Dirac Operators

Undergraduate Dissertation Dirac Operators Author: Diego Mojón Álvarez Supervisor: Jesús Antonio Álvarez López 2019/2020 UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Degree in Mathematics Undergraduate Dissertation Dirac Operators Author: Diego Mojón Álvarez Supervisor: Jesús Antonio Álvarez López July 2020 UNIVERSIDADE DE SANTIAGO DE COMPOSTELA Trabajo propuesto Área de Conocimiento: Geometría y Topología Título: Operadores de Dirac Breve descripción del contenido 1. Se empezará con una introducción a las variedades spin y geometría spin, con énfasis en los ejemplos. 2. También se hará una introducción a operadores diferenciales en variedades. 3. Se definirá el operador de Dirac y se estudiarán sus propiedades básicas. 4. Se culminará con el estudio espectral del operador de Dirac en variedades compactas. Se harán algunos cálculos concretos. Si llegase el tiempo, se probarían cotas inferiores del primer autovalor no nulo. Recomendaciones Es conveniente cursar la materia Variedades Diferenciables. En el caso de estudiantes del doble grado de Matemáticas y Física, es interesante cursar la materia Teoría Cuántica de Campos. Otras observaciones Esta propuesta de TFG es adecuada para ser complementada con alguna propuesta de TFG del Grado en Física. El operador de Dirac juega un papel importante en Mecánica Cuántica. iii Contents Abstract vii Preface ix 1 Preliminaries 1 1.1 Fiberbundles................................... 1 1.1.1 General fiber bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Vectorbundles .............................. 3 1.1.3 Principalbundles............................. 5 1.2 Connections.................................... 6 1.2.1 Connections on vector bundles . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Parallel transport and connections on principal bundles . . . . . . . 7 1.3 Fundamentals of Riemannian geometry . . . . . . . . . . . . . . . . . . . . . 8 1.4 Differentialoperators............................... 9 2 Spin manifolds and the spinor bundle 13 2.1 Spin groups and their representations . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Cliffordalgebras ............................. 13 2.1.2 The Pin and Spin groups . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.3 Spin representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Thespinorbundle ................................ 26 2.3 Connections on spinor bundles . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 The Dirac operator 29 3.1 Definition of the Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Basic properties of the Dirac operator . . . . . . . . . . . . . . . . . . . . . 30 3.3 The Schrödinger-Lichnerowicz formula . . . . . . . . . . . . . . . . . . . . . 35 3.4 Conformalcovariance............................... 38 v vi CONTENTS 4 The Dirac spectrum on compact manifolds 43 4.1 Explicit computations of spectra . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.1 Flattori.................................. 44 4.1.2 Spherical space forms . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Lower nonzero eigenvalue bounds . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.1 Bounds on closed manifolds . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.2 Brief note on bounds on manifolds with boundary . . . . . . . . . . 54 Bibliography 57 Index 59 Abstract This work is a mostly self-contained survey on Dirac operators. It starts by laying the fundamental building blocks at the heart of spin geometry, specifically its elemental geometrical and algebraic aspects. Thus, the concepts of vector and principal bundles over manifolds, Clifford algebras, the Pin and Spin groups, the spin representation and the spinor bundle are explored. A brief commentary on connections and linear differential operators on manifolds is also provided. Subsequently, the fundamental Dirac operator is presented, along with a review of its most important basic properties. The last section is devoted to a study of the Dirac spectrum on compact manifolds, including some explicit computations and bounds of the lower nonzero eigenvalue. Keywords— fiber bundles; connections; Clifford algebras; Spin group; spin representation; spinor bundle; Dirac operator; Dirac spectrum Resumen Este trabajo es un estudio esencialmente autocontenido de los operadores de Dirac. Empieza por establecer las bases fundamentales de la geometría espinorial, específicamente sus aspectos geométricos y algebraicos elementales. Así, se exploran los conceptos de fibrados vectoriales y principales, así como las álgebras de Clifford, los grupos Pin y Spin, la representación espinorial y el fibrado de espinores. También se proporciona un breve comentario sobre conexiones y operadores diferenciales en variedades. A continuación, se presenta el operador fundamental de Dirac, junto con un repaso a sus propiedades básicas más importantes. La última sección está dedicada al estudio espectral del operador de Dirac en variedades compactas, incluyendo algunos cálculos concretos y estimaciones de las cotas inferiores del primer autovalor no nulo. Palabras clave— fibrados; conexiones; álgebras de Clifford; grupo espinorial; representación espinorial; fibrado de espinores; operador de Dirac; espectro de Dirac vii 4CHAPTER 1. PRELIMINARIES example, let E, F be vector bundles over Mwith ranks r, q, and defining cocycles {Ui, gij} and {Ui, hij}, respectively (after refinement, the same open cover can be taken in both cocycles). Then E⊕Fis the vector bundle of rank r+qwith defining cocycle {Ui, kij}, where kij(x) = gij(x)⊕hij(x)∈GL(Rr⊕Rs)≡GL(r+s, R). Similarly, it is possible to define E⊗F,E∗,VE,EE, etc. Here, the notation is used for the symmetric tensor product (see [16] for more information on ). Note that E∗E∗is the vector bundle whose fibers consist of the symmetric bilinear forms on the fibers of E. Example 1.1.1 (Tangent bundle).Let {Ui, xi}be an atlas of a manifold Mof dimension n. The tangent space T M is the vector bundle over Mof rank ndefined by the cocycle given by the differential of the changes of coordinates, D(x−1 j◦xi) : Ui∩Uj→GL(n, R), p7→ D(x−1 j◦xi)(xi(p)). Its smooth sections are the (tangent)vector fields, and the notation X(M) = Γ(TM)is used. An Euclidean structure of a vector bundle Eof rank ris the choice of a positive definite scalar product gxon every fiber Exsuch that x7→ gxis a smooth section gof E∗E∗. If Eis equipped with a Euclidean structure, then it is called a Euclidean vector bundle; this is the same as reducing its structural group to O(r). An orientation of a vector bundle Eof rank ris given by a reduction of its structural group to SL(r, R), or to SO(r)after equipping it with some Euclidean structure. Not every vector bundle is orientable; for instance, the orientability of T M as a vector bundle means the orientability of Mas a manifold. All of the above concepts have obvious complex versions, changing Rto C. In this case, the conjugate bundle Ecan also be considered, whose fibers are the conjugate vector spaces of the fibers of E. Then a Hermitian structure on a complex vector bundle Eis the choice of a Hermitian product gxon every fiber Exdefining a smooth section of E∗E∗. If E is equipped with a Hermitian structure, then it is called a Hermitian vector bundle; this is the same as reducing its structural group to U(r). Any real/complex vector bundle admits a Euclidean/Hermitian structure [4, pp. 73-76]. For any Euclidean/Hermitian vector bundle Eon a Riemannian manifold M≡(M, g), with Riemannian density νg, there is a scalar product h,iL2on Γc(E)defined by hs1, s2iL2=ZMhs1(x), s2(x)iνg(x). The corresponding Hilbert space completion is denoted by L2(E). 1.1. FIBER BUNDLES 5 1.1.3 Principal bundles This is another major group of fiber bundles that will be of interest here. Let Gbe a Lie group. Its left translations form a subgroup of Diffeo(G)isomorphic to G. They can be characterized as the only transformations of Gthat are equivariant with respect to the right action on itself by right translations. A (smooth)principal bundle with structural group G, or a principal G-bundle, is a fiber bundle π:P→Mwith typical fiber Gwhose structural group is the group of left translations on G. By the above characterization of left translations, it follows that there is a smooth free right action of Gon Pwhose orbits are the fibers. Isomorphisms and triviality of principal G-bundles are particular cases of their analogues for general fiber bundles with a given structural group. A principal G-bundle Pis trivial if it has a section s:M→P: a trivialization M×G→Pis given by (x, g)7→ s(x)·g. The reciprocal property is obvious. Example 1.1.2 (Hopf fibration).The Hopf fibration π:S3→S2is a principal bundle with structural group S1. It can be defined by considering S3and S1as the standard unit spheres in C2and C, giving rise to an action of S1on S2by scalar multiplication. Then π:S3→S2 can be considered as the projection to the orbit space, S3→S3/S1≡CP1≈S2[4, pp. 244]. This principal bundle is nontrivial because the fundamental groups of the total spaces are different. With more generality, for any free right action of a compact Lie group Gon a manifold P, by the slice theorem, the orbit space P/G is a manifold and the canonical projection P→P/G is a principal G-bundle. Example 1.1.3 (Frame bundle).Let η:E→Mbe a C∞vector bundle of rank r. A frame vat a point x∈Xis an ordered basis for the vector space Ex. In other words, it is a linear isomorphism e:Rr−→ Ex. The set Fxof all frames at xhas a natural right action by the general linear group GL(r, R): every g∈GL(r, R)acts on the frame evia composition to give a new frame, e◦g:Rr−→ Ex. This action is free and transitive on Fx, since there is a unique invertible linear transformation sending one basis onto another in Rr. Then F(E) = G x∈M Fx becomes a GL(r, R)-principal bundle over M, so that the fiber over every x∈Mis Fx, and the local trivializations canonically induced by the local trivializations of E. This structure is known as the frame bundle of the vector bundle E[4, pp. 246-247]. Remark 1.1.4.Like in Example 1.1.3, on Euclidean (respectively, Hermitian) vector bun- 6CHAPTER 1. PRELIMINARIES dles, one can similarly define the principal bundle of orthonormal frames with structural group O(r)(respectively, U(r)). If the bundle is also orientable, the SO(r)-principal bundle of positively oriented orthonormal frames can also be built. The latter will be a central piece of this discussion. Example 1.1.5 (Vector bundle associated to a representation).Let Pbe a principal bundle with structural group Gand let ρ:G→GL(V)be a representation of Gon a vector space V. Define the following right G-action on P×V: (e, v)g= (eg, ρ(g−1)v). The quotient space P×ρVof P×Vby this action is a vector bundle over Mwith typical fiber V[6, p. 23]. It is known as the vector bundle associated to Pby ρ. The element of P×ρVrepresented by every (e, v)∈E×Vis denoted by [e, v]. 1.2 Connections The idea behind connections is extending the notion of the Euclidean directional derivative to a more general definition. This can be achieved for a wide array of mathematical structures. This work will keep its focus mostly on vector bundles, since it is the most useful case for the task at hand. However, some understanding of connections on principal bundles will be also needed, which will be given using the intuitive idea of parallel transport. 1.2.1 Connections on vector bundles Let π:E→Mbe a vector bundle over a manifold M. A connection on Eis a linear map ∇:X(M)⊗Γ(E)−→ Γ(E) such that the following properties hold for all X∈X(M),s∈Γ(E)and f∈ F := C∞(M): (i) ∇fX s=f∇Xs. (ii) ∇X(fs)=(Xf)s+f∇Xs(Leibniz rule). Remark 1.2.1.By (i) and (ii), for any p∈M, the value of (∇Xs)(p)only depends on the value of Xat pand the values of salong any given smooth path c: (−, )→M with c(0) = pand c0(0) = X(p). It follows that ∇can also be regarded as a linear map ∇: Γ(E)→Γ(T∗M⊗E)(the space of E-valued 1-forms) satisfying ∇(fs)=(df)·s+f∇s. This alternative definition is very useful in some cases. 1.2. CONNECTIONS 7 A connection on a Euclidean/Hermitian vector bundles Eover Mis said to be compatible with the metric or metric if, for every X∈X(M)and s, t ∈Γ(E), Xg(s, t) = g(∇Xs, t) + g(s, ∇Xt). Any Euclidean/Hermitian vector bundle admits a metric connection [4, pp. 73-76]. The concept of connection leads to several geometric ideas, the most important of which is the curvature operator. The curvature operator Kof a connection ∇on a vector bundle Eis a linear map K:X(M)⊗X(M)⊗Γ(E)−→ Γ(E) defined by K(X, Y )s≡KX,Y s=∇X∇Ys−∇Y∇Xs−∇[X,Y ]s . Since Kis F-linear (in all three arguments), it is defined pointwise. If K= 0, then ∇is said to be flat. A connection on the tangent bundle π:TM →Mof a manifold Mis more commonly referred to as an affine connection. The torsion of an affine connection ∇is the linear map T:X(M)×X(M)→X(M) defined by T(X, Y ) = ∇XY−∇YX−[X, Y ]. This operator is also F-linear (in its two arguments), and therefore it is defined pointwise. If T= 0, then ∇is said to be symmetric or torsion-free. 1.2.2 Parallel transport and connections on principal bundles Let π:E→Mbe a vector bundle equipped with a connection ∇. For every piecewise smooth path c: [a, b]→Mand v∈Ec(a), there is a unique piecewise smooth path ˜cv: [a, b]→Esuch that ˜cv(a) = v,π◦˜c=c(˜cis a lift of c) and ∇c0˜cv= 0. Moreover, for each t∈[a, b], the mapping v7→ ˜cv(t)defines a linear isomorphism Pc,t :Ec(a)→Ec(t), called the parallel transport along cup to t. This parallel transport Pdetermines ∇: ∇c0(a)s=d dtP−1 c,t sc(t)t=a∈Ec(a). Thus a connection can be also described by giving an abstract parallel transport satisfying certain properties. Going one step further, the connection on Einduces a so-called connection in the principal frame bundle F(E)(Example 1.1.3), which can be defined by extending the idea of parallel 8CHAPTER 1. PRELIMINARIES transport. For any piecewise smooth path c: [a, b]→M,t∈[a, b]and e:Rr→Ec(a)in Fc(a), let b Pc,t(e)be the composition Rre −−−−→ Ec(a) Pc,t −−−−→ Ec(t). This defines a GL(r, R)-equivariant diffeomorphism b Pc,t :Fc(a)→Fc(t), called the parallel transport in F(E)along cup to t. The above notion can be extended to define a parallel transport b Pon a principal G-bundle Pover M, assigning a G-equivariant diffeomorphism b Pc,t :Pc(a)→Pc(t)to any piecewise smooth path c: [a, b]→Mand t∈[a, b], satisfying certain properties. Such a parallel transport is a way to describe connections on principal bundles. Now, take a representation ρ:G→GL(V)and consider the associated vector bundle P×ρV(Example 1.1.5). Then b Pinduces a connection on P×ρV, whose parallel transport Pis given by Pc,t([e, v]) = [b Pc,t(e), v]. A more detailed description of connections on principal bundles is far from the main goals of this survey, so it is omitted. There is a large amount of literature regarding connections on principal bundles and associated vector bundles. The reader is referred to [4, Section 28], [17, Chapter 2]. 1.3 Fundamentals of Riemannian geometry The most basic notion that this work pulls from differential geometry is that of a Riemannian manifold. A Riemannian metric gon a manifold Mis a Euclidean structure on TM. Every manifold admits a Riemannian metric [4, p. 6]. An isometry between Riemannian manifolds is a diffeomorphism whose tangent map preserves the Riemannian metric at every tangent space. Local isometries are similarly defined using local diffeomorphisms. A particular, and greatly important, case of affine connection is the Levi-Civita connection of a Riemannian manifold M, which is the unique symmetric metric connection on M[15, p. 16-17], [6, p. 12]. Henceforth, Mwill be equipped with the Levi-Civita connection ∇. The curvature operator of the Levi-Civita connection is often referred to as the Riemann curvature operator, and denoted by R. In this case, in terms of a local frame (e1, . . . , en) of TM corresponding to local coordinates xi, the curvature operator can be put in a more physics-friendly, tensor form: Rej,ekel=Ri ljkei, 1.4. DIFFERENTIAL OPERATORS 9 where Einstein notation has been used. The Riemann curvature operator satisfies, for all X, Y, W, Z ∈X(M), RX,Y Z+RY,ZX+RZ,XY= 0 (the first Bianchi identity),(1.3.1) g(RX,Y Z, W) + g(RX,Y W, Z)=0. The symmetries of Rleave only one nontrivial contraction of the indices [15, p. 58-60], [6, p. 14]. This is known as the Ricci curvature tensor, and can be expressed, in components, as Ricab =Ri aib. It is symmetric. The last possible contraction is the scalar curvature, S=gabRicab . Example 1.3.1 (Flat Euclidean metric).The simplest and most well-known example of a Riemannian manifold is Rn≡(Rn, gc), where, using the standard coordinates on Rn, the metric gcis given by gc p((∂xi)p,(∂xj)p) = δij . This is often called the standard Riemannian structure on Rn, or the canonical Euclidean metric. This Riemannian manifold is flat. Reciprocally, any flat Riemannian manifold is locally isometric to Rn. Obviously, much more intricate Riemannian manifolds can be built. 1.4 Differential operators The theory of differential operators on manifolds provides most of the analytical tools one needs to study Dirac Operators. Notation 1.4.1.Amulti-index is an n-tuple α= (α1, . . . , αn)∈Nn. Set |α|=Pn k=1 αk. For each ξ∈Cn, let ξα=ξα1 1. . . ξαn n. Multi-index notation is used for differentiation with respect to local coordinates xion a manifold M: ∂|α| ∂xα≡∂|α| ∂(x1)α1. . . ∂(xn)αn≡∂|α| ∂α1 1. . . ∂αn n . Let Eand Fbe two smooth complex vector bundles over Mof ranks rand q, respectively. A (linear)differential operator of order kon Mis a linear map P: Γ(E)→Γ(F)satisfying the following property. For each point p∈M, take a neighborhood Uwith local coordinates xi, and local trivializations of Eand Fon U, giving rise to identities Γc(U, E)≡C∞ c(U, Cr) 10 CHAPTER 1. PRELIMINARIES and Γc(U, F)≡C∞ c(U, Cq). Then Pis required to satisfy P(Γc(U, E)) ⊂Γc(U, F)(Pis local), and the restriction P: Γc(U, E)≡C∞ c(U, Cr)→Γc(U, F)≡C∞ c(U, Cq)must be of the form P=X |α|≤k Aα(x)∂|α| ∂xα, where the local coefficients Aαare smooth maps U→Mr×q(C)≡Crq with Aα6= 0 for some αwith |α|=k. Example 1.4.2 (Connections).A connection ∇: Γ(E)→Γ(T∗M⊗E)on a vector bundle Eover Mis a differential operator of order one. In fact, it is universal in the sense that any differential operator of order one P: Γ(E)→Γ(F)on Mis a composition Γ(E)∇ −−−−→ Γ(T∗M⊗E)A −−−−→ Γ(F), for some zero order differential operator A. For a differential operator P: Γ(E)→Γ(F)of order kon M, its principal symbol σ(P) is a map that associates each point x∈Mand each covector ξ=ξjdxj∈T∗ xM(using Einstein notation), to a linear map σξ(P) = ikX |α|=k Aα(x)ξα:Ex−→ Fx. There is a coordinate-free way to define σ(P)as a section of (JkTM)⊗Hom(E, F)[8, p. 168]. It shows that the principal symbol does not depend on a particular choice of local coordinates or trivializations of the bundles. However, for the purposes of this work, the above definition of σ(P)will suffice as a means to analyze the Dirac operators. The invariance of the principal symbol under changes in local trivializations and coordinates allows for different classifications of differential operators, regarding what properties it possesses for certain classes of operators. The relevant class for this work is the following. It is said that Pis elliptic if, for every nonzero covector ξ∈T∗ xM, the principal symbol σξ(P) : Ex→Fxis an invertible mapping. Real differential operators and their symbols are defined in the same manner, only replacing Cwith R. Example 1.4.3 (Laplace-Beltrami operator).Let (Mn, g) be a Riemannian manifold. Consider the map ∆ : C∞(M)→C∞(M)on the space of smooth functions on Mgiven, in local coordinates xi, by ∆ = 1 √g∂j√g gjk ∂k=gjk ∂2 ∂j∂k +lower order terms, 1.4. DIFFERENTIAL OPERATORS 11 where Einstein notation has been used. Here, gjk dxjdxkis the metric tensor, g= det(gjk) and (gjk) = (gjk)−1. This mapping is known as the Laplace-Beltrami operator, and it is a second order differential operator. The principal symbol is easily determined for any cotangent vector ξ=ξkdxk: σξ(P) = −gjk ξjξk=−|ξ|2, which is invertible (as a complex linear map) for each ξ6= 0. The Laplace-Beltrami operator is hence elliptic. The following is a fundamental result for elliptic operators, which will be essential for the study of the Dirac operator in particular: Theorem 1.4.4. Let Mbe an n-dimensional compact Riemannian manifold, and let P: Γ(E)→Γ(E)be a self-adjoint elliptic differential operator of order k > 0. Then, the spectrum of Pis real and discrete. Each eigenspace of Pis finite-dimensional and consists of smooth sections. Moreover, the eigenspaces Eλ(λ∈Spec(P)) of Pprovide an orthogonal direct sum decomposition for the Hilbert space L2(E), L2(E) = M λ Eλ. Proof. It is somewhat technical and does not provide much insight on the topic of this survey. See [8, pp. 196-197].  Chapter 2 Spin manifolds and the spinor bundle The first step in the journey to build the Dirac operator is defining the structure on which it operates. As previously stated, this endeavor not only requires pulling tools from differential geometry, but also some algebraic notions. This chapter is devoted to exploring those notions and intertwining them with previous geometric aspects to further the reader’s understanding of spin geometry. 2.1 Spin groups and their representations This section presents the essential algebraic ideas at the heart of spin geometry. They provide the necessary tools to build the spinor bundle, which is essential in this discussion, since the Dirac operator acts on the space of its smooth sections. 2.1.1 Clifford algebras Definition 2.1.1. Let Vbe a vector space over a commutative field K. Let T(V) = ∞ M r=0 O r V denote the tensor algebra of V(see [7] for more information on tensor algebras). Suppose qis a quadratic form on V, and define Jq(V)to be the ideal in T(V)generated by the elements of the form v⊗v+q(v)1 (where 1is the unit of T(V)). The Clifford algebra of 13 20 CHAPTER 2. SPIN MANIFOLDS AND THE SPINOR BUNDLE Definition 2.1.14. Let Kbe a field of characteristic 6= 2. Then Kis spin if at least one of the equations t2=±acan be solved in Kfor each a∈K∗. Most of the more well-known fields are spin. For example, Rand Cboth verify the spin condition. The real case holds most of the practical interest for the purposes of this work. Finally, all the information contained in this section comes together for its main result. Theorem 2.1.15. Let V be a finite-dimensional vector space over a spin field K. Let qbe a nondegenerate quadratic form on V, and i=√−1. There are short exact sequences 1→F→Pin(V, q)ρ −→ O(V, q)→1, 1→F→Spin(V, q)ρ −→ SO(V, q)→1, where F=   Z2={1,−1}if i /∈K Z4={±1,±i}if i∈K. Proof. There are two steps in this proof. First, one has to characterize the kernel of ρand make sure that it is the image of the monomorphisms F→Pin(V, q)or F→Spin(V, q) that canonically maps elements of Fto their analogues in K(considered as a subset of the Pin or Spin groups). Then one must check the surjectivity of the mappings ρ. For the first stage, consider an element x=v1···vr∈Pin(V, q)such that x∈ker(ρ). By Proposition 2.1.10, x∈K∗. Moreover, x2=N(v1)···N(vr) = ±1. This characterizes ker(ρ)for both possible field types. For the surjectivity of ρ, consider the fact that the reflection ρvis the same as ρtv for every nonzero t∈K. Now, take the q-normalization q(tv) = t2q(v) = ±1. Since Kis spin, at least one of the equations t2=±q(v)−1can be solved in K. Coupling this fact with the Cartan-Dieudonné Theorem, it is clear that elements of O(V, q) or SO(V, q)can be written as suitable compositions of rreflections across the orthogonal subspaces of vectors of q-length 1. The surjectivity of ρimmediately follows.  Theorem 2.1.15 is the most important result this work has dealt with so far. It characterizes the Spin group in a way that will be essential in Section 2.1.3. Up until this point, the line of reasoning has been purposefully kept as general as possible, but now it becomes necessary to focus on the real case, K=R. 2.1. SPIN GROUPS AND THEIR REPRESENTATIONS 21 Let Vbe an n-dimensional real vector space endowed with a positive definite quadratic form q, so there is a basis for V∼ =Rnsuch that q(x) = x2 1+···+x2 n. For the sake of simplicity, the notation will be simplified to O(V, q)≡On, SO(V, q)≡SOn. Similarly, the corresponding Clifford algebra and the Pin and Spin groups will be named Cln, Pinn, Spinnrespectively. In particular, Spinnpossesses some interesting properties. Theorem 2.1.16. Let n∈N. The group Spinnsatisfies (i) For n≥2,Spinnis a connected group. (ii) For n≥3,Spinnis simply connected and ρ:Spinn→SOnis the universal covering of SOn. Proof. Consider the short exact sequence 1→Z2→Spinn ρ −→ SOn→1 given by Theorem 2.1.15. This establishes Spinnas a twofold cover of SOn(since ρx=ρ−x). To prove that Spinnis connected, it is enough [12, p. 16] to find a path connecting the elements ±1∈Spinn. To that end, choose two q-orthogonal vectors e1, e2∈Rnsuch that q(e1) = q(e2) = 1, and define a path γ: [0,1] →Spinnby γ(t) = cos(πt) + sin(πt)e1e2 =hcos π 2te1+ sin π 2te2ihsin π 2te2−cos π 2te1i, which satisfies γ(0) = 1,γ(1) = −1. The simple connectedness of Spinn,n≥3follows from standard computations in homotopy theory.  Basically, Theorem 2.1.16 states that Spinnis a nontrivial twofold cover of SOn. The reader should keep this fact in mind for future sections, since it helps to quickly characterize certain Spin groups. For example, the well-known diffeomorphism SO3≈RP3implies that Spin3≈S3. 2.1.3 Spin representation The application of Clifford algebras, and more particularly, of Spinn, to this work does not come in a direct, clean-cut way. A basic understanding of their representations is necessary. With this in mind, consider a quadratic vector space (V, q)over a field K. 22 CHAPTER 2. SPIN MANIFOLDS AND THE SPINOR BUNDLE Definition 2.1.17. Let Fbe a field such that K⊂F. An F-representation of Cl(V, q)is aK-algebra homomorphism λ:Cl(V, q)−→ EndF(W), where Wis a finite-dimensional vector space over F, and it is called a Cl(V, q)-module over F. The dimension of the representation is the dimension of W,dimF(W). One can define the operation λ(x)(w)between an element x∈Cl(V, q)and w∈W. When there is no confusion, the notation can be simplified to x·w. This product is referred to as a Clifford multiplication. For more information on representations, see [5]. As was the case in the previous section, this analysis will focus on Clnand its complex analogue, ClC n=Cln⊗RC. Definition 2.1.18. Consider the following complex vector space: ∆n:=C2k∼ =O k C2, n = 2k, 2k+ 1 . This structure is known as the vector space of complex n-spinors. Its elements are called complex spinors. It can be proved [12, p. 13] that ClC n∼ =   EndC(∆n)if nis even EndC(∆n)⊕EndC(∆n)if nis odd . As a result, one can define the spin representation δnof ClC n. In the even case, this is nothing but the isomorphism δn:ClC nEndC(∆n) ∼ =. The odd case is a little more convoluted, since one needs to compose the isomorphism with the projection onto the first component: δn:ClC nEndC(∆n)⊕EndC(∆n)EndC(∆n). ∼ =p1 Now, consider Clnas a subset of ClC n. By taking Spinn⊂Cln⊂ClC n, one can restrict the representation δnto obtain the spin representation of Spinn, δn:Spinn−→ Aut(∆n)≡GL(∆n). As usual, the same symbol is used to denote both the original mapping and its restriction. Proposition 2.1.19. The spin representation of Spinnis faithful (i.e. δnis injective). Proof. If nis even, then ClC n∼ =End(∆n), so the statement is trivial. Moving on to 2.1. SPIN GROUPS AND THEIR REPRESENTATIONS 23 n= 2k+ 1, as vector spaces, ∆2k= ∆2k+1, and Spin2kcan be thought of as a subset of Spin2k+1. Hence the diagram Spin2kGL(∆2k) Spin2k+1 GL(∆2k+1) i δ2k δ2k+1 commutes. This implies that the normal subgroup H:=ker(δ2k+1)satisfies H∩Spin2k={1}. Now, elements e A∈SO2kcan be identified with elements of the form diag(e A, 1) ∈SO2k+1. Given that ρis surjective, ρ(H)⊂SO2k+1 is normal. Moreover ρ(H)∩SO2k={I}. Choose A∈ρ(H)⊂SO2k+1. Its characteristic polynomial has odd degree, so either 1 or −1is a root. Since for an orientation-preserving matrix, both complex roots and -1 can only appear in pairs, 1 has to be a root3, so there is a unit vector vsuch that A(v) = v. Thus there is an element B∈SO2k+1 such that BAB−1∈SO2k. Since ρ(H)is normal, this implies BAB−1=I, hence ρ(H) = {I}. Since −1/∈H, this leaves only the option H={1}, so δ2k+1 is injective.  This proposition is not as unassuming as it might look. It actually implies that the spin representations do not arise as lifts of representations of SOn, since such a lift necessarily contains {±1}in its kernel. Going back to the concept of Clifford multiplications (Definition 2.1.17), fix a vector x∈ Rn⊂Cln⊂ClC n. Via the representation δn,xcan be considered as an endomorphism of ∆n. Since xis arbitrary, the Clifford multiplication of vectors and spinors can be defined as the linear map µ:Rn⊗R∆n−→ ∆n, where µ(x⊗ψ) = δn(x)(ψ)≡x·ψfor x∈Rn,ψ∈∆n. Furthermore, this map can be extended to the exterior algebra V•Rn. Let (e1, . . . , en)be a positively oriented orthonormal basis of Rn. For a multivector w∈V•Rn, w=X i1<···<ik wi1...ikei1∧···∧eik, 3This is clear by looking at the canonical form of orthogonal transformations. Geometrically, by the Cartan-Dieudonné Theorem, A∈SO2k+1 corresponds to the composition of, at most, 2kreflections across hyperplanes, so there is always a vector in the intersection of such hyperplanes that remains invariant by the reflections. 24 CHAPTER 2. SPIN MANIFOLDS AND THE SPINOR BUNDLE define µ(w⊗ψ) = w·ψ=X i1<···<ik wi1...ikei1···eik·ψ. It can be shown [12, pp. 21-22] that µis a homomorphism of representations of Spinn, and that for x∈Rnand w∈V•Rn, one has the equality x·(w·ψ)=(x[∧w)·ψ−(xyw)·ψ , where ydenotes the interior product. Since ψis arbitrary, the spinor is usually dropped and, making use of the isomorphism Cln∼ =V•Rn, the following equation is used to characterize the Clifford product by vectors x∈Rn⊂Cln, x· ≡ x[∧ −xy.(2.1.5) The reader should keep (2.1.5) in mind, since it will be used in future sections. This knowledge about the Clifford multiplication can be used to delve deeper into the properties of the spin representation. Take the so-called “complex volume form” of Rn, ωC n=ib(n+1)/2ce1···en. In the case n= 2k, the endomorphism f:=δ2k(ωC 2k):∆2k−→ ∆2k satisfies f(δ2k(g)(ψ)) = δ2k(g)(f(ψ)), since x·ωC 2k=ωC 2k·α(x), for every x∈Cln[8, p. 22] (so ωC 2kis central in Cl0 n). Hence it is an automorphism of the representation δn. Moreover (ωC 2k)2=i2k(−1)k(2k−1)e1···e2k·e2k···e1= (−1)k(−1)−k(−1)2k= 1 , so f2=id∆n. The spin representation decomposes into two subspaces, ∆2k= ∆+ 2k⊕∆− 2k,∆± 2k:={ψ∈∆2k:f(ψ) = ±ψ}. The spinors belonging to ∆± 2kare often referred to as positive and negative Weyl spinors, respectively. The representation δnsplits into δn=δ+ n⊕δ− n. The next step of this analysis is proving some of the properties of ∆2k+1 and ∆± 2k. This will culminate in the proof that the representations δ2k+1 and δ± 2kare irreducible. Proposition 2.1.20. The following affirmations regarding the spaces ∆nhold: (i) dimC(∆2k+1)=2k. (ii) dimC(∆± 2k) = 2k−1. (iii) Let x∈R2kand ψ±∈∆± 2k. Then x·ψ±∈∆∓ 2k, so Clifford multiplication induces homomorphisms µ:R2k⊗R∆± 2k→∆∓ 2k. 2.1. SPIN GROUPS AND THEIR REPRESENTATIONS 25 Proof. Property (i) is clear by the definition of ∆2k+1. Property (iii) follows from the following Clifford relation: x·e1···e2k=e1···e2k·α(x) = −e1···e2k·x, for x∈Rn\{0}. This implies that Clifford multiplication by any vector anti-commutes with f, thus it maps ∆± 2konto ∆∓ 2kin a bijective manner. Consequently, both subspaces have the same dimension, and dim(∆± 2k)=2k/2 = 2k−1, obtaining property (ii).  The irreducibility of these representations is a little trickier, since it requires some algebraic notions that, albeit well-known, have not been explored here. For the sake of simplicity, the following lemma, which is key to subsequent propositions, will be cited without proof. Lemma 2.1.21 (As seen in [12], p. 23).Let V, W be complex vector spaces such that dim(W)<dim(V). Then there are no nontrivial algebra homomorphisms End(V)→ End(W). Proposition 2.1.22. The representations δ± 2kare irreducible, i.e., there are no proper subspaces which are invariant by the representation. Proof. The treatment is analogous for both subspaces, so only the positive case will be explained. Proceed by contradiction. Assume that {0} 6=W(∆+ 2kis a proper invariant subspace. The Clifford products eiej(i < j)belong to Spin2k, so Wis invariant under Clifford multiplication by them (eiej·W⊂W). Additionally, the set {eiej:i < j} generates the even part of the complex Clifford algebra, (ClC 2k)0. Hence there is a complex (nontrivial) representation f: (ClC 2k)0→End(W). Given that (ClC 2k)0∼ =ClC 2k−1∼ =End(∆2k−1)⊕End(∆2k−1)and that dim(W)<dim(∆2k−1), by Lemma 2.1.21, fis trivial. This contradiction comes from assuming that Wis a proper invariant subspace.  Proposition 2.1.23. The representation δ2k+1 :Spin2k+1 →∆2k+1 is irreducible. Proof. The argument is analogous to the even case. Now, Spin2k+1 ⊂(ClC 2k+1)0⊂ClC 2k+1 ∼ =End(∆2k+1)⊕End(∆2k+1). Let {0} 6=W(∆2k+1 be a proper invariant subspace. As before, it follows that Wis also invariant by the action of (ClC 2k+1)0(since it is generated by elements of Spin2k+1), so there is a nontrivial representation f: (ClC 2k+1)0−→ End(W). 26 CHAPTER 2. SPIN MANIFOLDS AND THE SPINOR BUNDLE Now (ClC 2k+1)0∼ =ClC 2k∼ =End(∆2k). Since dim(W)<dim(∆2k+1) = dim(∆2k), by Lemma 2.1.21, fis once again trivial.  2.2 The spinor bundle In this section, all the previously discussed geometrical and algebraic notions come together to finally define the structure on which Dirac operators act. From now on, given an oriented Riemannian manifold (Mn, g), let FSO(TM)denote the SOn-principal bundle of its positively oriented orthonormal frames. Definition 2.2.1. Let (Mn, g)be an oriented Riemannian manifold of dimension n. A spin structure on (Mn, g)is a Spinn-principal bundle Spin(TM)→M, together with a twofold covering map Spin(T M)FSO(T M) ηcompatible with the respective group actions, i.e., such that the diagram Spin(TM)×SpinnSpin(TM) M FSO(T M)×SOnFSO(T M) η×ρη commutes. A spin manifold is an oriented Riemannian manifold with a spin structure. Not every manifold is spin. The conditions for the existence of a spin structure on a manifold (Mn, g)are of topological nature [13, p. 2]. Henceforth, when referring to a manifold M, it will be assumed that it is spin. Definition 2.2.2. The spinor bundle of a manifold Mis the complex vector bundle associated to Spin(TM)via the spin representation, ΣM=Spin(TM)×δn∆n. Recall that, for neven, ∆n= ∆+ n⊕∆− nand δn=δ+ n⊕δ− n. Hence the positive and negative spinor bundles can be defined as ΣM±=Spin(TM)×δ± n∆± n, so ΣM= ΣM+⊕ΣM−. The spinor bundle is thus a vector bundle with typical fiber ∆n (or ∆± n, in the positive and negative cases). Elements of ΣMare known as spinors. Just 2.3. CONNECTIONS ON SPINOR BUNDLES 27 as sections of the tangent bundle, X∈X(M), are called vector fields, sections of the spinor bundle, ψ∈Γ(ΣM), are called spinor fields4or, sometimes, just spinors. The Clifford multiplication of multivectors and spinors can be extended to a Clifford multiplication on the spinor bundle. One starts with the fiberwise action given by T∗M⊗ΣM−→ ΣM, X[⊗ψ7→ X·ψ:=δn(X)(ψ), such that X·Y·ψ+Y·X·ψ=−2g(X, Y )ψ. (2.2.1) Just like in Section 2.1.3, for a k-form w=Pi1<···<ikwi1...ike∗ i1∧···∧e∗ ik, (where (e1, . . . , en) is a positively oriented orthonormal basis of TM), the Clifford multiplication by ψis µ(w⊗ψ) = w·ψ=X i1<···<ik wi1...ikei1···eik·ψ, so the homomorphism extends to V•TM ⊗ΣM. Remark 2.2.3.Using the Riemannian metric on M, the orthonormal basis (e1, . . . , en)and its dual basis, (e∗ 1, . . . , e∗ n), can be identified. For the sake of simplicity, when there is no confusion, the superscript ∗will be dropped and eiwill refer to both each vector and its dual covector. Proposition 2.2.4. There is a Hermitian structure h·,·i on ΣMsuch that hX·ϕ, ψi=−hϕ, X ·ψi for every X∈X(M)and ϕ, ψ ∈Γ(ΣM). Proof. This holds because the spin representation is unitary; see e.g. [14, pp. 129-130].  The spinor bundle is a versatile and intricate structure, and a much more in-depth analysis of it is possible (see, for example, [8, Chapter II]). This very work will eventually dive deeper into its properties. For now, however, it will be put on the back burner until Section 2.3. 2.3 Connections on spinor bundles The description of a connection on the spinor bundle uses the concepts recalled in Section 1.2.2. If Mis a spin manifold, the Levi-Civita connection on TM induces a connection 4Spinor fields play an essential role in many fields of physics, such as particle physics or quantum mechanics. 28 CHAPTER 2. SPIN MANIFOLDS AND THE SPINOR BUNDLE on the SOn-principal bundle FSO(T M)of positively oriented frames. This connection can in turn be lifted to one on Spin(TM), via the covering map η:Spin(T M)→FSO(T M). Finally, one can canonically define an induced vector bundle connection on the associated spinor bundle. This connection is, at its core, induced by the Levi-Civita connection on TM, so it shall be called Levi-Civita spinorial connection. See e.g. [18, pp. 42-45] for a more detailed exposition of this process. From this point on, when encountering spinorial operators, tensors . . . , the reader should assume that they are spinorial Levi-Civita quantities. Thus the symbol ∇will be used both for the spinorial connection and the Levi-Civita connection on TM. The following results come from [14, p.140]: Proposition 2.3.1. Take an orthonormal basis of (σ1, . . . , σ2bn/2c)of ∆n, and a positively oriented local orthonormal basis (e1, . . . , en)of T M. Let (ψα)1≤α≤bn/2cbe any corresponding local spinorial frame, i.e. ψα= [es, σα]with η(es) = (e1, . . . , en). Then the following affirmations hold: 1. Locally, the spinorial covariant derivative is given by ∇ψα=1 4 n X i,j=1 g(∇ei, ej)ei·ej·ψα.(2.3.1) 2. The spinorial curvature tensor R∇can be expressed explicitly in terms of the Riemann curvature tensor Ron T M. This expression is R∇ X,Y ψ=1 4 n X i,j=1 g(RX,Y ei, ej)ei·ej·ψ. (2.3.2) 3. The spinorial covariant derivative is compatible with the Clifford multiplication “·” and the Hermitian structure h,i, i.e., for every X, Y ∈X(M)and ψ∈Γ(ΣM), Xhψ, ϕi=h∇Xψ, ϕi+hψ, ∇Xϕi, ∇X(Y·ψ)=(∇XY)·ψ+Y·∇Xψ . Chapter 3 The Dirac operator By this point, an impatient reader might have begun to wonder when he is finally going to come across the titular Dirac operators. The answer lies in this chapter. In fact, he might be surprised by the apparent simplicity of their definition, once all the preliminary pieces and tools have been put in place. This simplicity, however, is deceiving. They are very general operators, that can be defined in highly involved vector bundles, so their properties (such as their spectrum) can be extremely challenging to analyze. This chapter will try to do so for some of the more essential ones. 3.1 Definition of the Dirac operator For the sake of clarity, recap the three most important features associated to a spin structure on a Riemannian manifold (Mn, g): (i) The spinor bundle ΣM=Spin(TM)×δn∆nwith the fiberwise Clifford multiplication between k-forms and spinors (see Section 2.2). (ii) The natural Hermitian structure on ΣM(see Section 2.2). (iii) The Levi-Civita spinorial connection on ΣM(see Section 2.3). 29 36 CHAPTER 3. THE DIRAC OPERATOR which differs to the rough Laplacian ∆R=tr(∇2)by a sign. The titular formula is not particularly challenging to compute, but it does need some previous groundwork in the form of the two following lemmas. Lemma 3.3.3. Taking a synchronous local orthonormal frame (∇eiej)(x)=0at each x∈M, where (Mn, g)is a closed Riemannian spin manifold, yields ∆Bψ=− n X i=1 ∇ei∇eiψ for every smooth spinor field ψ∈Γ(ΣM). Proof. Choose ψ, ϕ ∈Γ(ΣM). By the definition of ∇∗, h∆Bψ, ϕiL2=h∇∗∇ψ, ϕiL2=h∇ψ, ∇ϕiL2= n X i=1h∇eiψ, ∇eiϕiL2. Now, proceeding as in Proposition 3.2.3, define two vector fields X1, X2∈X(M)by g(X1, X) + i g(X2, X) = hψ, X ·ϕi for all X∈TM. Thus for every x∈M, div(X1) + idiv(X2) = n X k=1 ekhψ, ek·ϕi. Hence n X k=1h∇ekψ, ∇ekϕi= n X k=1 [ekh∇ekψ, ϕi−h∇ek∇ekψ, ϕi] =div(X1) + idiv(X2)− n X k=1h∇ek∇ekψ, ϕi. Integration of this sum over Mmakes the divergence terms vanish, and the required condition for ∇∗to be the formal adjoint of ∇follows.  Lemma 3.3.4. The Ricci tensor, considered as a symmetric endomorphism of the tangent bundle Ric :TM →TM (see [12, p. 64]) of the Riemannian spin manifold (Mn, g)with local orthonormal frame (ei)1≤i≤nsatisfies, for every X∈T M and ψ∈ΣM, n X i=1 ei·R∇ X,eiψ=1 2Ric(X)·ψ, where Ric(X) = −Pn i=1 RX,eiei. 3.3. THE SCHRÖDINGER-LICHNEROWICZ FORMULA 37 Proof. Use the definition of the spinorial curvature and the first Bianchi identity to find n X i=1 ei·R∇ X,eiψ(2.3.2) =1 4 n X i,j,k=1 g(RX,eiej, ek)ei·ej·ek·ψ (1.3.1) =−1 4 n X i,j,k=1 g(Rei,ejX, ek)ei·ej·ek·ψ−1 4 n X i,j,k=1 g(Rej,Xei, ek)ei·ej·ek·ψ . Rearranging indexes on the second sum yields n X i=1 ei·R∇ X,eiψ=−1 4 n X i,j,k=1 g(RX,eiej, ek)(ej·ek·ei−ej·ei·ek)·ψ . Now, apply (2.2.1) to simplify the Clifford multiplication of the basis vectors ej·ek·ei−ej·ei·ek=−ej·ei·ek−2δikej+ei·ej·ek+ 2δijek =2ei·ej·ek+ 4δijek−2δikej, where δab is the Kronecker delta. Hence 3 n X i=1 ei·R∇ X,ei=− n X i,k=1 g(RX,eiei, ek)ek·ψ+1 2 n X i,j=1 g(RX,eiej, ei)ej·ψ = n X k=1 g(Ric(X), ek)ek·ψ+1 2 n X j=1 g(Ric(X), ej)ej·ψ=3 2Ric(X)·ψ .  With this knowledge, the main result of this section immediately follows. Theorem 3.3.5 (Schrödinger-Lichnerowicz formula).The Dirac operator Dof a closed Riemannian spin manifold (Mn, g)satisfies D2= ∆B+S 4id ,(3.3.1) where S=−Pn i=1 ei·Ric(ei)is the scalar curvature of (Mn, g). Proof. At each x∈M, choose a synchronous local orthonormal frame. Recall that this 38 CHAPTER 3. THE DIRAC OPERATOR means (∇eiej)(x)=0. For any smooth spinor field ψ, using (3.1.3) and (2.2.1), D2ψ= n X i,j=1 ei·∇ei(ej·∇ejψ) = n X i,j=1 (ei·∇eiej·∇ejψ+ei·ej·∇ei∇ejψ) = n X i,j=1 ei·ej·∇ei∇ejψ =− n X i=1 ∇ei∇eiψ+ n X i<j ej·ei·(∇ej∇ei−∇ei∇ej)ψ = ∆Bψ−1 2 n X i,j=1 ei·ej·R∇ ei,ejψ . The second sum can be simplified further by invoking Lemma 3.3.4, n X i,j=1 ei·ej·R∇ ei,ejψ=1 2 n X i=1 ei·Ric(ei)·ψ=−S 2ψ . Since ψ∈Γ(ΣM)was arbitrary, this completes the proof.  Remark 3.3.6.Since, to prove formal adjointness, it is enough to prove it for spinor fields with compact support on M, and which vanish on ∂M [18, p. 113], [13, p. 12], the Schrödinger-Lichnerowicz formula can be extended to general compact manifolds. The Schrödinger-Lichnerowicz formula comes into play in the proof of many more advanced results. In this work, it will be used for the proof of some lower nonzero eigenvalue bounds. 3.4 Conformal covariance Let (Mn, g)be a Riemannian spin manifold. It is natural to wonder how the Dirac operator changes under a conformal change of the metric g:=e2ug, where u∈C∞(M, R). The fixed spin structure on (Mn, g)induces a spin structure on (Mn, g). Explicitly, an isomorphism between the the two SOn-principal bundles is Gu:FSOg(TM)→FSOg(TM) {X1, . . . , Xn} 7→ {e−uX1, . . . , e−uX1}. Note that the usual notation eihas been changed to Xifor the sake of simplicity. The new spin structure is then defined by the following commutative diagram: 3.4. CONFORMAL COVARIANCE 39 Sping(TM)Sping(TM) FSOg(TM)FSOg(TM) ηg e Gu ηg Gu where all the arrows are compatible with the group action. This induces an isomorphism of the associated spinor bundles, given for a spinor ψ= [s, σ]∈ΣMby e Gu: ΣgM→ΣgM ψ7→ ψ= [ e Gu(s), σ]. It is easy to check that this map is an isometry with respect to the Hermitian product h,ion the spinor bundle. Combining this with the isometry TgM→TgMdetermined by X7→ X:=e−uXyields a relation between the Clifford multiplications “·” and “ ·”, X·ψ=X·ψ for each X∈X(M)and ψ∈Γ(ΣM). On the other hand, the induced spinorial covariant derivative follows from an elemental (but somewhat lengthy) calculation starting from the spinorial covariant derivative for g, given in local coordinates by (2.3.1): ∇Xψ=∇Xψ−1 2X·grad(u)·ψ−1 2X(u)ψ . (3.4.1) A more straightforward derivation of this equation (using connection forms) can be seen in [18, p. 70]. Using (3.4.1), the conformal covariance of the Dirac operator can be proved. Without going into much detail, suffice to say that a differential operator Pon the spinor bundle is conformally covariant if there exist a, b ∈Rsuch that P(eau ψ) = ebu Pψ for every conformally transformed metric gand every spinor field ψ∈Γ(ΣM). Proposition 3.4.1. Consider a Riemannian spin manifold (Mn, g)and its conformally modified analogue (Mn, g). Then, their respective Dirac operators, denoted as D≡Dgand D≡Dg, are related in the following manner: De−(n−1)u/2ψ=e−(n+1)u/2Dψ . Proof. Fix a local orthonormal basis (Xi)1≤i≤nof (Mn, g). It is obvious that taking 40 CHAPTER 3. THE DIRAC OPERATOR (Xi)1≤i≤n= (e−uXi)1≤i≤nprovides a local orthonormal basis of (Mn, g)as well. Hence D ψ = n X i=1 Xi·∇Xiψ= n X i=1 e−2uXi·∇Xiψ =e−2u n X i=1 Xi·∇Xiψ−1 2Xi·grad(u)·ψ−1 2Xi(u)ψ =e−u n X i=1 Xi·∇Xiψ−1 2Xi·Xi·grad(u)·ψ−1 2Xi(u)Xi·ψ. Now, knowing that X2 i·=−1·, and that Pn i=1 Xi(u)Xi=grad(u), it is clear that D ψ =e−uDψ +n−1 2grad(u)·ψ.(3.4.2) From this equation and Lemma 3.2.1 (i), it follows that De−(n−1)u/2ψ=e−uDe−(n−1)u/2ψ+n−1 2e−(n−1)u/2grad(u)·ψ =−e−un−1 2e−(n−1)u/2grad(u)·Dψ +e−ue−(n−1)u/2Dψ +e−un−1 2e−(n−1)u/2grad(u)·Dψ =e−(n+1)u/2Dψ .  Conformal covariance is another important feature of the Dirac operator. As seen in Section 4.2.1, it helps to improve certain usual lower eigenvalue bounds via the so-called “Hijazi inequality”. To that end, it is convenient to define a new, also conformally covariant, operator, which relates to Din a useful way. Definition 3.4.2 (Penrose operator).The Penrose (or twistor) operator of a Riemannian spin manifold (Mn, g)is a map P: Γ(ΣM)−→ Γ(T∗M⊗ΣM)−→ Γ(ker(·)) where the first arrow is given by the spinorial connection, and the second is given by the orthogonal projection on the kernel of the Clifford multiplication. In local orthonormal coordinates (ei)1≤i≤n, it takes the form Pψ = n X i=1 e∗ i⊗∇eiψ+1 nei·Dψ=∇ψ+1 n n X i=1 e∗ i⊗ei·Dψ . It is trivial to verify that Pψ ∈Γ(ker(·)). The Penrose operator can also be defined in the direction of a vector X∈X(M), as the map PX: Γ(ΣM)→Γ(ΣM)given by PXψ=∇Xψ+1 nX·Dψ . Remark 3.4.3.The Penrose operator is particularly useful because it allows for an optimal 3.4. CONFORMAL COVARIANCE 41 decomposition of the gradient of a spinor. It is obvious that, for every nonzero ψ /∈ker(D), 1 n n X i=1 e∗ i⊗ei·Dψ ∈ker(·)⊥, so by the Pythagorean Theorem: |∇ψ|2=|Pψ|2+1 n|Dψ|2. This equation is clearly also true when Dψ = 0, so it holds for all ψ∈Γ(ΣM). This decomposition will be used in a later section. Regarding the conformal behavior of the Penrose operator, the following statement holds. Proposition 3.4.4. Consider a Riemannian spin manifold (Mn, g)and its conformally modified analogue (Mn, g). Then, their respective Penrose operators, denoted as P≡Pg and P≡Pg, are related in the following manner: P(eu/2ψ) = eu/2Pψ . Proof. By definition, for any X∈X(M), PX(eu/2ψ) = ∇X(eu/2ψ) + 1 nX·D(eu/2ψ). The analysis is more tractable when one separates both terms in this sum. Thus ∇X(eu/2ψ) = eu/2∇Xψ+1 2X(u)ψ(3.4.1) =eu/2∇Xψ−1 2X·grad(u)·ψ, and, secondly, D(eu/2ψ) = eu/2D ψ +1 2e−u/2grad(u)·ψ (3.4.2) =e−u/2Dψ +n−1 2grad(u)·ψ+1 2e−u/2grad(u)·ψ =e−u/2hDψ +n 2grad(u)·ψi. Combining both terms and using X·ψ=euX·ψone gets PX(eu/2ψ) = eu/2∇Xψ−1 2X·grad(u)·ψ+e−u/2X·1 nDψ +1 2grad(u)·ψ =eu/2∇Xψ+1 nX·Dψ=eu/2PXψ .  With all the basic properties of the Dirac operator in mind, the reader is now ready to dive into a small survey of the Dirac spectrum on compact manifolds in Chapter 4. Chapter 4 The Dirac spectrum on compact manifolds Study of the Dirac spectrum in general spin manifolds is very complex. Most of them, in fact, do not allow for an explicit calculation of it, while others only admit an analytical determination of some of their eigenvalues. This chapter is devoted to the study of the Dirac spectrum and it is divided in two fronts. First, explicit calculations for some simple cases are presented. The second part is more general, and it looks over some of the most well-known lower nonzero eigenvalue bounds in compact manifolds. Nonetheless, aside from a brief commentary on compact manifolds with boundary in Section 4.2.2, all of the spin manifolds in this chapter will be closed (compact without boundary). Before tackling any particular examples, it is worthwhile to prove some general features of the Dirac spectrum, shared by all closed Riemannian spin manifolds. Theorem 4.0.1 (General properties of the Dirac spectrum).Let (Mn, g)be a closed Riemannian spin manifold. The following statements about the spectrum Spec(D)of the Dirac operator of Mhold: (i) Spec(D)is a discrete, unbounded on both sides, subset of R, and it is symmetric about the origen if n6= 3 mod(4). (ii) Each eigeinspace Eλof Dis finite-dimensional and consists of smooth spinor fields. (iii) The eigenspaces of Dprovide a complete orthonormal system for L2(ΣM), L2(ΣM) = M λ Eλ. 43 44 CHAPTER 4. THE DIRAC SPECTRUM ON COMPACT MANIFOLDS Proof. Since the Dirac operator is elliptic and essentially self-adjoint (Remark 3.2.4), Theorem 1.4.4 applies to the closure of D. Combining this with Corollary 3.2.6 and its subsequent remark proves all claims except for the unboundedness of Spec(D). Proof of this last statement is not very informative, but a version of it can be seen in [13, pp. 15-16].  Moving on, the following definition will be useful for some explicit calculations. Definition 4.0.2. AKilling spinor on a Riemannian spin manifold (Mn, g)is a spinor field ψ∈Γ(ΣM)which satisfies ∇Xψ=λX ·ψ for every X∈X(M), where λ∈Cis a constant known as the Killing number of ψ. If λ= 0,ψis said to be a parallel spinor. Remark 4.0.3.Note that every Killing spinor is an eigenspinor of the Dirac operator, but general eigenspinors need not be Killing spinors. In particular, every parallel spinor is also an eigenspinor of the eigenvalue 0, also known as a harmonic spinor. 4.1 Explicit computations of spectra In this section, some of the few closed spin manifolds whose spectrum can be completely or partially determined will be presented. 4.1.1 Flat tori Let Γbe a lattice, i.e. a discrete, cocompact, additive subgroup in Rn. The flat torus Tn is defined to be the compact quotient Tn= Γ\Rnby the action of Γ. This manifold can be endowed with a flat metric, induced by the flat Euclidean metric of Rn. Henceforth, let (γ1, . . . , γn)be a basis of Γ, and (γ∗ 1, . . . , γ∗ n)its dual basis of the dual lattice, Γ∗:={θ∈(Rn)∗:θ(Γ) ⊂Z}. It is a known fact [19] that there are 2ndifferent spin structures on Tn. They are given by the n-tuples (δ1, . . . , δn)where δi∈ {0,1}(i= 1, . . . , n), which in turn come from the 2n group homomorphisms δ1,...,δn: Γ → {±1}, defined by δ1,...,δn(γi) = (−1)δi. Furthermore, the spinor bundle ΣTnpreserves the Clifford multiplication and the covariant derivative from ΣRn, and the smooth sections of ΣTncan be identified with the 4.1. EXPLICIT COMPUTATIONS OF SPECTRA 45 Γ-equivariant smooth sections of ΣRn[13, p. 21], i.e. Γ(ΣTn)≡ {ψ∈Γ(ΣRn): ψ(γ·x) = γ·ψ(x),∀x∈Rn,∀γ∈Γ}, where the Γ-action on both Rnand ΣRnis denoted by “·”. The Dirac spectrum of Tncan now be computed explicitly. Theorem 4.1.1 (Dirac spectrum of flat tori).For any positive integer n, the spectrum of the Dirac operator of the flat torus Tnendowed with the induced flat metric and the (δ1, . . . , δn)-spin structure is given by Spec(D) = (±2πγ∗+1 2X i=1 δiγ∗ i:γ∗∈Γ∗) and each γ∗contributes multiplicity 2bn/2c−1if its corresponding eigenvalue is nonzero. In the case δ1=··· =δn= 0, the multiplicity of the eigenvalue 0 is 2bn/2c. Proof. Note thar the spinor bundle of Rnis trivial, i.e. isomorphic to Rn×∆n. For ψ∈Γ(ΣRn)≡C∞(Rn,∆n), the equivariance condition reads ψ(x+γk)=(−1)δkψ(x) for every x∈Rnand 1≤k≤n. For each γ∗∈Γ∗, denote by θγthe constant 1-form γ∗+1 2 n X k=1 δkγ∗ k∈(Rn)∗. Now, choose an arbitrary orthonormal basis (σl)1≤l≤2bn/2cof ∆n, which can be extended to Rnas elements of Γ(Rn×∆n)∼ =Γ(ΣRn). Define the following spinor field over Rn: φγ,l =e2iπθγσl. This spinor field satisfies the equivariance condition. To prove it, apply its definition, φγ,l(x+γk) = e2iπθγ(x+γk)σl(x+γk) = e2iπθγ(γk)φγ,l(x). Now, analize the exponential term: γ(γk) = γ∗(γk) + 1 2 n X j=1 δjγ∗ j(γk) = z+1 2δk, z ∈Z, so it follows that e2iπθγ(γk)φγ,l(x) = eiπδkφγ,l(x)=(−1)δkφγ,l(x). Moreover, for any X∈Rn ∇Xφγ,l =X(φγ,l)=2iπθγ(X)e2iπθγσl= 2iπθγ(X)φγ,l , 52 CHAPTER 4. THE DIRAC SPECTRUM ON COMPACT MANIFOLDS (ii) If λ∈Ris a nonzero eigenvalue of D, then λ2>1 4S0, S0:=infM(S). Proof. Consider a nontrivial spinor field ψ∈Γ(ΣM). By the Schrödinger-Lichnerowicz formula, one can write ZMhD2ψ, ψiνg=ZMh∇∗∇ψ, ψiνg+ZM 1 4Shψ, ψiνg. By definition, ∇∗is the formal adjoint of ∇, and since Dis formally self-adjoint, ZM|Dψ|2νg=ZM|∇ψ|2νg+ZM 1 4S|ψ|2νg.(4.2.1) Since S > 0,Dψ cannot be identically zero, so property (i) is satisfied. To prove property (ii), choose an eigenspinor ψassociated to the eigenvalue λ. Rearranging the previous equation yields ZM|Dψ|2νg−ZM 1 4S|ψ|2νg=ZM|∇ψ|2νg≥0. Hence λ2−1 4S0≥0, where the equality only holds if ∇ψ= 0. However, this would imply Dψ = 0, which is forbidden by (i). Therefore the inequality is strict.  Friedrich’s inequality The fact that the above inequality is strict is noteworthy. A first step toward sharper bounds could be trying to obtain one which allowed for the equality to be achieved. Indeed, this is what German mathematician T. Friedrich achieved with the following inequality, which bears his name. Theorem 4.2.2 (Friedrich’s inequality).On a closed Riemannian spin manifold (Mn, g) (n≥2), any eigenvalue λ∈Rof the Dirac operator satisfies λ2≥n 4(n−1)S0. Proof. It is obvious that the inequality holds if S0≤0, so only the case S > 0has to be explored. Choose an arbitrary ψ∈Γ(ΣM)and a local orthonormal basis (ei)1≤i≤n. Then, 4.2. LOWER NONZERO EIGENVALUE BOUNDS 53 using the Cauchy-Schwarz (CS) inequality, |Dψ|2= n X i=1 ei·∇eiψ 2 ≤ n X i=1 |ei·∇eiψ|!2 (3.1.1) = n X i=1 |∇eiψ|!2CS ≤n n X i=1 |∇eiψ|2=n|∇ψ|2. Combining this with (4.2.1), one gets 1 nZM|Dψ|2νg≤ZM|Dψ|2νg−ZM 1 4S|ψ|2νg, or, rearranging the integrals, 1−1 nZM|Dψ|2νg≥ZM 1 4S|ψ|2νg. For ψ∈Γ(ΣM)such that Dψ =λψ, one has λ2ZM|ψ|2νg≥n 4(n−1) ZM S|ψ|2νg≥n 4(n−1)S0ZM|ψ|2νg. Friedrich’s inequality trivially follows.  Hijazi’s inequality Friedrich’s inequality is not impervious to improvements. There are several ways one can go about finding new lower eigenvalue bounds, but the one that will be presented here makes use of the conformal covariance of the Dirac operator (see Section 3.4). This method is owed to O. Hijazi. Theorem 4.2.3 (Hijazi’s inequality).Let (Mn, g)be a closed Riemannian spin manifold with n≥2and u∈C∞(M, R). Let g:=e2ugbe a conformally transformed metric and denote the scalar curvature of (Mn, g)by S. Then any λ∈Spec(D)satisfies λ2≥n 4(n−1) infMSe2u. Proof. Recall the equation given by Proposition 3.4.1: De−(n−1)u/2ψ=e−(n+1)u/2Dψ . As a result of it, taking φ:=e−(n−1)u/2ψimplies Dψ =λψ =⇒D φ =λe−uφ . Now, since the Penrose operator can also be conformally transformed (Proposition 3.4.4), it is possible to take the decomposition |P φ|2=|∇φ|2−|D φ|2/n. Combining this with 54 CHAPTER 4. THE DIRAC SPECTRUM ON COMPACT MANIFOLDS the Schrödinger-Lichnerowicz integral (4.2.1) one gets a new integral equality: ZM|P φ|2νg=n−1 nZM|D φ|2νg−1 4ZM S|φ|2νg. Suppose that φcomes from an eigenspinor of Dassociated to the eigenvalue λ. Rearranging terms, n n−1ZM|P φ|2νg=ZMλ2−n 4(n−1)Se2ue−2u|φ|2νg≥0. Many more lower eigenvalue bounds exist. Methods for finding improvements on Friedrich’s inequality are varied, from using parallel forms to introducing the energy-momentum tensor, which is a concept that any physicist would immediately recognize. Nevertheless, for the purposes of this review, Hijazi’s inequality is enough to illustrate how sharper bounds can be determined. 4.2.2 Brief note on bounds on manifolds with boundary The case of general compact Riemannian spin manifolds with boundary is far more involved than that of closed manifolds. The principal reason for this is that now, boundary conditions must be taken into account. In particular, the most interesting boundary conditions are those called “elliptic”, which are pseudo-differential operators B:L2(Σ∂M)→L2(V), where Vis some Hermitian vector bundle over ∂M; that guarantee certain existence and smoothness solutions to the boundary value problem Dψ =ϕon M, B(ψ|∂M ) = χon ∂M. In fact, self-adjoint elliptic conditions restrict the Dirac spectrum to real values (see [13, p. 23ff.] for more information). There are uncountably many of these conditions, the four most often invoked being the usual and modified Atiyah-Patodi-Singer boundary conditions (gAPS and mgAPS), the conditions associated to a chirality operator (CHI), and the MIT boundary condition. Each one of them presents unique kinks, advantages and disadvantages. One of their many applications, and the one most closely related to this survey, is the fact that under any of them, some kind of modified Friedrich’s inequality holds. As an example to close out this survey, take the following theorem found in [13, p. 69]. Theorem 4.2.4. Let n≥2and (Mn, g)be a compact Riemannian spin manifold with nonempty boundary ∂M. If ∂M has nonnegative mean curvature with respect to the inner 4.2. LOWER NONZERO EIGENVALUE BOUNDS 55 normal, then any λ∈Spec(D)under the gAPS boundary condition satisfies λ2>n 4(n−1)S0. From this theorem, it is inferred that some boundary conditions can impose harsher bounds on the Dirac spectrum, since now the equality cannot be attained. Nonetheless, different boundary conditions yield different result, so they are a source of a highly varied and creative study of the Dirac operators. Bibliography [1] P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 117, no. 778, pp. 610–624, 1928. [2] M. F. Atiyah and I. M. Singer, “The index of elliptic operators: III,” Annals of Mathematics, vol. 87, no. 3, pp. 546–604, 1968. [3] L. W. Tu, An Introduction to Manifolds. Universitext, New York: Springer-Verlag, 2nd ed., 2011. [4] L. W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, vol. 275 of Graduate Texts in Mathematics. Springer International Publishing, 1st ed., 2017. [5] S. 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Milhorat, A. Moroianu, and S. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry. EMS Monographs in Mathematics, European Mathematical Society, 2015. [19] C. Bär, “Dependence on the spin structure of the Dirac spectrum,” in Global Analysis and Harmonic Analysis (Marseille- Luminy, 1999), Séminaires and Congrés, 4 (J. Bourguignon, T. Branson, and O. Hijazi, eds.), pp. 17–33, Société Mathématique de France, 2000. [20] S. F. Beitz et al., “Spectrum of generalized Hodge-Laplace operators on flat tori and round spheres,” Osaka Journal of Mathematics, vol. 57, no. 2, pp. 357–379, 2020. [21] F. Pfäffle, “The Dirac spectrum of Bieberbach manifolds,” Journal of Geometry and Physics, vol. 35, no. 4, pp. 367–385, 2000. [22] R. Miatello and R. Podestá, “The spectrum of twisted Dirac operators on compact flat manifolds,” Transactions of the American Mathematical Society, vol. 358, no. 10, pp. 4569–4603, 2006. [23] D. 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Index Bieberbach manifold, 47 bundle fiber, 1 principal, 5 vector, 3 Clifford algebra, 13 multiplication, 23 product, 14 codifferential, 30 complex volume form, 24 Conformal covariance, 38 connection flat, 7 Levi-Civita, 8 on a vector bundle, 6 curvature operator, 7 Riemann, 8 differential operator, 9 elliptic, 10 self-adjoint, 11 Dirac operator, 30 Elliptic boundary condition, 54 exterior differential, 30 Friedrich’s inequality, 52 Hijazi’s inequality, 53 isometry, 8 Killing spinor, 44 Laplace operator Bochner, 35 de Rham, 31 rough, 36 scalar, 31 parallel transport, 7 Penrose operator, 40 principal symbol, 10 Ricci curvature tensor, 9 Riemannian metric, 8 Schrödinger-Lichnerowicz formula, 37 section, 3 Spherical space form, 48 spin group, 16 representation, 22 spinor bundle, 26 field, 27 vector space, 22 spinorial covariant derivative, 28 symmetric connection, 7 59 60 INDEX torsion, 7 twisted adjoint representation, 17