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Dirac Operators in Geometry - Lecture Notes

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Dirac Operators in Geometry - Lecture Notes DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES IOANNIS CHRYSIKOS Abstract. These are notes based on a series of lectures with title “Dirac operators in geometry”, given by the author during August 2019 in the “Summer School on Geometry and Topology”, held in the University of Hradec Králové . Contents Introduction 2 1. Clifford Algebras and the spin group3 1.1. Clifford Algebras3 1.2. The universal construction.4 1.3. An appropriate basis of C`(V; q).5 1.4. The Z2-grading of C`(V; q).7 1.5. Real and complex Clifford algebras.8 1.6. The spin group Spin(n) as a subgroup of C`n. 11 2. Topology - The spin group Spin(n) as the universal covering of SO(n). 13 2.1. Material from the theory of fibre bundles 13 2.2. The topology of SO(n) and of Spin(n). 18 2.3. The Lie algebra of Spin(n). 20 3. Spin structures on Riemannian manifolds 21 3.1. Principal bundles 21 3.2. Reduction of the structure group 25 3.3. Orientable manifolds 26 3.4. Spin structures on oriented Riemannian manifolds 27 4. Spin representations and the spinor bundle. 31 4.1. Spin representations 31 4.2. The spinor bundle 34 5. The spinorial connection and spinor geometry 36 5.1. Connections on principal bundles 36 5.2. The spinorial connection 39 5.3. Spinorial Curvature 40 6. The Dirac operator on a Riemannian spin manifold 43 6.1. Definition and basic properties of the Dirac operator 43 6.2. The twistor operator 49 7. The Schrödinger-Lichnerowicz formula 51 7.1. The spinorial Laplace operator 52 1 7.2. The 2 -Ricci type formula and a proof of the SL-formula 52 8. Special spinor fields 54 8.1. Harmonic spinors 54 8.2. Atiyah-Singer index theorem 55 8.3. Parallel spinors 56 8.4. Killing spinors 58 1 2 IOANNIS CHRYSIKOS 9. Properties of twistor spinors and their relation with Killing spinors 61 9.1. Structural properties of twistor spinors 61 9.2. On the relation of twistor spinors with Killing spinors 64 10. Eigenvalues of the Dirac operator on compact manifolds 65 10.1. Lower eigenvalue estimates 65 10.2. Friedrich’s inequality 66 10.3. Compact manifolds with real Killing spinors 69 11. Appendix 72 11.1. Differential operators 72 11.2. Symplectic, complex and hypercomplex manifolds 74 11.3. Contact and Sasakian manifolds 80 11.4. G2-structures 82 11.5. Spin(7)-structures 83 References 85 Introduction The Dirac operator was discovered in 1928 by P. Dirac [Dir28], during his studies on the wave operator and on the relation of quantum mechanics with general relativity. The Dirac operator D= is a first order elliptic differential operator, acting on sections of the spinor bundle over a Riemannian spin manifold (M; g), and in such terms it was first introduced by M. F. Atiyah and I. Singer in their seminal papers on the index of elliptic operators, see [AS63]. Thus sometimes the Dirac operator is also called the Atiyah-Singer operator. The square of D= is related with the spinorial Laplacian ∆g and the scalar curvature Scalg of (M; g). Therefore, it turns out that D= is able to encode crucial geometrical or topological data of the underlying spin manifold. In particular, the Dirac operator lies in the heart of several central results in differential geometry, in topology, in analysis and in mathematical physics. Such results are realized for example from the famous Atiyah-Singer index theorem, or the keystone role of Dirac operators in non-commutative geometry. On the other hand, parallel spinors and Killing spinors form two very special classes of Dirac eigenspinors, which both have a strong interplay with central developments in supersymmetric string theory, in supergravity and in holonomy theory. These lecture notes form an elementary introduction to some of the most remarkable results related with the Dirac operator (and the twistor operator) on a Riemannian spin manifold. In fact, we put much more emphasis on geometrical results, rather than on analytical one. The notes has been designed and written for students having a profound background in differential geometry and topology, but no background in spin geometry is required. We begin with a smooth introduction to Clifford algebras, spin groups, spin representations and spin structures, which form the main building blocks of spin geometry and its relative calculus. For completeness, in this part we also discuss a few basic notions of the theory of fibre bundles, covering spaces, principal bundles and connections. Next we explicitly introduce the Dirac operator on the spinor bundle over a Riemannian spin manifold, and also the twistor operator, which is the complementary operator of D=. We analyse several of their features and prove the famous Schrödinger-Lichnerowicz formula. 1 For this, we are based on the so-called 2 -Ricci type formula, which is another useful spinorial identity having a series of strong applications. After that, special attention is given to parallel spinors, Killing spinors and twistor spinors. We also discuss harmonic spinors, the Atiyah-Singer index theorem and some vanishing theorems. We finally fix a closed Riemannian spin manifold and provide a short description of the important link between lower eigenvalue estimates of the Dirac operator and real Killing spinors. In particular, we present the seminal Friedrich’s estimate [Fr80], DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 3 and deduce with the classification of compact manifolds admitting (real) Killing spinors. In an appendix we include useful details related with several kinds of geometric structures, which have been used in the main part. 1. Clifford Algebras and the spin group The first section is devoted to Clifford algebras (associated to quadratic vector spaces) and the spin group (the double covering of the special orthogonal group). These objects, together with their representations, are the natural algebraic ingredients of the so-called spinorial calcucus, whose description is one of the basic aims of the present lecture notes. Later it will become more evident why the study of these objects is somehow indispensable for a better understanding of Dirac operators. Notation: In the following, K will be a field (of characteristic 6= 2) (usually K = R or K = C) and all K-vector spaces will be assumed to be finite dimensional, unless stated otherwise. By a K-algebra we understand a K-vector space A equipped with a bilinear associative operation • : A × A ! A (multiplication) which establishes A as a ring with unit 1A (multiplicative identity element 1A ≡ 1, i.e. 1 • a = a • 1 = a for any a 2 A ). A K-algebra homomorphism ' : A ! B (or just morphism) between two algebras A and B is a K-linear map which is also a ring homomorphism ('(xy) = '(x)'(y)) with '(1A ) = 1B. This is an isomorphism precisely when the linear map is a bijection. An algebra A with unit is called a division algebra, if all non-zero elements in A have a multiplicative inverse. It is easy to see that a finite dimensional algebra without divisors of zero is a division algebra. A subalgebra of a K-algebra A is a subspace a ⊂ A which contains 1 and which is closed under the operation •, i.e. x • y 2 a for any x; y 2 a. On the other hand, a left ideal (respectively right ideal) in A is a subspace I ⊂ A such that for all x 2 I and a 2 A we have a • x 2 I (respectively x • a 2 I). If I is both left and right, then we may say that I is a two-sided ideal of A (or just an ideal). 1.1. Clifford Algebras. Let V be a K-vector space with dimK V = n. Let B : V × V ! K be a symmetric bilinear form and denote by q : V ! K the corresponding quadratic form, i.e. q(x) = B(x; x), with 2B(x; y) = q(x+y)−q(x)−q(y) and q(cx) = c2q(x), for any x; y 2 V . For our purposes it suffices to assume that B is non-degenerate, so we have q(x) = 0 if and only if x = 0. Such a pair (V; q) is usually referred to as a quadratic vector space. As we will see below, Clifford algebras are algebras naturally associated to a quadratic vector space, introduced by W. K. Clifford during his short life (1845-1879). n Example 1.1. Consider the Euclidean space V = R endowed with the standard pseudo-Euclidean r;s n r+s product h ; ir;s, where r + s = n. We shall write R for (R = R ; h ; ir;s) and we shall denote the standard basis by fe1; : : : ; eng, such that hei; eiir;s = 1; (1 ≤ i ≤ r); hej; ejir;s = −1; (r + 1 ≤ j ≤ r + s = n) : Often, for the positive definite dot product h ; in;0 we shall just use the notation h ; i. The linear map Φr;s : V ! R, defined by r+s 2 2 2 2 X Φr;s(x) = x1 + ::: + xr − xr+1 − ::: − xr+s; x = xiei ; i=1 is the quadratic form corresponding to h ; ir;s. The pair (r; s) is called the signature of Φr;s. We define the pseudo-orthogonal group O(V ) = O(r; s) ⊂ GL(V ) as the group of isometries of Φr;s and the special pseudo-orthogonal group SO(V ) = SO(r; s) as the subgroup of O(p; q) consisting of isometries f with det(f) = 1. Exercise 1.2. Show that dim O(V ) = dim SO(V ) = n(n − 1)=2.
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