Chern Classes and the Hall Effect

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Chern Classes and the Hall Effect

Shin Komatsu

27th June 2020

Bachelor thesis Mathematics and Physics & Astronomy Supervisors: prof. dr. Eric Opdam, dr. Vladimir Gritsev

Institute of Physics

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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Abstract

This thesis aims to provide a link between the mathematics of Chern classes and the physics of the quantum Hall effect. The first step is to define connections on vector bundles in order to define Chern classes. Next, principal bundles and their connections are introduced and their interchangeability with connections on vector bundles is shown. The vector bundles are then looked at with ˇCech cohomology, and the integrality of the first Chern number for a compact orientable 2-manifold is proven.

Then the Hall effect is approached with different perspectives. First, we get the clas-sical Hall effect, using only clasclas-sical electromagnetism. We then take a typical quantum approach, where the eigenstates of the Hamiltonian for the electrons in the Hall effect are found. Finally, with the use of perturbation theory, an explicit calculation of the Hall conductivity is performed. This Hall conductivity is shown to be the first Chern number of some line bundle over a torus up to some constant, thus proving its integrality and its link to the Chern classes.

Title: Chern Classes and the Hall Effect

Author: Shin Komatsu, shin.komatsu2000@gmail.com, 11671785 Supervisors: prof. dr. Eric Opdam, dr. Vladimir Gritsev

Second graders: dr. Hessel Posthuma, prof. dr. Jean-S´ebastien Caux End date: 27th June 2020

Cover image: This image illustrates a Hermitian line bundle over manifold M and is taken from [1].

Institute of Physics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl

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1 Introduction

The Hall effect is the production of a voltage, the Hall voltage, in a conducting plate perpendicular to the current, when a magnetic field is applied perpendicular to the plate. The ratio of this Hall voltage and the current is called the Hall conductivity and was first discovered by Edwin Hall in 1879 [2].

In classical electromagnetism, this conductivity should increase linearly with the mag-nitude of the magnetic field. However, Klaus von Klitzing discovered a quantized Hall conductivity in 1980 and was awarded the Nobel prize for this discovery in 1985. Von Klitzing showed that at low temperatures and strong magnetic fields the Hall conduct-ivity would take on values of integer multiples of _2π~e2 [3]. This is called the integer quantum Hall effect and will be the primary focus of this thesis. To understand this phenomenon, we first take a look at the mathematics of Chern classes.

Chapter 2 explains the necessary background in differential geometry to study Chern classes. I define vector bundles and their transition maps and refresh the theory of the de Rham cohomology and the de Rham differential operator d.

In section 3.1 my goal is to define Chern classes and Chern numbers. I do this by extending some notions in manifolds, such as the differential and smooth forms, to vector bundles. The extension of the differential is the connection ∇. Instead of differentiating smooth functions, it differentiates smooth sections. Unlike the de Rham differential, the connection’s square is nonzero. This square of the connection is called the curvature. The curvature is used to define the Chern classes and Chern numbers.

I then introduce a new concept: principal bundles. This has a similar definition to vector bundles, but instead of vector spaces the fibers are Lie groups. We will see that vector bundles and principal bundles are closely related. I then give the definition of connections on a principal bundle, and here the similarities between vector bundles and principal bundles arise even more. The connection of the vector bundle that of the principal bundle are shown to be interchangeable with each other.

From this point on, we restrict ourselves to only look at line bundles: vector bundles of which the fibers are complex lines (one-dimensional vector spaces). For these, we have a strategy to calculate the Chern number. First, we introduce ˇCech cohomology by defining presheafs. Finally, we prove that the Chern number for a specific case (the one we need) is an integer up to some constant.

Chapter 4 discusses the physics of the Hall effect. First, I describe the classical Hall effect, what classical electromagnetism predicts the Hall conductivity to be. I then take an approach that is typical for quantum mechanics to find the energy states of the electrons in the Hall effect. Finally, I calculate the Hall conductance and show that it is actually equal to the first Chern number of a line bundle up to a constant, and we will have derived the integer quantum Hall effect.

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2 Preliminaries

This chapter aims to provide the necessary background in differential geometry to un-derstand the theory of Chern classes.

2.1 Vector bundles

Definition 2.1.1. Let M be a smooth manifold. A smooth complex vector bundle of rank n over M is a smooth manifold V with a smooth surjective map π : V → M such that:

(i) For each x ∈ M , Vx = π−1(x) has the structure of an n-dimensional complex

vector space.

(ii) For each x ∈ M , there is a neighbourhood U of p in M and a diffeomorphism Φ : π−1(U ) → U × Cn (called a trivialization of V over U ), such that:

• π_U◦ Φ = π (where π_U _{: U × C}n_{→ U is the projection on the first argument),}

• For each y ∈ U , Φ|_V_y : Vy → {y} × Cn∼= Cn is a vector space isomorphism.

π−1(U ) is called the vector bundle V restricted to U , and it is denoted by V |U. If

there exists a trivialization of V over M , also called a global trivialization, we call V the trivial vector bundle of rank n.

One can also define a smooth real vector bundle, by replacing Cn with Rn. Note that every complex vector bundle of rank n is a real vector bundle of rank 2n, by restriction of scalars. We will see that given a real vector bundle V , a complex vector bundle can be defined by using extension of scalars.

Example 2.1.2. For a smooth manifold M , the tangent vector bundle T M is a smooth real vector bundle.

Example 2.1.3. For vector bundles V, V0, of which the fibers are Vx, Vx0 respectively,

notions of vector spaces such as the dual space and tensor products can be defined. One can define a dual bundle V∗, of which the fibers are V_x∗, or a tensor product bundle V ⊗ V0, of which the fibers are Vx⊗ Vx0. Even though the ideas of these vector bundles

are simple, a lot more work is needed to prove that they can be defined. We can then take the tensor product of any real vector bundle V with the complex trivial bundle of rank 1 such that its fibers are C ⊗ Vx, making a complex vector bundle.

Example 2.1.4. Let π1,2 : V1,2 → M two smooth vector bundles over M . Then the

subspace V1×M V2 of V1× V2 defined by

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is a smooth vector bundle, with projection π(v1, v2) = π1(v1) = π2(v2).

Definition 2.1.5. Let π1,2 : V1,2 → M be smooth vector bundles. V1, V2 are isomorphic

if there exists a diffeomorphism f : V1→ V2 such that the diagram

V1 V2

M

π1

f

π2

commutes, and f restricted to π₁−1(x) is a vector space isomorphism for every x ∈ M . We will denote V1 ∼= V2 for vector bundle isomorphisms.

2.1.1 Sections and frames

Definition 2.1.6. For a smooth manifold M , define the smooth functions: C∞(M ) = {f : M → C : f is smooth}.

Definition 2.1.7. Let π : V → M be a smooth vector bundle. A smooth section of V is a smooth map σ : M → V such that π ◦ σ = idM. If σ : U → V |U for U ⊂ M is such

a map, we call σ a smooth local section on U.

The space of smooth sections over V is denoted by C∞(V ), and it is a C∞(M )-module. The space of smooth local sections on U is denoted by C∞(V |U).

Remark. When V is the trivial bundle of rank n, so V ∼= M × Cn, the space of smooth sections of V are isomorphic to smooth functions to Cn:

C∞(V ) ∼= {f : M → Cn : f smooth}.

Definition 2.1.8. Let π : V → M be a smooth vector bundle of rank n, and U an open subset of M . An n-tuple of smooth local sections on U (σ1, . . . , σn) is called a

smooth local frame on U if (σ1(x), . . . , σn(x)) is a basis of Vx for all x ∈ M . If U = M ,

(σ1, . . . , σn) is called a smooth global frame.

Proposition 2.1.9. Let π : V → M a smooth vector bundle, and U an open subset of M . Then the following statements are equivalent:

(i) There exists a trivialization over U , (ii) There exists a smooth local frame on U .

Proof. Given a trivialisation Φ : π−1(U ) → U ×Cn, the n-tuple (Φ−1(−, e1), . . . , Φ−1(−, en))

gives a smooth local frame, where (e1, . . . , en) is the standard basis of Cn.

Conversely, given a smooth local frame (σ1, . . . , σn) on U , the inverse of the map

(x, (v1, . . . , vn)) 7→ v1σ1(x) + · · · + vnσn(x)

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A Hermitian product can be defined for complex vector bundles (equivalently, an inner product for real vector bundles), as an extension of Hermitian products in vector spaces. Definition 2.1.10. Let π : V → M be a smooth complex vector field. A Hermitian product on V is a smooth map h−, −i : V ×M V → C such that for every x ∈ M , the

restriction h−, −i |Vx×Vx : Vx× Vx→ C is a Hermitian product. A vector bundle with a

Hermitian product is called a Hermitian bundle.

This makes every Vx not just a vector space, but a Hermitian vector space. We can

also define frames to be orthonormal if, they are an orthonormal basis in every Vx.

Using the Gram-Schmidt process, every frame can be made orthonormal in a Hermitian bundle. This has the following proposition as a consequence:

Proposition 2.1.11. Let π : V → M be a smooth Hermitian bundle, and U an open subset of M . Then te following statements are equivalent:

(i) There exists a trivialization over U ,

(ii) There exists an orthonormal smooth local frame on U .

2.1.2 Transition maps of vector bundles

For a vector bundle π : V → M , there is a cover {(Uα, Φα)}α∈A such that all (Uα, Φα)

satisfy the local trivialization condition. Such a cover is called a trivializing cover. Let {(Uα, Φα)}α∈A be such a trivializing cover. For α, β ∈ A,

hαβ := Φα◦ (Φβ)−1: Uα∩ Uβ× Ck→ Uα∩ Uβ× Ck

is a diffeomorphism that maps {x} × Ck to {x} × Ck for a x ∈ Uα ∩ Uβ, and the

restriction of hαβ to {x} × Ck gives a linear isomorphism to itself. This means that the

map gαβ : Uα∩ Uβ → GLn(C) given by

hαβ(x, v) = (x, gαβ(x)v)

is determined completely. Furthermore, because hαβ is smooth, so is gαβ.

So given a smooth complex vector bundle V of rank n over M , we get an open cover {Uα}α∈A of M and smooth maps {gαβ : Uα∩ Uβ → GLn(C)} that satisfy the following

conditions:

(i) gαα= In, because hαα = id,

(ii) gαβgβα= In, because hαβ ◦ hβα= id,

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All of these conditions are obvious if you write out hαβ = Φα◦ (Φβ)−1.

Conversely, Let {Uα}α∈A be an open covering of M with smooth maps {gαβ : Uα∩

Uβ → GLn(C)} that satisfy conditions (i)-(iii). Then there is a vector bundle V with

trivializations {(Uα, Φα)} such that

Φα◦ (Φβ)−1(x, v) = (x, gαβ(x)v),

for all (x, v) ∈ Uα∩ Uβ× Cn, and such a vector bundle is unique up to isomorphism [4].

In other words, only local data is needed to define the global vector bundle, given that the transitions satisfy (i)-(iii). Using this, we can extend the following notions of vector spaces, to vector bundles.

Example 2.1.12. Let π : V → M be a smooth vector bundle of rank n. V has trivializations {(Uα, Φα)}, but we have seen that this is the same as having local frames

{(U_α, (σα,1, . . . , σα,n))}. In every point, x ∈ Uα, (σα,1(x), . . . , σα,n(x)) is a basis, so we

can take the dual basis. Doing this in every point for every frame gets us local frames (σ∗_α,1, . . . , σ_α,n∗ ), which in turn gives us trivializations Φ∗_α, and transition maps g∗_αβ. Some basic linear algebra will show us that g_αβ∗ = (gβα)T, and these will satisfy (i)-(iii), because

gαβ satisfy those. So there is a vector bundle V∗ of which the fibers are precisely Vx∗.

Example 2.1.13. Given smooth vector bundles π : V → M and π0 : V0 → M , we can construct V ⊕ V0 and V ⊗ V0:

Given local trivializations {(Uα, Φα)}α∈A and {(U_β0, Φβ)}β∈A0, the trivializations for

V ⊕ V0and V ⊗ V0 are simply {(Uα∩ Uβ, (Φα, Φβ))} and {Uα∩ Uβ, Φα⊗ Φβ} respectively,

where we take the direct sum and tensor product in the second argument. Conditions (i)-(iii) follow from those of V and V0.

There are the following isomorphisms of C∞(M )-modules: C∞(V ⊕ V0) ∼= C∞(V ) ⊕ C∞(V0)

C∞(V ⊗ V0) ∼= C∞(V ) ⊗C∞_{(M )}C∞(V0).

Because of the construction of tensor products of vector bundles, we can make a complex vector bundle out of every real vector bundle π : V → M , by using extension of scalars, by taking the tensor product with the trivial complex line bundle. This way we get a complex vector bundle C ⊗ V , where the fibers are C ⊗ Vx.

From this point forward I will always assume that V is a complex vector bundle. Definition 2.1.14. For vector bundle V over M , the k-forms on V are defined as

Ωk(V ) = Ωk(M ) ⊗_C∞_{(M )}C∞(V ) ∼= C∞(

^k

T∗M ⊗ V ). Note that Ω0(V ) = C∞(M ) ⊗C∞_{(M )}C∞(V ) = C∞(V ).

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2.2 De Rham cohomology group

Definition 2.2.1. Let M be a smooth manifold. For a smooth function f ∈ C∞(M ), define the differential of f , df ∈ Ω1(M ) by

df (X) = Xf for X ∈ C∞(T M ). In local coordinates, it holds that df = P ∂f

∂xidxi, where xi : U → R is the i-th

component of the coordinate chart. This definition can be extended to k-forms on M as follows. Locally, ω ∈ Ωk_{(M ) can be written as}

ω = X0

I

ωIdxi1 ∧ · · · ∧ dxik

with ωI ∈ C∞(M ), whereP0 sums over all I = (i1, . . . , ik) with 1 ≤ i1 < · · · < ik ≤ m

with m being the dimension of M . We define dω locally as dω = X0

I

dωI∧ dxi1 ∧ · · · ∧ dxik.

It turns out that this definition doesn’t depend on local coordinates, which means that this local description actually defines a global map [5]. This gives a map d : Ωk(M ) → Ωk+1(M ) for every k. This can also be written as d : Ω•(M ) → Ω•+1(M ).

Definition 2.2.2. Let ω ∈ Ωk(M ). Then ω is closed if dω = 0. The space of closed k-forms is Ker(d : Ωk→ Ωk+1_).

Definition 2.2.3. Let ω ∈ Ωk(M ). Then ω is exact if there exists η ∈ Ωk−1(M ) such that dη = ω. The space of exact k-forms is Im(d : Ωk−1 → Ωk_{(M )).}

Some computation will show us that d2 = 0 [5]. In other words, every exact form is closed. So for each k, Im(d : Ωk−1_{(M ) → Ω}k_{(M )) ⊂ Ker(d : Ω}k_{(M ) → Ω}k+1_{(M )).}

Definition 2.2.4. The k-th de Rham cohomology group Hk(M ) is defined as

Hk(M ) =

Ker(d : Ωk_{(M ) → Ω}k+1_{(M ))}

Im(d : Ωk−1_{(M ) → Ω}k_{(M ))}.

Elements of the k-th cohomology group are equivalence classes of closed k-forms, where two closed k-forms ω, η are equivalent if their difference ω − η is exact.

Even though the closed forms and the exact forms are infinite dimensional spaces in most cases, the cohomology groups are always finite dimensional. The proof of this is given in [6].

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3 Chern classes of vector bundles

3.1 Connections and curvatures

In order to define Chern classes of a vector bundle π : V → M , we first need to define a connection. A connection can be viewed as differentiating sections of V in the same way that the de Rham differential d differentiates functions on M .

Definition 3.1.1. Let π : V → M be a vector bundle. A connection on V is defined as a linear map ∇ : C∞(V ) → Ω1(M ) ⊗C∞_{(M )}C∞(V ) such that for all sections σ ∈ C∞(V )

and all smooth functions f ∈ C∞(M ) a version of the Leibniz rule holds:

∇(f σ) = df ⊗ σ + f ∇(σ). (3.1)

As ∇(σ) gives an element in C∞(T∗M ) ⊗C∞_{(M )} C∞(V ), ∇ can also be seen as a

map C∞(V ) ⊗ C∞(T M ) → C∞(V ). In this case, because of the C∞(M )-linearity, the connection satisfies

∇f X(σ) = f ∇X(σ).

For the trivial bundle of rank n, a smooth section σ corresponds to a smooth function ˜

σ : M → Rn and the de Rham differential d is a connection. This means that on a local trivialization, there always exists a connection. Using a partition of unity, we see that there always exists a global connection on V .

A connection ∇ is a map Ω0(V ) → Ω1(V ). There is a way to extend this definition to a map Ωk(V ) → Ωk+1(V ).

Theorem 3.1.2. For a connection ∇ : C∞(V ) → Ω1(V ), there is an extension ∇ : Ωk(V ) → Ωk+1(V ).

Proof. Define ∇ : Ωk(M ) × C∞(V ) → Ωk+1(V ) by:

∇(ω, σ) = dω ⊗ σ + (−1)kω ∧ ∇(σ).

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check C∞(M )-bilinearity: ∇(f ω, σ) = d(f ω) ⊗ σ + (−1)kf ω ∧ ∇(σ) = df ∧ ω ⊗ σ + f dω ⊗ σ + f (−1)kω ∧ ∇(σ) = df ∧ ω ⊗ σ + f ∇(ω, σ) ∇(ω, f σ) = dω ⊗ f σ + (−1)kω ∧ ∇(f σ) = f dω ⊗ σ + (−1)kω ∧ df ⊗ σ + (−1)kω ∧ f ∇(σ) = f dω ⊗ σ + (−1)k(−1)kdf ∧ ω ⊗ σ + f (−1)kω ∧ ∇(σ) = df ∧ ω ⊗ σ + f ∇(ω, σ)

So there is a linear map ∇ : Ωk(V ) → Ωk+1(V ), with the following relation: ∇(f ω ⊗ σ) = df ∧ ω ⊗ σ + f ∇(ω ⊗ σ).

This is clearly an extension of ∇ : C∞(V ) → Ω1_{(V ).}

If we have two different connections ∇ and ∇0, requirement 3.1 gives us that their difference is C∞(M )-linear:

(∇ − ∇0)(f σ) = df ⊗ σ + f ∇(σ) − (df ⊗ σ + f ∇0(σ)) = f (∇ − ∇0)(σ). This means

∇ − ∇0 ∈ C∞(V∗) ⊗C∞_{(M )}C∞(V ⊗ T∗M ) = C∞(V∗⊗ V ⊗ T∗M ) = Ω1(End(V )).

Recall that the de Rham differential is a connection on local trivializations. We see that every connection ∇ on a trivializing Ui takes

∇ = d + Γi,

with Γi∈ Ω1(End(VUi)). Γi is a matrix with coefficients in the 1-forms, so it is a set of

n2 1-forms on Ui. When {(Ui, Φi)} is a trivializing cover of V , the data {(Ui, Φi, Γi)}

determine the connection completely.

Definition 3.1.3. For a connection ∇ on vector bundle V over M , define the curvature as ∇2: C∞(V ) → Ω2(V ).

Proposition 3.1.4. The curvature of a connection ∇ is C∞(M )-linear.

Proof. Let ∇ a connection on vector bundle V over M and f ∈ C∞(M ), σ ∈ C∞(V ). Then

∇(∇(f σ)) = ∇(df ⊗ σ + f ∇(σ))

= d2f ⊗ σ − df ∧ ∇(σ) + df ∧ ∇(σ) + f ∇(∇(σ)) = f ∇(∇(σ)).

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The previous proposition tells us that ∇2 ∈ C∞_(V∗_)⊗

C∞_{(M )}Ω2(M )⊗_C∞_{(M )}C∞(V ) ∼=

Ω2(M ) ⊗C∞_{(M )}C∞(End(V )). This means that the curvature is a matrix of global

2-forms.

As ∇2 is a matrix, one can take its characteristic polynomial. There are different conventions on what exactly the characteristic polynomial of a matrix is, but they all have the same coefficients up to order and signs. The convention that we will use is

p(∇2) = det I + t∇2 =

n

X

k=0

η2ktk.

Because the entries of ∇2 are 2-forms, the k-th coefficient of p, η2k, is a 2k-form.

A known fact for the characteristic polynomial of a matrix is that it is invariant under conjugation, so p(A) = p(gAg−1) for every A ∈ Mn(C) and every g ∈ GLn(C). A

polynomial with this property is called an invariant matrix.

Theorem 3.1.5. Let p be an invariant polynomial of a matrix (i.e. a polynomial in its coefficients), and π : V → M a smooth vector bundle.

(i) d(p(∇2)) = 0 for every connection ∇ (all coefficients are closed),

(ii) If ∇, ∇0 are connections, then p(∇2) − p(∇02) is exact (all coefficients are exact). Proof. This proof is given in [7].

This theorem tells us that the coefficients of the characteristic polynomial η2k are

closed, and that their cohomology class [η2k] ∈ H2k(M ) is independent of choice of

connection. This means that these cohomology classes are a property of the vector bundle itself.

Definition 3.1.6. The k-th Chern class ck(V ) of a smooth vector bundle π : V → M

is defined as:

ck(V ) = [η2k],

where η2k is the k-th coefficient of the characteristic polynomial of the curvature of a

connection ∇ on V .

Theorem 3.1.7. Let M be an oriented manifold of dimension 2n, and π : V → M be a smooth vector bundle. Then the Chern numbers are products of Chern classes with total degree 2n integrated over the manifold (e.g. when the dimension is 6, they are the integrals of c3₁, c1c2 and c3). Chern numbers of a vector bundle are integers.

We will prove this theorem for a compact manifold M with dim(M ) = 2, and V is a line bundle (i.e. the fibers are C).

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3.2 Lie groups and principal bundles

Although we study the connection in vector bundles, in physics the connection is usually defined in a principal bundle. Principal bundles are defined similarly to vector bundles, having Lie groups as fibers instead of vector spaces.

Definition 3.2.1. A Lie group is a group G that is also a smooth manifold, where the multiplication map G × G → G, (x, y) 7→ xy and the inversion map G → G, x 7→ x−1 are smooth.

Example 3.2.2. Matrix groups, such as GLn(R), GLn(C), U (n) are examples of Lie

groups.

Definition 3.2.3. For a Lie group G, define its Lie algebra g as Te(G).

Example 3.2.4. When G ⊂ GLn(C), the Lie algebra has a much more concrete

defini-tion:

g= {X ∈ Mn(C) : etX ∈ G for all t ∈ R},

Where we use the exponential map eX = In+ X +X

2

2 +

X3

6 + . . . .

We want to generalize this exponential map to general Lie algebras and Lie groups. Definition 3.2.5. A vector field K on G is called left-invariant if (dlg)(Kh) = Kgh for

every g, h ∈ G, where lg : h 7→ gh is the left multiplication. In other words, (lg)∗K = K.

Given an element X ∈ g, there is a unique left-invariant vector field LX _{defined as}

LX = (lg)∗X such that (LX)e= X.

This means that I can also view elements of g as left-invariant vector fields on G. The space of vector fields is equipped with the commutator bracket [·, ·]. For a diffeomorphism ψ, ψ∗[K, L] = [ψ∗K, ψ∗L]: [ψ∗K, ψ∗L](f ) = (ψ∗K)((ψ∗L)(f )) − (ψ∗L)((ψ∗K)(f )) = (ψ∗K)(L(f ◦ ψ) ◦ ψ−1) − (ψ∗L)(K(f ◦ ψ) ◦ ψ−1) = K(L(f ◦ ψ) ◦ ψ−1◦ ψ) ◦ ψ−1− L(K(f ◦ ψ) ◦ ψ−1◦ ψ) ◦ ψ−1 = [K, L](f ◦ ψ)ψ−1 = ψ∗[K, L](f )

So if K and L are both left-invariant, then (lg)∗[K, L] = [(lg)∗K, (lg)∗L] = [K, L], so

their commutator is also left-invariant. Using this, we obtain a Lie bracket for g: [·, ·] : g × g → g, (X, Y ) 7→ [X, Y ] := [LX, LY]0.

Definition 3.2.6. Define a one-parameter subgroup of G as a smooth homomorphism γ : R → G, so γ(s + t) = γ(s)γ(t). Note that γ(0) = e, as it is a group homomorphism, and so γ0(0) ∈ g. There is even a unique one-parameter subgroup γ for each X ∈ g, such that γ0(0) = X. We can obtain this by taking the flow of vector space LX, with initial condition γ(0) = e.

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Let X ∈ g, then define the exponential map etX as the unique one-parameter subgroup γ(t) with γ0(0) = X. For matrix algebras, this definition coincides with the usual exponential map.

Definition 3.2.7. Let G be a Lie group and g its Lie algebra. For g ∈ G, the map Conj_g : G → G, h 7→ ghg−1 is a diffeomorphism, so it induces a linear isomorphism deConjg : g → g. The map Adj : G → GL(g), g → deConjg is called the adjunct

representation.

In order for this map to be a representation, it has to be a homomorphism. Take g1, g2 ∈ G, and let X ∈ g. Using the chain rule and the fact that Conjg1◦ Conjg2 =

Conj_g₁_g₂, we see that:

Adj_g₁(Adj_g₂(X)) = deConjg1(deConjg2(X)) = de(Conjg1◦ Conjg2)(X) = deConjg1g2(X).

Definition 3.2.8. Let M be a smooth manifold and G a Lie group. A principal G-bundle over M is a smooth manifold P with a smooth right action P × G → P and a smooth surjective map ρ : P → M such that:

(i) G preserves the fibers, i.e. if p ∈ ρ−1(x) for a x ∈ M , then p · g ∈ ρ−1(x) for all g ∈ G.

(ii) The G-action on the fibers is:

• Free, i.e. if p · g = ph for a p ∈ P, g, h ∈ G, then g = h,

• Transitive, i.e. for all x ∈ M , given p ∈ ρ−1_{(x), then ρ}−1_{(x) = {p · g : g ∈ G}.}

(iii) For each x ∈ M , there is a neighbourhood U of x and a diffeomorphism (called the trivialization) Φ : ρ−1(U ) → U × G such that:

• πU◦ Φ = ρ (where πU : U × G → U is the projection),

• Φ is G-equivariant, with the obvious right action of G on U × G, i.e. Φ(p · g) = Φ(p) · g for p ∈ ρ−1(U ), g ∈ G.

Definition 3.2.9. Analogously to the case of vector bundles, for a principal bundle ρ : P → M we define a smooth section to be a smooth map s : M → P such that ρ ◦ s = idM.

For an open subset U ⊆ M , a smooth local section is defined as a smooth map s : U → ρ−1(U ) such that ρ ◦ s = idU.

However, whereas in the case of vector bundles a smooth global section always exists (the zero section), this is not the case with principal bundles, as is seen in the next theorem:

Theorem 3.2.10. Let M be a smooth manifold and G a Lie group. ρ : P → M is a smooth principal G-bundle. For an open subset U ⊆ M , the following are equivalent:

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(ii) There is a smooth local section s : U → ρ−1(U ).

Proof. Given a trivialization Φ, we can define a section s(x) = Φ−1(x, e). On the other hand, given a section s we can define the trivialization by Φ−1(x, g) = s(x) · g. The bijectivity of this trivial, as we reach all fibers with s(x), and the action of G is free and transitive on the fibers.

3.2.1 Transition maps of principal bundles

We saw that the definition of a vector bundle is equivalent to an open covering {Uα}α∈A

and a set of smooth transition maps {gαβ : Uα∩ Uβ → GLn(C)}, satisfying conditions:

(i) gαα= In,

(ii) gαβgβα= In,

(iii) gαβgβγgγα= In.

Similarly, a principal bundle is equivalent to an open covering {Uα} and a set of smooth

transition maps {gαβ : Uα∩ Uβ → G}, satisfying

(i) gαα= e,

(ii) gαβgβα= e,

(iii) gαβgβγgγα= e,

With e being the identity of G.

Using this, we can define a principal bundle ρ : P → M with group GLn(C) for a

vector bundle π : V → M , called the frame bundle. The fiber ρ−1(x) is then equal to all ordered bases of π−1(x). Note that this space is indeed in bijection with GLn(C),

because after choosing a basis, there is a unique element in GLn(C) to go to any other

basis. Because V and P are both equivalent to the same transition data, they contain all the information of each other.

Furthermore, if there is a Hermitian product on V , we can consider just the orthonor-mal frames. This gives the orthonororthonor-mal frame bundle, and has spaces diffeomorphic to U (n) as its fibers.

Given a principal G-bundle ρ : P → M , we can also construct a vector bundle, using a representation of G. Let ϕ : G → GL(F ) be a representation, where F is a finite dimensional vector field. G has a right action on the space P × F , namely

(p, f ) · g = (p · g, ϕ(g−1)f ).

Now take P ×ϕ F , the quotient space of P × F by this action of G, and define π :

P ×ϕF → M by π([(p, f )]) = ρ(p). Now let x ∈ M , and p ∈ P such that ρ(p) = x. The

fiber of x is:

ρ−1(x) = {[(p · g, f )] : f ∈ F, g ∈ G} = {[(p, ϕ(g)f )] : f ∈ F, g ∈ G} = {[(p, f )] : f ∈ F }.

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Because {ϕ(g)f : f ∈ F, g ∈ G} = {f : f ∈ F }. So we see there is a bijection between the fibers and F , and if we define λ[(p, f1)]+[(p, f2)] = [(p, λf1+f2)], we see that we have

a vector bundle with fibers isomorphic to F . This is called the vector bundle associated to P by the representation ϕ.

3.3 Connections on principal bundles

Let ρ : P → M be a principal G-bundle. Let p ∈ P and x ∈ M such that ρ(p) = x. ρ induces linear map dpρ : TpP → TxM . The vertical tangent vectors, VpP , are defined as

the kernel of this map. There is an exact sequence

0 VpP TpP TxM 0

dpρ

Because this is an exact sequence of vector spaces, it splits, and a choice can be made of a horizontal subspace Hp ⊂ TpP such that TpP = VpP ⊕ Hp. A connection on P is a

smooth choice of horizontal spaces in each element of P .

The map Rg : P → P with Rg(p) = p · g gives a diffeomorphism from P to itself. The

induced map dpRg : TpP → Tp·gP is a linear isomorphism. This isomorphism maps VpP

on Vp·gP : For v ∈ Vp, dpRg(v) is also vertical:

dp·gρ(dpRg(v)) = dp(ρ ◦ Rg)(v) = d(ρ)(v) = 0,

Because ρ ◦ Rg = ρ, as the right action of G preserves the fibers of P .

Now, I want to choose my horizontal subspaces in such a way that dpRg also maps Hp

on Hp·g.

Definition 3.3.1. A smooth subbundle H of T P is a connection on P if for ever p ∈ P Hp⊂ TpP is a horizontal subspace, and if for all p ∈ P, g ∈ G:

dpRg(Hp) = Hp·g.

3.3.1 The connection 1-form

Let G be a Lie group, and g its Lie algebra, and P a principal G-bundle. For all p ∈ P , I have a map ap: g → TpP given by

ap(X) = d dt _t=0p · e tX_.

We have the mapping σp : G → P given by σp(g) = p · g. Writing etX as γ(t), a smooth

curve with γ0(0) = X, I see that ap is just the differential of σp:

ap(X) = deσp(X).

σp is a diffeomorphism onto the fiber containing p, so it induces a linear isomorphism

deσp : g ∼

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Now let H be a connection on P . In every point p ∈ P , it holds that TpP = VpP ⊕ Hp,

so every v ∈ TpP has a unique decomposition v = vV + vH, with vV ∈ VpP and vH ∈ Hp.

Using this, we can make a linear map ωp : TpP → g, by taking

ωp(vV + vH) = a−1p (vV).

So ωp ∈ Tp∗P ⊗ g. We can even go further in saying that ω ∈ C∞(T P ⊗ g), where

T P ⊗ g is the vector bundle over P with fibers TpP ⊗ g. The smoothness follows from

the smoothness of H. We will also denote this as Ω1(P, g), and we call ω a g-valued 1-form. ω is also called the connection 1-form.

Theorem 3.3.2. Let H be a connection on principal bundle ρ : P → M . Then there is a unique g-valued connection 1-form ω on P satisfying these conditions:

(i) ωp(ap(X)) = X for all p ∈ P , X ∈ g

(ii) R∗_gω = Ad_g−1◦ω for all g ∈ G,

(iii) Ker(ωp) = Hp for all p ∈ P

Proof. This proof is given in [8].

Any g-valued 1-form satisfying the first two conditions is called a connection 1-form: Definition 3.3.3. Let ρ : P → M be a principal G-bundle, and ω ∈ Ω1(P, g) such that:

(i) ωp(ap(X)) = X for all p ∈ P, X ∈ g,

(ii) R∗_gω = Ad_g−1◦ω.

Then ω is called a connection 1-form.

Proposition 3.3.4. Given a connection 1-form Ω1(P, g), the subset H of T P defined pointwise by

Hp= Ker(ωp)

is a connection on T P .

Let ρ : P → M be a principal G-bundle, and {(Ui, Φi)} a trivializing cover. As in the

proof of theorem 3.2.10, these trivializations correspond to local sections si: Ui→ P .

For a connection 1-form ω ∈ Ω1(P, g), we can take the pullback of ω by si, and we obtain

a g-valued 1-form on Ui. We will denote this by:

ωi:= s∗iω.

Theorem 3.3.5. Let ρ : P → M be a principal G-bundle, {(Ui, Φi)} a trivializing

cover, and si the corresponding local sections. Let ω be a connection and ωi = s∗iω its

pullbacks. When i, j are such that Ui∩Uj 6= ∅, we have corresponding transition function

gij : Ui∩ Uj → G, and the following holds on Ui∩ Uj:

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Proof. This follows from a proof in [9].

The converse of this theorem is also true; given g-valued forms on ωi on all Ui of a

trivializing cover satisfying this condition, we know they are pullbacks of a connection 1-form:

Theorem 3.3.6. Let ρ : P → M be a principal G-bundle, {(Ui, Φi)} a trivializing cover,

and si the corresponding local sections.

Suppose that for each Ui, there is a g-valued 1-form ωi on Ui, such that on Ui∩ Uj:

ωj = g_ij−1ωigij+ g−1_ij dgij.

Then there exists a unique connection form ω ∈ Ω1(P, g) such that for all i, s∗_iω = ωi.

Proof. This proof can also be found in [9].

We saw that a connection ∇ on a vector bundle π : V → M of rank n on a local trivialization Ui is equal to d + Γi, where Γiis a 1-form with coefficients in Mn(C), which

is the Lie algebra of GLn(C). These Γi transform like in theorem 3.3.6 [9], so given a

connection ∇, there is a connection form ω on its frame bundle such that s∗_iω = Γi,

where si are the local sections corresponding to the trivializations on Ui.

We see that the connection form ω for a principal bundle is closely related to the connection ∇ of a vector bundle. Now we want to define the curvature for a principal bundle. To do this, we need two more definitions:

Definition 3.3.7. Let ρ : P → M be a principal G-bundle, and g the Lie algebra of G. We want to define the de Rham differential for g-valued P -forms. We do this by choosing a basis {e1, . . . , em} of g. Then we can write a ω ∈ Ωk(P, g) as:

ω =

m

X

i=a

ωaea

with ωa∈ Ωk_{(P ). Then we define}

dω =X

a=1

dωaea.

This definition turns out to be independent of choice of basis.

Definition 3.3.8. Let α ∈ Ωk(P, g) and β ∈ Ωl(P, g). Then define [α, β] ∈ Ωk+l(P, g) as

[α, β] =X

a,b

αa∧ βb[ea, eb].

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Definition 3.3.9. Let ρ : P → M be a principal G-bundle, and ω a g-valued connection 1-form on P . Then the curvature Ω is defined as the g-valued 2-form with

Ω = dω + 1 2[ω, ω].

On a trivializing open subset Ui ⊆ M , with corresponding local section si, the

curvature is a form on Ui, given by

Ωi = dωi+

1

2[ωi, ωi].

We have seen that the Γi of a connection ∇ on a vector bundle can be written as the

pullback of a connection form ω on its frame bundle. The curvature ∇2 is then given in local coordinates by:

∇2|Ui = Ωi = dΓi+

1

2[Γi, Γi] (3.2)

Definition 3.3.10. Let G be a Lie group, F a vector space, and ϕ : G → GL(F ) a represention. A F -valued k-form over a principal G-bundle P , α ∈ Ωk(P, g), is basic if:

(i) If X is a vertical field, then α(X, Y ) = 0 for any field Y , (ii) For all g ∈ G, R∗_gα = ϕ_g−1 ◦ α.

When F = g, we take ϕ = Ad.

Theorem 3.3.11. Let G be a Lie group, F a vector space, and ϕ : G → GL(F ) a represention, and ρ : P → M a principal G-bundle. Let E = P ×ϕF be the associated

vector bundle. Then there is an isomorphism of the basic F -valued forms on P , and E-valued forms on M :

hk: Ωkbas(P, F ) ∼

→ Ωk(M, E). For α ∈ Ωk(P, F ), this h is given for X1, . . . , Xk∈ TxM by

hk(α)x(X1, . . . , Xk) = [(p, αp( ˜X1, . . . , ˜Xk))],

where p, X1, . . . , Xk are such that ρ(p) = x and d ˜Xi= Xi.

Proposition 3.3.12. The curvature 2-form Ω of a connection ω is basic. Proof. This proof can be found in [10].

Definition 3.3.13. Let ρ : P → M be a principal G-bundle, and ω a connection form on P . Define the basic covariant derivative for basic g-valued forms on P :

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The proof that when α is basic, then dωα is basic, is a straightforward exercise that can be found in [10].

Using the exterior covariant derivative dω and the isomorphism h, we can construct a connection for a vector bundle.

Theorem 3.3.14. ρ : P → M is a principal G-bundle, and ω a connection form on P . Let E be the associated vector bundle of the Lie algebra g: E = P ×Ad g, dω be

the exterior covariant derivative, and h the isomorphism from theorem 3.3.11. Then ∇ω_{: C}∞_{(E) → Ω}1_{(M, E) defined by:}

∇ω = h1◦ dω◦ h−1₀

is a connection on E.

Furthermore, for the curvature form Ω,

h2(Ω) = (∇ω)2

3.4 Chern class of line bundles using ˇ

Cech cohomology

Having obtained the transition data of a line bundle, ˇCech cohomology gives us a very elegant way to describe the first Chern class.

Definition 3.4.1. For a topological space X, the open subset category Open(X) has objects {U : U ⊂ X open}, and morphisms {ι : U ,→ V : U ⊂ V }

Definition 3.4.2. A presheaf F on a topological space X is a contravariant functor from Open(X) to another category C. Usually C is the category of sets, groups, abelian groups, or commutative rings. In our case, we will use the abelian groups Ab.

In other words:

• For every open subset U of X, there is an object F (U ) in C.

• For every U, V such that U ⊂ V , with inclusion map ι : U ,→ V , there is a morphism, called the restriction morphism F (ι) : F (V ) → F (U ), such that:

– F (idU) : F (U ) → F (U ) = idF (U ),

– If U ⊂ V ⊂ W , then there are three inclusion maps, ι1 : U ,→ V, ι2 : V ,→

W, ι3 : U ,→ W , and of course ι3 = ι2◦ ι1. Then their restriction morphisms

satisfy F (ι3) = F (ι1) ◦ F (ι2).

Definition 3.4.3. Let X be a topological space. A good cover U = {Uα}α∈Aon X is an

open cover such that every U ∈ U and every intersection T

α∈IUα with I ⊂ A finite, is

contractible. Contractible means that the identity is homotopic to some constant map. In our case, X will be a smooth manifold, and we will require the sets to be differentially contractible, i.e. the identity is smoothly homotopic to some constant map. Every manifold has a good cover [6].

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Definition 3.4.4. Let X be a topological space and U an open cover. A q-simplex σ on U is an ordered q-tuple σ = (Ui)i∈{0,...,q} such that Ui∈ U for every i ∈ {0, . . . , q}, such

that the intersection of all those sets is non-empty. This intersection is denoted by |σ|. The j-th partial boundary of σ is defined as the q − 1-simplex, obtained by removing the j-th set from σ:

∂jσ = (Ui)i∈{0,...,q}\{j}.

And finally, the boundary of σ is the alternating sum of the partial boundaries:

∂σ =

q

X

j=0

(−1)j+1∂jσ,

where the sum is viewed in the free abelian group spanned by the simplices on U .

Let X be a smooth manifold, U a good cover of X, and F a presheaf of abelian groups on U . Then we can define a cochain complex, and therefore a cohomology:

Define the q-th cochain group Cq(U , F ) as follows:

Cq(U , F ) = Y

σ q-simplex in U

F (|σ|)

So f ∈ Cq(U , F ) associates an element f (σ) in the abelian group F (|σ|) to every q-simplex σ. We can then define a differential δq : Cq(U , F ) → Cq+1(U , F ). Let f ∈

Cq(U , F ), and τ a q + 1-simplex. δq(f )(τ ) = q+1 X j=0 (−1)jF (|τ | ,→ |∂jτ |) f (∂jτ ).

I claim that these operators satisfy δq+1◦ δq= 0.

Let f ∈ Cq(U , F ), and let τ be a q + 2-simplex. To calculate δq+1(δq(f ))(τ ), I will first

calculate δq(f )(∂jτ ).

Writing τ as (Ui)i∈{0,...,q+1}, we see that ∂jτ = (Ui)i∈{0,...,q+1}\{j}. For k ∈ {0, . . . , q},

removing the k-th set from ∂jτ is different for k < j and k ≥ j:

For k < j, ∂k(∂jτ ) = (Ui)i∈{0,...,q+1}\{j,k}, and for k ≥ j, ∂j(∂jτ ) = (Ui)i∈{0,...,q+1}\{j,k+1}.

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δq+1(δq(f ))(τ ) = q+2 X j=0 (−1)jF (|τ | ,→ |∂jτ |)δq(f )(∂jτ ) = q+2 X j=0 (−1)jF (|τ | ,→ |∂jτ |) q+1 X k=0 (−1)kF (|∂jτ | ,→ |∂k∂jτ |)f (∂k∂jτ ) = q+2 X j=0 q+1 X k=0 (−1)j+kF (|τ | ,→ |∂k∂jτ |)f (∂k∂jτ ) = q+2 X j=0   j−1 X k=0 (−1)j+kF (|τ | ,→ |∂_k∂jτ |)f (∂k∂jτ ) + q+1 X k=j (−1)j+kF (|τ | ,→ |∂_k∂jτ |)f (∂k∂jτ )   = q+2 X j=0   j−1 X k=0 (−1)j+kF (|τ | ,→ |∂_k∂jτ |)f (∂k∂jτ ) + q+1 X k=j (−1)j+kF (|τ | ,→ |∂_j∂k+1τ |)f (∂j∂k+1τ )   = q+2 X j=0   j−1 X k=0 (−1)j+kF (|τ | ,→ |∂_k∂jτ |)f (∂k∂jτ ) + q+2 X k=j+1 (−1)j+k−1F (|τ | ,→ |∂_j∂kτ |)f (∂j∂kτ )  

In this last sum, all terms with indices j, k ∈ {1, . . . , q + 2} satisfying j 6= k arise exacly once, and switching the indices gives opposing signs, so this sum is 0.

Because δq+1◦ δq = 0, this gives a cochain complex:

C•(U , F ) =

∞

M

q=0

Cq(U , F ),

and we can define the ˇCech cohomology groups:

Hk(X, U , F ) := Ker(δk) Im(δk−1)

.

Now let X be a smooth manifold, and π : V → X a smooth vector bundle of rank n. We have seen that this vector bundle is equivalent to a trivializing cover U = {Ui} of X

and smooth maps {gij : Ui∩ Uj → GLn(C)} satisfying conditions:

(i) gii= I for all i

(ii) gijgji = I for all i, j

(iii) gijgjkgki= I for all i, j, k

Let π : V → X be a smooth vector bundle. Take a good cover U of X. A vector bundle over a contractible space is trivial [9], so U is a trivializing cover, and has transition maps {g_ij}.

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Now consider the presheaf G with G(U ) = {g : U → GLn(C) smooth}. This is a group

with pointwise multiplication, and an abelian group if and only if n = 1. When U ⊂ V and ι : U ,→ V , the restriction morphism G(V ) → G(U ) is just (g 7→ g ◦ ι = g|U).

Then the set of transition data {gij} is an element in C1(U , F ) because for every

1-simplex of U , i.e. σ = (Ui, Uj) with Ui∩ Uj 6= ∅, {gij} gives exactly one element in

G(|σ|) = {g : Ui∩ Uj → GLn(C) smooth}.

{g_ij} is even a cocycle; δ₁({gij}) = In: Let τ = (Uk, Ul, Um) a 2-simplex, i.e. Uk∩ Ul∩

Um 6= ∅. Then δ1({gij})(τ ) = 2 X j=0 (−1)jF (|τ | ,→ |∂jτ |)({gij})(∂jτ )

= glm|Uk∩Ul∩Um(gkm|Uk∩Ul∩Um)

−1

gkl|Uk∩Ul∩Um

= (glmgmkgkl)|Uk∩Ul∩Um

= In,

So {gij} is a 1-cocycle, or {gij} ∈ Ker(δ1), meaning it gives an element in the cohomology

group:

[{gij}] ∈ H1(X, U , G)

Now, take n = 1, then GLn(C) = C∗. There is a short exact sequence of abelian

groups:

0 2πiZ C exp C∗ 1

Now consider for the three groups 2πiZ, C, C∗ the presheafs: 2πiZSh: U 7→ {f : U → 2πiZ : f smooth},

O : U 7→ {f : U → C : f smooth}, O∗ _{: U 7→ {f : U → C}∗ : f smooth}.

Theorem 3.4.5. This gives us a short exact sequence of the cochain complexes:

0 C•(U , 2πiZSh) C•(U , O) C•(U , O∗) 1

Proof. It suffices to prove that for every contractible open U ⊂ M , The sequence

0 2πiZSh(U ) O(U ) O∗(U ) 1

is exact. The only nontrivial part of this proof is that the map exp : O(U ) → O∗(U ) is surjective. In other words, we want for a smooth map f : U → C∗ to have a map log(f ) : U → C such that exp(log(f )) = f . Because exp : C → C∗ is a covering map and U is contractible and thus simply connected, there is a lift log(f ) [13]. We also know that these lifts are unique up to the choice of the pre-image of one point, and that two different lifts differ by a multiple of 2πi.

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For a line bundle π : V → X, its transition maps {gij : Ui∩ Uj → C∗} are an element

of the cohomology group H1(X, U , O∗). We can obtain an element in H2(X, U , 2πiZSh)

by a method called diagram chasing:

0 C0(U , 2πiZSh) C0(U , O) C0(U , O∗) 1 0 C1(U , 2πiZSh) C1(U , O) C1(U , O∗) 1 0 C2(U , 2πiZSh) C2(U , O) C2(U , O∗) 1 .. . ... ... ι0 exp0 ι1 exp1 ι2 exp2

We start by taking {gij} ∈ C1(U , O∗). We know that going down will give us 1;

we have calculated already that {gij} is a cocycle. We have seen that there are lifts

{log(gij)} ∈ C1(U , O) such that they are in the pre-image of {gij}.

Applying the differential operator δ1on {log(gij)} gives us {log(gij)−log(gik)+log(gjk)} =

{log(g_ij) + log(gjk) + log(gki)} ∈ C2(U , O).

Consider for i, j, k such that Ui∩Uj∩Uk6= ∅ such a function log(gij)+log(gjk)+log(gki).

Taking the exponential of this gives:

elog(gij)+log(gjk)+log(gki) _{= e}log(gij)_elog(gjk)elog(gki)_{= 1}

We see that {log(gij) + log(gjk) + log(gki)} ∈ Ker(exp2), so it is an element of Im(ι2),

meaning that {log(gij) + log(gjk) + log(gki)} ∈ C2(U , 2πiZSh).

Furthermore, we know that {log(gij) + log(gjk) + log(gki)} is in the image of δ1, and so

applying δ2 on it will give us 0. So we have found an element in H2(X, U , 2πiZSh):

[{log(gij) + log(gjk) + log(gki)}].

We will see that this cohomology class has a lot to do with the first Chern number.

3.5 The first Chern number of a line bundle on a compact

orientable 2-dimensional manifold

Theorem 3.5.1. Let M be a compact orientable 2-dimensional manifold, and π : V → M a line bundle. Then the first Chern number R_M∇2 _{is a multiple of 2πi.}

First, take a trivializing good cover {(Ui, Φi)}. Because M is compact, we can assume

this cover is finite. Now, fill M with triangles ∆1, . . . , ∆n, such that:

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(ii) ∆k∩ ∆l for k 6= l is either empty or a side of both triangles.

(iii) For all k, there is a Uik such that ∆k ∈ Uik.

A triangle is defined as a space homeomorphic to a compact space in R2 bounded by 3 distinct lines. The second condition means that when 2 triangles are neighbouring, one of each of their sides must coincide entirely, i.e. these situations aren’t allowed:

(3.3) Instead, it will look like this:

It is not trivial that this can always be done. I will show how it can be done for the 2-torus, and then prove the general case.

Consider the torus, which is a square with the sides glued together. Then we can do this:

· · ·

I claim that, if I keep doing this, after a finite amount of steps all my triangles are contained in a Ui.

Assume that this is not true. Then there are is a sequence of triangles F1 ⊃ F2 ⊃ . . . ,

where the triangles Fnhas diameter

√

2(1/2)n, which are all not contained in any Ui. A

decreasing sequence of bounded closed sets has nonempty intersection, so take x ∈ ∩nFn.

x ∈ Ui for some Ui, so there is an ε > 0 such that Bx(ε) ⊆ Ui. But there is a N such

that√2(1/2)N < ε/2, and for such a N , FN ⊆ Ui, so we have a contradiction.

So all the triangles will at some point be contained in a Ui at step N for some N .

Take those triangles to be my ∆k.

So for the torus, we see that we have an explicit algorithm to obtain these ∆k. In the

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Lemma 3.5.2. For every 2-manifold M , there are ∆1, . . . , ∆n such that:

(i) M = ∪k∆k

(ii) ∆k∩ ∆l for k 6= l is either empty or a side of both triangles.

This is called a finite triangulation of M .

Proof. It was proven by Tibor Rad´o in 1925 that every surface (2-dimensional manifold) has a (not necessarily finite) triangulation [11]. Furthermore, in [12] is a proof that for a compact M , every triangulation is finite.

We are almost there: we only need to prove that, given a finite triangulation, I can always divide it up in even smaller triangles such that every triangle is contained in some Ui.

Assume I have a triangulation, and one of the triangles ∆ is not contained in any Ui.

It does however, get covered by several Ui. Note that our triangle is homeomorphic the

standard triangle in R2:

O (1, 0)

(0, 1)

If we call the homeomorphism ϕ, then this standard triangle is covered by ϕ(Ui∩ ∆).

We can devide this triangle up into smaller pieces:

· · ·

We can give an analogous argument as with the torus that eventually, all of the triangles are contained in some ϕ(Ui ∩ ∆). Then, by taking the pre-image of these

triangles in R2 under ϕ, I have divided my triangle ∆ in smaller triangles such that every one of them is contained in some Ui.

Now, by dividing my triangle up in pieces, I don’t want situations like 3.3 to happen. This is easily solved: if a side of a triangle gets divided, then divide that triangle up in 2 pieces as well (with a minor step of taking a homeomorphism with a triangle in R2):

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We can apply this process for every one of the triangles of our original triangulation that wasn’t contained in a Ui. Because there were only finite triangles, we will be done

after a finite amount of steps, so we have found a finite triangulation ∆1, . . . , ∆n such

that for all k, there is a Uik such that ∆k∈ Uik.

When I have such a triangulation ∆1, . . . , ∆n, our integral is equal to:

Z M ∇2 ₌X k Z ∆k ∇2_.

On these ∆k, we can use equation 3.2:

∇2 = dΓik+ 1 2[Γik, Γik]. Note that Γik ∈ Ω 1_{(M ), so [Γ} ik, Γik] = 2Γik∧ Γik = 0. So we get: Z M ∇2=X k Z ∆k ∇2=X k Z ∆k dΓik

We can use the theorem of Stokes to get: Z M ∇2 =X k Z ∆k dΓik = X k Z ∂∆k Γik Note that ∂∆k= S l6=k

Uk∩ Ul, and when Uk∩ Ul1 ∩ Ul2 6= ∅ for different k, l1, l2, then

their intersection is 0-dimensional, so it doesn’t contribute in the integral of Γik. Thus

we get: Z M ∇2 =X k Z ∂∆k Γik = X k X l6=k ∆k∩∆l6=∅ Z ∆+_k∩∆l Γik

∆+_k ∩ ∆l is just ∆k∩ ∆l, but with its orientation in such a way that ∆k is on its left.

Note that, when ∆k∩ ∆l6= ∅ for k 6= l, both ∆+k ∩ ∆l as ∆+l ∩ ∆k contribute to the

integral. Because their orientations are opposite, we get: Z M ∇2 ₌X k X l6=k ∆k∩∆l6=∅ Z ∆+_k∩∆l Γik = X k<l ∆k∩∆l6=∅ Z ∆+_k∩∆l Γik− Γil.

Theorem 3.3.5 shows that for these Γik, Γil:

Γik− g

−1

ilikΓilgilik = g

−1 ilikdgilik

The rank of our vector bundle is 1, so everything commutes, and we get: Γik− Γil= g

−1 ilikdgilik

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This g_i−1

likdgilik looks like the derivative of a logarithm function: if a map log(gilik) :

Uik∩ Uil → C would exist such that exp(log(gilik)) = gilik, then d log(gilik) = g

−1 ilikdgilik.

Luckily, we have chosen a good cover, so Uik ∩ Uil is contractible, and thus simply

connected. Because C with the exponential map is a covering space of C∗, gilik then

has a lifting log(gilik), that is unique up to the choice of the pre-image of one point [13].

Furthermore, such a lifting is smooth, and different liftings differ an integer multiple of 2πi.

Because we are dealing with the differential of the logarithm, it doesn’t matter which lifting we choose: we just choose one for every ik, il. We now obtain

Z M ∇2= X k<l ∆k∩∆l6=∅ Z ∆+_k∩∆l Γik− Γil = X k<l ∆k∩∆l6=∅ Z ∆+_k∩∆l d log(gilik)

We can again use Stokes’ theorem to obtain: Z M ∇2 = X k<l ∆k∩∆l6=∅ Z ∆+_k∩∆l d log(gilik) = X k<l ∆k∩∆l6=∅ Z ∂∆+_k∩∆l log(gilik) = X k<l ∆k∩∆l6=∅ log(gilik)((∆ + k ∩ ∆l)final) − log(gilik)((∆ + k ∩ ∆l)initial). (3.4)

Let us look at an example, and see what this means:

P ∆1

∆2

∆3

The point P is in the boundary of ∆+₁ ∩ ∆₂, ∆+₁ ∩ ∆₃, and ∆+₂ ∩ ∆₃. The contribution of point P in equation 3.4 is:

− log(g21) + log(g31) − log(g32) = log(g12) + log(g23) + log(g31).

If we take the exponential of this, we see:

elog(g12)+log(g23)+log(g31)_{= e}log(g12)_elog(g23)_elog(g31)_{= g}

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We see that log(g12) + log(g23) + log(g31) ∈ 2πiZ, as we also saw earlier in section 3.4.

Now, of course it could be the case that more than 3 triangles come together in a point, even in our example of the torus, 6 triangles come together in each point. But no matter how many we have, it will come down to an equation that looks like:

gi1i2gi2i3· · · gim−1imgimi1,

and using the the three conditions of the {gij}, this will result in 1 (use gikilgilim =

g−1_i mik = gikim inductively): gi1i2gi2i3gi3i4· · · gim−1imgimi1 = gi1i3gi3i4· · · gim−1imgimi1 .. . = gi1imgimi1 = 1

So every point contributes to the sum in 3.4 with a multiple of 2πi, so we have our

final result: _Z

M

∇2 _{∈ 2πiZ.}

We have now proven for a compact orientable 2-dimensional manifold M that the first Chern number is a multiple of 2πiZ.

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4 The Quantum Hall Effect

4.1 Classical Hall Effect

The Hall effect, first discovered in 1879 by Edward Hall, is the phenomenon where a voltage is produced in a conducting plate perpendicular to the current.

Consider a conducting plate in the xy-plane, and put an electric field in the x-direction. This will produce a current in the x-direction, Ix. In the most common case, where the

charge carriers are electrons, the electrons will move in the −x-direction.

In the absence of a magnetic field, the electrons will approximately move in a straight line, but if there is a magnetic field perpendicular to the plane, they will experience a Lorentz force in the y-direction. If we take B = B ˆz, the force on the electrons is

F = −e(E + v × B).

In the steady state, F = 0, so −Fy/e = Ey + vxB = 0, so Ey = −vxB, there is an

electric field in the y-direction, and if the plate has width L, the voltage produced is Vy = −LvxB. This voltage is called the Hall voltage.

This effect can be used to determine whether the charge carriers have positive or negative charge. The Lorentz force acts on positive charge going in the x-direction and negative charge going in the −x-direction the same way, so the two produce opposing voltages.

If there is no electric field, only a magnetic field, the equation of motion for electrons is as follows:

mdv

dt = −ev × B.

Because the electrons can only move in the xy-plane, v = ( ˙x, ˙y, 0). The magnetic field is B = (0, 0, B).

This gives formulas m¨x = −eB ˙y and m¨y = eB ˙x. The general solution is that of circular motion:

x(t) = X − R sin(ωBt + ϕ)

y(t) = Y + R cos(ωBt + ϕ)

Where ωB = eB_m is called the cyclotron frequency.

If we add an electric field, we get the equation mdv

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but in reality the electrons can’t move totally freely. We have to add a sort of friction coefficient, −mv_τ , where τ can be seen as the scattering time, so the time between collisions of an electron. So our final equation is

mdv

dt = −eE − ev × B − mv

τ .

We are interested in when this system has reached an equilibrium, so when dv_dt = 0. Note that the current density, J, is proportional to the velocity of the charge carriers, v, with the relation J = −nev, where n is the electron density.

Setting dv_dt to zero, we get a set of two equations: ne2Ex = eBJy+ mJx τ ne2Ey = −eBJx+ mJy τ

If we write J and E as a 2-vector (as they both live in the xy-plane), we see that this gives a matrix equation:

1 ωBτ −ω_Bτ 1 J = e 2_nτ m E (4.1)

Ohm’s law states J = σE, where σ is the conductivity. In our case however, we see that σ is not a number, but it is a 2 × 2 matrix, which we can obtain by inverting the matrix in 4.1: σ = σDC 1 + ω_B2τ2 1 −ω_Bτ ωBτ 1 , where σDC = ne 2_τ

m would be the one-dimensional conductivity in the absence of a

mag-netic field.

The inverse of this matrix σ is the resistivity ρ, given by E = ρJ. We see that this matrix gives: ρ = σ−1= 1 σDC 1 ωBτ −ωBτ 1

The off-diagonal resistivity, ρxy, is given by:

ρxy = ωBτ σDC = (eB/m)τ ne2_{τ /m} = B ne

This resistivity has two nice properties. One is that it is independent of τ , the scattering time. This τ is a result of disorder in our plate, and the resistivity being independent of disorder, which in a real life experiment always exists.

The second property is that this resistivity coincides with the resistance R. Usually we don’t measure the resistivity, the ratio between the electric field and the current density, but the resistance, the ratio between the voltage and the current. The ratio between these two quantities is usually dependent on the size of the sample, but in this case, we see that this is not the case.

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Assume that the plate has length l in the y-direction. Then the off-diagonal resistance is: Rxy = Vy Ix = lEy lJx = Ey Jx = ρyx= −ρxy

This means that the calculated resistivity is the same as the measured resistance. For the resistivities, I have the values

ρxx= 1 σDC = m ne2_τ and ρxy = B ne. Plotted against the magnetic field B, they look like this [14]:

B ρxy

ρxx

4.2 quantum Hall effect

There are two types of quantum Hall effects: integer and fractional. In this thesis we will only discuss the integer quantum Hall effect, as the fractional Hall effect requires much more complicated mathematics to understand.

The integer quantum Hall effect occurs at low temperatures and strong magnetic fields. It was discovered by Klaus von Klitzing in 1980, and he was awarded the 1985 Nobel prize for this. The resistivities typically look like figure 4.1. Something very strange happens: the longitudinal resistivity, ρxx, is 0 except for some high peaks, and the

transverse resistivity, ρxy, remains unchanged for a certain range of the magnetic field,

with sudden jumps between these plateaus.

Experimental data showed that this transverse resistivity takes the form ρxy = 1 ν 2π~ e2 , where ν is an integer.

Note that the relation σ = ρ−1 gives that σxy = −ρxy ρ2 xx+ ρ2xy = − 1 ρxy ,

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Figure 4.1: longitudinal and transverse resistivities in quantum Hall effect [3]

because ρxx= 0, as we saw in figure 4.1. Thus the transverse conductivity always takes

the form

σxy = ν

e2 2π~, where ν is an integer.

In the following sections, I will show that this ν is the first Chern number of some vector bundle over a 2-torus, thus explaining why it is an integer.

4.3 Quantum states of the electrons

The classical Lagrangian of an electron in a vector potential is L = 1

2m ˙x

2_{− e ˙x · A}

The canonical momentum is:

p = ∂L

∂ ˙x = m ˙x − eA We call π = m ˙x = p + eA the mechanical momentum. We get that the Hamiltonian is

H = 1

2mπ

2 ₌ 1

2m(p + eA)

2

Typically momenta in different directions commute with each other, but now we have [πx, πy] = [px+ eAx, py+ eAy] = −i~e ∂Ay ∂x − ∂Ax ∂y = ie~B. The Hamiltonian is H = 1 2m(p + eA) 2 ₌ 1 2mπ 2₌ 1 2m(π 2 x+ π2y)

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Using these two equations, I can solve this Hamiltonian in the same way as the Harmonic oscillator. Introduce two operators, a and a†:

a = √ 1

2e~B(πx− iπy), a

†

= √ 1

2e~B(πx+ iπy) Some computation will show that [a, a†] = 1, and that

H = ~ωB a†a +1 2 .

This gives us exactly the energy eigenstates of the Harmonic oscillator: We get an eigenstate |ni for every n ∈ N0, with the following properties:

a |0i = 0

a |ni =√n |n − 1i for n ≥ 1

a†|ni =√n + 1 |n + 1i for n ≥ 0

|ni is an eigenstate of the Hamiltonian, with energy En = ~ωB n + 1₂. These energy

levels are called the Landau levels.

4.4 Obtaining the Hall conductance through perturbation

theory

In quantum mechanics, perturbation theory is used when there is a (small) perturbation added to a Hamiltonian of which the eigenstates are already known. We can view the magnetic field as a perturbation on the free electron, and by calculating hJi, the current induced by the perturbation, we can calculate the Hall conductance.

In our case, the Hamiltonian is

H = 1 2m(p + eA) 2 ₌ p2 2m+ ep · A m + e2A2 2m ≈ p2 2m+ ev · A A2 is negligibly small.

Remember that J = −nev. The electrons are spaced evenly, such that the difference between two neighbouring electrons in the x-direction is Lx, and in the y-direction it is

Ly. Then the electron density is n =_L_x1_L_y. This makes the perturbation −_L_x1_L_yJ · A.

We use perturbation theory using the Kubo formula. The Kubo formula tells us some-thing about when a perturbation ∆H(t) is added to a known Hamiltonian H0at a certain

time t0, so our Hamiltonian looks like

H0+ ∆H(t)ϑ(t − t0),

Where ϑ(t) = 1 if t > 0, and ϑ(t) = 0 otherwise.

If we have some operator O, then the time dependent version at time t0 is

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Now, assume we know that the system is in state |ψ(t0)i at time t0. We know that

i~dtd |ψ(t0)i = H0|ψ(t0)i. I want to know what our system will be at time t > t0, so I

want to find a time-evolution operator U (t, t0) such that U (t, t0) |ψ(t0)i = |ψ(t)i.

To find U , I just have to solve: i~d

dtU |ψ(t0)i = (H0+ ∆H)U |ψ(t0)i . This is satisfied exactly when i~_dtdU = ∆HU , so I find

U (t, t0) = T exp −i ~ Z t t0 ∆H(t0)dt0

We will now choose t0 to be −∞, and prepare the system in state |0i, as everything falls

back to the ground state when going to infinity. We will also write U (t) = lim

t0→−∞

U (t, t0).

We are now ready to calculate hJ(t)i.

hJ(t)i = h0(t)| J(t) |0(t)i = h0| U−1(t)J(t)U (t) |0i .

Because the perturbation is small we may assume (∆H)2 1. We can Taylor expand the U -operator, and we get:

U (t) ≈ T 1 − i ~ Z t −∞ ∆H(t0)dt0 U−1(t) ≈ T 1 + i ~ Z t −∞ ∆H(t0)dt0

Then, by using (∆H)2 1 again, we obtain hJ(t)i = h0| U−1(t)J(t)U (t) |0i = h0|

J(t) + i ~ Z t −∞ [∆H(t0), J(t)]dt0 |0i = h0| i ~ Z t −∞ [∆H(t0), J(t)]dt0|0i (4.2) h0| J(t) |0i is the current in the absence of a magnetic field, and so it is zero.

Let us assume (for now) that the electric field is alternating, of the form E(t) = E−iωt₀ . We will eventually take the limit ω → 0, so our result will hold for a constant electric field.

The second Maxwell law gives us that E = −∂A_∂t. This gives us that A = E0

iωe −iωt_.

Substituting ∆H = −_L1

xLyJ · A into 4.2, we get that:

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Let’s work this integrand out.

h0| [J_j(t0), Ji(t)] |0i = h0| Jj(t0)Ji(t) |0i − h0| Ji(t)Jj(t0) |0i

= h0| e−iE0t0/~_J j(t0)Ji(t)eiE0t 0_/~ |0i − h0| e−iE0t0/~_J i(t)Jj(t0)eiE0t 0_/~ |0i = h0| e−iH0t0/~_J j(t0)Ji(t)eiH0t 0_/~ |0i − h0| e−iH0t0/~_J i(t)Jj(t0)eiH0t 0_/~ |0i = h0| Jj(0)Ji(t − t0) |0i − h0| Ji(t − t0)Jj(0) |0i = h0| [Jj(0), Ji(t − t0)] |0i

And we get the following expression: hJi(t)i = 1 ~ωLxLy Z t −∞ dt0h0| [Jj(0), Ji(t − t0)] |0i E0,je−iωt 0

Now, I will use the substitution t00 = t − t0: hJi(t)i =

1 ~ωLxLy

Z ∞

0

dt00eiωt00h0| [Jj(0), Ji(t00)] |0i E0,je−iωt

= 1

~ωLxLy

Z ∞

0

dt00eiωt00h0| [Jj(0), Ji(t00)] |0i Ej(t)

Now note that the Hall conductance, σxy, is equal to _EJx_y. This gives us that

σxy(ω) = 1 ~ωLxLy Z ∞ 0 dt eiωth0| [Jy(0), Jx(t)] |0i

Remember that J(t) = V−1J(0)V , where V = e−iH0t0/~_{. Denote J(0) as just J. Using}

the complete basis of the unperturbed Hamiltonian, we obtain: σxy(ω) =

1 ~ωLxLy

Z ∞

0

dt eiωth0| [Jy(0), eiH0t/~Jx(0)e−iH0t/~] |0i

= 1 ~ωLxLy Z ∞ 0 dt eiωtX n

h0| Jy|ni eiEnt/~hn| Jxe−iE0t/~|0i

− eiE0t/~_{h0| J}

xe−iEnt/~|ni hn| Jy|0i

= 1 ~ωLxLy Z ∞ 0 dt eiωtX n

h0| J_y|ni hn| J_x|0i ei(En−E0)t/~

− h0| J_x|ni hn| J_y|0i ei(E0−En)t/~

We can integrate this expression by replacing ω with ω + iε where ε > 0, and then letting ε → 0. When we have done this, we get:

σxy(ω) = − i ωLxLy X n6=0 h0| Jy|ni hn| Jx|0i ~ω + En− E0 − h0| Jx|ni hn| Jy|0i ~ω + E0− En

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This is where we take ω → 0. We can Taylor expand 1

~ω+Ei−Ej, by:

1 ~ω + Ei− Ej = 1 Ei− Ej − ~ω (Ei− Ej)2 + O(ω2).

We see that the first terms in the expansions cancel out against each other, and we are left with only the second term, as the higher order ones vanish when ω → 0:

σxy = i~

LxLy

X

n6=0

h0| J_y|ni hn| J_x|0i − h0| J_x|ni hn| J_y|0i (En− E0)2

(4.3)

Our sample is invariant under translation in x-direction by Lx, or translation in

y-direction by Ly (this is actually not exactly true, details can be found in [14]). This

means that we can view our space as a torus with sides Lx and Ly. Note that there is

still a magnetic field perpendicular to the torus.1

We can take two solenoids and put them through the x- and y-cycle of the torus, with fluxes Φxand Φyin them. See figure 4.2. The magnetic field in the torus doesn’t change,

but the vector potential undergoes a gauge transformation [14]. Let us assume that we started with the Landau gauge, defined by:

Figure 4.2: Two fluxes are added to the torus

Ax = 0, Ay = Bx.

Then our new vector potential is: Ax = Φx Lx , Ay = Φy Ly + Bx. 1

This is not possible for an actual spacial torus without magnetic monopoles, so we see that it is not actually a torus, but we can view the problem as if it is.

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When our fluxes Φi are an integer multiple of Φ0 = 2π~/e, our quantum states are

unaffected [14]. This means that the flux space is also a 2-torus, with sides Φ0.

Adding these fluxes gives us a new perturbation of the Hamiltonian:

∆H0 = X

i=x,y

JiΦi

Li

. (4.4)

We now want to see how changes in the fluxes affect our ground state |0i. From this point forward we will denote the ground state as |ψ0i instead of |0i, as we will otherwise

get expressions like _∂Φ∂0

i.

Perturbation theory gives us that the ground state becomes: |ψ0i0 = |ψ0i + X n6=0 hn| ∆H0|ψ0i En− E0 |ni .

Substituting equation 4.4 and differentiating to Φi, we get:

∂ψ0 ∂Φi = − 1 Li X n6=0 hn| Ji|ψ0i En− E0 |ni .

Finally, we can substitute this in equation 4.3, and we get: σxy = i~ LxLy X n6=0 hψ₀| J_y|ni hn| J_x|ψ₀i − hψ₀| J_x|ni hn| J_y|ψ₀i (En− E0)2 = i~ ∂ψ0 ∂Φy ∂ψ0 ∂Φx − ∂ψ0 ∂Φx ∂ψ0 ∂Φy = i~ ∂ ∂Φy ψ0 ∂ψ0 ∂Φx − ∂ ∂Φx ψ0 ∂ψ0 ∂Φy

We then introduce dimensionless variables to replace the fluxes, ϑi. Let

ϑi= 2πΦi Φ0 = e ~Φi. As Φi ∈ [0, Φ0), we now have ϑi ∈ [0, 2π).

We can then write the conductivity as:

σxy = i~ ∂ ∂Φy ψ0 ∂ψ0 ∂Φx − ∂ ∂Φx ψ0 ∂ψ0 ∂Φy = ie 2 ~ ∂ ∂ϑy ψ0 ∂ψ0 ∂ϑx − ∂ ∂ϑx ψ0 ∂ψ0 ∂ϑy

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Let us specifically look at this part: F_xy := ∂ ∂ϑy ψ0 ∂ψ0 ∂ϑx − ∂ ∂ϑx ψ0 ∂ψ0 ∂ϑy

Fxy is called the Berry curvature, corresponding to the Berry connection:

Ai = hψ0|

∂ ∂ϑi

|ψ0i

In local coordinates of the 2-torus, this is equal to: ω = hψ0| d |ψ0i .

4.4.1 Why is this a connection?

The Berry connection is defined by hψ0| d |ψ0i. Note that the space in which the quantum

state |ψ0i lives is a principal U (1)-bundle over the 2-torus, as the phase is arbitrary. This

makes |ψ0i a section from the 2-torus to the bundle, as for every (ϑx, ϑy), it returns a

point in this bundle.

Let ω = hψ0| d |ψ0i. Let X be a tangent vector in the point |ψ0i. Then

ω(X) = hψ0| d |ψ0i (X) = hψ0| X |ψ0i .

In order for ω to be a connection 1-form, it should satisfy a few conditions, as in definition 3.3.3.

First of all, it should land in the Lie algebra of U (1). The Lie algebra of U (n) is {iH ∈ Mn(C) : H hermitian}, so ω(X) should be an imaginary number for all tangent

vectors X. As X is a derivation, it holds that

ω(X)+ω(X)†= hψ0| X |ψ0i+hψ0| X†|ψ0i = hψ0| (X |ψ0i)+(X hψ0|)ψ0i = X(hψ0|ψ0i) = 0,

as hψ0|ψ0i = 1. So we see that ω(X) ∈ iR.

Second, let iϕ ∈ iR an element in the Lie algebra. Then the tangent vector a|ψ0i(iϕ) =

d dt|t=0|ψ0i eitϕ. ωp(ap(iϕ)) = ω( d dt|ψ0i e itϕ_{) =} d dte itϕ _{= iϕ}

Third, let u ∈ U (1) and X a tangent vector in |ψ0i. Then

R∗_uω|ψ0i(X) = ω|ψ0i(dRu(X)) = (|ψ0i u)

†

X(|ψ0i u) = hψ0| u−1Xu |ψ0i = hψ0| X |ψ0i = ω|ψ0i(X).

So we see that ω = hψ0| d |ψ0i is a connection. Because ω is one dimensional, it holds

that ω ∧ ω = 0. This means that the curvature of ω is just dω. In the coordinates ϑi,

this is then equal to:

dω = ∂ ∂ϑy ψ0 ∂ψ0 ∂ϑx − ∂ ∂ϑx ψ0 ∂ψ0 ∂ϑy = Fxy.

Chern Classes and the Hall Effect