Homotopy Theory and Characteristic Classes · 2018-05-11 · Mombelli Stefania Homotopy Theory and Characteristic Classes ebruaryF 2018 1 Abstract The purpose of this proseminar was

Homotopy Theory and Characteristic Classes

Stefania Mombelli, BSc Physics

Proseminar in Theoretical Physics

ETH Zürich, spring semester 2018

Organiser: Prof. Matthias Gaberdiel

Supervisor: Dr. Blagoje Oblak

Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018

Contents

1 Abstract 2

2 Introduction 3

3 Homotopy Theory 3

3.1 Basic notions of topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 Homotopy of mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.3 Homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3.1 The fundamental group π1(Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3.2 Other homotopy groups πn(Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4 Topological quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Gauge Fields as Dierential Forms 9

4.1 Basic notions of dierential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.1 Fiber bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.2 Section and connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Gauge elds as dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2.1 Abelian gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2.2 Non-abelian gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Characteristic Classes 12

5.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3 Chern Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.4 Chern numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Conclusion 16

7 References 16

1


1 Abstract

The purpose of this proseminar was to nd a way to classify particular types of topological spaces called

ber bundles. In order to do this, we examined dierent areas of topology and dierential geometry,

namely homotopy theory and characteristic classes.

First of all, we studied the denitions of homotopy of mappings and homotpy groups and we analyzed

topological quantum numbers, which are important applications. Then, we made a short digression on

dierential geometry and gauge theories, which allowed us to understand the concept of characteristic

classes. We have seen how homotopy theory and characteristic classes are related with two examples,

regarding the cases of an abelian gauge theory on the plane and of a non-abelian gauge theory on the

Euclidean spacetime.

We have seen how the classication of ber bundles by means of homotopy theory and characteristic

classes leads to a quantization, which in some cases can also be measured physically. An example of this

appears in the Quantum Hall Eect, which we have not discussed in detail.

2


2 Introduction

The topic which we are going to discuss in this report is one of the rst ones presented in the proseminar

"Algebra, Topology and Group Theory in Physics". It constitutes an introduction to the mathematical

concepts which are necessary to explain some physical phenomena analyzed in the other proseminars.

We are going to refer to some of these applications without examining them in detail.

The main subjects of this work are homotopy theory and characteristic classes, which are two areas of

topology and dierential geometry. Our purpose is to use both these dierent and apparently unconnected

theories to classify particular types of mathematical spaces, called ber bundles, and collect them into

equivalence classes. In physics, this leads to the quantization of some observables associated with these

spaces, as we will see.

The report is divided into three sections. The rst one is about homotopy theory, which deals with

continuous deformations of mappings and topological spaces. The second part contains an introduction

in dierential geometry and gauge theories, which are necessary concepts in order to understand the last

part of the work, which is about characteristic classes. In particular, we are interested in Chern classes,

which are a type of characteristic classes, and we will see how these are closely related to homotopy

theory.

3 Homotopy Theory

In order to discuss homotopy theory, we rst need to understand some important concepts of topology.

3.1 Basic notions of topology

Topology is an area of mathematics which deals with the properties of space which are preserved under

continuous deformations. The most important notion in topology is the notion of topological space. The

following denition is not mathematically rigorous, but it is sucient for the purposes of this work.

Denition 3.1.1 (Topological space). A topological space is a mathematical space that allows for the

denition of concepts such as continuity, connectedness, and convergence.

In the following, we are going to deal with a particular type of mathematical spaces called manifols.

Denition 3.1.2 (Manifold). A manifold is a topological space which locally looks like the Euclidean

space. This means that each point on a n-dimensional manifold has a neighborhood which is homeomorphic

to Rn.

3.2 Homotopy of mappings

The starting point for dealing with homotopy theory is the following denition.

Denition 3.2.1 (Homotopic mappings). Let X, Y be topological spaces and let f, g: X −→ Y be two

continuous mappings. f and g are said to be homotopic if there exists a continuous mapping

h : X × [0, 1] −→ Y

(x, t) 7→ y

such that h(x, 0) = f(x) and h(x, 1) = g(x). This means that f is continuously deformable into g.

Example 3.2.1. Let X = S1 and Y = R2 \0. Let f, g be two mappings from X to Y, as shown in gure

1 . Then f is homotopic to the constant map, while g is not. It follows that f and g are not homotopic.

Statement 3.2.1. Homotopy is an equivalence relation.

3


Figure 1: [2]

Proof.

• Symmetry: if f is homotopic to g, then g is homotopic to f since the ow of the parameter t can

be reversed.

• Transitivity: f homotopic to g and g homotopic to k implies that f is homotopic to k, since

intervals can be adjoined and rescaled.

• Reexivity: f is homotopic to itself, since we can choose h(x, t) = f(x)∀t ∈ [0, 1].

The homotopy relation divides the set of continuous mappings from X to Y into equivalence classes,

called homotopy classes, which we denote by X,Y.

The denition of homotopic mappings is related to the concept of homotopic topological spaces.

Denition 3.2.2 (Homotopic topological spaces). Let X, Y be topological spaces. X and Y are said do

be homotopic if there exist mappings f : X −→ Y and g : Y −→ X such that f g : Y −→ Y and

g f : X −→ X are homotopic to the identity.

Example 3.2.2. The plane R2 is homotopic to a point P. In fact, we can construct the following map-

pings:

f : R2 −→ P

(x, y) 7→ P

and g : P −→ R2

P 7→ (0, 0).

The mapping f g : P −→ P is the identity. The mapping g f : R2 −→ R2 is homotopic to the identity.

The homotopy is given by the mapping:

h : R2 × [0, 1] −→ R2

((x, y), t) 7→ h((x, y), t) = (tx, ty).

We now report a statement which allows us to analyze the homotopy classes X,Y in a simple way.

Statement 3.2.2. Let X, Y1 and Y2 be topological spaces and let Y1 and Y2 be homotopic. Then there

exists a bijection between X,Y1 and X,Y2. Conversely, if X1, X2 and Y are topological spaces and

X1 and X2 are homotopic, there exists a bijection between X1,Y and X2,Y .

4


3.3 Homotopy groups

An interesting case consists in considering homotopy classes of mappings from the n-sphere Sn to a

topological space Y . In fact, in this particular case, it is possible to dene an operation between mappings

which induces an operation between homotopy classes and which confers to the set of classes a group

structure. In order to dene the group operation, the following denition is useful.

Denition 3.3.1 (Based maps). Let X and Y be topological spaces and let x0 ∈ X and y0 ∈ Y be xed

points. Maps f : X −→ Y with f(x0) = y0 are called based maps with base points x0 and y0.

We start from the case n = 1, that is we consider the homotopy classes of based maps from the circle

S1 to a space Y . The resulting group is called the fundamental group of Y and is denoted by π1(Y ).

Then, we will generalize this construction to mappings from the spheres Sn of higher dimensions n ≥ 2

to Y . In this case, we use the notation πn(Y ).

3.3.1 The fundamental group π1(Y )

We can view the based mappings f from S1 to Y as mappings from the interval [0,1] to Y , such that

f(0) = f(1) = y0. These maps are associated with paths in Y , beginning and ending at the point y0.

We can dene an operation f · g between two mappings f and g by taking the concatenation of the

corresponding paths. This means that we construct the path f · g by running rst along the path of f

and then along the one of g, as shown in gure 2.

Figure 2

Denition 3.3.2. Formally, the mapping f · g can be dened as

(f · g)(t) =

f(2t), 0 ≤ t ≤ 1/2

g(2t− 1), 1/2 ≤ t ≤ 1.

We now need to show that this operation is well dened, that is it induces an operation between the

homotopy classes of the considered mappings. In order to do this, it is enough to show the following

statement.

Statement 3.3.1. Consider based maps f, f ', g and g' from S1 to Y, starting and ending at y0, such

that f is homotopic to f ' and g is homotopic to g'. Then f ′ · g′ is homotopic to f · g.

Proof. Consider h1 : S1 × [0, 1] −→ Y to be the homotopy map between f and f ′ (h1(x, t = 0) = f(x)

and h1(x, t = 1) = f ′(x)) and consider h2 : S1 × [0, 1] −→ Y to be the homotopy map between g and g′

(h2(x, t = 0) = g(x) and h2(x, t = 1) = g′(x)). Then a homotopy map between f · g and f ′ · g′ is givenby h1 · h2.

5


This operation between homotopy classes is clearly associative.

In order to show that this operation allows to generate a group, we have to verify that some axioms,

namely the existence of the inverse element and of the identity element, are satised.

The identity element must be the class of mappings which are homotopic to the constant map c : S1 −→y0, since by concatenating a map with an element of this class we do not change the homotopy class of

the map.

The inverse element of a homotopy class is the class containing the maps associated with the paths covered

in the reverse direction, as shown in gure 3. In fact, by concatenating maps from the two classes, one

obtains a class of mappings which are homotopic to the constant map.

Figure 3

Finally, it is important to notice that the fundamental group is generally non-abelian. An example is

given by the fundamental group of the space Y shown in gure 4 , dened in polar coordinates by the

map s : [0, 2π] −→ R2, θ 7→ (θ, r = cos2(θ)). The fundamental group of Y is Z ∗ Z. This group is called

the free product of groups and consists of words whose letters are elements of the two groups, with the

multiplication of words induced by the multiplication of letters within their respective groups [4].

Figure 4: [15]

3.3.2 Other homotopy groups πn(Y )

The homotopy groups for n ≥ 2 are simply a generalization of the fundamental group. In order to dene

the group operation in this case, it is useful to notice that the n-sphere Sn is topologically equivalent

to a n-cube In with the boundary identied. Thus, because of statement 3.2.2, we are now allowed to

consider directly the homotopy classes of mappings from In to Y .

Denition 3.3.3. Consider two based maps f and g from In to Y, which map the boundary of the cube

to the base point y0 ∈ Y . We dene the operation f · g by the formula:

(f · g)(x1, xj) =

f(2x1, xj), 0 ≤ x1 ≤ 1/2

g(2x1 − 1, xj), 1/2 ≤ x1 ≤ 1

6


where j = 2,3,...,n and x1, x2, ..., xn are coordinates in the cube In. Since the boundary of the cube is

mapped to y0, we have f(1, xj) = g(0, xj) = y0. The operation is illustrated in gure 5: half of the n-cube

is mapped using the map f and the other half using the map g.

Figure 5: [2]

One can show that this operation is well dened in the same way as for the case of n = 1.

Again, we have to verify that the group axioms are satised.

The identity element is given by the class of mappings which map the whole n-cube to y0, like in the case

of the fundamental group.

The inverse element of a homotopy class of mappings f is the class containing the mappings f−1, where

f−1(x1, xj) = f(1− x1, xj).

In the case of n ≥ 2, the homotopy groups are always abelian. To show this, it is enough to show that

the mappings f · g and g · f from In to Y are homotopic. This can be done as follows. Consider the

n-cube of gure 5. We can now nd a family of mappings connecting f · g to g · f , as shown in gure

6 (the dark areas are mapped to y0). It is easy to see from this representation that this deformation is

allowed only for n ≥ 2.

Figure 6: [2]

3.3.3 Some examples

We would like now to give some important examples of homotopy groups.

Example 3.3.1. π1(Rd) = I. In fact, every path f : S1 −→ Rd is contractible (it is enough to parametrizeS1 with θ ∈ [0, 2π] and dene the homotopy as h(θ, t) = (1− t)f(θ)).

Example 3.3.2. π1(S1) = Z. A map from S1 to S1 can be dened as a continuous function f(θ) on

[0, 2π], where f(0) = 0 and f(2π) = 2πk, where k is an integer (because f is based and continuous).

k is called the winding number of f, since f(θ) goes k times around the target space S1 as θ goes once

around the domain. Maps f, g with the same winding number are homotopic, since one can dene a

family of mappings h(t, x) = (1 − t)f(x) + tg(x) (t ∈ [0, 1]), where h(t, 0) = 0 and h(t, 1) = 2πk, which

continuously connect them. Conversely, two homotopic maps have the same winding number (if this were

not the case, then the two maps would go dierent times around the target and it would not be possible to

7


construct a homotopy between them). It follows that the elements of π1(S1) can be labelled by integers,

and the concatenation of maps corresponds to the usual addition in Z. This means that π1(S1) and Z are

isomorphic.

An interesting application is that π1(U(1)) = Z (where U(1) = eiφ|φ ∈ R), since the Lie group U(1) is

homeomorphic to S1. We will see the importance of this example later.

Example 3.3.3. πn(Sn) = Z. The generator of the group is the class of the identity map from S1 to S1.

A representative of the k-th homotopy class is the map f : Sn −→ Sn, (rcos(θ), rsin(θ), x3, x4, ..., xn+1) 7→(rcos(kθ), rsin(kθ), x3, x4, ..., xn+1), where r is the radius of the n-sphere and θ ∈ [0, 2π].

An interesting application is that π3(SU(2)) = Z (where SU(2) = A ∈ GL(2,C)|A∗A = I, det(A) = 1),since the Lie group SU(2) is homeomorphic to S3. Again, we will see the importance of this example

later.

Example 3.3.4. πn(Sm) = I, for 1 ≤ n < m. In this case, the image of the map excludes at least one

point on the target sphere, which can be removed. In this way, we obtain the image of Sn in Rm, which

can be retracted to a point, as shown in example 3.3.1.

Example 3.3.5. The homotopy groups πn(Sm) for n > m ≥ 1 are dicult to compute. For example, we

have that πn(S1) = I ∀n ≥ 2, π3(S2) = Z, πn+1(Sn) = Z2 ∀n ≥ 3, πn+2(Sn) = Z2 ∀n ≥ 2. We will not

prove these statements.

Example 3.3.6. π1(Td) = Zd, where Td is the d-dimensional torus. In fact, π1(Td) is generated by d

loops which are not homotopic to each other.

3.4 Topological quantum numbers

Topological quantum numbers are quantities which take a discrete set of values due to topological con-

siderations, which are often related to the appearance of the homotopy group πn(Sn), whose elements

(homotopy classes) can be labelled by integers, as we have seen in example 3.3.3.

In order to compute these integers, it is useful to generalize the concept of winding number of a map

f : S1 −→ S1, also called topological degree, to higher dimensions. This will make the calculation of the

corresponding homotopy class easier.

Denition 3.4.1 (Topological degree (1)). Let X and Y be oriented closed manifolds of the same di-

mension d. Consider a map f : X −→ Y . We dene a volume form Ω = β(y)dy1 ∧ dy2 ∧ ... ∧ dyd on Y

with∫Y

Ω = 1 (Ω is normalized). The pull-back of Ω to X using the map f is f∗(Ω) = β(f(x))∂y1

∂xj dxj ∧

∂y2

∂xk dxk ∧ ...∧ ∂yd

∂xl dxl = β(f(x))J(x)dx1 ∧ dx2 ∧ ...∧ dxd where J(x) = det ∂y

i

∂xj is the Jacobian of the map

at x. We can now dene the topological degree of f by

degf =

∫X

f∗(Ω). (1)

One can show that the topological degree of a map is an integer and that it is a topological invariant

of f . It is important to notice that degf does not depend on the volume form Ω. In fact, if we choose

another normalized form Λ on Y , we have that∫Y

Λ−∫Y

Ω = 1− 1 = 0, which means that Λ−Ω = dχ is

an exact form. It follows that, when caluclating degf , the integral of the exact form vanishes, and hence

degf =∫Xf∗(Ω) =

∫Xf∗(Λ).

An important example is for X and Y being Sn. In this case, the degree of a map corresponds to the

integer number associated to the homotopy class of the map. If n = 1, the topological degree is equal to

the winding number of the map.

An example, whose importance will become clearer later, is the following.

Example 3.4.1. Consider a map from a three-dimensional manifold M to SU(2), represented by g(x).

The standard normalized volume form on SU(2) can be expressed as

Ω =1

24π2Tr(dgg−1 ∧ dgg−1 ∧ dgg−1). (2)

8


The topological degree of the map is obtained by integrating the pull-back of Ω to X, hence

deg g =1

24π2

∫X

Tr(dgg−1 ∧ dgg−1 ∧ dgg−1). (3)

There is a second, seemingly dierent way to dene the topological degree of a map.

Denition 3.4.2 (Topological degree (2)). Consider a map f : X −→ Y . For almost all points y ∈ Y ,the set of preimages of f (the points x(1), ..., x(M) in X mapped to y) is regular. We dene

degf =

M∑m=1

sign(J(x(m))) (4)

where sign(J(x(m)) is the sign of the Jacobian at x(m). degf counts the preimages of y with multiplicity

1 or −1, depending on whether f locally preserves or reverses the orientation.

The situation of the previous denition is well described in gure 7, in the case of a map f : S1 −→ S1.

Here, the equation f(x) = y for a regular point y may have either one or three solutions. In either case,

the sum (4) is equal to one.

Figure 7: [2]

One can show that degf = degf , which means that the two denitions are equivalent.

The notion of topological degree is very important for solitons. In fact, the topological aspect of a soliton

is often contained in the degree of a map associated with the soliton eld. Topological quantum numbers

also occur in quantum theory, as the name suggests. Here, quantum numbers are generated by some

topological constraints. An example of topological quantum numbers are Chern numbers. They will be

discussed in section 5.4.

4 Gauge Fields as Dierential Forms

The purpose of this section is to introduce gauge elds and to discuss them from a mathematical point

of view. The concept of gauge elds will turn out to be essential in order to dene characteristic classes

and Chern numbers.

4.1 Basic notions of dierential geometry

Before starting discussing gauge elds, we need to understand some basic concepts of dierential geometry,

namely the concepts of ber bundle, dierential form, section and connection.

9


4.1.1 Fiber bundles

Denition 4.1.1 (Fiber bundle). Let E, B and F be topological spaces and consider a mapping π :

E −→ B. We denote the space of the inverse images of the point b ∈ B as Fb (Fb = π−1(b)). If all Fb

are topologically equivalent to one another, we have a ber bundle. E is called the bundle space, B the

base space, F, to which all Fb are equivalent, is called ber and π is the bundle projection. The whole

construction is denoted by (E,B,F,π).

As one can deduce from the previous denition, a bundle space E is a topological space which locally

looks like the product space of its base space B and its ber F (E = B × F locally), i.e. for each point

b ∈ B, there exists a neighborhood U ⊂ B of b such that π−1(U) is homeomorphic to U × F . A ber

bundle is said to be trivial if the bundle space E is globally homeomorphic to the product space B × F .Here are some examples.

Example 4.1.1 (Cylinder). The surface of a cylinder is an example of a trivial ber bundle. The base

space is a circle and the ber is an interval. π is the orthogonal projection onto the base space.

Example 4.1.2 (Möbius strip). The Möbius strip is an example of a non trivial ber bundle. Also in

this case, the base space is a circle and the ber is an interval. But, contrary to the case of the cylinder,

the bers of the Möbius strip are twisted. So, the bundle space is not globally homeomorphic to a product

space any more.

Example 4.1.3 (Tangent bundle). Consider B to be a k-dimensional manifold in Rn. We imagine to

attach to every point b of the manifold the tangent space to that point, which is equivalent to Rk. This

construction is a ber bundle where the bundle space E is the set of all tangent vectors at all points of the

manifold B (if b and b' are dierent point in B, then the vectors tangent to B at b and b' are considered

to be dierent). The base space is the manifold B and the ber is Rk. The projection π maps every vector

of the tangent space to the point b to which it is attached.

Example 4.1.4 (Cotangent bundle). Consider a k-dimensional manifold B in Rn. If we imagine to

attach to every point of the manifold the dual space to the tangent space at that point, we construct a

ber bundle where the bundle space is the space of all linear forms λ : Rk −→ R (if λ1 and λ2 are dened

in dierent points of B, they are considered to be dierent), the base space is again the manifold B and

the ber F is Hom(Rk,R) ' Rk. The projection π maps every linear form of the cotangent space to the

point b to which it is attached.

Example 4.1.5 (d-th exterior power of the cotangent bundle). The last example can be generalized as

follows. Consider again a k-dimensional manifold B. To each point of B, we imagine to attach the space∧d(Rk) of all alternating d-linear maps λ : Rk × Rk × ... × Rk −→ R, also called dierential d-forms.

Again, we obtain a ber bundle.

Example 4.1.6 (Principal bundle). Consider a manifold B in Rn and a Lie group G. Then we can

construct a ber bundle with base space is B and ber G. The projection π maps the ber to the point to

which it is attached. This construction is called principal bundle.

In the following sections, we will be interested in the last two examples. In order to understand the

link between ber bundles and gauge elds, we still need to dene a couple of concepts, namely the

concepts of section and connection.

4.1.2 Section and connection

Denition 4.1.2 (Section). Consider a ber bundle (E,B,F,π). A section of (E,B,F,π) is a continuous

map σ : B −→ E such that π(σ(b)) = b, ∀b ∈ B.

A section of a ber bundle is simply the generalization of the concept of function in a product space. In

fact, in the product space B×F , σ is a mapping which maps every point b ∈ B to the point (b, f(b)) ∈ E,where f is some function on B.

10


Denition 4.1.3 (Connection). Let G be a Lie group. A connection on the principal bundle (E,B,G,π)

is a dierential 1-form with values in the Lie algebra g of G.

A connection can be seen as a device which allows to compare bers over nearby points.

4.2 Gauge elds as dierential forms

Now, we are ready to interpret physical gauge elds as dierential forms.

Consider B to be the euclidian spacetime R4 and consider a Lie group G. We want to construct three

types of ber bundles with base space B: a principal bundle with group G, the cotangent bundle and

the second exterior power of the cotangent bundle. It is clear from the considerations made in the

course on electrodynamics [5] that the vector potential A(x) = Aµ(x)dxµ and the eld strength F (x) =12Fµν(x)dxµ∧dxν can be interpreted as sections of the cotangent bundle and of the second exterior powerof the cotangent bundle, respectively. The interesting thing is that we can also see the vector potential

as a connection on the principal bundle, if we consider its components to be valued in the Lie algebra g

of G.

A gauge transformation is a locally dened transformation of the elds which leaves the Lagrangian

of the eld conguration invariant. It can be represented as a section of the principal bundle eiΛ(x)

(x ∈ R4,Λ(x) ∈ g), also called gauge parameter.

In the following two sections, we are going to discuss the gauge transformations of these elds, namely

the transformations which leave the Lagrangian (and the action) of the eld conguration invariant. We

will see that the elds transform in dierent ways depending on whether the group G is abelian or not.

4.2.1 Abelian gauge theory

Consider a principal bundle with base space the Euclidean spacetime R4 and ber an abelian Lie group

G, for example G = U(1), whose Lie algebra g is R. This is the case of electrodynamics.Consider A(x) = Aµ(x)dxµ, Aµ(x) ∈ g, to be a connection on the principal bundle, associated with some

eld strength F (x) = 12Fµν(x)dxµ ∧ dxν (F = dA), whose dynamics is contained in the following action

S[A] = −1

4

∫d4xFµνF

µν . (5)

This action is clearly invariant under the global transformation A(x) 7→ A(x)′ = A(x) +dΛ, where Λ ∈ g.

Consider φ ∈ C to be a eld dened on R4 (a section of a ber bundle with base space R4 and ber C),whose dynamics is described by the action

S[φ] =

∫d4x(

1

2∂µφ

∗∂µφ− m2

2φ∗φ) (6)

and on which the group G acts as a global symmetry (φ(x) 7→ gφ(x) where g = eiΛ, Λ ∈ g, is a

representation of G). If we require the global transformation g to become a gauge transformation (g =

g(x) and Λ = Λ(x)), then we need to modify the action (6) in order to make it gauge invariant. The

resulting action is the following.

S[φ,A] =

∫d4x[

1

2(∂µ − iAµ)φ∗(∂µ + iAµ)φ− m2

2φ∗φ− 1

4FµνF

µν ] (7)

which is invariant under the local gauge transformations

φ(x) 7→ φ′(x) = g(x)φ(x) (8)

Aµ(x) 7→ A′µ(x) = Aµ(x) + ∂µΛ(x) (9)

= Aµ(x)− ie−iΛ(x)∂µeiΛ(x) (10)

= Aµ(x)− ig−1(x)∂µg(x). (11)

The eld strength F is also clearly invariant under these transformations.

11


4.2.2 Non-abelian gauge theory

Let us move now to the more complicated case of a non-abelian gauge theory. Consider again a principal

bundle with base space the Euclidean spacetime R4 and ber a non-abelian Lie group, for example SU(2),

whose Lie algebra is the space iXaσa|X ∈ R (where the σa are the Pauli matrices).Consider A(x) = iAaµσadx

µ = iAµ(x)dxµ, iAµ(x) ∈ g , to be a connection on the principal bundle.

Consider φ ∈ C2 to be a eld dened on R4 described by an action of the form

S[φ] =

∫d4x(∂µφ

†∂µφ−m2φ†φ− λ(φ†φ)2), (12)

where λ ∈ C and on which the group SU(2) acts as a global symmetry (φ(x) −→ gφ(x), g ∈ SU(2)). If

we require the transformation g to be a gauge transformation (g = g(x)), we need to modify the action,

analogously to the abelian case. Again, it is necessary to introduce a gauge potential A in order to make

the action (12) gauge invariant. We obtain

S[φ,A] =

∫d4x[

1

2((∂µ +Aµ)φ)†(∂µ +Aµ)φ−m2φ†φ− λ(φ†φ)2] (13)

It follows that the gauge transformations are the following.

φ(x) 7→ φ′(x) = g(x)φ(x) (14)

Aµ(x) 7→ A′µ(x) = g(x)Aµ(x)g(x)−1 − ig−1(x)∂µg(x). (15)

We see that these transformations are slightly dierent from the abelian case. In fact, under global

transformations, the abelian gauge potential does not change, while non-abelian potential transforms

non-trivially according to the adjoint representation of G,

Aµ(x) 7→ A′µ(x) = gAµ(x)g−1. (16)

We would like now to construct the action for the eld Aµ analogously to (5). For this, we rst need to

nd the eld strength F corresponding to A. We require the following:

• F must contain a term ∂µAν − ∂νAµ, by analogy with the abelian gauge theory,

• F transforms according to the adjoint representation for all gauge transformations, that is

Fµν(x) −→ F ′µν(x) = g(x)Fµν(x)g−1(x).

In order for both conditions to be satised, one can show that the eld strength must have the following

form

Fµν(x) = ∂µAν − ∂νAµ + [Aµ, Aν ]. (17)

We can now choose the invariant action to be

S[A] = −1

4g2

∫d4xTr(FµνF

µν), (18)

where g is the gauge coupling constant, since the trace Tr is invariant under cyclic permutations.

5 Characteristic Classes

Given a ber F , a base space B and a structure group G, we are interested in knowing how many dierent

ber bundles we can construct and how they dier from the trivial bundle. Two ber bundles are con-

sidered dierent if they cannot be continuously deformed one into the other. A continuous deformation

of a ber bundle means a continuous deformation of the base space and of the bers. For example, the

surface of a cylinder and the surface of a conical frustum 1 are equivalent (and also trivial), while the

1A conical frustum is the portion of a cone which lies between two parallel planes cutting the cone [16].

12


cylinder and the Möbius strip are not, since the bers of the Möbius strip are twisted.

Homotopy theory already provides a way to measure the twisting of the bers of a ber bundle. For

example, an SU(2) bundle over S3 can be classied by the homotopy group π3(SU(2)) = Z, where theelements of the homotopy classes are the gauge parameters (sections of the principal bundle). The inte-

gers associated with the homotopy classes indicate the degree of twisting of the bers.

Another tool which allows to measure the non triviality of twisting of a ber bundle are characteristic

classes. In order to dene this concept, we rst need to know a couple of notions from algebraic topol-

ogy, namely the notion of de Rham cohomology and of invariant polynomials. We will understand the

importance of homotopy theory and gauge elds for characteristic classes later in this section.

5.1 Basic notions

We start with the denition of de Rham cohomology.

Denition 5.1.1. Let M be a m-dimensional manifold and let ω be a closed dierential k-form (dω = 0)

dened on M. The de Rham cohomology class of ω is dened as

[ω] = ω + dχ|χ is a (k − 1)− formonM. (19)

We denote by Ek(M) the set of exact k-forms on M and by Ck(M) the set of closed k-forms on M. The

k-th de Rham cohomology group of M is dened as

Hk(M) = [ω]|dω = 0 = Ck(M)/Ek(M) (20)

The second important denition which we need is the denition of invariant polynomials.

Let M be a m-dimensional manifold and consider a principal bundle on M with structure group G.

Let ω = ωη, with ω ∈ g and η ∈∧k

(M), be a dierential k-form dened on M . Let P be a polyno-

mial of degree r which takes its values in g. With the notation P (ω) we indicate the dierential form

P (ω) η ∧ η ∧ ... ∧ η︸︷︷︸r

.

Denition 5.1.2 (Invariant polynomials). A polynomial P (ω) is said to be invariant if it is invariant

with respect to the adjoint representation of G, namely if

P (ω) = P (gωg−1) (21)

for g ∈ G.

In the following, we are interested in polynomials of the form P (F ), where F is the eld strength (also

called curvature 2-form) associated to some gauge potential A (connection). We notice that P (F ) must

be a gauge invariant quantity, since, as we have seen, in the worst case (the case of a non-abelian gauge

theory) the eld strength F transforms according to the adjoint representation of the gauge group.

5.2 Characteristic Classes

The importance of invariant polynomials and de Rham cohomology resides in the following theorem,

which we state without proof.

Theorem 5.2.1 (Chern-Weil). Let P be an invariant polynomial and F a curvature 2-form dened on a

manifold M. Consider a principal bundle on M with structure group G. Then P(F) satises:

• dP(F) = 0

• Let F and F' be curvature two-forms corresponding to dierent connections A and A' on E. Then

the dierence P(F')-P(F) is exact.

13


The Chern-Weil theorem states that it is possible to associate to an invariant polynomial P a coho-

mology class of M , which does not depend on the connection chosen. Since a variation of the connection

implies a continuous deformation of the bers of the principal bundle (and of all other bundles dened on

M which are associated to the principal bundle), topologically equivalent ber bundles can be associated

to the same cohomology class of M , represented by P . This leads to the denition of characteristic class.

Denition 5.2.1 (Characteristic class). Let P be an invariant polynomial and consider a ber bundle

with a manifold M as a base space. The cohomology class of M corresponding to P is called characteristic

class of the ber bundle (corresponding to P).

It follows from the considerations above that characteristic classes are topological invariants of the

ber bundle, in the sense that topologically equivalent bundles have the same characteristic classes. This

implies that ber bundles with dierent characteristic classes must be inequivalent. Hence, the compu-

tation of characteristic classes provides a way of classifying ber bundles.

There are many dierent kinds of characteristic classes, depending on the choice of the invariant poly-

nomials. In the following, we are going to consider a type of characteristic classes called Chern classes,

which have some applications in physics.

5.3 Chern Classes

In order to dene Chern classes, we need to give some representative invariant polynomials.

Denition 5.3.1 (Total Chern class). Consider a ber bundle on a manifold M whose ber is Ck and

with structure group G. Consider the gauge potential A and the eld strength F to take their values in g.

The total Chern class is dened by

C(F ) = det(1 +iF

2π). (22)

Since F is a 2-form, C(F) is a direct sum of forms of even degrees,

C(F ) = 1 + C1(F ) + C2(F ) + ... (23)

where Ci(F ) ∈∧2i

(M) is an invariant polynomial which represents the i-th Chern class. By expanding

the determinant in (22), we obtain the following explicit formulae.

C0(F ) = 1 (24)

C1(F ) =i

2πTr(F ) (25)

C2(F ) =1

2(i

2π)2[Tr(F ) ∧ Tr(F )− Tr(F ∧ F )] (26)

...

Ck(F ) = (i

2π)kdet(F ). (27)

In the case of G being abelian, this denition reduces to

C0(F ) = 1 (28)

C1(F ) =1

2πF (29)

C2(F ) =1

8π2(F ∧ F ) (30)

...

Ck(F ) = (i

2π)k F ∧ F ∧ ... ∧ F︸︷︷︸

k

. (31)

14


5.4 Chern numbers

As we have seen, ber bundles can be classied using both characteristic classes (Chern classes) and

homotopy theory. One possible link between the two is represented by Chern numbers, which are an

example of topological quantum numbers. In fact, they allow us to label the characteristic classes of the

ber bundle with integer numbers corresponding to the topological degree of a map from the base space

to the ber. This will become clearer later with some examples.

Chern numbers have a direct application in physics, since they are topological quantum numbers. An

example is given by the Hall conductivity in the Quantum Hall Eect, which is quantized and is a Chern

number.

Chern numbers are dened in the following way.

Denition 5.4.1 (Chern numbers). Given a ber bundle on a 2i-dimensional manifold M, the i-th Chern

number ci is dened to be the integral of the i-th Chern form over M.

ci =

∫M

Ci. (32)

Chern numbers can also be obtained by integrating wedge products of Chern forms Ci1 , Ci2 , ..., Cin (with

2i = 2(i1 + i2 + ...+ in)) on M. One can show that Chern numbers are always integers.

The following examples will show how Chern numbers and homotopy theory are related.

Example 5.4.1 (Chern numbers of abelian gauge elds). Let us consider a U(1) gauge eld in the plane

R2. Consider F to be smooth and to decay to zero rapidly as ‖x‖ −→ ∞. This implies that the gauge

potential at innity can be written as a pure gauge A(x) = dΛ(x), where Λ(x) ∈ R. We also assume the

gauge parameter eiΛ(x) ∈ U(1) to be single valued, that is, in polar coordinates, eiΛ(0,r) = eiΛ(2π,r) for

every r ∈ [0,∞] (this can be justied by coupling the gauge potential to a scalar eld φ, since the quantity

|φ|2 is a physical observable and hence globally well dened). It follows that the dierence Λ(2π, r)−Λ(0, r)

must be 2πN , where N is an integer corresponding to the winding number of the gauge parameter. We

want to calculate the rst Chern number of this conguration.

c1 =1

2π

∫R2

F (33)

=1

2π

∫ ∞−∞

∫ ∞−∞

F12 dx1dx2 (34)

=1

2π

∫S1∞

A (35)

=1

2π

∫ 2π

0

Aθ dθ (36)

=1

2π

∫ 2π

0

∂θΛ dθ (37)

=1

2π(Λ(2π)− Λ(0)) (38)

= N (39)

The third step follows from Stokes theorem and the fth follows from the boundary conditions which we

have imposed.

We can see that the rst Chern number of this conguration corresponds to the topological degree of a

map S1 −→ U(1) (the gauge parameter).

The same example can be made in higher dimensions with a non abelian gauge theory.

Example 5.4.2 (Chern numbers of non-abelian gauge elds). Consider a SU(2) gauge eld in the

Euclidean spacetime R4. Consider F to decay suciently fast as ‖x‖ −→ ∞. Then, the gauge potential

15


at innity can be written as a pure gauge A(x) = −dg(x)g(x)−1, where g(x) ∈ SU(2). We can now

compute the second Chern number of this conguration.

c2 =

∫R4

C2 (40)

=1

8π2

∫R4

(Tr(F ∧ F )− TrF ∧ TrF ) (41)

=1

8π2

∫R4

d[Tr(F ∧A− 1

3A ∧A ∧A)] (42)

=1

8π2

∫S∞

Tr(F ∧A− 1

3A ∧A ∧A) (43)

= − 1

24π2

∫S∞

Tr(A ∧A ∧A) (44)

=1

24π2

∫S∞

Tr(dgg−1 ∧ dgg−1 ∧ dgg−1) (45)

The third step follows from the Stokes theorem while the fth one follows from the boundary conditions

which we imposed.

This is the formula which we had obtained in example 3.4.1. for the degree of a map (the gauge parameter

g) from S3 −→ SU(2). Thus, the second Chern number of this conguration is an integer.

We know from homotopy theory that the topological degree of a mapping is invariant under continuous

deformations. Since, as we have seen, the Chern classes of a ber bundle are also invariant under

continuous deformations, we are allowed to label these classes with the Chern numbers.

6 Conclusion

We have seen that ber bundles can be classifyed using dierent methods, namely homotopy theory and

characteristic classes, which are strictly related one to the other. Topologically equivalent ber bundles

can be collected into equivalence classes, which are topological invariants of the bundle. Passing from

a class to another implies deformations of the bundle which are not continuous. These classes can be

labelled by integers, which correspond to the topological degree of sections of the ber bundle. In physics,

this is associated with the quantization of some quantities related to the bundle.

7 References

Books

[1] N. Manton, P. Sutclie, "Topological Solitons", Cambridge University Press, 2004.

[2] V. Rubakov, "Classical Theory of Gauge Fields", Princeton University Press, 2002.

[3] M. Nakahara, "Geometry, Topology and Physics", IOP publishing, 1990.

[4] A. Hatcher, "Algebraic topology", Cambridge University Press, 2002.

Lecture notes

[5] M.R. Gaberdiel, "Klassische Elektrodynamik", ETH Zürich, 2017.

Web pages

[6] Wikipedia, The Free Encyclopedia, "Topological quantum number", https://en.wikipedia.org/

wiki/Topological_quantum_number, 05.03.2018.

[7] Wikipedia, The Free Encyclopedia, "Section (ber bundle)", https://en.wikipedia.org/wiki/

Section_(fiber_bundle), 05.03.2018.

[8] Wikipedia, The Free Encyclopedia, "Connection (principal bundle)", https://en.wikipedia.org/

wiki/Connection_(principal_bundle), 07.03.2018.

[9] Wikipedia, The Free Encyclopedia, "Principal bundle", https://en.wikipedia.org/wiki/Principal_

16

https://en.wikipedia.org/wiki/Topological_quantum_number

https://en.wikipedia.org/wiki/Topological_quantum_number

https://en.wikipedia.org/wiki/Section_(fiber_bundle)

https://en.wikipedia.org/wiki/Section_(fiber_bundle)

https://en.wikipedia.org/wiki/Connection_(principal_bundle)

https://en.wikipedia.org/wiki/Connection_(principal_bundle)

https://en.wikipedia.org/wiki/Principal_bundle



bundle, 07.03.2018.

[10] Wikipedia, The Free Encyclopedia, "Dierential form", https://en.wikipedia.org/wiki/Differential_

form, 07.03.2018.

[11] Wikipedia, The Free Encyclopedia, "Topological space", https://en.wikipedia.org/wiki/Topological_

space, 10.03.2018.

[12] Wikipedia, The Free Encyclopedia, "Characteristic class", https://en.wikipedia.org/wiki/Characteristic_

class, 09.02.2018.

[13] Wikipedia, The Free Encyclopedia, "Chern class", https://en.wikipedia.org/wiki/Chern_class,09.02.2018.

[14] Wikipedia, The Free Encyclopedia, "Topology", https://en.wikipedia.org/wiki/Topology,10.03.2018.

[15] Wolfram Alpha, http://www.wolframalpha.com/widgets/view.jsp?id=8d8e2c27bcaa121d6ee0de4b98774bb4,

14.03.2018.

[16] Wolfram Math World, http://mathworld.wolfram.com/Frustum.html, 26.03.2018

17



https://en.wikipedia.org/wiki/Differential_form

https://en.wikipedia.org/wiki/Differential_form

https://en.wikipedia.org/wiki/Topological_space

https://en.wikipedia.org/wiki/Topological_space

https://en.wikipedia.org/wiki/Characteristic_class

https://en.wikipedia.org/wiki/Characteristic_class

https://en.wikipedia.org/wiki/Chern_class

https://en.wikipedia.org/wiki/Topology

http://www.wolframalpha.com/widgets/view.jsp?id=8d8e2c27bcaa121d6ee0de4b98774bb4

http://mathworld.wolfram.com/Frustum.html

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Homotopy Theory and Characteristic Classes · 2018-05-11 · Mombelli Stefania Homotopy Theory and Characteristic Classes ebruaryF 2018 1 Abstract The purpose of this proseminar was