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ATIYAH-SINGER INDEX THEOREM FROM SUPERSYMMETRIC QUANTUM MECHANICS

HEATHER LEE

Master of Science in Mathematics Nipissing University

2010 ATIYAH-SINGER INDEX THEOREM FROM SUPERSYMMETRIC QUANTUM MECHANICS

HEATHER LEE

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS

NIPISSING UNIVERSITY SCHOOL OF GRADUATE STUDIES NORTH BAY, ONTARIO

Heather Lee August 2010

I hereby declare that I am the sole author of this Thesis or major Research Paper.

I authorize Nipissing University to lend this thesis or Major Research Paper to other institutions or individuals for the purpose of scholarly research.

I further authorize Nipissing University to reproduce this thesis or dissertation by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research.

Abstract

Some preliminary knowledge for understanding the Atiyah-Singer theorem, including

differential manifolds, Cartan’s exterior algebra, de Rham cohomology, Hodge

theory, Riemannian manifolds and K¨ahler manifolds, fibre bundle theory, and

characteristic classes is reviewed. The Atiyah-Singer index theorems for four

classical elliptic complexes, including the Gauss-Bonnet theorem of the de Rahm complex, Hirzebruch Signature theorem of the signature complex, Reimann-Roch theorems of the Dolbeault complex, and the index theorem for A-roof genus of spin complex, are introduced. Quantum mechanics is formulated from classical mechanics using the canonical and path integral quantizations. Supersymmetric quantum mechanics is introduced including the 0+1-dimensional supersymmetric nonlinear sigma model. By representing supersymmetric quantum theory as a graded Hilbert space, the identification between a supersymmetric quantum mechanical system and a certain elliptic complex is established. The Witten index is shown to be identified with the index of classical elliptic complexes. The Gauss-Bonnet theorem,

Hirzebruch Signature theorem, the index theorem for A-roof genus, and the

Reimann-Roch theorem are then derived in great detail using supersymmetric quantum mechanics. Finally, a mathematical conjecture is proposed about the mysterious cancelations between the terms in the proof of the Atiyah-Singer index theorem in the heat equation approach.

iv. Acknowledgment

First and foremost, I would like to thank my supervisor Dr.Wenfeng Chen for the invaluable instruction and guidance he has provided. In addition, thanks to Dr.Vesko

Valov and Dr. Alexandre Karassev for the knowledge they have passed on to me throughout the M.Sc. program. I would also like to acknowledge my second reader

Dr. Murat Tuncali, and external examiner Dr. Achim Kempf. This paper is dedicated to my family and friends, for their continual and everlasting support.

v. Contents

1 Introduction and Background 5

2 Atiyah-Singer Index Theorem for Four Classical Elliptic Complexes 9

2.1 Calculus on A Diﬀerential Manifold ...... 9

2.1.1 Diﬀerential Manifold ...... 9

2.1.2 Tangent Space, Cotangent Space and Diﬀerential Forms ...... 10

2.1.3 Riemannian Manifold ...... 12

2.1.4 Hodge Theorem and de Rham Cohomology ...... 15

2.1.5 K¨ahlerManifold ...... 19

2.2 Fibre Bundle Theory ...... 22

2.2.1 Topology of Fibre Bundle ...... 22

2.2.2 Geometry of Vector Bundle ...... 25

2.3 Characteristic Classes of Vector Bundles ...... 27

2.3.1 Chern Class ...... 28

2.3.2 Pontrjagin Class ...... 30

2.3.3 Euler Class ...... 31

2.3.4 Stiefel-Whitney Class ...... 32

2.4 Classical Elliptic Complexes and Index Theorems ...... 33

2.4.1 Elliptic Complexes and Index of Elliptic Diﬀerential Operator ...... 33

2.4.2 de Rham Complex and Gauss-Bonnet Theorem ...... 36

2.4.3 Signature Complex and Hirzebruch Signature Theorem ...... 38

2.4.4 Dolbeault Complex and Riemman-Roch Theorem ...... 39

2.4.5 Spin Complex and Index Theorem for A-roof Genus ...... 39

1 3 Some Aspects of Supersymmetric Quantum Mechanics 41 3.1 Lagrangian Formulation and Hamiltonian Formulation of Classical Mechanics . . 41 3.1.1 Lagrangian Mechanics ...... 41 3.1.2 Hamiltonian Mechanics and Poisson Bracket Operation ...... 42 3.2 Construction of Quantum Mechanics from Classical Mechanics ...... 43 3.2.1 Canonical Quantization ...... 44 3.2.2 Path Integral Quantization ...... 48 3.3 Supersymmetric Quantum Mechanics ...... 54 3.3.1 Supersymmetry Algebra of Supersymmetric Quantum Mechanical System 54 3.3.2 An Example of N=1 supersymmetric Quantum Mechanics ...... 54 3.3.3 General Graded Structure of the Hilbert Space of Supersymmetric Quan- tum Mechanics and Witten Index ...... 58

4 Derivation of Atiyah-Singer Index Theorem from Supersymmetric Quantum Mechanics 61 4.1 Introducing 0+1-dimensional Supersymmetric Non-Linear Sigma Model . . . . . 61 4.2 Gauss-Bonnet Theorem from 0+1-dimensional N=1 Supersymmetric Sigma Model 62 4.2.1 Identiﬁcation between Quantum N=1 Supersymmetric Sigma Model and de Rham Complex ...... 62 4.2.2 Derivation of Gauss-Bonnet Theorem ...... 64 4.3 Hirzebruch Signature Theorem from 0+1-dimensional N=1 Supersymmetric Sigma Model ...... 66 4.3.1 Identify Signature Complex from Quantum N=1 Supersymmetric Sigma Model ...... 66 4.3.2 Derivation of Hirzebruch Signature Theorem ...... 68 4.4 Index Theorem for A-roof Genus from 0+1-dimensional N=1/2 Supersymmetric Sigma Model ...... 72 4.4.1 Designing Spin Complex from Quantum N=1/2 Supersymmetric Sigma Model ...... 72 4.4.2 Derivation of Index Theorem for A-roof Genus ...... 74 4.5 Riemann-Roch Theorem from 0+1-dimensional N=2 Supersymmetric Sigma Model 75 4.5.1 Making Dolbeault Complex from Quantum N=2 Supersymmetric Sigma Model ...... 75 4.5.2 Derivation of Riemann-Roch Theorem ...... 78

5 Summary and A Mathematical Conjecture 79

A N¨otherTheorem in a Mechanical System 81 B Grassmann Variable Calculus 83

C Cliﬀord Algebra and Spinor 87

Chapter 1

Introduction and Background

The Atiyah-Singer index theorem, proved by M. Atiyah and I. Singer in 1963 [1, 2, 3, 4], is one of the most remarkable discoveries in modern mathematics. It states that the analytical index of an elliptic differential operator on a compact differential manifold is equal to the topological index. The analytical index counts the number of independent solutions of a partial differential equation obtained from the zero-eigenvalue equation of the elliptic differential operator. Meanwhile, the topological index characterizes the non-trivial topology of the vector bundles constructed on the compact manifold and reflects the global topological features of the differential manifold. Further, the topological index can be expressed as an integral of a certain characteristic class of vector bundles over the differential manifold. Therefore, this theorem brings together several distinct branches of modern mathematics including analysis, geometry and topology. Because of the Atiyah-Singer index theorem, the topological features of a differential manifold can be studied by means of mathematical analysis. Conversely, the existence, and even the number, of linearly independent solutions of a partial differential equation can be investigated by observing the global topology of a differential manifold on which the elliptic differential operator is defined. The original proof by Atiyah and Singer relies on cobordism theory [1]. Later, Atiyah and Singer made use of K-theory to reprove the theorem and its numerous variants and generaliza- tions [2, 3]. Both approaches are hard to understand. In 1973, Atiyah, Bott, and Patodi gave an excellent proof by using the heat equation [3, 5]. This method is based on the spectrum theory of a positive definite self-adjoint operator. The basic idea is to first use the inner product to define the adjoint operator of an elliptic differential operator. Then, construct two self-adjoint non-negative definite operators from the elliptic differential operator and its adjoint. The two self-adjoint positive-definite operators are usually called heat operators because their eigenvectors can furnish an orthonormal basis in which the solution to a heat equation, defined by the operators, admits an expansion. Despite two heat operators having paired non-zero eigenvalues, their eigenspaces corresponding to the zero eigenvalue can have different dimensions. There- fore, the index of the elliptic differential operator can be evaluated by calculating the difference between the traces of the exponentials of two heat operators [5]. This method has actually provided inspiration for theoretical physicists to use supersymmetric quantum mechanical systems to derive the Atiyah-Singer index theorem. The idea of using supersymmetric quantum mechanics to prove the Atiyah-Singer index theorem was originally proposed by Witten when he investigated the dynamical supersymmetry breaking in various quantum field models possessing supersymmetry [6, 7]. In physics, quantum theory can be constructed from classical theory by a standard procedure called quantization. Classical mechanics has two equivalent descriptions: Lagrangian mechanics and Hamiltonian

5 6 Introduction mechanics. There are two equivalent quantization methods corresponding to the two classical formulations; that is, canonical quantization based on the Hamiltonian formulation, and path integral quantization based on the Lagrangian formulation. The mathematical description of a quantum theory constructed by means of canonical quantization is the theory of the operator and of the Hilbert space in functional analysis [11, 12]. A physical state is represented by a vector in the Hilbert space, while a physical quantity is represented by a self-adjoint (or Hermitian) operator. According to the statistical interpretation of quantum mechanics, once a basis of the Hilbert space, such as coordinate representation, is specified, the component of the state vector is relevant to the probability density for a physical phenomenon to take place. The measured value of a physical quantity is the average value of the operator representing it which is defined by the inner product of the Hilbert space. Further, the Hilbert space for a physical system has a special feature - it has at least one vector which can make the average value of the Hamiltonian operator minimal, termed in physics as the vacuum state vector. A symmetry in a classical mechanical system is the invariance of the classical action under certain transformations of the variables in the Lagrangian. Classical action is defined as the integration of the Lagrangian over time. According to the celebrated Nöthertheorem, the generator of a symmetry transformation must be a conservative physical quantity. At a quantum level, the conservative quantity is a time-independent operator. A classical theory possessing supersymmetry consists of two types of variables in either the Lagrangian or the Hamiltonian, the usual commutative variable and the anticommutative variable, called the Grassmann variable. In physics, the commutative and anticommutative variables are called bosonic and fermionic variables, respectively. Supersymmetry transformation is an exotic transformation that converts a bosonic variable into a fermionic variable and vice-versa. Therefore, the Hilbert space of a supersymmetric quantum theory is a graded space [13]. The generators of a supersymmetry transformation map the bosonic sector into the fermionic sector and vice-versa. An operator (−1)F can be defined to distinguish the bosonic sector from the fermionic sector. Because of the supersymmetry algebra, the dynamical breaking of supersymmetry depends on the existence of the vacuum state vectors that make the Hamiltonian take a zero eigenvalue. Hence, Witten defined the trace Tr(−1)F to characterize the dynamical supersymmetry breaking, and realized it is actually identical to the index of an elliptic differential operator — the generator of supersymmetry transformations [7]. Subsequently, Alvarez-Gaumé[15, 16, 17], and Friedan and Windey [18] independently worked out detailed proofs. Further, Getzler gave a more mathematically rigorous description [19]. There are a number to advantages of using supersymmetry quantum mechanics to prove the Atiyah-Singer index theorem over other approaches. First, an explicit isomorphism between an elliptic complex and a supersymmetric quantum mechanical system can be established. Second, functional integration can be used to calculate the index straightforwardly. This evaluation method is much simpler and easier than the heat equation method [5]. In particular, it is believed that the supersymmetry can explain the mysterious cancelations between a large number of terms arising from the small parameter expansion used in the heat equation approach. The disadvantage of the supersymmetry approach is the lack of mathematical rigor in some aspects including the quantization procedure, operator ordering problem, and the mathematical definition of path integral etc. The layout of this research paper is as follows: In Chapter 2, the Atiyah-Singer index theorem for four classical elliptic complexes and the relevant preliminary materials is reviewed, which covers calculus on differential manifolds (including Cartan’s exterior algebra, de Rahm cohomology theory, Hodge decomposition theorem, Stokes’ theorem, Riemannian and Kählermanifolds), the global topology and local differential geometry of fibre bundle theory, characteristic classes Atiyah-Singer Index Theorem 7 for vector bundles, and elliptic complexes. In Chapter 3, some necessary knowledge in quantum mechanics is introduced. This includes two equivalent descriptions of classical mechanics and the corresponding quantization methods of constructing quantum theory, supersymmetry algebra, and the structure of the Hilbert space of a supersymmetric mechanical system. Chapter 4 is the main part of the paper. It is shown in great detail how supersymmetric quantum mechanical systems can be identified with four classical elliptic complexes. Hence, four classical index theorems are derived using the functional integration method. Finally, in Chapter 5, a summary is given and a mathematical conjecture relevant to the proof of the Atiyah-Singer index theorem in the heat equation approach is proposed. 8 Introduction Chapter 2

Atiyah-Singer Index Theorem for Four Classical Elliptic Complexes

2.1 Calculus on A Diﬀerential Manifold

2.1.1 Diﬀerential Manifold

A differential manifold is a generalization of the Euclidean space we are familiar with. Locally, an n-dimensional differential manifold looks like Rn or Cn. By definition [8], an n-dimensional differential manifold M is a Hausdorff topological space together with a family of pairs {(Uα, ϕα)} such that

∪ • The Uα’s are open sets covering M. i.e., M = Uα. α

• The ϕα’s are homeomorphisms such that ϕα : Uα → Vα where Vα is an open neighborhood in Rn or Cn.

• ∩ ̸ ∅ ≡ ◦ −1 ∩ → ∩ Whenever Uα Uβ = , then the map ϕβα ϕβ ϕα : ϕα(Uα Uβ) ϕβ(Uα Uβ) is inﬁnitely diﬀerentiable (C∞).

n n Because ϕα maps each patch Uα on the manifold to R (or C ), a local coordinate system can thus be established on Uα, and each point P on the manifold can be represented by an ordered n-tuple of numbers (x1, x2, ··· , xn). This last point is essential when defining a coordinate system on the manifold. ϕβα relates two coordinate systems ϕα and ϕβ in the overlapping region Uα ∩ Uβ. A pair (Uα, ϕα) is called a chart on M and the family {(Uα, ϕα)} is called an atlas [14]. Therefore, one can define local smooth functions on M and use Calculus to study the geometry, and further the topology, of a differential manifold.

9 10 Atiyah-Singer Index Theorem

2.1.2 Tangent Space, Cotangent Space and Diﬀerential Forms

1. Tangent Space

Just like in calculus, the local shape of a plane curve can be studied by its tangent line at each point. The geometrical feature of a differential manifold Mn in the neighborhood of a point P can be studied using a tangent space TP (M) — that is a vector space formed( by tangent) vectors of M. Suppose a point P ∈ M is represented by the local coordinate x = x1, x2, ··· , xn . Then one can define a smooth curve x(t) ≡ (x1(t), x2(t), ··· , xn(t)) on M passing through P = x(0). We can consider a smooth function f(x) on M. When f(x) is defined on the curve, it’s directional derivative along the curve at P is

∑n µ d dx ∂f VP f = f[x(t)] = . (2.1) dt dt ∂xµ t=0 µ=1 P P

From now on, in this paper we shall adopt Einstein’s convention that repeated indices are summed and the summation symbol will be omitted. The viewpoint in modern mathematics is that a tangent vector of M is just the diﬀerential operator appearing in (2.1),

∂ V = V µ(x) . (2.2) ∂xµ

A tangent space TP (M) is the set of all the tangent vectors V at P ; hence it is a vector space µ with basis {∂/∂x }, µ = 1, 2, ··· , n. Therefore, TP (M) has the same dimension as M.

2. Cotangent Space

According to functional analysis, there always exists a dual space associated with a vector space. The dual space of a vector space is deﬁned as the set of linear functionals on the vector ∗ ∈ space. The cotangent space, TP (M) at a point P M is the dual vector space to TP (M). That ∈ ∗ → ∗ is, if ω TP (M), then ω : TP (M) R. The basis vectors of TP (M) are actually the diﬀerential line elements dxµ, which are uniquely determined by the requirement ( ) ∂ ∂ dxµ = xµ = δµ . (2.3) ∂xν ∂xν ν ∈ ∗ Hence, a dual vector ω TP (M) can be written as

µ ω = ωµ(x)dx . (2.4)

It is easy to see that V and ω are actually contravariant and covariant vectors in the traditional µ theory of tensor analysis. Because of the x-dependence of the components V (x) and ωµ(x), V and ω are called contravariant and covariant vector fields, respectively. Further, we can construct a larger tensor field space by taking the tensor product of k tangent ∗ k spaces TP (M) and l cotangent spaces TP (M), whose elements Tl (M) are called type-(k,l) mixed tensor fields with l covariant and k contravariant indices, ∂ ∂ k ≡ µ1...µk ⊗ · · · ⊗ ⊗ ν1 ⊗ · · · ⊗ νl Tl Tν ...ν (x) dx dx . (2.5) 1 l ∂xµ1 ∂xµk

3. Diﬀerential Form and Cartan’s Exterior Algebra Diﬀerential Form 11

A completely antisymmetric covariant tensor field, that is, a tensor field which changes sign under the interchange of any two pairs of arguments, is called a differential form.A p-form at the point P takes the following form,

µ1 ∧ · · · ∧ µp ωp = ωµ1···µp (x) dx dx . (2.6)

In deﬁnition (2.6), dxµ1 ∧ · · · ∧ dxµp is the antisymmetrization of the tensor product dxµ1 ⊗ · · · ⊗ dxµp , ie. 1 ∑ dxµ1 ∧ · · · ∧ dxµp = sgn(σ) dxµσ(1) ⊗ · · · dxµσ(p) , (2.7) p! σ∈Sp where Sp is a permutation group of p elements, sgn(σ) = 1 for an even permutation, and sgn(σ) = −1 for an odd permutation. Obviously, there exists 0 ≤ p ≤ n. When p = 0, the 0-form is just a local function on M ; when p = n, dx1 ∧ · · · ∧ dxn is the usual oriented volume element of M, which is usually called a top form. Let Λp(x) denote the set of p-forms at x. Then Λp(x) is a vector space of dimension n!/[p!(n − p)!]. Further, deﬁne a 2n-dimensional graded linear space Λ∗ formed by the following direct sum,

Λ∗ = Λ0 ⊕ Λ1 ⊕ · · · ⊕ Λn. (2.8)

∗ p Then we introduce a wedge product operation on Λ as follows: given a p-form αp ∈ Λ , and a q q-form βq ∈ Λ , a (p + q)-form is created in the following way [14],

pq αp ∧ βq = antisymmetrizing αp ⊗ βq = (−1) βq ∧ αp. (2.9)

Clearly Λ∗ is closed under the wedge product ∧. Therefore, Λ∗ is a graded algebraic system under the operation of the wedge product ∧, which is called Cartan’s exterior algebra.

4. Exterior Diﬀerentiation

Assume that all the differential forms in Λ∗ are smooth. Then there exists another well- defined operation on Λ∗, the exterior differential operator d which is a mapping d :Λp −→ Λp+1 defined by

µ µ ∂fµ1,···,µp ν µ µ dω = d(f ··· dx 1 ∧ · · · ∧ dx p ) = dx ∧ dx 1 ∧ · · · ∧ dx p . (2.10) p µ1, ,µp ∂xν Note that the 1-form dxν is always inserted at the start of the wedge product. It can easily be proved that d has the following properties:

1. d is nilpotent,

d2 = 0. (2.11)

2. The rule for d to diﬀerentiate the wedge product of a p-form αp and a q-form βq is

p d (αp ∧ βq) = dαp ∧ βq + (−1) αp ∧ dβq. (2.12)

5. Closed Form and Exact Form 12 Atiyah-Singer Index Theorem

Next, we introduce two special types of diﬀerential forms that will be used when discussing the de Rham cohomology. p A p-form αp ∈ Λ is called a closed p-form if it satisﬁes

dαp = 0. (2.13)

p A p-form αp ∈ Λ is called an exact p-form if it satisﬁes

αp = dβp−1, (2.14)

p−1 for a βp−1 ∈ Λ . Clearly, because of Eq. (2.13), an exact form must be a closed form, while the reverse is not true.

2.1.3 Riemannian Manifold

1. Deﬁnition of Riemannian Manifold

Let M be a smooth differential manifold. If there exists a non-degenerate positive-definite symmetric second-rank covariant tensor field G at each point on M where,

µ ν G = gµν(x)dx ⊗ dx , gµν = gνµ, and det [gµν(x)] ≠ 0, (2.15) then M is called a Riemannian manifold. If it is not required that (gµν) is positive-deﬁnite, M is called a generalized Riemannian manifold. This type of Reimannian manifold was used by Einstein to describe gravitational theory. We discuss only Riemannian manifolds in this paper. gµν(x) is called a metric on M, from which the distance ds between two inﬁnitesimally nearby points, represented by the coordinates (xµ) and (xµ + dxµ), can be evaluated as,

2 µ ν µ 2 ds = gµν(x)dx dx = ||dx || . (2.16)

2. Vierbein (or Local Orthonormal Frame) on Riemannian Manifold

a As a symmetric tensor, the metric tensor gµν(x) can be decomposed into vierbeins e µ as [10],

a b ab µν a b gµν = δabe µ(x)e ν(x), and δ = g e µ(x)e ν(x), (2.17)

µν where g is the inverse of gµν(x),

µν µ g gνρ = δ ρ. (2.18) gµν can be decomposed into the following form,

µν ab µ ν g = δ Ea Eb . (2.19)

µ a Ea is the inverse of e µ deﬁned by

µ µν b Ea = δabg e ν, (2.20) which obeys

µ b b µ ν Ea e µ = δa , and gµνEa Eb = δab. (2.21) Diﬀerential Form 13

In the above expressions, a = 1, 2, ··· , n is the index of the local orthornormal frame. a µ a The meanings of e µ(x) and Ea are as follows: e µ(x) is the orthogonal matrix which { µ} { a} µ transforms the basis dx of the cotangent space into an orthonormal basis e , while Ea µ transforms the basis ∂/∂x of the tangent space Tx(M) into an orthonormal basis Ea where, ∂ ea = ea (x)dxµ, and E = E µ . (2.22) µ a a ∂xµ Each ea is called a vierbein one-form.

3. Geometry of Riemannian Manifold

Riemannian geometry is usually studied by introducing a connection which can define parallel transportation of geometric objects on the manifold. Like a simple curve or a surface in R3, the geometry of a differential manifold is described by curvature and torsion tensor fields, both of which can be calculated using parallel transportation of geometric objects. Only a Riemannian manifold has the following two special features:

1. It is torsion-free.

2. The norm of a vector should be preserved in the parallel transport, which is usually called metricity.

The connection satisfying these two conditions is called a Riemannian connection. There are two types of Riemannian connections on a Riemannian manifold: one is the Riemannian spin connection, and the other is the Levi-Civita connection (or called the Christoﬀel symbol in physics). Both connections describe the same Riemannian geometry.

1. Riemannian geometry in terms of spin connection a a µ We start from a more general aﬃne spin connection 1-form ω b = ω bµdx . Then the parallel transportation of the orthonormal frames ea on the manifold yield the torsion tensor. The parallel transportation of the aﬃne spin connection itself on the manifold gives the curvature tensor. Both of these are described by Cartan’s structure equation as: [8] 1 dea + ωa ∧ eb = T a = T a eb ∧ ec, (2.23) b 2 bc 1 dωa + ωa ∧ ωc = Ra = Ra ec ∧ ed. (2.24) b b b b 2 bcd

Then the Riemannian spin connection can be obtained by imposing( two requirements) listed above. From Eqs. (2.16), (2.17), (2.22) and (2.24), the metricity d ||dxµ||2 = 0 leads to

ωab = −ωba, (2.25)

c where ωab = δacω b. Further, from Eq. (2.23), the torsion-free requirement yields a a a ∧ b T = de + ω b e = 0. (2.26)

a a Once e is speciﬁed, ω b can be determined. Then the Riemmanian curvature can be found by means of (2.24) and the geometry of the Riemannian manifold is fully determined. 14 Atiyah-Singer Index Theorem

2. Riemannian geometry in terms of Levi-Civita Connection µ Expressed in terms of the Levi-Civita connection Γνρ, the metricity and torsion-free requirements take the following forms [8], − λ − λ Metricity: Dρgµν = ∂ρgµν Γρµgλν Γρνgµλ = 0; (2.27) ( ) 1 Torsion-free: T λ = Γλ − Γλ = 0. (2.28) µν 2 µν νµ The Riemannian curvature tensor 2-form is deﬁned by 1 Rλ = Rλ dxµ ∧ dxν = dΓλ + Γλ ∧ Γσ , (2.29) ρ 2 ρµν ρ σ ρ λ λ ν where Γ ρ = Γ νρdx . Eqs. (2.27) and (2.28) can uniquely determine the Levi-Civita connection in terms of the metric, 1 Γλ = gλρ (∂ g + ∂ g − ∂ g ) . (2.30) µν 2 µ ρν ν µρ ρ µν Then the Riemannian curvature tensor can be calculated from (2.29) by σ σ − σ σ α − σ α R νλρ = ∂λΓ νρ ∂ρΓ νλ + Γ λαΓ νρ Γ ραΓ νλ, σ Rµνλρ = gµσR νλρ. (2.31)

Rµνλρ has the following symmetric properties:

Rµνλρ = −Rνµλρ = −Rµνρλ = Rλρµν, and

Rµνλρ + Rλµνρ + Rνλµρ = 0. (2.32)

3. Relation between Two Descriptions a The curvature 2-form R b can be decomposed as 1 1 Ra = Ra ec ∧ ed = Ra dxµ ∧ dxν. (2.33) b 2 bcd 2 bµν Then there is the relation

λ λ b a R ρµν = Ea e ρR bµν. (2.34) The torsion-free requirement (2.26) yields ( ) a ∧ b a b µ ∧ ν − b ν − b µ ∧ ν ω b e = ω bµe νdx dx = d e νdx = ∂µe νdx dx ( ) − a − λ a µ ∧ ν ≡ − b µ ∧ ν = (∂µe ν Γµνe λ)dx dx Dµe ν dx dx . (2.35)

Hence we obtain ( ) a b − b − a − λ a ω bµe ν = Dµe ν = (∂µe ν Γµνe λ), (2.36)

which gives the relation between the Riemannian spin connection and the Levi-Civita connection, ( ) λ λ a a b Γµν = Ea ∂µe ν + ω bµe ν . (2.37)

These relations will be used in Sect. 4.4. Diﬀerential Form 15

2.1.4 Hodge Theorem and de Rham Cohomology

Once we work on a Riemannian manifold M, the metric structure brings more operations and structures on Λ∗(M).

1. Hodge ∗ operation

The Hodge ∗ operator is a map ∗ :Λp −→ Λn−p. That is, it converts p-forms into(n−p)-forms in Λ∗. On a Riemannian manifold it is deﬁned by

µ µ 1 1/2 µ1···µp ν ν ν ∗(dx 1 ∧ · · · ∧ dx p ) = g ε ··· dx p+1 ∧ dx p+2 ∧ · · · ∧ dx n (n − p)! νp+1 νn

1 1/2 µ ν µ ν ν ν ν = g g 1 1 ··· g p p ε ··· ··· dx p+1 ∧ dx p+2 ∧ · · · ∧ dx n , (2.38) (n − p)! ν1 νpνp+1 νn where g = det(gµν) and εµ ···µn is the n-dimensional totally antisymmetric tensor [8],  1  0, any two indices repeated; εµ ···µ = +1, (µ1, ··· , µn)=even permutation of (1, ··· , n); (2.39) 1 n  −1, (µ1, ··· , µn)=odd permutation of (1, ··· , n).

It is a straightforward proof to show that the Hodge ∗ operator has the following property when acting on a p-form ωp, p(n−p) ∗ ∗ ωp = (−1) ωp. (2.40)

2. Inner Product deﬁned with ∗

Using the Hodge operator we can deﬁne an inner product on Λ∗ as follows, ∫

(αp, βp) = αp ∧ ∗βp. (2.41) M This inner product has the following properties:

(αp, βp) = (βp, αp) = (∗αp, ∗βp) ,

(αp, αp) ≥ 0, and “=” stands only when αp = 0. (2.42)

3. Codiﬀerential Operator Coclosed and Coexact Forms

The adjoint of the exterior differential operator d can be defined by using the inner product (2.41). Define the adjoint of the exterior differential operator d, denoted by δ, as,

(αp, dβp−1) = (δαp, βp−1). (2.43) Clearly, δ is a mapping from Λp to Λp−1, δ :Λp −→ Λp−1. (2.44) It can be veriﬁed that when δ acts on a p-form, δ = (−1)np+n+1 ∗ d ∗ . (2.45) It follows straightforwardly from either (2.43) or (2.45) that, δ2 = 0. (2.46)

Recall that a p-form ωp is called 16 Atiyah-Singer Index Theorem

• exact if ωp = dαp−1, where αp−1 is a (p − 1)-form,

• closed if dωp = 0.

Similar, a p-form ωp is called

• coexact if ωp = δαp+1, where αp+1 is a (p + 1)-form,

• coclosed if δωp = 0.

4. Laplacian Operator and Harmonic Form

The Laplacian operator can be deﬁned from d and δ as,

∆ = (d + δ)2 = dδ + δd. (2.47)

It takes the following form when acting on a p-form,

∆ωp = dδωp + δdωp, (2.48)

p which is a map from Λ to itself. A p-form αp is called a harmonic form if it satisﬁes

∆αp = 0. (2.49)

The set of all harmonic p-forms is denoted by Harmp(M). It can be proved that ∆ has the following properties:

1. ∆ commutes with ∗, d and δ,

[∆, d] = [∆, δ] = [∆, ∗] = 0. (2.50)

2. ∆ is a self-adjoint positive deﬁnite operator. From the inner product (2.41), we have

(∆αp, βp) = (αp, ∆βp) , and

(αp, ∆αp) = (αp, dδαp) + (αp, δdαp) = (δαp, δαp) + (dαp, dαp) ≥ 0. (2.51)

Further, Eq. (2.51) gives

∆αp = 0 if and only if dαp = 0 and δαp = 0. (2.52)

This means a harmonic form must be both closed and coclosed.

6. Stokes’ Theorem

Stokes’ Theorem is the generalization of the Fundamental Theorem of Calculus to a diﬀeren- tial form. It states that given an n-dimensional, oriented manifold M with non-empty boundary and an (n − 1)-form, ωn−1, there exists ∫ ∫

dωn−1 = ωn−1. (2.53) M ∂M

7. de Rham Cohomology Diﬀerential Form 17

The de Rham cohomology tells us that closed differential forms can reflect topological in- formation of a differential manifold through a duality established by Stokes’ theorem. Further, the equivalent classes of closed forms classified by exact forms constitutes a finitely generated Abelian group. Therefore, the de Rham cohomology is a link between differential topology and algebraic topology. Let Zp denote the set of all closed p-forms on M, and let Bp denote the set of all exact p-forms on M so that:

p p Z (M) = {ωp ∈ Λ : dωp = 0} = Ker dp, (2.54) p p B (M) = {ωp ∈ Λ : ωp = dαp−1} = Im dp. (2.55) Because of d2 = 0, we have

Bp ⊆ Zp. (2.56)

Therefore, all elements in Zp can be classiﬁed by the equivalence relation deﬁned as follows: for p p αp ∈ Z and βp ∈ Z , p αp ∼ βp if and only if αp − βp = dθp−1 ∈ B . (2.57)

p Consequently, all the elements ωp of Z can be classified into different equivalent classes, i.e., the cosets of Bp, 1 1 p ··· k k p ··· [ωp] = ωp + B , , [ωp ] = ωp + B , . (2.58) The quotient space { } p p p 1 ··· k ··· p HDR(M) = Z (M)/B (M) = [ωp], , [ωp ], ,B (2.59) p is called the de Rham cohomology group. HDR usually has finite order. p In the following we show that the dual of HDR(M,R) is just the homology group Hp(M,R) from algebraic topology [9].

Let Zp denote the set of p-chains with no boundary, which are called p-cycles, and let Bp denote the set of p-chains which are boundaries of p + 1-chains:

Zp(M) = {Cp| ∂Cp = 0} ,

Bp(M) = {Cp| Cp = ∂Cp+1} . (2.60)

2 Then Bp ⊆ Zp because the boundary operator is nilpotent ∂ = 0. The simplicial homology group with real coeﬃcients is the quotient group [9],

Hp(M,R) = Zp(M)/Bp(M). (2.61)

p Let Cp ∈ Zp and ωp ∈ Z . Then the dual map ωp : Cp −→ R is deﬁned by ∫

⟨ωp,Cp⟩ = ωp. (2.62) Cp By using Stokes’ theorem, we have ∫ ∫ ∫ ∫ ∫ ∫

(ωp + dαp−1) = ωp + dαp−1 = ωp + αp−1 = ωp, and Cp ∫ ∫Cp ∫Cp ∫ Cp ∂Cp Cp

ωp = ωp + dωp = ωp. (2.63) Cp+∂Cp+1 Cp Cp+1 Cp 18 Atiyah-Singer Index Theorem

Therefore, the above dual map can be expressed as, ∫

⟨[ωp], [Cp]⟩ = [ωp]. (2.64) [Cp] Consequently, the de Rham cohomology group and the simplicial homology group are naturally isomorphic, p ∼ HDR(M,R) = Hp(M,R). (2.65) The pth Betti number of M is deﬁned as p bp = dim Hp(M,R) = dim HDR(M,R), (2.66) ∑n p and the Euler characteristic is χ(M) = (−1) bp. p=0

8. Hodge Decomposition Theorem

Hodge’s decomposition theorem is a very useful result in the theory of partial differential equations (PDE). It provides a sufficient and necessary condition for the existence of the solu- p tion(s) to the differential equation ∆ωp = ηp in Λ .

Hodge’s decomposition theorem: Let M be a compact Riemannian manifold without p boundary. Then any p-form, ωp ∈ Λ admits the following unique decomposition,

ωp = ∆θp + γp, (2.67) where γp is a harmonic p-form, i.e., ∆γp = 0.

From Eq. (2.43), we have (∆θp, γp) = (θp, ∆γp) = 0. So Hodge’s decomposition theorem can also be formally expressed as Λp(M) = Harmp(M) ⊕ ∆Λp(M) = Harmp(M) ⊕ [Harmp(M)]⊥. (2.68) Eq. (2.67) can be rewritten in the following form with Eq. (2.47),

ωp = (dδ + δd)θp + γp ≡ dαp−1 + δβp+1 + γp. (2.69) This shows that a p-form can be uniquely decomposed as a sum of exact, co-exact and harmonic forms. The Hodge decomposition theorem implies that the equation ∆ωp = ηp has a solution in ⊥ Λp(M) if and only if ωp ∈ [Harmp(M)] .

9. Isomorphism between Harmp(M) and Hp(M,R)

p The Hodge decomposition theorem can lead to an important result. If ωp ∈ H (M,R), then by Eqs. (2.52) and (2.69), we have

dωp = dδβp+1 = 0, and (β, dδβ) = (δβ, δβ) = 0.

So δβ = 0 and hence ωp = dαp−1 + γp. Therefore, there exists p ∼ p ∼ Harm (M) = H (M,R) = Hp(M,R). (2.70) Diﬀerential Form 19

2.1.5 K¨ahlerManifold

A Kählermanifold is a differential manifold that allows three compatible structures to be defined on it: a complex structure, a metric structure, and a symplectic structure. Hence, it is usually defined as a complex manifold with a Hermitian metric and a Kählerform.

1. Complex Manifold and Complex Tangent and Cotangent Spaces

An n-dimensional complex manifold is locally isomorphic to the complex vector space Cn, α i.e., there exist ϕi : Ui ⊂ M −→ z = {z }, α = 1, ··· , n. Further, the local coordinate ≡ ◦ −1 ∩ −→ ∩ transformations defined by ϕji ϕj ϕi : ϕi (Ui Uj) ϕj (Ui Uj) must be holomorphic, i.e., z′β = f β(z) depend only on z, not on its complex conjugate z. An n-dimensional complex manifold can be considered as a 2n-dimensional real differential manifold with a complex structure. Hence, the complex coordinates can be represented by 2n real coordinates as zα = xα + iyα, where yα ≡ xn+α. The complex conjugate coordinates are defined in the usual way,

zα = xα − iyα.

Then we have ( ) ( ) ∂ 1 ∂ ∂ ∂ 1 ∂ ∂ = − i , = + i ; ∂zα 2 ∂xα ∂yα ∂zα 2 ∂xα ∂yα dzα = dxα + idyα, dzα = dxα − idyα. (2.71)

The complex tangent and cotangent spaces at each point of M are deﬁned in terms of their local bases [8]:

α α Tc(M) = {∂/∂z } , T c(M) = {∂/∂z } , ∗ { α} ∗ { α} Tc (M) = dz , T c (M) = dz . (2.72)

2. Complex Structure, [p, q]-diﬀerential Forms and Exterior Diﬀerential Operator

A complex structure J on a 2n-dimensional real differential manifold is a smooth (1, 1) type 2 tensor field satisfying J = −I, and it is a linear automorphism on the tangent space TP (M2n) at each point P , defined by ∂ ∂ ∂ ∂ ∂ ∂ J = ≡ , and J = J = − . ∂xα ∂xn+α ∂yα ∂xn+α ∂yα ∂yα From the dual relation (2.3) between tangent and cotangent vectors, dxµ(J∂/∂xν) = (Jdxµ)(∂/∂xν), we obtain Jdxα = −dyα, and Jdyα = −dxα. Consequently, we have ( ) ∂ 1 ∂ ∂ ∂ J α = α + i α = i α , ∂z 2 (∂y ∂x ) ∂z ∂ 1 ∂ ∂ ∂ J = − i = −i , (2.73) ∂zα 2 ∂yα ∂xα ∂zα 20 Atiyah-Singer Index Theorem and

J dzα = −dyα + idxα = idzα, J dzα = −dyα − idxα = −idzα. (2.74)

Therefore, the vectors in Tc(M) and T c(M) are called [1,0] and [0,1]-type vector ﬁelds, respec- ∗ ∗ tively, and those in Tc (M) and T c (M) are called [1,0] and [0,1]-type one-forms. Subsequently, we have complex-valued [p, q]-type smooth diﬀerential forms as,

α1 αp β1 βq ωp,q = f ··· ··· (z, z)dz ∧ · · · ∧ dz ∧ dz ∧ · · · ∧ dz . (2.75) α1 αpβ1 βq

p,q All the ωp,q form a set Λ , which is a module over a complex number ring.

The exterior diﬀerential operator on a sequence {Λp,q} can be deﬁned as,

∂ ∂ ∂ ∂ d = dxα + dyα = dzα + dzα ≡ ∂ + ∂. (2.76) ∂xα ∂yα ∂zα ∂zα

Clearly, there exist

∂ :Λp,q −→ Λp+1,q, ∂ :Λp,q −→ Λp,q+1, (2.77) and

2 ∂2 = ∂ = ∂∂ + ∂∂ = 0. (2.78)

2. Hermitian Metric, Hodge ∗ and Three Types of Laplacian Operators

A Hermitian metric is the counterpart of the Riemannian metric on a complex manifold. It can be expressed as the following form in terms of local coordinates {zα},

α ⊗ β h = hαβdz dz , (2.79) ( ) × where hαβ is an n n non-degenerate positive-deﬁnite matrix, i.e., it is a Hermitian matrix where

( ) ( )T h = det (hαβ) > 0, and hαβ = hαβ = (hβα) . (2.80)

Further, the real part of h gives the usual Riemannian metric, while its imaginary part is a [1,1]-type diﬀerential form i.e., a simplectic structure in the context of a real 2n-dimensional manifold, ( ) 1 ds2 = h dzα ⊗ dzβ + dzβ ⊗ dzα = h dzαdzβ, 2 αβ αβ ( ) i K = h dzα ⊗ dzβ − dzβ ⊗ dzα = h dzα ∧ dzβ = K. (2.81) 2 αβ αβ A complex manifold with a Hermitian metric is called a Hermitian manifold. Diﬀerential Form 21

Subsequently, in the same way as on a Riemannian manifold, the complex Hodge ∗ operator can be deﬁned on a [p, q]-form as, ( ) ∗ dzα1 ∧ · · · ∧ dzαp ∧ dzβ1 ∧ · · · ∧ dzβq

1 1 n n(n−1)/2+np α γ α γ β δ β δ ≡ i (−1) h 1 1 ··· h p p h 1 1 ··· h q q ϵ ··· ··· (n − p)! (n − q)! γ1 γpσ1 σn−p × λ1 ∧ · · · ∧ λn−q ∧ σ1 ∧ · · · ∧ σn−p ϵδ1···δqλ1···λn−q dz dz dz dz . (2.82)

The inner product between two [p, q]-forms is deﬁned as, ∫

(αp,q, βp,q) = αp,q ∧ ∗βp,q. (2.83)

Using the inner product, we can introduce the adjoint operator of d, the co-diﬀerential operator δ, which is deﬁned by (dαp,q, βp+1,q+1) = (αp,q, δβp+1,q+1) and found to be the following form, ( ) ∗ δ = − ∗ d∗ = − ∗ ∂ + ∂ ∗ = ∂∗ + ∂ , δ :Λp,q −→ Λp−1,q−1, δ2 = 0, (2.84) where

∂∗ = − ∗ ∂∗ :Λp,q −→ Λp−1,q; ∗ ∂ = − ∗ ∂∗ :Λp,q −→ Λp,q−1. (2.85)

Clearly there exist

∗2 ∗ ∂ = 0, (∂αp,q, βp+1,q) = (αp,q, ∂ βp+1,q); ∗2 ∗ ∂ = 0, (∂αp,q, βp,q+1) = (αp,q, ∂ βp,q+1). (2.86)

Then there are three Laplacian operators that can be deﬁned as,

∆ = (d + δ)2 = dδ + δd, ∆′ = 2(∂ + ∂∗)2 = 2(∂∂∗ + ∂∗∂) ∗ ∗ ∗ ∆′′ = 2(∂ + ∂ )2 = 2(∂∂ + ∂ ∂). (2.87)

All of them are self-adjoint operators such that,

′ ′′ (Oαp,q, βp,q) = (αp,q, Oβp,q),O = ∆, ∆ , ∆ .

Further, for a compact Hermite manifold without boundary, there exists a generalized Hodge decomposition theorem,

∗ ′ ωp,q = ∂αp−1,q + ∂ βp+1,q + γp,q, ∆ γp,q = 0.

3. Geometry of K¨ahlerManifold

When the diﬀerential form K deﬁned in Eq. (2.81) is closed, ie.

dK = 0, (2.88) 22 Atiyah-Singer Index Theorem then the Hermitian manifold becomes a Kählermanifold and K is called a Kählerform. It can be proved that on a Kählermanifold, the three Laplacian operators listed in (2.87) are identical, ∆ = ∆′ = ∆′′.

Similar to the Riemannian manifold, a K¨ahlermanifold has vanishing torsion. The non- vanishing connections are only

∂h ∂h Γα = hαδ βδ , and Γα = Γα = hαδ δβ . (2.89) βγ ∂zγ βγ βγ ∂zγ The curvature is a [1,1]-type form [8]

Rα = Rα dzα ∧ dzδ,Rα = Rα , (2.90) β βγδ β β and the non-vanishing components of the curvature tensor ﬁelds are only

∂Γα Rα = − βγ , and Rα = −Rα . (2.91) βγδ ∂zδ βγδ βγδ

2.2 Fibre Bundle Theory

The theory of fibre bundles studies differential geometry and topology of differential manifolds by observing a twisted topological product of two differential manifolds. It a powerful mathematical tool using local differential geometry data to study global topology of a differential manifold.

2.2.1 Topology of Fibre Bundle

1. Deﬁnition of Fibre Bundle

A (diﬀerentiable) ﬁbre bundle is a mathematical structure made up of six parts: (E, M, F, π, ϕ, G), each of which can be explained as follows:

• M is an n-dimensional manifold called a base. It is assumed that M is covered by a set of open coordinate neighborhoods Uα;

• F is called a fibre, which is a k-dimensional manifold and is located at each point of M. The dimension of a fibre bundle usually means the dimension of the fibre F ;

• The topological space E is called the total space which is locally like Uα × F in each neighborhood Uα ⊆ M;

• π is a surjective projection map such that π : E → M, which shrinks each ﬁbre to a point of M;

• { } −1 → × ϕ = ϕα is a set of homeomorphisms such that ϕα : π (Uα)( Uα F . ϕα is called) a µ µn i i local trivialization. Because of ϕα, a local coordinate (x; z) ≡ x 1 ··· x ; z 1 , ··· z k can −1 be assigned to each point of E. Further, ϕα satisﬁes πϕα (x; z) = x. Fibre Bundle Theory 23

• ∩ ◦ −1 In the overlap Uα Uβ, ϕα ϕβ is an invertible continuous map of the form ◦ −1 ∩ × −→ ∩ × ϕα ϕβ :(Uα Uβ) F (Uα Uβ) F. ∈ ∩ ◦ −1 For a ﬁxed x Uα Uβ, ϕαβ(x) = ϕα ϕβ (x) are transformation matrices in a ﬁbre space Fx located at x, which are called transition functions. Clearly, the transition functions satisfy ϕαα(x) = Ik×k, −1 ϕαβ(x) = [ϕβα(x)] for x ∈ Uα∩Uβ, ϕαβ ◦ ϕβγ(x) = ϕαγ(x) for x ∈ Uα∩Uβ∩Uγ.

Therefore, the transition functions form a group G ≡ {ϕαβ(x)}, which is a transformation group on the ﬁbre space Fx. G is usually called the structure group of the ﬁbre bundle E.

When E = M × F and the transition function is an identity, the bundle E is called a trivial fibre bundle. Another important notion in fibre bundle theory is the cross section, or simply the section, of a fibre bundle. A section of a fibre bundle is a one-to-one smooth map S : U ⊂ M −→ π−1(U), which assigns a function S(x) ∈ Fx for every point x ∈ M. Note that a local section is defined only in a neighborhood U of M [14]. Only a trivial bundle can have a well-defined global section. A simple example of a non-trivial fibre bundle is the Mobius strip. The Mobius strip is made up of the base manifold S1 together with the fibre F being a line segment I = [−1, 1]. S1 is covered by two coverings, namely two arcs which are greater than the semicircle with structure group G = {e, g}.

2. Some Typical Fibre Bundles

The following is a list of some typical ﬁbre bundles:

• An k-dimensional real vector bundle E is a fibre bundle whose fibre F is a real vector space of dimension k. The structure group is GL(k, R). If a Riemannian metric is defined on F , then GL(k, R) reduces to its subgroup O(k), the orthogonal group. Further, if F is oriented, then O(k) reduces to its subgroup SO(k), the special orthogonal group [8]. Let (e1, e2, ··· , ek) be a basis of F . Then the local section S(x) at a point x ∈ M is a function on M, and also a vector in F

∑k i S(x) = ei(x)z (x). (2.92) i=1 For example, a tangent bundle T (M), whose ﬁbres are tangent spaces at each point of a diﬀerential manifold M, is a typical vector bundle. So is the cotangent bundle T ∗(M).

• A complex vector bundle is similar to a real vector bundle but has the ﬁbre F with the structure of Ck, and transition functions belonging to GL(k, C). When an inner product is deﬁned on F , GL(k, C) reduces to its subgroup U(k), the unitary group. Further, if F is oriented, then U(k) reduces to its subgroup SU(k), the special unitary group.

• A spin bundle is a bundle whose ﬁbre is a space of spinors (see Appendix C), and the structure group is the spin group Spin(k). Spin(k) is a covering group of SO(k). For example, Spin (3)=SU(2) is a double covering of SO(3). 24 Atiyah-Singer Index Theorem

• A principle bundle, denoted P , is made up of a ﬁbre that is a Lie group G with transition functions belonging to G [8].

We will focus on vector bundles and spin bundles for later use.

3. Dual Bundle, Direct Sum and Tensor Product of Vector Bundles

New bundles can be constructed from some existing ﬁber bundles. For later use, we introduce only the dual bundle of a vector bundle, the line bundle, the direct sum and tensor product of vector bundles.

• Dual Bundle Let E be a vector bundle whose ﬁber at each point x ∈ M is a k-dimensional vector space V with transition functions of k × k matrices Φ(x). Then the dual bundle E∗ is a vector bundle whose ﬁbre is the dual space V ∗ with transition functions of k × k matrices [ ]− Φ(x)t 1.

• Line Bundle A line bundle is a vector bundle whose ﬁbre is a one-dimensional vector space. It is a family of real lines R or complex number C located at each point of the base manifold M. The structure group of a line bundle is GL(1,R) if it is real or GL(1,C) if it is complex. Both GL(1,R) and GL(1,C) are Abelian groups.

• Direct Sum of Vector Bundles Let A and B be two vector bundles over a differential manifold M. Suppose that the fibre of A is a j-dimensional vector space V with basis {e1, ··· , ej} and the transition functions are j × j matrices X, while the fibre of B is a k-dimensional vector space W with basis {f1, ··· , fk} and the transition functions composed of k × k matrices Y . Then the direct sum A ⊕ B can be obtained by taking both the direct sum of V and W at each point x ∈ M, along with the direct sum of X and Y where, [ ] X 0 X ⊕ Y = , 0 Y

V ⊕ W = span {e1, ··· , ej; f1, ··· , fk} . (2.93)

Thus the direct sum A⊕B is a j + k-dimensional vector bundle with ﬁbre V ⊕ W , whose transition functions are (j + k) × (j + k) matrices.

• Tensor Product of Vector Bundles Taking the tensor product of V and W and the direct product of X and Y at each point x ∈ M as,

V ⊗ W = span {ep ⊗ fq : p = 1, ··· , j; q = 1, ··· , k} ,

X × Y = Y × X = (Xpp′ Y ) = (Yqq′ X), (2.94)

we obtain a tensor product A ⊗ B, which is a tensor bundle of jk dimensions with ﬁbers being V ⊗ W and transition functions being jk × jk matrices. The generalization to the tensor product of many bundles is straightforward. Fibre Bundle Theory 25

2.2.2 Geometry of Vector Bundle

The local differential geometry of a vector bundle is characterized by its curvature, which can be detected by parallel transport of the section of the fibre bundle along a path in the base manifold. Actually, curvature measures how much a section has changed when it has been parallelly transported along a path. This parallel transportation of the section is defined by the connection of the bundle. This is because the section of a fibre bundle, e.g., a vector field in a vector bundle, is an extension of a function defined on the base manifold. One can directly calculate the differentiation of a smooth scalar function, whereas to differentiate a section one must define how to compare the sections of a fibre located at one point with the section of a fibre at a point nearby. This comparison requires a connection on the bundle. Let X be a tangent vector at a point x ∈ M and let S(x) be a section of a vector bundle E. The connection, denoted by ∇, is a rule that defines the directional derivative ∇X S(x) of S(x) and compares it to another section of E at a neighborhood point along the curve whose tangent vector is X. According to the definition of the directional derivative along a vector, we have

∇X S(x) = X(∇S(x)) = ⟨X, ∇S(x)⟩. (2.95)

Let Γ(E) denote the set of sections of E. Then from (2.95), ∇ should be a map deﬁned as,

∇ : Γ(E) −→ Γ[E ⊗ T ∗(M)], S(x) −→ ∇S(x), [ ] i i i ∇S(x) = ∇ ei(x)z (x) = ei(x) ⊗ dz (x) + [∇ei(x)] ⊗ z ≡ ⊗ i j ⊗ i ei(x) dz (x) + ej(x)ω i(x) z , (2.96) where the action of ∇ on the basis {ei} is deﬁned by ∇ j ei(x) = ej(x)ω i(x), (2.97) and

j j µ ω i(x) = ω µidx (2.98) is the one-form connection of the vector bundle E. With the above connection, a parallel transportation can be deﬁned as follows: Let x(t) = (x1(t), x2(t), ··· , xn(t)) be a curve in M. Then the tangent vector to x(t) isx ˙(t) = dx(t)/dt. When S(x) takes a parallel transportation along x(t), the component of S(x) in the direction of the tangent vectorx ˙(t) should remain constant. That is, the change ∇S(x) should be orthogonal tox ˙(t). Hence, the directional derivative of S(x) in the direction of the tangent vector is zero, [ ] i ∇x˙ S(x) = ⟨x,˙ ∇S(x)⟩ = ∇d/dt ei(x)z (x) = 0.

This leads to [8] ( ) j j i µ ∂µz + ω µ iz x˙ ej = 0. As in Eq. (2.35), we denote

Dµ ≡ ∂µ + ωµ, (2.99) which is called a covariant derivative. 26 Atiyah-Singer Index Theorem

The curvature then arises as a matrix-valued 2-form, deﬁned as 1 Ωi = dωi + ωi ∧ ωk = (Dω)i = Ωi dxµ ∧ dxν. (2.100) j j k j j 2 jµν From (2.98), the components of the curvature tensor ﬁeld are i i − i i − i Ω jµν = ∂µωνj ∂νωµj + [ωµ, ων] j = (Dµων Dνωµ) j . (2.101)

Applying exterior diﬀerentiation to the matrix form of the curvature (2.100),

dΩ = dω ∧ ω − ω ∧ dω = (Ω − ω ∧ ω) − ω ∧ (Ω − ω ∧ ω) = Ω ∧ ω − ω ∧ Ω, (2.102) we obtain the Bianchi identity

DΩ = dω + ω ∧ Ω − Ω ∧ ω = 0, (2.103) which plays an essential role in constructing characteristic classes of fibre bundles. If the vector bundle is a tangent or cotangent bundle, torsion also plays a role in the local geometry. However, since we consider only the case where the base manifold M is a Riemannian manifold, the Levi-Civita connection is used to define the parallel transportation and so the torsion tensor vanishes. The following shows the transformations of connection and curvature under the action of transition functions. Let g(x) ≡ ϕαβ(x) be transition functions on Uα ∩ Uβ. Then the transformations of the basis {ei(x)} on a fibre F at x ∈ Uα ∩ Uβ, and on the coordinate components {zi(x)}, are

′ −1 ′i j ei(x) = ei(x)gij (x), and z (x) = gij(x)z (x), (2.104) so that the section S(x) is invariant, ie.

i ′ ′i ′ S(x) = eiz = eiz (x) = S (x).

In (2.104), gij(x)(i, j = 1, ··· , k) are matrix elements of g. Requiring that ∇ (S(x)) is invariant under the basis change (2.104), and using (2.97) where,

∇S(x) = ∇′S′(x) ⊗ i j ⊗ i = ei(x) dz (x) + ej(x)ω i(x) z ′ ⊗ ′i ′ ′j ⊗ ′i = ei(x) dz (x) + ej(x)ω i(x) z , (2.105) one can ﬁnd the connections (2.98) and (2.100) transform as follows:

′i k −1 −1 ′i k −1 ω j = gikω lglj + gikdgkj , Ω j = gikΩ lglj . (2.106) In matrix form, the above transformations can be expressed as

ω = gωg−1 + gdg−1, and Ω = gΩg−1. (2.107)

Finally, we brieﬂy mention the parallel transportation of the section of a spin bundle based on a Riemannian manifold. The section ψ(x) of a spin bundle is a spinor function with 2[n/2] [n/2] components (see Appendix C): ψ(x) = (ψα), α = 1, ··· , 2 . The parallel transportation of Characteristic Classes 27

ψ(x) ω can only be done by a spin connection, which is a 1-form on the base manifold but takes the value in the Lie algebra of a spin group , ( ) α α α µ i µ i α µ ω β = (ωµ) β dx = ωµab [γa, γb] dx = ωµab (γaγb) β dx , 4 β 2 a a µ ··· ··· [n/2] γ = e µγ , a, µ = 1, , n, α, β = 1, , 2 . (2.108)

Note that i/4 [γa, γb] is the spinor representations for the generators of the spin group Spin(n). The covariant derivative on the section deﬁned by the spin connection is

α Dµψα(x) = ∂µψα(x) + ω µβψβ(x). (2.109) We use frequently the positive deﬁnite Dirac operator, ( ) ∂ i D = γaEµ D = γaEµ + ω [γ , γ ] , (2.110) a µ a ∂xµ 4 µab a b µ a ··· [n/2] × [n/2] where Ea is given in (2.20), and γ , a = 1, 2, , n are 2 2 γ-matrices introduced in Appendix C.

2.3 Characteristic Classes of Vector Bundles

In general, a characteristic class is a closed differential form on the base manifold of a fibre bundle. It is usually made up of a symmetric polynomial of the curvature which is invariant under the transformation (2.107). For a non-trivial fibre bundle, the characteristic class cannot be written as an exact form and so its integration over a compact base manifold with no boundary yields a non-zero topological number. Therefore, a characteristic class is a cohomology class of the base manifold. It provides a way of using topological invariants to distinguish between inequivalent fibre bundles [8] and to measure the deviation of a fibre bundle from the corresponding trivial bundle. Let P (Ω) be a symmetric polynomial of the matrix-valued curvature 2-form Ω satisfying the following invariance under the transformation (2.107),

P (g−1Ωg) = P (Ω) (2.111) for any g ∈ G, where G is the structure group of a real or complex vector bundle. Without a loss of generality, we can simply choose [17]  

  m Pm(Ω) = Tr Ω| ∧ Ω ∧{z · · · ∧ Ω} ≡ Tr (Ω ) (2.112) m to illustrate the features of a symmetric invariant polynomial. P (Ω) has the following properties:

1. Pm(Ω) is a closed 2m-form on M: ( ) [ ] m−1 m−1 d Pm(Ω) = mTr dΩΩ = mTr (dω + ω Ω − Ω ω)Ω = 0, (2.113)

where we have used (2.12), the Bianchi identity (2.103), and the following cyclic property for matrix-valued pi-diﬀerential forms, ( ) ··· ( ) ∧ · · · ∧ ∧ − p1+p2+ +pk ∧ ∧ · · · ∧ Tr αp1 αpk−1 αpk = ( 1) Tr αpk αp1 αpk−1 . 28 Atiyah-Singer Index Theorem

2. The integration of Pm(Ω) over the base manifold M2m is independent of the choice of the connection ω and is a topological invariant. This fact can be shown as follows [8, 17]:

Choose two arbitrary connections ω0 and ω1, and deﬁne the interpolation

ωt = ω0 + (ω1 − ω0) t, t ∈ [0, 1], Ωt = dωt + ωt ∧ ωt.

One can ﬁnd ( [ ]) d P (Ω ) = m d Tr (ω − ω ) ∧ Ωm−1 . dt m t 1 0 t Then ∫ (∫ ) 1 1 [ ] d − − ∧ m−1 dt Pm (Ωt) = Pm (Ω1) Pm (Ω0) = md dt Tr (ω1 ω0) Ωt . 0 dt 0 Hence by Stokes’ theorem, on a 2m-dimensional compact base manifold M with no boundary, we have ∫ ∫

Pm (Ω1) = Pm (Ω0) . (2.114) M M

2.3.1 Chern Class

1. Chern Form

The Chern classes are the characteristic classes defined on a k-dimensional complex vector bundle E with transition functions from GL(k, C). The curvature 2-form Ω takes its value from the Lie algebra of GL(k, C). The total Chern form is defined as ( ) i c(E) ≡ det I + Ω = 1 + c (Ω) + c (Ω) + ... (2.115) 2π 1 2 From this definition, each Chern form is a symmetric invariant polynomial in Ω and is composed of the basic polynomial Pm(Ω) defined in (2.112). For example, [ ( ) ] i 1 c = 1, c = TrΩ, c = Tr Ω2 − (TrΩ)2 , · ·· , 0 1 2π 2 8π2 where c(Ω) is always a finite sum since cj = 0 when 2j > n = dim M. From Eqs. (2.113) and th (2.114), each Chern form cj(Ω) defines the 2j cohomology class of M,

2j cj(Ω) ∈ H (M), (2.116) and the integration on a 2j-cycle γ2j in M with integer coefficients gives an integer, the Chern number, regardless of the connection used. From the definitions of dual vector bundle E∗ and the direct sum A ⊕ B of vector bundles A and B, one can easily find [8]

∗ C(E ) = 1 − c1(Ω) + c2(Ω) − · · · (2.117) and

C(A ⊕ B) = C(A) ∧ C(B). (2.118) Characteristic Classes 29

2. Chern Character

The Chern character is also an invariant polynomial deﬁned by [ ( )] ( ) i ∑ 1 i j ch(E) = Tr exp Ω = Tr α 2π j! 2π j [ ] 1 = k + c (Ω) + c (Ω)2 − 2c (Ω) + ··· . (2.119) 1 2 1 2 The Chern character is convenient for studying the direct sum and tensor product of vector bundles. For two vector bundles A and B based on the same manifold M, the following hold [8]:

ch(A ⊕ B) = ch(A) + ch(B), and ch(A ⊗ B) = ch(A)ch(B). (2.120)

3. Splitting Principle

The splitting principle is widely used to manipulate algebraic identities involving characteristic classes. It allows us to perform characteristic class operations as if every bundle is a direct sum of 1-dimensional line bundles. When a complex vector bundle E is split into a direct sum of k complex line bundles Lj, each having curvature Ωj, ie.

⊕k E = Lj (2.121) j=1 then i c(L ) = 1 + x , x = Ω , (2.122) j j j 2π j and the total Chern form is ( ) ∏k i ∏k c(E) = 1 + Ω = (1 + x ). (2.123) 2π j j j=1 j=1

In comparison with Eq. (2.115), it is equivalent to diagonalize the k × k matrix 2-form Ω into a direct sum of k curvature two-forms Ωj (j = 1, ··· , k) of complex line bundles,   Ω  1  ( ) ⊕k  Ω  i  2  Ω = Ωj =   . j  ..  j=1 . Ωk

Each Chern class takes an elegant form,

∑k ∑ c1(E) = xj, c2(E) = xixj, ··· , ck(E) = x1x2 ··· xk. (2.124) j=1 i

⊕k ∗ ∗ ∗ − For the dual bundle E , there exists E = Lj . Since c(Lj ) = 1 xj, so we obtain j=1

∏k ∗ C(E ) = (1 − xj) . (2.125) j=1

With the splitting principle, characteristic classes are conveniently deﬁned by their generating functions and∏ expressed as combinations of Chern classes. For example,∑ the total Chern class is x generated by (1 + xj) and the Chern character generated by e j . Another class is the Todd class which is generated by

∏k xj 1 1 2 ·· · td(E) = − = 1 + c1(E) + (c1 + c2)(E) + . (2.126) 1 − e xj 2 12 j=1

The Todd class has the following property: for two vector bundles A and B, there exists

td(A ⊕ B) = td(A)td(B). (2.127)

2.3.2 Pontrjagin Class

The Pontrjagin classes are defined for a k-dimensional real vector bundle E with transition functions from GL(k, R). As mentioned in Sect. 2.2, when a metric is introduced on the fibre, GL(k, R) reduces to the orthogonal group O(k). In fact, it can be proved that the characteristic classes corresponding to GL(k, R) and O(k) differ just by an exact form [8], so they lead to the same cohomology class. Therefore, we just take the structure group G = O(k), consequently the curvature 2-form satisfies Ω = −Ωt since Ω is a real 2-form taking value in the Lie algebra of O(k). The total Pontrjagin class is defined by the invariant polynomial ( ) 1 p(E) = det I − Ω = 1 + p (Ω) + p (Ω) + ··· 2π 1 2 [ ( ) ] 1 1 1 2 1 = 1 − TrΩ2 + TrΩ2 − TrΩ4 + ·· · , (2.128) 8π2 (2π)4 8 4 ( ) where we have used Tr Ω2p+1 = 0 for an antisymmetric matrix Ω. Clearly,

4j pj(Ω) ∈ H (M), (2.129) and when 4j > n or 2j > k, pj(Ω) = 0. In the following we derive the generating function of P (E). First, the Pointrjagin class can be expressed in terms of the Chern class by complexifying a real bundle E as Ec ≡ E ⊗C. Then comparing P (E) with C(Ec) where, ( ) i C(E ) = det I + Ω = 1 + c (Ω) + c (Ω) + ··· = 1 − p (Ω) + p (Ω) − · · · , (2.130) c 2π 1 2 1 2 we have

j pj(E) = (−1) c2j(Ec). (2.131) Characteristic Classes 31

Conversely, a k-dimensional complex vector bundle E can be considered as a 2k-dimensional complex vector bundle Er by leaving out the complex structure of E. Complexify Er as (Er)c = ⊗ Er C,(Er)c becomes a complex bundle of 2k complex dimensions. Let E denote the complex conjugate bundle of the original complex bundle E. Then E is isomorphic to E∗ [8], the dual bundle of E. Hence     ⊕k ⊕k ⊕ ⊕ ∗   ⊕  ∗ (Er)c = E E = E E = Lj Lj . (2.132) j=1 j=1

From Eqs. (2.123), (2.117), (2.118), (2.130) and (2.132), we have

C[(Er)c] = 1 − p1(Er) + p2(Er) − ... = C(E) ∧ C(E) ∏k ( ) ∧ ∗ − 2 = C(E) C(E ) = 1 xj . (2.133) j=1

Hence we obtain the following generating function of pj(Er),

∏k ( ) ··· 2 P (E) = P (Er) = 1 + p1(Er) + p2(Er) + = 1 + xj , j=1 ∑ ∑ 2 2 2 ··· 2 2 ··· 2 p1(E) = xi , p2(E) = xi xj , pk(E) = x1x2 xk. (2.134) i i

Two other characteristic classes can be deﬁned from the following generating functions, both of which are related the Pointrjagin class:

1. Hirzebruch L-polynomial: ∏ x L(E) = j (2.135) tanh x j j

2. Ab genus polynomial:

∏ x /2 Ab(E) = j (2.136) sinh(x /2) j j

2.3.3 Euler Class

The Euler class is deﬁned for an oriented, real vector bundle E of k = 2r dimensions. The transition functions of the vector bundle belong to SO(k), and the curvature 2-form Ω takes value in the Lie algebra of SO(k). Hence Ω is a k × k antisymmetric matrix in the ﬁbre space,

1 Ω = −Ω = Ω dxµ ∧ dxν. (2.137) ij ji 2 µνij The Euler class e(E) can be deﬁned as a k-form that satisﬁes

e(E) ∧ e(E) = pk/2(Ω). (2.138) 32 Atiyah-Singer Index Theorem

In this way, we obtain an SO(k)-invariant polynomial in terms of Ω. Clearly, because pk/2(M) ∈ 2k H (M,R), k = 2r, and d pk/2(Ω) = 2 [de(E)] ∧ e(E) = 0 we have,

e(E) ∈ Hk(M,R). (2.139)

The SO(k) invariance of e(E) can be observed in the following way [8]: use Ωij to formally construct a “two-form” in the fibre space, 1 α = Ω dzi ∧ dzj, (2.140) 2 ij with zi, i = 1, · ·· k = 2r being the coordinates of fibre space. Then define the r = k/2-fold wedge product to get e(E) ( ) 1 α r 1 1 = Ω ··· Ω dzi1 ∧ dzj1 ∧ · · · ∧ dzir ∧ dzjr r! 2π r! (4π)r i1j1 irjr 1 1 ··· = Ω ··· Ω ϵi1j1 irjr dz1 ∧ · · · ∧ dzk = e(E)dz1 ∧ · · · ∧ dzk. (2.141) r! (4π)r i1j1 irjr

For a k × k transition matrix g(x) ∈ SO(k), det g(x) = 1 so (2.141) shows that e(Ω) is explicitly SO(k) invariant. Using the splitting principle, we put Ω into a standard form,   0 x  1   .  t  −x 0 ..  Ω = −Ω −→  1  .  0 xr  −xr 0

Then from Eq. (2.140) and (2.141), we obtain

1 2 2r−1 2r α = x1dz ∧ dz + ··· + xrdz ∧ dz , and hence

k/∏2 1/2 e(E) = x1x2 ··· xr = xj = (pk) . (2.142) j=1

Note that the Euler class can only be calculated with the Riemannian connection since it is SO(k) invariant rather than GL(k, R) invariant. The Euler class e(M) for a diﬀerential manifold M represents the Euler class of the tangent bundle T (M). Also e(E)=0 for odd-dimensional bundles. Like the Pontrajagin class, the Euler class for a direct sum of vector bundles A and B satisﬁes e(A ⊕ B) = e(A) ∧ e(B).

2.3.4 Stiefel-Whitney Class

The Stiefel-Whitney (SW) classes for a real bundle E with a k-dimensional ﬁbre over an n- dimensional base manifold M is the Z2 cohomology classes of M:

i wi ∈ H (M; Z2), i = 1, ··· , n − 1, (2.143) Elliptic Complexes 33 and

n wn ∈ H (M; Z). (2.144) The total Stiefel-Whitney class is

w(E) = 1 + w1 + ··· + wn−1 + wn. (2.145) Note that the SW classes in general cannot be expressed as polynomials of the curvature. The Stiefel-Whitney classes are very useful for showing that how the topological property of the base manifold can determine the geometric structure. Some examples of the applications are [8] :

• w1[T (M)] = 0 if and only if M is orientable.

• Spinors are well-deﬁned on M and M is a spin-manifold if and only if w1[T (M)]=w2[T (M)]= 0.

• If w2[T (M)] ≠ 0, M does not admit a spin structure.

2.4 Classical Elliptic Complexes and Index Theorems

2.4.1 Elliptic Complexes and Index of Elliptic Diﬀerential Operator

1. Elliptic Diﬀerential Operator

Let us begin this section by letting M be a compact, smooth, n-dimensional manifold without boundary, with local coordinates {xj}. Let E and F be two vector bundles over M of dimension k and l, and let Γ(E) and Γ(F ) denote the sets of the sections of E and F . Then, in terms of the coordinates at each point x ∈ M we have, {( )} Γ(E) = χ1(x), ··· , χk(x) , and {( )} Γ(F ) = η1(x), ··· , ηl(x) . (2.146)

A diﬀerential operator D : Γ(E) → Γ(F ) can be expressed as the following form in terms of local coordinates, ( ) ( ) D χ1(x), ··· , χk(x) = η1(x), ··· , ηl(x) ,

∑ p1+···+pn i i j i ∂ j η (x) = D jχ (x) = a j,p1···pn (x) p1 pn χ (x), ∂x ·· · ∂xn j,p1,···pn 1 p1 + ··· + pn ≤ p, i = 1, ··· , k, j = 1, ··· , l. (2.147) By performing the Fourier transform of the section χ(x), ∫ i(k1x1+···+knxn) χ(x) = dk1 ··· dkne χ(k), we can deﬁne from Eq. (2.147) the symbol σ(D) of D, ∫ · ··· η = Dχ = dnk eik x ip1+ +pn σ(D)(x, k)χ(k), ∑ p1 ··· pn σ(D)(x, k) = (ap1···pn (x)) k1 kn . (2.148) p1+···+pn≤p 34 Atiyah-Singer Index Theorem

In σ(D)(x, k), the terms with the highest order in k form the leading symbol of D, denoted by De(x, k), ∑ e p1 ··· pn D(x, k) = ap1···pn (x) k1 kn . (2.149) p1+···+pn=p

D is called an elliptic operator if dim Γ(E) = dim Γ(F ) and D˜(x, k) is invertible for k ≠ 0, i.e., e det D(x, k) ≠ 0 for k1, ··· , kn ≠ 0.

2. Analytic Index of Elliptic Diﬀerential Operator and Elliptic Complex

The kernel, image and cokernel of D are deﬁned as follows:

KerD = {χ ∈ Γ(E): Dχ = 0}, (2.150) ImD = {η ∈ Γ(F ): η = Dχ, χ ∈ Γ(E)}, (2.151) CokerD = Γ(F ) − ImD ≡ Γ(F )/ImD. (2.152)

Then the analytic index of D is given by

Index(D) = dim Ker(D) − dim Coker(D). (2.153)

Further, because D : Γ(E) −→ Γ(F ) is a linear invertible operator from the vector space Γ(E) to the vector space Γ(F ) we have from elementary algebra [10]

dim Γ(E) = dim KerD + dim ImD, and dim Γ(F ) = dim CokerD + dim ImD. (2.154)

Hence from (2.153)

Index(D) = dim Γ(E) − dim Γ(F ). (2.155)

When there exist metric structures in both ﬁbre spaces of E and F , an adjoint operator D† : Γ(F ) −→ Γ(E) can be deﬁned with the inner product given by the metrics, ( ) (Dχ, η) = χ, D†η . (2.156)

Then two non-negative self-adjoint elliptic operators can be constructed as,

† ∆E = D D : Γ(E) −→ Γ(E), and † ∆F = DD : Γ(F ) −→ Γ(F ). (2.157)

There exists the following result: Theorem 2.4.1: Let

Γλ(E) ≡ {χ ∈ Γ(E)| ∆Eχ = λχ} , and

Γλ(F ) ≡ {η ∈ Γ(F )| ∆Fη = λη} . (2.158)

Then for λ ≠ 0, Γλ(E) is isomorphic to Γλ(F ), and further, there exist

† Γ0(E) = KerD, and Γ0(F ) = KerD . (2.159) Elliptic Complexes 35

† Proof: Let χλ ∈ Γλ(E), λ ≠ 0. Then by deﬁnition of Γλ(E), ∆Eχλ = D Dχλ = λχλ. Hence

† ∆F(Dχλ) = DD Dχλ = λ(Dχλ).

Thus Dχλ ≡ ηλ ∈ Γλ(F ) and moreover, Dχλ ≠ 0. Otherwise, there would arise ∆Eχλ = 0 and this contradicts with λ ≠ 0. Therefore, D :Γλ(E) −→ Γλ(F ) is a one-to-one correspondence † ∼ for λ ≠ 0. Similarly, D :Γλ(F ) −→ Γλ(E) is also a bijective map. Hence Γλ(E) = Γλ(F ), and dim Γλ(E) = dim Γλ(F ).

Theorem 2.4.1 implies

dim ImD = dim ImD†. (2.160)

Similar to dim Γ(E) in (2.154), there exists

dim Γ(F ) = dim KerD† + dim ImD† = dim KerD† + dim ImD = dim CokerD + dim ImD. (2.161)

Hence dim CokerD = dim D†. Therefore, from Eq. (2.153), vector bundles which have metric structures in their ﬁbre spaces have the following analytic index of the elliptic operator,

Index(D) = dim Ker(D) − dim KerD†. (2.162)

3. Elliptic Complex (The Sequence Formulation)

Let {Ep} be a finite sequence of vector bundles over M. Also, let {Dp} be the sequence Dp : Γ(Ep) → Γ(Ep+1) of differential operators as defined above. The pair (E,D) = ({Ep}, {Dp}) is † → a complex if Dp+1Dp = 0. Let Dp : Γ(Ep+1) Γ(Ep) be the adjoint operator of D. Define the corresponding Laplacian as

△ † † p = DpDp + Dp−1Dp−1. (2.163)

(E,D) is an elliptic complex if ∆p is an elliptic operator on Γ(Ep). Then the cohomology groups for (E,D) are given by

p KerDp ∼ H (E,D) = = Ker∆p. (2.164) ImDp−1

The analytic index of (E,D) is ∑ ∑ p p p Index(E,D) = (−1) dim H (E,D) = (−1) dim Ker∆p. (2.165) p p

4. Two-grade formulation of Elliptic Complex

An elliptic complex has another equivalent two-grade formulation. Deﬁne the even and odd bundles as ⊕ ⊕ e o E = E2p, and E = E2p+1. (2.166) p p 36 Atiyah-Singer Index Theorem

Then combine the operators as ⊕ ( ) ⊕ ( ) O † O∗ † = D2p + D2p−1 , and = D2p + D2p+1 . (2.167) p p

Clearly, there exists mappings O : Γ(Ee) −→ Γ(Eo) and O : Γ(Eo) −→ Γ(Ee). The associated Laplacian operators are ⊕ ⊕ ∗ ∗ ∆e = O O = ∆2q, and ∆o = OO = ∆2q+1. (2.168) q q

Consequently, the complex is (E, O) ≡ ({Ee,Eo}, {O, O∗}), and the index of the complex is the same as that described in the sequence form ∑ p Index (E, O) = dim Ker∆e − dim Ker∆o = (−1) dim Ker∆p = Index(E,D). (2.169) p

5. Atiyah-Singer Index Theorem for Elliptic Complex

Now we are able to deﬁne the general form of the Atiyah-Singer index theorem for elliptic complexes. Let ch(Ep) be the Chern character of Ep, td[T (M)] be the Todd class of the tangent bundle, and e(M) be the Euler class of T (M). Then the Atiyah-Singer index theorem [8] states ∫ ( ) ⊕ ⊗ n(n+1)/2 p td[T (M) C] Index(E,D) = (−1) ch (−1) Ep . (2.170) M p e(M)

This general form is applied to the four classical elliptic complexes in the subsequent subsections.

2.4.2 de Rham Complex and Gauss-Bonnet Theorem

The de Rham complex follows from decomposing the exterior algebra Λ∗(M) into the following two-grade (even and odd) forms:

Λeven = Λ0 ⊕ Λ2 ⊕ ..., Λodd = Λ1 ⊕ Λ3 ⊕ ... . (2.171)

Denote the de Rahm complex as (Λeven/odd, d + δ) with operator d + δ such that

d + δ : Γ(Λeven) → Γ(Λodd). (2.172)

p This gives the de Rham cohomology group H (M,R) = Kerdp/Imdp−1 with the analytic index of (Λeven/odd, d + δ) to be the Euler characteristic

∑n Index(Λeven/odd, d + δ) = (−1)p dim Hp(M,R) = χ(M). (2.173) p=0

Next apply the general form (2.170) to the de Rham complex. In the de Rahm complex, ∧ p ∗ p ∗ ∧ · · · ∧ ∗ Ep = Λ = [T (M)] = |T (M) {z T (M}) . (2.174) p Elliptic Complexes 37

By the splitting principle,

⊕n ∗ ∗ T (M) = Lj . (2.175) j=1

Hence from (2.174),   p ∧ ∧ ⊕n ⊕ ( ) p ∗ p  ∗ ⊗ · · · ⊗ Λ = [T (M)] = Lj = Lj1 Ljp . (2.176) j=1 1≤j1

Using the property of the Chern character for the direct sum and tensor product of vector −x bundles given in (2.120) and the result for a dual line bundle, ch(Lj) = e j , we obtain   ⊕ ( ) p  ⊗ · · · ⊗  ch (Λ ) = ch Lj1 Ljp 1≤j

1≤j1

The result (2.177) leads to ( ) ⊕ ∑n ∑ p p p − x +x +···+x ch (−1) Λ = (−1) e ( j1 j2 jp )

p p=0 1≤j1

Further, from Eq. (2.126), (2.127) and (2.142), [ ] td[T (M) ⊗ C] = td T (M) ⊕ T (M) = td [T (M) ⊕ T ∗(M)] n/∏2 [ ] ∗ xj −xj = td [T (M)] td [T (M)] = − , and 1 − e xj 1 − exj j=1 n/∏2 e(M) = xj. (2.179) j=1

Substituting (2.178) and (2.179) into (2.170), we obtain the Gauss-Bonnet theorem, ∫ ( ) ⊕ td[T (M) ⊗ C] Index(Λeven/odd, d + δ) = χ(M) = (−1)n(n+1)/2 ch (−1)pΛp M p e(M) ∫ n/∏2 ∫ n(n+1)/2 n/2 = (−1) (−1) xj = e(M), (2.180) M j=1 M where n is an even number. 38 Atiyah-Singer Index Theorem

2.4.3 Signature Complex and Hirzebruch Signature Theorem

The signature complex is based on a diﬀerent splitting of the exterior algebra Λ∗(M) on an n = 4k-dimensional Riemannian manifold. Let ω be an operator action on p-forms deﬁned by

ω = ip(p−1)+n/2 ∗ . (2.181)

It can easily be proven that

ωd = −δω, ωδ = −dω, and ω2 = 1. (2.182)

Hence there arises

ω(d + δ) = −(d + δ)ω. (2.183)

When n = 4k, ω = ∗ on Λ2k. Letting Λ± be the ±1 eigenspaces of ω, we have

p+ p p− p Λ = {αp ∈ Λ : ωαp = αp} , Λ = {αp ∈ Λ : ωαp = −αp} , Λ+ = ⊕Λp+, Λ− = ⊕Λp−, Λ = Λ+ ⊕ Λ−. (2.184)

From Eq. (2.183), there exists

d + δ :Λ+ −→ Λ−. (2.185)

(Λ±, d + δ) is called the signature complex. The analytic index of the signature complex is ± 2k − 2k Index(Λ , d + δ) = dim H+ (M, R) dim H− (M, R) + − − = b2k b2k = τ(M), (2.186) which is contributed to only by the harmonic p-forms with p = n/2 = 2k. We apply the general form (2.170) of the Atiyah-Singer index theorem to the signature + − complex, for which {Ep} = {Λ , Λ }. It can be found in a similar way as deriving (2.178) that ( ) ⊕ ( ) ( ) n/∏2 ( ) + − −xj xj ch Ep = ch Λ − ch Λ = e − e . (2.187) p j=1

Substituting (2.178) and (2.187) into (2.170), we obtain we obtain the Hirzebruch signature theorem ∫ ( [ ] [ ]) td[T (M) ⊗ C] τ(M) = (−1)n(n+1)/2 ch Λ+ − ch Λ− M e(M) [ ] ∫ n/2 − ∏ xj − xj − n(n+1)/2 e e xj xj = (−1) − x 1 − e xj 1 − exj M j=1 j ∫ n/∏2 x −x n/∏2 xj (e j − e j ) xj cosh(xj/2) = − = xj /2 − xj /2 2 sinh(x /2) M j=1 (e e ) j=1 j ∫ n/2 ∫ ∏ x /2 = 2n/2 j = L(M), (2.188) tanh(x /2) M j=1 j M where L(M) is the Hirzebruch L-polynomial given in Eq. (2.135) (up to a constant factor). If n ≠ 4k, then τ(M) = 0. Elliptic Complexes 39

2.4.4 Dolbeault Complex and Riemman-Roch Theorem

Let M be a complex manifold with real dimension n = 2k. From Subsection 2.1.5, the exterior algebra splits into the sets Λp,q and there exists the antiholomorphic exterior diﬀerential operator, ( ) ∂ ∂ = dzα : Γ (Λp,q) −→ Γ Λp,q+1 . (2.189) ∂zα ( ) The Dolbeault complex is {Λ0,q}, ∂ , and the analytic index is

∑n/2 Index(∂) = (−1)q dim H0,q(M). (2.190) q=0

0,q For the Dolbeault complex, Ep is played by Λ . In a similar way as deriving (2.178), we can ﬁnd (⊕ ) n/∏2 ( − ) ch (−1)qΛ0,q = 1 − e xj . (2.191) j=1 Substituting (2.178) and (2.191) into (2.170), we obtain the Riemann-Roch theorem: ∫ ( ) ⊕ td[T (M) ⊗ C] Index(∂) = (−1)n(n+1)/2 ch (−1)qΛ0,q M e(M) [ ] ∫ n/2 − ∏ − xj − n(n+1)/2 1 e xj xj = (−1) − x 1 − e xj 1 − exj M j=1 j ∫ n/∏2 ∫ xj = − = td[Tc(M)], (2.192) 1 − e xj M j=1 M where Tc(M) is the complex tangent space deﬁned in (2.72).

2.4.5 Spin Complex and Index Theorem for A-roof Genus

The spin complex is deﬁned as follows: First, as mentioned in Sect. 2.2 and shown in Appendix C, the sections of the spin bundle are spinor functions ψ(x), and the set of sections can decompose into two subspaces composed of chiral spinor functions,

S = S+ ⊕ S−, + S = {ψ+(x) ∈ S : γn+1ψ+(x) = ψ+(x)} , − S = {ψ−(x) ∈ S : γn+1ψ−(x) = −ψ−(x)} . (2.193)

The operator acting on the sections of spin bundles is the Dirac operator deﬁned in (2.110),

a µ D = γ Ea Dµ. (2.194)

Because γn+1γa = −γaγn+1, there arises

γn+1D = −Dγn+1, (2.195) D : Γ(S+) −→ Γ(S−), and D† : Γ(S−) −→ Γ(S+). (2.196) 40 Atiyah-Singer Index Theorem

(D,S±) is the spin complex and the analytic index of the spin complex is

± † Index(S ,D) = dim KerD − dim KerD = n+ − n−, (2.197) where n± are the numbers of spinor functions with chirality ±1. These are eigenfunctions of the Dirac operator corresponding to the zero eigenvalue, respectively. We apply the general form of the Atiyah-Singer index theorem with dimM = n = 4k on the ± spin complex, and {Ep} = {S }. Using the results

n/∏2 ( ) ( ) ( −) − ch S+ − ch S = exj /2 − e xj /2 , (2.198) j=1 with the same td[T (M)] and e(M) as shown in Eqs. (2.187) and (2.191), we obtain the index theorem for Ab genus, ∫ [ ( ) ( )] td[T (M) ⊗ C] Index(S±,D) = (−1)n(n+1)/2 ch S+ − ch S− M e(M) ∫ ( ) n/∏2 x /2 −x /2 e j − e j xj −xj = − x 1 − e xj 1 − exj M j=1 j ( ) ∫ n/2 ∫ ∏ x /2 = j = Aˆ(M), (2.199) sinh(x /2) M j=1 j M where Aˆ(M) is the A-roof genus of tangent bundle on M given by

n/2 ∏ x /2 1 1 Aˆ(M) = i = 1 − p [T (M)] + (7p2 − 4p )[T (M)] + ·· · . sinh(x /2) 24 1 5760 1 2 i=1 i Chapter 3

Some Aspects of Supersymmetric Quantum Mechanics

Classical Mechanics describes the motion of a physical body. The most fundamental physical principles in classical mechanics are Newton’s three laws. With the invention of calculus of variations in mathematics, Newtonian mechanics has been developed into analytical mechanics: namely Lagrangian and Hamiltonian formulations. These two formulations have not only re- vealed the elegant algebraic and geometric structures of classical mechanical systems, but have also provided two equivalent routes to construct quantum theory from classical mechanics.

3.1 Lagrangian Formulation and Hamiltonian Formulation of Classical Mechanics

3.1.1 Lagrangian Mechanics

The fundamental principle underlying Lagrangian mechanics is the least action principle, while the mathematical foundation is the calculus of variation. Lagrangian mechanics is deﬁned using a generalized position coordinate system q ≡ {qa} = {q1, q2, ··· qN } of a mechanical system. The Lagrangian function L[q(t), q(˙t)] is a function of the generalized position coordinates q and their derivativesq ˙ = dq/dt, the generalized velocities. The classical action is deﬁned as ∫ t2 S[q] = dt L[q(t), q˙(t)]. (3.1) t1

Geometrically, as t changes, q(t) describes a trajectory (or a path) in a conﬁguration space {qa}. The least action principle states that among all the possible trajectories joining q1 at time t1 to q2 at time t2, the actual physical one should be the trajectory that make S[q(t)] minimal, i.e.,

δS = 0. (3.2) δqa(t)

41 42 Supersymmetric Quantum Mechanics

From ∫ [ ] t2 ∂L ∂L δS = dt δq + δq˙ ∂q a ∂q˙ a ∫t1 { a( )a [ ( )] } t2 d ∂L ∂L d ∂L = dt δq + − δq dt ∂q˙ ∂q dt ∂q˙ ∫t1 [ a ( )] a a t2 ∂L d ∂L = dt − δqa, (3.3) t1 ∂qa dt ∂q˙a we obtain the Euler-Lagrange equations describing the motion of the mechanical system,

δS[q] d ∂L ∂L = − = 0. (3.4) δq(t) dt ∂q˙a ∂qa

In Eq. (3.3) we have used

d δq˙ (t) = δq (t), and δq(t ) = δq(t ) = 0. (3.5) a dt a 1 2

For example, consider a particle of mass m moving in R3 governed by a potential ﬁeld V (x, y, z). The generalized position coordinates are q = (x, y, z). The corresponding Lagrangian function is 1 L(x, y, z, x,˙ y,˙ z˙) = T − V = m(x ˙ 2 +y ˙2 + z˙2) − V (x, y, z). 2 The Euler-Lagrangian equation (3.4) gives

∂V ∂V ∂V mx¨ + = 0, my¨ + = 0, and mz¨ + = 0. ∂x ∂y ∂z

Written in vector notation, this represents ma = −∇V = F, which is exactly the equation of motion determined by Newton’s second law.

3.1.2 Hamiltonian Mechanics and Poisson Bracket Operation

In Hamiltonian mechanics, we use a Hamiltonian to describe a classical mechanical system. To construct the Hamiltonian, we first find canonical conjugate momenta pa conjugating the generalized position coordinates qa, defined by ∂L[q, q˙] pa = . (3.6) ∂q˙a

In physics, the 2N variables (q, p) = (qa, pa) form a set that is called a phase space.

A Hamiltonian H[q(t), p(t)] is a function of qa and pa obtained from the Lagrangian function through a Legendre transformation

H(qa, pa) = paq˙a(p, q) − L[q, q˙a(p, q)]. (3.7)

Then the classical action (3.1) takes the form ∫ ∫ t2 t2 S[q, p] = dt L[q, q˙] = dt [padqa − H(p, q)] dt. (3.8) t1 t1 Quantization 43

The least action principle gives ∫ [ ( ) ( )] t2 ∂H dδq ∂H δS[q, p] = dt δp q˙ − + p a − δq dt a a ∂p a dt ∂q a ∫t1 [ ( a ) ( ) a ] t2 ∂H ∂H = dt δpa q˙a − + −p˙a − δqa dt = 0, (3.9) t1 ∂pa ∂qa and hence we obtain a set of Hamilton’s equations, ∂H ∂H q˙a = , andp ˙a = − . (3.10) ∂pa ∂qa

For the above example considered in the Lagrangian formulation, the conjugate momenta are ∂L ∂L ∂L p = = mx,˙ p = = my,˙ and p = = mz.˙ x ∂x˙ y ∂y˙ z ∂z˙ The corresponding Hamiltonian is 1 H = p x˙ + p y˙ + p z˙ − L = (p2 + p2 + p2) + V (x, y, z). x y z 2m x y z Then from Hamilton’s equations we have ∂H p ∂H p ∂H p x˙ = = x , y˙ = = y , z˙ = = z , ∂px m ∂py m ∂pz m and ∂H ∂V ∂H ∂V ∂H ∂V p˙ = − = − , p˙ = − = − , p˙ = − = − . x ∂x ∂x y ∂y ∂y z ∂z ∂z It is clear that these equations represent the classical equations ma = F = −∇V . In the following we introduce an important algebraic operation in phase space, the Poisson bracket. Let us consider a function f(q, p) in phase space. Then

df ∂f ∂f ∂H ∂f − ∂H ∂f = q˙a + p˙a = = [H, f]PB (3.11) dt ∂qa ∂pa ∂pa ∂qa ∂qa ∂pa where the operation of the Poisson bracket on two functions f(qa, pa) and g(qa, pa) on phase space is deﬁned by ∂f ∂g ∂f ∂g [f, g]PB = − . (3.12) ∂qa ∂pa ∂pa ∂qa It is easy to see that there exist

[qa, pb]PB = δab, and [qa, qb]PB = [pa, pb]PB = 0. (3.13)

3.2 Construction of Quantum Mechanics from Classical Me- chanics

Quantum mechanics can be constructed from classical mechanics by a procedure called quantization. There are two equivalent quantization approaches which are called canonical quantization and path integral quantization. The former is based on the Hamiltonian formulation of classical mechanics, and the latter on the Lagrangian formulation. 44 Supersymmetric Quantum Mechanics

3.2.1 Canonical Quantization

Under canonical quantization, the motion state of a mechanical system is represented by a vector |ψ⟩ in a Hilbert space. The coordinate q, momentum p, and hence any function deﬁned in phase space are changed into the operators,

q → q,b p → p,b f(q, p) → fb(q,b pb), (3.14) in the Hilbert space. The Poisson bracket (3.12) is replaced by the Lie bracket between two operators, b b b [f, g]PB −→ [f, gb] ≡ fgb − gbf. (3.15)

Hence, the fundamental Poisson bracket operations listed in (3.13) become

[qba, pbb] = iδab, and [qba, qbb] = [pba, pbb] = 0. (3.16)

The change of a physical state is governed by the Schro¨dinger equation ∂ i |Ψ(t)⟩ = Hb (q,b pb) |Ψ(t)⟩ . (3.17) ∂t In physics, we usually consider the conservative mechanical system. In this case,

|Ψ(t)⟩ = e−iEt|ψ⟩, (3.18) and |ψ⟩ is the eigenvector of Hb with eigenvalue E,

Hb |ψ⟩ = E|ψ⟩. (3.19)

The eigenvalue equation (3.19) is also called the time-independent Schr¨odingerequation. Eq. (3.16) represents formal operations between abstract operators. To do explicit calcula- tions, we must choose concrete representations for abstract operators and state vectors. Only two representations need to be introduced for later use in this paper: coordinate representation and particle number representation.

1. Coordinate representation Coordinate representation is deﬁned by choosing the eigenvectors {|q⟩} of the coordinate operator qb as the basis of the Hilbert space. Then every vector in the Hilbert space can be expanded in terms of {|q⟩}. As a basis of the Hilbert space, {|q⟩} must be complete, ie. ∫ dq|q⟩⟨q| = 1, (3.20)

where 1 denotes the unit matrix of inﬁnite rank. Then every vector |Ψ(t)⟩ admits the expansion ∫ ∫ |Ψ(t)⟩ = dq|q⟩⟨q|Ψ(t)⟩ = dq|q⟩Ψ(q, t), (3.21)

where the coordinate component

Ψ(q, t)≡⟨q|Ψ(t)⟩ (3.22) Quantization 45

is usually called a wave function. In physics, |Ψ(q, t)|2 = Ψ(q, t)Ψ(q, t) is interpreted as the probability density function that a particle appears in the position q at time t. As a consequence, in coordinate representation,

qb = q. (3.23)

Then the operator relations (3.16) determine that pb should be a partial derivative operator, ∂ pb = −i . (3.24) ∂q By deﬁnition, the Hilbert space H is composed of square-integrable complex-valued functions ψ(q), { ∫ ∞ } H = (ψ(q)) 0 ≤ dq |ψ(q)|2 < ∞ . (3.25) −∞ ∫ When |ψ(q)⟩ is normalized, dq|ψ(q)|2 = 1. For example, the expansion of |q⟩ is ∫ ∫ |q⟩ = dq′|q′⟩⟨q′|q⟩ = dq′ψ(q′, q)|q′⟩, (3.26)

and the wave function corresponding to |q⟩ is the Dirac delta function [11]

ψ(q′) = ⟨q′|q⟩ = δ(q − q′), (3.27)

As for the momentum operator pb, if |p⟩ is an eigenvector of the momentum operator pb with eigenvalue p, then it admits the expansion in the basis {|q⟩} [11], ∫ ∫ |p⟩ = dq|q⟩⟨q|p⟩ = dq|q⟩ψ(q, p), 1 ψ(q, p) ≡ ⟨q|p⟩ = exp (ipq) , and (2π)1/2 ∫ ∂ ∂ pb|p⟩ = −i |p⟩ = −i dq ψ(q, p)|q⟩ = p|p⟩. (3.28) ∂q ∂q The Schr¨odingerequation becomes a partial diﬀerential equation, ∂ i Ψ(q, t) = H(q, −i∂/∂q)Ψ(q, t). (3.29) ∂t The solution to the above equation takes the form,

Ψ(q, t) = e−iEtψ(q), and Hψ(q) = Eψ(q). (3.30)

All the solutions of (3.30) form the Hilbert space. In fact, from Eqs. (3.20), (3.27) and (3.28), one can derive ∫ dp |p⟩⟨p| = 1, (3.31)

so the set of eigenvectors {|p⟩} is also complete. We can also choose {|p⟩} as the basis of the Hilbert space, which in physics is usually called the momentum representation. Further, Eqs. (3.27), (3.28) and (3.31) yield the identity, ∫ ∫ 1 [ ] δ(q − q′) = ⟨q′|q⟩ = dp⟨q′|p⟩⟨p|q⟩ = dp exp ip(q′ − q) . (3.32) 2π 46 Supersymmetric Quantum Mechanics

2. Particle number representation The particle number representation is defined as follows: First, define operators a and the adjoint a† as 1 a = √ (qb + ipb), and (3.33) 2 1 a† = √ (qb − ipb). (3.34) 2 It is easy to see [a, a†] = 1. (3.35) The Hamiltonian operator can be expressed as Hb (q,b pb) = Hb (a, a†). (3.36) Second, find a normalized eigenvector of Hb such that Hb has a minimal eigenvalue, which is usually denoted as |0⟩. Then the Hilbert space H can be constructed as { } 1 H = span |n⟩| |n⟩ ≡ √ (a†)n|0⟩ . (3.37) n!

The following uses the theory of the one-dimensional harmonic oscillator to illustrate the representation theory and construction of the corresponding Hilbert space [11]. The Hamiltonian is 1 Hˆ = (pb2 + qb2). (3.38) 2 1. Coordinate representation In coordinate representation, from Eqs. (3.23) and (3.24), the Hamiltonian (3.38) is a second-order differential operator, 1 d2 1 Hb = − + q2, (3.39) 2 dq2 2 and the time-independent Schrödingerequation (3.19) is a second-order ordinary linear differential equation, ( ) 1 d2 1 − + q2 ψ(q) = Eψ(q). (3.40) 2 dq2 2 Eq. (3.40) can be converted into the Hermite equation corresponding to different discrete eigenvalues ( ) 1 E = n + , n = 0, 1, 2, ··· , (3.41) n 2 with solutions [11] ( ) 1 1 ψ (q) = √ exp − q2 H (q), (3.42) n (2nn! π)1/2 2 n

where Hn(q) is the Hermite polynomial [20], ( ) 2 d 2 H (q) = (−1)neq e−q . (3.43) n dq Quantization 47

2. Particle number representation In particle representation, the Hamiltonian operator becomes 1 1 Hb = (aa† + a†a) = N + (3.44) 2 2 where

N = a†a. (3.45)

Now the eigenvalue problem of H reduces to constructing the eigenvector of N. Let |ν⟩ be an eigenvector of N with corresponding eigenvalue ν,

N|ν⟩ = ν|ν⟩. (3.46)

Notice that from (3.35) and (3.45), [ ] [N, a] = −a, and N, a† = a†, (3.47)

so there exist

N (a|ν⟩) = a(N − 1)|ν⟩ = (ν − 1)a|ν⟩, ( ) N a†|ν⟩ = a†(N + 1)|ν⟩ = (ν + 1)a†|ν⟩. (3.48)

This means that a|ν⟩ is an eigenvector of N with eigenvalue ν−1 and a†|ν⟩ is an eigenvector of N with eigenvalue ν + 1. Using mathematical induction proves that for n ∈ Z+,

N (an|ν⟩) = (ν − n)an|ν⟩, ( ) N a†n|ν⟩ = (ν + n)a†n|ν⟩. (3.49)

Hence we have obtained the set of eigenvectors of N

a|ν⟩, a2|ν⟩, ··· , ap|ν⟩, ··· ,

corresponding to the eigenvalues ν−1, ν−2, ···, ν−p, ···. Similarly, the set of eigenvectors

a†|ν⟩, a†2|ν⟩, ··· , a†p|ν⟩, ··· ,

corresponds to eigenvalues ν + 1, ν + 2, ···, ν + p, ···. Because N is a self-adjoint (or Hermitian) operator, N = N †, all its eigenvalues must be non-negative. This means the minimal eigenvalue of N should be zero, and the decreasing sequence of eigenvectors should stop at a certain step. Therefore, ν = n and

a|0⟩ = 0. (3.50)

This means the collection of all eigenvalues of N is equal to the set of non-negative integers,

N|n⟩ = n|⟩, n ∈ Z, n ≥ 0. (3.51)

Requiring that all the eigenvectors are normal,

⟨n|n⟩ = 1, n = 0, 1, 2, ··· , (3.52) 48 Supersymmetric Quantum Mechanics

one can prove

a†|n⟩ = (n + 1)1/2|n + 1⟩, a|n⟩ = n1/2|n − 1⟩, 1 |n⟩ = a†n|0⟩, ⟨m|n⟩ = δ . (3.53) (n!)1/2 mn

Hence we have obtained the Hilbert space in the particle number representation, { } 1 H = span |n⟩| |n⟩ = a†n|0⟩, a|0⟩ = 0, ⟨0|0⟩ = 1, n ∈ Z, n ≥ 0 . (3.54) (n!)1/2

In particle number representation, all the operators have inﬁnite dimensional matrix representations realized on |n⟩: ( ) 1 Hb = (H ) ,H = n + δ , mn mn 2 m,n † † a = (amn), a = (amn), 1/2 † 1/2 amn = n δm,n−1, amn = (n + 1) δm,n+1, 1 † qb = (qmn) , qmn = √ (a + amn), 2 mn i † pb = (pmn) , qmn = √ (a − amn). (3.55) 2 mn

Eqs. (3.49), (3.51) and (3.53) show why this representation is called particle number representation. In physics, a is called a destruction operator, while a† is called a creation operator.

This example shows that different choices of the representation can make the theory super- ficially different, but physical results like eigenvalues are identical.

3.2.2 Path Integral Quantization

The way of using path integral quantization to construct quantum theory is based on the La- grangian formulation of classical theory. In classical physics, when a particle moves from the initial position q1 at time t1, to a final position q2 at time t2, the Euler-Lagrangian equation (or equivalently the least action principle) only chooses one trajectory. In quantum theory, the particle can take any path to evolute from an initial state |Ψ(t1)⟩ at time t1 to a final state |Ψ(t2)⟩ at time t2. Each path has equal probability. Thus, the inner product ⟨Ψ(t2)|Ψ(t1)⟩, which in physics is called the transition probability amplitude for a quantum system from initial state |Ψ(t1)⟩ to evolve into the state |Ψ(t1)⟩, should be given by summing up all the possible paths. In semi-classical approximation, we can think of the path determined by the Euler-Lagrangian equation as the most probable one, while other paths are quantum fluctuations around classical trajectory.

1. Minkowskian Path Integrals in Quantum Mechanics Quantization 49

Starting with the Schr¨odingerequation (3.17), because Hb is time-independent it has the formal solution, [ ∫ ] [ ] t b b |Ψ(t)⟩ = exp −i dtH |Ψ(t0)⟩ = exp −iH(t − t0) |Ψ(t0)⟩ ≡ U(t, t0)|Ψ(t0)⟩. (3.56) t0

In physics, U(t, t0) is called the evolution operator. Once U(t, t0) is evaluated, all physical states Ψ(t) at each time t can be determined. In coordinate representation, the state vector |Ψ(t)⟩ is represented by the wave function Ψ(q, t) deﬁned by ∫ ∫

Ψ(q, t) = ⟨q|Ψ(t)⟩ = dq0⟨q|U(t, t0)|q0⟩⟨q0|Ψ(t0)⟩ = dq0⟨q|U(t, t0)|q0⟩Ψ(q0, t0). (3.57)

The main task in path integral quantization is to calculate [21] b −iH(t−t0) ⟨q|U(t, t0)|q0⟩ = ⟨q|e |q0⟩. (3.58)

The calculation procedure is as follows: First, divide the time-interval [t, t0] into n small intervals, t − t ∆t = 0 , n and deﬁne

tk = t0 + k∆t, k = 0, 1, ··· , n, tn ≡ t.

Then we have [ ] { } b b exp −iH(t − t0) = exp −iH [(tn − tn−1) + ··· + (t1 − t0)] ∏n [ ] b = exp −iH(tk − tk−1) k=1 and

− b − ⟨ | iH(t t0)| ⟩ q e ∫ q0 b b = lim dqn−1 ··· dq1⟨q|qn−1⟩⟨qn−1| exp[−iH∆t]|qn−2⟩···⟨q1| exp[−iH∆t]|q0⟩ n→∞ ∫ n∏−1 ∏n [ ] b = dqj ⟨qk| exp −iH∆t |qk−1⟩, (3.59) j=1 k=1 where qn ≡ q. In Eq. (3.59) we have used (3.20), the completeness of the basis {|q⟩} of the Hilbert space. Because ∆t is small, we consider the approximation to the ﬁrst order of ∆t, [ ] b b 2 ⟨qk| exp −iH∆t |qk−1⟩ = ⟨qk|qk−1⟩ − i∆t⟨qk|H|qk−1⟩ + O(∆t ) b 2 = δ(qk − qk−1) − i∆t⟨qk|H|qk−1⟩ + O(∆t ), (3.60) where we have used Eq. (3.27). For later use, consider only the Hamiltonian of the following form for commutative variables, 1 Hb = pb2 + V (qb). (3.61) 2 50 Supersymmetric Quantum Mechanics

Hence we have

b 1 2 ⟨qk|H|qk−1⟩ = ⟨qk|pb |qk−1⟩ + ⟨qk|V (qb)|qk−1⟩ ( 2 ) ( ) 2 2 −1 ∂ ⟨ | ⟩ −1 ∂ − = 2 + V (qk) qk qk−1 = 2 + V (qk) δ (qk qk−1) . (3.62) 2 ∂qk 2 ∂qk

Substituting (3.62) into (3.60) and making use of (3.32), we obtain [ ( )] [ ] 2 b 1 ∂ ⟨q | exp −iH∆t |q − ⟩ = 1 − i∆t − + V (q ) ⟨q |q − ⟩ k k 1 2 ∂q2 k k k 1 ∫ { [ k ]} dpk − − 1 2 = exp [ipk(qk qk−1)] 1 i∆t pk + V (qk) ∫ 2π 2 dpk = exp [ipk(qk − qk−1)] [1 − i∆tH(pk, qk)] ∫ 2π dpk ≈ exp [ipk(qk − qk−1) − i∆tH(pk, qk)] ∫ 2π { [ ]} dpk 1 2 = exp −i ∆tp − p (q − q − ) + ∆tV (q ) 2π 2 k k k k 1 k { [ ] [( ) ]} ∫ 2 2 dp ∆t q − q − q − q − = k exp −i p − k k 1 + i∆t k k 1 − V (q ) 2π 2 k ∆t ∆t k { [ ( ) ]} 2 1 1 qk − qk−1 = √ exp i∆t − V (qk) . (3.63) 2πi∆t 2 ∆t

In deriving Eq. (3.63), we have used the Gaussian integration formula (3.76).

Substituting (3.63) into (3.59), we obtain

b −iH(t−t0) ⟨q|e |q0⟩ { ( [ ( ) ])} ∫ n∏−1 ∑n 2 1 dqk 1 qk − qk−1 = lim √ √ exp i ∆t − V (qk) n→∞ 2 ∆t 2πi∆t k=1 2πi∆t k=1 ∫ [ ∫ ( )] q(t)=q t 1 = [dq] exp i dt q˙2 − V (q) q(t )=q t 2 ∫ 0 0 [ ∫ 0 ] ∫ q(t)=q t q(t)=q = [dq] exp i dtL(q, q˙) = [dq]eiS, (3.64) q(t0)=q0 t0 q(t0)=q0 where

− 1 n∏1 dq [dq] ≡ lim √ √ k (3.65) n→∞ 2πi∆t k=1 2πi∆t is called the path integral measure.

In a similar way, for a mechanical system described by the Grassman variable χ with the following Lagrangian [21],

i L = iχ∗χ˙ − V (χ∗, χ) = (χ∗χ˙ − χ˙ ∗χ) − V (χ∗, χ) , (3.66) 2 Quantization 51 we can derive the path integral formula, ∫ [ ∫ ] ∫ χ∗(t)=χ∗ χ∗(t)=χ∗ ∗ − b − ∗ ∗ ⟨χ |e iH(t t0)|χ⟩ = [dχ dχ] exp i dtL = [dχ dχ] exp [iS] , (3.67) χ(t0)=χ0 χ(t0)=χ0 where the path integral measure for Grassmann variables is deﬁned by

n∏−1 ∗ ∗ ∗ ∗ [dχ dχ] = lim dχkdχ , χk = χ(tk), χ = χ (tk), n→∞ k k k=1 t − t t = t + k∆t, ∆t = 0 , k = 0, 1, ··· , n, (3.68) k 0 n and especially, H = V (χ, χ∗).

2. Euclidean Path Integrals in Quantum Mechanics

Later in Chapter 4 we shall evaluate the trace (3.124) in supersymmetric quantum mechanics, [ ] Tr (−1)F exp(−βHb ) where β is a positive number, Hb is the Hamiltonian of the system, F is the so-called fermion number, F = 0 mod 2 on commmutative quantum states, and F = 1 mod 2 on anticommutative quantum states. Therefore, we have [ ] ∑ [ ] Tr (−1)F exp(−βHb ) = (−1)F exp(−βHb ) nn ∑n [ ] F b = ⟨ψn| (−1) exp(−βH) |ψn⟩. (3.69) n

We can consider β = β − 0 and then divide [β, 0] into small intervals to use the path integral to calculate the trace. However, there are two distinct features from Eqs. (3.64) and (3.67). First, the trace is a summation taken over diagonal matrix elements, so the initial state and ﬁnal state vectors should be the same. Furthermore, there is the presence of (−1)F . So the functional integral over commutative variables should be performed with periodic boundary conditions (P.B.C.), q(β) = q(0), while over the Grassmann variable with anti-periodic boundary conditions (A.B.C),

χ(β) = −χ(0), and χ∗(β) = −χ∗(0).

Second, the evolution operator is exp(−βHb ) rather than exp(−iβHb ). Thus, we should make the replacement t −→ −it = exp(−iπ/2) t in Eqs. (3.64) and (3.67) to evaluate the above trace. In physics, this operation is called Wick rotation. Therefore, in a supersymmetric quantum mechanical system composed of both commutative and Grassmann variables, we have [17, 21] [ ] ∫ ∫ F b ∗ Tr (−1) exp(−βH) = [dq] [dχ dχ] exp (−SE) . (3.70) P.B.C A.B.C

In (3.70), SE denotes the Euclidean action, ∫ ∗ ∗ SE = dtLE,LE = −L[q, iq˙; χ, iχ,˙ χ , iχ˙ ]. (3.71) 52 Supersymmetric Quantum Mechanics

3. Calculating Path Integral with Semi-classical Approximation

Most of the functional integrals cannot be calculated exactly except a special case called Gaussian integrals in which the Lagrangian is at most quadratic in variables. In the case that the interaction in the physical system is very weak, we can use semi-classical approximation to evaluate the functional integration. The basic idea is to first find a solution to the classical equation of motion, i.e., a classical trajectory, and to expand the theory around the classical solution. Then calculate this solution using the Gaussian integral, and use the perturbation method to deal with higher order terms. That is, consider a classical physical system represented by a classical (Euclidean) action S[ϕ], ϕ = {ϕi} representing either a set of real commutative, or a set of Grassmann variables. Suppose that ϕcl is the solution to the classical equation of motion, δS[ϕ] ∂L d ∂L = − = 0. δϕi ∂ϕi dt ∂ϕ˙i Then we define

ϕ = ϕcl + φ, (3.72) and expand S[ϕ],

S[ϕ] = S[ϕcl + φ] ∫ δS 1 δ2S O 3 = S[ϕcl] + φ + dt φi(t)φj(t) + (φ ) δϕi ϕ=ϕ 2 δϕjδϕj ∫ cl ϕ=ϕcl 1 = S[ϕ ] + dtφ a φ + O(φ3), (3.73) cl 2 i ij j

2 where aij ≡ δ S/(δϕjδϕj) . Hence up to the quadratic term, we obtain ϕ=ϕcl ∫ ∫ [ ∫ ] 1 [dϕ] exp (−S[ϕ]) ≈ exp (−S[ϕ ]) [dφ] exp − dtφ a φ . (3.74) cl 2 i ij j Using the formula (3.82) for real commutative variables, or equation (B.29) for real Grassmann variables, we have { ∫ [ ∫ ] − 1 (det a) 1/2 , φ = real commutative variables; [dφ] exp − dtφiaijφj = (3.75) 2 ± (det a)1/2 , φ = real Grassmann variables.

4. Gauss Functional Integration Formulas

Finally we display some Gaussian integral formulas for commutative variables which have been used earlier, or will be used later, and the counterparts for Grassmann variables are shown in Appendix B. First, consider the following formulas from Calculus, ∫ ( ) +∞ dx x2 √ exp − = 1, and −∞ 2π 2 ( ) ∫ ( ) √ +∞ dx 1 b2 √ exp −ax2 + bx = exp − , (3.76) −∞ 2π 2a 4a where a and b can be either real or complex numbers, and Re|a⟩ > 0.. Quantization 53

To evaluate the Gaussian functional integration, ∫ [ ] [ ∫ ] dϕ(t) 1 ∞ √ exp − dt ϕ2(t) , 2π 2 −∞ we usually use the following approach: ﬁrst assume t ∈ [−T/2,T/2] and impose the periodic boundary condition (P.B.C.),

ϕ[−T/2] = ϕ[T/2], (3.77) and take the limit T → ∞ in the ﬁnal stage. The P.B.C. implies that ϕ(t) admits the following Fourier expansion, ∞ ( ) ∑ 1 2πkt ϕ(x) = √ ϕ exp i , k ∈ Z. (3.78) k T k=−∞ T

For convenience of discussion, choose ϕ0 = 0. Later we shall see the case when ϕk ≠ 0 is highly nontrivial. Hence we have   ∫ [ ∫ ] ∫ +∞ ∏ ∑ 1 2 dϕk  1  [dϕ(t)] exp − dt ϕ (t) = √ exp − ϕkϕ−k 2 −∞ ̸ 2iπ 2 ̸   k=0 k=0 ∫ ∏ ∑ ∏ ∫ dϕk dϕ−k   dϕk dϕ−k = √ √ exp − ϕkϕ−k = √ √ exp (ϕkϕ−k) k>0 2iπ 2iπ k>0 k>0 2iπ 2iπ ∏ = 1 = 1. (3.79) k>0 The orthogonal relations ∫ ( ) ( ) T/2 2πkt 2πlt exp i exp i = T δk,−l, (3.80) −T/2 T T and the deﬁnition (3.65) of path integral measure, along with the integration formula ∫ ∫ ∞ ∞ dx dy √ √ exp (−xy) −∞ −∞ ∫ 2iπ( 2iπ ) ∫ ( ) ∞ du 1 ∞ dv 1 = i √ exp − u2 √ exp v2 = 1, −∞ 2π 2 −∞ 2π 2 1 1 u = √ (x + y), u = √ (x − y). (3.81) 2 2 were used in deriving (3.79). From (3.79) we can ﬁnd the Gaussian functional integration over a set of real commutative variables {ϕi}, ∫ [ ∫ ] 1 1 [dϕ] exp − dt ϕ a ϕ = √ . (3.82) 2 i ij j det(aij)

Further, the Gaussian functional integration over a set of complex conjugate commutative variables can be obtained from (3.82) by replacing the complex conjugate pair by two real variables, ∫ [ ∫ ] ∗ − ∗ 1 [dϕ dϕ] exp dt ϕi aijϕj = . (3.83) det(aij) 54 Supersymmetric Quantum Mechanics

3.3 Supersymmetric Quantum Mechanics

Supersymmetric quantum mechanics describes particles with half-integer spins moving in a particular potential well. It was originally proposed by Witten as toy models to study dynamical supersymmetry breaking [6, 7, 13]. It turned out that during the past 30 years, these types of quantum mechanical models, especially a special class of them called 0+1-dimensional supersymmetric non-linear models, have presented rich mathematical structures. Two of the remarkable branches in diﬀerential topology, the Atiyah-Singer index theorem and Morse inequalities, can be derived from 0+1-dimensional supersymmetric non-linear models [7, 15, 16, 17, 22].

3.3.1 Supersymmetry Algebra of Supersymmetric Quantum Mechanical Sys- tem

The construction of a supersymmetric quantum mechanical system is dominated by the following graded algebra [6],

[Qi,H] = 0, {Qi,Qj} = 2Hδij, i, j = 1, ··· , 2N, (3.84) where H is the Hamiltonian of the system, and the Qi are generators of supersymmetry. At the classical level, Qi can be obtained by the N¨othertheorem from the invariance of classical action under a supersymmetry transformation. At the quantum level, the Qi are operators in the Hilbert space of the system.

The supersymmetry algebra (3.84) means

2 { } ̸ Qi = H, Qi,Qj = 0 for i = j. (3.85)

In the case when N = 1, it is convenient to deﬁne [7]

1 ∗ 1 Q = √ (Q1 + iQ2) , and Q = √ (Q1 − iQ2) . (3.86) 2 2

Consequently, the N=1 supersymmetry algebra takes the following form,

Q2 = Q∗2 = 0, {Q, Q∗} = H, and [H,Q] = [H,Q∗] = 0. (3.87)

3.3.2 An Example of N=1 supersymmetric Quantum Mechanics

Consider the following example using N = 1 supersymmetry algebra to reveal the structure of the Hilbert space for an N = 1 supersymmetric quantum mechanics. The model is described by the following Lagrangian [6, 23, 24],

1 1 1 L = x˙ 2 + iψ∗ψ˙ − W 2(x) − [ψ∗, ψ] W ′(x), (3.88) 2 2 2 where W (x) is an arbitrary function of x and,

dx ∂W x = x(t), x˙ = ,W ′(x) = , (3.89) dt ∂x Supersymmetric Sigma Model 55 with Grassman variables ψ(t) and ψ∗(t) (see Appendix B), dL ψ˙ = , {ψ, ψ} = {ψ∗, ψ∗} = 0. (3.90) dt The classical action ∫ [ ] S = dt L x, x,˙ ψ, ψ˙ (3.91) is invariant under the following supersymmetric transformations,

δx = i (ψϵ + ϵ∗ψ∗) , δψ = ϵ∗x˙ − iϵ∗W, δψ∗ = ϵx˙ + iϵW, (3.92) since ∫ ∫ [ ] d 1 1 δS = dtδL = dt − iϵψ (x ˙ − iW ) + iϵ∗ψ∗ (x ˙ + iW ) = 0. (3.93) dt 2 2 In (3.92) ϵ and ϵ∗ are constant Grassmann parameters of the transformation. By means of the N¨oethertheorem introduced in Appendix A, we ﬁnd the generators of supersymmetry transformations (up to a normalization constant factor) to be, 1 1 Q = √ (x ˙ + iW ) ψ∗, and Q∗ = √ (x ˙ − iW ) ψ. (3.94) 2 2

To ﬁnd the Hamiltonian of the model, according to the procedure stated in Sect. 3.1, choose the coordinates

qα = (x, ψ) . (3.95)

The conjugate momenta are ∂L ∂L pα = (p, π) , p = =x, ˙ and π = = iψ∗. (3.96) ∂x˙ ∂ψ˙ The Hamiltonian is given by the Legendre transformation, [ ] 1 1 H (x, ψ, p, ψ∗) = p2 + W 2(x) + [ψ∗, ψ] W ′(x). (3.97) 2 2

According to the procedure of canonical quantization shown in Sect. 3.2, all the quantities should be replaced by operators, and the following canonical commutation and anticommutation relations should be imposed, { } [x,b pb] = i, ψb∗, ψb = 1, { } { } [x,b xb] = [p,b pb] = 0, ψ,b ψb = ψb∗, ψb∗ = 0. (3.98)

To construct the Hilbert space of the quantum mechanical system, we should choose representations for operators. For the commutative operators xb and pb, the usual coordinate representation, ∂ xb = x, and pb = −i , (3.99) ∂x 56 Supersymmetric Quantum Mechanics is chosen. For the anticommutative operators, it is convenient to use the “particle number” representation. We can choose [ ] [ ] b 1 0 0 b∗ 1 0 1 ψ = (σ − iσ ) ≡ σ− = , and ψ = (σ + iσ ) ≡ σ = . (3.100) 2 1 2 1 0 2 1 2 + 0 0

Further, there exists [ ] [ ] 1 0 ψb∗, ψb = σ = , (3.101) 3 0 −1 where σi, i = 1, 2, 3, are the Pauli matrices [12] [ ] [ ] [ ] 0 1 0 −i 1 0 σ = , σ = , σ = . (3.102) 1 1 0 2 i 0 3 0 −1

It can easily be checked that the representations (3.99) and (3.100) indeed satisfy the operator relations (3.98). Hence, the supercharge and the Hamiltonian operators take the forms ( ) [ ] 1 ∗ 1 ∂ 0 D Q = √ (p + iW ) ψ = √ −i + iW σ+ = , 2 2 ∂x 0 0 ( ) [ ] ∗ 1 1 ∂ 0 0 Q = √ (p − iW ) ψ = √ −i − iW σ− = ∗ , 2 2 ∂x D 0 H = Q∗Q + QQ∗ [ ] [ ] [ ] 2 ∗ 1 ∂ 2 1 ′ DD 0 H+ 0 = − + W (x) + W (x)σ3 = ∗ = , (3.103) 2 ∂x2 2 0 D D 0 H− where ( ) ( ) 1 ∂ 1 ∂ D = √ −i + iW ,D∗ = √ −i − iW , 2 ∂x 2 ∂x [ ] 1 ∂2 H = DD∗ = − + W 2(x) + W ′(x) , + 2 ∂x2 [ ] 2 ∗ 1 ∂ 2 ′ H− = D D = − − W (x) + W (x) . (3.104) 2 ∂x2

Then it can be veriﬁed by a straightforward calculation that the N = 1 superalgebra (3.87) is indeed realized in this model. In the following, observe the structure of the Hilbert space of this supersymmetric quantum mechanical model. First, according to one of the fundamental principles of quantum mechanics [11], a quantum state is speciﬁed by a common eigenstate of a complete set of commutative ∗ Hermitian operators. For the model at hand, the Hamiltonian H and ψ ψ = σ3 constitute a complete set since there exists

∗ [H, ψ ψ] = [H, σ3] = 0. (3.105) Supersymmetric Sigma Model 57

Because σ3 has two linearly independent eigenvectors, [ ] [ ] [ ] [ ] 1 1 0 0 σ = and σ = − , (3.106) 3 0 0 3 1 1 the Hilbert space H of the model is the eigenspace spanned by the two eigenvectors. Hence a quantum state must be of the form of two-component spinor wave functions, [ ] [ ] [ ] 1 0 Ψ+(x) Ψ(x) = ⟨x|Ψ⟩ = Ψ+(x) + Ψ−(x) = , (3.107) 0 1 Ψ−(x) where Ψ+(x) and Ψ−(x) can further be determined by the eigenvalue equation of Hamiltonian operators

HΨ(x) = EΨ(x), (3.108) i.e., [ ][ ] [ ] H 0 Ψ (x) Ψ (x) + + = E + , 0 H− Ψ−(x) Ψ−(x)

H+Ψ+(x) = EΨ+(x),H+Ψ−(x) = EΨ−(x). (3.109)

2 2 From Eq. (3.85), H = Q1 = Q2, hence there must exist E ≥ 0. (3.110)

Eqs. (3.106), (3.107) and (3.109) imply that the Hilbert space H can be decomposed into a direct sum of two subspaces graded by the eigenvalues of σ3, [ ] HB H = HB ⊕ HF = , HF { [ ]} { [ ]} Ψ+(x) 0 HB = |ΨB⟩ = , HF = |ΨF ⟩ = 0 Ψ−(x)

σ3|ΨB⟩ = |ΨB⟩, σ3|ΨF ⟩ = −|ΨF ⟩. (3.111)

In physics terminology, HB and HF are called the “bosonic” and “fermionic” sectors of the Hilbert space, respectively. Using the explicit representations of Q∗, Q and H listed in Eq. (3.103), we have [ ][ ] [ ] ∗ 0 0 Ψ+(x) 0 Q ΨB(x) = ∗ = ∗ ; D 0 0 D Ψ+(x) [ ][ ] [ ] 0 D 0 DΨ−(x) QΨF (x) = = , 0 0 Ψ−(x) 0 [ ][ ] [ ] H+ 0 Ψ+(x) H+Ψ+(x) HΨB(x) = = ; 0 H− 0 0 [ ][ ] [ ] H+ 0 0 0 HΨF (x) = = . (3.112) 0 H− Ψ+(x) H−Ψ−(x) 58 Supersymmetric Quantum Mechanics

These results imply the following mappings between HB and HF ,

∗ Q : HB −→ HF ,

Q : HF −→ HB,

H : HB −→ HB,

HF −→ HF . (3.113)

In physics, σ3 is usually expressed in terms of a so-called “fermion number” operator F [7, 13],

F σ3 = (−1) , [ ] 1 1 0 0 F ≡ (1 − ψ∗ψ) = (1 − σ ) = , 2 2 3 0 1

F |ΨB⟩ = 0,F |ΨF ⟩ = |ΨF ⟩,

F F (−1) |ΨB⟩ = ΨB, (−1) |ΨF ⟩ = −|ΨF ⟩. (3.114)

We can easily show from Eqs. (3.103) and (3.114) that [ ] H, (−1)F = 0, (−1)F Q + Q(−1)F = 0, and (−1)F Q∗ + Q∗(−1)F = 0. (3.115)

3.3.3 General Graded Structure of the Hilbert Space of Supersymmetric Quantum Mechanics and Witten Index

The features of a Hilbert space shown in the above example are universal for any supersymmetric quantum mechanical system. Due to the underlain supersymmetry algebra, the Hilbert space H of a supersymmetric quantum physical system is always decomposed into the direct sum of two graded subspaces,

H = HB ⊕ HF , (3.116) where HB and HF are called the spaces of “bosonic” and “fermionic” states, respectively. A supersymmetric quantum theory is deﬁned as a quantum theory with symmetry operators Qi, i = 1, ··· , 2N in the Hilbert space, which can map HB into HF and vice-versa,

Qi : HB −→ HF or Qi : HF −→ HB. (3.117)

Further, the two subspaces are distinguished by the operator (−1)F , which is deﬁned as follows [7, 22]:

F F (−1) |λ⟩ = |λ⟩ for |λ⟩ ∈ HB, or (−1) |χ⟩ = −|χ⟩ for |χ⟩ ∈ HF . (3.118)

Then we have following two results:

Theorem 3.3.1 There exist the following commutation and anticommutation relations

F F F [H, (−1) ] = 0, and (−1) Qi + Qi(−1) = 0. (3.119) Supersymmetric Sigma Model 59

This result can be easily proved using supersymmetry algebra (3.85), supersymmetry mapping (3.117), and the deﬁnition (3.118) of (−1)F .

Theorem 3.3.2 The eigenvectors with non-zero eigenvalues of the Hamiltonian operator H always arise in pairs from the bosonic sector HB and the fermionic sector HF [7, 15]. | ⟩ ∈ H | ⟩ | ⟩ ̸ 2 Proof: Let λ B, and let H λ = E λ and E = 0. From Eq. (3.85), H = Qi , so E > 0. Then deﬁne a state vector 1 |χ⟩ = √ Qi|λ⟩. (3.120) E

Obviously, |χ⟩ ∈ HF since Q maps a bosonic state into a fermionic state. Using [H,Qi] = 0, we have 1 H|χ⟩ = √ QiH|λ⟩ = E|χ⟩. (3.121) E

So both |χ⟩ ∈ HF and |λ⟩ ∈ HB are the eigenvectors of H with the same eigenvalue. Further, 2 from H = Qi , there exists √ √ Qi|λ⟩ = E|χ⟩, and Qi|χ⟩ = E|λ⟩. (3.122)

Therefore, all the state vectors in the Hilbert space H corresponding to non-zero eigenvalues of the Hamiltonian operator appear in boson-fermion pairs, furnishing a two-dimensional representation of supersymmetry algebra (3.84) or (3.85) for each value E allowed by physics.

Theorem 3.2.2 works only for those state vectors corresponding to non-zero eigenvalues of 2 | ⟩ H, while the states corresponding to E = 0 may not be paired. Since Qi = H, if H λ = 0 for |λ⟩ ∈ HB (or H|ψ⟩ = 0 for |ψ⟩ ∈ HF ), then Q|λ⟩ = 0 (or H|ψ⟩ = 0). Hence, these state vectors form trivial one-dimensional representations of the supersymmetry algebra. As Witten pointed out [7], in general, depending on a concrete physical model, there may be an arbitrary number E=0 E=0 nB of bosonic state vectors with E = 0, and an arbitrary number nF of fermionic state vectors with E = 0. When the parameters in a physical model, such as masses of particles, coupling E=0 E=0 constants of particle interteraction, and the volume of space vary, the numbers nB and nF may change since some state vectors corresponding to E ≠ 0 may become vectors with E = 0 and vice-versa. However, as long as supersymmetry does not break, according to Theorem 3.3.2, H ̸ E=0 − E=0 the state vectors in with E = 0 always arise in pairs, so the diﬀerence nB nF always stays constant. Therefore, there exists the following result introduced by Witten [7], which is a direct consequence of Theorem 3.2.2:

E=0 E=0 Theorem 3.3.3 Let nB and nF be the numbers of bosonic and fermionic eigenvectors of the Hamiltonian operators corresponding to E = 0 in the Hilbert space H, respectively. Then there exists [7]

E=0 − E=0 − F nB nF = Tr( 1) , (3.123) where the trace is taken over all the state vectors in the Hilbert space H.

This result is obvious from Theorem 3.2.2 and the definition (3.118) of (−1)F . The state F vectors corresponding to E ≠ 0 do not contribute to Tr(−1) because for every state |λ⟩ ∈ HB 60 Supersymmetric Quantum Mechanics √ with E ≠ 0 that contribute +1 to the trace, there is a state |χ⟩ = 1/ EQi|λ⟩ ∈ HF with E ≠ 0 that contribute −1 and cancel the contribution from |λ⟩. Therefore, Tr(−1)F can only E=0 − E=0 be contributed to by the state vectors with E = 0 and hence equals nB nF . Tr(−1)F is nowadays termed as the Witten index. Witten initially introduced this notion mainly for observing dynamical supersymmetry breaking [7]. If Tr(−1)F ≠ 0, there must exist at least one state vector |φ⟩ (either in HB or HF ) such that H|φ⟩ = 0, and hence Qi|φ⟩ = 0. Therefore, the supersymmetry does not break. In the meantime, he realized that Tr(−1)F is the index for the operator of supersymmetry generator. Further, he argued that the 0+1-dimensional supersymmetric sigma model should lead to the Atiyah-Singer index theorem [7]. From a mathematical viewpoint, Tr(−1)F is ill defined because the Hilbert space is infinitely dimensional and the infinite summation over all the state vectors is in general not absolutely convergent. Therefore, when calculating Tr(−1)F in a supersymmetric physical system, one usually adopts the regularized version as [ ]

Tr(−1)F ≡ Tr (−1)F exp(−βH) , (3.124) R where β is an arbitrary positive parameter, and H is the Hamiltonian. Certainly Tr(−1)F exp(−βH) is independent of β because the state vectors with E ≠ 0 do not contribute. Finally, consider the correspondence between a supersymmetric quantum mechanical system and the de Rham Cohomology and Hodge Theory established on the exterior algebra Λ∗. The details will be shown in Chapter 4. Comparing the supersymmetry algebra (3.87), the decomposition (3.116) of the Hilbert space, the definition (3.117) of supersymmetry transformation and the definition (3.118) of the grading operator (−1)F with Eqs.(2.11), (2.46) and (2.50) in Sect. 2.1., we have the following identifications:

H ←→ ∗ { p} Λ = ⊕Λ , e p HB ←→ Λ = Λ , p=2k ⊕ o p HF ←→ Λ = Λ , p=2k+1 Q ←→ d, Q∗ ←→ δ, H = Q∗Q + QQ∗ ←→ ∆ = dδ + dδ, (−1)F ←→ (−1)k, k = even, odd. (3.125) Chapter 4

Derivation of Atiyah-Singer Index Theorem from Supersymmetric Quantum Mechanics

4.1 Introducing 0+1-dimensional Supersymmetric Non-Linear Sigma Model

In physics, the supersymmetric non-linear sigma model in 0+1 dimension describes a certain physical object (e.g., a superstring) with a tower of spins (up to spin n/2) moving an n-dimensional manifold Mn. Here we consider Mn to be a compact Riemannian manifold without boundary for deriving the Atiyah-Singer index theorem. The Lagrangian of the model is [7, 15, 17] µ ν ν 1 dx dx i µ Dψ 1 µ ν L = g (x) + g (x)ψ γ0 + R (x)ψ ψλψ ψρ, 2 µν dt dt 2 µν dt 12 µνλρ µ, ν, λ, ρ = 0, 1, ··· , n − 1. (4.1)

The various quantities appearing in L are listed as follows: xµ(t)(µ = 0, 1, ··· , n − 1) are local coordinates on the Riemannian manifold Mn, gµν(x)(µ, ν = 0, 1, ··· , n) is the metric of the manifold, Rµνλρ is the curvature of the manifold given in Eq. (2.31). ψµ is a two-component real spinor with, [ ] ψµ (t) ψµ(t) = 1 , ψµ = ψµ∗ , r = 1, 2; ψµ (t) r r 2 [ ] µ 0 −i ψ = ψµT γ0, γ = σ = , 0 2 i 0 Dψν dψν dxν = + Γν ψρ, (4.2) dt dt λρ dt ∫ σ where Γ λρ is the Christoﬀel symbol shown in (2.30). The action S = dtL is invariant under the following supersymmetry transformations,

δxµ = ϵψµ, µ − 0 µ − µ ν ρ δψ = iγ x˙ ϵ Γ νρ (ϵψ ) ψ . (4.3)

61 62 Derivation of Index Theorem

Above,x ˙ ν = dxν/dt, and ϵ is a real two-component constant anticommuting spinor, [ ] ϵ1 T ϵ = , ϵ = ϵ γ0. (4.4) ϵ2

In the following sections, we shall use this model and its variants to derive the Atiyah-Singer index theorems for four classical complexes introduced in Sect. 2.4.

4.2 Gauss-Bonnet Theorem from 0+1-dimensional N=1 Super- symmetric Sigma Model

4.2.1 Identiﬁcation between Quantum N=1 Supersymmetric Sigma Model and de Rham Complex

To establish the identiﬁcation between the quantum N=1 supersymmetric sigma model and the de Rham complex, ﬁrst recombine two real components of the spinor,

1 1 χµ = √ (ψµ + iψµ) , and χ∗µ = √ (ψµ − iψµ) . (4.5) 2 1 2 2 1 2

Then the original Lagrangian (4.1) can be expressed as,

1 Dχν 1 L = g (x)x ˙ µx˙ ν + ig (x)χ∗µ − R χ∗µχ∗νχλχρ. (4.6) 2 µν µν dt 4 µνλρ The supersymmetry transformations (4.3) become

µ ∗µ − ∗ µ δx = ηχ( η χ , ) µ µ − µ ∗ν ρ δχ = η ix˙ Γ νρχ χ , and ( ) ∗µ ∗ − µ − µ ν ∗ρ δχ = η ix˙ Γ νρχ χ , (4.7) where

1 ∗ 1 η = √ (ϵ1 + iϵ2) , and η = √ (ϵ1 − iϵ2) . (4.8) 2 2

Using the N¨other theorem, we obtain the generators of supersymmetry transformations, ( ) ( ) − ∗ µ − µ ∗ν ρ ∗ µ − µ ∗ν ρ Q = iχµ ix˙ Γ νρχ χ , and Q = iχµ ix˙ Γ νρχ χ . (4.9)

Choose canonical coordinates (xµ, ψµ), the corresponding canonical conjugate momenta are ( ) µ ∂L µν λ ∗σ ρ p = = g x˙ ν + iΓ νρgλσχ χ , and ∂x˙ µ

µ ∂L µν ∗ π = = ig χ˙ ν. (4.10) ∂χ˙ µ Gauss-Bonnet Theorem 63

Now we use canonical quantization to construct quantum theory. The formal canonical commutation and anticommutation relations are

ν ν µ ν [xµ, p ] = iδµ , [xµ, xν] = [p , p ] = 0, {χµ, χ∗ν} = gµν(x), {χµ, χν} = {χ∗µ, χ∗ν} = 0. (4.11)

Similar to the example shown in Sect. 3.3, choose the coordinate representation for the commutative variable, and the particle number representation for the Grassmann variable. Then µ ∗µ µ pµ = −i∂/∂x , χ acts as a creation operator of a fermion, and χ acts as an destruction operator of a fermion. Therefore, the Hilbert space of the quantum theory is { ∗ ∗ ∗ } H | ⟩| | ⟩ µ1 µ2 ··· µp | ⟩ ··· = Ψp Ψp = ωµ1µ2···µp (x)χ χ χ Ω , p = 1, 2, , n , (4.12) where |Ω⟩ is called the fermionic vacuum deﬁned by [7, 15, 16]

ψµ|Ω⟩ = 0. (4.13)

In physics, |Ψp⟩ is called a quantum state vector representing p fermions. From Eq. (4.11), ∗µ ∗ν ∗ν ∗µ χ χ = −χ χ . Therefore, |Ψp⟩ corresponds to a p-form on the Riemannian manifold Mn, ∗ ∗ ∗ | ⟩ µ1 µ2 ··· µp | ⟩ ←→ µ1 ∧ µ2 ∧ · · · ∧ µp Ψp = ωµ1µ2···µp (x)χ χ χ Ω ωp = αµ1µ2···µp (x)dx dx dx . (4.14)

From Eqs. (4.9) and (4.10), the supersymmetry generators are

∗µ ∗ µ Q = iχ pµ, and Q = −iχ pµ. (4.15) It can easily be veriﬁed with Eq. (4.11) that ∂2 Q2 = −χ∗µp χ∗νp = −χ∗µχ∗ν = 0, µ ν ∂xµ∂xν ∂2 Q∗2 = −χµp χνp = −χµχν = 0. (4.16) µ ν ∂xµ∂xν Further, [ ] ∂ ∗µ ∗µ ∗µ ∗µ Q|Ψ ⟩ = ω ··· (x) χ χ 1 χ 2 ··· χ p |Ω⟩ ∼ |Ψ ⟩; p ∂xµ µ1µ2 µp p+1 ∗ ∂ µ ∗µ ∗µ ∗µ ∗µ µ ∗µ ∗µ ∗µ Q |Ψ ⟩ = − ω ··· (x) [{χ , χ 1 } χ 2 ··· χ p − χ 1 {χ , χ 2 } χ 3 ··· χ p p ∂xµ µ1µ2 µp ∗ ∗ ∗ ∗ + ··· + (−1)pχ µ1 χ µ2 ··· χ µp−1 {χµ, χ µp }] |Ω⟩

∂ ∗µ ∗µ = −p ω ··· (x) χ 2 ··· χ p |Ω⟩ ∼ |Ψ − ⟩. (4.17) ∂xµ µµ2 µp p 1 So there arise identiﬁcations

Q ←→ d, Q∗ ←→ δ, ∗ ∗ H = QQ + Q Q ←→ ∆P = dδ + δd, and (−1)F ←→ (−1)p, (4.18) where F is the operator of the fermion number. Therefore, we obtain [ ] ∑n F F −βH p Tr(−1) = Tr (−1) e = (−1) Bp = χ(M). (4.19) R p=0 64 Derivation of Index Theorem

4.2.2 Derivation of Gauss-Bonnet Theorem

Once we have identiﬁed the physical quantity Tr(−1)F exp(−βH) as the index of the de Rham complex,[ the following] step is to employ the standard quantum mechanical method to calculate Tr (−1)F exp(−βH) , and hence to derive the Gauss-Bonnet formula. By means of Eq. (3.70) in Sect. 3.2, we have ∫ ( ∫ ) [ ] β Tr (−1)F exp(−βH) = [dxµ(t)dχ(t)dχ∗(t)] exp − dtL [x, χ, χ∗] , (4.20) P.B.C 0 where L is shown in Eq. (4.6). Because of the periodic boundary conditions,

xµ(0) = xµ(β), χµ(0) = χµ(β), and χ∗µ(0) = χ∗µ(β), (4.21) we have the following Fourier series expansions with frequency 2πk/β, k ∈ Z, ∞ ( ) ∑ 2kπ xµ = xµ exp i t , k β k=−∞ ∞ ( ) ∑ 2kπ χµ = χµ exp i t , k β k=−∞ ∞ ( ) ∑ 2kπ χ∗µ = χ∗ exp −i t . (4.22) k β k=−∞ µ ∗µ In (4.22) χk and χk are the Grassmann numbers. The term with frequency 2πk/β is usually called a k-mode. ∫ β With Eq. (4.22), the action SE = dtL becomes a sum of infinite number of modes due to 0 the orthogonality among complex exponential functions (see Eq. (4.23) below). The functional µ µ ∗µ integral (4.20) converts into an infinite number of multiple integrations over xk , χk , and χk . Further, as shown above, the index of the de Rham complex is independent of β, so we can evaluate the functional integral (4.20) in the small-β limit. When β is small, only the constant (i.e.,k = 0) modes in exp [−SE] make dominant contributions to the integration, while the non-constant modes, which depend on 2π/β, are terms with very small values. In physics terminology, β plays a role of the coupling parameter of interactions. The constant modes are strongly coupled, while the non-constant modes are weakly coupled. Thus the functional integral splits into an integral over constant bosonic (or commutative) and fermionic (or Grassmann) configurations which are finite dimensional, and an integral over non-constant configurations. The latter can be calculated by means of a perturbation approach, which actually vanishes due to supersymmetry [17]. The concrete procedure begins by using the expansion (4.22) and the orthogonal relation ∫ ( ) ( ) β 2kπ 2lπ dt exp i t exp i t = βδk,−l, (4.23) 0 β β to obtain ( ) 1 ∑ 2kπ 2 ∑ 2kπ S = β g (x )xµxν − β g (x )χ∗µχν E 2 β µν 0 k k µν 0 β k=0̸ k=0̸ 1 ∗ − βR (x )χ µχ∗νχλχρ + O(β2). (4.24) 4 µνλρ 0 0 0 0 0 Gauss-Bonnet Theorem 65

Hence  [ ] [ ] ∫  ∏n µ [ ] dx ∗ 1 ∗ Tr (−1)F exp(−βH) = 0 dχµdχ µ exp − βR (x )χ µχ∗νχλχρ  1/2 0 0 4 µνλρ 0 0 0 0 0 µ=1 (2π)   ( ) ∫ ∏ ∑ 2 × µ 1 2kπ µ ν dxk exp βgµν(x0) xk xk ̸ 2 ̸ β k=0  k=0  ∏ ∫ ∑  ∗ 2kπ × dχµdχ µ exp −βg (x ) χ∗µχν  k k µν 0 β  k=0̸ k=0̸ ∏ [ ] 2kπ/β ∫ ∏n µ [ ] dx ∗ 1 ∗ ̸ 0 µ µ − µ ∗ν λ ρ √k=0 = dχ0 dχ0 exp βRµνλρ(x0)χ0 χ0 χ0 χ0 ∏ (2π)1/2 4 2 µ=1 (2kπ/β) k=0̸ [ ] ∫ ∏n µ [ ] dx ∗ 1 ∗ = 0 dχµdχ µ exp − βR (x )χ µχ∗νχλχρ . (4.25) 1/2 0 0 µνλρ 0 0 0 0 0 µ=1 (2π) 4

Further, rescaling the constant fermionic modes by a factor of β−1/4 and absorbing β,

µ −→ −1/4 µ ∗µ −→ −1/4 ∗µ χ0 β χ0 , χ0 β χ0 , (4.26) we obtain [ ] Tr (−1)F exp(−βH) ∫ ∏n ∫ ∏n [ ] 1 1 ∗ = dxµ dχνdχ∗ν exp − R (x )χ µχ∗νχλχρ . (4.27) n/2 0 0 0 µνλρ 0 0 0 0 0 (2π) µ=1 ν=1 4

In the case that Mn is an even dimensional manifold, i.e., n = 2k, expand the exponential function of the Grassmann variables such that, [ ] 1 ∗ 1 ∗ exp − R (x )χ µχ∗νχλχρ = 1 − R (x )χ µχ∗νχλχρ + ··· 4 µνλρ 0 0 0 0 0 4 µνλρ 0 0 0 0 0 ( ) n/2 [ ] 1 1 ∗ n/2 + − R (x )χ µχ∗νχλχρ . (4.28) (n/2)! 4 µνλρ 0 0 0 0 0

According to the integration property of the Grassmann variables, only the last term gives a non-vanishing contribution, thus we have [ ] Tr (−1)F exp(−βH) ∫ ∫ { 1 1 = d(vol) dχ1dχ∗1dχ2dχ∗2 ·· · dχndχ∗n (2π)n/2 0 0 0 0 0 0 (n/2)! ( ) } n/2 [ ] 1 ∗ n/2 × − R (x )χ µχ∗νχλχρ 4 µνλρ 0 0 0 0 0 ∫ n/2 (−1) ··· ··· = d(vol)ϵµ1ν1 µnνn ϵλ1ρ1 λnρn R · ·· R , (4.29) n n/2 µ1ν1λ1ρ1 µnνnλnρn 2 (n/2)!π M

1 ··· n where d(vol) = dx0 dx0 is the volume element of M. 66 Derivation of Index Theorem

In the case that M is a odd-dimensional manifold, i.e., n = 2k+1, it is impossible to saturate the Grassmann integrals with any term appearing in the expansion of the exponential functions of the Grassman variables. Therefore, by the integration property of the Grassmann variables introduced in Appendix C, we obtain [ ] Tr (−1)F exp(−βH) = 0 when n = 2k + 1. (4.30)

Eqs. (4.29) and (4.30) are the Gauss-Bonnet-Chern-Avez formula.

4.3 Hirzebruch Signature Theorem from 0+1-dimensional N=1 Supersymmetric Sigma Model

4.3.1 Identify Signature Complex from Quantum N=1 Supersymmetric Sigma Model

Use the same model described by the Lagrangian (4.1) or (4.6) to identify the signature complex. It is easy to see that when n is even, the Lagrangian (4.6) has a discrete symmetry: it is invariant under the following exchange of the anti-commutative variables,

χµ −→ χ∗µ, χ∗µ −→ −χµ, µ = 0, 1, ··· , n − 1. (4.31)

At the quantum level, both χµ and χ∗µ become operators. This implies that there must exist a unitary operator in the Hilbert space to implement the discrete symmetry transformation. Denoting the operator as U5, we have [15, 16, 17] U µU −1 ∗µ U ∗µU −1 − µ 5χ 5 = χ , 5χ 5 = χ , (4.32) and the Hamiltonian is invariant under the transformation,

U U −1 5H 5 = H, (4.33) which can be veriﬁed explicitly. From Eqs. (4.11) and (4.32), we obtain

U U −1 ∗ U ∗U −1 − 5Q 5 = Q , and 5Q 5 = Q, (4.34) that is,

∗ ∗ U5Q = Q U5, and U5Q = −QU5. (4.35)

Eq. (4.31) or (4.32) shows that the four successive actions of U5 should be an identity operator, U 4 5 = I. (4.36)

In the following, observe the operation of U5 on a state |Ψp⟩ in the Hilbert space to ﬁnd ∗µ the geometric correspondence. First, from Eq. (4.32), U5 converts a creation operator χ of fermions into a destruction operator χµ, and vice-versa, while |Ω⟩ is a vacuum state with no fermion. Therefore, the action of U5 should lead to a fully-ﬁlled fermionic state, i.e.,

∗1 ∗2 ∗n U5|Ω⟩ = |Ψn⟩ = χ χ ··· χ |Ω⟩. (4.37) Hirzebruch Signature Theorem 67

Then the action of U5 on |Ψp⟩ is

∗ ∗ ∗ U | ⟩ U µ1 µ2 ··· µp | ⟩ 5 Ψp = 5 ωµ1µ2···µp (x)χ χ χ Ω ∗ − ∗ − ∗ − U µ1 U 1U µ2 U 1U ···U µp U 1U | ⟩ = ωµ1µ2···µp (x) 5χ 5 5χ 5 5 5χ 5 5 Ω µ µ µ ∗1 ∗2 ∗n = ω ··· (x) χ 1 χ 2 ··· χ p χ χ ··· χ |Ω⟩ µ1µ2 µp { } ∗ ∗ ∗ µ1 µ2 ··· µp 1 2 ··· n | ⟩ ∼ | ⟩ = ωµ1µ2···µp (x) χ χ χ , χ χ χ Ω Ψn−p . (4.38)

This means that U5 converts |Ψp⟩ into |Ψn−p⟩. Hence, it can be identiﬁed with the Hodge star operator in the diﬀerential manifold Mn,

U5 ←→ ∗. (4.39)

e Second, when U5 acts on a p-fermion state |Ψp⟩, deﬁne an operator U5 such that

e n/2+p(p−1) U5 = i U5. (4.40)

Note that there exists e U5 = U5 when n = 4k, p = n/2 = 2k. (4.41)

Then we ﬁnd that when n = 4k (k ∈ Z+),

Ue2 5 = I. (4.42)

Thus the Hilbert space when n = 4k can be decomposed into two subspaces with respect to the e eigenvalues of U5:

H H ⊕ H = { + −, } e H+ = |Ψp⟩| U5|Ψp⟩ = |Ψp⟩, 0 ≤ p ≤ n , { } e H− = |Ψp⟩| U5|Ψp⟩ = −|Ψp⟩, 0 ≤ p ≤ n . (4.43)

Third, using Eq. (4.35), we can show

e ∗ e e ∗ e U5Q = −Q U5, and U5Q = −QU5. (4.44)

Hence

e ∗ ∗ e U5 (Q + Q ) = − (Q + Q ) U5, e ∗ ∗ e U5 (Q − Q ) = (Q − Q ) U5. (4.45)

∗ Thus Q + Q maps H+ into H− and vice-versa,

∗ Q + Q : H± −→ H∓. (4.46)

Then from (4.45), if E ≠ 0, (Q + Q∗)|Ψ(E, ±1)⟩ is a state vector with the same E ≠ 0 but e opposite eigenvalue of U5, since

H (Q + Q∗) |Ψ(E, ±1)⟩ = (Q + Q∗) H|Ψ(E, ±1)⟩ = E (Q + Q∗) |Ψ(E, ±1)⟩, e ∗ ∗ e ∗ U5 (Q + Q ) |Ψ(E, ±1)⟩ = − (Q + Q ) U5|Ψ(E, ±1)⟩ = ∓ (Q + Q ) |Ψ(E, ±1)⟩. (4.48) e Thus all state vectors with E ≠ 0 appear in pairs of opposite U5 eigenvalues, and only the state e vectors with E = 0 need not to appear in U5 pairs. Therefore, we can identify

∗ (H±,Q + Q ) (4.49) as the signature complex in diﬀerential form theory when n = 4k, and p = n/2 = 2k. The index for the complex is

E=0 E=0 Tr (U5) = n (U5 = +1) − n (U5 = −1) = τ(M), (4.50)

E=0 where n (U5 = ±1) denotes the number of p = 2k-fermion states in H± with E = 0. F Finally, it should be emphasized that like Tr (−1) , the trace Tr (U5) is ill-defined due to the infinite dimensionality of the Hilbert space. Therefore, when calculating Tr (U5) we use the regularized definition, ( ) U | U −βH Tr ( 5) R = Tr 5 e , β > 0. (4.51)

The result should be independent of β since its only contribution is from the states with E = 0.

4.3.2 Derivation of Hirzebruch Signature Theorem ( ) −βH In the following we calculate Tr U5 e to derive the Hirzebruch signature theorem. Adopting the Lagrangian (4.1) rather than the Lagrangian (4.6), we have from Eq. (4.5),

1 i ψµ = √ (χµ + χ∗µ) , and ψµ = −√ (χµ − χ∗µ) . (4.52) 1 2 2 2

Hence the discrete symmetry given in Eq. (4.31) leads to the following symmetry transformations µ µ on ψ1 and ψ2 , µ −→ − µ µ −→ µ ψ1 ψ1 , ψ2 ψ2 . (4.53) ( ) −βH To be compatible with this symmetry, when we use the functional integral to evaluate Tr U5 e , µ µ ψ1 (t) must be integrated over with anti-periodic boundary condition (A.B.C.), and ψ2 (t) integrated over periodic boundary conditions (P.B.C.) in β. The integration over the commutative variables xµ(t) is always taken with periodic boundary conditions. Therefore, we have ∫ ∫ ( ∫ ) ( ) β U −βH µ µ µ − Tr 5 e = [dx dψ2 ] [dψ1 ] exp dtL [x, ψ1, ψ2] . (4.54) P.B.C. A.B.C 0

µ µ µ The P.B.C for ψ2 (t) and x (t) and the A.B.C. for ψ1 (t), ie. µ µ µ µ µ − µ x (0) = x (β), ψ2 (0) = ψ2 (β), and ψ1 (0) = ψ1 (β), (4.55) Hirzebruch Signature Theorem 69 mean that there exist the following Fourier series expansions, ∞ ( ) ( ) ∑ 2kπ ∑ 2kπ xµ(t) = xµ exp i t = xµ + xµ exp i t k β 0 k β k=−∞ k=0̸ ≡ µ µ x0 + u (t), ∞ ( ) ( ) ∑ 2kπ ∑ 2kπ ψµ(t) = ψµ exp i t = ψµ + ψµ exp i t 2 2k β 2,0 2,k β k=−∞ k=0̸ ≡ ψµ + ηµ(t), 2,0 ( ) ∑ (2k + 1)π ψµ(t) = ψµ exp i t = 0 + ψµ(t), (4.56) 1 1,k β 1 k=0̸ where k ∈ Z. Note that because all variables are real,

µ ∗µ µ ∗µ µ ∗µ x (t) = x (t), ψ2 (t) = ψ2 (t), and ψ1 (t) = ψ1 (t), (4.57) and there should exist

∗µ µ ∗µ µ ∗µ µ xk = x−k, ψ1,k = ψ1,−k, and ψ2,k = ψ2,−k. (4.58)

In the following we evaluate the functional integral (4.54) around the constant conﬁguration ( ) µ µ µ µ µ (x , ψ1 , ψ2 ) = x0 , 0, ψ2,0 . (4.59)

It is easy to see from the Lagrangian (4.1) that on the constant conﬁguration (4.59),

µ µ L[x0 , 0, ψ2,0] = 0. (4.60) Substituting the expansions (4.56) around the background of the constant conﬁguration (4.59) µ µ into the Lagrangian (4.1), we expand the Lagrangian to the second order of u (t), ψ1 (t), and ηµ(t), [17]

L = L(2) + O(β), 1 duµ duν i du L(2) = g (x ) + R ψµ ψν uλ 2 µν 0 dt dt 4 µνλρ 2,0 2,0 dt i dψν i i dην + g (x )ψµ 1 + R ψµ ψν ψλψρ + g (x )ηµ . (4.61) 2 µν 0 1 dt 4 µνλρ 2,0 2,0 1 1 2 µν 0 dt In deriving L(2), we have discarded some total derivative terms due to the periodic bound- µ µ ary conditions, and made use of the classical equations of motion for xµ, ψ1 , and ψ2 . Note that the constant configuration (4.59) trivially satisfies the classical equations of motion, and µ µ µ consequently, the terms which are linear in u , η and ψ1 vanish. We define ( ) a a µ µ µ µ µ µ X = e µ(x0)X , and X = u , η , ψ1 , ψ2,0 , (4.62)

a where eµ(x) are vierbein shown in Eq. (2.17). The Lagrangian (4.61) becomes

1 dua dub i dub i d L(2) = δ + R ψc ψd ua + δ ηa ηb 2 ab dt dt 4 abcd 2,0 2,0 dt 2 ab dt i d 1 + δ ψa ψb + R ψc ψd ψaψb. (4.63) 2 ab 1 dt 1 4 abcd 2,0 2,0 1 1 70 Derivation of Index Theorem

Hence ∫ ∫ { [ ] β β (2) (2) 1 a − d i c d b SE = dtL = dt u 2 + Rabcd(x0)ψ2,0ψ2,0 u 0 0 [ 2 ] dt 2[ ] } 1 d 1 d 1 + ηa iδ ηb + ψa iδ + R (x )ψc ψd ψb . (4.64) 2 ab dt 2 1 ab dt 2 abcd 0 2,0 2,0 1 Therefore by the formulas (3.82) and (B.29), the index of the signature complex from Eq. (4.54) is ( ) ∫ [ ] ∏n dxµ ∏n Tr U e−βH = 0 dψν [du dηdψ ] exp −S(2) 5 1/2 2,0 1 E µ=1 (2π) ν=1 ( )  1/2 ∫ n µ n ′ ab ∏ dx ∏ det iδ d/dt det (iδabd/dt + Mab) = 0 dψµ   , (4.65) 1/2 2,0 ′ − ab 2 2 µ=1 (2π) µ=1 det ( δ d /dt + iMabd/dt) where 1 M ≡ R (x )ψc ψd . (4.66) ab 2 abcd 0 2,0 2,0 In (4.65), det′ means that the zero modes have been taken away and that the determinant is evaluated over the conﬁgurations satisfying the P.B.C, whereas the determinant, det (iδabd/dt + Mab), is calculated with the A.B.C. t Because Mab form an n × n antisymmetric matrix, M = −M . According to linear algebra [10], Mab can be converted into a standard skew diagonal form by a similar transformation, −1 (Mab) = QSQ (4.67) where   0 x  1   −   x1 0     0 x2     −x2 0  S =   . (4.68)  .   ..     0 xn/2  −xn/2 0 Then we have the following: [ ( )] ∫ [ ∫ ( ) ] d 1/2 1 β d det′ iδab = [dηa] exp − dtηa iδab ηb dt 2 0 dt   ∫ ∏ ∏n ∑ a − a a  = dψ2,k exp πkψ2,kψ2,−k ̸ a=1 ̸ k=0 k=0   ∫ ∏ ∏n [ ] ∑ a a − a a  = dψ2,k dψ2,−k exp 2πkψ2,kψ2,−k k>0 a=1 k>0 ∏n ∏ ∫ [ ] a ∗a − a ∗a = dψ2,k dψ2,k exp 2πkψ2,kψ2,k a=1 k>0 ∏n ∏ ∏ = (2πk) = (2πk)n . (4.69) a=1 k>0 k>0 Hirzebruch Signature Theorem 71

[ ( )] [ ( ) ] d 1/2 d 1/2 det iδ + M = det Q iδ + S Q−1 ab dt ab ab dt ab [ ( )] ∫ [ ∫ ( ) ] 1/2 β d −1 a d b = det iδab + Sab = [dψ1] exp dtψ1 iδab + Sab ψ1 dt 2 0 dt ( ) ∫ ∏n ∏ ∑ a − a b = dψ1,k exp ψ1,kψ1,−k [(2k + 1)πδab + Sab] a,b=1 k  k  ∫ n/∏2 ∏  ∑ ∑n/2 [ ] p − p p 2 2 2 = dψ1,k exp  ψ1,kψ1,−k (2k + 1) π + xp  p=1 k k p=1 { } n/∏2 ∫ ∏ ∑ [ ] p p − p p 2 2 2 = dψ1,kdψ1,−k exp 2 ψ1,kψ1,−k (2k + 1) π + xp p=1 k≥0 k n/∏2 ∏ ∫ { [ ]} p ∗p − p ∗p 2 2 2 = dψ1,kdψ1,k exp 2ψ1,kψ1,k (2k + 1) π + xp p=1 k≥0 n/∏2 ∏ [ ] 2 2 2 = 2 (2k + 1) π + xp p=1 k≥0 [ ] [ ] n/2 ∏ n/2 ∏ ∏ (x /2)2 = 2n/2 (2k + 1)2π2 1 + p . (4.70) π2(k + 1/2)2 k≥0 p=1 k≥0

[ ( )] [ ( )] 1/2 d −1/2 d2 d det′ iδ + M = det′ −δ + iS ab dt ab ab dt2 ab dt ∫ [ ∫ ( ) ] β 2 −1 a − d d b = [du] exp dtu δab 2 + iSab u 2 0 dt dt   ∫ [ ] ∏n ∏ dxa ∑ = k exp − xaxb (2πk)2δ + 2πkS  1/2 k −k ab ab ̸ (2π) ̸ a,b=1 k=0  k=0  n/∏2 ∫ ∏ p p  ∑ [ ] dx dx− = k k exp −2 xpxp (2πk)4 + (2πk)2x2 2π  k −k p  p=1 k>0 k>0 n/∏2 ∏ ∫ p ∗p { [ ]} dx dx ∗ = k k exp −2xpx p(2πk)2 (2πk)2 + x2 2π k k p p=1 k>0 [ ] n/2 n/2 ∏ ∏ 1 ∏ 1 ∏ ∏ (x /2)2 = [ ] = 1 + p . (4.71) 2 2 2 2n/2(2πk)2n π2k2 p=1 k>0 2(2πk) (2πk) + xp k>0 p=1 k>0

The multiple variable integral formulas for commutative variables and the Grassmann ones have been used in calculating these determinants.

Substituting the results (4.69), (4.70), and (4.71) into (4.65) and discarding some irrelevant 72 Derivation of Index Theorem factors that can be absorbed into the normalization factor of the functional integral, we obtain ( ) n/2 ∏ ∏ (x /2)2 2 1 + p ( ) ∫ ∏n µ ∏n π2(k + 1/2)2 U −βH dx0 µ p=1 k>0 Tr 5 e = dψ2,0 ( ) 1/2 n/∏2 ∏ 2 µ=1 (2π) µ=1 (x /2) 1 + p π2k2 p=1 k>0 n/∏2 2 cosh (xp/2) ∫ ∏n ∫ ∏n 1 p=1 = dxµ dψµ n/2 0 2,0 n/2 (2π) µ=1 µ=1 ∏ [sinh (xp/2) /(xp/2)] p=1 ∫ n/2 ∫ 1 ∏ x /2 = d(vol) p = L(M ) = τ(M ). (4.72) n/2 tanh (x /2) n n (2π) p=1 p Eq. (4.72) used the product representations of cosh x and sinh x [28], the identiﬁcation 1 1 M = R (x )ψcψd ←→ Ω = R ec ∧ ed, (4.73) ab 2 abcd 0 0 0 ab 2 abcd and Eqs. (4.67) and (4.68),

n/2 ⊕ n (Mab) = (xpϵp,q) , a, b = 1, ··· , n, p, q = 1, ··· , . p=1 2 Eq. (4.72) is the Hirzebruch signature theorem reproduced from the supersymmetric quantum mechanical model (4.1).

4.4 Index Theorem for A-roof Genus from 0+1-dimensional N=1/2 Supersymmetric Sigma Model

4.4.1 Designing Spin Complex from Quantum N=1/2 Supersymmetric Sigma Model

To get the spin complex from supersymmetric quantum mechanics, we need to slightly modify the model (4.1). Impose the following condition on the Lagrangian (4.1), 1 ψµ = ψµ ≡ √ χµ. (4.74) 1 2 2 Then, due to the symmetric property of the Riemmannian curvature listed in (2.32), the term containing the curvature vanishes and the Lagrangian becomes 1 dxµ dxν i Dχν L = g (x) + g (x)χµ . (4.75) 2 µν dt dt 2 µν dt

Correspondingly,√ in the supersymmetry transformations (4.3), one should take ϵ1 = −ϵ2 = −1/ 2 η. Then the supersymmetry transformations reduce (4.3) to

δxµ = iϵχµ, and δψµ = −x˙ µϵ. (4.76) Index Theorem for Ab-roof genus 73

By the N¨othertheorem, we can obtain one (real) supersymmetry generator Q from the Lagrangian (4.75),

µ ν Q = χ gµνx˙ . (4.77)

This type of supersymmetry is usually called N = 1/2 supersymmetry since 2N = 1. The Hamiltonian is

H = Q2. (4.78)

As in Sects. 4.2 and 4.3, use the standard canonical quantization procedure to construct quantum theory. The coordinates are (xµ, χµ) and the corresponding canonical conjugate momenta are ( ) µ ∂L µν i λ σ ρ p = = g x˙ ν + Γ νρgλσχ χ , and ∂x˙ µ 2

µ ∂L i µν π = = g χ˙ ν. (4.79) ∂χ˙ µ 2 Then we have the canonical commutation relation,

[xµ, pν] = igµν(x), (4.80) and anticommutation relation,

{χµ, χν} = gµν(x). (4.81)

µ µ For the commutative variables x and p , choose the coordinate representation, then pµ = −i∂/∂xµ. To find the representations for the anti-commutative variables χµ, we need to express them in terms of the orthonormal vierbein defined at each point of the Riemannian manifold. The reason for this is that the isometry group for the Riemannian manifold is GL(n, R), which has no spinor representation. If we express various vectors such as the metric, connection and various vectors on M in the vierbein formalism, the isometry group is the local Lorentz group SO(n), which admits a spinor representation. Therefore, we define a ≡ a µ χ e µ(x)χ . (4.82) Then the anticommutation relation (4.81) becomes { } χa, χb = δab, (4.83) which is precisely the fundamental relation of the Clifford algebra. Thus we can choose the representation γa χa = √ , (4.84) 2 with γa being 2n/2 × 2n/2 matrices, as shown in Appendix C. Consequently, the conjugate momenta operator pµ becomes ∂ i p = −i = g (x)x ˙ ν + ω [χa, χb] µ ∂xµ µν 4 µab 1 = g (x)x˙ ν − ω [χa, χb], (4.85) µν 8 abµ 74 Derivation of Index Theorem

µ where we have used the relation between the Christoffel symbol Γ νρ and the spin correction ωµab given in Eq. (2.37). Using (4.85) we can express the supersymmetry generator (4.77) as, ( ) i µ ∂ 1 a b i µ i Q = −√ γ + ωabµ[γ , γ ] = −√ γ Dµ ≡ −√ D, (4.86) 2 ∂xµ 8 2 2 µ ≡ a µ where γ γ Ea (x). Eq. (4.86) shows that the supersymmetry generator is identified with the Dirac operator. Then the representation of the Hamiltonian operator is H = 1/2 D2. The following is a construction of the Hilbert space based on the above representations of µ the operator. Choose the coordinate representation for x and pµ, and choose the γ-matrix representations for ψµ since they satisfy the Clifford algebra whose representation space is the space of spinor functions. Therefore, the Hilbert space can be identical to the spinor space formed by the spinor functions Ψ(x),     ψ (x)    1    ψ (x)  H |  2  = Ψ(x) Ψ(x) =  .  . (4.87)   .    ψn/2(x)

Therefore, we have obtained the following identiﬁcation between the supersymmetric quantum mechanics (4.75) and the spin complex:

F (−1) ←→ γn+1, + HB ←→ S , − HF ←→ S , Q ←→ iD. (4.88)

Hence, the index for the spin complex can be obtained from supersymmetric quantum mechanics described by the Lagrangian (4.75), and it is is [ ] Index of iD = Tr (−1)F e−βH , (4.89) where, as above, β is an arbitrary positive parameter and H is the Hamiltonian operator.

4.4.2 Derivation of Index Theorem for A-roof Genus [ ] The procedure of computing Tr (−1)F e−βH to get the index of the Dirac operator is identical to the procedure of calculating Tr (U5 exp [−βH]). That is, start from the Lagrangian (4.75) and expand it to the second order of variables around the trivial constant solution to the classical equations of motion. Then, calculate the functional integration over (xµ, χµ). Note that the functional integral should be evaluated with both the variables satisfying periodic boundary conditions. Therefore, the index of the spin complex is ∫ [ ∫ ] [ ] β Ind(iD/) = Tr (−1)F exp (−βH) = [dx][dχ] exp − dtL P.B.C. 0 ( )  1/2 ∫ ′ ab 1 ∏n ∏n det iδ d/dt = dxµ dψµ   . (4.90) n/2 0 2,0 ′ − ab 2 2 (2π) µ=1 µ=1 det ( δ d /dt + iMabd/dt) Riemman-Roch Theorem 75 ( ) As shown in Eq. (4.70), det′ iδabd/dt is a constant factor, which can be canceled by the normalization factor of the functional integral. Thus, only the determinant in the denominator contributes to the index. Using the result displayed in Eq. (4.71), we obtain

∫ ∫ n/2 1 ∏n ∏n ∏ x /2 Ind(iD/) = = dxµ dχµ p n/2 0 0 sinh (x /2) (2π) µ=1 µ=1 p=1 p ∫ n/2 ∫ 1 ∏ x /2 = d(vol) p = Ab(M ). (4.91) n/2 sinh (x /2) n (2π) p=1 p

Eq. (4.91) is the index theorem for the spin complex.

4.5 Riemann-Roch Theorem from 0+1-dimensional N=2 Super- symmetric Sigma Model

4.5.1 Making Dolbeault Complex from Quantum N=2 Supersymmetric Sigma Model

To make the Dolbeault complex from a supersymmetric quantum mechanical system, we need to modify the model (4.1). The ﬁrst step is to consider the n-dimensional manifold M as a K¨ahler µ µ manifold (n = 2k). Second, write the real coordinates x , and the real Grassmann variables ψr , into two complex counterparts according to the standard form of the complex structure shown in Sect. 2.1, ( ) ( ) (xµ) = x1, ··· , xk, x1+k, ··· , x2k −→ (zα; zα) = z1, ·· · , zk; z1, ··· , zk , ( ) ( ) ( ) µ 1 ·· · k 1+k ··· 2k −→ α α α ··· α 1 ··· k (ψr ) = ψr , , ψr , ψr , , ψr ψr ; ψr = ψr , , ψr ; ψr, , ψr , µ = 1, 2, · ·· , n, α = 1, 2, ··· , k = n/2, r = 1, 2, (4.92) where ( ) ( ) 1 1 zα = √ xα + ixα+k , zα = √ xα − ixα+k ; 2 2 ( ) ( ) 1 α 1 ψα = √ ψα + iψα+k , ψ = √ ψα − iψα+k . (4.93) r 2 r r r 2 r r

Further, deﬁne

1 1 χα = √ (ψα + iψα), χ∗α = √ (ψα − iψα), 2 1 2 2 1 2 1 α α 1 α α χα = √ (ψ + iψ ), χ∗α = √ (ψ − iψ ). (4.94) 2 1 2 2 1 2

Then the Lagrangian (4.1) is converted into the following form [26, 27],

dzα dzβ D D 1 L = g (z, z) + ig (z, z)χ∗α χβ + ig (z, z)χ∗α χβ + R χαχ∗βχ∗γχδ. (4.95) αβ dt dt αβ dt αβ dt 4 αβγδ 76 Derivation of Index Theorem

The real two-component anti-commutative spinor parameter in (4.4) should be promoted a µ complex counterpart and be combined in the same way as ψr shown in (4.93) and (4.94),

1 ∗ 1 η = √ (ϵ1 + iϵ2) , η = √ (ϵ1 − iϵ2) , 2 2 1 ∗ 1 η = √ (ϵ1 + iϵ2) , η = √ (ϵ1 − iϵ2) . (4.96) 2 2

Consequently, the supersymmetry transformations for the model (4.95) are listed as follows, δzα = iη∗χα + iηχ∗α, δzα = iη∗χα + iηχ∗α, α − ∗ α − ∗β α γ δχ = η z˙ iηχ Γ βγχ , δχα = −η∗z˙ α − iηχ∗βΓα χγ, βγ ∗α − α − ∗ β α ∗γ δχ = ηz˙ iη χ Γ βγχ , δχ∗α = −ηz˙ α − iη∗χαΓα χ∗γ, (4.97) βγ where Γα and Γα are given in Eq. (2.89). βγ βγ Choose the canonical coordinates (zα, zα, χα, χα), the corresponding conjugate momenta are, respectively, ( ) ∂L p = = g z˙ β − Γβ χ∗γχδ , α ∂z˙α αβ γδ ( ) ∂L β β ∗γ δ p = = gαβ z˙ − Γ χ χ , α ∂z˙ α γδ ∂L ∂L π = = ig (z, z)χ∗β, and π = = ig (z, z)χ∗β. (4.98) α ∂χ˙ α αβ α ∂χα αβ

Using the N¨other theorem, we can ﬁnd four generators of supersymmetry transformations corresponding to the four parameters η, η, η∗, and η∗,

∗α β α β Q = igαβχ p , Q = −igαβχ p , ∗ ∗α β ∗ − α β Q = igαβχ p , Q = igαβχ p . (4.99)

The quantum theory can be constructed by canonical quantization. First deﬁne the following operator commutation relations, [ ] α α α α [z , pβ] = iδ β, z , pβ = iδ ; [ ] [ ] [ ] β zα, zβ = zα, zβ = zα, zβ = 0, [ ] [ ] [pα, pβ] = pα, pβ = pα, pβ = 0, [ ] α α z , pβ = [z , pβ] = 0; (4.100) and anticommutation ones, { } { } ∗ ∗ χα, χβ = g , χα, χβ = gαβ, {αβ } { } { } { } ∗ ∗ ∗ χα, χβ = χα, χβ = χα, χβ = χα, χβ = 0, { } { } { } { } ∗ ∗ ∗ ∗ ∗ χα, χβ = χα, χβ = χα, χβ = χα, χβ = 0. (4.101) Riemman-Roch Theorem 77

For the commutative operators, choose the coordinate representation, and for the commutative ones, choose the particle number representation. Then we have ∂ ∂ p = −i , and p = −i , (4.102) α ∂zα α ∂zα where χα = destruction operator of holomorphic fermions, χ∗α = creation operator of holomorphic fermions, χα = destruction operator of anti-holomorphic fermions, χ∗β = creation operator of anti-holomorphic fermions. (4.103)

The Hilbert space is composed of the state vectors which can be called (p, q)-fermion states, { } ∗α1 ∗αp ∗β1 ∗βq H = |Ψp,q⟩| |Ψp,q⟩ = ω ··· ··· (z, z)χ ··· χ χ ··· χ |Ω⟩, 0 ≤ p, q ≤ k (4.104) α1 αpβ1 βq where |Ω⟩ is the (p, q)-fermion vacuum state vector, deﬁned by χα|Ω⟩ = χα|Ω⟩ = 0. (4.105)

As in Sect. 4.2, we can easily show

2 ∗2 2 ∗2 Q = Q( = Q = Q ) = 0,( ) ∗ ∗ H = 2 QQ + Q Q = 2 QQ∗ + Q∗Q , (4.106) and further, ∗ Q|Ψp,q⟩ ∼ |Ψp+1,q⟩, Q |Ψp,q⟩ ∼ |Ψp−1,q⟩; ∗ Q|Ψp,q⟩ ∼ |Ψp,q+1⟩,Q |Ψp,q⟩ ∼ |Ψp,q−1⟩. (4.107) So we have the identiﬁcations ∗ Q ←→ ∂, Q ←→ ∂; ∗ Q ←→ ∂∗,Q∗ ←→ ∂ ; ⊕k p,q H = {|Ψp,q⟩} ←→ Λ(M) = Λ (M), { } { } p,q=0 { } ∗ ∗ H = Q, Q = Q, Q∗ ←→ ∆ = {d, δ} = {∂, ∂∗} = ∂, ∂ , (−1)Fp ←→ (−1)p, (−1)Fq ←→ (−1)q, (4.108) where Fp and Fq are the operators of p-fermion and q-fermion numbers. Now take p = 0 and consider a subspace of the Hilbert space which is composed only of q-fermion states,

HA = {|Ψ0,q⟩, 0 ≤ q ≤ k} . (4.109) ( ) ( ) 0,q Then Q, HA is identiﬁed with the Dolbeault complex and ∂, ⊕Λ . Therefore, the analytic index of the Dolbeault complex is [ ] ∑n/2 − Index of ∂ = Tr (−1)Fq e βH = (−1)qb0,q, (4.110) q=0 where β > 0 and H is the Hamiltonian operator. 78 Derivation of Index Theorem

4.5.2 Derivation of Riemann-Roch Theorem

We brieﬂy state how the Riemann-Roch theorem for the Dolbeault complex can be derived since the procedure is identical to that[ for the spin complex] and the signature complex. We only need to restrict the calculation of Tr (−1)F exp(−βH) to the anti-holomorphic part of the Hilbert space. The computation is exactly the same as for that of the Dirac operator, and the non-trivial contribution to the index of ∂ comes only from the integration over zα and zα. Finally we obtain

∫ n/∏2 ∫ n/∏2 [ ] n/∏2 1 ∗ ω /2 Ind(∂) = [dzα dzα] dχ αdχδ eωα/2 α n/2 0 0 0 0 sinh (ω /2) (2iπ) α=1 α=1 α=1 α ∫ n/2 ∫ 1 ∏ ω = d (vol) α = td (M), (4.111) n/2 − − c (2iπ) M α=1 1 exp( ωα) M where iωα is the eigenvalue of the antisymmetric matrix

i ∗ Ω = R χ γχδ. (4.112) αβ 2π αβγδ 0 0 Eq. (4.111) is the Riemann-Roch theorem for the Dolbeault complex. Chapter 5

Summary and A Mathematical Conjecture

So far the Atiyah-Singer index theorem for four classical elliptic complexes and the relevant preliminary knowledge has been introduced. It has been shown in great detail how the Hilbert spaces of 0 + 1-dimensional supersymmetric nonlinear sigma models can be identical to the sections of fibre bundles for elliptic complexes, and how the generators of supersymmetry can be identified with differential operators. Further, the index theorems have been derived using powerful physical approaches – path integration and using structures of supersymmetric quantum theory. As Witten realized in the early 1980s [6, 7, 22], some supersymmetric quantum theories have provided a natural physical framework for studying the Atiyah-Singer index theorem and relevant mathematical problems. First, the Hilbert space of a supersymmetric quantum theory presents a graded structure. Hence an isomorphism with the exterior algebra can be established. Second, the generators of supersymmetry in a quantum mechanical system are nilpotent and transform the “bosonic” sector of the Hilbert space into the “fermionic” sector and vice versa. Hence, they can be identified as exterior differential operators and the relevant variants in various complexes. According to supersymmetry algebra, the Hamiltonian of the system, which in general determines all the physical processes, is identified with the Laplacian operator in elliptic complexes. Third, supersymmetry can make infinity reduce to finiteness. It is well known in physics that supersymmetry enforces the quantum contributions to a physical process coming from the paired states in the graded sectors to cancel. Therefore, the Hilbert space can effectively become a finite dimensional space composed of only non-paired states, which are eigenstates corresponding to the zero-eigenvalue of the Hamiltonian operator. The creation of finite dimensional reduction from an infinite dimensional Hilbert space is the most useful property of supersymmetry for mathematics.

However, the derivation of the Atiyah-Singer theorem by supersymmetric quantum mechanics relies on some ideas of physics. Because of this, it is somehow not mathematically rigorous. As mentioned in Chapter 1, the proof of the Atiyah-Singer index theorem using the approach of heat equations [4, 5] is based on positive-self adjoint operator theory and has been accepted as a mathematically rigorous proof. In this approach, the derivation of the index theorem is actually the calculation

Index (D) = hE(t) − hF (t), (5.1)

79 80 Summary where E and F denote two vector bundles over a diﬀerential manifold M, and ∑ − − h(t) = e tλn = Tre t∆, (5.2)

λn where ∆ is the Laplacian operator on the vector bundle E or F . h(t) is called the heat function, which satisﬁes the heat equation ∂ − h = ∆h. (5.3) ∂t Hence, the calculation on the index becomes the problem of solving the heat equation (5.3). One conveniently uses the Green function method to solve the equation, which gives ∫ √ h(t) = K(t, x, x) gdnx, x = (x1, ··· , xn), (5.4) M where K(t, x, y) is determined by ( ) ∂ + ∆ K(t, x, y) = 0,K(0, x, y) = δ(n)(x − y). (5.5) ∂t x 2 µ For the Laplacian operator in Euclidean space, ∆0 = ∂ = ∂ ∂µ, one can easily ﬁnd [ ] 1 (x − y)2 K (t, x, y) = exp − . (5.6) 0 (4πt)n/2 4t

But for a Laplacian operator deﬁned on a general vector bundle, it is impossible to solve (5.5) exactly. One can only ﬁnd the asymptotic expansion form of K(t, x, y) at t → 0+ [5], ∑∞ √ (i−n)/2 K(t, x, y) = t fi(x) g. (5.7) i=0

Substituting (5.7) into the heat equation, we can obtain a recursive formula for fi(x). From Eqs. (5.1), (5.4), and (5.7), one can ﬁnd that as t → 0+, the pole and regular terms (i.e., i ≠ n) cancel, and hence ∫ [ ] √ E − F n Index(D) = Tr fn (x) fn (x) gd x. (5.8) M E − F Eq. (5.8) shows that fn (x) fn (x) plays the role of characteristic classes. In fact, the asymptotic expansion at t → 0+ is very complicated. It is just the large numbers of cancelations between the i ≠ n terms that make it possible to consider only the i = n terms (see Eq. (5.8)). It is believed that the cancelations should be caused by a hidden supersymmetry- like algebraic structure in the heat equation approach. Hence, the following conjecture has been proposed.

Theorem: Let f(t) and g(t) be two solutions to a system of heat equations ∂ − f = ∆f, and ∂t ∂ − g = ∆†g, ∂t where ∆ = D†D is an non-negative elliptic operator. Then there exists a graded algebra structure that determines solutions. Appendix A

N¨other Theorem in a Mechanical System

The N¨othertheorem establishes a direct relation between symmetry and the conservative law in a classical physical system. If the classical action is invariant under any transformations characterized by time-independent parameters, then the classical mechanical system has one conservative (i.e. time-independent) physical quantity corresponding to each parameter. Be- cause successive transformations can form a Lie group, the N¨othertheorem provides a way to analyze the algebraic features of physical systems. In quantum theory, the conservative physical quantities correspond to the generators of a Lie group.

Let us consider a classical mechanical system described by a Lagrangian L[ϕµ(t), ϕ˙µ(t)]. The classical action is ∫

S = dtL[ϕµ(t), ϕ˙µ(t)], µ = 1, 2, ··· , n, (A.1) where ϕµ can be either commutative variables or Grassmann variables. Assume that the action is invariant under certain transformations whose inﬁnitesimal version is

δϕµ = ϵifiµ(ϕ), i = 1, 2, ··· , p (A.2) where ϵi are inﬁnitesimal constant transformation parameters. Then we have ∫ ∫ ( ) ∂L ∂L δS = dtδL = dt δϕµ + δϕ˙µ ∂ϕµ ∂ϕ˙µ ∫ [ ( ) ( )] ∂L d ∂L d ∂L = dt δϕµ − + δϕµ ∂ϕµ dt ∂ϕ˙µ dt ∂ϕ˙µ ∫ [ ] d ∂L = dt ϵi fiµ(ϕ) = 0, (A.3) dt ∂ϕ˙µ where we have used the Euler-Lagrange equation (3.4). In addition, when ϕµ are Grassmann variables (see Appendix B), the anticommutative property must be taken into account in the calculation. Because ϵi are arbitrary parameters, Eq. (A.3) yields conservative laws [ ] d ∂L fiµ(ϕ) = 0, (A.4) dt ∂ϕ˙µ

81 82 N¨otherTheorem and we obtain p conservative (or time-independent) quantities corresponding to ϵi, ∂L Qi = fiµ(ϕ) = πµfiµ(ϕ), (A.5) ∂ϕ˙µ where πµ is the conjugate momentum of ϕµ (see Eq. (3.6)). By Poisson brackets (3.13), the symmetry transformations (A.2) can be expressed as

− δϕµ = [ϵiQi, ϕµ]PB . (A.6)

At a quantum level, the variable becomes an operator, the Poisson bracket is replaced by a Lie bracket, and (A.6) becomes [ ] b b b δϕµ = −i ϵiQi, ϕµ . (A.7)

The ﬁnite version of Eq. (A.7) is [ ] [ ] b b b b ϕµ −→ exp −iϵiQi ϕµ exp iϵiQi . (A.8) [ ] b Therefore, the successive transformations form a continuous Lie group since exp iϵiQi is a b representation of a group element of the Lie group with generators Qi. Appendix B

Grassmann Variable Calculus

Grassmann variables are anticommutative functions, and they are essential dynamical variables in a theory with supersymmetry. The following introduces the calculus on Grassmann variables, and shows distinct features from usual commutative variables.

1. Grassmann algebra

Grassmann variables are deﬁned by the algebra they satisfy. A Grassmann algebra A is an algebra over the ﬁeld R or C constructed from a set of generators θi under anticommutative product [21]:

{θi, θj} ≡ θiθj + θjθi = 0, i, j = 1, ··· , n. (B.1)

2 A n Obviously, Eq. (B.1) implies θi = 0. It is easy to see that is a vector space of dimension 2 on R or C, with a basis of monomials of degree k,

··· ≤ ≤ θi1 θik , 0 k n. (B.2)

2. Grassmann Parity

Grassmann parity is a useful notion for defining differentiation and integration over Grass- mann variables. On the algebra A, there exists a simple automorphism P which is a reflection defined by

P (θi) = −θi. (B.3)

Then under the action of P the basis listed in (B.2) behaves as,

··· − k ··· P (θi1 θik ) = ( 1) θi1 θik . (B.4)

3. Grassmann diﬀerentiation

83 84 Grassmann Variables

The diﬀerentiation operator D on A is a linear mapping D : A → A such that

D(λ1f1 + λ2f2) = λ1D(f1) + λ2D(f2), (B.5) for λ1,λ2 ∈ R or C and f1, f2 ∈ A, satisfying the condition

D(f1f2) = D(f1)f2 + P (f1)D(f2). (B.6)

D can be explicitly represented by its action on the generators θi [21], ∂ θj = δij. (B.7) ∂θi

The diﬀerential operators ∂/∂θi, as a representation of D realized on A, can be understood as derivatives with respect to θi as the case of commutative variables. It is easy to ﬁnd ∂/∂θi, together with θi considered as operators acting on A by left-multiplication, satisfying the following anticommutation relations [21],

∂ ∂ ∂ ∂ ∂ ∂ θiθj + θjθi = 0, + = 0, and θi + θi = δij. (B.8) ∂θi ∂θj ∂θj ∂θi ∂θj ∂θj

Thus these diﬀerential operators ∂/∂θi, and generators θi, form a Cliﬀord algebra (see Appendix C).

4. Grassmann Integration

The integration operator I is a linear operator acting on A such that

I(λ1f1 + λ2f2) = λ1I(f1) + λ2I(f2) (B.9) satisfying

ID = 0,DI = 0, (B.10) and

I(fg) = I(f)g if D(g) = 0. (B.11)

The ﬁrst condition in Eq. (B.10) means that the integration of a total derivative term should vanish in the absence of boundary terms. The second condition implies that once a Grassmann variable has been integrated over, the value obtained by the integration no longer depends on this variable. Finally the condition in Eq. (B.11) implies that a factor whose derivative with respect to a Grassmann variable vanishes can be taken out of the integration.

The explicit representation on the∫ generators θi of the integral operator I that satisﬁes the above three conditions, denoted as dθi, can be chosen to be ∂/∂θi: ∫ ∂ dθi f ≡ f, ∀f ∈ A. (B.12) ∂θi This is one of the most distinct features of Grassmann integration from the integration over commutative variables. Grassmann Variables 85

5. Change of Variables in A Grassmann Integral

We have the following result on the change of variable in multiple variable integrations over Grassmann variables: Theorem B.1. Consider the following multiple variable integration over a number of Grass- mann variables, ∫

dθ1 ··· dθnf(θ). (B.13)

If there is a change of variables,

′ θi = θi(θ ), (B.14) then there exists ∫ ∫ ··· ′ ··· ′ ′ ′ dθ1 dθnf(θ) = dθ1 dθn J(θ ) f(θ ), (B.15) with the Jacobian determinant, ( ) ∂θi J = det ′ . (B.16) ∂θj

Use the following simple example to verify the result. Making the change of variables by a linear transformation constructed from commutative numbers,

′ θi = aijθj, (B.17) and using the identity

′ ′ ′ ′ ′ ′ ··· ··· ··· ··· ··· ··· θn θ1 = anin θin a1i1 θi1 = anin a1i1 ϵin i1 θn θ1 = det(aij) θn θ1, (B.18) and substituting (B.18) into the following integration ∫ ∫ ··· ··· ′ ··· ′ ′ ··· ′ 1 = dθ1 dθn θn θ1 = dθ1 dθn θn θ1 ∫

= dθ1 ··· dθnJ det(aij)θn ··· θ1, (B.19)

′ we obtain J = det(aij) = det(∂θ/∂θ ).

6. Gaussian Integrals with Grassmann Variables

1. Gaussian integral over a pair of complex conjugated Grassmann numbers To introduce Gaussian integrals over Grassmann variables, we need to understand the Gaussian integral over a pair of complex conjugated Grassmann numbers.

First, given two real Grassmann numbers θ1 and θ2, deﬁne two conjugate complex Grass- mann numbers, 1 1 η = √ (θ1 + iθ2) , and η = √ (θ1 − iθ2) . (B.20) 2 2 86 Grassmann Variables

It is easy to verify

η2 = η2 = 0, and ηη + ηη = 0. (B.21)

Then with the deﬁnition 1 1 dη = √ (dθ1 − idθ2) , and dη = √ (dθ1 + idθ2) , (B.22) 2 2 we obtain ∫ ∫ ∫ ∫ ∫ ∫ dηη = dηη = 1, dηη = dηη = 0, and dη = dη = 0. (B.23)

Then we have ∫ ∫ ∫ dηdη exp (−ηη) = dηdη (1 − ηη) = dηdηηη = 1. (B.24)

2. Gaussian integral over n pairs of complex conjugated Grassmann numbers Let us consider a Gaussian integral over n pairs of complex conjugate Grassmann numbers: ∫ ··· − I(a) = dη1dη1dη2dη2 dηndηn exp ( ηiaijηj) . (B.25)

Using the change of variables

′ aijηj = ηi, (B.26)

and the properties shown in (B.15) and (B.16),

··· ′ ··· ′ dη1 dηn = (det a) dη1 dηn, a = (aij) , (B.27)

we have ∫ ∫ ( ) ′ ··· ′ − ′ I(a) = det a dη1dη1 dηndηn exp ηiηi = det a. (B.28)

Further, using the result (B.28), (B.20) and (B.22) we obtain the Gaussian integral over n real Grassmann variables, ∫ ( ) { √ 1 ± det a, n=even positive integer; dθ ··· dθ exp − θ a θ = (B.29) n 1 2 i ij j 0, n=odd positive integer. Appendix C

Cliﬀord Algebra and Spinor

Spinors are elements of a complex vector space, and are represented by complex row or column vectors. They furnish a representation for the Cliﬀord algebra.

1. Cliﬀord Algebra and Matrix Representation

A Cliﬀord algebra with n generators γµ, µ = 1, ··· , n is deﬁned by the following relations,

{γµ, γν} = γµγν + γνγµ = 2δµν, µ, ν = 1, 2, ··· , n. (C.1)

By deﬁning

1 γ ··· = γ γ ··· γ , (C.2) µ1µ2 µp p! [µ1 µ2 µp] we can construct a ﬁnite group Cn under multiplication,

{± ± ± ··· ± } Cn = 1, γµ, γµ1µ2 , , γµ1···µn . (C.3)

··· Because γµ1µ2···µp is completely antisymmetric with respect to the indices µ1, µ2, , µp, there are ( ) n p

distinct γµ1µ2···µp ’s. Hence the order of Cn is ( ) ∑n n |C | = 2 = 2 × 2n = 2n+1. (C.4) n p p=0

In the following we work out the matrix representation of the Cliﬀord algebra in the case that n is even, i.e., n = 2k, since it is the only case we need in this paper. First, consider the following list of results in the theory of a ﬁnite group [29]:

87 88 Spinor

1. Theorem C.1. The conjugation class [a] of a group element a ∈ G is deﬁned by { } [a] = gag−1, ∀g ∈ G . (C.5)

Then the number of irreducible representations of any ﬁnite group G equals the number of its conjugation class.

2. Theorem C.2. The commutator group C(G) of a group G is deﬁned by { } C(G) = aba−1b−1, ∀a, b ∈ G . (C.6)

Let N denote the number of inequivalent one-dimensional representations of a ﬁnite group G. Then |G| N = , (C.7) |C(G)|

where |G| and |C(G)| are orders of G and C(G).

3. Theorem C.3. Let |G| denote the order of a ﬁnite group G. If G has p irreducible inequivalent representations of dimension di, i = 1, ··· , p, then

∑p 2 |G| = (di) . (C.8) i=1

The following applies the above three results to Cn. First, by using Eq. (C.1) it is easy to n show that Cn has 2 + 1 conjugation classes:

− ··· [+1], [ 1], [γµ], [γµ1µ2 ], , [γµ1µ2···µn ]. (C.9)

n Hence by Theorem C.1, Cn has 2 + 1 irreducible representations. Second, for n = 2k, we have

C(Cn) = {+1, −1} , (C.10) and hence the number of inequivalent one-dimensional irreducible representations is

|C | 2n+1 N = n = = 2n. (C.11) |C(Cn)| 2

From above, the total number of irreducible representations is 2n +1, leaving only one irreducible representation whose dimension is greater than 1.

According to Theorem C.3., we have

2n+1 = 122n + d2, (C.12) where d denotes the dimension of the only irreducible representation whose dimension is greater than 1. Hence

d = 2n/2. (C.13) Spinor 89

n/2 n/2 Therefore, γµ can be represented by 2 × 2 matrices. Consequently, a spinor ψ is column vector with 2n/2 complex components,   ψ  1     ψ2  ψ = [ψα] =  .  . (C.14)  . 

ψ2n/2

The space formed by spinor provides a matrix representation space for γµ.

2. Decomposition of Spinor Space S

When n = 2k, deﬁne a matrix from the n γ-matrices,

γn+1 = γ1γ2 ··· γn. (C.15)

A straightforward calculation shows

2 − n(n−1)/2 γn+1 = ( 1) I, (C.16) where I is an 2n/2 × 2n/2 identity matrix. This means

2 γn+1 = I for n = 0 mod 4; 2 − γn+1 = I for n = 2 mod 4. (C.17)

Subsequently, for n = 2 mod 4, we deﬁne 1 ψ± = (I ± γ )ψ, and γ ψ± = ±ψ±. (C.18) 2 n+1 n+1 For n = 4 mod 4, we deﬁne

2 γen+1 = iγn+1, γe = 1, 1 ψ± = (I ± γe )ψ, γe ψ± = ±ψ±. (C.19) 2 n+1 n+1

ψ± are called chiral (or Weyl) spinors with chirality ±1. Therefore, the spinor space can decompose a direct sum of two subspaces composed of chiral spinors as,

+ − + + S = S ⊕ S ,S = {ψ+} ,S = {ψ−} . (C.20) 90 Spinor Bibliography

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