On Chern-Simons invariant of 3-manifolds

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UNIVERSIT `

A DEGLI STUDI DI PISA

Department of Physics

Master Thesis in Physics

2018/2019

On Chern-Simons invariant of 3-manifolds

Andrea Bevilacqua

Advisors: Prof. Riccardo Benedetti, Prof. Enore

Guadagnini

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Abstract

This master thesis is centered on the study of the Chern-Simons (CS) theory and on the calculation of invariants of 3-manifolds. There are several reasons why this is an interesting topic from both a physical and a mathematical point of view.

From a mathematical point of view, there are still many unanswered questions regarding the quantization of the CS theory. More pragmatically, the study of this topic is also an introduction to several technical tools of great importance in contemporary topology and mathematical physics.

From a more physical point of view, one of the main reasons of interest in Chern-Simons theory lies on the fact that it is a non-trivial quantum field theory of topological type, meaning that it is a quantum field theory that exhibits gen-eral covariance. Contrary to canonical Quantum Field Theory (QFT), which is formulated in the context of a fixed flat Minkowski space-time background, it can give some insight on the structure of a theory which unifies concepts from Gen-eral Relativity (GR) and QFT. On more practical side, CS theory is intimately linked to the calculation of manifolds and knots invariants. In this context, it can be linked to the Reshetikhin-Turaev construction for the calculation of invariants [RT91], which believed to constitute an abstraction of the CS procedure; further details on this topic can be found in [Wit89] and especially in [GMT17]. In par-ticular, the formalism used in [RT91]to solve the Yang-Baxter equation allowed in recent times to mathematically formulate the so called Yangian symmetry, which is at the basis of a renewed interest in the reformulation of scattering amplitudes in a fundamentally new way [AHT14].

In this thesis we will start with a mathematical introduction to Rn and its features, defining calculus in Rn_{. This will be the starting point for the definition}

of manifolds, fiber bundles and their properties. In particular, we will define tangent bundles and principal bundles, which will play a fundamental role in the rest of the thesis. After this, we will also introduce important constructions on manifolds and fiber bundles, such as forms, vector fields and connections. In particular, we will discuss the properties of connections, and we will motivate the fact that we will usually describe them as 1-forms.

Starting from this background, we will introduce the Chern Simons Lagrangian, both in the abelian case and in the non-abelian one. We will study the properties of the CS action under gauge transformation in both cases, and we will discuss its properties. Furthermore, we will investigate the definition of the CS invariant for manifolds with and without boundary. In this context, we will show two different methods for actually calculating the CS invariant. In particular, in the case of

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the structural group SU (N ), we will present the approach based on the so called Heegaard splitting developed in [GMT17], and in the case of structural group SL(2, C) the simplicial approach revisited in [Mar10]. In the two cases, we will introduce the necessary additional mathematical notions needed in order to fully understand them.

After the examples of calculation of invariants of 3-manifolds, we will inves-tigate the issue of quantization in the context of the Chern-Simons theory. In particular, in this context, we will clarify the (physical) assumptions made related with the skein relations in [Wit89]. In this discussion, we will also use [Gua94]. In this way, the thesis will lead toward a deeper comprehension of this theory and its features. We will conclude with some remarks about some open questions related to the CS theory.

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Acknowledgment

This thesis has been the result of long periods of darkness with periodical short periods of light, which were mainly due to my discussions with my two advisors, Prof. R. Benedetti and Prof. E. Guadagnini. Because of this, and for giving me the possibility to study such an interesting and deep topic, I would like to thank them both. Furthermore, nothing in this thesis would have happened if not for A. B., which has always helped me in many ways, guiding me to what I am now.

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Chapter 1 Manifolds and fiber bundles

In this chapter we will introduce the essential definitions that we will use through the text. In particular, we will try to approach each object both from a completely rigorous mathematical point of view, and from a more intuitive and physical one. In this way, we hope to get the advantages of both points of view by gaining intuitive understanding of precise mathematical concepts.

We will first introduce the concept of manifold and of fiber bundle, which will lead us to the definition of the tangent and cotangent bundles as well as to the definition of gauge transformations. After introducing the notions of pullback and pushforward we will introduce the connection and its curvature. For the material in this chapter, our main reference has been [Ben19].

As a matter of notation, we will denote the end of a definition with the symbol •, and the end of a proof with the symbol . Notice, however, that in this chapter we will omit most proofs since this is just a brief introduction to the subject; this does not come at the expense of formality. For the proofs of the statements, we refer the reader to the provided references.

1.1 Manifolds

One of the most important concepts that we will need in the present work is that of a manifold. There are several definitions of what a manifold is, but it can be intuitively defined as a sufficiently well-behaved subset of Rn_{that locally looks like}

Rk with 0 ≤ k ≤ n [WM19][BM95]. The precise mathematical definition has to define what the expressions ”well-behaved” and ”looks like” mean. We will at first give a more concrete definition, and we will then give a more abstract one.

1.1.1 _{Some properties of R}

n

We will start by briefly mentioning some facts about Rn which are surely familiar to the reader, but that will allow us to introduce our notation. The proofs of the statements in the following subsections can be found in any textbook on analysis or differential topology, and we will omit them since they are outside the scope of this section.

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Rn has several properties. It is a vector space of dimension n, each vector is written as a n × 1 matrix, i.e. a column vector, and it has a positive definite scalar product (∗, ∗) given by (x, y) := n X i=1 xiyi = xµδµνyν x, y ∈ Rn (1.1.1)

where in the last step we used the Einstein convention for summation of repeated indices and δµν is the Kronecker delta. Furthermore, it is a complete metric space

with the euclidean distance dn(∗, ∗) given by

dn(x, y) = p (x − y, x − y) = " _n X i=1 (xi− yi)2 #1₂ . (1.1.2)

Using this distance we can define open sets1 _{in R}n_{. A set U ∈ R}n _{is said to be}

open if for each u ∈ U there is a ball

Bn_{(u, r) := {x ∈ R}n : d(x, u) < r} (1.1.3) which is contained in U . Notice that such balls are themselves open sets. With these open sets, Rn is a topological space. We remember that a topological space [RM74] is defined to be a pair (X, T ) where X is a set and T is a topology on X, i.e. a collection of subsets of X which has the following properties:

1) ∅ ∈ T and X ∈ T ;

2) Given a finite family of subsets in T , their intersection is also in T ; 3) Given any family of subsets in T , their union is also in T .

Any subset in this collection T is called open subset of X, and this definition in general does not require the existence of a distance, in particular there are topological spaces2 which however are not metric spaces. A basis of open sets of T is given by open sets {Wi}i∈I such that any open set T ∈ T can be written as

union of sets in the basis. In particular, a basis of open sets of the topology of Rn is given by the balls Bn(p, q) with p ∈ Qn, q ∈ Q, i.e. open balls with rational radius and center in Qn, so that Rn has a countable basis3. In general, we will consider topological spaces with countable basis.

A neighbour of a point x ∈ X is a set U such that there is an open set A ∈ T and x ∈ A ⊆ U . In terms of neighbours, we can define what it means for a topological space to be ”well behaved”, a feature that we described as being one of the key features of the manifolds that we will be considering in this work. 1_{The definition of open sets by using the distance is possible in any metric space, not}

nec-essarily Rn. Furthermore, notice that there is nothing special about the Euclidean distance in terms of the discussions in this subsection. For further details, see [Ben19].

2_{In general when one talks about a topological space it is customary to omit mentioning the}

name T altogether, so one just talks about the topological space X.

3_{Which in this case means that the set of indices I in {W}

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In particular, by well behaved topological space we mean a Hausdorff space: a topological space X is said to be a Hausdorff space if for every two distinct points x, y ∈ X there are two neighbours Ux and Uy (one for each of them) with empty

intersection, Ux∩ Uy = ∅.

From the definition of topological space, it is possible to see that if X ⊂ Rn _is

open, then one can define a topology on X with

T ∩ X = {U ∩ X, U ∈ T } (1.1.4)

where T is a topology of Rn_{; the couple (X ⊂ R}n_{, T ∩ X) is then a topological}

subspace of the topological space (Rn, T ).

One can also define the notion of continuous function and homeomorphism by using open sets. In particular, a function f : X ! Y between two topological subspaces X ⊂ Rnand Y ⊂ Rnis said to be continuous if for every open set U ∈ Y the inverse image f−1(U ) = {x ∈ X : f (x) ∈ U } is open. A homeomorphism is defined as a continuous function f : X ! Y which is also bijective and with continuous inverse f−1 : Y ! X.

The last step that we need before introducing the definition of a manifold is the definitions of smooth function and diffeomorphism. We will introduce differential calculus in the case of Rn.

As a matter of nomenclature, a function f = (f1, . . . , fm) : U ⊂ Rn! V ⊂ Rm

with U and V open sets is a C0 _{function if it is continuous. Adopting the same}

notation as [Ben19], let us call L(Rn, Rm) the space of all the linear mappings L : Rn_{! R}m_{, which can be identified with the space of m × n matrices which we}

call M (m, n, R). Notice that there is a canonical isomorphism between M (m, n, R) and Rnm obtained by a lexicographic order of the matrix entries. We can now define differentiable maps.

A function f : U ⊂ Rn _{! V ⊂ R}m _{with U and V open is said to be}

dif-ferentiable in x ∈ U if there exist a unique linear map dxf ∈ L(Rn, Rm) such

that

lim

h!0

||f (x + h) − f (x) − dxf ||

||h|| = 0 (1.1.5)

where x + h ∈ U and where ||.|| is the Euclidean norm defined in eq.(1.1.2). If a function is differentiable for every x ∈ U , we say that the function f is globally differentiable. In this case, we can define the differential map

df : U ! M(m, n, R) (1.1.6)

with df (x) = dxf . At this point we can define a C1 function as a continuous

function such that it is differentiable and the differential df is itself continuous4_.

In an inductive manner, we can define a Cr _{function as a function f such that its}

differential df is a Cr−1 function. It is now natural to define a smooth function as a C∞_{function, i.e. f is C}r _{∀r. A diffeomorphism f is defined as a homeomorphism}

in which both f and f−1 are smooth functions. 4

Remember that there is a canonical relation between L(Rn

, Rm

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To conclude this section, we notice that the definition of a Cr function can be more concretely defined in terms of partial derivatives. In particular, a function f = (f1, . . . , fm) : U ⊂ Rn ! V ⊂ Rm is a Cr function if it is defined the partial

derivative function

∂Jfi

∂j1x

1. . . ∂jnxn

: U ! R (1.1.7)

which must be continuous, where |J | := j1+ · · · + jn≤ r and i = 1, . . . , m.

1.1.2 Concrete definition of smooth manifolds

We now have all the tools to define an embedded smooth manifold. In the following, we will drop the adjectives ’smooth’ or ’differential’ in front of ’manifold’, we will assume them to be implicit unless otherwise stated.

Definition 1. Embedded manifold (without boundary) [Ben19] [WM19] A topological subspace M ⊆ Rn_{is called an embedded k-manifold (0 ≤ k ≤ n)}

if for all p ∈ M there exist the subsets W ⊂ M (which is taken to be a neighbour of p) and U ⊆ Rk together with a diffeomorphism φ : W ! U between them. • For all the subsets Wi ⊂ M , the functions φi : Wi ⊂ M ! U ⊆ Rk are

called charts and the set of all charts is called the atlas AM of the embedded

manifold. The inverse functions ψi = φ−1i : U ⊆ Rk ! Wi ⊂ M are called local

parametrizations of the embedded manifold. There is a natural extension of this definition to the case with boundary. It is just sufficient to replace Rk _{in the}

definition with the so called half space Hk defined as

Hk = {x ∈ Rk : xk≥ 0} (1.1.8)

where the definitions of chart, atlas and parametrization are left unchanged. The boundary ∂M is defined as those points in M which get sent to ∂Hk = {x ∈ Hk : xk _{= 0} under some chart.}

Before going to a more abstract definition of a manifold, we first introduce the important definition of a submanifold, which can be thought as a subset of a manifold which is itself a manifold.

Definition 2. Submanifold [Ben19]

Given two manifolds X ⊂ Y , we say that X is a submanifold of Y . •

1.1.3 Abstract definition of smooth manifolds

We now give a more abstract definition of the same object5. In particular, this definition will involve the same concepts defined above, but a priori.

5_{As a side note, there have always been two main distinct points of view about whether}

mathematical objects are simply discovered or are created by mathematicians. This is an in-teresting philosophical question that we are however not addressing here, because we will prove that the two definitions are equivalent.

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Definition 3. Abstract manifold (without boundary) [Ben19]

A topological space M is called a smooth m-manifold if it has the following properties:

1) M is a Hausdorff space and it has a countable base; 2) On M a smooth atlas can be defined, i.e.

2a) there is an open covering {Wi}i∈I of M ;

2b) the charts are homeomorphisms, i.e. φi : Wi ! Ui ⊂ Rn with Ui open

∀i ∈ I and n ≥ m;

2c) the transition functions φj◦ φ−1i : φi(Ui∩ Uj)! φj(Ui∩ Uj) are

diffeo-morphisms.

Any smooth atlas A, as defined above, is contained in a maximal smooth atlas

AM. •

As before, the inverse map of a chart is a local parametrization, and we can use the same notation. The case of an abstract manifold with boundary can again be obtained as before.

Notice, however, that now it does not make any sense to say that charts are diffeomorphisms because we have not defined a differential calculus in something which is not Rnfor some n. With the same reasoning, it does make perfect sense to impose that the transition functions are diffeomorphisms since they are functions between open subsets of Rn_.

It can be proved that these two definitions are equivalent, namely [Ben19] if M is an abstract manifold, then there is some embedded manifold ˜_{M ⊂ R}n _{for some}

n together with a diffeomorphism f : M ! M0. In the following, we will omit using the adjectives ’abstract’ and ’embedded’, since it will usually be clear from the context to which of the two we are referring, and in any case one can switch from an abstract one to an embedded one through the diffeomorphism f . In order to give a more concrete intuition about manifolds, we give a few examples:

• The easiest example is given by Rk _{for some k with charts given by the}

identity map.

• Another easy example of manifold is given by the graph of a function. For the sake of the argument, suppose we have a smooth function f : R2 _{! R.}

Its graph will be a two dimensional surface in R3. Open sets on the graph can be defined through the use of (1.1.4), and diffeomorphisms between open sets W on the graph and subsets U ∈ R2 _{can be defined to be the}

projections of the graph on R2.

Two example of 2-manifolds and an example of non-Hausdorff manifold are given in the figures below.

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Using manifolds as our starting point, we now introduce another important quantity in what follows, namely the concept of fiber bundle.

1.2 Fiber bundles

Having defined manifolds, the most relevant non-trivial construction on manifolds that we will employ during the rest of this work is given by the fiber bundles.

Again, we will first begin with a simple case and then we generalize to more interesting situations.

1.2.1 Intuitive introduction: the tangent bundle

The easiest and more direct way to understand what is a fiber bundle is by using the notion of differential map, through which one can define the so called tangent bundle of a manifold M . Intuitively, the tangent bundle assigns to each point x ∈ M a vector space TxM tangent (in the intuitive meaning of the word) in x to

M . To be a little more precise, one first needs to introduce the notion of category of open sets in Rn, and of functor between categories.

Definition 4. Category [Awo10]

A category is composed by the following entities: • Objects, to which we can assign names like A, B, . . . ;

• Arrows, which we can call f, g, . . . . These arrows are sometimes called morphisms,

such that the following properties hold:

• each arrow has a domain object dom(f ) and a codomain object cod(f ), and one writes f : dom(f )! cod(f). In general one never use this notation, and it is customary to call both the domain and the codomain of an arrow by their object names, so one has A := dom(f ), B := cod(f ) and f : A! B; • For each object A there is an identity arrow 1A: A! A;

• Given the arrows f : A ! B and g : B ! C, there exist another arrow g ◦f : A! C which we call the composition of f and g, and the composition is associative, which means that h ◦ (g ◦ f ) = (h ◦ g) ◦ f for every f : A! B, g : B ! C, and h : C ! D. More formally, one can say that the following diagram commutes6

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A B C D f g◦f g h◦g h • f ◦ 1A= 1B◦ f = 1 for all f : A! B. • Definition 5. Functor [Awo10]

A (covariant) functor F between two categories C and D is defined by the following mappings7

• Any object A of C is sent to an object F (A) of D;

• Any arrow f : A! B of C is sent to an arrow F (f) : F (A) ! F (B) of D, such that, for all objects in C one has

• F (1A) = 1F (A);

• F (g ◦ f ) = F (g) ◦ F (f ) (this is the property that defines the covariance of the functor).

• It is fairly easy to give examples of categories and of functors between cate-gories, but this will lead us too far. We simply describe one example of category, which is the one we need in order to quickly introduce the tangent bundle; imme-diately after, we will give the definition of the tangent bundle as a functor.

Example: Category of open subsets of Rn _{for some fixed n.}

In this category, which we can call for example Sub(Rn), the objects are the subsets U ⊂ Rn _{and the arrows can be defined to be diffeomorphisms between}

them. It is easy to see that the composition of diffeomorphisms is itself a diffeo-morphism because, using the well known chain rule, we have dx(f ◦ g) = dyf ◦ dxg

with y = f (x). Furthermore, the identity function (which exists for every subset of Rn_{) is a diffeomorphism and the composition rule for functions ensures that}

the composition is associative. As a matter of notation, for every couple of open subsets (U, V ) of Rn_{, we call the arrows between U and W by C}∞_{(U, W ).}

Starting from this example, one can immediately give an example of a functor. Example: Tangent functor of Rn.

This is defined as a functor from the category Sub(Rn_{) to Sub(R}2n_{), T :}

Sub(Rn₎! Sub(R2n_{). More explicitly, we have}

U ⊂ Rn T−! T (U) := U × Rn ⊂ Rn× Rn (1.2.1) f ∈ C∞(U, W )−! T (f) ∈ CT ∞(T (U ), T (W )) (1.2.2) 7_{We are here ignoring the subtleties given by the fact that we are giving the same name to}

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with the definition

T (f )(x, v) := (f (x), dxf (v)) ∈ T (W ), (x, v) ∈ T (U ) (1.2.3)

or, in other words,

(x, v) ∈ T (U ) := U × Rn T (f )−−! (f(x), dxf (v)) ∈ T (W ). (1.2.4)

With this definition, it is easy to verify that T (g ◦ f ) = T (g) ◦ T (f ) and T (1U) =

1T (U ), so that we have indeed defined a functor. As a side note, notice that one

can define naturally a projection function

πU : T U ! U, πU(x, v) := x (1.2.5)

where we used the lighter notation T U instead of T (U ). We can now give the definition of the tangent bundle.

Definition 6. Tangent bundle (concrete) [Ben19]

The collection (T U, πU) is called the tangent bundle of U . For each point

x ∈ U , the set π_U−1 := TxU is called fiber over the point x and it has a natural

identification with Rn _{seen as a vector space}8_. _•

Notice that the nature of the definition of the tangent bundle is analogous in concept to the case of a function and the tangent lines to its graph, which by the way represents an example of the above construction. The following picture, with two examples of tangent fibers, should help clarify the construction.

Obviously, the construction of the tangent bundle implies that there is a fiber for each point, without fibers intersecting, but this would be too difficult to represent.

1.2.2 General definition of fiber bundles

The way in which a tangent bundle has been constructed is indicative of the way in which a general fiber bundle is built. In particular, we assign to each point x of a manifold M another manifold (which in general depends on the point x), which we call fiber over x. In the example of the tangent bundle, it was easy to start

8_{Notice that technically T}

xU = {x} × Rn and only F := Rn is called fiber. However, since

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from the construction of the differential in Rn for some n in order to explicitly formulate in which way one can assign a fiber to each point of another manifold. Furthermore, the fibers that we chose were of a particular kind, namely again copies of Rn. However, there is no reason to restrict our attention only to such cases, and we want to be able to generalize the same kind of construction of the tangent bundle. This gives rise to the following definition.

Definition 7. Embedded smooth fiber bundle [Ben19]

An embedded smooth fiber bundle consists of the collection (E, X, π, F ) where E, X and F are embedded smooth manifolds respectively called total space, base space and fiber. The function π : E ! X is a surjective submersion9 and it is called projection. Furthermore, the fiber bundle is localy trivial, which means that for each x ∈ X there is an open neighbour U such that the following diagram commutes U × F π−1(U ) U projU φ π

where projU : U × F ! U with proj(x, v) = x is the canonical projection operator

on a Cartesian product, and where φ is a diffeomorphism. • Notice that the tangent bundle, as defined before, is locally trivial.

From a physical point of view, a nice intuitive way of understanding a fiber bundle is to imagine the base space as a space-time, so that if a particle is on a point x in the base space, then the fiber over x represents all the possible internal states of the particle. Notice that the condition of local triviality only means that we can consider every fiber bundle locally as a cartesian product, even though this needs not to be the case globally. A good example of a locally trivial bundle which is not globally trivial is given by the M¨obius strip seen as a fiber bundle with base space the circle. On the other hand, the cilinder is an easy example of a globally trivial fiber bundle, again with base space the circle. In the following picture, we represent both the cilinder and the M¨obius strip together with some fibers represented as line segments. In both cases, the base manifold is the red circle.

When one talks about fiber bundles one rarely refers to them as the collection of the above manifolds, instead a briefer description is used. In particular, we will frequently refer to a fiber bundle as E −! X or simply as E.π

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1.2.3 Principal bundles

One thing to notice about this definition is that, even though we have defined it in terms of embedded quantities, it can be easily generalized to different cases, by allowing different kinds of manifolds or imposing different conditions on the functions φ and π. The definition given above is already quite general for our purpose, because we will only need to use calculus on manifolds. Furthermore, since we only defined E, X, and F to be smooth manifolds, nothing prevents (some of) them to also have some kind of additional structure. In fact, we will be interested in fiber bundles in which there is such an additional structure in some of its components.

For example, it can happen that the fiber F has some structure which is preserved by the action of Aut(F ) on the fiber10_{, i.e. by the action of the}

repre-sentation ρ : G! Aut(F ) of G, where G is some group. In this case, we would say that the fiber bundle is a G-bundle, referring to the group G in the definition of ρ. We first describe how to define a G-bundle, and then we introduce the principal bundle as a special case.

Example of G-bundle: Abstract definition of tangent bundle

The process of constructing a G-bundle can be illustrated by again using the tangent bundle, in particular by defining it again in a more general and abstract context. In the concrete case, one can immediately define the fibers thanks to the differential of the charts on the manifold. In the abstract case, however, the charts are only homeomorphisms, and only the transition functions are diffeomorphisms. Hence we can only use the differential in the transition functions. In order to (re)define the tangent bundle we will need to take the set of all local trivializations andh glue them together using the transition functions in the appropriate way, so that we obtain something with the same properties of the concrete tangent bundle defined in def.(6).

To be more precise, we first take the base manifold X together with its maximal smooth atlas AX := {Ui, φi}i∈I where I is a set of indices. At this point, every

time that Uij = Ui ∩ Uj 6= ∅, we can define the following set of functions

µij : Uij ! GL(n, R) µij(x) = dφi(x)(φ ◦ φ

−1

) (1.2.6)

for i, j ∈ I. These functions must be smooth and they must satisfy several condi-tions in order to be well defined. In particular

• µii= 1n ∀i ∈ I, ∀x ∈ Uii;

• µij = µ−1ji ∀i, j ∈ I, ∀x ∈ Uij;

• µijµjkµki = 1n ∀i, j, k ∈ I, ∀x ∈ Uijk = Ui∩ Uj∩ Uk

The set of the µij is called smooth cocycle on the open covering A with values in

GL(n, R). Before commenting on these conditions, we will first complete the def-inition of the tangent bundle. We now only need to introduce local trivializations

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and glue them using the functions defined in (1.2.5). We assume that the local trivializations are given by

T = {Ui× Rn× I} (1.2.7)

where I is discrete, and therefore it is endowed with the discrete topology. In order to glue them together, we first introduce an equivalence relation ∼ between points in different trivialization. Our tangent bundle will therefore be T / ∼, which represents the set of local trivializations in which every two points related by the relation ∼ are identified.

Two points (x, v, i) and (y, u, j) in two trivializations Ui× R×I and Uj× R×I

are equivalent, i.e. (x, v, i) ∼ (y, u, j) if the following holds

x = y u = µij(x)v. (1.2.8)

By taking the quotient of T and ∼, these points are identified, and we call the result

T (x) := T

∼. (1.2.9)

It is possible to prove that this construction yields a Hausdorff manifold with countable basis, and in general it has the correct properties needed for a fiber bundle, while at the same time being an equivalent definition of the concrete tangent bundle defined above [Ben19].

Notice that the definition of the abstract tangent bundle relies on the definition of local pieces (the local trivializations) that are then glued thanks to the action of a representation of some group on the fibers, which above was given by the µij’s. This is a general procedure, and one does not need to be restricted to the

case in which the fibers are copies of Rn_.

The principal bundle is a particular case of G-bundle. One can again repeat the procedure above. However, contrary to the case of the tangent bundle in which the fibers had a familiar structure so that one could easily identify an explicit representation of some group on them, this time we abstractly define the µij as

µij : Uij ! G for some group G. They respect exactly the same conditions as

before. In order to let them act on the fibers, we also need a representation of the group G on the fibers, i.e. we need ρ : G! Aut(F ), with action on the fibers defined as G × F ! F with (g, x) 7! ρ(g)(x). The composition of them respects the conditions given above, and we obtain a smooth G-bundle over the base space. Definition 8. Principal G-bundle [Ben19] [Bas19]

A principal G-bundle is defined to be a smooth G-bundle such that the fibers F are the structure group itself, i.e. F = G. In this case, the group acts on the fibers by left multiplication, i.e.

G × G! G (g, h)7! Lg(h) = gh. (1.2.10)

As a matter of convention, the base space of a G-bundle is still called M , while the total space is called P . Furthermore, we require that P/G is diffeomorphic to M , where P is the total space, G is the fiber and M the base space. •

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Before proceeding, we notice that principal bundles are not merely a mathe-matical curiosity, and their usefulness goes way beyond their use in the Chern-Simons theory that we will underline below. Once we will introduce what a connection is, we will see that gauge fields in Yang-Mills theories like QCD or QED are described by connections on principal bundles, where the base space of the principal bundle in this case is given by space-time.

1.2.4 Sections and bundle morphisms

Now that we have introduced fiber bundles and principal bundles, it is natural to ask what are the constructions that can be built with them. The first intuitive one is given by the concept of section of a bundle, which amounts to picking a single element from each fiber. We give a more precise definition.

Definition 9. Section of a bundle (global) [Ben19]

A section of a fiber bundle E −! X is defined by the continuous functionπ X −! E, which is a right inverse of the projection, i.e. π(σ(x)) = x ∀x ∈ X. •σ A simple example of a section of a bundle can be give when the bundle is a tangent bundle. Since every fiber of a tangent bundle is a vector space, picking an element of each fiber through the function σ amounts to assigning a vector to each point x ∈ X, thus obtaining a vector field on the base space X. We will expand on this notion a bit later, because we will need them when we will talk about pullback and pushforward.

Notice, however, that not all fiber bundles admit a section defined globally. A simple example [Ste99] is given by the M¨obius strip intended as a bundle over a circle with fibers R − {0}; a global continuous section would be forced to pass at least once through a zero of some fiber, which however is not part of the domain. It is therefore convenient to define also what we mean by local section. This is easily obtained by substituting X −! E with U ⊂ Xσ −! E, with the property thats π(s(x)) = x ∀x ∈ U .

Sections of fiber bundles linked to the triviality of the fiber bundle itself. Since we are considering locally trivial bundles, it is easy to see that sections exist at least locally. In fact, thanks to the local trivializations U × F −! πφ −1(U ) one can easily define U ! U × F −! πφ −1(U ), which is a right inverse of the projection π. The question of whether a local section can be extended to a global one is not trivial. There is a branch of mathematics called obstruction theory that deals with this kind of questions. In this work we will not need many results in this sense since we are mainly interested in principal bundles. In this case, there is an interesting theorem which links triviality of the bundle to the existence of a section.

Theorem 1. The triviality of a principal G-bundle P −! M is equivalent to theπ existence of a global section s : M ! P .

The question now arises of when does a principal bundle admit a global section, or equivalently when is it trivializable. In this sense, there is another important theorem that gives us some answers:

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Theorem 2. [Fre95] Given a principal G-bundle P −! M, if G is simply con-π nected11_{and if dim(M ) ≤ 3, then the principal G-bundle admits a global section,}

hence it is trivializable.

This theorem is of interest to us because, as we will see later, we will mainly work with base manifolds of dimension 3.

Another fundamental construction in what follows will be the notion of bundle morphism. We already know how to define morphisms between manifolds12_{. To}

define a morphism between bundles, we have to take a bit more care because there are different manifolds involved, but the definition is not a difficult one.

Definition 10. Bundle morphisms [Ben19] Given two smooth fiber bundles E −! X and Eπ 0 π

0

−! X0_{, we can define a bundle}

morphism as the following commuting diagram

E E0 X X0 ˜ f π π0 f

where f and ˜f are diffeomorphisms. •

In the particular case in which the two bundles are principal bundles and they coincide, and if f is the identity function, the morphism are called gauge trans-formations [Fre95]. This terminology comes from physics where, as said above, principal bundles are related to gauge fields, so that automorphisms of principal bundles are naturally related to gauge transformations. Before introducing the concept of connection, we first introduce what pullback and pushforward are; they will be needed when talking about connections. In order to do so, we first explore in more detail the definition of vector fields and p-forms, to which the pullback and pushforward will be applied in the next sections.

Vector fields and differential forms

Both vector fields and forms can be described as sections of some appropriate bundles. Starting from vector fields, one can define them as sections

V : U ! T U, V (x) = (x, vV(x)). (1.2.11)

If we have a diffeomorphism ϕ : U ! W , then we can define a section ϕ∗V : W !

T W by using the section V and the diff. ϕ by composition of V with T ϕ defined in (1.2.2). In particular, we would write

ϕ∗V (y) = (y, dxϕ(vV(x))) (1.2.12) 11_{Intuitively speaking, a set is simply connected if every loop in the set can be continuously}

deformed to a point without leaving the set to which it belongs at any step of the continuous transformation. A simple example of a simply connected set is the open disk.

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where y = ϕ(x). We call Γ(T M ) the set of all the vector fields defined in the tangent bundle of a manifold M ; this set is actually a module, more precisely a module over C∞_{(U < R) [Ben19]:}

Definition 11. Module over C∞(M ) [AF92] [BM95]

A (left) module R is defined as an abelian group G, in our case G = (Γ(M ), +), together with an operation C∞× G! G such that the following properties hold:

• f (v + w) = f v + f w; • (v + w)f = vf + wf ; • (f g)v = f (gv); • 1v = v

for all f ∈ C∞(M ) and v, w ∈ Γ(T M ) with U ⊂ M . A right module can be defined in the same way, but with multiplication defined on the right, i.e. G × C∞! G. A submodule is defined as a subset of R which is by itself a module. • An interesting property about vector fields is given by the fact that one can write them locally in terms of a basis of (locally well defined) vectors in the following way. If we call Ei = (x, ei) with x ∈ V and where ei = (0, . . . , 1, . . . , 0)

with the 1 in the i-th component belongs to the canonical basis of Rn, then we can write

vV =

X

i

v_Vi ei. (1.2.13)

Using a more physical notation, sometime the basis vectors eiare called ∂i = ∂/∂xi

so that, using Einstein summation notation and the Greek indices, one can write v = vµ_∂

µ.

In a similar manner, we can define 1-forms [Ben19]. In particular, we can define them as sections of the dual of the tangent bundle; if we define T∗U = U ×M (1, n) where M (1, n) is the set of 1 × n matrices, i.e. the set of functionals on the space Γ(T U ), then we can define a cotangent bundle

T∗U π

∗

−! U (1.2.14)

which assigns to each x ∈ U the fiber T_x∗U . A 1-form is a section as before of the cotangent bundle just defined

w : U ! T∗U x7! (x, dxf ) =: (x, ω(x)) (1.2.15)

where f ∈ C∞(U ). The set of all 1-forms defined on a manifold M is denoted by Ω1_{(M ). In the same way as vector fields, forms can be expressed in terms of a}

basis. In particular, if we call ei _{the functionals such that e}i_(e

j) = δij where ej is

the relative local basis for the vector fields, then one can write

ω = X

i

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Again, in a notation more common to physicists, one usually writes ei = dxi so that, with the same conventions as before, one gets ω = ωµdxµ.

The construction of 1-forms ca be generalized to p-forms. In order to introduce them we need the definition of wedge product, also called external product, of elements of T M . We will introduce it in the context of vector spaces, but its definition will be immediately generalized to allow us to apply it to forms. In order to define it, we first need to introduce what is called a tensor bundle of order k, which we will name Tk_{M . For example it can be constructed in the same}

way as a (concrete) tangent bundle, but instead of assigning to each point x ∈ M the tangent space TxM , we assign the tensor spaceNk_i=1(TxM )i. A section of this

bundle is given by

vT : M ! TkM vT(x) = v1(x) ⊗ · · · ⊗ vk(x). (1.2.17)

Again, the set of sections of the tensor bundle is actually a (left and right) module over C∞(M ), which we call13 _{N R. Furthermore, we also need the concept of two}

sided ideal or simply ideal. We will not go too deep in this definition; for our purposes, it is sufficient to note that a two sided ideal J is a submodule of N R; however, we will only need the submodule I generated by v ⊗ v [LB99]. We can now give the definition of the exterior product.

Definition 12. Wedge product [LB99]

Given a tensor bundle Tk_{M with base space M whose fibers are given by}

Nk

i=1(TxM )i, we define a new set of fibers k ^ TxM = Nk i=1(TxM )i I . (1.2.18)

We call the fiber bundle with base space M and fibers Vk

TxM by VkT M . The

wedge product between two elements v and w of TxM can be therefore defined by

the canonical map

˜ ∧ : Tk_M _! k ^ T M ∧(v ⊗ w) := v ∧ w.˜ (1.2.19) • By using the same construction for the wedge product using the fibers T_x∗M we can define the bundle Vk

T∗M , whose sections are p-forms. The set of all p-forms on M is called Ωp_{(M ), while the set of all forms regardless of their order is called}

Ω(M ). The wedge product is defined with the same procedure. From this point onward, we will only use the wedge product in the context of forms.

A more intuitive description of the wedge product can be immediately given by describing the properties of the wedge product just defined. It is distributive and associative like the tensor product. Furthermore, the fact that we factored by I means that we are identifying v ∧ v with 0, which in turn implies that the

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wedge product is anti-symmetric for the exchange of two nearby 1-forms. In fact, 0 = (v +w)∧(v +w) = v ∧w +w ∧v +v ∧v +w ∧w = v ∧w +w ∧v. More in general, given a p-form v and a q-form, it is possible to prove that v ∧ w = (−1)pq_{w ∧ v.}

The wedge product of a p-form v and a q-form gives a (p + q)-form, so the wedge product represents a way to obtain higher order forms starting from lower order ones. Notice, however, that it is not the only way in which we can obtain higher order forms. In this aspect, it is important to introduce the exterior derivative. Definition 13. Exterior derivative

The exterior derivative is defined as the unique mapping d : Ωi_{(M )}_{! Ω}i+1_{(M )}

such that the following properties are satisfied;

• d : Ω0_{(M ) = C}∞_{(M )} _{! Ω}1_{(M ) is the differential of functions, i.e. f ∈}

C∞(M )7! df;

• d(v + w) = dv + dw for all v, w ∈ Ω(M ), d(αv) = αdv for all α ∈ Rn _and

v ∈ Ω(M );

• d(v ∧ w) = dv ∧ w + (−1)p_{v ∧ dw for all v ∈ Ω}p_{(M ) and w ∈ Ω(M );}

• d(dv) = 0 for all v ∈ Ω(M ).

• In component notation and using Einstein convention, a p-form can be written explicitly as w = wµ1...µpdx

µ1 _{∧ · · · ∧ dx}µp_{. Furthermore, any p-form ω can be}

applied to p vectors obtaining a scalar; for the simple case of a 1-form applied to a vector field v, the resulting scalar is indicated by ω(v). However, in the next sections and chapters, we will be interested in yet another kind of forms, namely Lie-algebra-valued p-forms. Everything that has been said about forms up to now still stands, with the caveat that now a form is defined as a section of the bundle g⊗Vk

T∗M , where g is the Lie algebra. In this case the exterior product can be defined by [Bas19]

. ∧ . : Ωp(g, M ) ⊗ Ωq(g, M ) ! Ωp+q(g ⊗ g, M ) (1.2.20) which is explicitly given by the following mapping

AIdxI ∧ BJdxJ ! AI⊗ BJdxI∧ dxJ (1.2.21)

where AIdxI = (Aµ1...µpdx

µ1 _{∧ · · · ∧ dx}µp_{) and B}

JdxJ = Bν1...νqdx

ν1 _{∧ · · · ∧ dx}νq_.

Furthermore, we will also need the following modified definition for the exterior product of Lie-algebra-valued forms, in which instead of taking the tensor product of the coefficient we take the Lie bracket [Bas19]

[. ∧ .] : Ωp(g, M ) ⊗ Ωq(g, M )! Ωp+q(g, M ) (1.2.22) which is explicitly given by

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with the same meaning of the terms. When we apply this second product to forms and then we apply them to vectors, the result of the calculation is given by [Bas19]

[A ∧ B](v1, . . . , vp+q) = X

σ∈Sp+q

sgn(σ)[A(vσ(1), . . . , vσ(p), B(vσ(p+1), . . . , vσ(p+q))] (1.2.24) where A ∈ Ωp_{(g, M ) and B ∈ Ω}q_{(g, M ).}

All this formalism is rather abstract, and one may wonder if it is actually any useful in physics. It is fairly simple to show that one can write, for example, Maxwell equations and Yang-Mills equation in a simple way using forms. We will come back to this topic once we have introduced the concept of connection and of curvature of the connection.

1.2.5 Pullback ans pushforward

The pullback and its dual, the pushforward, arise from the need to ”transport” some quantities defined in some fiber bundle to another one. In a more phisical notation, these two notions are related to contravariant and covariant quantities [BM95]. We will mainly use them when talking about sections, and in particular vector fields and p-forms, but we first give a general definition of their meaning and then we will apply them more concretely to see how they work. Intuitively, if we have two (embedded) fiber bundles E −! M and Eπ 0 −! N and if there isρ some function ϕ : M ! N, we expect to be able to relate also the total spaces of the two bundles through some construction involving ϕ. It is easy to realize this construction.

Definition 14. Pullback bundle [Ben19]

Given an embedded fiber bundle E −! M and a deffeomorphism ϕ : N ! M,π we can define the following:

• ϕ∗_{E := {(p, y) ∈ N × E : ϕ(p) = π(y)};}

• ϕ∗ _{: ϕ}∗_E _{! E with ϕ}∗_{(p, y) = y;}

• ϕ∗_{π : ϕ}∗_E _{! N with ϕ}∗_{π(p, y) = p.}

These definitions can be summarized in the following commutative diagram

ϕ∗E E

N M.

ϕ∗

ϕ∗_π _π

ϕ

If we call ξ := E −! M, then if we call ϕπ ∗ξ := ϕ∗E ϕ

∗_π

−−! N, we have that ϕ∗_{ξ is}

the pullback of ξ via ϕ. •

Theorem 3. [Ben19] If ξ := E ! πM is an embedded smooth bundle with fiber F , then also ϕ∗ξ := ϕ∗E ϕ

∗_π

−−! N is an embedded smooth bundle with the same fiber. Furthermore, the following properties can be verified:

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• (ϕ ◦ ψ)∗_{ξ = ψ}∗_(ϕ∗_ξ)

• (ϕ ◦ ψ)∗ _{= ϕ}∗_{◦ ψ}∗_.

Notice that, if the bundle ξ has some additional properties, these properties are still present in the pullback bundle.

In this way we have obtained another fiber bundle ϕ∗ξ as a pullback of the original bundle ξ. A more intuitive description of the pullback on fiber bundles can be immediately given. If we start from the original fiber bundle ξ := E −! Mπ and if we have a diffeomorphism ϕ : N ! M, then ϕ∗ξ is a fiber bundle with base manifold N and with fiber obtained from those of ξ in the following way: if Ey is

the fiber of ξ such that π(Ey) = y with y = ϕ(x), then one can define the fiber

of ϕ∗ξ over the point x ∈ N simply by (ϕ∗E)x = Eϕ(x) = Ey. From this point of

view, we can say that the fibers of ϕ∗ξ have been ”pulled back” from those of ξ.

ϕ∗E E N M ϕ∗ ϕ ϕ∗π _π x y = ϕ(x) (ϕ∗E)x Ey

The name pullback comes from the intuitive fact that we are going ”against the direction of ϕ : N ! M”, meaning that we have something defined on M to obtain something defined on N [BM95]. It is therefore intuitively clear that the pushforward should do the opposite thing. In particular, the role of the pushforward (among other things) is to allow the definition of vector fields and forms on M given a vector field on N .

Our main interest will be to define pullback and pushforward for p-forms and vector fields, establishing how these behave when applied to wedge product of forms or to differentials.

It is immediate to define the pushforward of a vector field, since we already used it (without explicitly naming it) in the definition of a vector field itself. Definition 15. Pushforward of a vector field [Ben19]

Given a vector field V : U ! T U, V (x) = (x, vV(x)) and a diffeomorphism

ϕ : U ! W , then the pushforward of V by ϕ is denoted by ϕ∗V : W ! T W with

the definition

ϕ∗V (y) = (y, dxϕ(vV(x))). (1.2.25)

• In a similar way, it is also easy to define the pushforward of a form. For simplicity, we give the definition for the case of 1-forms interpreted as sections of the fiber bundle T∗M .

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Definition 16. Pushforward of a form [Ben19]

Given a local section of the bundle T∗U , i.e. a 1-form Ω(x) = (x, ω(x)), and given a diffeomorphism ϕ : U ! W , we can define the pushforward of Ω by ϕ as a local section of the bundle T∗W , defined as

ϕ∗Ω(y) = (y, ω(dyϕ−1(v))) (1.2.26)

where v ∈ TyW and y = ϕ(x). •

The pullback of a a form can be now defined in a simple way. In order to lighten our notation, we will refer to a form simply as ω and not as Ω = (x, ω(x)). Definition 17. Pullback of a form [BM95]

Given a 1-form ω defined on the bundle T∗W and a diffeomorphism ϕ : U ! W , then the pullback of ω by ϕ is defined by the following relation:

ϕ∗ω(v) = ω(ϕ∗v) (1.2.27)

where v is a vector field defined in T U . Furthermore, the following properties for the pullback can be proved:

1) ϕ∗(ω1∧ · · · ∧ ωp) = ϕ∗ω1∧ · · · ∧ ϕ∗ωp;

2) ϕ∗dω = dϕ∗ω.

• These definitions will be useful when talking about connections and the Chern-Simons theory. Notice that the forms used in the definitions can very well be Lie-algebra-valued forms, the definitions are the same. Furthermore, the pullback of a form can be defined also in another way using integrals of forms. We will introduce this important alternative definition in the next section.

1.2.6 Integration of forms

One of the most important things that we will need when talking about the Chern-Simons Theory is the concept of integration of forms. We first give a brief and intuitive motivation of the fact that forms are natural objects to be integrated. We will summarize the presentation in an article14 by T. Tao, to which we will refer to as [TTform]. We will only intuitively motivate and describe the integral on forms, and then we will state (without proof) the Stokes theorem as well as several properties of the integral of forms that will be useful for us in the next sections.

Starting from the case of an integral of a continuous function of one variable, an integral is generally defined by the limit of the usual Riemann sums15

Z f (x)dx = lim ∆xi!0 n−1 X i=0 f (xi)∆xi (1.2.28) 14_{https://www.math.ucla.edu/ tao/preprints/forms.pdf}

15_{We are not reviewing the theory of the integral, so we are ignoring the more formal procedure}

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when the RHS exists and is bounded and where ∆xi = xi+1− xi. The important

feature to notice of this construction is that it basically is a limit of a sum of numbers. In this case, therefore, the integrand computed at each xi is just a

number.

If we now try to generalize this notion, we could try to integrate some object ω on a one dimensional curve γ in, for example, R3_{; the point we will try to}

intuitively present is that such an object ω can naturally be conceived to be a 1-form.

If we try to again define the Riemann sum, we would immediately find out that this time ∆xi is not a scalar quantity like before, but it is a vector. However,

since the integral is a generalization of the sum of scalars, each ω(xi) should be a

functional so that ω(xi)∆xi is a scalar. We can therefore define

Z γ ω = lim ∆xi!0 n−1 X i=0 ω(xi)∆xi (1.2.29)

again when the RHS exists and is bounded. The integrand, therefore, assigns a covector to each point x of the integration domain in a continuous way, and this is the definition of a 1-form. This kind of reasoning can be generalized to surfaces and different integration domains. Furthermore, notice that p-forms are to be integrated in p-dimensional manifolds.

To define more precisely the integral of forms, we first need to specify what does it means for a manifold to be oriented. There are several definitions of orientability for a manifold. One of the most intuitive is given in terms of its tangent bundle [Spi71]. We know from the definition of abstract tangent bundle that the transition functions (1.2.6), which are obtained from the transition functions of the base manifold through differentiation, are elements of GL(n, R). If the determinant of these transition functions µij(x) ∈ GL(n, R) is different than zero, it can be

either positive (in which case the transition function is said to be orientation preserving) or negative (the transition functions are orientation reversing). A manifold in which all the transition functions of the tangent bundle are for example all orientation preserving is said to be an orientable manifold. In the following figure there are two simple examples of manifolds which are orientable (left) and non-orientable (right). In particular, we have represented a single axis of a local frame of reference, which has a well defined orientation in the orientable example, while it does not in the non-orientable one.

The integration over an oriented domain of a p-form can now be defined. We first define it for the case of integrals in Rn_{, which is obviously integrable.}

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Definition 18. Integral of a form in Rn [Sad16] Given a p-form α = αµ1...µp(x

ν_)dxµ1 ∧ · · · ∧ dxµp _{defined over a compact}16

domain A ⊂ Rn_{, we can define}

Z A α := Z A αµ1...µp(x ν_)|dxµ1_{. . . dx}µp_| _(1.2.30)

where the RHS is a Riemann integral, |dxµ1_{. . . dx}µp_{| is usually denoted by d}p_x

and the choice of the ordering of the indices µ1, . . . , µp corresponds to a choice of

orientation of the integration domain. •

At this point, it is possible to show the following theorem

Theorem 4. [Sad16] Given an orientation preserving diffeomorphism U −! Vf where U, V ⊂ Rn _{are compact sets. Then}

Z V ω = Z U f∗ω (1.2.31) where ω is an n-form.

This property of integral of forms on subsets of Rn _{can actually be used to}

define the integral of forms in a more general setting.

Definition 19. Integral of a form in a manifold [Sad16]

Let M be an orientable manifold of dimension p, and ω a p-form defined on it with a compact support V ⊂ M . Because of the definition of a manifold, we can define a local parametrization (see def.(1)) ψi : Ui ! V with U ⊂ Rn which

is orientation preserving. Then we can define Z V ω := Z Ui ψ∗_iω. (1.2.32) • This procedure using the pullback can be generalized without restricting one of the two sets to belong to Rn. In particular, some authors even use it to define the pullback.

Definition 20. Pullback of a form [TTform]

Given a diffeomorphisms U −! V between two (sub)manifolds and a form ωf on V , the pullback form f∗ω is the unique form such that

Z f (U ) ω = Z U f∗ω. (1.2.33) • 16_{As stated in the cited article, this assumption is not fundamental and it is usually included}

in the definition to avoid talking about problems related to a non-compact domain. In any case, this will be sufficient in our case.

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Notice that the integral of forms satisfies the same properties of a Riemann integral, in particular for sum of domains or orientation reversing of the domain. We now state a theorem which will play a fundamental role in the next chapters. Theorem 5. Stokes’ theorem [Spi71] Given an n-dimensional manifold M with boundary ∂M . Then we have

Z M dω = Z ∂M ω (1.2.34)

where ω is a (n − 1)-form and where the orientation of ∂M is induced by that of M through the restriction of the charts Ui ⊂ M

φi

−! Hn _{to the boundary.}

The explicit formulation of this theorem in dimensions 2 and 3 are common in physics, since they allow for the simplification of several integrals. One notable example is the Gauss theorem for the integral of a divergence on a volume. These notions will be sufficient in the remaining chapters.

1.3 Connections and curvature

We now have the basics of manifolds, fiber bundles and bundle morphisms, to-gether with vector fields and forms. We now need to develop the concept of parallel transport on such structures. When dealing with concrete smooth manifolds it was easy to introduce and understand the concept of smoothness of a function on a manifold since a manifold was merely an open subset of Rn_{for some n. However,}

with the introduction of bundles and sections, it is not immediate to take deriva-tive of sections, since those are collections of objects on different fibers, without any a priori identification. In order to compare them, we will need to introduce the concept of connection.

There are several definitions of what a connection is. The most intuitive one is given in terms of horizontal and vertical tangent spaces T P to a principal bundle P . Intuitively speaking, we need to be able to consider vectors which are tangent both to the fibers of the principal bundle and to the base manifold. In the picture below, we represent a bundle with some vertical (green) and a horizontal (orange) vectors. These can be divided into two subbundles of the tangent bundle T P . Definition 21. Subbundle

Given a tangent bundle17 _{T M , a subbundle T U of T M is defined by a}

diffeo-morphism i : T U ! T M which is iniective. •

Intuitively speaking, a subbundle is basically a subset of a tangent bundle which is by itself also a tangent bundle. Using this notion, we can divide the tangent bundle of a manifold M in two pieces, considering the tangent vector spaces to the fibers and those tangent to the base manifold.

17_{This definition applies more generally to vector bundles, i.e. fiber bundles whose fibers are}

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Definition 22. Horizontal and vertical tangent spaces [Bas19]

Given the tangent bundle T P of a principal bundle P , we call TpP the vector

space at each point p ∈ P . Each vector space can be decomposed into a part which is tangent to the fibers G of P , which we call TpPx where x ∈ M , and a

part which is tangent to M , which we call HpPx. The vectors in TpPx are called

vertical, while the vectors in HpPx are called horizontal. We call HP constructed

in this way the horizontal subbundle, and its complement in T P is called the vertical subbundle. Furthermore, the horizontal space is said to be equivariant, i.e. it satisfies the following relation:

R_{g ∗}HpP = HpgP (1.3.1)

where Rg : G × G ! G is the action of the group G on p ∈ P from the right.

Intuitively speaking, right multiplication let us move ”along the base manifold”. • The following picture can help clarifying the choice of nomenclature.

A connection on a principal bundle can be immediately defined18. Definition 23. Connection [Bas19]

A connection A of a principal bundle P , also called principal connection, is

defined to be the horizontal subbundle HP of T P . •

The intuitive idea of this definition is the following. If we were to have a path γ on the base manifold M , then the tangent vectors to γ will be horizontal by definition. However, since the horizontal subbundle HP is defined on all the fiber bundle P , then we can ”lift” in a unique way the horizontal path γ to a horizontal path Γ in the fibers; a visualization of this fact appears in the following picture. Notice that we only represent some of the relevant fibers.

M

γ Γ

18_{The following definition applies also to generic bundles, but in the following chapters we}

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This means that now we have defined a trajectory linking different fibers, so that we can now define a parallel transport operator Γ : PΓ0 ! PΓ1, where Γ is

parametrized by a parameter s ∈ [0, 1]. This allows us to compare mathematical objects defined on different fibers, which was our original goal.

Alternatively, one can define a connection as a 1-form. Contrary to the previ-ous definition which can be applicable to any fiber bundle, it is not always possible to define the connection as a 1-form. From now on, we will consider to always be possible to use the definition of connection in terms of 1-forms; each time we will specify what make it possible to use this definition, but we will not enter in too much detail now.

If we were to use the language of forms, we can give another intuitive definition of a connection. Before doing so, we need to introduce a few preliminary concepts. First, in a principal G-bundle one can define a diffeomorphism between the group G and the fibers Px, let us call it φp : G ! Px [Bas19] with the definition

g ∈ G 7! pg ∈ Px. Its differential dφ : g! TpPx sends a ∈ g 7! Xa ∈ TpPx, i.e.

sends elements of the Lie algebra g of G into vertical vectors Xa. We can now

define the connection.

Definition 24. Connection [Bas19]

Given a principal G-bundle, a connection ω is defined as a Lie-algebra-valued 1-form (see eq.(1.2.20)), i.e. a section of the bundle g ⊗ T∗M , such that

1) ω(Xa) = a, where Xa has been defined above;

2) R∗_gω = Ad_g−1ω,

where AdgB = _dtd g e(tB)g−1|t=0. •

Notice that this definition is not completely independent from the previous one, because HpP = Ker(ωp).

Furthermore, because of the new definition, the restriction of the connection to the fibers can be identified with what is called Maurer-Cartan form, defined by θ : TgG! g Xg 7! (Lg−1)_∗X_g ∈ g (1.3.2)

where we have used the fact (which we will not prove) that the elements of the Lie algebra g of a group G are the elements of the tangent vector space to the identity of G. In order to lighten the terminology, we will use just the term ”connection” to indicate both Lie-algebra-valued connections and C∞-valued connections.

Having defined the connection, it is now possible to define another funda-mental quantity, namely the curvature. We will define it using the definition of connections in terms of 1-forms.

Definition 25. Curvature [BM95] [Bas19]

Given a connection ω, the curvature Ω is defined as Ω = dω + 1

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where we defined [. ∧ .] in eq.(1.2.22) and eq.(1.2.23). Furthermore, if we denote by ix : Px ! P the inclusion map, then

i∗_xΩ = 0 (1.3.4)

• Taking the exterior derivative of eq.(1.3.3) and using the properties of the exterior derivative listed in def.(13), it is possible to show that

dΩ + [ω ∧ Ω] = 0. (1.3.5)

As a final note, the curvature of the Maurer-Cartan form is zero, i.e. we have dθ + 1

2[θ ∧ θ] = 0. (1.3.6)

1.3.1 Gauge transformations of connection and curvature

A key role for the Chern Simons theory is played by the transformation rules for both the connection and the curvature. We recall (see def.(10)) that a gauge transformation is given by a bundle morphism of the following type

P P

M

ϕ

π π (1.3.7)

where P is our principal bundle. Notice that, sometimes, gauge transformations are indicated simply by ϕ : P ! P or P −! P . It is possible to show (see [Bas19]ϕ for more details) that, under the gauge transformation ϕ, the connection and the curvature transform under the following rules

ω−! ϕϕ ∗ω = Adg−1ω + g∗θ (1.3.8)

where g∗θ is just a short-hand notation for g−1dg which is a notation most com-monly used in physics, and

Ω−! ϕϕ ∗Ω = Adg−1Ω. (1.3.9)

1.3.2 Example in Physics

We now give a brief physical example of the use of forms in order to familiarize with the mathematical tools introduced, which means that we will be more explicit in our calculations. The classical example is given by the field strength tensor of Maxwell theory of electromagnetism as well as the Yang Mills field strength. In both cases we introduce a gauge field Aµ, however in order to use the formalism

of forms we define A = Aµdxµ, which is a Lie-algebra-valued 1-form since Aµ =

Aa

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is easy to see that the curvature F = dA +1₂[A ∧ A] corresponds to the usual Fµν

in a non Abelian theory. In fact, if we apply both sides to two arbitrary vector fields V and W , and using eq.(1.2.24)

F (V, W ) = dA + 1 2[A ∧ A] (V, W ) = d(Aνdxν)(V, W ) + 1 2[Aµ, Aν] dx µ_{∧ dx}ν_{(V, W )} = ∂µAν + 1 2[Aµ, Aν] dxµ∧ dxν_{(V, W )} = 1 2(∂µAν − ∂νAµ) + 1 2[Aµ, Aν] dxµ∧ dxν_{(V, W )} := 1 2Fµνdx µ_{∧ dx}ν_{(V, W )} _(1.3.10)

where we used the antisymmetry of the wedge product, so that the curvature in components reads

Fµν = (∂µAν − ∂νAµ) + [Aµ, Aν] (1.3.11)

which, up to i factors, is the canonical field strength tensor for a non Abelian gauge field theory19. The gauge transformations for the gauge fields using g∗θ = g−1dg becomes A−! ϕϕ ∗A = Adg−1A + g−1dg = g−1Ag + g−1dg = (g−1Aµg + g−1∂µg)dxµ (1.3.12) or equivalently Aµ ϕ − ! g−1 Aµg + g−1∂µg (1.3.13)

which is the usual non Abelian gauge transformation for the gauge field. In the same way, the curvature transforms as

F −! ϕϕ ∗F = Adg−1F =

1 2(g

−1

Fµνg) dxµ∧ dxν (1.3.14)

which again, considering only the components, gives the correct transformation for the field strength tensor. The Abelian case, i.e. for example the Maxwell case, the calculations are the same, but the term [A ∧ A] = 0; furthermore, using g = eiα(x)_{one gets the usual gauge transformations for the fields and the strength}

tensor.

Furthermore, we can also get a more intuitive understanding of the differential operator d by performing a few explicit calculations. Notice that there is a subtlety in the discussion below which we will describe at the end.

19_{Notice that the missing i in front of the commutator can be easily recovered by defining}

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We already saw that df = ∂µf dxµ so that the differential of a function

imme-diately gives us the (coeficients of the) gradient of the function. We can calculate now the curl of a vector field ω ≡ (ωx, ωy, ωz) in R3. We can write it as a 1-form

in R3 in the following way

ω = ωµdxµ = ωxdx + ωydy + ωzdz (1.3.15)

so that the exterior derivative dω can be written as20

dω = dωx∧ dx + dωy ∧ dy + dωz∧ dz. (1.3.16)

where the ωi are functions in C∞(R), hence we can write dωi = ∂µωidxµ, i.e.

dwx = ∂xωxdx + ∂yωxdy + ∂zωxdz (1.3.17)

dwy = ∂xωydx + ∂yωydy + ∂zωydz (1.3.18)

dwz = ∂xωzdx + ∂yωzdy + ∂zωzdz. (1.3.19)

Substituting these into eq(1.3.16) and using dµ ∧ dµ = 0 for µ ∈ Ω1(M ) we get dω = (∂yωz− ∂zωy)dy ∧ dz + (∂zωx− ∂xωz)dz ∧ dx + (∂xωy− ∂yωx)dx ∧ dy

(1.3.20) which is basically the curl21 _{of the vector function ω ≡ (ω}

x, ωy, ωz).

In the same way, if we write a tensor field f as a 2-form F

F = fxydx ∧ dy + fyzdy ∧ dz + fzxdz ∧ dx (1.3.21)

then, by the same reasoning as before,

dF = df ∧ dy ∧ dz = (∂xfyx+ ∂yfzx+ ∂zfxy)dx ∧ dy ∧ dz (1.3.22)

so that we get back the divergence of f . Summarizing, we can write that

• d : Ω0_(R3₎_{! Ω}1_(R3₎ _=⇒ _Gradient

• d : Ω1_(R3₎! Ω2_(R3₎ _=⇒ _Curl

• d : Ω2_(R3₎_{! Ω}3_(R3₎ _=⇒ _Divergence.

The differential operator, therefore, allows us to write in a coordinate indepen-dent way the gradient, curl and divergence, and it generalizes easily to manifolds different than R3_.

In order to be more precise, we have to notice that this kind of reasoning relies on the identification of the tangent and the cotangent bundle through the use of

20_{Here we are using the fact that terms like ω}

1d2x1= 0 since d2= 0.

21_{We would need to talk more extensively about the differential operator and forms to}

con-cretely justify this assertion. However, since we only want to gain a more intuitive understanding of the differential, we will not go into more mathematical details.

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a metric defined on the base manifold, which in Rn is always possible thanks to the Riesz representation theorem, see for example [Rud87]. In fact, for example, eq.(1.3.20) is not a curl, but (in Rn_{) one can always find a way to build a curl}

out of this 2-form. As a simple example of this identity, if one has a vector in Rn represented by a column of its components, there is a bijective relation with functionals on vector which is given by transposition of the column. In this way, we get covectors as rows of their components22. However, the differential operator d does not need the existence of a metric to be well defined, and if there is no identification of the tangent and cotangent bundles then we are not allowed to do this intuitive ”switch” which is always possible in Rn while still being able to use the differential.

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Chapter 2 Chern Simons theory

We now have all the tools to tackle the Chern Simons action and its properties. In the discussion of the Chern Simons theory, we will be more explicit about the calculations as well as the proofs of the statements.

Let h.i : g ! C be a linear form which is also Ad-invariant1_{, i.e. hAd} ga ∧

Adgbi = ha ∧ bi. We will call this form as ”trace”. It can be shown [Bas19] that

Ad-invariance implies

h[c ∧ a] ∧ bi = −ha ∧ [c ∧ b]i (2.0.1)

Let A be a connection, the Chern Simons Lagrangian can be written as α(A) = hA ∧ dA + 1

3A ∧ [A ∧ A]i (2.0.2)

= hA ∧ Ωi − 1

6hA ∧ [A ∧ A]i (2.0.3)

where Ω is the curvature defined in terms of the connection A. In the continuation of this chapter we will be more explicit with the proofs and the derivations.

2.1 Chern Simons action and its properties

It is possible to show that the Chern Simons (CS) Lagrangian satisfies a few important properties.

Theorem 6. [Bas19] [Fre95] Given the Chern Simons Lagrangian α(A) = hA ∧ Ωi −1₆hA ∧ [A ∧ A]i, the following relations hold:

1) i∗_xα(A) = −1

6hθ ∧ [θ ∧ θ]i, where ix : Px! P is the inclusion map;

2) dα(A) = hΩ ∧ Ωi; 3) α(ϕ∗A) = ϕ∗α(A);

1_{Usually this bilinear form is also indicated by the notation T r(.). Notice that here we are}

using the term form as a means to indicate an object which takes values on the Lie algebra and gives back complex numbers.

On Chern-Simons invariant of 3-manifolds

UNIVERSIT `

A DEGLI STUDI DI PISA

Department of Physics

Master Thesis in Physics

On Chern-Simons invariant of 3-manifolds

Andrea Bevilacqua

Advisors: Prof. Riccardo Benedetti, Prof. Enore

Guadagnini

Abstract

Acknowledgment

Contents

Chapter 1

Manifolds and fiber bundles

1.1

Manifolds

1.1.1

Some properties of R

1.1.2

Concrete definition of smooth manifolds

1.1.3

Abstract definition of smooth manifolds

1.2

Fiber bundles

1.2.1

Intuitive introduction: the tangent bundle

1.2.2

General definition of fiber bundles

1.2.3

Principal bundles

1.2.4

Sections and bundle morphisms

1.2.5

Pullback ans pushforward

1.2.6

Integration of forms

1.3

Connections and curvature

1.3.1

Gauge transformations of connection and curvature

1.3.2

Example in Physics

Chapter 2

Chern Simons theory

2.1

Chern Simons action and its properties

_{Some properties of R}