Uppsala University

Department of Physics and Astronomy Division of Theoretical Physics

Supersymmetric Quantum Mechanics and the Gauss-Bonnet Theorem

Supervisor: Matthew Author: Magill Rikard Olofsson Subject Reader: Joseph Minahan

July 10, 2018 Abstract

We introduce the formalism of supersymmetric quantum mechanics, including super- symmetry charges, Z2-graded Hilbert spaces, the chirality operator and the Witten index. We show that there is a one to one correspondence of fermions and bosons for energies different than zero, which implies that the Witten index measures the dif- ference of fermions and bosons at the ground state. We argue that the Witten index is the index of an elliptic operator. Quantization of the supersymmetric non-linear sigma model shows that the Witten index equals the Euler characteristic of the un- derlying Riemannian manifold over which the theory is defined. Finally, the path integral representation of the Witten index is applied to derive the Gauss-Bonnet theorem. Apart from this we introduce elementary mathematical background in the subjects of topological invariance, Riemannian manifolds and index theory.

Sammanfattning

Vi introducucerar formalismen f¨orsupersymmetrisk kvantmekanik, d¨aribland super- symmetryladdningar, Z2-graderade Hilbertrum, kiralitetsoperatorn och Wittenin- dexet. Vi visar att det r˚aderen till en-korrespondens mellan fermioner och bosoner vid energiniv˚aerskillda fr˚annoll, vilket medf¨oratt Wittenindexet m¨aterskillnaden i antal fermioner och bosoner vid nolltillst˚andet. Vi argumenterar f¨oratt Wittenin- dexet ¨arindexet p˚aen elliptisk operator. Kvantisering av den supersymmetriska icke-linj¨arasigmamodellen visar att Wittenindexet ¨arEulerkarakteristiken f¨orden underliggande Riemannska m˚angfald¨over vilken teorin ¨ardefinierad. Slutligen applicerar vi v¨agintegralrepresentationen av Wittenindexet f¨oratt h¨arledaGauss- Bonnets sats. Ut¨over detta introduceras ocks˚agrundl¨aggandematematisk bakrund i ¨amnenatopologisk invarians, Riemmanska m˚angfalderoch indexteori.

1 Contents

1 Introduction 3

2 Mathematical Preliminaries 5 2.1 Homotopy invariance and the Euler characteristic ...... 5 2.1.1 de Rham cohomology ...... 6 2.2 Fibre Bundles ...... 8 2.2.1 Affine connections ...... 11 2.2.2 Curvature ...... 13 2.3 Riemannian Manifolds ...... 15 2.3.1 The Hodge star and adjoint of the exterior derivative . . . . . 18 2.3.2 Hodges theorem ...... 20 2.4 Elliptic Operators ...... 22 2.4.1 Indices of elliptic operators and elliptic complexes ...... 23

3 Supersymmetric Quantum Mechanics 28 3.1 General formalism ...... 28 3.1.1 The Witten index as the index of an elliptic operator . . . . . 30 3.2 The sigma model ...... 31 3.2.1 Quantization ...... 33

4 Path integral computation of T r(−1)F e−βH 35 4.1 Path integral formulation of Quantum Mechanics ...... 35 4.1.1 Fermionic path integrals ...... 37 4.2 Gaussian integrals ...... 40 4.3 The Gauss-Bonnet Theorem ...... 41

5 Discussion and conclusions 45

6 Acknowledgements 46

7 References 47

2 1 Introduction

Supersymmetry was discovered in the early seventies within the framework of quan- tum field theory. It is a symmetry between the two fundamental types of quantum particles; bosons and fermions. For each bosonic particle there is, according to su- persymmetry, a fermionic so called superpartner and vice versa. Applied to the standard model this symmetry implies the existence of corresponding superparti- cles to each earlier known particle; the selectron as the bosonic superpartner to the electron, the higgsino as the fermionic superpartner to the Higgs boson, and so on. Supersymmetry is a very attractive idea and major area of research in modern physics. However, experimental confirmation is yet to be found. The ideas of supersymmetry was later applied to quantum mechanics (which might be regarded as a field theory in 0 + 1 spacetime dimensions) in hope to gain better understanding of the concept by studying the symmetry in a simpler setting. This application proved successful in many ways. For example, by using the language of supersymmetry, the solutions to the Schr¨odinger equation for the hydrogen atom could be obtained with greater elegance and less effort than the standard approach using ordinary quantum mechanical methods. Even more remarkably, based on ideas by E. Witten supersymmetry was able to provide proofs for index theorems [9]. These theorems are, loosely speaking, formulas to calculate a certain topological invariant known as the index of an operator. The goal of this thesis is illustrate this idea, by deriving the Gauss-Bonnet the- orem as an example. This theorem, in its original formulation which dates back to 1848, relates the Euler characteristic of a Riemannian manifold to an integral over the Guassian curvature. The theorem has seen some generalisation over the years, and might today be regarded as the special case of the Atiyah-Singer index theorem applied to the de Rham complex. Mathematically, a supersymmetric quantum mechanical Hilbert space admits a direct sum decomposition into two subspaces, called fermionic and bosonic, respec- tively. What subspace a particle belongs to is determined by an operator called the chirality operator denoted (−1)F . If the eigenvalue of the chirality operator is −1 the state is called fermionic, and belongs to the fermionic Hilbert subspace. If the eigenvalue is +1 the state is called bosonic, and belongs to the bosonic subspace. An operator Q known as supersymmetry charge or generator is taken to anticom- mute with the chirality operator, which implies that it maps fermions to bosons and bosons to fermions. Further, there is a supersymmetry algebra that relates Q to the Hamiltonian operator H, with the implication of an one to one correspondence of fermions and bosons for energies different than zero. Because of this, the trace of the chirality operator (known as the Witten index) measures the difference between the number of bosonic and fermionic states at zero energy. A specific model describing a supersymmetric QM system is known as the non- linear sigma model (in 0 + 1 dimensions). The model is defined on a compact Riemannian manifold (M, g). The supersymmetric extension of the action is in- variant under certain supersymmetry transformations, which gives rise to conserved charges via Noethers theorem. After quantizing the theory these conserved charges corresponds to the supersymmetry charges and the quantum mechanical version of the Hamiltonian is obtained from the supersymmetry algebra. The Hamiltonian is proportional to the Laplacian acting on forms and the Witten index is the Euler

3 characteristic of the underlying manifold. Apart from canonical quantization there is another way to calculate the Witten index, namely the path integral formalism of Quantum mechanics. This approach to quantum theory was first introduced by R. Feynman in the forties. As opposed to the canonical quantization method which uses the Hamiltonian of the system as the starting point, the path integral quantization procedure uses the Lagrangian. This formulation states that the transition amplitude of going from an initial state to a final state is given by the integral over all possible paths between the initial and final position weighted by the factor iS . This is indeed a conceptually deep ~ statement. The equivalence of these different approaches is what allows us to derive the Gauss-Bonnet theorem. We start this thesis by introducing the necessary mathematical tools and back- ground. We later proceed by discussing the general structure of a supersymmetric quantum mechanical system. We examine the sigma model and show that the Wit- ten index is indeed the Euler characteristic. Lastly, we briefly review the path integral formalism and calculate the path integral resulting in the Gauss-Bonnet theorem.

4 2 Mathematical Preliminaries

In this section we introduce the necessary mathematical tools and language in order to understand the rest of this thesis. The notion of topological invariance is of great importance, as it is one of the most interesting and relevant features of the Witten index, one of the main quantities of study. de Rham cohomology is a great tool and an important example of topological invariance, and provides a definition for the Euler Characteristic. We proceed by defining Fibre bundles, loosely speaking a way to ”glue” together pieces that each is homeomorphic to a product space. This notion is of big importance for the rest of the mathematical discussion, as for example the space of smooth differential forms on a manifold is defined in terms of bundles. Next, we define Riemannian manifolds; one of the key words in the sigma model describing the physical theory of interest. Also, only on a Riemannian manifold are we able to state the very important Hodges theorem which connects the Witten index to the topology of the underlying manifold. Lastly we discuss the very basics of index theory in order to better appreciate the structure of supersymmetric quantum mechanics. i In any linear context, repeated indices like X Xi are summed over.

2.1 Homotopy invariance and the Euler characteristic We start by a brief discussion about topological invariance and the Euler character- istic. This section will mainly follow Lee ([3]) and Nash ([2]). Let Map(X,Y ) be the set of all continuous maps between the topological spaces X and Y . f, g ∈ Map(X,Y ) are said to be homotopic if there exists a continuous map F : X × [0, 1] → Y such that:

F (x, 0) = f and F (x, 1) = g

This intuitively means that f can be continuously deformed to g. We denote ”f homotopic to g” by f ' g. Homotopy defines equivalence classes of continuous functions, called homotopy classes. The homotopy class of a continuous function f is denoted [f]:

[f] = {g ∈ Map(X,Y )|g ' f}

We define the space of homotopy classes [X,Y ] = map(X,Y )/ ', such that [f] = [g] if f ' g

Example 2.1: Any two continuous functions f, g : Rn → Rn are homotopic. To see this, define F : Rn × [0, 1] → Rn by F (x, s) = sf(x) + (1 − s)g(x). Then F (0, x) = g(x) and F (1, x) = f(x). F (x, s) is continuous, since it is a composition of continuous functions. This implies that [Rn, Rn] contains only one homotopy class.

5 Two topological spaces M and N are said to be of the same homotopy type if there exists maps f and g:

f : M −→ N g : N −→ M such that f ◦ g ' idN and g ◦ f ' idM .

Example 2.2: Rn is of the same homotopy type as a point. Without loss of generality we let the point be described by the space {0}. Let g : {0} → Rn and n n n f : R → {0}. Then g ◦ f = id{0} and f ◦ g : R → R , which from the previous example is homotopic to idRn .

Some properties of topological spaces are the same for spaces of the same homo- topy type. In the next section we encounter an important example of homotopy invariance in a brief discussion of de Rham theory.

2.1.1 de Rham cohomology Let M be a smooth manifold of dimension m. The space of smooth k-forms on M is denoted by Ωk(M). (See Example 2.8 in the next section.) For convenience the space of smooth 0-forms is defined to be the smooth functions on M, denoted by C∞(M).

Definition 2.1: The exterior algebra of M is the triple (Ω∗(M), +, ∧), where Ω∗(M) is the direct sum

m M Ω∗(M) = Ω0(M) ⊕ Ω1(M) ⊕ ... ⊕ Ωm(M) = Ωi(M) i=0 The degree of the forms cannot extend the dimension of the manifold, so we only sum up to m = dim M. We define the even and odd subspaces of the exterior algebra M Ω+(M) = Ω2i(M) i M Ω−(M) = Ω2i+1(M) i

Elements of Ω+(M) commutes under the algebraic operation i.e. the wedge product, while elements of Ω−(M) anticommutes under the wedge product. The total algebra thus admits what is called a Z2-grading: Ω∗(M) = Ω+(M) ⊕ Ω−(M).

This feature is of importance when discussing the structure of a supersymmetric Hilbert space.

6 In a local coordinate chart with coordinates (x1, x2 . . . xm) we can expand a k- form ω:

X i1 ik I ω = ωi1,...,ik dx ∧ ... ∧ dx = ωI dx

i1<...im...

m dim Ωk(M) = . k

The exterior derivative d :Ωk(M) → Ωk+1(M) acting on ω takes the local coordinate form: ∂ω dω = I dxi ∧ dxI ∂xi Proposition 2.1: The exterior derivative is nilpotent, d(d(ω)) = 0.

Proof. We prove this using local coordinates: ∂ω  d(dω) = d I dxi ∧ dxj ∂xi ∂2ω = I dxk ∧ dxi ∧ dxI ∂xi∂xk  ∂2ω  = I dxk ∧ dxi ∧ dxI = 0 ∂xi∂xk

 The last equality follows from the antisymmetry of the wedge product, and the symmetry of partial differentiation of (continuous, which the components ωI are by definition) functions. This nilpotent property of the exterior derivative is of great importance in de Rham-theory. A k-form ω is said to be exact if there exists a k − 1 form λ such such that ω = dλ. A form σ is called closed if dσ = 0. In terms of the kernel and image of linear maps, we might write:

Space of exact k-forms = im dk−1

Space of closed k-forms = ker dk

The fact that the exterior derivative is nilpotent trivially implies im dk−1 ⊂ ker dk. Now consider the following chain complex:

d d 0 −→d0 Ω0(M) −→d1 ... −−→i−2 Ωi−1(M) −−→i−1 ... d ... → Ωi(M) −→di Ωi+1(M) −−→i+1 ... −→d 0 and recall the definition of a linear quotient space: for two vector spaces U and V with V ⊂ U, the quotient U/V is the space of equivalence classes [u], such that

7 [u1] = [u2] ⇔ u1 − u2 = v, for u1, u1 ∈ U and v ∈ V .

Definition 2.2:([2]) The kth de Rham cohomology group is defined to be the quotient space k ker dk  HdR(M) = im dk−1 (1)

This definition relies on the fact that that im dk−1 is a subspace of ker dk. Two rep- k resentatives of [ω], [η] ∈ HdR(M) are equivalent if they differ by an exact k −1-form. ω and η are then said to be cohomologous with each other. We now state a very important theorem:

Theorem 2.1:([4]) (Homotopy invariance of de Rham cohomology) If k ∼ k M and N are manifolds of the same homotopy type, then HdR(M) = HdR(N) for all k.

The proof of this might be found in [4] page 455. Since the spaces are isomor- phic, they have the same dimension. This gives rise to a definition of a well known topological invariant:

Definition 2.3:([4]) The Euler characteristic is defined to be the alternating sum: X r r χ(M) = (−1) dim HdR(M) (2) r From theorem 2.1 it follows that if M and N are manifolds of the same homotopy type, χ(M) = χ(N), which means that this quantity is a topological invariant.

Example 2.3: The de Rham cohomology groups of the n-torus T n = S1×S1 ...×S1 (n Cartesian products) are given by:[1]

k n (n) H (T ) = R k 0 2 1 2 2 For the special case of two dimensions, we get HdR(T ) = R, HdR(T ) = R , and 2 2 HdR(T ) = R. This gives the Euler characteristic:

2 X r r 2 χ(T ) = (−1) dimHdR(T ) = 0 r

Example 2.4: The two dimensional sphere S2 have cohomology groups H0(S2) = R, H1(S2) = 0 and H2(S2) = R ([4]), yielding the Euler characteristic χ(S2) = 2.

The intuition of the topological invariance is that all surfaces that can be con- tinuously deformed to each other have the same Euler characteristic. We note that the sphere cannot be continuously deformed to the torus.

2.2 Fibre Bundles For the rest of this thesis the notion of fibre bundles and smooth sections becomes important, and are thus introduced in this section. Fibre bundles are an important

8 concept that occurs in many places in mathematics and physics. It is a topological space that locally, but in general not globally, looks like a product space. A trivial example is the cylinder, which (even globally) is the product space R × S1. For non trivial cases we rather require local pieces of the space to be homeomorphic to a product space and then ”glue” together the local pieces. This is made precise in the following definition. The discussion mostly follows Nakahara([1]) and Lee([3]).

Definition 2.4:([1]) A fibre Bundle (E, π, B, F, G) consists of the following ele- ments: (i) A manifold E called the total space. (ii) A manifold B called the base space. (iii) A manifold F called the fibre. (iv) A surjective map π : E → B called the projection −1 ∼ such that π (p) = Fp = F . Fp is called the fibre at p. (v) A group G called the structure group, which acts on F from the left. (vi) A set of open coverings Ui of B with homeomorphisms −1 φi : π (Ui) → Ui × F such that π ◦ φi(p, f) = p. The homeomorphisms are called the local trivialisations since φi maps the local pieces of the bundle to the product U × F (vi) On Ui ∩ Uj 6= ∅ we require that the transition functions tij defined by −1 (φi ◦ φj )|p : F → F satisfy the following properties:

tii(p) = identity (p ∈ Ui) −1 tij(p) = tji (p)(p ∈ Ui ∩ Uj)

tij(p) ◦ tjk(p) = tjk(p)(p ∈ Ui ∩ Uj ∩ Uk) and are elements of the group G.

The properties of the transitions function is a way to say how to properly ”glue” together the local pieces of the bundle. For a trivial bundle E = B ×F all transition functions can be taken to be the identity. For the rest of this thesis, we will only consider smooth manifolds. The definition of a differenitable fibre bundle is the same as above except that all manifolds involved are smooth, the group G is a Lie group and the homeomorphisms are replaced by diffeomorphisms.

Example 2.5: As mentioned earlier, the cylinder can be thought of as a trivial fibre bundle. In terms of the above definition, B = S1 and E = S1 × R with projec- tion π : S1 × R → S1. Since the cylinder globally is a product space, for any subsets U, V ⊂ S1 , with p ∈ U ∩ V , we have π−1(U) = U × R and π−1(V ) = V × R. The −1 transition functions (φU ◦ φV )|p : R → R can be taken to be identity. The next definition is an example of an in general non trivial fibre bundle.

Definition 2.5: The cotangent bundle T ∗M is the disjoint union of cotangent spaces: ∗ G ∗ T M = Tp M p∈M

9 ∗ ∗ with projection: π : T M → M defined by for p ∈ M and tp ∈ Tp M, (p, tp) 7→ p ∈ M. For any p ∈ M there is a subset U of M containing p, such that we can locally −1 ∗ trivialise: φ : π (U) → U × Tp M.

Let n be the dimension of M. Since the cotangent bundle locally looks like U ×T ∗M we have dim T ∗M = 2n. To find out what the structure group G of the cotangent bundle is, we consider a −1 ∗ point p ∈ U and a local trivialisation φ : π (U) → U × Tp M. In a local coordinate i chart (U, x) we write ω|p = ωidx |p and:

i φ(ωidx |p) = (p, (ω1, . . . ωn))

Now consider another chart (V, y) also containing p such that U ∩ V 6= ∅. In this i −1 chart we instead write ω =ω ˜idy . We denote a local trivialisation of π (V ) by ψ. On the intersection U ∩ V , we then have φ ◦ ψ−1 : π−1(U ∩ V ) → π−1(U ∩ V ), or, restricted to p:

−1 ∗ ∗ (φ ◦ ψ )|p : Tp M → Tp M

(ω1, . . . , ωn) 7→ (ω ˜1,..., ω˜n)

∂xj It is well known that 1-form components transform byω ˜i = ∂x˜i ωj. This is a linear ∂xj transformation and ∂x˜i can be thought of as matrix components. Thus the structure group G of the cotangent bundle is the general linear group GL(n, R), assuming the manifold is real. The tangent bundle TM can be constructed in complete analogy.

Definition 2.6: A smooth section s is a map from the base space B to the fi- bre E such that π◦s = idB The space of all smooth sections on B is denoted Γ(B,E)

Smooth sections come in as a very useful notion, allowing us to express several already well known concepts in a unified way. Below are some examples.

Example 2.6: The space of smooth vector fields X(M) on a smooth manifold M are the smooth sections Γ(M,TM). At every point p ∈ M this assigns an element σ(p) ∈ TpM, i.e assigns a vector to each point on the manifold. Similarly, the space of covector fields are the sections Γ(M,T ∗M). In the case of tangent and cotangent fields, we might simplify the notation to Γ(TM), Γ(T ∗M), as the base space is still to be understood as M from the notation of tangent and cotangent bundle.

p Example 2.7: The space of smooth tensor fields Tq M is defined as the tensor product p q p  O   O ∗  Tq M := Γ(TM) ⊗ Γ(T M) i i ∂ i Since ∂xi and dx are local bases of the tangent and cotangent bundles respectively, we can locally expand a (p, q)-tensor T :

 ∂ ∂  i1...ip j1 jq T = Tj ...j ⊗ ... ⊗ ⊗ (dx ⊗ ... ⊗ dx ) 1 q ∂xi1 ∂xip

10 Example 2.8: The space of smooth k-forms Ωk(M) on M is defined as the space of smooth sections from M to the kth anti symmetric power of T ∗M:Ωk(M) = Γ(M, ∧kT ∗M). This means the (0, k)-tensor field ω ∈ Ωk(M) should satisfy:

ω(X1,X2, ..., Xk) = sgn(π)ω(Xπ(1),Xπ(2), ..., Xπ(k)) where π is the permutation group. Because of antisymmetry, any form of degree higher than dim M is necessarily 0.

Definition 2.7: Two bundles (E, π, B) and (E0, π0,B) are said to be isomorphic if there exists a diffeomorphism f such that the following diagram commutes:

f E0 E

π0 π B idB B Definition 2.8:([1]) Let f : N → M be a smooth map, and U a fibre bundle over ∗ M with projection π. The pullback bundle (N, πN , f U) is a bundle over N with the same fibres as U over M. The total space is defined by:

f ∗U := (n, u) ∈ N × U | f(n) = π(n)

To summarise, we have the following commutative diagram:

f ∗U U

πN π f N M Theorem 2.2:([1]) Let M and N be smooth manifolds, f and g be homotopic smooth maps from M to N, and π : E → N a smooth fibre bundle over N. Then f ∗E and g∗E are isomorphic bundles over M.

Definition 2.9:([4]) A vector bundle is a fibre bundle whose fibre at each point is a vector space. The transitions functions are required to be linear maps in order to respect the vector space structure.

Example 2.7: The tangent bundle and cotangent bundle both are vector bun- ∗ dles since the fibres TpM and Tp M both admits a vector space structure. Also, the cylinder with fibre R is a vector bundle, provided the usual vector space structure on R.

2.2.1 Affine connections Connections on fibre bundle involves much theory and here we will only consider a so called affine connection (or covariant derivative) ∇, which will provide a notion of derivative of vector fields on manifolds, and later naturally extended to general (p, q)-tensor fields. It can be thought of as a way of connecting the tangent spaces

11 at different points, and thus providing a well defined notion of derivatives of vector fields, as they are elements of the smooth sections from the manifold to the tangent bundle. Let M be a smooth manifold of dimension m.

Definition 2.10: An affine connection ∇ is a map ∇ : Γ(TM) × Γ(TM) → Γ(TM), (X,Y ) 7→ ∇X Y such that the following properties hold:

(i) ∇fX Y = f∇X Y

(ii) ∇X (fY ) = (Xf)Y + f∇X Y

(iii) ∇(X+Y )Z = ∇X Z + ∇X Z

(iv) ∇Z (X + Y ) = ∇Z X + ∇Z Y

We choose a local coordinate chart (U, x) in which the vector fields X,Y ∈ Γ(TM) i i can be expressed in local coordinates as X = X ∂i and Y = Y ∂i. The single ”del”- ∂ symbol ∂i is a shorthand notation for (where i = 1, . . . m), the basis for the ∂xi tangent bundle. We write the covariant derivative of X in direction Y as:

j i ∇X Y = ∇(X ∂i)(Y ∂j)  ∂Y j  ∂ = Xi + Y jXi(∇ ∂ ) ∂xi ∂xj ∂i j  ∂Y k  ∂ = Xi + XiY jΓk ∂xi ij ∂xk In the last step there have been a relabelling of the summation indices and the k connection coefficients Γij have been implicitly defined: ∂ Γk = ∇ ∂ ij ∂xk ∂i j

k k The local kth component is defined by dx (∇X Y ), and we write this as (∇X Y ) :

k k (∇X Y ) : = dx (∇X Y ) ∂Y k = Xi + XjY iΓk ∂xi ij For notational simplicity the covariant derivative in the ith direction is defined by ∇i = ∇∂i Given a connection on the tangent bundle we can extend it to act on the entire p tensor space Tq M by also requiring the properties:

∞ ∇X f = Xf for f ∈ C (M)

∇X (T ⊗ S) = (∇X T ) ⊗ S + T ⊗ (∇X S) (Leibniz rule)

Which reproduces (ii) for a (0, 0)-tensor field i.e a smooth function. We can express the covariant derivative of a (p, q)-tensor fields T by expanding T :

i ,...i T = T 1 p ∂ ⊗ ... ⊗ ∂ ⊗ dxj1 ⊗ ... ⊗ dxjq j1,...jq i1 ip

12 i and applying the Leibniz rule. We first need to find an expression for ∇X (dx ). This i i ∞ is easily done by applying the covariant derivative on dx (∂k) = δk ∈ C (M):

i i ∇X (dx (∂k)) = X(δk) = 0 i i ⇒ 0 = dx (∇X (∂k)) + ∇X (dx )(∂k) l i l i l i ⇒ X ∇l(dx )(∂k) = −X dx (∇l∂k) = −X Γlk i i ⇒ ∇l(dx )(∂k) = −Γlk

l So we can express the covariant derivative (in direction X = X ∂l) of a 1-form i ω = ωidx by:

l l i k ∇X ω = X (∂lωk − X Γlkωi)dx

The only difference from the covariant derivative of a vector field is the minus sign in front of the connection coefficient and the indices that are being contracted. The kth component is:

k (∇X ω) : = (∇X ω)(∂k) l i l = X ∂lωk − ΓlkX ωi

Finally, we can write the covariant derivative of a general (p, q)-tensor field T as:

i ...i ∇ T = Xl(∇ T ) 1 p ∂ ⊗ ... ⊗ ∂ ⊗ dxj1 ⊗ ... ⊗ dxjq X l j1,...jq i1 ip with

(∇ T )i1...ip =∂ T i1...ip + Γi1 T ki2...ip + ... + Γip T i1...ip−1k + ... l j1...jq l j1,...jq lk j1...jq lk j1...jq ... −Γk T i1...ip − ... − Γk T i1...ip lj1 kj2...jq ljq j1...jq−1k The connection provides a notion of parallel transport, which in turn provides a notion of the straightest curve between two points, and also a notion of curvature on the manifold. Given a smooth manifold there are many different choices of connections and therefore also different notions of straight curves, curvature, and shape off the manifold. On a Riemannian manifold however, there is a natural choice for the connection, as explained in the next section.

2.2.2 Curvature Let γ : I → M be a curve on a smooth manifold M. We define the vector field ∂ X :=γ ˙ i(λ) ∂xi

i  d  whereγ ˙ (λ) = proji dλ γ(λ) , and λ is a parameter that takes values in I. A vector i field Y is said to be parallel transported along γ if ∇X Y =γ ˙ (λ)∇iY = 0. If

13 ∇X X = 0, the curve γ is called autoparallel. By straight forward calculation:

0 = ∇X X j i = ∇(γ ˙ ∂i)(γ ˙ ∂j) ∂γ˙ j = γ˙ i
∂ + (γ ˙ iγ˙ jΓk )∂ ∂xi j ij k k i j k = (¨γ +γ ˙ γ˙ Γij)∂k

k i j k We find Γijγ˙ γ˙ +γ ¨ = 0, for all k. The solutions to this equation has the interpre- tation of the ”straightest curve” between point specified by boundary conditions. For future reference we define the following two tensors, without discussing their geometrical meaning:

Definition 2.11: The Riemann curvature tensor R : Γ(TM) × Γ(TM) × Γ(TM) → Γ(TM) is defined by:

R(X,Y )Z = ∇X ∇Y Z − ∇X ∇Y Z − ∇[X,Y ]Z (3)

i The local components Rljk can be computed:

i i Rljk = dx (R(∂j, ∂k)∂l) i = dx (∇j∇k∂l − ∇k∇j∂l − ∇[∂j ,∂k]∂l) i = dx (∇j∇k∂l − ∇k∇j∂l) i a b = dx (∇j(∂k∂l + Γkl)∂a) − ∇k(∂j∂l + Γjl)∂b) i a b a a b a = dx ((∂jΓkl + ΓklΓjb)∂a − (∂kΓjl + ΓjlΓkb)∂a) a b a a b a i = (∂jΓkl + ΓklΓjb − ∂kΓjl − ΓjlΓkb)δa i i b i b i = ∂jΓkl − ∂kΓjl + ΓklΓjb − ΓjlΓkb

We used the fact that the basis vectors ∂1 . . . ∂m commutes. Loosely one can say that this gives a measure for how much a vector does not return to itself while being parallel transported around a loop.

Definition 2.12: The Torsion tensor is defined as:

T (X,Y ) = ∇X Y − ∇Y X − [X,Y ]

k k Similarly, the local components Tij = dx (T (∂i, ∂j)):

k k Tij = dx (T (∂i, ∂j)) k = dx (∇i∂j − ∇j∂i) k l l = dx (∂i∂j − ∂j∂i + (Γij − Γji)∂l) k k = Γij − Γji If the torsion tensor vanishes the connection is said to be torsion free, and implies that the connection coefficients is symmetric in its lower two indices.

14 2.3 Riemannian Manifolds A Riemannian Manifold is a smooth manifold that contains extra structure in the form of a symmetric (0, 2)-tensor field g, i.e. a map g : TM × TM → M, defined pointwise by g|p : TpM × TpM → M. Riemannian manifolds are important for the rest of this thesis as it is the mathematical notion on which the relevant physical model is defined. It is also of importance for the rest of the mathematical discussion. We follow references ([4]), ([1]) and ([2]).

Definition 2.13 ([4]) A Riemannian manifold (M, g) is a pair containing a smooth manifold M and a (0, 2)−tensor field g called a (Riemannian) metric on M that satisfies:

(i) g(X,Y ) = g(Y,X) (symmetric) (ii) g(X,X) ≥ 0 and g(X,X) = 0 ⇔ X = 0 (Positive definite)

The second property might be exchanged with non degeneracy, in which case M is called a pseudo-Riemannian manifold. Those types of spaces are important in the theory of relativity, but there will be no further references to pseudo-Riemannian manifolds in this work. In a local coordinate chart (U, x) we can write

 ∂ ∂  g(X,Y ) = g Xi ,Y j (4) ∂xi ∂xj  ∂ ∂  = XiY jg , (5) ∂xi ∂xj i j = X Y gij (6)

The first equality follows from the bilinearity of g, and in the last equality we have defined gij := g(∂i, ∂j)

Example 2.8: Rd is a manifold of dimension d. A global coordinate chart is d (R , idRd ), so we can label points on this manifold by the standard Cartesian coor- dinates (x, y, z). With respect to these coordinates we define the inner product:

d X i i i j X · Y = X Y = δijX Y i=1

Obviously this is symmetric and positive definite, so the pair (Rd, ·) is a Riemannian manifold. Comparing with (6) the metric components gij is identified with δij.

Example 2.9: (Sphere embedded in R3.) Let (R3, δ) denote three dimensional euclidean space, with δ being the metric from the previous example in three dimen- sions. An embedding f : S2 → R3 of the 2-sphere is given locally by (θ, φ) 7→ f k(θ, φ) = (sin θ cos φ, sin θ sin φ, cos θ)k (7)

We can turn S2 into a Riemannian manifold by equipping it with the induced pullback

15 metric g := f ∗δ, with local components

∗ f δ(∂i, ∂j) = δ(f∗∂i, f∗∂j) k l = dx (f∗∂i)dx (f∗∂j)δ(∂k, ∂l) k l = ∂if ∂jf δkl Where we in the first and last equality applied the definitions of the push forward and pullback. By computing the partial derivatives of (7) we find, now representing the components gij in matrix form, 1 0  {g } = (8) ij 0 sin2 θ which is seen to be symmetric and positive definite. Thus the pair (M, g) with g = f ∗δ is a Riemannian manifold.

Proposition 2.2: The metric induces an isomorphism TM ∼= T ∗M.

Proof. Consider the map:

g˜ : TM → T ∗M (p, X) 7→ (p, g(X, −)) = (p, g˜(X)(−))

This map is injective, sinceg ˜(X)(−) = 0 implies g(X,X) = 0 ⇒ X = 0. Since ∗ dim TM = dim T M the map bijective, i.e an isomorphism. 

i l i In local coordinates we can expandg ˜(X) asg ˜(X)(∂i)dx = gliX dx . Often one uses the terminology of obtaining covector fields from vector fields and vice verse by raising and lowering indices:

j Xi = gijX i ij X = g Xj

ij ik i Where g is the inverse of gij, namely g gkj = δj. The metric also provides a notion of the speed of a curve. Given a curve γ we i can define a vector field X ∈ Γ(TM), as X =γ ˙ (λ)∂i in local coordinates. (where γ˙ (λ) again denotes the derivative of γ with respects to its argument λ). We define its norm to be phX,Xi, analogous with standard euclidean geometry. We can, without loss of generality, choose the curve λ to be parameterised between 0 and 1. The length of γ is defined to be: Z 1 Z 1 p p i j S[γ] = hX,Xidλ = gijγ˙ γ˙ dλ (9) 0 0

p i j i.e. a functional integral of γ with Lagrangian L = gijγ γ . The shortest curve is obtained from the Euler-Lagrange equations: d ∂L ∂L − = 0 dλ ∂γ˙ i ∂γi

16 (Strictly speaking, the Euler Lagrange equations only gives the extremal of the action, and whether or not it is a minimum or maximum requires one to consider the second variation.) From the previous section, the connection induced notion of straightest curves are those who satisfy the equation: d2γi(λ) dγj(λ) dγk(λ) + Γi (10) dλ2 jk dλ dλ We have now independent notions of straight curves and short curves. These coin- cides if and only if the connection coefficients are taken to satisfy 1   Γk = gkl ∂ g + ∂ g − ∂ g (11) ij 2 i kj j ik l ij This is known as the Levi-Civita connection. It is equivalent with ∇g = 0

∇X Y − ∇Y X = [X,Y ] (Torsion free) To show the implication in one direction, one write down the components of the covariant derivative of the metric. By cyclic permutation of the indices and using the symmetry in the lower two indices of the connection coefficients (which follows from torsion free connection), the connection coefficients is seen to be (11). Given the Levi-Civita connection we refer to equation (10) as the geodesic equa- tion and its solution are called geodesics.

On a Riemannian manifold one can lower one index of the local components of the Riemann curvature tensor, using the metric

a Rijkl = giaRjkl (12) This appears later in a term of the Lagrangian describing the relevant physical the- ory for this thesis.

Example 2.10: if we again look at d-dimensional euclidean space (Rd, g) with metric components gij = δij, all the connection coefficients vanishes as they all con- tains derivatives of gij. This implies that the components of the Riemann curvature tensor also vanishes, having the natural interpretation that the manifold is ”flat”. k i With Γij = 0 for all i, j, k, equation (10) reduces toγ ¨ = 0, with the solutions being a straight line between two points once boundary conditions are specified, as expected.

Example 2.11: For the 2-sphere S2 we use (11) to compute the connection co- efficients using the metric (8) from example 2.9. The indices runs from 1 = θ and 2 = φ. Since the metric is diagonal the inverse is easily found, and the components −2 are g21 = g12 = 0, g11 = 1 and g22 = (sin θ) . Because of symmetry and the fact that only the 22-component admits coordinate dependence, the only non vanishing connection coefficients are 1 1 ∂ Γ2 = g22∂ g = (sin2 θ) = cot θ 12 2 1 22 2 sin2 θ ∂θ 1 1 ∂ Γ1 = − g11∂ g = − (sin2 θ) = − sin θ cos θ 22 2 1 22 2 ∂θ

17 Using these equations we can solve (10) to fins the geodesics on the sphere. The solutions are called great circles.

Around a point p ∈ M for a Riemannian manifold M of dimension m, there exists so called Riemann normal coordinates x1, . . . xm, such that([1]):

gij(p) = δij k Γij(p) = 0 i i i Rljk(p) = ∂jΓkl − ∂kΓjl ∀ijkl

2.3.1 The Hodge star and adjoint of the exterior derivative Let (M, g) be a Riemannian manifold of dimension m without boundary, i.e ∂M = ∅, and Γ(M, ∧kT ∗M) = Ωk(M) the smooth k-forms of M. The Hodge star a linear map: ∗ :Ωk(M) → Ωm−k(M) In local coordinates the action of the Hodge star is defined on the basis dxi1 ∧ dxi2 ... ∧ dxik , i = 1, . . . , m: √ g dxi1 ∧ ... ∧ dxik 7→ i1,...ik dxk+1 . . . dxim (m − k)! ik+1,...im ∗(ω) ∈ Ωm−k(M) is then expanded: √ g i1,...ik k+1 im ∗(ω) = ωi ,...,i  dx . . . dx (m − k)! 1 k ik+1,...im g here denotes the determinant of the metric, somewhat confusingly. To make sense of this definition it is helpful to consider for example the Rieman- nian manifold (R3, g) from example 2.8. With respect to the Cartesian coordinates (x, y, z) the metric can be expressed in matrix form: 1 0 0 {gij} = 0 1 0 0 0 1

The determinant clearly is 1. dx ∧ dy, dx ∧ dz, dy ∧ dz spans Ω2(R3), and we use the multi-index notation (dxi1 ∧ dxi2 ) in order to apply the definition of the Hodge stars action on the basis. Since (m − k)! = 1:

2 3 1 3 ∗ :Ω (R ) → Ω (R )

dxi1 ∧ dxi2 7→ i1i2 dxi3 i3 ⇒ ∗(dx ∧ dy) = dz ∗(dx ∧ dz) = −dy ∗(dy ∧ dz) = dx

18 and (dz, −dy, dx) is indeed a basis for Ω1(R3). We now consider ω, η ∈ Ωk(M), and let dim M = m. Then ω ∧ ∗η is an m-form. In local coordinates: ω ∧ ∗η = √ g j1...jk i1 jm ik jk+1 jm = ωi ...i ηi ...j  dx . . . dx ∧ . . . dx ∧ dx . . . dx (m − k)! 1 k 1 k jk+1...jm √ g = η ωi1...ik dx1 ∧ ... ∧ dxm (m − k)! i1...1k Which is seen to be symmetric. We might therefore define the L2-inner product: Z hω, ηi = ω ∧ ∗η (13) M (This is indeed a inner product, as it can be proved to be positive definite.)

Theorem 2.3: The operator d∗ = (−1)km+m+1∗d∗ is the adjoint to d with respect to the inner product on forms. For ω ∈ Ωk(M) and η ∈ Ωk+1(M): hdω, ηi = hω, d∗ηi Proof. This proof follows Nakahara [1]. Since ∂M = ∅, We have Z Z 0 = ω ∧ ∗η = d(ω ∧ ∗η) ∂M M by the generalised Stokes’s theorem. Employing the product rule of the exterior derivative: Z Z Z d(ω ∧ ∗η) = dω ∧ ∗η + (−1)k ω ∧ d(∗η) = 0 (14)

M M M Z Z ⇒ dω ∧ ∗η = (−1)k+1 ω ∧ d(∗η) (15)

M M Now, since η ∈ Ωk+1(M), we have: d∗ω = (−1)m(k+1)+m+1∗d∗ = (−1)2m(−1mk+1)∗d∗ = = (−1)mk+1∗d∗ Using this we can express the right hand side of the theorem as Z hω, d∗ηi = (−1)mk+1 ω ∧ ∗∗d∗η

M There is a useful identity of the double Hodge star: ∗∗θ = (−1)p(m−p)θ, for θ ∈ Ωp(M). In this case the double Hodge star acts on the (m − k)-form d∗η, giving a factor of (−1)(m−k)(m−(m−k)) in front of the integral: Z hω, d∗ηi = (−1)mk+1(−1)(m−k)(m−(m−k)) ω ∧ d(∗η) =

M Z = (−1)1−k2 ω ∧ d(∗η)

M

19 From (15) we have: Z Z hdω, ηi = dω ∧ ∗η = (−1)k+1 ω ∧ d(∗η)

M M And, by noting that (−1)k+1 = (−1)1−k2 for all positive integers k we have hdω, ηi = ∗ hω, d ηi, proving the theorem. 

Forms in the kernel of the adjoint of the exterior derivative is refered to as coexact. It immediately follows that d∗ is nilpotent. For any ω ∈ Ωk(M): hd2ω, ωi = hω, d∗2ωi = 0 ⇒ d∗ω = 0

2.3.2 Hodges theorem In this section we introduce the laplacian on forms and prove from Hodge decom- position that its kernel is isomorphic to the de Rham cohomology groups, providing an important link between topology and analysis generalised in the next section.

k k Definition 2.14: The Laplacian on k-forms ∆k :Ω (M) → Ω (M) is defined by ∗ ∗ ∆k = (dd + d d)k This indeed maps k-forms to k-forms since

∗ d∗d :Ωk(M) −→d Ωk+1 −→d Ωk(M) And similarly for the second term dd∗. Forms in ker ∆ are called harmonic. If a form is both exact and co-exact, it trivially follows that it is also harmonic, since d∗(dω) + dd∗(ω) = 0 We now prove the converse. Let ω be a k-form in ∗ ∗ ker ∆k ⇒ (dd + d d)kω = 0. It then follows that

∗ ∗ ∗ ∗ h(dd + d d)kω, ωi = hdd ω, ωi + hd dω, ωi = 0 from which hd∗ω, d∗ωi = 0 ⇒ d∗ω = 0 hdω, dωi = 0 ⇒ dω = 0

∗ We see that ω ∈ ker ∆k implies that ω ∈ ker dk−1 and ω ∈ ker dk+1 This is realised in the following theorem:

Theorem 2.4: ([1]) (Hodge Decomposition) Let M be a compact Rieman- nian manifold without boundary. There exists a unique L2-orthogonal direct sum decomposition: Ωk(M) = im d ⊕ im d∗ ⊕ ker ∆

20 The index of the operators are now dropped but it is clear that d acts on k − 1- forms, d∗ on k + 1-forms and ∆ on k-forms. The proof of this is highly nontrivial and require much analysis, and are left out. This theorem allows us to write any k-form as dα + d∗β + γ, where γ is an harmonic form. k We now recall the de Rham cohomology groups HdR(M) = ker dk/im dk−1.A consequence of Hodge decomposition is another theorem:

Theorem 2.5:(Hodge’s Theorem) On a compact Riemannian manifold there is an unique harmonic k-form in each de Rham cohomology class. The two vector spaces are isomorphic:

∼ k ker ∆k = HdR(M) (16) k Proof. Let [ω] be a representative for HdR(M). It follows from the definition of de Rham cohomology groups that dω = 0. From Hodge decomposition: ω = dα + d∗β + γ ⇒ dω = dd∗β = 0 Since d2 = 0 and γ ∈ ker ∆ it follows that γ ∈ ker d. Now: dd∗β = 0 ⇒ hdd∗β, βi = hd∗β, d∗βi = 0 ⇒ d∗β = 0

k So every representative [ω] in HdR(M) can be written in the form ω = dα + γ, and since two representatives of the de Rham cohomology groups are equivalent if they differ by an exact form we have [ω] = [γ]. Consider another representative [η], decomposed η = dσ + ξ, where ξ ∈ ker ∆k. Then, by definition: ω = η + dλ = dσ + ξ + dλ From the uniqueness of Hodge decomposition, it follows that γ=ξ. Thus each coho- mology class has an unique harmonic representative. We conclude:

k HdR(M) ⊂ ker ∆k

Now, for all γ ∈ ker ∆k we have dγ = 0 ⇒ γ ∈ ker d. Also:

im dk−1 ∩ ker ∆k = ∅ from Hodge decomposition. This implies:

k ker dk  ker ∆k ⊂ im dk−1 := HdR(M)

k We define the projection operator P : HdR(M) → ker ∆k by [ω] 7→ P([ω]) = γ and from the above arguments concludes that this projection defines an isomorphism k ∼ HdR = ker ∆k. 

This is a very important theorem and will see some generalisation in the next section about indices of elliptic operators.

21 2.4 Elliptic Operators In this section generalise what have been discussed so far about topological invariants by briefly discussing the theory of indices of elliptic operators. The discussion mostly follows Nash ([4]) and Gilkey ([7]). Let M be a smooth manifold and U,V vector bundles over M. We consider a linear differential operator:

L : Γ(M,U) −→ Γ(M,V ) which we might express in local coordinates:

X α L = aα(x)D (17) |α|≤m

We use the multi index notation:

 α1  αn α |α| ∂ ∂ D = (−i) ... n ∂x1 ∂x α = (α1, α2, ..., αn) X |α| = αi

The factor of i is conventional. m is called the order of L. Special cases of (17) are for example the Laplacian operator ∆ and the Dirac oper- µ ator iγ ∂µ, for which the order m is 2 and 1, respectively.

Definition 2.15 The symbol of the operator L is the fourier transform:

L X α σ (x, ξ) = aα(x)ξ (18) |α|≤m

k Here ξ = ξkdx denotes the Fourier transform variable. The domain of ξ is the cotangent bundle π : T ∗M → M ([2]). We can thus think of the symbol as a map between the pullback bundles π∗U and π∗V . The leading or principal symbol L σm(x, ξ) is defined to be the highest order terms of the symbol, i.e. |α| = m in (17).

Definition 2.16 ([2]): A linear differential operator L : Γ(M,U) → Γ(M,V ), of order m, is said to be elliptic if the leading symbol

∗ ∗ σm : π U −→ π V is a bundle isomorphism off the zero section of T ∗M

It follows from this that rank of U most equal the rank of V .

Proposition 2.3: ([2]) If a smooth function f solves the equation Lu = f, where L is an elliptic operator, then u is also smooth.

22 We demonstrate this with the Cauchy-Riemann operator ∂¯ : C → C defined on the complex plane as: 1 ∂ ∂  ∂¯ = + i 2 ∂x ∂y The equation satisfied by u can be rewritten using a Green function:

LG(z) = δ(z) with Z u(x) = G(z − v)f(v)dv for some v ∈ C C The solution to this is: 1 1 G(x, y) = = π(x + iy) πz The fact that G(z) is C∞ away from z = 0 implies that u is C∞. The symbol ∂¯ ∗ ∗ 1 σ k : π C → π C is given by 2 (k1 + ik2). The pullback bundle is the trivial bundle C × C. For k 6= 0 this is clearly an isomorphism.

2.4.1 Indices of elliptic operators and elliptic complexes The cokernel of an operator O is defined as the following quotient space:

coker O := V /imO

(See section 2.1.2). A property of elliptic operators is that the kernel and cokernel are finite-dimensional.

Definition 2.17:([2]) The (analytical) index of the elliptic operator L is the in- teger index(L) = dim ker L − dim coker L (19) Given metrics on the vector bundles U and V we can define the adjoint operator L∗ : Γ(M,V ) → Γ(M,U) which is also an elliptic operator. There is an isomorphism ker L∗ ∼= coker L, so that the index of L might equivalently be given by index(L) = kerL − dim kerL∗. In terms of indices, the Euler characteristic can be regarded as a special case. We can generalise the Euler Characteristic by extending the notion of ellipticity to complexes, such as the de Rham complex. We consider complexes of the form:

d0 d1 di−2 di−1 di 0 −→ Γ(M,E0) −→ ... −−→ Γ(M,Ei−1) −−→ Γ(M,Ei) −→

di di+1 d ... −→ Γ(M,Ei+1) −−→ ... −→ 0

Where d is some operator that satisfy di ◦ di−1 = 0. If we choose the sections to be Γ(M, ∧kT ∗M) the above reproduces the de Rham complex. In order to define ellipticity of complexes we first give the following definition:

23 Definition 2.18: An exact complex is a collection of (for our purposes) vec- tor spaces {E}i with linear operators {di}i:

d0 d1 di−2 di−1 di 0 −→ E0 −→ ... −−→ E1 −−→ Ei −→ Ei+1 ... d −−→i+1 ... −→d 0 such that di−1 ◦ di = 0 and im di−1 = ker di.

Proposition 2.4: An exact complex of length one, 0 −→d0 A −→d B −→d1 0, is an isomorphism A ∼= B.

Proof. From exactness, im d0 = 0 = ker d, so the kernel of d is empty. Also, im d = ker d1 = B, i.e this is an isomorphism. We are now ready to define an elliptic complex:

i Definition 2.19: ([7]) (Elliptic complex) An elliptic complex (E , di) over a n manifold M is a finite collection {E}i of vector bundles over M with a differential operator d : Γ(M,Ei) → Γ(M,Ei+1) of order m > 0 for each i = 0, . . . , n − 1 satis- fying: (i) di−1 ◦ di = 0 (ii) For each x ∈ M and ξ ∈ T ∗M the associated symbol complex

σm(d0) σm(di−1) ∗ σm(di) ∗ σm(di+1) σm(d) 0 −−−−→ ... −−−−−→ π Ei −−−−→ π Ei+1 −−−−−→ ... −−−→ 0 is exact off the zero section of T ∗M

From proposition 2.4 it is clear that this definition reduces to the one given for ∗ d ∗ ellipticity of a single elliptic operator given above for the special case π E1 −→ π E2. We divide the vector bundles into even and odd parts:

+ E = E0 ⊕ E2 ⊕ ... ⊕ E2i ⊕ ... − E = E1 ⊕ E3 ⊕ ... ⊕ E2i+1 ⊕ ... and define the operator: X ∗ D = (d2i + d2i+1) i where d∗ is the adjoint of d. With these definitions we can rewrite the complex and its associated symbol complex as the length one complexes: D : Γ(M,E+) → Γ(M,E−) σD : π∗E+ → π∗E− So that the requirement of exactness of the symbol complex becomes just that off D being an isomorphism of the zero section of T ∗M. In what is to come we suppress the ◦-symbol between the operators.

i Proposition 2.5: ([7]) Given an elliptic complex (E , di) one can construct el- liptic operators: ∗ ∗ ∆i = (dd + d d)i

24 ∆i Proof. To show that ∆i is elliptic is equivalent with showing that σ is invertible. Consider θ ∈ E such that

∆ d d∗ d∗ d σ i θ = (σ i σ i + σ i+1 σ i+1 )θ = 0 Then (now the subscript is suppressed for notational simplicity:)

d d∗ d∗ d h(σ σ i + σ σ )θ, θi = 0 ⇒ hσd∗ θ, σd∗ θi + hσdθ, σdθi = 0 ⇒ σdθ = σd∗ θ = 0

di From the exactness of the symbol complex, σ θ = 0 ⇒ ∃ ω ∈ Ei−1 such that θ = σdω ⇒ σd∗ σdω = 0. Then: hσd∗ σdω, ωi = 0 ⇒ hσdω, σdωi = 0 ⇒ σdω = 0 ⇒ θ = 0

d Hence σ is invertible, and ∆i is elliptic. 

We now define:

+ ∗ X ∗ ∗ ∆ := D D = (d2id2i + d2i+1d2i+1) i − ∗ X ∗ ∗ ∆ := DD = (d2id2i + d2i+1d2i+1) i Proposition 2.6: ker D = ker ∆+ and ker D∗ = ker ∆−

Proof. If ω ∈ ker D, then obviously ω ∈ ker D∗D = ker ∆+. And, if ω ∈ ker D∗D, then: D∗Dω = 0 ⇒ hD∗Dω, ωi = hDω, Dωi = 0 ⇒ Dω = 0 ⇒ ω ∈ ker D We now have ω ∈ ker D ⇒ ω ∈ ker ∗D, and ω ∈ ker D∗D → ω ∈ ker D, so we + − ∗ conclude ker = ker D. By the same argument we also have ker ∆ = ker D . 

Definition 2.20:([2]) The index of an elliptic complex is index D = dim ker D − dim ker D∗ = dim ker ∆+ − dim ker ∆− M M = dim ker ∆2i − dim ker ∆2i+1 i i X i = (−1) dim ker ∆i i

25 which is well defined since the ∆i’s are elliptic operators. We define the cohomology i i i groups H (M) of an elliptic complex (E , di) as H (M) = ker di/im di−1

Theorem 2.6:([4]) (Generalised Hodge’s theorem) For an elliptic complex i i (E , di) over a compact manifold M the cohomology groups H (M; E) are isomor- ∗ ∗ phic to the kernels of the Laplacians ∆i = (dd + d d)i:

i ∼ H (M; E) = ker ∆i

The proof will not be showed. There is also corresponding generalised decomposition theorems. We define the Generalised Euler-Characteristic as: X χ(M; E) = (−1)r dim Hr(M; E) r We now have:

X i index D = (−1) dim ker ∆i = i X = (−1)i dim Hi(M; E) = χ(M; E) i For a complex of length one, this reduces to (19). Consider the complex

d0 d d1 0 −→ E0 −→ E1 −→ 0

d ∗ ∗ From proposition 2.4 ellipticity now means that σ : π E0 → π E1 is an isomor- phism off the zero section. The dimension of the cohomology groups reduces to the dimension of the kernel and cokernel of d.   0 . dim H (M) = dim ker d {0} = dim ker d     1 dim H (M) = dim ker d1 /im d = dim { 0 } /im d = dim coker d

Yielding index d = dim ker d − dim coker d, reproducing (19). (The higher/lower cohomology groups have dimension zero.)

Example 2.12: As already mentioned the De Rham complex is an example of i ∗ an elliptic complex, for which the spaces of sections Γ(M,Ei) are Γ(M, ∧ T M) al- though the actual proof of this is technical. It is illustrated with a example in [7]. The index of this complex clearly is the usual Euler characteristic.

The index, as it has now been defined, depends on analytical data. For example, the Laplacian on forms requires the structure of a Riemannian metric. However, as in the case of de Rham theory, the generalised cohomology groups of elliptic complexes are invariant under homotopy. This suggest the possibility of computing the index only using topological data, and that is indeed the case; the topological index is as the name suggest an alternative definition of the index purely in terms of

26 topological information. The Atiyah-Singer Index Theorem is the celebrated proved statement that the topological index and the analytical index are equal. However, the required mathematical background for stating this theorem have not been intro- duced, so instead we will focus on the special case when the theorem is applied to the de Rham complex, called the Gauss-Bonnet Theorem. (Sometimes the Gauss- Bonnet Chern theorem or the generalised Gauss-Bonnet Theorem).

Theorem 2.7: (Gauss-Bonnet Theorem)([2]) Let (M,g) be a d-dimensional compact Riemannian manifold without boundary, and D = d + d∗ be the underlying operator of the de Rham complex. The index of D is given as the integral of the Euler class over the manifold: Z χ(M) = indexD = e(M) M The Euler class can be computed by using the formula:([1])

 R  (−1)d/2 X e(M) = P f = sgn(π)R ... R = 2π (4π)d/2( d )! π(1)π(2) π(n−1)π(n) 2 π

Where d = 2n, and Rij are the local components of the curvature 2-form: Rij = 1 k l 2 Rijkldx ∧ dx .

Example 2.13: In two dimensions the theorem the theorem reduces to 1 Z χ(M) = − (R12 − R21) 4π M 1 Z = − R12 2π M by the antisymmetry of the curvature 2-form. The unit sphere from example and example has the local curvature 2-from component R12 = −R21 = − sin θdθ ∧ dφ ([1]). For the two dimensional sphere, we thus find the Euler characteristic

2π π 1 Z Z Z χ(M) = sin θdθ ∧ dφ = dφ sin θdθ = 2 2π M 0 0 In agreement with example 2.4.

In local coordinates the theorem takes the form, in term of the Riemann tensor: 1 (−1)d/2 Z χ(M) = dV i1j1...injn ... × (2d)πd/2 d
2 ! M k1l1...knln ... ×  Ri1j1k1l1 ...Rinjnknln This is the formula that we will deduce using path integral formalism in section 4.

27 3 Supersymmetric Quantum Mechanics

3.1 General formalism Supersymmetry is a mathematical symmetry between fermions and bosons, in quan- tum mechanical terms implying the existence of an operator that interchange fermions and bosons. These operators are taken satisfy a certain algebra with the implication of fermion-boson superpartners for energies different than zero. We now give a definition of what it means for a quantum mechanical system to be supersymmetric:

(i) The Hilbert space H admits a Z2-decomposition (for our purposes) H = Hf ⊕ Hb, where index f refers to a fermionic subspace and index b refers to a bosonic subspace. Their respective state vectors are called fermionic and bosonic. (ii) There is an operator (−1)F such that: ( |Ψi , if |Ψi ∈ Hb (−1)F |Ψi = (20) − |Ψi , if |Ψi ∈ Hf called the chirality operator. ∗ (iii) There are operators Qi, Qi , for i = 1,...N, called (non-Hermitian) super- symmetry charges or generators which anticommutes with the chirality operator. Because of this anticommutation, they maps element of Hb to Hf and the other way around. That is, they map fermions to bosons and vice versa. These operators satisfy the algebra: ∗ {Qi, Qj } = 2δijH and ∗ ∗ {Qi, Qj} = {Qi , Qj } = 0 which is called the supersymmetry algebra.

In order to examine the implications of this algebra we define the Hermitian supersymmetry charges:

1 ∗ Q1 = √ (Qi + Q ) i 2 i i ∗ Q2 = √ (Qi − Q ) i 2 i From the supersymmetry algebra we find: 1 Q 2 = (Q2 + Q∗Q + Q Q∗ + Q2) = H (21) 1i 2 i i i i i i i2 Q 2 = (Q2 − Q∗Q − Q Q∗ + Q2) = H (22) 2i 2 i i i i i i for all i. From this, it trivially follows that [H,Q1i] = [H,Q2i] = 0. One also finds, by simple computation: i {Q ,Q } = {(Q + Q∗), (Q − Q∗)} 1i 2j 2 i i j j i   = {Q , Q } + {Q , Q∗} − {Q , Q∗} − {Q∗, Q∗} = 0 2 i j i j i j i j

28 The fact that the charges commutes with the Hamiltonian is of great importance as it allows us to find simultaneous eigenstates of H and Qi. (From now on it is sufficient to consider any i and therefore the index will be dropped for notational 2 simplicity). We let |q1,E1i denote the eigenstate to Q1. From (21), we have E = q1. We define |q2,E2i = Q2 |q1,E1i. Employing the derived properties of Q1 and Q2:

Q1 |q2,E2i

= Q1Q2 |q1,E1i

= −Q2Q1 |q1,E1i

= −Q2q1 |q1,E1i

= −q1 |q2,E2i i.e |q2,E2i is also an eigenstate to Q1, with eigenvalue −q1. The energy corresponding 2 2 to this state is E2 = −q1 = q1 = E1. Making use of standard results from linear algebra:

•| q1,E1i and |q2,E2i have different eigenvalues, unless q1 = q2 = 0, to the operator Q, which implies that they are linearly independent. This of course implies that they are different states.

• H = Q2. so the energy spectrum is positive.

That is, for every energy eigenvalue different from zero with eigenstate |q1,Ei we can find another state |q2,Ei = Q2 |q1,Ei with the same energy, a so called ∗ superpartner. Since the operators Q1 and Q2 are linear combinations of Q and Q the states |q1,Ei and |q2,Ei belongs to different subspaces of the Hilbert space H = Hf ⊕ Hb. If the state |q1,Ei is bosonic, |q2,Ei is a fermionic superpartner with the same energy eigenvalue. We thus have a one to one correspondence of fermions and bosons for energies different from zero. We define the Witten index (after Edward Witten) as the trace of the chirality operator, T r(−1)F . Since the Hilbert space can be written as a decomposition of fermionic and bosonic subspaces we might represent the state kets as column vectors in the form Ψ  |Ψi = b (23) Ψf In this basis it is clear that the chirality operator takes the form

I 0  (−1)F = 0 −I

(The diagonal identity matrices is generally not of the same size despite the nota- tion). It is from this clear that the trace of the chirality operator counts the number of bosonic states minus the number of fermionic states. And, because of the one to one correspondence of fermionic and bosonic states from energies different than zero, we might as well ”cancel them out” and are left with the difference of bosonic states and fermionic states of zero energy:

F E=0 E=0 T r(−1) = nb − nf (24)

29 From now on we will rather consider the regularised trace T r(−1)F e−βH . Since the trace only counts the difference between the number of zero energy eigenstates, this quantity is independent of β, and for β = 0 reproduces the earlier trace. The regularised quantity is of special interest in this thesis as it yields a path integral representation. (There are also different advantages, see for example Witten ([4])). Remarkably, the Witten index can also be associated to the index of an elliptic operator. We can calculate this quantity in two ways; the first way is by canonically quantizing the theory under consideration. The second way is by going to the path integral formalism of quantum mechanics. The equivalence of these different approaches is what allows us to prove index theorems using ”physical” methods.

3.1.1 The Witten index as the index of an elliptic operator We consider the self adjoint charge Q. In a basis where the state kets take the form (21) we represent the operator in matrix form:

Q Q  Q = 11 12 Q21 Q22

From the anticommutation with (−1)F one quickly realises that the charge Q need ∗T ∗ to be off diagonal. Also from Hermiticity, Q = Q ⇒ Q12 = Q21. We find:  0 D∗ Q = (25) D 0

Where D is some linear operator and D∗ its adjoint with respect to the Hilbert space inner product. By inspecting this operators action on a state:

 0 D∗ Ψ  D∗Ψ  Q |Ψi = b = f D 0 Ψf DΨb we find that D∗ maps fermions to bosons and D maps bosons to fermions, i.e.

∗ D : Hf → Hb

D : Hb → Hf

From the supersymmetry algebra:

D∗D 0  H = Q2 = (26) 0 DD∗

The states of zero energy are those satisfying H |Ψ0i = 0, i.e the kernel of the Hamiltonian. In matrix form we have

 ∗    D DΨ0b 0 H |Ψ0i = ∗ = DD Ψ0f 0

The numbers of fermions/bosons at zero energy are given by the kernels of DD∗ D∗D. Since D∗is the adjoint of D with respect to the Hilbert space inner product,

30 the kernel ker DD∗ = ker D∗ and ker D∗D = ker D. This claim is equivalent with proposition 2.6. We now have

E=0 ∗ nb = dim ker D D = dim ker D E=0 ∗ ∗ nf = dim ker DD = dim ker D This identification leads to the very important result:

F E=0 E=0 T r(−1) = nb − nf = dim ker D − dim ker D∗ = index D The operators D and D∗ is related to the non-Hermitian supersymmetry charge, by definition: √ √  0 D∗ Q + Q∗ = 2Q = 2 D 0 We see that √ (Q + Q∗)| = 2D (27) Hb √ ∗ ∗ (Q + Q )|Hf = 2D (28)

3.2 The sigma model The term ”sigma model” appears in many different contexts throughout physics and in this text we will only be concerned with the non linear sigma model. Let (M, g) be a compact Riemannian manifold of dimension m, and consider the curve:

φ : R → M ”R” might be exchanged to various different more general manifolds depending on context, but here we will consider a quantum mechanical system, and the domain R is to represent the values of the time variable t. (If we were to consider field the- ories the domain would rather be the entire space-time). In some local coordinates x1, . . . xm we write the ith component of φ as φi := xi ◦ φ. We define the vector field ˙ ˙i φ = φ ∂i ∈ Γ(TM). For an interval [0, β] ⊂ R, the action of the bosonic sigma model is given by:

β β 1 Z 1 Z S = g(φ,˙ φ˙)dt = g (φk)φ˙iφ˙jdt 2 2 ij 0 0 (We have emphasised that the metric is in general dependent of φk ). The su- persymmetric version of this Lagrangian is obtained from adding terms containing fermionic degrees of freedom, represented by spinors ΨiΨ¯ i, such that the Lagrangian is invariant under transformations of the form: Bosonic degrees of freedom → Fermionic degrees of freedom Fermionic degrees of freedom → Bosonic degrees of freedom

31 The extended Lagrangian is: (Alvarez Gaume ([6])) 1 i 1 L = g (φk)φ˙iφ˙j + g (φk)Ψ¯ iγ (∇ Ψ)j + R (φk)Ψ¯ iΨjΨ¯ kΨl (29) 2 ij 2 ij 0 t 12 ijkl

R β i And the action is given by S = 0 Ldt.Ψ is a two component real spinor, and ¯ i T Ψ = Ψ γ0, with γ0 = σ2. Since the spinors are to describe fermions, they are Grassman valued (see next section) and anticommutes with each other. The operator ∇t is defined by

i (∇tΨ) = (∇φ˙ ψ) ˙l i i ˙l k = φ ∇lΨ + Γlkφ Ψ i i ˙l k = ∂tΨ + Γlkφ Ψ

i Where the last equality follows from the chain rule. Γlk are the Levi-Civita con- nection coefficients (11), and Rijkl is the contracted Riemann curvature tensor men- tioned in section 2.3. The supersymmetry transformations are

δφi = ¯Ψi (30) i i i k l δΨ = −iγ0φ  − Γkl¯Ψ Ψ (31) Where  and ¯ are infinitesimal Grassman valued spinors. We now transform to a complex basis, where  ψi  1 0  Ψi = γ = ψ∗i 0 0 −1 where ψ∗i is the complex conjugate of ψi. Since the spinors now are complex valued i † Ψ = Ψ γ0 In this basis the Lagrangian (29), by using the fact that the fermions anticommute and the symmetries of the Riemann tensor: 1 i 1 L = g φ˙iφ˙j + g (ψ∗i(∇ ψ)j − (∇ ψ)∗iψj)) − R ψ∗iψ∗jψkψl (32) 2 ij 2 ij t t 4 ijkl The canonical momenta are:

∂L ˙j i ∗j k l i k l∗ j pi = = gijφ + gjkψ Γ ψ − gjkΓ ψ ψ (33) ∂φ˙i 2 il 2 il

∂L i ∗j πi = = gijψ (34) ∂ψ˙ i 2

∗ ∂L i i πi = i = − gijψ (35) ∂ψ˙∗ 2 When we qunatize this system we impose the commutation/anticommutation rela- tions

l l [φ , pk] = iδk (36) i {ψl, π } = δl (37) k 2 k i {ψ∗l, π∗ } = − δl (38) k 2 k

32 From this we find

{ψj, ψ∗i} = gij (39) {ψi, ψj} = {ψ∗i, ψ∗j} = 0 (40)

The Noether charges are given by [11]

i ∗j Q = igijφ ψ (41) ∗ i j Q = −igijφ ψ (42) Using the canonical momentum (34) we rewrite this as i i Q = i(p − g ψiΓk ψ∗l + g ψ∗jΓk ψl)ψi∗ = ip ψi∗ (43) i 2 ik il 2 jk il i i i Q∗ = −i(p − g ψiΓk ψ∗l + g ψ∗jΓk ψl)ψi = −ip ψi (44) i 2 ik il 2 jk il i We used the Grassman property {ψi, ψj} = {ψi∗, ψj∗} = {ψi, ψj∗} . The Γ-term vanishes because of the symmetry in the lower two indices.

3.2.1 Quantization

∗ k ∗ We represent the Hilbert space by the exterior algebra Ω (M) = ⊕kΓ(M, ∧ T M) of the manifold, with the inner product Z hω, ηi = ω ∧ ∗η M That is, the inner product on forms. The anticommutation relations (32) and (33) allows us to interpret ψi and ψ∗i as fermionic annihilation and creation operators. We construct the Hilbert space by letting the creation operator ψ∗i act on the ground state, |0i which we represent as a bosonic state 1 ∈ Ω0(M) = C∞(M). We create a state with one fermion by letting ψ∗i act on the on the vacuum as multiplication with dxi∧:

|0i 7→ 1 state containing zero fermions ψ∗i |0i 7→ dxi ∧ 1 = dxi state containing one fermion ψ∗iψ∗j |0i 7→ dxi ∧ dxj state containing two fermions

And so on. The Hilbert space of states with one fermion is thus Ω1(M), the space of two fermions is Ω2(M), and so on. To see that this description agrees with the anticommutation relation (40), we consider a state of k fermions represented by a k-form ω:

∗i i i1 ik ψ |Ψi 7→ dx ∧ (ωi1,...ik dx ∧ ... ∧ dx )

∗k ∗i i k i1 ik ψ ψ |Ψi 7→ dx ∧ dx ∧ (ωi1,...ik dx ∧ ... ∧ dx ) Since dxi ∧ dxk = −dxk ∧ dxi this agrees with the anticommutation relation. The annihilation operator ψi removes a fermion and thus maps ωk(M) to ωk−1(M). We

33 i il thus represent the operator ψ with g i∂l , where i∂l is the interior product in basis direction l, in general defined on a k-form ω by:

iX ω = ω(X, −,..., −) i.e plugging the vector field X into the first slot of ω and leaving the remaining k − 1 slots left empty; thus producing a k − 1-form. By definition, iX f = 0, for f ∈ C∞(M), so the annihilation operator acting on the ground state is zero, as it i j should be. Consider a state |Φi containing two fermions, represented by ηijdx ∧dx . The action of ψi is then:

l lk i j ψ |Φi 7→ g i∂k (ηijdx ∧ dx ) lk i j = g ηijdx (∂k) ∧ dx lk i li j = g ηijδk = g ηijdx which is indeed a state of one fermion. Finally, we quantize the bosonic canonical momentum pi by a covariant derivative in direction φi, and the position operator φi as multiplication with position: D p 7→ i Dφi φk 7→ φk×

The superscharges then are, from (43): D Q = ig ψi∗φj 7→ dxi ∧ = d ij Dφi Q∗ = d∗

To demonstrate this, we for simplicity consider the action of the supercharge on a i state containing one fermion, A = Aidx :

i∗ k Q |Ψi 7→ piψ (Akdx ) D = dxi ∧ (A dxk) Dφi k i j l = dx ∧ (∂iAl − ΓilAj)dx i = (dx ∧ ∂i)A = dA

j i l (The last equality follows from Γildx ∧ dx = 0, because of the antisymmetry of the wedge product.) This can be showed to work analogously for forms of higher degree, as the only difference are a higher amount of terms with connection coefficients, but they all vanish because of symmetry in their lower indices. Since Q and Q∗ maps fermionic and bosonic Hilbert spaces into each other, and since the ground state is defined to be bosonic, we find that the bosonic and fermionic subspaces are represented by the even and odd parts of the exterior algebra respectively, namely:

+ Hb = Ω (M) − Hf = Ω (M)

34 Which is reasonable since the elements of the odd part anticommutes under the wedge product. 1 ∗ 1 The Hamiltonian is H = 2 {Q, Q } = 2 ∆. (We changed notation from Q to Q, since from now on we will not consider the self adjoint operators earlier called Q.) The zero energy Hilbert space is the subspace ker ∆ ⊂ H . The number of fermions and bosons at zero energy thus are:

E=0 nb = dim ker ∆2i E=0 nf = dim ker ∆2i+1 from Hodges theorem it then follows that the Witten index is the Euler character- istic:

T r(−1)F e−βH = X X = dim ker ∆2i − dim ker ∆2i+1 i i X i X i i = (−1) dim ker ∆i = (−1) dim HdR(M) i i = χ(M)

This result is fascinating, as it directly connects the physics of supersymmetry with the topology of the underlying Riemannian target manifold. By modifying the Lagrangian describing the theory, one can also relate T r(−1)F e−βH to other topological invariants. However, we focus on this case and in the next section demonstrates how the Gauss-Bonnet theorem can be derived from the Path integral representation of the Witten index.

4 Path integral computation of T r(−1)F e−βH

4.1 Path integral formulation of Quantum Mechanics We begin this section by a brief introduction of the path integral (or functional integral) formulation of quantum mechanics. We follow Nakahara [1], Srednicki [10] and Bastianelly [9]. As opposed to canonical quantization, which uses the Hamiltonian as the starting point, the path integral quantization relies on the Lagrangian of the theory, obtained from the Hamiltonian via the Legendre transform:

L = pq˙ − H

The path integral formalism, except from being of practical use in modern physics, is also of conceptual interest. Roughly speaking, it gives the interpretation of the probability amplitude, or the transition amplitude, as the sum over all posible paths

35 iS going from the initial state to the final state weighted by the factor e ~ , where S is the classical action: t Z f S = L(q, q˙; t)dt

ti and ti and tf are the initial and final time under consideration. In the classical limit ~ → 0 the dominating contributions will be exactly those that are the solution to the Euler-Lagrange equations. Previously in this work we have set ~ = 1 and will do so from now on also in this section. The transition amplitude of going from an initial state |qi, tii to a final state |qf , tf i is calculated by the inner product hqf , tf |qi, tii. Since the time evolution of a state is given by |q, ti = eiHt |q, t = 0i we can rewrite the transition amplitude iH(ti−tf ) as hqf , t0| e |qi, t0i, where t0 := t(0). We define β := tf − ti, allowing us to express: −iHβ hqf , tf |qi, tii = hqf , t0| e |qi, t0i To find the path integral representation of this quantity we first assume that the 1 2 Lagrangian is of the form L = 2 q˙ − V (q). We divide the time interval β into N + 1 β R slices; δt = N+1 . By inserting N complete set of states dq |qi hq| = 1 into the transition amplitude: Z Z Z −iHβ iHδt iHδt hqf , t0| e |qi, t0i = dq1 dq2 ... dqN hqf , t0| e |q1i hq1, t0| e |q2, t0i ...

−iHδt ... × hqN , t0| e |qi, t0i

The above already looks like an integral over different paths from qi to qf . After some computation it can be rewritten to read: ([10]) Z N Y Y dpj hq , t | e−iβH |q , t i = dq exp{ip (q − q )} exp{−iH(p, q¯)δt} i 0 f 0 i 2π j j+1 j i j 1 Whereq ¯ := 2 (qj + qj+1). After taking the limit δt → 0 and integrating out the momenta, the transition amplitude can be computed as the path integral: Z −iHβ iS hqf , t0| e |qi, t0i = Dq(t)e (45)

The measure Dq(t) symbolically denotes the integration over all paths q(t) with q(ti) = qi and q(tf ) = qf . We can cast this expression in a different form by performing a Wick rotation, which is a rescaling of the time variable t → −iτ, where τ ∈ R. The time derivatives scales as: d 1 d d → − = i dt i dτ dτ and the action: tf ! τf Z 1dq(t)2 Z  1 dq(τ)  S = dt − V (q(t)) = −i dτ − − V (q(τ)) 2 dt 2 dτ ti τi τf Z 1  = i dτ q˙(τ) + V (q(τ)) 2 τi

36 The last integral is usually referred to as the Euclidean action, denoted SE. The path integral (45) can now be written

0 Z −Hβ −SE hqf , t0| e |qi, t0i = Dq(t)e (46)

0 Where β is the scaled time interval τf − τi = −iβ.

Definition 4.1: The partition function Z(β) is defined as the trace:

Z(β) = T r(e−βH )

The trace can be calculated in a basis of eigenstates of the Hamiltonian, i.e |Ψni such that H |Ψni = En |Ψni. In this basis, the trace is the sum of the diagonal −βH −βEn matrix elements hΨn| e |Ψni = e :

−βH X −βH X −βEn X −βEn T r(e ) = hΨn| e |Ψni = e δnn = e n n n Which is nothing but the well known partition function from statistical mechanics. We can also express the partition function by ”summing” up the diagonal matrix el- ements in the position basis, hq00| e−βH |q0i- Of course, these elements are not discrete but continuous, and so the sum is replaced by the integral: Z Z(β) = dq hq| e−βH |qi

The integrand of the above expression can be expressed using the path integral, by integrating over all paths satisfying qi = qf . That is, all the loops in configuration space. We denote this by PBC, for ”periodic boundary conditions” Z Z Z Z(β) = dq Dq(t)e−SE = Dq(t)e−SE PBC PBC

The first integral is absorbed by the path integral measure Dq(t).

4.1.1 Fermionic path integrals The classical analogue of a fermion is ill defined. When quantizing a bosonic theory, we impose commutation relations. The most famous example is probably [x, p] = i. When quantizing fermions on the other hand, one instead imposes anticommutation relations; for some fermionic variables, say ψ and ψ¯, the canonical quantization re- lation takes the form {ψ, ψ¯} = i. This suggest that the fermionic variables should anticommute before being quantized, as the bosonic variables commutes before quan- tization. We once again use the language of differential forms.

Definition 4.2: Let V be a complex vector space of dimension m, and θ1, . . . θm a basis for V . The Grassman algebra Gr(V ) is defined as the exterior algebra Ω∗(V ).

37 Elements of Gr(V ) are called Grassman numbers or Grassman variables, but math- ematically they are nothing but elements in the exterior algebra. From the definition of exterior algebra given in section 2.1.1 it follows that ele- ments of the Grassman algebra are expanded as

m X X ω = ωi1,...im θi1 . . . θim

i=0 i1

The algebraic operation is the wedge product, θiθj = θi ∧ θj. Clearly the anticom- mutation relation {θi, θj} = 0 holds, leading to:

θi1...in = i1,...in θ1 . . . θn We define differentiation on Grassman numbers by the rules

∂θi j = δi ∂θj ∂ ∂θi ∂θj (θiθj) = θj − θi ∂θk ∂θk ∂θk It follows that the derivatives anticommutes. Integration is defined to be equivalent with differentiation: Z ∂ dθi = ∂θi Z ∂ ∂ dθ1 . . . dθN = ... ∂θ1 ∂θN such that: Z Z i dθiθj = δj dθi1 = 0 ,∀ij

This differentiation and integration may first appear strangely defined, but proves to be the correct way to approach fermionic path integrals. In particular, we can taylor expand any functions f(θ) of one Grassman variable with the result being a linear polynomial in θ due to anticommutation. Then the ”boundary” terms vanishes: Z ∂f dθ = 0 ∂θ allowing a nice way to do integration by parts.

Example 4.1: The fermionic creation and annihilation a and a∗ operators sat- isfy {a, a} = {a∗, a∗} = 0 and {a, a∗} = 1. In analogy with the bosonic case the number operator is defined N = a∗a. Using the anticommutation relations:

N 2 = a∗aa∗a = a∗1a − a∗a∗aa = N ⇒ N 2 − N = N(N − 1) = 0

This implies that the eigenvalues of the number operator is either zero or one. This is the Fermi exclusion principle for fermions.

38 The eigenvectors of the annihilation and creation operators are called coherent states. We can expand them in a Hilbert space spanned by the two eigenvectors of the num- ber operator, which we denote |0i and |1i. θ and θ∗ are crassman valued coefficients.

|θi = |0i + |1i θ hθ| = h0| + h1| θ∗ ⇒ |θi a = |0i θ = |θi θ hθ| a∗ = h1| θ∗ = hθ| θ∗

The last equality in the third and fourth lines follows from the anticommutativity of Grassman numbers. In terms of the coherent states we have a completeness relation: Z ∗ −θ∗θ dθ dθ |θi hθ| e = I. (47)

To see this we plug in the definition of the coherent states: Z Z dθ∗dθ |θi hθ| e−θ∗θ = dθ∗dθ(|0i h0| + |1i θ∗ h0| + |0i θ h1| + |1i θ∗θ h1|)e−θ∗θ

The factor e−θ∗θ is defined by its taylor expansion, which is of first order because terms with three or more Grassman numbers are zero due to anticommutation. We are thus left with: Z dθ∗dθ(|0i h0| + |1i θ∗ h0| + |0i θ h1| + |1i θ∗θ h1|)(1 − θ∗θ) Z = dθ∗dθ(|0i h0| − |0i h0| θ∗θ − |1i θ∗θ h1|)

= |0i h0| + |1i h1| = I Where in the last equality we have taken into account the order of the integration measure. Using coherent states, the trace of an bosonic operator A (meaning that it com- mutes with θ∗ and θ) can be written

Z T r(A) = dθ∗dθe−θ∗θ h−θ| A |θi (48)

This is easily proved: Z Z dθ∗dθe−θθ∗ h−θ| A |θi = dθ∗dθ(1 − θ∗θ)(h0| − h1| θ∗)A(|0i + |1i θ) = Z = dθ∗dθ(−θ∗θ h0| A |0i − θ∗θ h1| A |1i)

= h0| A |0i + h1| A |1i = T r(A)

Since the action of the chirality operator is to change the sign on fermionic states, the above calculation also shows that we can wright: Z F ∗ −θ∗θ T r(−1) A = dθ θ.e hθ| A |θi

39 For the trace of e−βH , there is as in the bosonic case a path integral representation. By dividing time into slices and making use of the identity (47), we can in analogy with the bosonic case derive an expression very similar to the earlier presented path integral. When calculating the trace, we integrate over anti-periodic boundary conditions, because of the opposite signs of the coherent states appearing in (48). (Anti-periodic means that the configurations should satisfy θ(ti) = −θ(tf ). However, by multiplying with (−1)F we exchange the anti-periodic boundary conditions to periodic ones, and the remaining path integral is independent of (−1)F . Extending to supersymmetry, we want to integrate over both bosonic and fermionic degrees of bosonic degrees of freedom. The path integral representation of the Witten index thus is: Z T r(−1)F e−βH = DφDψDψ∗e−SE PBC (The path integral measures might vary depending on situation; the above expression is however the case for the model under consideration).

4.2 Gaussian integrals

i j Usual terms appearing in path integrals are bilinear forms Aijx x , in both fermionic and bosonic cases. We first consider the bosonic case when Aij are the components of a real symmetric matrix. We want to deal with integrals like: Z  1  I = dnx exp − A xixj 2 ij For a real symmetric matrix A. Such matrices can be diagonalised by an orthogonal transformation J: A = J T DJ D = JAJ T Where D is the m × m-matrix:   a11 0 ··· D =  0 a22   . .  . ..

Where aii are the eigenvalues of A. We can now rewrite the integral: Z  1  Z  1  I = dnx exp − A xixj = dny| det(J i)| exp − a (yk)2 2 ij j 2 k

i Since the transformation is orthogonal we have |det Jj | = 1. The above integral 2 contains n Gaussian integrals of the form R e−ai(yi) , yielding: Z  1  I = dnx exp − A xixj = (2π)n/2(a . . . a )−1/2 = (49) 2 ij 1 n 2πn/2 = (50) pdet(A)

40 P There is a similar expression for fermions. Consider a biliniear form i θiAijχi, i = 1, . . . n The matrix A is antisymmetric as opposed to the example in the bosonic case, because only the antisymmetric part of any matrix survives the summation over antisymmetric indices. θi and χi are two different set of Grassman numbers. We need a method to calculate integrals like

( n ) Z X I = dθ1 . . . dθndχ1 . . . dχn exp − θiAijχj j,i=1

0 We first perform the change of variable in one set of Grassman numbers χi = P i Aijχj, which yields: Z ( n ) 0 0 X 0 I = det(Aij) dθ1 . . . dθndχ1 . . . dχn exp − θiχi i=1

Z 0 0 0 0 χ1θ1 χnθn = det(Aij) dθ1 . . . dθndχ1 . . . dχne . . . e

By Taylor expanding the exponential, terms of order higher than one vanish because since χ2 = θ2 = 0, so we are left with eθχ = 1 + θχ. The resulting integral is Z n 0 0 Y I = det(Aij) dθ1 . . . dθndχ1 . . . dχn (1 + χiθi) i

= det(Aij)

4.3 The Gauss-Bonnet Theorem As stated in previous section, the regularised trace of the chirality operator yields the following path integral representation: Z T r(−1)F e−βH = DφDψ∗Dψe−SE PBC The action is integrated over the interval [0, β], so the periodic boundary conditions reads:

φi = φi(t + 2πβ) ψi = ψi(t + 2πβ) ψi∗ = ψ∗i(t + 2πβ)

As we saw in section (3.2.2) T r(−1)F e−βH = χ(M) for the Lagrangian (32). This equivalent up to a total time derivative with 1 1 L = g φ˙iφ˙j + ig ψ∗i(∇ ψ)j − R ψ∗iψj∗ψkψl 2 ij ij t 4 ijkl

41 To obtain the Euclidean action we perform a wick rotation: t → −it dt → −idt 1 1 g φ˙iφ˙j → − g φ˙iφ˙j 2 ij 2 ij ∗i j ∗i ˙ j igijψ (∇tψ) → −gijψ (∇tψ) The Riemann curvature term is unchanged since it contains no time derivatives. The Euclidean action is: β Z 1  S = g φ˙iφ˙j + g ψi(∇ ψ)j + R ψi∗ψj∗ψkψl dt E 2 ij ij t ijkl 0 (There is an extra minus sign coming from the time rescaling.) The path integral to evaluate now reads: β Z ( Z F −βH ∗ 1 ˙i ˙j ∗i j T r(−1) e = DφDψDψ exp − dt gijφ φ + igijψ ∇tψ + PBC 2 0 ) 1  + R ψ∗iψ∗jψkψl 4 ijkl

Because we have periodic boundary condition we Fourier expand the bosonic and fermionic variables with frequencies ωn = 2πnt/β,

i i X i i2πnt/β φ (t) = φ0 + φne n6=0 i i X i i2πnt/β ψ (t) = ψ0 + ψne n6=0 i∗ i X ∗i i2πnt/β ψ (t) = ψ0 + ψn e n6=0 We expand the first term in the Lagrangian in terms of non constant modes only, as it only contains time derivatives, we find the that the action is proportional to 1/β

β Z (i2π)2nm X  2π2 X S = dt φ˙i φ˙i e(2πi(n+m)t)/β = n2φ˙i φ˙i E1 β2 n m β n n 0 n,m6=0 n6=0 Since the Witten index is independent of β, we might consider the limit of β → +0. In this limit, the above term diverge unless the non constants modes approaches zero, and thus the contribution from the constant modes will dominate this limit. With this in mind, we choose Riemann normal coordinates at φ0 and taylor expand the metric and Levi-Civita connection coefficients 1 g = g | + (∂ g )| δφl + (∂ ∂ g )| δφlδφk + ... ij ij φ0 l ij φ0 2 l k ij φ0 1 Γk = Γk | + (∂ Γk )| δφl + (∂ ∂ Γk )| δφlδφm + ... ij ij φ0 l ij φ0 2 m l ij φ0

42 k l l l P l Where gij|φ0 = δij and Γij = 0. By definition, δφ := φ − φ0 = φn exp{i2πnt}. In the small β limit the correction terms approaches zero, and the second part of the action simply reads:

β Z  1  S = dt ψ∗iψ˙ j + R (φk)ψ∗iψj∗ψkψl (51) E2 4 ijkl 0 0 (We still implicitly sum over repeated indices.) The first term in (51) is expanded as    ∗i X ∗i iωnt X i iωmt L2 = ψ0 + ψn e 2πimψme = n6=0 m6=0  X 2πim X 2πim  = ψ∗i ψi eiwπmt/β + ψ∗ψ e(i2π(n+m)t)/β 0 β n β n m m6=0 n,m6=0 Because we integrate over an entire period the only terms that will survive the integral are those where n + m = 0, because the exponent contains an integer times 2π. Therefore, after rescaling the constant fermionic modes with a factor β−1/4 and performing the integration over time, we are left with

2π2 X X 1 S = n2φ˙i φ˙i + 2πi nφ∗iφi + R ψ∗iψ∗jψkψl + O(β) E β n n n n 4 ijkl 0 0 0 0 n6=0 n6=0

We throw away the O(β)-term. The path integral measures are ([13])

d i 1 Y d i dV Y d φn Dφ = d d φn = d d (2π) 2 2 n (2π) n6=0 (2π) Y d i d i Y d i Dψ = d ψn = d ψ0 d ψn n n6=0 Y d i∗ d ∗i Y d ∗i Dψ = d ψn = d ψ0 d ψn n n6=0 Where d is the dimension of the manifold. The integration over all constant paths 1 d is equivalent with integration over the manifold, so we write dV = dφ0 . . . dφ0. The integral splits to integrals over constant and non-constant modes: ( ! Z X 2π2n2 T r(−1)F e−βH = DφDψDψ∗ exp φ˙i φ˙i + 2πinφ∗iφi + β n n n n PBC n6=0 ) 1 − R ψ∗iψ∗jψkψl = 4 ijkl 0 0 0 0

Z d ( 2 ) Z ( ) Y d φn 2πn i i Y d ∗ d ∗i i = d exp φnφn d ψnd ψn exp 2πinψn ψn × 2 β n6=0 (2π) n6=0 Z Z ( ) dV d ∗ d 1 ∗i ∗j k l × d d φ0d φ0 exp − Rijklψ0 ψ0 ψ0 ψ0 (2π) 2 4

43 The integral over non-zero modes can be calculated with perturbation theory, and the first term is one due to the so called supersymmetric ward identity [6]. To calculate the integral over constant fermionic modes, we taylor expand  1 ∗i j∗ k l exp − 4 Rijklψ ψ ψ ψ . The only power of the expansion surviving the integration is the one of order d/2, as a consequence of the integration defined for Grassman numbers: Z dψψ = 1 Z dψ1 = 0 since that is the only term that contains a factor d factors of ψ∗ and d factors of 1 1 d/2 ψ. That term contains a factor (d/2)! (− 4 ) from the Taylor series representation of the exponential function. Obviously there is no such term if the dimension of the manifold is odd, which is in agreement with what we expect from the Euler Characteristic. In the even dimensional case we now have

d (−1) 2 Z χ(M) = T r(−1)F e−βH = dV × d d
(2 ) 2 ! M Z d d d ∗i ∗j k l
2 × d ψ0d ψ0 Rijklψ0 ψ0 ψ0 ψ0

d 1 1 2 d/2 d ∗ ∗1 ∗2 ∗d/2 With d ψ0 = d ψ0dψ0 . . . dψ0 and d ψ = dψ0 dψ0 . . . dψ0 , we rerwite the inte- gral over non constant configurations (we now put n = 2d): Z ∗1 ∗2 ∗d/2 1 1 2 d/2 dψ0 dψ0 . . . dψ d ψ0dψ0 . . . dψ0 Ri1j1k1l1 Ri2j2k2l2 × ...

∗i1 ∗j1 k1 l1 ∗i2 j2 k2 l2 ... × Rinjnknln (ψ0 ψ0 ψ0 ψ0 )(ψ0 ψ0 ψ0 ψ0 ) × ... ∗in ∗in in in ... × (ψ0 ψ0 ψ0 ψ0 ) = i1j1...injn k1l1...knln =   Ri1j1k1l1 ...Rinjnknln

In the last step we have changed the order of the integration measures and picked up a bunch of minus signs encoded in the Levi-Civita symbols. (integration is equivalent to differentiation for Grassman variables and thus the measures anticommute.) We have the following result:

d (−1) 2 Z χ(M) = T r(−1)F e−βH = dV i1j1...injn ... × d d
d/2 (2 ) 2 !π M k1l1...knln ... ×  Ri1j1k1l1 ...Rinjnknln

Which is the Gauss-Bonnet theorem 2.7.

44 5 Discussion and conclusions

In the section 2 we introduced the Language of Riemannian manifolds and the Hodge star, which allowed us to qunatize the sigma model and identify the Witten index with the Euler characteristic. We discussed the definition of this object in terms of de Rham cohomology and concluded that it is an topological invariant, by theorem 2.1, and in this way we find that the topology of the underlying Riemannian manifold is connected to the difference between bosonic and fermionic zero energy states. In section 4 we introduced the Path integral formalism and argued that the Witten index has a path integral representation as an integral over periodic boundary conditions. By computing this integral we derived the Gauss Bonnet theorem, as was the goal of this project. This is a truly remarkable result, considering that the Gauss-Bonnet theorem is a purely mathematical result. The fact that it can be reproduced in the physicist language of supersymmetry, quantization and path integrals is very fascinating. It is worth briefly mentioning how this method can be extended to derive index theorems for different complexes. By letting the spinors Ψi in Lagrangian (29) take the form 1 ψi Ψi = √ 2 ψi The Lagrangian is expanded to read: ([6]) 1 i L = g φ˙iφ˙ + g ψi(∇ ψ)j 2 ij 2 ij t (The Riemann curvature tensor vanishes because of the Bianchi identity.) The canonical quantization relation now takes the form {ψi, ψj} = 2gij. This is the algebra of the γ-matrices, the Clifford algebra. By identifying the ψi with γi the conserved charge is nothing but the Dirac operator. By using the path integral formalism one can now derive the index theorem for the Dirac complex. For the interested reader, this is done in detail in e.g Nakahara ([1]). Finally, some recommended further reading: the paper by Alvarez-Gaume ([5]) discusses how one can further modify the Lagrangian in order to compute the index density for yet different complexes. A nice discussion of index theory in general, suitable for physicists with not to much mathematical background is found in Nash ([2]). The paper by Witten ([4]) contains the original claim that the Witten index is the Euler characteristic for Lagrangian (32). A more rigorous and mathematical treatment of elliptic operators, elliptic complexes and index theory is found in Gilkey. ([7])

45 6 Acknowledgements

First of all i would like to thank my supervisor Matthew for introducing me to this interesting branch of physics and mathematics, and being of great help during the entire process. Secondly i would like to thank my fellow bachelor students Jim and Erik for interesting discussions and relevant inputs during this work. I would also like to thank my friend William for being a great private teacher in topology and geometry.

46 7 References

[1 Mikio Nakahara. Geometry, Topology and Physics, Second Edition. Taylor Fran- cis group, LCC, 2003.

2 Charles Nash. Differential Topology and Quantum Field Theory. ACADEMIC PRESS, 1991.

3 John M. Lee Introduction to Smooth Manifolds, Second Edition. Springer, 2003.

4 E. Witten. Constraints on Supersymmetry Breaking. Nucl. Phys. 253-316, 1982.

5 A. Mostafazadeh. Supersymmetry, Path Integration and the Atiyah Singer-Index Theorem. The University of Texas at Austin, 1994.

6 L. Alvarez-Gaume. Supersymmetry and the Atiyah-Singer Index Theorem. Com- mun. Math. Phys. 161-173, 1983.

7 Peter B. Gilkey. The Index and the Heat Equation, Mathematical Lecture Series 4. Publish or Perish, inc, 1974.

8 P.D. Jarvis. Supersymmetric Quantum Mechanics and the Index Theorem. Uni- versity of Tasmania.

9 P. Bastianelli. Path integrals for fermions, SUSY Quantum Mechanics, etc... University of Bologna, 2012.

10 Mark Srednicki. Quantum Field Theory. Cambridge University Press, 2007.

11 S. Li. Supersymmtric Quantum Mechanics and Lefschetz fixed-point Formula. University of Science and Technology of China, Shanghai Institute for Advanced Studies, 2005.

12 Kentaro Hori, Sheldon Katz, Albrecht Pandharipande, Richard Thomas, Cum- run Vafa, Ravi Vakil, Eric Zaslow. Mirror Symmetry. American Mathematical Society, 2003.

13 M. Loon. Path Integral Methods in Index Theorems. Merton collage, Uni- versity of Oxford, 2015.

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