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Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu

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Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. Amazingly, most elliptic operators which are relevant in geometry are of Dirac type. The index theorem for these operators contains as special cases a few celebrated results: the Gauss-Bonnet theorem, the Hirzebruch signature theorem, the Riemann-Roch-Hirzebruch theorem. In this course we will be concerned only with the index problem for the Dirac type elliptic operators. We will adopt an analytic approach to the index problem based on the heat equation on a manifold and Ezra Getzler’s rescaling trick. + Prerequisites: Working knowledge of smooth manifolds, and algebraic topology (especially cohomology). Some familiarity with basic notions of functional analysis: Hilbert spaces, bounded linear operators, L2-spaces. + Syllabus: Part I. Foundations: connections on vector bundles and the Chern-Weil construction, calculus on Riemann manifolds, partial differential operators on manifolds, Dirac operators, [21]. Part II. The statement and some basic applications of the index theorem, [27]. Part III. The proof of the index theorem, [27]. + About the class There will be a few homeworks containing routine exercises which involve the basic notions introduced during the course. We will introduce a fairly large number of new objects and ideas and solving these exercises is the only way to gain something form this class and appreciate the rich flavor hidden inside this theorem. ii Notations and conventions • K = R; C. • For every finite dimensional K-vector space V we denote by AutK(V ) the Lie group of K-linear automorphisms of V . • We will orient the manifolds with boundary using the outer normal first convention. • We will denote by gl ( ) o(n), so(n), u(n) the Lie algebras of the Lie groups GL ( ) and r K r K respectively U(n), O(n), SO(n). Contents Introduction i Notations and conventions ii Chapter 1. Geometric Preliminaries1 x1.1. Vector bundles and connections1 1.1.1. Smooth vector bundles1 1.1.2. Principal bundles 10 1.1.3. Connections on vector bundles 12 x1.2. Chern-Weil theory 21 1.2.1. Connections on principal G-bundles 21 1.2.2. The Chern-Weil construction 22 1.2.3. Chern classes 26 1.2.4. Pontryagin classes 29 1.2.5. The Euler class 34 x1.3. Calculus on Riemann manifolds 36 x1.4. Exercises for Chapter 1 42 Chapter 2. Elliptic partial differential operators 45 x2.1. Definition and basic constructions 45 2.1.1. Partial differential operators 45 2.1.2. Analytic properties of elliptic operators 55 2.1.3. Fredholm index 58 2.1.4. Hodge theory 63 x2.2. Dirac operators 67 2.2.1. Clifford algebras and their representations 67 2.2.2. Spin and Spinc 76 2.2.3. Geometric Dirac operators 84 x2.3. Exercises for Chapter 2 90 Chapter 3. The Atiyah-Singer Index Theorem: Statement and Examples 93 iii iv Contents x3.1. The statement of the index theorem 93 x3.2. Fundamental examples 94 3.2.1. The Gauss-Bonnet theorem 94 3.2.2. The signature theorem 99 3.2.3. The Hodge-Dolbeault operators and the Riemann-Roch-Hirzebruch formula 103 3.2.4. The spin Dirac operators 119 3.2.5. The spinc Dirac operators 130 x3.3. Exercises for Chapter 3 136 Chapter 4. The heat kernel proof of the index theorem 139 x4.1. A rough outline of the strategy 139 4.1.1. The heat equation approach: a baby model 139 4.1.2. What really goes into the proof 141 x4.2. The heat kernel 142 4.2.1. Spectral theory of symmetric elliptic operators 142 4.2.2. The heat kernel 146 4.2.3. The McKean-Singer formula 153 x4.3. The proof of the Index Theorem 154 4.3.1. Approximating the heat kernel 154 4.3.2. The Getzler approximation process 158 4.3.3. Mehler formula 165 4.3.4. Putting all the moving parts together 166 Bibliography 167 Index 169 Chapter 1 Geometric Preliminaries 1.1. Vector bundles and connections 1.1.1. Smooth vector bundles. The notion of smooth K-vector bundle of rank r formalizes the intuitive idea of a smooth family of r-dimensional K-vector spaces. Definition 1.1.1. A smooth K-vector bundle of rank r over a smooth manifold B is a quadruple (E; B; π; V ) with the following properties. (a) E; B are smooth manifolds and V is a r-dimensional K-vector space. −1 (b) π : E ! B is a surjective submersion. We set Eb := π (b) and we will call it the fiber (of the bundle) over b. (c) There exists a trivializing cover, i.e., an open cover U = (Uα)α2A of B and diffeomorphisms −1 Ψα : E jUα = π (Uα) ! V × Uα with the following properties. (c1) For every α 2 A the diagram below is commutative. Ψ E j α w V × U Uα ' α '') [ : π [^[proj Uα (c2) For every α; β 2 A there exists a smooth map gβα : Uβα := Uα \ Uβ ! Aut(V ); u 7! gβα(u) such that for every u 2 Uαβ we have the commutative diagram V × fug ΨαjEu [[] [ Eu ' gβα(u:) '') u Ψβ jEu V × fug 1 2 Liviu I. Nicolaescu B is called the base, E is called total space, V is called the model (standard) fiber and π is called the canonical (or natural) projection.A K-line bundle is a rank 1 K-vector bundle. Remark 1.1.2. The condition (c) in the above definition implies that each fiber Eb has a natural structure of K-vector space. Moreover, each map Ψα induces an isomorphism of vector spaces Ψα jEb ! V × fbg: ut Here is some terminology we will use frequently. Often instead of (E; π; B; V ) we will write π −1 E ! B or simply E. The inverses of Ψα are called local trivializations of the bundle (over Uα). The map gβα is called the gluing map from the α-trivialization to the β-trivialization. The collection n o gβα : Uαβ ! Aut(V ); Uαβ 6= ; is called a (Aut(V ))-gluing cocycle (subordinated to U) since it satisfies the cocycle condition gγα(u) = gγβ(u) · gβα(u); 8u 2 Uαβγ := Uα \ Uβ \ Uγ; (1.1.1) where ”·” denotes the multiplication in the Lie group Aut(V ). Note that (1.1.1) implies that −1 gαα(u) ≡ 1V ; gβα(u) = gαβ(u) ; 8u 2 Uαβ: (1.1.2) Example 1.1.3. (a) A vector space can be regarded as a vector bundle over a point. (b) For every smooth manifold M and every finite dimensional K-vector space we denote by V M the trivial vector bundle V × M ! M; (v; m) 7! m: (c) The tangent bundle TM of a smooth manifold is a smooth vector bundle. π π (d) If E ! B is a smooth vector bundle and U,! B is an open set then E jU ! U is the vector bundle π−1(U) !π U: 1 2 1 (e) Recall that CP is the space of all one-dimensional subspaces of C . Equivalently, CP is the 2 quotient of C n f0g modulo the equivalence relation 0 ∗ 0 p ∼ p () 9λ 2 C : p = λp. 2 For every p = (z0; z1) 2 C n f0g we denote by [p] = [z0; z1] its ∼-equivalence class which we view as the line containing the origin and the point (z0; z1). We have a nice open cover fU0;U1g of 1 CP defined by Ui := f[z0; z1]; zi 6= 0g: The set U0 consists of the lines transversal to the vertical axis, while U1 consists of the lines transver- sal to the horizontal axis. The slope m0 = z1=z0 of the line through (z0; z1) is a local coordinated over U0 and the slope m1 = z0=z1 is a local coordinate over U1. On the overlap we have m1 = 1=m0: Let 2 1 y z1 E = f(x; y;[z0; z1]) 2 C × CP ; = ; i:e: yz0 − xz1 = 0g x z0 2 1 1 1 The natural projection C ×CP ! CP induces a surjection π : E ! CP .
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