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Lecture Notes on Spin Geometry

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Lecture Notes on Spin Geometry Lecture Notes on Spin Geometry Konstantin Wernli November 25, 2019 arXiv:1911.09766v1 [math.DG] 21 Nov 2019 Contents Introduction 4 DiracOperators..................................... 4 IndexTheorem ..................................... 6 CourseOutline ..................................... 7 1 Algebra 9 1.1 Superalgebra ................................... 9 1.2 Cliffordalgebras.................................. 12 1.3 PinandSpingroups ............................... 17 1.4 Classification of Clifford algebras over R and C ................ 21 1.4.1 Classification of real algebras . 21 1.4.2 Classification of complex Clifford algebras . 25 1.5 Representations.................................. 26 c 1.5.1 Representations of the complex Clifford algebra Cln .......... 26 1.5.2 Thevolumeelement ........................... 28 1.5.3 RepresentationsoftheSpingroup. 29 1.5.4 Someremarksontherealcase. 32 1 2 Geometry 34 2.1 Vector bundles, principal bundles, connections . ..... 34 2.1.1 Vectorbundles .............................. 34 2.1.2 Principalbundles ............................. 39 2.1.3 Connectionsonvectorbundles. 42 2.1.4 Connections on principal bundles . 45 2.1.5 Metrics and metric compatibility . 50 2.2 Chern-Weiltheory ................................ 54 2.2.1 Ad-invariant polynomials . 54 2.2.2 Chern-Weiltheorem ........................... 55 2.2.3 Characteristic classes of U(n)-bundles.................. 57 2.2.4 Characteristic classes of O(n)-bundles.................. 59 2.3 Spinstructures,spinors,Diracoperators . ..... 60 2.3.1 Spinstructures .............................. 60 2.3.2 Digression: Cechcohomologyˇ ...................... 62 2.3.3 Spinors................................... 68 2.3.4 Spinconnections ............................. 70 2.3.5 Diracoperators .............................. 72 2.4 CliffordModules ................................. 72 3 Analysis 79 3.1 EllipticOperators................................. 79 3.1.1 Definitions................................. 80 2 3.1.2 The symbol of a partial differential operator . 82 3.1.3 Formaladjoints .............................. 83 3.2 Fredholmindex .................................. 85 3.3 DiracOperators.................................. 86 4 The index theorem and its applications 88 4.1 IndextheoremforspinDiracoperators . ... 88 4.2 IndextheoremforCliffordModules . 89 4.3 Chern-Gauss-Bonnettheorem . .. 90 4.3.1 Definitions................................. 90 4.3.2 The Clifford module and its index . 91 4.3.3 TherelativeCherncharacter. 92 4.4 Ontheheatkernelproofoftheindextheorem . .... 96 4.4.1 SomepropertiesofHeatkernels . 97 4.4.2 McKean-Singerformula . 100 4.4.3 Asymptoticsoftheheatkernel. 101 3 Introduction The subject of Spin Geometry has its roots in physics and the study of spinors. However, once adapted to a mathematical framework, it beautifully intertwines the realms of algebra, geometry and analysis. When combined with the Atiyah-Singer index theorem - one of the most remarkable results in twentieth century mathematics - it has far-reaching applications to geometry and topology. This course has three main goals. The first goal is to understand the concept of Dirac operators. The second is to state, and prove, the Atiyah-Singer index theorem for Dirac operators. The last goal is to apply these concepts to topology: A remarkable number of topological results - including the Chern-Gauss-Bonnet theorem, the signature theorem and the Hirzebruch-Riemann-Roch theorem - can be understood just by computing the index of a Dirac operator. Before starting a serious discussion, let us try to illuminate the central concepts in a leisurely way. What are Dirac operators? Dirac operators were introduced first by Paul Dirac (of course) in an attempt to understand relativistic quantum mechanics. A simplified version of the problem goes as follows. Consider 3 a particle of mass m moving in R with momentum p =(p1,p2,p3). Relativity tells us that the particle has energy1 E = m2 + p2 (1) 1In units where the speed of light c = 1. p 4 2 2 2 2 2 where p = p1 + p2 + p3. Passing to quantum mechanics , the particle is described by a 2 3 ∂ ∂ time-dependent “wave function” ψt L (R ) and we replace E i ,pj i , which ∈ → ∂t → − ∂xj formally leads to the equation ∂ i ψ = √m2 + ∆ψ (2) ∂t t t 3 ∂2 where we define the Laplacian ∆ = j=1 ∂x2 . To understand this equation, Dirac looked − j for a first order differential operator withP constant coefficients 3 ∂ D = γ + γ (3) j ∂x 0 j=1 j X such that D2 = m2 + ∆. (4) Given such an operator D, we can solve equation (2) as ψ = exp( itD)ψ . However, it t − 0 turns out that (4) cannot be solved if the coefficients γi are real or complex numbers. If we accept for the moment that the γi do not commute with each other, equation (4) leads to the set of equations γ γ + γ γ =0 i = j i j j i 6 γ2 =1 (5) 0 γ2 = 1 j =1, 2, 3 j − which, introducing the anti-commutator a, b = ab+ba and the standard Minkowski metric { } on R4, η = diag( 1, +1, +1, +1), can be summarized as ij − γ ,γ = 2η . (6) { i j} − ij We call equation (6) the Clifford Relation. It leads to very interesting algebra that we will study in Chapter 1. Another implication is that the wave function Ψ is not real valued, rather, it is a section of a certain “spinor bundle”. This will be discussed in 2. In chapter 3, after discussing some analytic preliminaries, we will finally introduce Dirac Operators and their index. 2For us this is an entirely formal operation, and we will not discuss it any further. 5 What is the index (theorem)? The index, and the index theorem, are statements about the solutions to equations. Let us illuminate this by means of two simple examples3. First, let V, W be vector spaces and consider a linear map A: V W . We are interested → in the equation Av = 0. The number of independent solutions is dim ker A. A priori, we do not have any information about this number on its own. However, what we can consider is the rank-nullity theorem rk(A) + dim ker A = dim V. (7) This equation can be rewritten as dim ker A (dim W rk(A)) = dim V dim W (8) − − − or, introducing the cokernel of A as coker(A)= W/imA, as dim ker A dim cokerA = dim V dim W. (9) − − In other words, while we cannot say anything about dim ker A, we know exactly what dim ker A dim cokerA (in fact it is independent of A). If we define the index − ind(A) := dim ker A dim cokerA (10) − then we can interpret equation (9) is a first incarnation of the index theorem. As a second example, consider the family of differential operators 1 1 D : C∞(S ) C∞(S ) λ → df D f = 2πiλf λ dx − Then, the kernel of D is nonzero if and only if λ Z, and in this case is spanned by ∈ f(x)= e2πiλx. Hence, the dimension of the kernel is given by 0 λ / Z dim ker Dλ = ∈ (11) 1 λ Z ∈ 3It may be worth to note that the operators appearing here are not Dirac operators, but the index theorem still holds. 6 in particular it does not vary continuously in λ. On the other hand, let us consider the index 4 d of Dλ. By a general fact , we have cokerDλ = ker Dλ∗, where Dλ∗ = dx +2πiλ is the adjoint of Dλ. Hence, we see that ind(D ) = dim ker D dim ker D∗ 0 (12) λ λ − λ ≡ which does vary continuously in λ. The observation that the index of an elliptic operator is invariant over continuous families led to the discovery of the topological index - a number associated to an elliptic operator D, denoted top ind(D) that can be computed from − topological data - and the celebrated Atiyah-Singer index theorem: Theorem 0.0.1. Let E, F be vector bundles over a manifold M and D : Γ(E) Γ(F ) an → elliptic differential operator. Then the index of D equals the topological index of D: ind(D) = top ind(D). (13) − The central goal of this course are to understand the statement and proof of this theorem for Dirac Operators and study some of its applications. This requires that apart from understanding Dirac Operators and their index, we define their topological index. This is done via the theory of characteristic classes, that will be introduced in Chapter 2. The precise analytical definitions will be given in 3. Finally, in Chapter 4, we will be able to state and prove the index theorem and consider some applications. Course outline Before we begin, let us briefly discuss the outline of the course. As mentioned in the begin- ning, the subject of Spin Geometry draws from Algebra, Geometry, and Analysis, and the first three chapters will each be devoted to studying the necessary preliminaries in each of these areas. The last and main chapter is reservers for the index theorem, its proof, and applications. Chapter 1 deals with Algebra. After a brief introduction to superalgebra, we will discuss the essential notions of Clifford Algebras, Spin and Pin Groups, and their representations. 4We will deal with these technicalities in more detail later. 7 Chapter 2 deals with Geometry, and consists of three main sections. First, we will recall preliminaries, such as vector bundles, principal bundles and connections. Then we will be concerned with the Chern-Weil theory of characteristic classes. Lastly, we will take the algebra of Chapter 1 and promote it to spin structures and spinor bundles over manifolds. In Chapter 3 we discus Analysis. After introducing some technical machinery on differential operators, we give a precise definitions of the index. We then proceed to define Dirac operators and Dirac bundles. Finally, in Chapter 4, we give the precise statement of the index theorem for Dirac operators. We will a corollary - the Chern-Gauss-Bonnet Theorem - and sketch the ideas of the proof. Further reading There are a number of notable omissions in these lecture notes - e.g. several other applications of the index theorem, Pin structures and associated Dirac operators, and the generalization to elliptic operators in terms of K-theory.
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