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
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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 de nitions 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 classi cation 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.
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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 de nition is not mathematically rigorous, but it is su cient for the purposes of this work.
De nition 3.1.1 (Topological space). A topological space is a mathematical space that allows for the de nition 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.
De nition 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 de nition.
De nition 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.
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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.
• Re exivity: 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 de nition of homotopic mappings is related to the concept of homotopic topological spaces.
De nition 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:
2 2 f : R −→ P and g : P −→ R (x, y) 7→ P 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:
2 2 h : R × [0, 1] −→ R ((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 }.
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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 de ne 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 de ne the group operation, the following de nition is useful.
De nition 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 1 S 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 de ne 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
De nition 3.3.2. Formally, the mapping f · g can be de ned as f(2t), 0 ≤ t ≤ 1/2 (f · g)(t) = g(2t − 1), 1/2 ≤ t ≤ 1.
We now need to show that this operation is well de ned, 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.
1 Statement 3.3.1. Consider based maps f, f', g and g' from S to Y, starting and ending at y0, such that f is homotopic to f' and g is homotopic to g'. Then f 0 · g0 is homotopic to f · g.
1 0 Proof. Consider h1 : S × [0, 1] −→ Y to be the homotopy map between f and f (h1(x, t = 0) = f(x) 0 1 0 and h1(x, t = 1) = f (x)) and consider h2 : S × [0, 1] −→ Y to be the homotopy map between g and g 0 0 0 (h2(x, t = 0) = g(x) and h2(x, t = 1) = g (x)). Then a homotopy map between f · g and f · g is given by h1 · h2.
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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 satis ed. 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 , de ned 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 de ne 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 identi ed. Thus, because of statement 3.2.2, we are now allowed to consider directly the homotopy classes of mappings from In to Y . De nition 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 de ne the operation f · g by the formula: f(2x1, xj), 0 ≤ x1 ≤ 1/2 (f · g)(x1, xj) = g(2x1 − 1, xj), 1/2 ≤ x1 ≤ 1
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where j = 2,3,...,n and x1, x2, ..., xn are coordinates in the cube In. Since the boundary of the cube is j j mapped to y0, we have f(1, x ) = g(0, x ) = 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 de ned in the same way as for the case of n = 1. Again, we have to verify that the group axioms are satis ed.
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. d . In fact, every path 1 d is contractible (it is enough to parametrize π1(R ) = I f : S −→ R S1 with θ ∈ [0, 2π] and de ne the homotopy as h(θ, t) = (1 − t)f(θ)).
Example 3.3.2. 1 . A map from 1 to 1 can be de ned as a continuous function on π1(S ) = Z S S f(θ) [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 de ne 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
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1 construct a homotopy between them). It follows that the elements of π1(S ) can be labelled by integers, and the concatenation of maps corresponds to the usual addition in . This means that 1 and are Z π1(S ) Z isomorphic. An interesting application is that (where iφ ), since the Lie group U(1) is π1(U(1)) = Z U(1) = {e |φ ∈ R} homeomorphic to S1. We will see the importance of this example later.
Example 3.3.3. n . The generator of the group is the class of the identity map from 1 to 1. πn(S ) = Z S S 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 (where ∗ ), π3(SU(2)) = Z 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.
m Example 3.3.4. πn(S ) = 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.
m Example 3.3.5. The homotopy groups πn(S ) for n > m ≥ 1 are di cult to compute. For example, we have that 1 , 2 , n , n . We will not πn(S ) = I ∀n ≥ 2 π3(S ) = Z πn+1(S ) = Z2 ∀n ≥ 3 πn+2(S ) = Z2 ∀n ≥ 2 prove these statements.
Example 3.3.6. d d, where d is the d-dimensional torus. In fact, d is generated by d π1(T ) = Z T π1(T ) 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- n siderations, which are often related to the appearance of the homotopy group πn(S ), 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.
De nition 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 de ne a volume form Ω = β(y)dy1 ∧ dy2 ∧ ... ∧ dyd on Y with R ( is normalized). The pull-back of to X using the map f is ∗ ∂y1 j Y Ω = 1 Ω Ω f (Ω) = β(f(x)) ∂xj dx ∧ ∂y2 k ∂yd l 1 2 d where ∂yi is the Jacobian of the map ∂xk dx ∧ ... ∧ ∂xl dx = β(f(x))J(x)dx ∧ dx ∧ ... ∧ dx J(x) = det ∂xj at x. We can now de ne the topological degree of f by Z degf = f ∗(Ω). (1) X 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 on , we have that R R , which means that is Λ Y Y Λ − Y Ω = 1 − 1 = 0 Λ − Ω = dχ an exact form. It follows that, when caluclating degf, the integral of the exact form vanishes, and hence R ∗ R ∗ . degf = X f (Ω) = X f (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
1 Ω = T r(dgg−1 ∧ dgg−1 ∧ dgg−1). (2) 24π2
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The topological degree of the map is obtained by integrating the pull-back of Ω to X, hence Z 1 −1 −1 −1 (3) deg g = 2 T r(dgg ∧ dgg ∧ dgg ). 24π X There is a second, seemingly dierent way to de ne the topological degree of a map.
De nition 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 de ne
M X degf = sign(J(x(m))) (4) m=1 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 de nition 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 de nitions 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 de ne 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.
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4.1.1 Fiber bundles
De nition 4.1.1 (Fiber bundle). Let E, B and F be topological spaces and consider a mapping π : −1 E −→ B. We denote the space of the inverse images of the point b ∈ B as Fb (Fb = π (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 de nition, 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 k (if and are de ned λ : R −→ R λ1 λ2 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 Vd k k k k (R ) of all alternating d-linear maps λ : R × R × ... × R −→ 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 de ne a couple of concepts, namely the concepts of section and connection.
4.1.2 Section and connection
De nition 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.
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De nition 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 µ course on electrodynamics [5] that the vector potential A(x) = Aµ(x)dx and the eld strength F (x) = 1 µ ν can be interpreted as sections of the cotangent bundle and of the second exterior power 2 Fµν (x)dx ∧dx of 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 de ned transformation of the elds which leaves the Lagrangian of the eld con guration 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 con guration 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. µ Consider A(x) = Aµ(x)dx , Aµ(x) ∈ g, to be a connection on the principal bundle, associated with some eld strength 1 µ ν ( ), whose dynamics is contained in the following action F (x) = 2 Fµν (x)dx ∧ dx F = dA 1 Z S[A] = − d4xF F µν . (5) 4 µν
This action is clearly invariant under the global transformation A(x) 7→ A(x)0 = A(x) + dΛ, where Λ ∈ g. Consider φ ∈ C to be a eld de ned on R4 (a section of a ber bundle with base space R4 and ber C), whose dynamics is described by the action
Z 1 m2 S[φ] = d4x( ∂ φ∗∂µφ − φ∗φ) (6) 2 µ 2 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.
Z 1 m2 1 S[φ, A] = d4x[ (∂ − iA )φ∗(∂µ + iAµ)φ − φ∗φ − F F µν ] (7) 2 µ µ 2 4 µν which is invariant under the local gauge transformations
φ(x) 7→ φ0(x) = g(x)φ(x) (8) 0 (9) Aµ(x) 7→ Aµ(x) = Aµ(x) + ∂µΛ(x) −iΛ(x) iΛ(x) = Aµ(x) − ie ∂µe (10) −1 = Aµ(x) − ig (x)∂µg(x). (11)
The eld strength F is also clearly invariant under these transformations.
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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 a (where the are the Pauli matrices). {iX σa|X ∈ R} σa Consider a µ µ, , to be a connection on the principal bundle. A(x) = iAµσadx = iAµ(x)dx iAµ(x) ∈ g Consider φ ∈ C2 to be a eld de ned on R4 described by an action of the form Z 4 † 2 † † 2 S[φ] = d x(∂µφ ∂µφ − m φ φ − λ(φ φ) ), (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
Z 1 S[φ, A] = d4x[ ((∂ + A )φ)†(∂µ + Aµ)φ − m2φ†φ − λ(φ†φ)2] (13) 2 µ µ It follows that the gauge transformations are the following.
φ(x) 7→ φ0(x) = g(x)φ(x) (14) 0 −1 −1 (15) Aµ(x) 7→ Aµ(x) = g(x)Aµ(x)g(x) − ig (x)∂µg(x).
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,
0 −1 (16) Aµ(x) 7→ Aµ(x) = gAµ(x)g .
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 − ∂ν Aµ, by analogy with the abelian gauge theory,
• F transforms according to the adjoint representation for all gauge transformations, that is 0 −1 . Fµν (x) −→ Fµν (x) = g(x)Fµν (x)g (x) In order for both conditions to be satis ed, one can show that the eld strength must have the following form
Fµν (x) = ∂µAν − ∂ν Aµ + [Aµ,Aν ]. (17)
We can now choose the invariant action to be 1 Z S[A] = − g2 d4x T r(F F µν ), (18) 4 µν where g is the gauge coupling constant, since the trace T r 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 Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 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 bundle over 3 can be classi ed by the homotopy group , where the SU(2) S π3(SU(2)) = Z elements 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 de ne 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 de nition of de Rham cohomology.
De nition 5.1.1. Let M be a m-dimensional manifold and let ω be a closed dierential k-form (dω = 0) de ned on M. The de Rham cohomology class of ω is de ned as
[ω] = {ω + dχ| χ is a (k − 1) − form on M}. (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 de ned as
Hk(M) = {[ω]|dω = 0} = Ck(M)/Ek(M) (20)
The second important de nition which we need is the de nition 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 ∈ Vk(M), be a dierential k-form de ned 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. | {z } r De nition 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 de ned on a manifold M. Consider a principal bundle on M with structure group G. Then P(F) satis es:
• 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.
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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 de ned 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 de nition of characteristic class.
De nition 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 de ne Chern classes, we need to give some representative invariant polynomials.
De nition 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 de ned by
iF C(F ) = det(1 + ). (22) 2π 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)
V2i where Ci(F ) ∈ (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) i C (F ) = T r(F ) (25) 1 2π 1 i C (F ) = ( )2[T r(F ) ∧ T r(F ) − T r(F ∧ F )] (26) 2 2 2π ... i C (F ) = ( )kdet(F ). (27) k 2π In the case of G being abelian, this de nition reduces to
C0(F ) = 1 (28) 1 C (F ) = F (29) 1 2π 1 C (F ) = (F ∧ F ) (30) 2 8π2 ...
i k Ck(F ) = ( ) F ∧ F ∧ ... ∧ F . (31) 2π | {z } k
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5.4 Chern numbers
As we have seen, ber bundles can be classi ed 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 de ned in the following way.
De nition 5.4.1 (Chern numbers). Given a ber bundle on a 2i-dimensional manifold M, the i-th Chern number ci is de ned to be the integral of the i-th Chern form over M. Z ci = Ci. (32) M Chern numbers can also be obtained by integrating wedge products of Chern forms , , ..., (with Ci1 Ci2 Cin 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 kxk −→ ∞. This implies that the gauge potential at in nity 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 justi ed by coupling the gauge potential to a scalar eld φ, since the quantity |φ|2 is a physical observable and hence globally well de ned). 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 con guration.
1 Z c1 = F (33) 2π 2 R Z ∞ Z ∞ 1 1 2 = F12 dx dx (34) 2π −∞ −∞ 1 Z = A (35) 2π 1 S∞ 1 Z 2π = Aθ dθ (36) 2π 0 1 Z 2π = ∂θΛ dθ (37) 2π 0 1 = (Λ(2π) − Λ(0)) (38) 2π = 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 con guration 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 su ciently fast as kxk −→ ∞. Then, the gauge potential
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at in nity 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 con guration. Z c2 = C2 (40) 4 R Z 1 (41) = 2 (T r(F ∧ F ) − T rF ∧ T rF ) 8π 4 ZR 1 1 (42) = 2 d[T r(F ∧ A − A ∧ A ∧ A)] 8π 4 3 ZR 1 1 (43) = 2 T r(F ∧ A − A ∧ A ∧ A) 8π S∞ 3 Z 1 (44) = − 2 T r(A ∧ A ∧ A) 24π S∞ Z 1 −1 −1 −1 (45) = 2 T r(dgg ∧ dgg ∧ dgg ) 24π S∞ 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 con guration 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 Mombelli Stefania Homotopy Theory and Characteristic Classes February 2018 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
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