strong assumption, namely that we considered the bundle to be the trivial bundle (the cartesian product), but the fundamental observation here is that this may not be always the case ("It might be a product in many ways"). For instance, one could consider

iλ(x) α1(x) = (x, e ψ1(x)) with ψ1 : X → C and λ(x) a fixed real-valued function on the base manifold X, the whole picture would change, we have thus to consider arbitrary line bundles, the trivial one doesn’t suffice. It is important to note that one wants "U(1)-isomorphisms" (those preserving the structure group), not only for purely mathematical purposes, but also for physical reasons, to preserve the physical interpretation of |ψ(x)| as a probability distribution, as one wants in the Quantum Mechanical description of systems. With all this seen we arrive to the fundamental conclusion, that quantum mechanics should deal with cross-sections of line bundles, not just with the treat- ment usually given about "complex-valued functions". This geometric interpretation has been heavily emphasized in the work of Kostant’s and Souriau’s work on the program that goes under the name of Canonical Quantization. Physicist call this whole apparatus "gauge invariance". What this illus- trates is the deep notion that the "theory of gauge fields" is identical with the theory of fiber spaces!

0.6.6 Line Bundle defined by a closed 2-form on a manifold In this section we will further explore the fundamental relationship we es- tablished between physics and the theory of fiber spaces, and further explore the geometry behind all this. One of the key ideas of the theory of connections we have been using is that a connection has an associated curvature tensor! This curvature tensor is a "2-differential form" on the base manifold of our bundle of a more general type (it takes values in a vector bundle). Thus we introduce the fundamental concepts adapted to this situation, we start with a vector bundle π : E → X with real vector spaces as fibers. We consider the space of cross-sections Γ(E) as a C∞(X)-module. For a certain connection on the vector bundle ∇ : Γ(TM) × Γ(E) → Γ(E), we can associate its curvature tensor K defined as follows:

K(V V )(σ) = ∇ ∇ (σ) − ∇ ∇ (σ) − ∇ (σ) 1, 2 V1 V2 V2 V1 [V1,V2] , for every pair of vector fields V1, V2 and every section σ. Notice that K is C∞(X) − bilinear (a tensor field). Now denote by End(E) the vector bundle over X whose typical fiber is End(Ex), the space of linear maps (denoting by Ex the vector fiber of the bundle π : E → X at the point x) Ex → Ex. Having this in mind we talk in the following terms about this object

K is a 2-form on X with values in End(E) (in the bundle associated to the endomorphisms). Or in shorthand notation, K ∈ Ω2(X, End(E))

it is a fundamental view on curvature, that relates deeply to the classical notion. The problem one faces in this subject is that many similar concepts described in apparently different ways turn out to be describing the same notion, and one has to be ready to accept the differences and to try to incorporate them into a deeper theoretical perspective, and much work has been done in this area both in mathematics and physics. We have thus seen that we have a deep "functorial" relationship between

(vector bundles, connections) → 2-forms on the base manifold X of a certain type.

Being able to correctly characterize the image of this mapping, thus knowing when a certain tensor field of a certain type is the curvature tensor of a certain vector bundle. This is a highly non-trivial task in general, however we can further study a special case which is extremely important, the case of line bundles. We suppose now that the fibers of E are 1-dimensional complex vector spaces, namely that E is a (complex) line bundle. Then notice that we can clearly multiply each section σ ∈ Γ(E) with the imaginary unit i, thus Γ(E) becomes a complex vector space. Notice that in this case, we have a bigger space acting on the sections, we have that C∞(X) ⊗ C (the algebra of complex-valued smooth functions on X) acts naturally on Γ(E). Since we are in the complex case, we further suppose that the connection ∇ satisfies the following condition

∇V (iσ) = i∇V, for every vector ﬁeld V ∈ Γ(TM) and every section σ ∈ Γ(E). We can equivalently say this by imposing ∇ to be a connection relative to C∞(X) ⊗ C, this meaning that it satisﬁes

∇V ( f σ) = X( f )σ + f ∇V σ, for every smooth function f ∈ C∞(X) ⊗ C. Using the deﬁnition of the curvature tensor given before we thus get the following property

K(V V )(iσ) = ∇ ∇ (iσ) − ∇ ∇ (iσ) − ∇ (iσ) = iK(V V )σ 1, 2 V1 V2 V2 V1 [V1,V2] 1, 2 .

This just means that we can interpret every K(V1, V2) (for two concrete ∞ vector ﬁelds V1, V2) as a C (X) ⊗ C linear map

K(V1, V2) : Γ(E) → Γ(E). Now we will heavily use the fact that we are in a line bundle. Since the ﬁbers of the vector bundle E are 1-dimensional complex vector spaces and K(V1, V2) is complex-linear, we thus conclude that there exists a certain complex-valued 2-form ω ∈ Ω2(X, C), that is just the curvature we have,

K(V1, V2)(σ) = ω(V1, V2)σ, for every pair of vector ﬁelds V1, V2 ∈ Γ(TM). Now we will see a very deep result, that heavily connects with the ideas we developed in previous sections on symplectic geometry.

Theorem 0.6.6.1. The 2-form ω is a closed form on X (furthermore, it is locally exact), dw = 0.

Proof. Recall that we are in a context where π : E → X is a line bundle, not necessairly trivial, but are about to see how can we bypass this non-triviality. This result is purely local, and hence it sufﬁces to show that it is true locally. What this means is that if our bundle is not the trivial bundle, we can work on a sufﬁciently small open neighborhood of the base manifold X so that we can restrict our bundle appropriately and consider a trivial bundle (at ∞ least locally). This means that Γ(E) has a basis σ0 as a C (X) ⊗ C-module. Consider θ to be the complex-valued 1-form on the base manifold X such that

∇V σ0 = θ(V)σ0 is satisfied for every vector field V ∈ Γ(TX). Now consider V1, V2 to be two vector fields in the base manifold X that commute,

[V1, V2] = 0. Thus we can write the curvature 2-form ω in the following way:

( )( ) = ∇ ∇ − ∇ ∇ ω V1, V2 σ0 V1 V2 σ0 V2 V1 σ0 = ∇ ( ( ) ) − ∇ ( ( ) ) V1 θ V2 σ0 V2 θ V1 σ0 = V1(θ(V2))σ0 + θ(V2)θ(V1)σ0 − V2(θ(V1))σ0 − θ(V1)θ(V2)σ0

And if we write this in a more elegant form using the exterior derivative we have seen that

ω = dθ

and hence we have that dω = 0, by the deﬁning property of the exterior derivative (d2 = 0). Let us summarize what have we seen here. We have a complex line bundle

π : E → X.

We can understand the connection ∇ in any of the views we already have about them, we will prefer the interpretation that it is a linear covariant derivative that satisﬁes ∇V (iσ) = i∇V σ for every vector ﬁeld X ∈ Γ(TX) and every section σ ∈ Γ(E). We end this section with the following theorem that carries information about the difference between two connections (this is very important if one tries to study dynamics in this space of connections!).

Theorem 0.6.6.2. Let π : E → X be a complex line bundle and ∇, ∇0 be two different connections on the bundle satisfying the previous mentioned condition. Denote by ω, ω0 the curvature 2-forms of these connections. Then the theorem asserts that ω and ω0 belong to the same de-Rahm cohomology class, recall this means that there is a 1-form η such that ω − ω0 = dη.

Proof. Start by considering the "difference" of the vector bundle connections:

0 τ(X)(σ) = ∇Xσ − ∇Xσ.

Due to the cancellation of the non-tensorial terms, τ is C∞(X)-bilinear. It also satisﬁes that τ(X)(iσ) = iτ(X)σ, and hence in the same way we did before, τ is determined by a 1-form η, in the sense that τ(X)σ = η(X)σ, and from this we get the result we wanted.

0.6.7 Quantization using Complex Linear Connections in Com- plex Vector Bundles

In this section and the ones that follow we will talk brieﬂy about a fundamental concept in modern theoretical physics, quantization. This is a general notion that includes a variety of frameworks, the general idea that is common to all of them is to better understand the relationship between the classical description of the physical universe and the Quantum Mechanical description. Quantization can be described thus as a certain process that allows us to describe the transition between both by developing a framework in which to quantize a classical theory. What we want to emphasize is the deep idea that we can try to understand quantization geometrically. The process of Quantization usually takes the following path: We start with a given smooth manifold X. We know from previous sections that the algebra of C∞(T∗X) functions on the cotangent bundle admits a Poisson Bracket structure coming from the canonical 2-form that lives in the cotangent bundle. With a given volume element form dx on X we put a Hilbert space structure on C∞(X) ⊗ C Z < ψ1|ψ2 >= ψ1(x) ∗ ψ2(x)dx, X this works nicely if the manifold X is compact, if it isn’t, one should restrict adequately the space of functions to be considered, but we will not enter in those analytic details in this work, suppose for the time being that X is compact to see how the whole machinery works. Now the Holy grail of this whole process is being able to assign Hermitian Operators (the fundamental object of Quantum Mechanics) to functions f ∈ C∞(T∗X) in such a way that the Poisson Bracket goes onto operator commutator, and thus viewing (in a certain sense) classical information in the quantum setting, thus we are quantizing the classical system. We want to generalize the case of the line bundle, and there are heavy topological restrictions to this whole process (restrictions that Dirac was able to intuitively deduce in his work [?]). This view allows us to obtain "quantum" effects for elementary particles in a natural geometric way.

0.6.8 Hermitian Vector Bundles and Connections In this sections we will further investigate the notions introduced in the last section in an important setting that is very relevant to physics. We consider the same framework we have been working on, namely a (complex) vector bundle π : E → X. We further suppose that for every point x ∈ X, the ﬁber π−1(x) has a Hermitian symmetric inner product that we denote by < | >. We suppose further that this inner product varies smoothly in x, the geometric object we are considering is usually called a Hermitian Vector Bundle.

Deﬁnition 0.6.8.1. A linear connection ∇ for a Hermitian Vector Bundle is a connection (in the Ehresmann sense, or thought of as a covariant derivative) that can be characterized by the following algebraic properties:

1. ∇V (iσ) = i∇V σ

2. V(< σ1|σ2 >) =< ∇V σ1|σ2 > + < σ1|∇V σ2 > that need to be satisﬁed for every vector ﬁeld V and every pair of sections σ1, σ2 ∈ Γ(E). Thus we see that this is the natural generalization of the concepts we developed for line bundles.

Now we suppose that we have a Hermitian connection ∇. Now we want to introduce a similar structure (product) as the one we defined on sections of the line bundle that we defined in the previous section, recall that we have to be careful with the analytical details if X is not compact. To deal with this analytic difficulty, we will just consider cross-sections σ : X → E that have compact support (this just means that they vanish outside of a compact set of X), this will allow us to integrate without analytical inconsistencies. We denote the space of cross-sections with compact support by Γ(0)(E). Consider dx to be a volume element form for X. With this structure in hand we can define an inner product on Γ(0)E that makes it into a pre-Hilbert space Z (σ1, σ2) = < σ1(x)|σ2(x) > dx. X We note here that this may not define a Hilbert space since this space may not be complete (in the sense that every Cauchy sequence has limit in the space). In the physics literature there is usually no distinction made, but it is an important thing to note from a mathematical point of view. We will now consider a notion of divergence in our context.

Definition 0.6.8.2. For a vector field V ∈ Γ(TM), we define div(V) to be the smooth function such that

LV (dx) = (divV)dx.

With this definition in hand we will define the following object acting on Γ(0)(E): i h (σ) = i∇ (σ) + (divV)σ. V V 2 The definition of this object is essentially motivated by the following theorem,

Theorem 0.6.8.1. (Theorem 20.1 in [9]). The object hV : Γ(0)(E) → Γ(0)(E) we de- ﬁned is (with respect to the previously deﬁned inner product on Γ(0)(E)) a symmetric ("Hermitian", in physics language) operator.

We can also consider the following object, related to this one. For a smooth function f ∈ C∞(X), we deﬁne

h f σ = f σ.

The remarkable fact is that h f is also a symmetric operator! And also satisﬁes the following relation: [hV, h f ] = ihV( f ) (Theorem 20.2 in [9]). And thus we have introduced Quantum Mechanical structures into our geometric structure, and modulo some technical details of functional analysis (since we should generate an actual Hilbert Space to deal correctly with Quantum Mechanics), we deﬁned a "quantization" procedure. And one can see that both operators h f and hV are essentially self-adjoint, and this shows us that we have gone deeply into the domain of quantum mechanics, since self-adjoint operators on a Hilbert space are the fundamental objects of quantum mechanics. This is one of the fundamental ideas behind the Kostant-Souriau program for elementary particle physics. This view however introduces an important consequence into the whole picture:

The existence of the Hermitian bundle imposes topological conditions!

And with this program in hand they are able to geometrically explain the quantization of Dirac’s magnetic monopole! The concrete computations relating to this example can be found in [9]. For the concrete case of Line Bundles, for the interested reader, the relevant topological restriction is that The curvature of a Hermitian Line Bundle belongs to an integral cohomology class This is proven by Hermann in [9], and it is, for the author, a very deep result. We quote his own words:

This result is beautifully simple, once one understands the machinery, but is amazingly powerful for "quantization" purposes [...] it gives precisely Dirac’s quantization condition for the magnetic monopole!

We thus end up with this deep and fundamental result in the geometrization of physics. Now we will give some ideas on how to treat some spaces of fundamental importance in modern theoretical physics, Spaces of connections. Some of the ideas related to Spaces of Connections been a major topic of research for more than decades, there is a famous paper written by Atiyah- Bott on the subject, where they study Yang-Mills theories for Surfaces [?]. In these paper the authors introduce the deep observation that the curvature works as a moment map for the Gauge Group action (fiber-preserving transformations) on the space of connections of a principal bundle over a Riemannian surface. But bear in mind that for all this talk we would need the notion of an infinite-dimensional manifold (since spaces of connections are usually infinite-dimensional). Due to this fact and that we have already given a brief description of this process of geometrization of physics, this will be the end of this work. But it is an open end, since History is still being written, and there are still many things to understand in this area of geometry that deeply connects to theoretical physics. We will thus end this work by referring the reader to the description that the Clay Mathematics institute gives of the state of the art of Yang-Mills theory http: //www.claymath.org/millennium-problems/yang-mills-and-mass-gap

.1 Tensors, a brief description

Tensor Product of Modules

Let R be a ring and A, B, C ∈ R-Mod. A map Phi : A × B → C is said to be R-linear if ∀a ∈ A the map Φa : B → C, b 7→ Φ(a, b) and ∀b ∈ B the map Φb : A → C, a 7→ Φ(a, b) are both R-linear. Now we want to construct a R-Module A ⊗R B and the inclusion (bilinear) map ⊗ : A × B → A ⊗R B, namely the tensor product of A and B, such that R-bilinear maps A × B → C are in one-to-one correspondence with R-linear maps A ⊗R B → C. In the language of modern mathematics, that for every R-bilinear mapping Φ, ∃!Φˆ (linear map) that makes the following diagram commute:

⊗ A × B A ⊗R B

Φˆ Φ C

As a sidenote, formally we should write below every tensor product the base Ring we are using to define it, but that is only useful to do in the first definitions and when one can not deduce that from context. This module exists (via an explicit construction using a quotient on a certain free R-Module) and it is unique up to isomorphism (in the appropiate Category). The fundamental thing to keep in mind about the tensor product is the defining Universal Property, this is what is essential. Note too that if (ai)i∈I, (bj)j∈J are generators of A and B respectively, then the elements (ai ⊗ bj)i∈I,j∈J generate A ⊗ B. We note that we have introduced the tensor product for bilinear maps but obviously this can be iterated and is in fact equivalent to starting with multilinear mappings Φ : A1 × ... × As → C linear in each variable. Following the same proof as before we end up with a tensor product A1 ⊗ ... ⊗ As generated by all products (a1 ⊗ ... ⊗ as) with ai ∈ Ai. This tensor product satisfies, just as before, that it is unique up to isomorphism. Now there are some so-called "canonical isomorphisms", some of which are the following. Recall R is a ring and A, B, C ∈ R-Mod

1. A ⊗ B → B ⊗ A

2. (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) → A ⊗ B ⊗ C

3. (A ⊕ B) ⊗ C → (A ⊗ C) ⊕ (B ⊗ C)

4. R ⊗ A → A that correspond respectively to the following maps,

1. a ⊗ b 7→ b ⊗ a

2. (a ⊗ b) ⊗ c 7→ a ⊗ (b ⊗ c) 7→ a ⊗ b ⊗ c

3. (a, b) ⊗ c 7→ (a ⊗ c, b ⊗ c)

4. r ⊗ a 7→ r · a Note that many maps between tensor products get deﬁned on (rank one) simple tensors, and then extender linearly (this is common practice when talking about tensors, and we will do it in what comes, not always mentioning the fact that we have to be careful and extend the map to the whole tensor product, not just on simple tensors, but that is just a linear extension that can be performed with no problem). Now we have a fairly good understanding of tensor products of Modules and its elements, is there any sense in which we can talk about tensor products of maps? Does this tensor map ⊗ have any functorial propertie? The answer to this questions is yes, and works as follows. Let ϕ : A → A0, ψ : B → B0 be linear maps of R-Modules. Deﬁne Ξ : A × B → A0 ⊗ B0, (a, b) 7→ Ξ(a, b) = ϕ(a) ⊗ ψ(b). We quickly see that this map Ξ is R-bilinear and therefore, by the universal property of tensor products, it induces a R-Module homomorphism:

ϕ ⊗ ψ : A ⊗ B → A0 ⊗ B0 (1) satisfying (ϕ ⊗ ψ)(a ⊗ b) = ϕ(a) ⊗ ψ(b) (2) This deﬁnition of tensoring maps turns out to be very useful to describe suc- cessfully properties (operators) of the composite state of a Quantum Mechani- cal system (that we previously said was nothing more than the tensor product ⊗ of the underlying spaces).

Specialising to Tensor products of vector spaces

The first concept we will have to introduce in tensors is rank. Let {Ai}i∈I N be a collection of vector spaces, we say a tensor T ∈ i∈I Ai is of rank one if ∃αi ∈ Ai such that T = ⊗i∈I αi, if we restrict to finite tensor products and ∗ say that Ai = Vi for i = 1, ..., n, i.e that each Ai is the dual space of a certain vector space Vi. Rank one means now that the tensor T has the following = ⊗ ⊗ ( ) ∈ n form T α1 ... αn and that it acts on vectors v1, ..., vn ×i=1 Vi in the following fashion, T(v1, ..., vn) = α1 ⊗ ... ⊗ αn(v1, ..., vn) = α1(v1)...αn(vn). Note that having rank one is something independent of the basis one is using. N Now we say a tensor T ∈ i∈I Ai is of rank Rank(T) = r if this number r is the minimum number r such that T = Σu=1Zu with each Zu of rank one. A good property to have in mind (that we passed through in the algebraic description) is that for a tensor T ∈ A1 ⊗ ... ⊗ An, Rank(T) ≤ ∏i(dimAi). So the rank has a finite number of possible values. Note that the rank of a tensor is invariant under a change of basis of the ⊗d Nd vector spaces Ai. From now on we call V = i=1 V, the tensor product of d copies of V. Observe now that we get a natural action of the group GL(V) on V⊗d. This ⊗d action on rank one elements is, for g ∈ GL(V) and v1 ⊗ ... ⊗ vd ∈ V ,

g · (v1 ⊗ ... ⊗ vd) = (g · v1) ⊗ ... ⊗ (g · vd) we obtain the action on the whole V⊗d just by extending this linearly. In a similar way, GL(V1) × ... × GL(Vn) acts on V1 ⊗ ... ⊗ Vn. Now we give some examples of invariant tensors, namely tensors that, viewed as multilinear maps, commute with the action of the change of bases group (they are invariant with respect to the group action, matrix multiplication is one of those). Contractions of tensors: We get a natural bilinear map, ∗ Con : (V1 ⊗ ... ⊗ Vn) ⊗ (Vn ⊗ U1 ⊗ ... ⊗ Um) → V1 ⊗ ... ⊗ Vn−1 ⊗ U1 ⊗ ... ⊗ Um given by,

(v1 ⊗ ... ⊗ vn, α ⊗ b1 ⊗ ... ⊗ bm) 7→ α(vk)v1 ⊗ ... ⊗ vk−1 ⊗ b1 ⊗ ... ⊗ bm. This operation is called contraction, recall that we used this before for deﬁning the Ricci tensor, it is a good exercise to convince oneself that they are in fact the same operation. Matrix multiplication as a tensor: Let A, B, C be vector spaces of dimensions a,b,c, consider the matrix multiplication operator Ma,b,c that composes a linear map A → B with a linear map B → C giving rise to a linear map ∗ ∗ ∗ A → C. Now let V1 = A ⊗ B, V2 = B ⊗ C, V3 = A ⊗ C, so really the matrix ∗ ∗ multiplication operator is an element Ma,b,c ∈ V1 ⊗ V2 ⊗ V3. It works in the following way: ∗ ∗ ∗ Ma,b,c : (A ⊗ B) × (B ⊗ C) → A ⊗ C that on rank one tensors looks like, (α ⊗ b) × (β ⊗ c) 7→ β(b)α ⊗ c Another GL(V)-invariant tensor: The space V ⊗ V ⊗ V∗ ⊗ V∗ = End(V ⊗ V), in addition to the identity map IdV⊗V, has another GL(V)-invariant tensor. As a linear map we can write it: σ : V ⊗ V → V ⊗ V that looks like, a ⊗ b 7→ b ⊗ a

Symmetric and skew-symmetric 2-tensors

In this section we introduce a crucial object for us, namely symmetric and skew-symmetric tensors, since they are very important theoretically and prac- tically (in Quantum Mechanics or General Relativity for example). ⊗2 Consider a vector space V and V = V ⊗ V with basis {ei ⊗ ej}1≤i,j≤dimV we deﬁne the two subspaces (symmetric and skew-symmetric 2-tensors respectively): 2 S V = span(ei ⊗ ej + ej ⊗ ei, 1 ≤ i, j ≤ dimV) = span(v ⊗ v, v ∈ V) = {X ∈ V ⊗ V|X(α, β) = X(β, α)∀α, β ∈ V∗} = {X ∈ V ⊗ V|X ◦ σ = X} V2 V = span(ei ⊗ ej − ej ⊗ ei, 1 ≤ i, j ≤ dimV) = span(v ⊗ w − w ⊗ vkv, w ∈ V) = {X ∈ V ⊗ V|X(α, β) = −X(β, α)∀α, β ∈ V∗} = {X ∈ V ⊗ V|X ◦ σ = −X}

With the convention that we are considering X as a map X : V∗ ⊗ V∗ → C. The second description of these spaces implies that both S2V and V2V are invariant under linear changes of coordinates (if T ∈ S2V, g ∈ GL(V) then g · T ∈ S2V, and similarly for V2V), so they can ve viewed as GL(V)-submodules of V⊗2. Now we define to crucial operations for the study this spaces of tensors, the symmetric product and the skew-symmetric product: Let v1, v2 ∈ 1 2 1 V, we define v1v2 : = 2 (v1 ⊗ v2 + v2 ⊗ v1) ∈ S V and v1 ∧ v2 : = 2 (v1 ⊗ v2 − V2 v2 ⊗ v1) ∈ V. From these definitions follow a bunch of properties for this spaces and operations that we need to mention (Exercises 2.6.2. in [?]): 1. V ⊗ V = S2V ⊕ V2V (with the above said, this direct sum decomposition is invariant under the GL(V) action. One says that V⊗2 decomposes as a GL(V)-module to V2V ⊕ S2V.

2. No proper linear subspace of S2V is invariant under the action of GL(V), or in other words, that S2V is an irreducible GL(V)-submodule of V⊗2, the same is satisﬁed for V2V.

3. Deﬁne maps

⊗2 ⊗2 πS : V → V 1 X 7→ (X + X ◦ σ) 2 ⊗ ⊗ π∧ : V 2 → V 2 1 X 7→ (X − X ◦ σ) 2 ⊗2 2 ⊗2 V2 We have that πS(V ) = S V and π∧(V ) = V. Let’s now extend this maps to more than 2-tensors (what we have deﬁned).

Symmetric and skew-symmetric tensors in general

d ⊗d ⊗d Start with Symmetric tensors S V. Let πS : V → V be the map defined on rank one tensors by 1 π (v ⊗ ... ⊗ v ) = v ⊗ ... ⊗ v , S 1 d d! ∑ τ(1) τ(d) τ∈Sd where Sd denotes the group of permutations of d elements. Introduce the d ⊗d notation v1v2...vd : = πS(v1 ⊗ v2 ⊗ ... ⊗ vd). We thus define S V : = πS(V ), the d-th symmetric power of V. From this definition we can extract some properties (Exercises 2.6.3.3,2.6.3.4,2.6.3.5 in [?]): 1. In bases, if u ∈ SpCr, v ∈ SqCr, the symmetric tensor product uv ∈ Sp+qCr is 1 (uv)i1,...,ip+q = uI vJ, ( + ) ∑ p q ! I,J

where the summation is over I = i1, ..., ip with i1 ≤ ... ≤ ip and analo- gously for J.

2. SdV ⊂ V⊗d is invariant under the action of the group GL(V).

( ) ( ) d 3. If e1, ..., ev is a basis of V, then ej1 ej2 ...ejd 1≤j1≤...≤jd≤v is a basis of S V. d v v+d−1 We can thus conclude that dimS C = ( d ). Now we want to understand (connecting to concepts one uses in Algebraic Geometry) SkV∗ as the space of homogeneous polynomials of degree k on V. Recall the space SkV∗ was the space of symmetric k-linear forms on V. We want to see now that it can also be considered as the space of homogeneous polynomials of degree k on V. Given a multilinear form P, the map x 7→ P(x, ..., x) is a polynomial mapping of degree k. The process of passing from a homogeneous polynomial to a multilinear form is called polarization. As an example, let P be a homogeneous polynomial with degP = 2 in V, deﬁne the following bilinear form P by the equation

1 P(x, y) = (P(x + y) − P(x) − P(y), 2 for general multilinear forms, the polarization identity is, using standard multi-index notation: 1 P(x , ..., x ) = (−1)k−|I|Q( x ) 1 k k! ∑ ∑ i φ6=I⊂{1,...,k} i∈I

Note that one can thus understand P and P as the same object, and thus we generally do not distinguish them by different notation (useful, but one has to keep in mind that they are different beasts). Also note that we will concerned with homogeneous polynomials (for reasons the Algebraic Geometer should be aware of) and there is no loss of generality in this (Section 2.6.5 in [?]). From this perspective, the contraction map we talked about before is

V∗ × SdV → Sd−1V (α, P) 7→ P(α, ·)

Observe that is one fixes α ∈ V∗, this is just the partial derivative of P in the direction of α (∂αP). One can see this by choosing coordinates to work locally (derivatives are a local operation) and taking α = x1. Now let’s turn to Alternating tensors. Define the following map (defined as always for rank one tensors and extended linearly): ⊗ ⊗ π∧ : V k → V k 1 v ⊗ ... ⊗ v 7→ v ∧ ... ∧ v : = (e(τ))v ⊗ ... ⊗ v , 1 k 1 k k! ∑ τ(1) τ(k) τ∈Sk where Sk is as before the symmetric group of permutations and e(τ) = ±1 Vk is the sign of the permutation τ. Denote its image by V = Im(π∧), it is usually called the space of alternating k-tensors, note it agrees with the previous definition we had of V2V when k = 2, so it is in fact a generalization of the concept. We have in particular that:

Vk ⊗d V = {X ∈ V |X ◦ τ = e(τ)X, ∀τ ∈ Sk} . This space satisfies an interesting property, that we will use later and has Vk great influence on the structure of V, namely that an element v1 ∧ ... ∧ vk = 0 ⇔ v1, ..., vk are linearly dependent. Now we introduce some abstract algebras (tensor algebra, symmetric algebra, exterior algebra) one has to keep in mind. Let V be a vector space, we define the following objects:

⊗ L ⊗k 1. V : = k V , the tensor algebra of V.

V L Vk 2. V : = k V, the exterior algebra of V.

L d 3. S V : = d S V, the symmetric algebra of V. with multiplications deﬁned respectively (and extending linearly if needed) as: N 1. v1 ⊗ ... ⊗ vs w1 ⊗ ... ⊗ wt to be v1 ⊗ ... ⊗ vs ⊗ w1 ⊗ ... ⊗ wt

Vs Vt 2. α ∧ β : = π∧(α ⊗ β) for α ∈ V, β ∈ V s t 3. αβ : = πS(α ⊗ β) for α ∈ S V, β ∈ S V So we get in fact (checking the proper technical properties we are not proving) three algebras. Now we turn up again to an old friend, namely the operation we have introduced several times now called contraction. What we want to emphasize is that contractions preserve symmetric and skew-symmetric tensors, a fact that practitioners just accept like obvious (since they work in coordinates.) Recall the contraction

V∗ × V⊗k → V⊗k−1

(α, v1 ⊗ ... ⊗ vk) 7→ α(v1)v2 ⊗ ... ⊗ vk.

We could deﬁne as well contractions in any of the factors, we are not restricted to that obviously. This contraction preserves the subspaces of symmetric and skew-symmetric tensors (Exercise 2.6.10 (3) in [?]). This is an important fact, and we have more on this, namely Remark 2.6.9.1 in [?] which says that due to The ﬁrst fundamental theorem of invariant theory [25] we know that the only GL(V)-invariant operators are of the form of the contraction just above, and that the only SL(V)-invariant operators are these and contractions with the volume form. (SL(V) being the group of invertible endomorphisms of determinant one).

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