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 ∈ A ). A K-algebra homomorphism ϕ : A → B (or just morphism) between two algebras A and B is a K-linear map which is also a ring homomorphism (ϕ(xy) = ϕ(x)ϕ(y)) with ϕ(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 ∈ a for any x, y ∈ a. On the other hand, a left ideal (respectively right ideal) in A is a subspace I ⊂ A such that for all x ∈ I and a ∈ A we have a • x ∈ I (respectively x • a ∈ 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 ∈ 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 {e1, . . . , en}, 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. 4 IOANNIS CHRYSIKOS

P∞ ⊗k Let T (V ) = k=0 V = K ⊕ V ⊕ V ⊗ V ⊕ · · · be the tensor algebra of V , with unit element 1. Definition 1.3. The Clifford algebra C`(V, q) of the quadratic vector space (V, q) is the associative algebra with unit defined as the quotient C`(V, q) = T (V )/I(V, q) , where I(V, q) is the two-sided ideal in T (V ) generated by all elements of the form x ⊗ x − q(x)1, x ∈ V . We shall denote the product in C`(V, q) by “·” (Clifford product).

Remark 1.4. There is a natural embedding jq : V → C`(V, q) which is deﬁned by the composition of the projection map π : T (V ) → T (V )/I(V, q), with the natural inclusion i : V → T (V ), i.e. jq = π ◦ i. π T (V ) / C`(V, q) O 9 i jq V 2 By construction, jq(u) = q(u) · 1 and we can view V as a subset of C`(V, q). Thus, C`(V, q) can be considered as the associative algebra with unit, generated multiplicatively by V (and the unit 1) subject to the conditions x · x = q(x) · 1 = B(x, x) · 1 , ⇐⇒ x · y + y · x = 2B(x, y) · 1 .

Exercise 1.5. Prove that jq : V → C`(V, q) is injective (hence one can identify the image jq(V ) with V , as we will usually do). 1.2. The universal construction. There is a very useful characterization of Cliﬀord algebras, in terms of the so-called universal property. Consider a quadratic vector space (V, q) over K, as above. Problem A: Find a pair (C , j), where C is a commutative algebra with unit and j : V → C is a 2 homomorphism satisfying j(u) = q(u) · 1C , such that for any K-algebra A with a homomorphism 2 α : V → A with α(u) = q(u)·1A (u ∈ V ), there exists a unique algebra homomorphism α˜ : C → A which makes the following diagram commutative, i.e. α =α ˜ ◦ j, i.e.

j V / C

α˜ α A

Theorem 1.6. Problem A admits a unique, up to isomorphism, solution (C , j) which is given by the Cliﬀord algebra C`(V, q) associated to (V, q), while j coincides with the natural embedding jq : V → C`(V, q). Proof. We will show that C`(V, q) = T (V )/I(V, q) is a solution and then the uniqueness follows easily. Indeed, let α : V → A be a linear map such that α(u)2 = q(u) · 1 for any u ∈ V , where A is a K-algebra. Since the tensor algebra T (V ) satisﬁes the universal property, α factorizes (uniquely) through T (V ) to an algebra homomorphism α¯ : T (V ) → A such that α =α ¯ ◦i, where i : V → T (V ). Then α¯ inherits the property (¯α(u))2 = q(u) · 1, and consequently α¯(u ⊗ u − q(u) · 1) = (¯α(u))2 − q(u)¯α(1) = (¯α(u))2 − q(u) · 1. This means that α¯ vanishes in the ideal I(V, q). Therefore it factorizes (uniquely) through C`(V, q) to an algebra homomorphism α˜ :C`(V, q) → A, such that α =α ˜ ◦ jq =α ˜ ◦ π ◦ i, i.e. the following DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 5 diagram commutes. α V / A 6= α¯ i α˜ jq=π◦i T (V )

π C`(V, q) = T (V )/I(V, q) To see the uniqueness of α˜, notice that π is onto and α¯ is unique algebra homomorphism which makes the above diagram commutative. The universality property has several consequences, for example

Corollary 1.7. Let f :(V, q) → (W, r) be an isometry of two quadratic vector spaces over K, i.e. q(u) = r(f(u)) for any u ∈ V . Then there is a homomorphism fe :C`(V, q) → C`(W, r), satisfying fe(jq(u)) = jr(f(u)). If f is an orthogonal isomorphism, then fe is an algebra isomorphism. This result implies that the universal construction of C`(V, q) is functorial: There is a functor from the category of q-compatible linear maps on a quadratic vector space (V, q), to that of Cliﬀord algebras C`(V, q) related with (V, q). Moreover, it shows that there is an embedding of the orthogonal group O(V, q) = {L ∈ GL(V ): L∗q = q} corresponding to (V, q), inside the group of automorphisms of C`(V, q), i.e. O(V, q) ,→ Aut(C`(V, q)) . Below we shall describe further applications based on the universal construction of Cliﬀord algebras.

1.3. An appropriate basis of C`(V, q). Let {ei} be a q-orthogonal basis of (V, q), i.e. q(ei, ej) = 0, whenever i 6= j. Set I for the following set of indices I = {1 ≤ i1 < i2 . . . < ik ≤ n = dimK V }, with 1 ≤ k ≤ n. It is clear that eI := ei1 · ei2 ··· eik is an element of C`(V, q). We shall write |I| = k in the sequel and we deﬁne the length of eI as being |I|. Proposition 1.8. A basis of the Cliﬀord algebra C`(V, q) associated to a quadratic vector space

(V, q) is given by {1 = e∅, eI := ei1 · ei2 ··· eik }, such that

ei · ej + ej · ei = 2B(ei, ej) · 1 .

The cardinality of the set {1, eI } is n n 1 + n + + ··· + = 2n , 2 n thus dim C`(V, q) = 2n. Exercise 1.9. Prove Corollary 1.7 and Proposition 1.8. Note that Proposition 1.8 admits several diﬀerent proofs. Deﬁnition 1.10. For an additive group G and an algebra A over a commutative ring, we say that L A is G-graded, when there is a (vector space) direct sum decomposition A = g∈G Ag, such that the multiplication respects the grading, that is, if a ∈ Ag and b ∈ Ah, then ab ∈ Ag+h. When G = Z we speak for a Z-grading, while for G = Z2, i.e. a Z2-grading, A is often referred to as a superalgebra. Remark 1.11. Recall that the quotient of T (V ) by the ideal L(V ) generated by all {u ⊗ w + w ⊗ u : u, w ∈ V } (or equivalently, by all u ⊗ u with u ∈ V ) is the exterior algebra (or the Grassmann algebra), denoted by V•(V ) := T (V )/L(V ), with the usual wedge (or exterior) product 6 IOANNIS CHRYSIKOS

φ ∧ ψ := φ ⊗ ψ(mod L(V )). Since L(V ) is a graded ideal, the exterior algebra inherits a Z-grading from the tensor algebra T (V ), • k ^ M ^ (V ) = (V ) , k∈Z where Vk(V ) = T k(V )/Lk(V ), with Lk(V ) = T k(V ) ∩ L(V ) and T k(V ) = V ⊗k. This means ω ∧ ψ ∈ Vk+`(V ) whenever ω ∈ Vk(V ) and ψ ∈ V`(V ). Moreover, ω ∧ ψ = (−1)k`ψ ∧ ω , V0 V1 and thus the wedge product is graded anti-commutative. Clearly (V ) = K and (V ) = V , so we can view V as a subspace of V•(V ). Therefore, V•(V ) is an associated algebra which is linearly generated by V , subject to the condition u ∧ w + w ∧ u = 0. In other words, V•(V ) is equipped with a linear map j : V → V•(V ) with the property j(u)2 = 0, for any u ∈ V , under the requirement that the universal property is satisﬁed: If A is any K-algebra and f : V → A is a linear map such that f(u)2 = 0, or equivalently f(u) ∧ f(w) + f(w) ∧ f(u) = 0, for any u, w ∈ V , then f extends uniquely to a morphism of algebras f˜ : V•(V ) → A such that f˜◦ j = f. If dim V = n Vk and {e1, . . . , en} is a basis, then (V ) is spanned by the so-called homogeneous (or decomposable) elements eI = e∗ ∧ · · · ∧ e∗ for all ordered subsets I = {i , . . . , i } of {1, . . . , n} (for k = 0, set i1 ik 1 k n e∅ = 1), where e∗ is the dual of e . We see that dim Vk(V ) = and thus i i k

• n ^ X n dim (V ) = = 2n . k k=0 Remark 1.12. Although, C`(V, q) and V•(V ) have the same dimension, be aware that as algebras n 2 2 V• n they differ. For example, comparing C`n := C`(R , −x1 − ... − xn) and (R ), we see that there is a natural vector space isomorphism given by (see also [LM89]) • ^ n F : (R ) → C`n , ei1 ∧ · · · ∧ eik 7→ ei1 ····· eik . However, observe that the Clifford multiplication is different than the exterior multiplication,

ei · ei = −1 , but ei ∧ ei = 0 . Example 1.13. The Cliﬀord algebra of a vector space V with the trivial quadratic form q ≡ 0, is just the exterior algebra V•(V ).

2 Example 1.14. Let V = R (over K = R) and q = −x1. Then the Clifford algebra C`1 := 2 2 C`(R, −x1) is generated by {1, e1} (where 1 is now the unit of C`1), under the condition e1 = −1. The linear map C`1 3 1 7→ 1 ∈ C, e1 7→ i ∈ C defines an algebra isomorphsim C`1 ' C. Here the injection jq : R → C`1 is given by x 7→ ix, i.e. R sits inside C`1 = C as the imaginary part. 2 2 2 Example 1.15. Let V = R (over K = R) and q = −x1 − x2. Then the Clifford algebra C`2 := 2 2 2 2 C`(R , −x1 − x2) is the algebra of quaternions H. Indeed, a basis of C`(R , q) is given {1, e1, e2, e1 · 2 2 2 2 e2}, and thus C`(R , q) is generated by e1, e2 ∈ R under the conditions e1 = e2 = −1 and e1 · e2 = −e2 · e1. By setting i := e1, j := e2 and k := e1 · e2 we obtain the fundamental relations of the division algebra of quaternions H = h1, i, j, ki, i.e. i2 = j2 = k2 = ijk = −1 . DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 7

Figure 1. The multiplication rules of quaternions.

These examples clarify that the Cliﬀord algebra is a natural object to consider and moreover that the algebraic structure is intrinsically linked with the geometric structure given by the (non-degenerate) symmetric bilinear form on the vector space.

1.4. The Z2-grading of C`(V, q). The Clifford algebra C`(V, q) does not inherit a Z-grading from 2 T (V ). Indeed, the ideal I(V, q) is not homogeneous; e.g. e1 + 1 is in I(V, q), but the homogeneous 2 2 1 parts of e1 +1, e1 and 1, clearly cannot be in I(V, q). However, C`(V, q) admits a Z2-grading (hence C`(V, q) is a superalgebra). This means that one can decompose C`(V, q) into a direct sum C`(V, q) = C`0(V, q) ⊕ C`1(V, q) , where C`0(V, q) (respectively C`1(V, q)) is the span of all products of an even (respectively odd) number of elements of V . Because the orthogonal group O(V, q) of the quadratic vector space (V, q) acts on the Clifford algebra via automorphisms, Corollary 1.7 says that given some invertible orthogonal transformation f : V → V such that q(u) = q(f(u)), there exists an automorphism β :C`(V, q) → C`(V, q). Moreover: Proposition 1.16. Any Clifford algebra C`(V, q), has a unique automorphism β :C`(V, q) → 2 C`(V, q) such that β = IdC`(V,q) and β(jq(u)) = −jq(u) for any u ∈ V . Proof. Apply Corollary 1.7 to the orthogonal transformation f(u) = −u, u ∈ V ; there exists an automorphism β :C`(V, q) → C`(V, q) such that jq(−u) = β ◦ jq(u), or equivalently β ◦ jq(u) = −jq(u) for all u ∈ V . Now, by construction

(β ◦ β ◦ jq)(u) = β(−jq(u)) = −β(jq(u)) = jq(u) , 2 so β is the identity on the set jq(V ) ⊂ C`(V, q). Since jq(V ) generates C`(V, q) (any element x ∈

C`(V, q) is written by x = xi1 ··· xik for some k, with xij ∈ jq(V )), we obtain β ◦ β = IdC`(V,q). The map β is called the parity automorphism. In fact, β :C`(V, q) → C`(V, q) can be deﬁned also as follows: k |I| β :C`(V, q) → C`(V, q), eI = ei1 ··· eik 7→ (−1) ei1 ··· eik = (−1) eI . (1.1) The direct sum C`(V, q) = C`0(V, q) ⊕ C`1(V, q) is obtained by considering the subspaces C`0(V, q) and C`1(V, q) as the eigenspaces of β: C`0(V, q) = {x ∈ C`(V, q): β(x) = x} , C`1(V, q) = {x ∈ C`(V, q): β(x) = −x} .

1Recall that an ideal I ⊂ A of a graded algebra (or graded ring), is called homogeneous if for any element a ∈ I the homogeneous part of a belongs also in I. For a homogeneous ideal I the quotient A/I is also a graded algebra. For Z2-graded algebras, homogeneous elements are either of degree 0 (even) or 1 (odd), with respect to the grading. 8 IOANNIS CHRYSIKOS

Because C`i(V, q) · C`j(V, q) ⊆ C`(i+j)mod2(V, q), it is obvious that C`0(V, q) · C`0(V, q) ⊂ C`0(V, q) , C`0(V, q) · C`1(V, q) ⊂ C`1(V, q) , C`1(V, q) · C`1(V, q) ⊂ C`0(V, q) . We shall call C`0(V, q) the even part of C`(V, q) and C`1(V, q) the odd part of C`(V, q), respectively. Notice that V ⊂ C`1(V, q) and C`0(V, q) ⊂ C`(V, q) is a subalgebra. Remark 1.17. Working with a non-degenerate bilinear form B and considering some B-orthonormal basis {ei} of V , it is clear that elements eI ∈ C`(V, q) of length |I| = 1, generate the vector space V ⊂ C`1(V, q) ⊂ C`(V, q). Then, the center Z(C`(V, q)) of C`(V, q) consists of those elements that commute with all u ∈ V . For n even, the center is given by K, while for n odd, it is

Z(C`(V, q)) = K ⊕ K[e1, . . . en]. For example, let eI = eii ··· eik be an element in C`(V, q). For |I| j∈ / I, we have eI ej = (−1) ejeI , and thus the length |I| of eI must to be even for eI to commute |I|−1 with ej. On the other hand, for j ∈ I, we have eI eij = (−1) eij eI , so that |I| needs to be odd for a commutation. For the general case see [Fr00]. 1.5. Real and complex Cliﬀord algebras. Recall that given a quadratic vector space (V,B) (with quadratic form q), its complexiﬁcation

C (V = V ⊗R C,BC) is a complex vector space, where BC is deﬁned as

BC(u1 ⊗ z, u2 ⊗ w) = B(u1, u2)zw . 2 C The complexiﬁed quadratic form is given by qC(u ⊗ z) = z q(u) and hence (V ,BC) becomes a complex quadratic vector space. Similarly, the complexiﬁcation of a real algebra A is given by A ⊗R C. Endowed with the product

(a1 ⊗ z)(a2 ⊗ w) := (a1a2) ⊗ (zw) ,

A ⊗R C becomes a complex algebra. Let us pass now to the associated Cliﬀord algebras C`(V, q). C It is natural to ask for the relation of C`(V, q) and C`(V , qC). The answer is given as follows. C Proposition 1.18. C`(V , qC) = C`(V, q) ⊗R C. C Proof. Consider the R-linear map map φ : V → C`(V, q) ⊗R C given by

φ(u, z) := jq(u) ⊗ z = u ⊗ z , where jq : V → C`(V, q) is the associated injective map. Because 2 2 2 2 2 φ(u ⊗ z) = (u ⊗ z) = u ⊗ z = B(u, u)z 1 ⊗ 1 = BC(u ⊗ z, u ⊗ z) · 1 , C it follows that φ extends to an algebra homomorphism φe :C`(V , qC) → C`(V, q) ⊗R C. Counting dimensions we conclude that φe is an isomorphism. Later we are going to use some theory of Clifford algebras on Riemannian manifolds (but we will not examine the pseudo-Riemannian setting). Hence below we will pay attention more to the positive-definite case, and we shall not treat Clifford algebras associated to quadratic forms of general r,s signature, e.g. C`r,s = C`(R , Φr,s). We fix the following notation (see also [BFGK89, Fr00], and for a different notation we refer to [LM89]) n 2 2 C`n := C`(R , −x1 − · · · − xn), 0 n 2 2 C`n := C`(R , x1 + ··· + xn). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 9

r,s If we set C`r,s = C`(R , Φr,s), where Φr,s is the quadratic form given in Example 1.1, then C`n = 0 n n C`0,n and C`n = C`n,0. Moreover, if {ei}i=1 is an orthonormal basis of R with respect to q = 2 2 0 2 2 0 −x1 −· · ·−xn (respectively q = x1 +···+xn), then C`n (respectively C`n) is generated by e1, . . . , en subject to the relation

eiej + ejei = −2δij, (respectively eiej + ejei = 2δij). n For the complex case V = C and since a (non-degenerate) complex quadratic form has no signature, we set C n 2 2 C`n := C`(C ; z1 + ··· + zn) 2 2 2 2 which is the Cliﬀord algebra for both q1 = z1 + ··· + zn, or q2 = −z1 − · · · − zn. Note that the 2 2 2 2 complexiﬁcations of x1 + ··· + xn and −x1 − · · · − xn are equivalent, in particular by Proposition 1.18 we obtain the following algebra isomorphisms.

C 0 Corollary 1.19. C`n = C`n ⊗RC = C`n ⊗RC. Let us now describe some further algebra isomorphisms.

Proposition 1.20. The Clifford algebra C`n is isomorphic to the even part of C`n+1, i.e. 0 C`n ' C`n+1 . n n+1 Proof. Let {e1, . . . , en} and {e1, . . . , en+1} denote the canonical bases of R and R , respectively, n 0 2 and define the linear map f : R → C`n+1 by f(ei) = ei·en+1. By definition of f we get f(ei) = −1, 0 so f extends to an algebra homomorphism fe :C`n → C`n+1. Clearly fe is an injective map between vector spaces of the same dimension, thus fe is an isomorphism. 0 Proposition 1.21. As algebras over R, there are isomorphisms C`1 ' C, C`2 ' H, C`1 ' R ⊕ R, 0 and C`2 ' M2(R). Proof. The first two cases were described in Examples 1.14 and 1.15, respectively. For the relation 0 0 2 2 C`1 = R⊕R, note that the Clifford algebra C`1 ' C`(R, x1) is generated by {1, e1}, such that e1 = 1 2 with e1 ∈ R. Set p± := (1/2)(1 ± e1). Then p+ + p− = 1, p+p− = 0 and p± = p±. This decomposes 0 the Clifford algebra and the map xp+ + yp− 7→ (x, y) is the desired isomorphism C`1 ' R ⊕ R. For 0 the relation C`2 ' M2(R), where M2(R) denotes the space of 2 × 2 matrices with real entries, we 0 2 2 2 0 start with C`2 = C`(R , x1 + x2). By its definition, C`2 is generated by {1, e1, e2, e1 · e2}, subject 2 2 to the conditions e1 = e2 = 1 and e1 · e2 = −e2 · e1. Hence, in this case the isomorphism is given by 1 0 1 0 0 1 1 0 1 7→ , e 7→ , e 7→ , e · e 7→ . 0 1 1 0 −1 2 1 0 1 2 0 −1 In other words, for any x, y, z, w ∈ R we have 0 x + y z + w C` 3 x1 + ye + ze + w(e · e ) 7→ ∈ M ( ). 2 1 2 1 2 z − w x − y 2 R C C Proposition 1.22. As algebras over C we have C`1 ' C ⊕ C and C`2 ' M2(C) ≡ gl(2, C). In particular, C ⊗R C ' C ⊕ C, C ⊗R H ' M2(C). Proof. For a basis of gl(2, C) we choose the matrices 1 0 0 −i i 0 0 i E = ,T = , g = , g = . 0 1 i 0 1 0 −i 2 i 0

2 2 C C We have g1 = −E = g2, g1g2 + g2g1 = 0 and thus C`2 ' M2(C). On the other hand C`2 = C C`2 ⊗RC = H ⊗R C, hence we conclude. For the ﬁrst relation we have C`1 = C`1 ⊗RC = C ⊗R C ' 10 IOANNIS CHRYSIKOS

1 C ⊕ C where the last isomorphism C ⊕ C ' C ⊗R C is obtained by (1, 0) 7→ 2 (1 ⊗ 1 + i ⊗ i) 1 and (0, 1) 7→ 2 (1 ⊗ 1 − i ⊗ i). Thus, for the complex Clifford algebras the results follow by the isomorphisms C 0 C`n = C`n ⊗RC = C`n ⊗RC,Mn(R) ⊗ K ' Mn(K), K ⊗ (R ⊕ R) ' K ⊕ K, where in the last two cases K ∈ {C, H}. Proposition 1.23. There are algebra isomorphisms 0 0 0 C`n+2 ' C`n ⊗R C`2, C`n+2 ' C`n ⊗R C`2 . Here the tensor product is the usual one for algebras. Proof. We will prove the first relation, and the second one is obtained by exactly the same way. Again our strategy is based on the universal property; we want to construct a linear function n+2 0 f : R → C`n ⊗R C`2 0 which factorizes through C`n+2 and thus induces and isomorphism fb :C`n+2 → C`n ⊗R C`2. Let n+2 0 {e1, . . . , en+2} be the standard orthonormal basis of R . Both Clifford algebras C`n and C`n are generated by the first n vectors of the above basis, but when these vectors are considered as 0 0 0 n+2 0 elements of C`n, they will be denoted by {e1, . . . , en}. We define a map f : R → C`n ⊗R C`2 by f(e1) = 1 ⊗ e1, f(e2) = 1 ⊗ e2 and 0 f(ei) = ei−2 ⊗ e1 · e2, (3 ≤ i ≤ n + 2). 0 Then, in C`n ⊗R C`2 the following relations hold: 2 2 2 2 f(e1) = (1 ⊗ e1)(1 ⊗ e1) = 1 ⊗ e1 = −1, f(e2) = (1 ⊗ e2)(1 ⊗ e2) = 1 ⊗ e2 = −1, 2 0 0 0 2 f(ei) = (ei−2 ⊗ e1 · e2)(ei−2 ⊗ e1 · e2) = (ei−2) ⊗ e1 · e2 · e1 · e2 = 1 ⊗ e1 · e2 · e1 · e2 = −1, and for mixed terms f(ei) · f(ej) = −f(ej) · f(ei). Therefore, there exists a unique algebra homomorphism 0 fb :C`n+2 → C`n ⊗R C`2, n+2 such that f = fb◦ j, where j : R → C`n+2 is the associated embedding. But fb preserves the n+2 n 2 0 0 Z2-grading and dimR C`n+2 = 2 = 2 · 2 = dimR C`n · dimR C`2 = dimR(C`n ⊗R C`2), thus fb is an algebra isomorphism.

Lemma 1.24. The following is an isomorphism of real associative algebras, H ⊗R H ' M4(R). C 0 Proof. Again we make use of C`n = C`n ⊗RC = C`n ⊗RC. We deﬁne a (vector space) isomorphism φ : H ⊗R H → EndR(H) ' M4(R) by φ(q1 ⊗ q2)q := q1qq2. One can easily prove that this is an algebra homomorphism. Because dimR(M4(R)) = 16 = dimR(H⊗R H) and φ is surjective, the result follows (see [Hus96, p. 162] for details). Corollary 1.25. There exist the following algebra isomorphisms: C C (1) C`n+2 ' C`n ⊗CM2(C). 0 0 0 0 0 (2) C`n+4 ' C`n ⊗R C`4 and C`n+4 ' C`n ⊗R C`4, where C`4 ' C`4 ' C`2 ⊗ C`2 ' M2(H). 0 0 (3) C`n+8 ' C`n ⊗M16(R) and C`n+8 ' C`n ⊗M16(R). Proof. (1) By Proposition 1.23 we obtain: C C`n+2 = C`n+2 ⊗RC 0 = (C`n ⊗R C`2) ⊗R C 0 = (C`n ⊗RC) ⊗C (C`2 ⊗C) C C = C`n ⊗C C`2 C = C`n ⊗CM2(C). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 11

0 0 (2) Let us apply Proposition 1.23 for n = 2: It is C`4 ' C`2 ⊗R C`2 ' C`4 where 0 C`2 ⊗R C`2 ' M2(R) ⊗R H ' M2(H). Thus, 0 C`n+4 = C`n+2 ⊗R C`2 0 = (C`n ⊗R C`2) ⊗R C`2 0 = C`n ⊗R(C`2 ⊗R C`2) = C`n ⊗R C`4 . The second relation is treated similarly. (3) For the third isomorphism we write for example C`n+8 = C`(n+4)+4, and we apply (2). We use M2(H) ⊗R M2(H) = (H ⊗R M2(R)) ⊗R (H ⊗R M2(R)) = M4(R) ⊗R M4(R) = M16(R). Remark 1.26. Part (3) of Corollary 1.25 indicates a periodicity of order 8 for the Cliﬀord algebras 0 C`n and C`n. This is the famous Cartan-Bott periodicity. Besides detecting spinors in 1913, É. Cartan discovered in 1908 the above periodicity of 8, by identifying the Cliﬀord algebras as matrix algebras with entries in R, C, R ⊕ R, H ⊕ H. Periodicity 8 was re-discovered by Atiyah, Bott and Shapiro in 1964 ([ABS64]), with respect to the graded tensor product.

C C By applying the relation C`n+2 = C`n ⊗M2(C) (2-periodicity) and based on the low-dimensional 0 C results given in Proposition 1.21, one can easily obtain all Cliﬀord algebras C`n, C`n and C`n , for n = 1, 2,..., 8. We list them below. n 1 2 3 4 5 6 7 8 C`n C H H ⊕ H M2(H) M4(C) M8(R) M8(R) ⊕ M8(R) M16(R) 0 C`n R ⊕ R M2(R) M2(C) M2(H) M2(H) ⊕ M2(H) M4(H) M8(C) M16(R) C C`n C ⊕ C M2(C) M2(C) ⊕ M2(C) M4(C) M4(C) ⊕ M4(C) M8(C) M8(C) ⊕ M8(C) M16(C)

1.6. The spin group Spin(n) as a subgroup of C`n. Next we introduce the spin group Spin(n) as a subspace of the Clifford algebra C`n. Generally speaking, one can define spin groups of the form Spin(V, q), as appropriate subspaces of general Clifford algebras C`(V, q) over arbitrary quadratic vector spaces (V, q). Such a description is given for example in [LM89]. n P 2 2 Consider R endowed with B := hx, yi0,n = −hx, yi = − i xiyi and q := qn = −x1 − ..., −xn. n 2 2 Any vector x ∈ R satisfies x = x · x = −||x|| · 1 and has an inverse inside C`n, given by −1 2 x = −x/||x|| = x/qn(x) . (1.2) n−1 n Obviously, the elements of the unit sphere S ⊂ R ⊂ C`n are invertible in C`n. × Definition 1.27. Let us denote by C`n the multiplicative group of units in C`n, that is × −1 −1 −1 C`n = {φ ∈ C`n : ∃φ with φ · φ = φ · φ = 1} . n −1 × n Since any non-zero x ∈ R is such that x = x/qn(x), C`n contains all vectors x ∈ R with × qn(x) 6= 0. The group C`n is open in C`n. To see this one can embed C`n into EndR(C`n). Then, × since C`n is finite dimensional we see that an element φ ∈ C`n lies in C`n if and only if φ is invertible in EndR(C`n). However, the last condition is equivalent to the open condition det φ 6= 0. × n In particular, C`n is a Lie group of dimension 2 . Now, the Clifford algebra C`n acts by algebra automorphisms and hence there is a homomorphism ∗ −1 Ad : C`n → Aut(C`n), ϕ 7→ Adϕ :C`n 3 σ 7→ Adϕ(σ) := ϕ · σ · ϕ ∈ C`n . ∗ Hence, Ad is the adjoint representation of C`n. 12 IOANNIS CHRYSIKOS

n Lemma 1.28. Let x ∈ R some non-zero vector. Then hx, yi − Ad (y) = y − 2 x, x kxk2 n n for any y ∈ R . Hence, − Adx y is the reﬂection through the hyperplane perpendicular to 0 6= x ∈ R . Proof. By (1.2) it follows that x−1kxk2 = −x and hence 2 2 −1 2 −1 −kxk Adx(y) = −kxk x · y · x = x · y · (−kxk x ) = x · y · x = x · − x · y − 2hx, yi = −x2 · y − 2hx, yix = kxk2y − 2hx, yix .

Recall that any element of the group SO(n) is the product of an even number of such reflections (Cartan-Dieudonné Theorem). Hence, the special orthogonal group must be the image via Ad of a certain subgroup of invertible elements in C`n. First it is useful to get rid of the negative sign ∗ appearing above. For this, one may consider the so-called twisted adjoint representation of C`n, which is the linear group homomorphism defined by ∗ −1 Adf : C`n → Aut(C`n), ϕ 7→ Adf ϕ :C`n 3 σ 7→ Adf ϕ(σ) := β(ϕ) · σ · ϕ , where β :C`n → C`n is the parity automorphism. Note that 0 Lemma 1.29. For any element ϕ ∈ C`n we have the identification Adf ϕ = Adϕ. n n −1 Lemma 1.30. The map Adf x : R → R , y 7→ Adf x(y) = β(x) · y · x is a reflection through the hyperplane orthogonal to x. n Proof. As in the proof of Proposition 1.28 and since β(x) = −x for any x ∈ R , we see that 2 2 −1 2 −1 kxk Adf x(y) = kxk β(x) · y · x = x · y · (−kxk x ) = x · y · x = x · − x · y − 2hx, yi = −x2 · y − 2hx, yix = kxk2y − 2hx, yix, and the results follows. n With this definition in mind, we see that for x1, . . . , xm ∈ R , Adf x1·...·xm is a composition of reflections. × Definition 1.31. (a) The pin group Pin(n) ⊂ C`n ⊂ C`n is the subgroup of invertible elements of n−1 C`n which is multiplicatively generated by all vectors x ∈ S , that is n Pin(n) = {x1 ··· xm : m ∈ N, ||xi|| = 1, xi ∈ R } . (b) The spin group, Spin(n), is defined as 0 Spin(n) = Pin(n) ∩ C`n, n−1 thus its elements are products of an even number of xi ∈ S , that is n Spin(n) = {x1 ··· x2k : k ∈ N, ||xi|| = 1, xi ∈ R } . n Remark 1.32. Considering V = R and C`(V, q) = C`n, we can regard Spin(n) as the subgroup of t t all elements a ∈ Pin(n) such that a · γ(a) = a · a = 1, where γ = ( ) :C`n → C`n is the transpose. t In particular, the inverse of an element a = x1 ··· x2m ∈ Spin(n) is given by γ(a) = a = x2m ··· x1. The main conclusion here is the following one (for two slightly different approaches to this result we refer to [LM89, Fr00]).

n n −1 Proposition 1.33. For ϕ ∈ Pin(n) the map Ad(f ϕ): R → R , Ad(f ϕ)x = β(ϕ) · x · ϕ , deﬁnes a surjective homomorphism Adf : Pin(n) → O(n) with Ad(f Spin(n)) = SO(n) and ker Adf = Z2. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 13

2. Topology - The spin group Spin(n) as the universal covering of SO(n). In order to understand deeper the relation between SO(n) and Spin(n) and in particular, Propo- sition 1.33, it is useful to collect a few good definitions and facts related with the theory of fibre bundles and universal covering spaces. For this part, and for more information, we refer the reader to [MilS74, LM89, Hus96, MimT91, GOV94, B04]. 2.1. Material from the theory of fibre bundles. Definition 2.1. A smooth fibre bundle with standard fiber F (or a locally trivial fibre bundle), is a quadruple (E, π, M, F ) consisting of three smooth manifolds E, M, F and a continuous surjective S map π : E → M satisfying the following: M has an open covering, M = i∈I Ui and for each i ∈ I −1 there is a homeomorphism φi : π (Ui) → Ui × F which preservers the fibers, i.e. the following diagram commutes − φ 1 −1 Ui × F / π (Ui) ⊂ E

proj 1 π w Ui −1 or in other words π(φi (x, f)) = x for any x ∈ Ui, f ∈ F . E and M are called the total space and the base space, respectively, while for any x ∈ M, the preimage π−1(x) is homeomorphic to −1 F . In particular, since π is a submersion, each fibre Ex := π (x) is a regular submanifold of E, named the fibre of E over the point x ∈ M. The pair (Ui, φi) is called a local trivialization, while the collection {(Ui, φi)}i∈I will be referred to as an atlas of bundle charts. −1 Notice that fibre Ex = π (x) of π : E → M over x ∈ M has the topology induced by the natural inclusion j : F,→ E and clearly every point E lies in exactly one fiber, i.e. G [ E = Ex = {x} × Ex . x∈M x∈M This is the precise meaning of the statement that, locally, a fibre bundle looks like a direct product. Now, assuming that the intersection Ui∩Uj of two different local trivialization domains is non-empty,

Figure 2. The intersection of two local trivializations of a ﬁbre bundle π : E → M. it makes sense to consider the composition −1 φij = φi ◦ φj :(Ui ∩ Uj) × F → (Ui ∩ Uj) × F. This transition map between two local trivilizations, realizes a smooth map

ϕij : Ui ∩ Uj → Diff(F ),Ui ∩ Uj 3 x 7→ ϕij(x) ∈ Diff(F ) , x x −1 defined by ϕij(x) := φi ◦ (φj ) : F → F , where Diff(F ) is the group of diffeomorphisms of F and x for any x ∈ Ui ⊂ M we set φ := proj ◦ φi : Ex → {x} × F (this is just the restriction of the i 2 Ex 14 IOANNIS CHRYSIKOS

x trivialization φi in the ﬁber Ex, in particular φi is a diﬀeomorphism). The functions ϕij are called transition functions and satisfy the following cocycle conditions:

ϕik ◦ ϕkj = ϕij , if Ui ∩ Uk ∩ Uj 6= ∅ , ϕii = IdF .

Usually, a family of smooth functions {ϕik : Ui ∩ Uk → Diff(F )}i,k∈I satisfying the previous conditions is called a cocycle of the bundle. Example 2.2. The trivial bundle over a smooth manifold M with fibre F is the bundle π : M ×F → M, where π is the projection of M × F to M. It should be emphasized that a trivial fibre bundle admits different trivializations E ' M ×F , which differ each other in surjections E → F . Moreover, it is not hard to see that any fibre bundle over a contractible manifold M is always trivial. Example 2.3. Let G be a Lie group and K ⊂ G be a closed subgroup. Then π : G → G/K is a fibre bundle with fibre the Lie group K, total space the Lie group G and base space the homogeneous space G/K. Definition 2.4. A real (respectively complex) vector bundle over M is a fibre bundle π : E → M −1 whose each fibre Ex := π (M) is a vector space over R (respectively C), such that the corresponding local trivializations −1 n φ : π (U) → U × R −1 n (respectively π (U) → U × C ) are linear homeomorphisms in each fibre, i.e. x φ n proj. n φ : Ex −−→{x} × K −−−−→ K is a K-vector space isomorphism, for any x ∈ M, where K ∈ {R, C}.

When dimK V = n, we often say that π : E → M is a real/complex vector bundle of rank n, depending on K ∈ {R, C}. As before, given two local trivializations (φi,Ui) and (φj,Uj), the map −1 n n φij = φi ◦ φj :(Ui ∩ Uj) × K → (Ui ∩ Uj) × K x x −1 induces the transition functions ϕij : Ui ∩ Uj → GLn K. These are deﬁned by ϕij(x) := φi ◦ (φj ) : n n K → K and satisfy

ϕij ◦ ϕjk ◦ ϕki = IdKr , on Ui ∩ Uj ∩ Uk, ϕii = IdKr on Ui , where here the composition is to be understood as a matrix product. S Remark 2.5. Consider a smooth manifold M, with an open countable covering M = i Ui and ∞ assume that for each non-empty intersection Ui ∩ Uj we have assigned a C -function

ϕij : Ui ∩ Uj → GLn K , satisfying the above compatibility conditions, where we are always working with K ∈ {R, C}. Then, ˜ F n one can construct a vector bundle π : E → M over M, as follows. Set E = i Ui × K and deﬁne the equivalence relation

(x, u) ∼ (y, w), if and only if y = x and w = ϕij(x)u , r r ˜ where (x, u) ∈ Uj ×K and (y, w) ∈ Ui ×K . Then, E := E/ ∼ equipped with the quotient topology, becomes a vector bundle over M, with projection sending the equivalence class [x, u] corresponding r to (x, u) ∈ Uj × K into the ﬁrst coordinate x. Example 2.6. The most familiar example of a vector bundle is the tangent bundle G [ TM = TxM = {x} × TxM = {(x, u): x ∈ M, u ∈ TxM} x∈M x∈M over a smooth manifold M. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 15

Figure 3. The tangent bundle of the Lie group G = S1 is obtained by considering all the tangent spaces (right), and joining them together in a smooth and non-overlapping manner (left). This ensure that for any two points g, e ∈ S1, the 1 1 corresponding tangent spaces Te S and Tg S have no common vector.

n n n Example 2.7. Recall that the real projective space RP is the quotient space RP = S / ∼ where n n+1 1 x ∼ −x and S ⊂ R denotes the n-dimensional sphere. We deﬁne the canonical line bundle γn n over RP , by 1 1 n n n n+1 o γn := P (γn), RP , π = ({±x}, u) ∈ RP × R : u ∈ span{x} .

1 n −1 The projection π : P (γn) → RP is given by π ({±x}, u) := {±x}. Thus, each fibre π ({±x}) n+1 can be identified with the line through x and −x in R , and so it inherits a vector space structure. n 0 n Let U ⊂ S be an open set which contains no pair of antipodal points, and let U ⊂ RP be its n n 0 −1 0 image under the quotient map q : S → RP . Then, the map h : U × R → π (U ) given by h({±x}, t) = ({±x}, tx) is a homeomorphism and a vector space isomorphism when restricted to 1 n the fibers. Thus, the line bundle γn over RP is a simple example of a real vector bundle. Let us now discuss coverings over smooth manifolds. Roughly speaking, coverings are locally trivial fibre bundles with discrete fibre.

Definition 2.8. A smooth map π : Mf → M between two connected manifolds is called a covering map, if π is surjective and for any point x ∈ M there is a connected neighbourhood U of x such that −1 ` the restriction of π to each connected component Uα of π (U) = α∈I Uα be a diffeomorphism onto U, i.e. π Uα : Uα ' U (here I is an indexing set). The manifold Mf is called a covering space of M. It immediately follows that π−1(U) ' U × π−1(x). Thus, any smooth covering map between two manifolds is a local diffeomorphism. Note that for each x ∈ M, the fiber π−1(x) over x is a discrete set in Mf. In particular, for any x, y ∈ M, π−1(x) and π−1(y) have the same cardinality (since there exists a bijection π−1(x) ' π−1(y)), and thus we often use the terminology that Mf is a n-fold covering space of M (where we have assume that π−1(x) consists of n points). Moreover, if Mf is simply connected, i.e. π1(Mf) = 1, then Mf is called the universal covering of M. It is well-known that every connected smooth manifold admits a unique (up to equivalence) universal covering space. An important class of mappings, useful in the study of covering spaces, are the so-called lifts (or liftings).

Deﬁnition 2.9. Let π : Mf → M be a covering of a manifold M.A lift of a map f : N → M, is a smooth map fe : N → Mf such that π ◦ fe = f. Liftings have various interesting properties and it will be useful to list the most important of them in a proposition (see also [MimT91, B04]). 16 IOANNIS CHRYSIKOS

Proposition 2.10. Let π : Mf → M be a covering of a connected manifold M. Fix y0 ∈ Mf and x0 ∈ M such that x0 = π(y0). Then: (α) If σ : I → M is a path in M with σ(0) = x0, then there exists a path σe : I → Mf with σe(0) = y0. (β) For any path homotopy F : I × I → M with F (0, 0) = x0, there is a lifting Fe : I × I → Mf with Fe(0, 0) = y0 which is also a path homotopy. (γ) Let fe0 : N → Mf be a lift of a map f : N → M. Then any homotopy F : N × I → M of f0, i.e. F (x, 0) = (π ◦ fe)(x) = f(x), lifts uniquely to a homotpy Fe : N × I → Mf of fe0, i.e. Fe(x, 0) = fe(x). Thus π ◦ Fe = F . Given two Lie group G and K, roughly speaking, a smooth surjective homomorphism f : G → K is called a covering homomorphism if it satisfies any of the following equivalent conditions (see [AVL91, Ons93, GOV94, B04]) 1) f maps diffeomorphically some neighbourhood of the identity of G onto a neighbourhood of the identity of K; 2) The kernel of f is discrete; 3) f is a covering map in the topological sense, i.e. a locally trivial fibre bundle with discrete fibre; 4) The differential dfe is an isomorphism of tangent spaces (Lie algebra isomorphism). Here, we prove a similar fact which is given as follows: Lemma 2.11. Suppose that f : G → K is a smooth map of Lie groups of the same dimension with K connected. If the kernel of f is discrete, then f is a covering map. Proof. As a Lie group homomorphism, f has constant rank. The rank is equal to the co-dimension of the kernel which by assumption is given by dim G − 0 = dim G = dim K. Hence f has full rank and is in particular a local diffeomorphism. Thus a neighbourhood of the identity in G maps diffeomorphically to a neighbourhood of the identity of K. But K is connected and so it is generated by a neighbourhood of the identity, so f is surjective. Since its kernel is discrete, f must be a covering map. Theorem 2.12. For any connected Lie group G there exists a unique (up to isomorphism) connected simply connected cover Ge which carries the structure of a Lie group also. This is the so-called universal covering of G and the covering map π : Ge → G is a homomorphism of Lie groups. Proof. The covering map π : Ge → G satisfies the homotopy lifting property. Thus, given a smooth map f : M → G from a smooth connected simply connected manifold M to G with x0 ∈ M −1 and y0 ∈ π (f(x0)) ∈ Ge, then there exists a unique lift fe : M → Ge satisfying π ◦ fe = f and fe(x0) = y0. In this way the Lie group structure on G lifts to a Lie group structure on Ge, making π a homomorphism (see also [MimT91, Ons93, Kn96, B04]). For a given connected Lie group G the description of the covering homomorphisms reduces to the description of discrete subgroups of G. Lemma 2.13. If Γ is a discrete normal subgroup of a connected Lie group G, then Γ is a central subgroup, which means that it is contained in the center Z(G) of G. Proof. Fix an element γ ∈ Γ and consider the continuous map G → Γ, g 7→ gγg−1. Then, the −1 subgroup Cγ = {gγg : g ∈ G} is connected as the continuous image of G. Also, normality of Γ implies that Cγ ⊆ Γ, hence because Γ is discrete and Cγ connected, Cγ must be a single point. Since γ is clearly in Cγ, it is Cγ = {γ}, so γ is central. By using basic homotopy theory and Lemmas 2.11 and 2.13 we can prove the following theorem. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 17

Theorem 2.14. ([MimT91, Ons93, Kn96, B04]) Let G be a connected Lie group and let π : Ge → G be its universal covering. Then (α) The kernel Ze = ker π is a discrete normal (and thus central) subgroup of Ge. (β) The covering map π induces an isomorphism G ' G/e Ze. (γ) The natural inclusion π1(G) ,→ Ge is an homomorphism of Lie groups. In particular, −1 π ({e}) = π1(G) = Ze and thus π1(G) is abelian for any connected Lie group G. Remark 2.15. A sequence of groups and their homomorphisms

fn+1 fn · · · → Gn+1 → Gn → Gn−1 → · · · f is called exact if ker fn = Im fn+1 for all n. Note that a sequence 0 → G → K → 0 is exact, if and only if f is an isomorphism. 1 ∗ Example 2.16. The unit circle S = {z ∈ C : |z| = 1} = U(1) = SO(2) is a n-fold covering 1 1 n ∗ over itself, given by p : S → S , z 7→ z , where C := C\{0}. In polar coordinates we have 1 −1 p(1, θ) = (1, nθ) and the fibre over 1 ∈ S is p (1) = Z/nZ; so p wraps the circle around itself n times. Let U be an open arc on S1 subtended by an angle with 0 ≤ θ ≤ 2π and containing a point x. Then p−1(U) consists of n open arcs, each containing the n-th root of x. 1 The universal covering over the circle S is the real line R, which is contractible and hence simply connected. In particular, the covering map is the exponential map 1 2iπt ρ := exp : R → S , t 7→ ρ(t) = cos 2πt + i sin 2πt = e −1 with fibre ρ (1) = Z. The pre-image of a little open arc in the circle is a collection of open intervals 1 ∗ in the real line, offset by multiples of 2π. Thus, if we give S ⊂ C the group structure which inherits ∗ 1 from C , then the universal covering map ρ : R → S is a homomorphism and from Theorem 2.14 1 1 1 we may identify its kernel ker ρ = Z with the fundamental group of S , i.e. π1(S ) = Z. Thus S is not simply connected (for example, the path α(t) = (cos 2πt, sin 2πt) with 0 ≤ t ≤ 1 cannot be contracted to a point). This result also follows by considering the sequence of homotopy groups 1 associated to the covering ρ : R → S with fiber j : Z ,→ R. Because πk(R) = 0 for any k ≥ 0 and πk(Z) = 0 for any k ≥ 1, we obtain the short exact sequence (see Proposition 2.17) ∂ j# ρ# 1 ∂ j# ··· → π1(Z) = 0 → π1(R) = 0 → π1(S ) → π0(Z) = Z → π0(R) = 0, 1 1 which immediately determines the isomorphism π1(S ) = Z. In general, it is πk(S ) = 0 whenever k ≥ 2 (see also [MimT91]). Proposition 2.17. ([Hus96]) Consider a fibre bundle π : E → M over a smooth manifold M. Then, the natural inclusion j : F,→ E of the fiber F in E induces a homomorphism of homotopy groups ∂ : πk(M) → πk−1(F ), such that the sequence of the induced homomorphisms

∂ j# π# ∂ j# π# ∂ · · · → πk+1(M) → πk(F ) → πk(E) → πk(M) → πk−1(F ) → πk−1(E) → πk−1(M) → ... (2.1) is an exact sequence of groups. Assume that G is a Lie group and let K ⊂ G be a closed subgroup. Since π : G → G/K is a ﬁbre bundle with ﬁbre K, by applying (2.1) we obtain the exact sequence (see also [Ons93, B04]) 0 · · · → π2(G/K) → π1(K) → π1(G) → π1(G/K) → π0(K) = K/K → π0(G) → π0(G/K) → 0 . Here, K0 denote the identity component of K. Corollary 2.18. (α) If K and G/K are both connected, then so is G. (β) If π1(G/K) = π2(G/K) = 1, then π1(G) ' π1(K). 0 (γ) If G is connected and simply connected then π1(G/K) = K/K . Moreover, if K is connected, then G/K is simply connected. 18 IOANNIS CHRYSIKOS

Exercise 2.19. Provide a proof of Corollary 2.18. Classical examples of homogeneous spaces satisfying Corollary 2.18 (β) are spheres Sr with r ≥ 3 n (recall that πn(S ) = Z). Note that the condition π2(G/K) = 0 means that any continuous path of a two-dimensional sphere into G/K is contractible to a point.

2.2. The topology of SO(n) and of Spin(n). Based on the tools established in the previous paragraph, we shall now focus on the special orthogonal Lie group SO(n) in order to understand some of its very basic topological features. Corollary 2.20. The special orthogonal Lie group SO(n) is connected. Proof. We proceed by induction on n. The case n = 1 corresponds to SO(1) = {1}, which is trivial. The induction hypothesis says that SO(n − 1) is connected and we will prove the statement for n−1 n SO(n). Indeed, for n ≥ 2 consider the natural action of SO(n) on the unit sphere S ⊂ R , deﬁned by the matrix multiplication, where as usual n n−1 n X 2 n S := {(x1, . . . , xn) ∈ R : xi = 1} ⊂ R . i=1 This action is transitive, and for n ≥ 2 the stabilizer K = {g ∈ SO(n): gN = N} of the north pole N = (1, 0,..., 0) is SO(n − 1). Thus we obtain the ﬁbre bundle SO(n − 1) → SO(n) → Sn−1 ' SO(n)/ SO(n − 1)

Because spheres are connected, by Corollary 2.18 our assertion follows. Consider the 3-dimensional Lie groups t n z w o SU(2) = {A ∈ M : AA = I, det A = 1} = : |z|2 + |w|2 = 1, z, w ∈ 2C −w z C ∗ ∗ and Sp(1) = {g ∈ GL1 H ' H : gg = I}, where g denotes the quaternionic conjugate of g. Then, SU(2) and Sp(1) coincide each other, in particular the space of unit quaternions H1 = {u ∈ H : 3 4 2 2 2 2 4 |u| = 1}' Sp(1) is diﬀeomorphic with S = {(a, b, c, d) ∈ R : a + b + c + d = 1} ⊂ R , and so connected. We summarize as follows Proposition 2.21. The Lie groups SU(2) and Sp(1) are isomorphic each other and they are simply connected. In particular, SU(2) ' Sp(1) is the universal covering of the Lie group SO(3) and we will denote by Ad : SU(2) → SO(3) the double covering map. Moreover, π1(SO(3)) = Z2. Remark 2.22. One observes that the Lie group homomorphism Ad is equivalent with the adjoint representation of SU(2), Ad : SU(2) → Aut(su(2)), where su(2) is the Lie algebra of SU(2) (see [Kn96, B04]). At the level of Lie algebras we have the isomorphisms sp(1) ' su(2) ' so(3), which means that the diﬀerential ρ = (d Ad)e : su(2) → so(3) has an inverse, say ρ−1. However, Ad : SU(2) → SO(3) does not have an inverse since its kernel −1 π1(SO(3)) is non-trivial. Therefore, ρ : so(3) → su(2) is an example of a Lie algebra isomorphism which does not correspond to a Lie group isomorphism. Proposition 2.21, shows that the obstruction of such a lifting lies in the non simply connectedness of the group of rotations SO(3). Exercise 2.23. Provide a proof of Proposition 2.21. Moreover, and in a similar manner with the proof of Corollary 2.20, show that the Lie groups SU(n) and Sp(n) are connected. Hint: Use Proposition 2.21. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 19

Exercise 2.24. Show that 1) SU(2) × SU(2) ' S3 × S3 is the double covering of SO(4). Thus su(2) ⊕ su(2) ' so(4). 2) Sp(2) is the double covering of SO(5), i.e. Sp(2)/{±1}' SO(5). Thus sp(2) ' so(5). 3) SU(4) is the double covering of SO(6), i.e. SU(4)/{±1}' SO(6). Thus su(4) ' so(6). More general, one can prove that Proposition 2.25. The orthogonal group SO(n) is not simply connected, Z if n = 2, π1(SO(n)) = Z/2Z = Z2 if n ≥ 3. 1 Proof. For n = 2, SO(2) ' S , thus π1(SO(2)) = Z. For n = 3 the result follows from Proposition 2.21. By the proof of Corollary 2.20 we know that SO(n) is a ﬁbre bundle over the sphere Sn−1 = j SO(n)/ SO(n − 1), i.e. SO(n − 1) → SO(n) →π Sn−1 . This bundle induces the following exact sequence of homotopy groups:

n−1 ∂ j# π# n−1 · · · → π2(S ) → π1(SO(n − 1)) → π1(SO(n)) → π1(S ). n−1 n−1 Since π2(S ) = π1(S ) = 0, it follows from Corollary 2.18 that for n ≥ 4, the map

j# π1(SO(n − 1)) → π1(SO(n)) is an isomorphism. But π1(SO(3)) = Z2, hence π1(SO(n − 1)) ' π1(SO(n)) = Z2 for all n ≥ 4 Exercise 2.26. Prove that the Lie groups SU(n) and Sp(n) are simply connected for any n. These conclusions are in a line with the statement of Proposition 1.33. In particular, by recalling 0 the inclusion Spin(n) ⊂ C`n, one has Adf |Spin(n) = Ad |Spin(n). Hence, from now on we will identify the restriction of Adf to Spin(n) with the double covering Ad : Spin(n) → SO(n). −1 Proposition 2.27. For n ≥ 2, the spin group Spin(n) = Adf (SO(n)) is connected and for n ≥ 3 it is simply connected, in particular Adf ≡ Ad : Spin(n) → SO(n) is the universal covering of SO(n). Proof. Since ker Adf = {1, −1}, to show that Spin(n) is connected, is enough to ﬁnd a continuous path in Spin(n) which joins (−1) ∈ Spin(n) with 1 ∈ Spin(n). Such a path is given by

c(t) := − cos(πt) − sin(πt)e1 · e2 with t ∈ [0, 1] (inside C`n we have π π π π c(t) = cos( t)e + sin( t)e · cos( t)e − sin( t)e , 2 1 2 2 2 1 2 2 thus c(t) ∈ Spin(n)). Now, by Proposition 2.25, for n ≥ 3 we have that π1(SO(n)) = Z2. Hence, Ad Z2 → Spin(n) → SO(n) induces the short exact sequence 1 → π1(Spin(n)) → π1(SO(n)) = Z2 → 1, and so π1(Spin(n)) is trivial. Consequently, one can adopt the following definition. Definition 2.28. For n ≥ 3, the spin group Spin(n) is defined to be the the universal covering group of SO(n). We list the following isomorphisms between low dimensional spin groups and certain classical groups: Spin(3) ' Sp(1) ' SU(2), Spin(5) ' Sp(2), Spin(4) ' Sp(1) × Sp(1) ' SU(2) × SU(2), Spin(6) ' SU(4) . 20 IOANNIS CHRYSIKOS

2.3. The Lie algebra of Spin(n). Since for n ≥ 3, Spin(n) is the universal covering of SO(n), at the level of Lie algebras we get an isomorphism

(dAdf |Te Spinn )e ≡ d Ade := ad : spin(n) → so(n), with dim Spin(n) = dim SO(n) = n(n − 1)/2. Next our aim is to examine the above isomorphism in more details. We begin by ﬁxing some notation.

Notation. In the following, we shall denote by Di,j the (n × n)-matrix having 1 in the (i, j)-entry and zeros elsewhere. We set 0 ...... 0 0 ... 0 0 ...... −1 ... 0 . . . . . . Ei,j := −Di,j + Dj,i = . . . . ... 1 ... .   0 ...... 0 0 ...... 0 V2 n Observe that there is a natural isomorphism (R ) ' so(n) induced by associating to any u, w ∈ n R the skew-symmetric endomorphism u ∧ w, deﬁned by (u ∧ w)(x) := hu, xiw − hw, xiu, (2.2) and then extending by universality. Hence, and since Ej,i = −Ei,j, the 2-forms ei ∧ ej corresponds to the elementary matrices Ei,j. In particular, the matrices {Ei,j : 1 ≤ i < j ≤ n} generate the Lie algebra so(n) and form an orthonormal basis with respect to the scalar product −(1/2) tr AB. It is a classical fact that for any algebra A the associated group of invertible elements, say A×, is a Lie group whose Lie algebra, say a×, coincides with A (see [LM89]): × × a = T1(A ) ' A. × × In our case, A is the Lie group C`n of all invertible elements in A = C`n, and its Lie algebra, say × Cln , coincides with C`n equipped with the Lie bracket [x, y] = x · y − y · x, for any x, y ∈ C`n. The exponential map × × exp : Cln → C`n P∞ n × is given by the standard series exp(a) = n=0(a /n!). Because Spin(n) ⊂ C`n , in order to describe × × the Lie algebra spin(n) ⊂ Cln , it is suﬃcient to describe the tangent space T1(Spin(n)) ⊂ T1(C`n ) = C`n. Proposition 2.29. The Lie algebra of the spin group Spin(n) ⊂ C`× is a Lie subalgebra of n C`n, [ , ] given by spin(n) = spanR{ei · ej : 1 ≤ i < j ≤ n}. Moreover, ad(ei · ej) = 2(ei ∧ ej) = 2Eij.

Proof. The Lie algebra of Spin(n) is the vector subspace of C`n spanned by the tangent vectors to n Spin(n) at the identity 1. Fix an orthonormal basis {ei} of R . As usual, we identify the space V2 n (R ) with a vector subspace of C`n, via the map

ei ∧ ej 7→ ei · ej, (1 ≤ i < j ≤ n).

This subspace lies in spin(n), since ei · ej is the tangent vector at t = 0 to the curve in Spin(n) given by (notice that c(0) = 1) t t t t c(t) := sin( )e − cos( )e · (sin( )e + cos( )e 2 i 2 j 2 i 2 j = cos(t) + sin(t)ei · ej, DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 21

d V2 n i.e. dt c(t) t=0 = ei · ej. Therefore, spin(n) contains the vector subspace spanR{ei ∧ ej} = (R ). V2 n V2 n Comparing dimensions, and since dimR spin(n) = dimR so(n) = dimR (R ), we get (R ) ' spin(n). Now, by the deﬁnition of Ad : Spin(n) → SO(n), we get Ad(c(t))(y) = c(t) · y · c−1(t) and hence d −1 d (c (t)) = − c(t) = −ei · ej. dt t=0 dt t=0 Thus d d −1 ad(ei · ej)(y) = Ad(c(t))(y) = c(t) · y · c (t) dt t=0 dt t=0 0 −1 0 = c (0) · y · c(0) + c(0) · y · (c ) (0) = ei · ej · y − y · ei · ej = ei · ej · y − − ei · y − 2hy, eii · ej = ei · ej · y + ei · y · ej + 2hy, eii · ej = ei · ej · y + ei · − ej · y − 2hy, eji + 2hy, eii · ej (2.2) = ei · ej · y − ei · ej · y − 2hej, yi · ei + 2hy, eii · ej = 2(ei ∧ ej)(y) . 0 Example 2.30. For n = 1, Spin(1) ⊂ C`1 ' C`0 ' R. In R we have just two unit vectors, ±1, which sit inside C as ±i and generate Pin(1), which is isomorphic to Z4. Hence, Spin(1) ' Z2 = {±1} and the connected double covering is nothing than the square map 2 Ad : Spin(1) ' Z2 → SO(1) '{1}, t 7→ t . 0 1 For n = 2 we have Spin(2) ⊂ C`2 ' C`1 ' C. Thus we identify S ' U(1) ' Spin(2) via the map iθ e 7→ cos(θ) + sin(θ)e1 · e2 described before. The connected double covering Ad : Spin(2) → SO(2) is given by cos(2θ) − sin(2θ) eiθ 7→ . sin(2θ) cos(2θ) The right hand side deﬁnes a rotation and because SO(2) ' S1 this is again the square map. 0 For n = 3 it is Spin(3) ⊂ C`3 ' C`2 ' H, where

1, e1, e2, e3, e1 · e2, e1 · e3, e2 · e3, e1 · e2 · e3 ∗ is a basis of C`3 ' H ⊕ H. Thus Spin(3) is identified with a subgroup of H as we have already seen in Proposition 2.21, Spin(3) ' SU(2) ' Sp(1). The universal covering is given by Ad : Spin(3) ' Sp(1) → SO(3), q 7→ (x ∈ Im(H) 7→ qxq¯) . 3. Spin structures on Riemannian manifolds In this section we shall examine spin structures on Riemannian manifolds. 3.1. Principal bundles. We begin we preliminaries on G-principal bundles. For details that we omit we refer to [LM89, MimT91, AVL91, Hus96]. Definition 3.1. Let G be a Lie group. A G-principal bundle P over a smooth manifold M is a fibre bundle π : P → M together with a continuous free right action of G on P which preserves the −1 fibers and acts transitively on them. Thus, the fibers Fx = π (x)(x ∈ M) are exactly the orbits of G, i.e. Fx = G (since the action is transitive). Moreover, for any x ∈ M there exists an open neighborhood U ⊂ M of x and a homeomorphism −1 ΦU : π (U) → U × G −1 with ΦU (p) = (π(p), φU (p)), where φU : π (U) → G is a G-equivariant map, that is φU (pg) = −1 φU (p)g, for all p ∈ π (U) and g ∈ G. We shall denote a G-principal bundle over M by ξ = (P, π, M, G). 22 IOANNIS CHRYSIKOS

Example 3.2. Consider a smooth manifold M and let π : M˜ → M be its universal covering. Then, π is a π1(M)-principal bundle over M.

Example 3.3. Let M be a n-dimensional smooth manifold and let TxM be its tangent space at a point x ∈ M. We denote by

Fx(TM) = {ux = {u1, . . . , un} ∈ TxM : ux basis of TxM} the set of all linear frames (or bases) ux = {u1, . . . , un} ∈ TxM at x ∈ M. Then, the disjoint union G F(M) = Fx(TM), x∈M forms the so-called frame bundle, which is GLn R-principal bundle over M. The projection π : F(M) → M maps a linear frame ux onto x. The right free action of GLn R to F(M) is given by i the matrix multiplication: If ux = {u1, . . . , un} ∈ F(M) and A = (Aj) ∈ GLn R, then set X j uxA := wx = {w1, . . . wn}, wi := Ai uj. j Example 3.4. Let G be a Lie group and K ⊂ G be a closed subgroup. Then, the projection π : G → G/K deﬁnes an K-principal bundle over G/K. There is a natural construction which allows us to go back from a principal bundle to a vector bundle, as follows. Consider a n-dimensional manifold M n and let π : P → M be a G-principal bundle over M, where G is a Lie group. Any G-representation ρ : G → GL(V ) on a (ﬁnite dimensional) vector space V (real or complex), induces an action (P × V ) × G → P × V of G on P × V , given by (p, u), g 7→ pg, ρ(g−1)u, for p ∈ P , u ∈ V and g ∈ G. The quotient (P × V )/ ∼, where ∼ is the equivalence relation (p, u) ∼ (pg, ρ(g−1)u), is the so called associated bundle, which we shall denote by

E := (P × V )/ ∼ = P ×G V = P ×ρ V → M. Next we will write [p, u] for the equivalence class corresponding to (p, u). The associated bundle comes with a natural projection πρ : E = P ×ρ V → M, given by πρ([p, u]) = π(p). In particular, −1 πρ([pg, ρ(g )u]) = π(pg) = π(p) = πρ([p, u]) and hence the projection πρ is well-defined and asserts to E the structure of a smooth fiber bundle −1 over M with fiber Ex := πρ (x) ' V , for any x ∈ M. Thus each fibre of E = P ×ρ V has the structure of a (real or complex) vector space. In particular, for any point x ∈ M there is an open set U 3 x of M and a map φ : π−1(U) → U × V which in linear on each fibre. This establishes E as a vector bundle (real or complex, depending on V ). Note that if P is defined via the transition functions ϕij : Ui ∩ Uj → G, for some members Ui,Uj of an open cover of M, then P ×ρ V is defined via the transition functions

ρ ◦ ϕij : Ui ∩ Uj → GL(V ) . n n Example 3.5. Let M be a smooth manifold and let (F(M), π, M , GLn R) be its frame bundle. n Assume that ρ is the natural representation of GLn R to R , given by the matrix multiplication. Then the associated vector bundle is isomorphic to the tangent bundle via the well-deﬁned map n P F(M) ×ρ R → TM given by f(u1, . . . , un; c1, . . . , cn) := uici. Thus n F(M) ×ρ R ' TM. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 23

Let ξ = (P, π, M, G) be a principal G-bundle. A smooth map s : M → P is called a cross-section −1 or just section, if π ◦ s = IdM , i.e. s(x) ∈ π (x) for all x ∈ M. If s is defined on an open subset U ⊂ M, it is called a local section. Clearly, local sections always exist. We will denote by Γ(P,M), or just by Γ(P ), the set of sections of P . Example 3.6. On the product bundle M × G cross-sections are just the graphs of functions f : M → G. Indeed we can always define a section s : M → M × G by setting s(x) = (x, fs(x)) for some map fs : M → G to the fibre G. Thus, Γ(M × G) coincides with the set of maps fs : M → G. Lemma 3.7. Consider a G-principal bundle π : P → M, a representation ρ : G → Aut(V ) and the associated vector bundle E = P ×ρ V with projection πρ : E → M. Then, the space of smooth (local) sections of E, i.e.

Γ(E,U) = {σ : U ⊂ M → E : πρ ◦ σ = IdU } is identified with the set of G-equivariant maps f : π−1(U) ⊂ P → V , Γ(E,U) ' Map(π−1(U),V )G := {f : π−1(U) ⊂ P → V : f(pg) = ρ(g−1)f(p)} . Proof. Given such an equivariant map f : π−1(U) ⊂ P → V we define a local section σ : U → E by the rule σ(x) = [p, f(p)] , −1 −1 for some p ∈ π (x) where π (x) ' Px is the fiber over x ∈ U ⊂ M. By the equivariance property it follows that [pg, f(pg)] = [pg, ρ(g−1)f(p)] = [p, f(p)] = σ(x) , so this is well-defined. Conversely, given a section σ : U → E, we define a map f : π−1(U) → V by f(p) = u, where σ(π(p)) = [p, u]. It follows that f(pg) = ρ(g−1)f(p) since σ(π(pg)) = σ(π(p)) = [p, u] = [pg, ρ(g−1)u] , for any g ∈ G. It is clear that passing from σ to f and vice versa are inverse smooth operations. Let us recall now when we can identify two principal bundles. Definition 3.8. (a) Two G-principal bundles ξ = (P, π, M, G) and η = (P 0, π0,M,G) over the same base M, are called isomorphic, ξ ' η, if there exists a G-equivariant diffeomorphism f : P → P 0 such that π = π0 ◦ f. (b) A morphism from a G-bundle ξ = (P, π, M, G) to a G0-bundle η = (P 0, π0,M 0,G0) is a pair of maps (f 0, f) with f 0 : P → P 0 and f : M → M 0, such that π0 ◦ u = f ◦ π. In particular, f 0 is G-equivariant, which implies that it takes fibres to fibres. A principal bundle which is isomorphic to the product bundle, is called trivial. Proposition 3.9. A principal G-bundle ξ = (P, π, M, G) is trivial, if and only if it admits a section s : M → P . Proof. Necessity was described in Example 3.6. Given a section s : M → P , then the map h : M × G → P defined by h(x, g) = s(x)g, for any x ∈ M and g ∈ G, is a morphism of principal bundles. Thus h may regarded as an isomorphism from the product bundle to ξ. On the other hand, for real vector bundles we have Proposition 3.10. A n-dimensional (real) vector bundle is trivial, if and only if it admits n smooth sections which are everywhere linearly independent. Exercise 3.11. Provide a proof for Proposition 3.10. 24 IOANNIS CHRYSIKOS

1 n Example 3.12. The canonical line bundle γn over RP , described in Example 2.7, is not trivial n 1 for any n ≥ 1. Indeed, assume that s : RP → P (γn) is a section and consider the composition n n 1 S → RP → P (γn) n 1 which maps each x ∈ S to a pair {±x}, ψ(x)x ∈ P (γn), for some real-valued function ψ(x) with ψ(−x) = −ψ(x). Because the sphere Sn is connected, from the intermediate value theorem n it follows that ψ(x0) = 0 for some x0 ∈ S . Hence s({±x0}) = ({±x0}, 0) and the canonical line 1 bundle γn has no nowhere vanishing sections. Example 3.13. The Möbius strip is a 1-dimensional real vector bundle over the circle S1 and is the 1 simplest example of a non-trivial vector bundle. This bundle is arising from γn for n = 1. Indeed,

Figure 4. The Möbius strip.

1 for n = 1, the total space of γ1 is given by 1 P (γ1 ) = {±(cos θ, sin θ)}, t · (cos θ, sin θ) : θ ∈ [0, π], t ∈ R . 1 1 1 Another formal definition is given by P (γ1 ) ' S ×GR, where G = {±1} = Z2 acts on S by 1 (Ad, z) 7→ Ad z. One can show that P (γ1 ) can be obtained from the strip [0, π]×R in the (θ, t)-plane by identifying the left hand boundary {0}×R with the right one {π}×R, under the correspondence (0, t) 7→ (π, −t). Definition 3.14. A smooth manifold M n is called parallelizable, if it admits n vector fields (sections of the tangent bundle TM) which are linearly independent at each point. This is equivalent to say that the tangent bundle TM of M is trivial as a vector bundle. Thus, for a parallelizable manifold M its frame bundle F(M) has a global section and it is a trivial GLn R- principal bundle. Important examples of parallelizable manifolds are Lie groups. We mention that the Euler characteristic χ(M) of a parallelizable manifold M vanishes (since the existence of a nowhere zero vector field is equivalent with the vanishing χ(M) = 0). n n+1 Example 3.15. Remember that S ⊂ R admits a nowhere zero vector field if and only if n is odd, say n = 2k + 1, given by

X(x1, . . . , x2k+2) = (−x2, x1, −x4, x3,..., −x2k+2, x2k+1). Indeed, if Sn admits a nowhere zero field, we can use it to build a homotopy between the identity map in Sn and the antipodal point map. Given such one, both maps must have the same degree and we have to solve the equation (−1)n+1 = 1, which amounts to n being odd (see [Hus96]). Therefore, if n is even, Sn is not parallelizable, and then for these values the tangent bundle n n n+1 T S = {(x, u) ∈ S ×R : x ⊥ u} is not trivial (since the trivial bundle always has a non-vanishing section). It is well-known that the only parallelizable unit spheres are S1, S3 and S7 (and thus only these three spheres can be DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 25 considered as Lie groups). Note that S2 does not belong to this list, since its Euler characteristic χ(S2) = 2 6= 0. 3.2. Reduction of the structure group. Let us now describe the “change” of the structure group of a principal bundle, i.e. the notion of reduction. Definition 3.16. Let π : P → M be a G-principal bundle and λ : K → G be a smooth group homomorphism, where K,G are Lie groups. A pair (P 0, f) consisting of a K-principal bundle π0 : P 0 → M and a mapping f : P 0 → P is said to be a λ-reduction of P , if π0 = π ◦ f and f(p0k) = f(p0)λ(k) for any p0 ∈ P 0, k ∈ K. In other words, a λ-reduction defines the following commutative diagram

P 0 × K / P 0 π0 f×λ f π P × G / P / M Whenever λ is the embedding of a subgroup K ⊂ G, then one talks briefly about a K-reduction. Be aware that reductions may not always exist, and if they exist they are not necessarily unique. Definition 3.17. Let M n be a n-dimensional manifold. A Riemannian metrric on M is defined to be a metric on the tangent bundle TM of M, i.e. a symmetric bilinear form g on TM such that gx(u, u) > 0 for all non-zero tangent vectors u ∈ TxM at some point x ∈ M, and which varies smoothly along M. This means that for any two vector fields X,Y ∈ Γ(TM) on M the assignment x → gx(Xx,Yx) is a smooth function on M. Similarly we define a Riemannian bundle metric on a vector bundle π : E → M as a map ρ which assigns to each x ∈ M an inner product ρx( , ) on −1 the fiber Ex = π (x) over x ∈ M. Again we require the assignment to be smooth, which means that if σ1, σ2 are two smooth sections of E, then ρ(σ1, σ2) is a smooth function on M. A vector bundle E together with a Riemannian bundle metric is called a Riemannian bundle. Obviously, these definitions extend to pseudo-Riemannian metrics of signature (r, s) with r + s = n. Remark 3.18. Given a maximal compact Lie subgroup K of G, it is well-known that G/K is homeomorphic to an Euclidean space. Using this fact one can show that any principal G-bundle P → M over a smooth n-dimensional manifold M (or more general over a CW-complex M), admits a reduction to K (see [Hus96]). Example 3.19. For example, consider a real vector bundle E → M of rank n over a smooth n manifold M. We can express E as the associated bundle E = F(E)×GLn R R of the GLn R-principal bundle F(E) (i.e., the bundle whose fibre at x ∈ M is the set of all bases of the vector space Ex). O(n) is a maximal compact Lie subgroup of GLn R, in particular GLn R/ O(n) is homeomorphic to the n(n+1)/2 group of upper triangular matrices with positive diagonal entries and thus GLn R/ O(n) ' R . Then, Remark 3.18 guarantees the existence of an O(n)-reduction or equivalently a Riemannian bundle metric. For example, on a smooth manifold M, the existence of the metric tensor is equivalent to the reduction of the structure group GLn R of the natural frame bundle F(M) over M to O(n), and such a metric always exists (under the assumption that M is paracompact). More different types of reductions on smooth manifolds, are shortly described in the Appendix, see Sections 11.2, 11.3, 11.4 and 11.5. In contrast with a Riemannian structure on a real vector bundle π : E → M, an orientation on E, i.e. an orientation continuously defined on the fibers, may not always exist. In particular, E is orientable if and only if its structure group can be reduced from O(n) to SO(n), i.e. if we can reduce the non-connected structure group to a connected one. If we denote the bundle of Riemannian metrics on E by M(E), then the bundle of orientations in E must be the quotient M(E)/ SO(n), 26 IOANNIS CHRYSIKOS where two bases of Ex are identified if the orthogonal matrix transforming one to the other has determinant +1. Note that G/K = O(n)/ SO(n) = {±1} = Z2, and thus M(E)/ SO(n) → M is a 2-fold covering of M, or in other words a Z2-principal bundle over M. Then, one can prove that E is orientable if and only if this covering is the trivial one.

Remark 3.20. As we said above, a Z2-principal bundle ξ = (P, π, M, Z2) over a connected manifold M (or more general, topological space M) is just a 2-fold covering π : P → M of M. Such a covering is described completely by a homomorphism

f : π1(M) → Z2 and we have P = M ×f(π1(M)) Z2. When f is trivial we get the trivial bundle P = M × Z2 and ker f = π1(M). Otherwise, we have that the kernel ker f ⊂ π1(M) is a normal subgroup of index 2, i.e. π1(M)/ ker f ' Z2. Because Z2 = O(1) is abelian, one can show that the set of isomorphism classes of Z2-bundles is given by (see [LM89, Hus96]) π1(M) 1 PZ2 (M) = Hom(π1(M), Z2) = Hom , Z2 = Hom H1(M, Z), Z2 = H (M; Z2) . [π1(M), π1(M)]

Note that the third equality follows from the so-called Hurewicz isomorphism ϕn : πn(M) → Hn(M; Z), which for the special case n = 1, induces the isomorphism between the abelianization of 1 π1(M) and H1(M; Z). Any Z2-principal bundle is classiﬁed by an element w1(ξ) = w1 ∈ H (M; Z2), the so-called ﬁrst Stiefel-Whitney class (see for example [MilS74, Hus96]). In terms of Remark 3.20 we obtain that

Theorem 3.21. ([MilS74, Hus96]) A real vector bundle E is orientable, if and only if w1(E) = 0. 3.3. Orientable manifolds. Let us now examine orientable manifolds. In general, one can pose several different but equivalent definitions of an orientation on M. Definition 3.22. A smooth manifold M n is said to be orientable, if there exists a smooth atlas for M such that the Jacobians of all the transition functions are positive. Let Ωp(M) := Γ(Vp(T ∗M)) be the space of smooth differential p-forms on M. Then, it is easy to show that Proposition 3.23. A n-dimensional smooth manifold is orientable if there exists a nowhere vanish- n ing n-from ω ∈ Ω (M), i.e. ωx 6= 0 at any x ∈ M. Usually, such a n-form is called the orientation form on M n. n 1 n Example 3.24. On the Euclidean space R we have the orientation form dx ∧ ... ∧ dx and this n represents the standard orientation of R . n Given a smooth manifold M with orientation form ω, a basis {u1, . . . , un} of TxM is said to be positively, or negatively oriented (with respect to ω), depending on whether ωx(u1, . . . , un) ∈ R is a positive or negative real number, respectively. If ω, θ are two orientation forms on M n, then θ = f · ω, for a uniquely determined smooth function on M, with f(x) 6= 0 for any x ∈ M. We say that ω and θ determine the same orientation at x, if f(x) > 0 (which means that these orientation forms must induce the same positively oriented bases on TxM). Now, if M is connected, then f has constant sign on M, so Lemma 3.25. On a connected orientable smooth manifold there are precisely two orientations. Proposition 3.26. Consider an oriented Riemannian manifold (M n, g). Then, M admits a uniquely g determined orientation form, say dv ≡ volM , with g dv (e1, . . . , en) = 1 DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 27 for any positively oriented orthonormal basis of T M. This is the so-called volume form of M. For x √ coordinate functions (x1, . . . , xn) on a local chart on M, we can abbreviate dvg by gdx, where dx := dx1 ∧ ... ∧ dxn. For a smooth manifold M the Stiefel-Whitney classes are defined to be the Stiefel-Whitney classes of the tangent bundle TM → M of M, i wi(M) ≡ wi(TM) ∈ H (M; Z2) . Then, by Corollary 3.21 it follows that (see also [MilS74, Hus96, LM89]). Corollary 3.27. A smooth manifold M is orientable, if and only if its tangent bundle is orientable, i.e. w1(M) = 0. n n+1 n Example 3.28. Consider the unit spehere S ⊂ R . Then, the tangent space of S at any point x ∈ Sn, consists of all vectors perpendicular to this point. Hence the tangent bundle is given by n n+1 n+1 TS = {(x, u) ∈ R × R : kxk = 1, hx, ui = 0} . It follows that n+1 n n T R = T S ⊕N S , n n n n+1 where N S denotes the normal bundle of S . But both N S and T R are trivial bundles, hence n+1 n n n n 0 = w1(T R ) = w1(T S ) + w1(N S ) = w1(T S ) ≡ w1(S ) . Consequently, Sn is orientable. 3.4. Spin structures on oriented Riemannian manifolds. Let (M, g) be a connected, oriented Riemannian manifold. From now on let us denote by Pg := SO(M, g) the SO(n)-principal bundle of positively oriented g-orthonormal frames, i.e., Pg ≡ SO(M, g) = { x, {e1, . . . en} : {ei} is a positively oriented ON-frame of TxM, x ∈ M} .

Hence an element u ∈ Pg over x ∈ M may be regarded as an isometry n n X R 3 u = (u1, . . . , un) 7→ uiei ∈ TxM. i=1

We shall denote by π : Pg → M the projection map. The tangent bundle of M can be viewed as the n associated vector bundle to Pg via the standard representation of SO(n) on R , say ρn : SO(n) → n Aut(R ), i.e. n n TM = Pg ×ρn R = Pg ×SO(n) R . ∗ n ∗ ∗ Similarly, and by considering the dual representation ρn : SO(n) → Aut((R ) ) we see that T M = ∗ n ∗ ∗ Pg ×ρn (R ) . In fact, in the presence of the metric g, we have ρn = ρn and hence we can identify ∗ p TM and its dual, TM ' T M. Now ρn induces a series of representations, for example ∧ ρn : Vp n SO(n) → Aut( R )(p = 0, 1, . . . , n), deﬁned by p ∧ ρn(A)(u1 ∧ ... ∧ up) := ρn(A)(u1) ∧ ... ∧ ρn(A)(up) ,A ∈ SO(n) (and extending by linearity to more general elements). Hence, applying the associated bundle construction it follows that the vector bundle of p-forms on (M n, g) is given by p p p ^ ^ ∗ ^ n ∗ p (M) ≡ (T M) := Pg ×∧ ρn (R ) . Roughly speaking, a spin structure on an oriented Riemannian manifold (M n, g) is a reduction of the structure group SO(n) of the orthonormal frame bundle Pg to its simply connected double covering, i.e. the spin group Spin(n). In particular, 28 IOANNIS CHRYSIKOS

Deﬁnition 3.29. A spin structure on a connected, oriented Riemannian manifold (M n, g) is a Spin(n)-principal bundle Peg := SOf (M, g) over M, together with a 2-fold covering Λg : Peg → Pg such that Λg(˜pa) = Λg(˜p) Ad(a), for any p˜ ∈ Peg and a ∈ Spin(n), which equivalently means that the following diagram commutes action Peg := SOf (M, g) × Spin(n) / Peg π0 Λg×Ad Λg action π Pg := SO(M, g) × SO(n) / Pg / M. Thus, by a Riemannian spin manifold we understand a connected, oriented Riemannian manifold (M n, g) endowed with a spin structure.

1 2 n Deﬁnition 3.30. Two spin structures Λ1 : Peg → Pg and Λ2 : Peg → Pg of (M , g) are called 1 2 1 2 equivalent, if there is a Spin(n)-equivariant map f : Peg → Peg such that Λg = Λg ◦ f, i.e. the following diagram is commutative 1 f 2 Peg / Peg

Λ1 Λ2 Pg

Remark 3.31. For n = 2, a spin structure is defined by replacing Spin(2) by SO(2) ' S1 and for n = 1 a spin structure is just a 2-fold covering on M. In particular, on M = R there exists a unique spin structure. Example 3.32. The circle S1 =∼ SO(2) admits two spin structures. The frame bundle is trivial, 1 1 1 Pg ' S , and hence we immediately get a spin structure Peg = S × Spin(1) with Spin(1) ' Z2. Notice that the spinor bundle associated to this spin structure is also trivial and 1-dimensional since ∆1 ' C (see the next section for the spinor bundle). Hence spinors are simply C-valued functions 1 1 on S (which we can regard as 2π-periodic functions on R). In fact, S admits two inequivalent spin structures which coincide with the two-fold coverings of S1 (see also Example 2.16) 1 1 1 1 2 λ1 : S ×Z2 → S , (z, c) 7→ cz, λ2 : S → S , z 7→ z , with c ∈ {1, −1} = Z2. The second “non-trivial” spin structure is defined by 2 Peg = ([0, 2π] × Spin1)/ ∼ where ∼ identifies 0 with 2π while it interchanges the two elements of Spin(1) ' Z2. Spinors in this case correspond to 2π-anti-periodic complex values on R, ϕ(t + 2π) = −ϕ(t). As we will see below, the existence of two inequivalent spin structures on S1 fits with the fact that S1 is not simply connected, in particular 1 1 1 H (S , Z2) = HomZ(H1(S ; Z), Z2) ' Z2 . For the 2-torus T 2 = S1 × S1 the bundle of positively oriented orthonormal frames is canonically 2 1 1 2 trivialized, Pg ' T × S . Moreover, it is H (T , Z2) = Z2 × Z2 and one can prove that there are 4 distinct spin structures on T 2. More general, there are 2n inequivalent spin structures on the n n n n n n-torus T = R /Zn ' R /Γ, where Γ is a lattice in R (in fact Γ ' H1(T , Z)). If {β1, . . . , βn} n is a basis of Γ, then spin structures on T can be classified by n-tuples (α1, . . . , αn) where each αi ∈ {0, 1} indicates whether or not the spin structure is twisted in the direction of βi (1 ≤ i ≤ n). For example (0,..., 0) defines the trivial spin structure. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 29

Remark 3.33. On a fixed oriented Riemannian manifold (M n, g) the existence of a spin structure does not depend on the metric tensor g, or on the orientation of M, but it has only a topological nature. In particular, the existence of a spin structure on M relates the question whether the transition functions ϕij : Ui ∩ Uj → SO(n) have a lift ϕeij into Spin(n), i.e. Spin(n) 9 ϕeij Ad Ui ∩ Uj / SO(n) ϕij such that they satisfy the cocycle condition ϕeij ◦ ϕejk ◦ ϕeki = 1. This topological task is analysed in Theorem 3.34. It follows that not every oriented Riemannian manifold is spin. For example, the n complex projective space M = CP for n = even is not spin (since w2(M) 6= 0, see Theorem 3.34 below). The similar result is valid also for Lorentzian manifolds [Mor07]. However, it fails for other types of signature (cf. [B81, p. 78] and [Kar68]). Theorem 3.34. An orientable Riemannian manifold (M n, g) is a spin manifold if and only if 2 the second Stiefel–Whitney class w2(M) ≡ w2(TM) ∈ H (M, Z2) vanishes. In this case, the spin 1 structures are classified by elements in H (M, Z2) = Hom(π1(M), Z2). Proof. Given a Lie group G, recall that equivalence classes of principal G-bundles over a manifold M are in 1-1 correspondence with elements in H1(N,G), see [MilS74, Hus96]. Consider for example Z2-bundles (P, π, M, Z2) over M, i.e. 2-fold coverings π : P → M of M. Already in Remark 3.20 1 we explained that such principal bundles are classified by H (M, Z2). Since we know the situation 1 with S and R, we may now assume that n ≥ 2. By definition, a spin structure on M coincides with a 2-fold covering Λg : Peg → Pg of the orthonormal frame bundle Pg = SO(M, g) which is non-trivial on each fibre of the projection map π : Pg → M. From above we deduce that the set of Z2-principal 2 π bundles over Pg must coincide with H (Pg, Z2). Consider now the SO(n)-fibration Pg −→ M. This induces the following short exact sequence of cohomology groups

1 π∗ 1 i∗ 1 ∂ 2 0 → H (M, Z2) −−−→ H (Pg, Z2) −−→ H (SO(n), Z2) = Z2 −−→ H (M, Z2) . (3.1) Then, the second Stiefel-Whitney class of TM coincides with the image under ∂ of the non-trivial 1 element of H (SO(n), Z2) = Z2, i.e. (see for example [Hus96, LM89, Fr00]) 2 w2(M) ≡ w2(TM) = ∂(1) ∈ H (M, Z2) . 1 On the other hand, spin structures can be interpreted as elements f ∈ H (Pg, Z2) with non-zero image under i∗, see [Fr00]. In particular, (M, g) is spin if and only if such an element exists, that is, if and only if i∗ is surjective. Thus, from the relation ∂ ◦ i∗ = 0 we see that this is equivalent to ∂ = 0, i.e. to the vanishing of the second Stiefel-Whitney class of M,

w2(M) = 0 , which gives a proof for the ﬁrst claim. Based again on (3.1) one also deduces that when w2(M) = 0, i.e. when (M, g) is spin, then the set of spin structures identiﬁes with the non-zero coset in 1 ∗ 1 1 H (Pg, Z2)/π H (M, Z2) , which has the same cardinality as H (M, Z2).

Exercise 3.35. Show that the 2-fold coverings of Pg = SO(M, g) can be identiﬁed with elements 1 f ∈ H (Pg, Z2) such that ∗ 1 0 6= i (f) ∈ H (SO(n), Z2) = Z2, ∗ 1 1 where i : H (Pg, Z2) → H (SO(n), Z2) = Hom(Z2, Z2) = Z2. Let us describe now basic constructions of spin manifolds and some examples. 30 IOANNIS CHRYSIKOS

Proposition 3.36. The Cartesian product of two spin manifolds is a spin manifold. Any submanifold of a spin manifold with a spin structure on its normal bundle, is a spin manifold. Proposition 3.37. Any parallelizable manifold, in particular any Lie group G, is spin. Example 3.38. Any oriented 3-manifold (M 3, g) is spin, since it is parallelizable.

n n+1 Example 3.39. For n ≥ 2, the unit sphere S ⊂ R admits a unique spin structure. Indeed, n n+1 n recall that Tx S is the orthogonal complement of the line in R through the origin and x ∈ S . n ⊥ n Hence, an oriented orthonormal basis for Tx S is an oriented orthonormal frame for x ' R . This induces a bijective correspondence with the points in SO(n). Adding x itself we obtain an n+1 oriented orthonormal frame for R and hence an element of SO(n + 1) (the element which takes the standard orthonormal basis to the specific one). Conversely, we have a map SO(n + 1) → Sn n+1 sending any matrix A ∈ SO(n + 1) to its first column, which is a unit vector on R , say x. We deduce that the corresponding fibre consists of the remaining n columns, which form an oriented frame in the n-dimensional subspace orthogonal to x. In other words, Pg = SO(n + 1) where g here we mean the canonical metric. The double covering Peg is precisely the spin group Spin(n + 1) and n n since π1(S ) is trivial for n > 1, we conclude that there is a unique spin structure on S (n ≥ 2).

n Example 3.40. As we said above, CP admits a spin structure if and only if n is odd. On the n other hand, the real projective space RP is spin if and only if n = 3(mod 4), while the quaternionic n projective space HP is always spin. Exercise 3.41. Write the details of a proof for the statements in Example 3.40, see also [LM89, Hus96]. Let (M,J) be an almost complex manifold, i.e. J : TM → TM is a tensor field of type (1, 1) 2 which satisfies J = − Id. Such a structure turns the tangent space TxM (x ∈ M) into a complex vector space, hence it implies that the dimension of M is even. Moreover, it induces an orientation on M (see Appendix, Section 11.2 for a short review of almost complex manifolds). We can view the tangent bundle TM as a complex vector bundle. Then, one can define the Chern classes

2j cj ≡ cj(M) := cj(TM,J) ∈ H (M; Z) 2n of (TM,J). When M is assumed to be compact, then we have the isomorphism H (M; Z) ' Z due to the natural orientation. Moreover, the second Stiefel-Whitney class w2(M) of M is the reduction mod 2 of the ﬁrst Chern class c1(M) (cf. [LM89, p. 82]), i.e.

c1(TM,J) = w2(TM) (mod 2). (3.2) Hence we obtain: Proposition 3.42. ([At71]) Let (M 2n,J) be a compact (almost) complex manifold. Then M admits a spin structure if and only if the ﬁrst Chern class

2 c1(M) = w2(TM) (mod 2) ∈ H (M, Z) 2 is even, i.e. it is divisible by 2 in H (M, Z). Exercise 3.43. A full ﬂag manifold is a compact homogeneous space of the form G/T , where G is a compact simple Lie group and T ⊂ G a maximal torus. Prove that M = G/T is always spin. (Hint: Compute the ﬁrst Chern class of an invariant almost complex structure J on M and show its divisibility by two. Then apply Proposition 3.42). What is the dimension of the related space of spin structures on M? DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 31

4. Spin representations and the spinor bundle. Let us now discuss spin representations and introduce over a Riemannian spin manifold the so- called spinor bundle. The space of sections of this complex vector bundle, is the space where the Dirac operator is defined. 4.1. Spin representations. We begin with the classification of complex Clifford algebras. The 2-periodicity discussed in the first section, induces the following fundamental result related with the classification of complex Clifford algebras. Theorem 4.1. (see [Fr00, Hus96]) (Classification Theorem of Complex Clifford Algebras) (1) For n = 2k there is an algebra isomorphism

k C ' 2 2 2 Φ2k :C`2k −→ M2(C) ⊗ · · · ⊗ M2(C) = End(C ⊗ · · · ⊗ C ) = End(C ) , | {z } | {z } k times k times explicitly given by e 7→ E ⊗ · · · ⊗ E ⊗ g ⊗ T ⊗ · · · ⊗ T ,  2j−1 1  | {z }  (j−1) times (4.1) e2j 7→ E ⊗ · · · ⊗ E ⊗ g2 ⊗ T ⊗ · · · ⊗ T , | {z }  (j−1) times  for j = 1, . . . , k. (2) For n = 2k + 1 there is an algebra isomorphism

k k C ' 2 2 Φ2k+1 :C`2k+ −→ M2(C) ⊗ · · · ⊗ M2(C) ⊕ M2(C) ⊗ · · · ⊗ M2(C) = End(C ) ⊕ End(C ) , | {z } | {z } k times k times  ej 7→ Φ2k(ej), Φ2k(ej) (1 ≤ j ≤ 2k)  . (4.2) e2k+1 7→ iT ⊗ · · · ⊗ T, −iT ⊗ · · · ⊗ T  Here we have used the notation 1 0 0 −i i 0 0 i E = ,T = , g = , g = . 0 1 i 0 1 0 −i 2 i 0 Next we will describe a fundamental representation κ of the spin group Spin(n), which is not C the lift of a representation of SO(n). Since Spin(n) ⊂ C`n ⊂ C`n , it is natural to obtain κ as C a restriction of a suitable representation of C`n . Hence, as a matter of fact we only consider complex finite dimensional representations of Spin(n), i.e. Lie group homomorphisms of the form κ : Spin(n) → Aut(∆), where ∆ is a finite-dimensional C-vector space. Let us discuss first the representations of the Clifford algebra. Consider a quadratic vector space (V, q) over a field K (ch(K) 6= 2) and let C`(V, q) be the associated Clifford algebra. Definition 4.2. A Clifford module, or simply a C`(V, q)-module, is a finite-dimensional vector space E together with a K-algebra homomorphism ρ :C`(V, q) → EndK(E) between C`(V, q) and the algebra of endomorphisms of E over K. Thus, a Clifford module is a representation of a Clifford algebra and we shall often write ρ(φ)(w) ≡ φ · w , for any φ ∈ C`(V, q) and w ∈ E. The product φ·w will often referred to as the Clifford multiplication (see also below). A Clifford module E is called irreducible if there exists no non-trivial proper subspace E0 ⊂ E with φ · E0 ⊂ E0, for any φ ∈ C`(V, q). Moreover, two representations ρj : 32 IOANNIS CHRYSIKOS

C`(V, q) → EndK(Wj)(j = 1, 2), are said to be equivalent if there exists a K-linear isomorphism −1 f : W1 → W2 such that f ◦ ρ1(x) ◦ f = ρ2(x), for any x ∈ C`(V, q). We shall be mainly interested in equivalence classes of representations. Exercise 4.3. Prove that the simplest example of a Clifford module is the Clifford algebra C`(V, q) itself, endowed with the left multiplication. Moreover, show that the exterior algebra V•(V ) is also a C`(V, q)-module. Which is the action in this case? (Hint: See Remark 1.12). Regarding finite dimensional modules, we have the following result:

Proposition 4.4. ([LM89, Prop. 5.4]) Every K-representation ρ of the Cliﬀord algebra C`(V, q) decomposes into a direct sum ρ = ρ1 ⊕ · · · ⊕ ρq of irreducible representations, i.e. it is completely reducible.

Example 4.5. For K = {R, C, H}, the natural representation ρnat of the ring Mn(K) on the vector n space K , is up to equivalence, the only irreducible real representation of Mn(K). On the other hand, the algebra Mn(K) ⊕ Mn(K) has exactly two equivalence classes of real irreducible representations n acting on K , given by ρ1(x, y) = ρnat(x) and ρ2(x, y) = ρnat(y). C Let us focus now on C`n . C Definition 4.6. For n = 2k, the spin representation of the Clifford algebra C`2k is the isomorphism k C ∼ 2 κ2k ≡ Φ2k :C`2k −→ End(C ) , C defined by (4.1), while for n = 2k + 1, the spin representation of C`2k+1 consists of the isomorphism k k pr k C ∼ 2 2 1 2 κ2k+1 = pr1 ◦ Φ2k+1 :C`2k+1 −→ End(C ) ⊕ End(C ) −→ End(C ) , where the first isomorphism is given by (4.2). For n = 2k, 2k + 1, the obtained representation space n 2k 2[ 2 ] ∆n := C = C 2 is called the spinor module. The elements of ∆n are called complex spinors (or Dirac spinors).

C Thus, ∆n turns into a C`n -module. By the classiﬁcation theorem of complex Cliﬀord algebras (see Theorem 4.1) we deduce that

k C Theorem 4.7. (1)For n = 2k, there exists a unique irreducible module of dimension 2 over C`2k, namely C κ2k :C`2k → End(∆2k) . (2) For n = 2k + 1, there exist exactly two inequivalent irreducible modules of dimension 2k over C C`2k+1, which we denote by C C κ2k+1 :C`2k+1 → End(∆2k+1), and κb2k+1 :C`2k+1 → End(∆b 2k+1) , respectively.

Proof. By the Burnside’s Theorem, the matrix algebra MN (C) admits a unique (up to equivalence) N [ n ] irreducible representation which is the canonical representation of C . In our case it is N = 2 2 and the result follows by the Classiﬁcation Theorem 4.1. n C Note that any vector x ∈ R ⊂ C`n ⊂ C`n ' End(∆n) is viewed as an endomorphism of ∆n and n this leads to the Cliﬀord multiplication of vectors x ∈ R and spinors, encoded by the linear map n µ : R ⊗R ∆n → ∆n, µ(x ⊗ ψ) = κn(x)(ψ) = x · ψ .

2 n The notation [ 2 ] means the integer part of n/2. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 33

The Cliﬀord multiplication extends to exterior forms ωk = P ω e ∧· · ·∧e ∈ V•( n) i1<···

× C Because Spin(n) ⊂ C`n ⊂ C`n ⊂ C`n , one can restrict the spin representation of the Clifford C algebra C`n to obtain a representation κ of the spin group. In particular, every irreducible representation of the Clifford algebra restricts to a representation of Spin(n), and as we will see, there is only one such representation (up to isomorphism) of Spin(n) in any dimension, which is called the spin representation of Spin(n). Definition 4.10. The spin representation κ of the spin group Spin(n) is defined by the restriction

κ := κn Spin(n) : Spin(n) → Aut(∆n) .

Remark 4.11. The spin representation κ : Spin(n) → Aut(∆n) is faithful (see [Fr00]). Therefore, the spin representation does not arise as a lift of a SO(n)-representation, since the kernel of such a lift has to contain {±1}.

Proposition 4.12. ([Fr00, LM89]) For n = 2k + 1 odd, the spin module ∆2k+1 is irreducible as a Spin(2k + 1)-representation. Example 4.13. For n = 3, recall that Spin(3) = SU(2). It is well-known that SU(2) has only n one irreducible representation πn on the (n + 1)-dimensional vector space V2 of homogeneous polynomials of degree n, in two complex variables (see [Kn96, B04]). The spinor representation 2 κ : Spin(3) → AutC(∆3) is a 2-dimensional complex irreducible representation (∆3 = C ), thus it must be equivalent to π1. On the other hand, for n = 2k the endomorphism k k f := i κ(ω) = i κ(e1 · ... · e2k) : ∆2k → ∆2k is an automorphism of the spinor representation, i.e. f(κ(g)ψ) = κ(g)f(ψ), for any g ∈ Spin(n) 2 k and ψ ∈ ∆2k. Because ω := e1 · ... · e2k satisﬁes ω = (−1) it follows that f is an involution, i.e. 2 3 f = Id∆2k . Thus the spinor representation ∆2k decomposes into the eigenspaces of f: + − ± ∆2k = ∆2k ⊕ ∆2k, ∆2k = {ψ ∈ ∆2k : f(ψ) = ±ψ} . ± Deﬁnition 4.14. The spin representations ∆2k are called half-spin representations. Spinors be- ± longing to ∆2k are called positive or negative Weyl spinors. Moreover, by the discussion above it follows that (see [Fr00, LM89, Gi09] for details)

3 The n-form ω = e1 · ... · en ∈ C`n is the so-called volume element of C`n and is independent of the choice of the n 0 basis, and so well-defined if we fix an orientation on R . For n = 2k, ω belongs to the center of C`2k and satisfies 2 k 0 the relation ω = (−1) . Hence ω commutes with any element in C`2k (and so in Spin(2k)) and anti-commutes with 1 0 1 C`2k. For n = 2k + 1 the volume element ω commutes with both C`2k+1 and C`2k+1. 34 IOANNIS CHRYSIKOS

Proposition 4.15. For n = 2k even, the spin module ∆2k decomposes as a Spin(2k)-representation + − + − into two irreducible submodules, κ = κ ⊕ κ and ∆2k = ∆2k ⊕ ∆2k, of complex dimension ± k−1 2k ± ± ± ∓ dimC ∆2k = 2 . Moreover, for any x ∈ R , ψ ∈ ∆2k it is x · ψ ∈ ∆2k. 4.2. The spinor bundle. Consider now a connected Riemannian spin manifold (M n, g) and denote by (Peg, Λg) the corresponding spin structure, where Λg : Peg → Pg is the double covering of Pg. Definition 4.16. The spinor bundle over (M n, g) is the complex vector bundle g Σ M := Peg ×κ ∆n = Peg ×Spin(n) ∆n , associated to the Spin(n)-principal bundle πe : Peg → M via the spin representation κ : Spin(n) → Aut(∆n). g g g By the definition of Σ M, the fiber ΣxM of Σ M over any x ∈ M, consists of equivalence classes of pairs [˜p, φ] with p˜ ∈ (Peg)x and φ ∈ ∆n, subject to the condition [˜p, φ] = [˜pa, κ(a−1)φ], ∀a ∈ Spin(n). Sections of ΣgM are called spinor fields; locally such sections are given by

ϕ|U := [e,e σ] , for some smooth (local) section e : U → Peg deﬁned on an open set U ⊂ M, and σ : U → ∆n a g smooth ∆n-valued function. In particular, by Lemma 3.7 we see that sections ϕ ∈ Γ(Σ M) can be identiﬁed with smooth functions (and hence we will use the same notation)

ϕ : Peg → ∆n, −1 obeying ϕ(pae ) = κ(a )ϕ(pe) for any a ∈ Spin(n) and pe ∈ Peg. g g Lemma 4.17. When n = 2k, then Σ M decomposes into two subbundles Σ±M associated to the ± spin representations κ± : Spin(2k) → Aut(∆n ), g g g g ± Σ M = Σ M+ ⊕ Σ−M, Σ± = Peg ×κ± ∆2k. Remark 4.18. It is worth clarifying that since the spinor representation does not descend to a + representation of SO(n) (or of GLn R), the spinor bundle cannot be defined without specifying a metric, and this is the reason that in differential geometry we usually define spin structures for (pseudo) Riemannian vector bundles. To summarize, the definition of spinor fields depends on g, in contrast to tensor fields and differential forms. Fix a Riemannian spin manifold (M n, g) and let ΣgM the related spinor bundle. We can extend the notion of Clifford multiplication discussed before to the tangent bundle of M n. Definition 4.19. The Clifford multiplication between vector fields and spinor fields is defined by a complex linear vector bundle homomorphism µ : TM × ΣgM → ΣgM,X ⊗ ψ 7→ X · ψ , such that X · Y · ψ + Y · X · ψ = −2g(X,Y )ψ , (4.3) for any X,Y ∈ Γ(TM) and ψ ∈ Γ(ΣgM). The Clifford multiplication extends to a multiplication between exterior forms and spinor fields, as follows: k ^ g g X µ : M ⊗ Σ M → Σ M, ω ⊗ ϕ 7→ ω · ϕ := ωI eI · ϕ, I where eI := ei1 · ... · eik and ωI = ωi1,...,ik := ω(ei1 , . . . , eik ), for some ordered set of indices I = {1 ≤ i1 < . . . < ik ≤ n} ⊆ {1, . . . , n}. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 35

Lemma 4.20. For any X ∈ Γ(TM), ω ∈ Ωp(M) and ϕ ∈ Γ(ΣgM) we have that [ X · ω · ϕ = (X ∧ ω) · ϕ − (Xyω) · ϕ p [ , (4.4) ω · X · ϕ = (−1) (X ∧ ω + Xyω) · ϕ where as usual, given a vector ﬁeld X we denote by X[ the dual 1-form, i.e. X[ = g(X, −). Exercise 4.21. Prove Lemma 4.20 Because the spin representation κ is a representation of a compact group in a complex vector space, there exists some Spin(n)-invariant Hermitian scalar product in ∆n (see for example [Kn96] for general theory of representations of compact Lie groups). In particular

Theorem 4.22. The space ∆n of n-spinors admits a Hermitian scalar product ( , ) such that (x · ψ, φ) + (ψ, x · φ) = 0 , n for any x ∈ R , and φ, ψ ∈ ∆n. Thus, the spinor representation κ : Spin(n) → AutC(∆n) is a unitary representation with respect to ( , ). Proof. Because Spin(n) is compact, any ﬁnite dimensional representation is unitary with respect to a suitable inner product. If the representation is irreducible, this product is determined uniquely up to a scalar factor. Consider the Lie algebra

spin(n) = spanR{ei · ej : 1 ≤ i < j ≤ n} ⊂ C`n n of the spin group and set g = R ⊕ spin(n). The commutator [w1, w2] := w1 · w2 − w2 · w1 makes g a Lie algebra and it suﬃces to show that g is compact. Let f : g → C`n+1 the map given by

f|spin(n) = Id, f(ei) = ei · en+1 (1 ≤ i ≤ n) . n This is the restriction of F : R → C`n+1, F (ei) = ei · en+1 and thus f maps bijectively g onto the Lie algebra of Spin(n + 1). In particular, it is an isomorphism of Lie algebras and g is compact. By the general theory of compact Lie groups (see [Kn96]), we conclude that any complex representation κ : g → End(∆) of g = spin(n + 1) carries a positive definite Hermitian scalar product ( , ) such that (κ(x)w1, w2) + (w1, κ(x)w2) = 0 , for any x ∈ g, w1, w2 ∈ ∆. Obviously, the spinor bundle admits a Spin(n)-invariant Hermitian metric ( , ), unique up to scale, defined fiberwise as the natural Hermitian inner product ( , ) on ∆n. Lemma 4.23. The Hermitian metric ( , ) on the spinor bundle ΣgM over a Riemannian spin manifold (M n, g), satisfies the following relations (X · ϕ, ψ) = −(ϕ, X · ψ) , (4.5) (ω · ϕ, ψ) = (−1)p(p+1)/2(ϕ, ω · ψ) , (4.6) for any vector field X ∈ Γ(TM), p-form ω ∈ Ωp(M) and spinor field ϕ, ψ ∈ Γ(ΣgM). In addition, the induced real scalar product ( , )R := Re( , ) is such that 2 (X · ϕ, Y · ϕ)R = g(X,Y )|ϕ| . (4.7) Remark 4.24. Note that one can extend the Hermitan metric ( , ) of ΣgM to sections of T ∗M ⊗ ΣgM, as follows: ∗ g ∗ g ∞ ( , ) : Γ(T M ⊗ Σ M) × Γ(T M ⊗ Σ M) → C (M; C) , α ⊗ ϕ, β ⊗ ψ 7→ (α ⊗ ϕ, β ⊗ ψ) := g(α, β)(ϕ, ψ) . Here, g acts to covectors by means of the identification T ∗M ' TM. Then, for some (smooth) ∗ g Pn sections Θ, Φ ∈ Γ(T M ⊗ Σ M) we get the formula (Θ, Φ) = i=1(Θ(ei), Φ(ei)). 36 IOANNIS CHRYSIKOS

5. The spinorial connection and spinor geometry In this section our aim is to present the basic setting of spin geometry. In particular, we will discuss the spinorial connection and the spinorial curvature. We begin by recalling the notion of a connection on a principal bundle and on a vector bundle. 5.1. Connections on principal bundles. Consider a principal G-bundle π : P → M over a n-dimensional smooth manifold M. For any left-invariant vector field X ∈ g we denote by d X˜ := {q exp(tX) | dt t=0 the so-called fundamental vector field induced by the right G-action on P . Then, the map X 7→ X˜(p) ≡ X˜p induces an isomorphism g 'Vp , where Vp is the tangent space to the orbit through p ∈ P . As we know, the orbits are the fibres of π, and Vp can be viewed as the vertical space through p. This coincides with the kernel of the differential of π at p ∈ P ,

Vp = {X ∈ Tp(P ):(π∗)p(X) = 0} = ker(π∗)p . Thus, there is a short exact sequence

(π∗)p 0 → Vp → Tp(P ) −→ Tπ(p)M → 0 F and dim Vp = dim Tp(P ) − dim Tπ(p)M = dim G. The disjoint union V := p∈P Vp is called thee vertical subbundle of TP . Deﬁnition 5.1. A (principal) connection on a G-prrincipal bundle π : P → M over M n, is a G-invariant distribution of tangent n-dimensional planes H = {Hp : p ∈ P, dim Hp = n} ⊂ TP such that the linear map

(π∗)|Hp : Hp → Tπ(p)M is a linear isomorphism for all p ∈ P . H is called the horizontal distribution, or horizontal subbundle. Thus, a connection on (P, π, M, G) is a smoothly varying choice of horizontal subspaces

H : P 3 p 7→ Hp ⊂ Tp(P ) which are complementary to the vertical subspaces Vp,

TpP = Vp ⊕ Hp , and invariant under the right action of G on P , i.e. (Rg)∗Hp = Hpg for any p ∈ P and g ∈ G. Then, at each p ∈ P , Hp deﬁnes a linear projection

prp : TpP → Vp .

Hence, and due to the isomorphism Vp ' g which allows to trivialize the vertical bundle VP by the fundamental vector ﬁelds X˜, we may use the above vertical projection to deﬁne a so-called connection 1-form, which is a 1-form on P with values in g, Z : TP → g, or in other words at p ∈ P

prp ' Zp : TPp = Vp ⊕ Hp −→ Vp −→ g . ∗ −1 Thus, Z satisfies Z(X˜) = X for all X ∈ g and moreover (Rg) Z = Ad(g )Z for any g ∈ G, where here Ad is the adjoint representation of G (hence we say that Z is of type Ad, see also below). Conversely, given a g-valued 1-form ω ∈ Ω1(P, g) on P which satisfies the previous two conditions, we can define a connection on P by setting

Hp = {X ∈ TpP : ω(X) = 0} = ker ωp . DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 37

Definition 5.2. Let (P, π, M) be a principal G-bundle and ρ : G → GL(V ) a G-representation. A k-form on P with values in V , i.e. an element ω ∈ Vk(P,V ), is called tensorial of type ρ, whenever the following two conditions are satisfied: 1) ωp(X1,...Xk) = 0 if some vector Xi ∈ TpP is vertical, i.e. Xi ∈ Vp, ∗ −1 2) Rgω = ρ(g )ω, for any g ∈ G. Example 5.3. Let π : P → M be a G-principal bundle and ρ : G → Aut(V ) be a representation of a Lie group G. Consider the associated bundle E = P ×ρ V . Then, the tensorial k-forms of type ρ on P are in natural correspondence with E-valued k-forms on M. We shall denote the space of E-valued k-forms on M by Vk(M; E) and write V(M; E) := Γ(M, V T ∗M ⊗ E) for the space of differential forms on M with values in E. Definition 5.4. Let w ∈ Vk(P,V ) be a k-form on π : P → M with values in V . The absolute differential of w is the linear mapping

k k+1 ^ ^ D : (P ; V ) → (P ; V ), (Dω)p(X1,...,Xk+1) := dω(hX1, . . . , hXk+1) , where h : TpP → Hp is the projection to the horizontal subspace. Proposition 5.5. ([KN69, p. 76, I], [Fr00, p. 166]) 1) If ω is a k-form of type ρ, then Dω is a tensorial (k + 1)-form of type ρ. Vk 2) Let ω ∈ (P ; V ) a tensorial form of type ρ. Then Dω = dω +ρ∗(Z)∧ω, where ρ∗ : g → End(V ) and k X j (ρ∗(Z) ∧ ω)(X0,...,Xk) := (−1) ρ∗(Z(Xj))ω(X0,..., Xˆj,...,Xk) . j=0 Let us recall now the notion of the covariant derivative or connection on vector bundle π : E → M over a smooth manifold M. Definition 5.6. Consider a vector bundle (E, π, M) and let Γ(E) be the space of sections. A covariant derivative (or connection) on E is a linear map ∇ : Γ(E) → Γ(T ∗M ⊗ E) satisfying the Liebniz rule ∇(fσ) = df ⊗ σ + f∇σ for any f ∈ C∞(M) and σ ∈ Γ(E). In other words, a connection is a map ∇ : Γ(TM)×Γ(E) → Γ(E) ∞ such that ∇X σ is linear over C (M; R) in X, ∇X σ is linear over R in σ and ∇ satisfies the rule ∞ ∇X (fσ) = f∇X σ + X(f)σ, for any X ∈ Γ(TM) and f ∈ C (M; R), where we always identify the space of smooth sections Γ(TM) of TM with the Lie algebra of smooth vector fields X (M) of M. ∞ Here, X(f) denotes the directional derivative of a smooth function f ∈ C (M; R).

Fix a vector field X ∈ Γ(TM) ≡ X (M) and evaluate ∇X σ at a point x ∈ M. Then, it is easy to prove that (∇X σ)x only depends on the vector Xx ∈ TxM and the value of the section σ in an open neighbourhood U ⊂ M of x. Note also that the 1-form ∇σ is nothing than (∇σ)(X) = ∇X σ, with X ∈ Γ(TM). Definition 5.7. Consider a vector bundle (E, π, M) and let ∇ be a connection on E. Given a section σ ∈ Γ(E) and some vector fields X,Y on M, we define the curvature of ∇ via the rule ∇ R (X,Y )σ = ∇X ∇Y σ − ∇Y ∇X σ − ∇[X,Y ]σ .

∇ ∇ ∞ R is a R-multilinear map such that R (X,Y )σ is C (M; R)-linear in all three arguments. Thus, R is deﬁned pointwise and induces a (1, 3)-tensor ﬁeld on M. Note that R∇(X,Y ) = −R∇(Y,X). 38 IOANNIS CHRYSIKOS

Example 5.8. Assume that π : E → M is the tangent bundle TM of a smooth manifold M. Then, a connection on E = TM is a so-called linear or affine connection on M, i.e. a R-linear map ∇ : Γ(TM)×Γ(TM) → Γ(TM), with ∇X Y ∈ Γ(TM) for any two vector fields X,Y ∈ Γ(TM) = X (M), ∞ which is C (M; R)-linear in the variable X and satisfies the Leibniz rule ∇X (fY ) = X(f)Y + f∇X Y . Consider the induced (linear) map Γ(TM) 3 X 7→ ∇X ∈ End(Γ(TM)), where we view both Γ(TM) = X (M) and End(Γ(TM)) as Lie algebras. Then, the curvature R∇ of ∇ measures the deviation of the map X 7→ ∇X from being a Lie algebra homomorphism. In addition to the curvature tensor R∇ an affine connection on M has another related tensor, the so-called torsion tensor, defined by ∇ T (X,Y ) = ∇X Y − ∇Y X − [X,Y ] . ∇ ∞ ∇ In particular, T if C (M; R)-linear in both variables X,Y and thus T is a vector-valued bilinear form on M which induces a (1, 2)-tensor field on M. It is easy to see that T ∇(X,Y ) = −T ∇(Y,X). Definition 5.9. Consider a Riemannian bundle π : E → M with Riemannian bundle metric g.A connection ∇ on E is said to be metric (or compatible with g) if ∇g = 0, which is equivalent to say that ∇X g(σ1, σ2) ≡ Xg(σ1, σ2) = g(∇X σ1, σ2) + g(σ1, ∇X σ2) , for any vector field X of M and any two sections σ1, σ2 ∈ Γ(E). Proposition 5.10. ([KN69, LM89, Hus96, Fr00]) Let (P, π, M) a G-principal bundle with a connection 1-form Z. Consider a finite-dimensional representation ρ : G → GL(V ) of G on a vector space V and let E = P ×ρ V be the associated vector bundle. Then, the absolute differential 1 ^ D : Γ(E) → (M,E) = Γ(T ∗M ⊗ E) is a covariant derivative, which we shall call the covariant derivative in E associated with Z. Definition 5.11. Let (P, π, M) be a G-principal bundle. The curvature of a principal connection on P defined by a connection 1-form Z : TP → g, is defined by Ω := DZ. Lemma 5.12. The curvature Ω is a 2-form on P with values in g, which is tensorial of type Ad 1 and satisfies Ω = dZ + 2 [Z,Z]. Thus, we may define the curvature 2-form of a principal connection on P directly by the rule 1 Ω(X,Y ) = dZ(X,Y ) + [Z(X),Z(Y )] , ∀ X,Y ∈ Γ(TP ) . 2 Exercise 5.13. Prove the previous lemma. Remark 5.14. Let π : P → M be a G-principal bundle over a smooth manifold M n. Consider a connection Z : TP → g and let ρ : G → GL(V ) be a G-representation. The curvature form Z 1 Z Ω = dZ + 2 [Z,Z] defines a 2-form ρ∗(Ω ) with values in the space of endomorphisms of the associated vector bundle E = P ×ρ V . Recall that any section σ ∈ Γ(E) is identified with a G- equivariant map σ : P → V , i.e. σ(pg) = ρ(g−1)σ(p). Consider the absolute differential of σ, DZ σ. This is a tensorial 1-form of type ρ and we can regard it as a 1-form on M with values in E. The induced covariant derivative on E is given by Z Z h ∇X σ := D σ(X ) , where Xh is the horizontal lift of X. Then, one can prove (see [Fr00, p. 62] or [LM89, Prop. 4.7]) the following relation between the curvature RZ induced by ∇Z and the curvature 2-form ΩZ , Z Z R (X,Y )σ = ρ∗(Ω (X,Y ))σ . This identity can be applied for the computation of the curvature tensor on the spinor bundle over a Riemannian spin manifold. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 39

5.2. The spinorial connection. Consider a connected oriented Riemannian manifold (M n, g) and let us denote by ∇g the Levi-Civita connection associated to g, ∇g : Γ(TM) → Γ(T ∗M ⊗ TM) .

Viewed as a connection on the SO(n)-principal bundle Pg = SO(M, g) of positively oriented orthonormal frames on M, it corresponds to a connection 1-form Z on Pg, i.e. a so(n)-valued 1-form

Z : TPg → so(n) .

Consider a local section e : U ⊂ M → Pg of Pg and write e = (e1, . . . , en) in terms of some orthonormal frame ﬁeld {ei} deﬁned on the open set U ⊂ M. The local connection 1-form e ∗ e X ω ≡ Z := e (Z): TU → so(n), ω ≡ Z = ωijEi,j i

(ei ∧ ej)(X) = g(X, ei)ej − g(X, ej)ei , for any vector field X of M. On the other hand, the connection forms ωij are defined by g ωij := g(∇ ei, ej) , such that ωij = −ωji. On a connected Riemannian spin manifold (M n, g) any connection defined on the SO(n)-principal bundle Pg, lifts to a connection on the Spin(n)-principal bundle Peg = SOf (M, g) via the double g covering Λg. In particular, the Levi-Civita connection ∇ lifts to a spinorial Levi-Civita connection on Peg, and this in turn induces a covariant derivative on the space of sections of the spinor bundle, ∇g : Γ(ΣgM) → Γ(T ∗M ⊗ ΣgM) . Here, and in the following, we maintain the same notation for ∇g and its lift onto ΣgM (it will be also clear from the text which covariant derivative is used). Definition 5.15. The covariant derivative ∇g on ΣgM is usually called the spinorial Levi-Civita connection on (M n, g) (for the purposes of these notes it is sufficient to refer to ∇g just by the term spinorial connection). So, the covariant derivative ∇g on TM descends to a covariant derivative on ΣgM. To illustrate its local picture, consider a simply connected open subset U ⊂ M. Then any local section e ∈ ΓU (Pg) lifts to a section e ∈ ΓU (Peg), i.e. the following diagram commutes:

Peg ; e e Λg e U ⊂ M / Pg .

1 1 On the other hand, the connection 1-form Z ∈ Ω (Pg, so(n)) lifts to a 1-form Ze ∈ Ω (Peg, spin(n)) over the Spin(n)-principal bundle Peg, which means that the following diagram is commutative:

Ze T Peg / spin(n) 9 e∗ dΛg ad=d Ad

e∗ Z TU ⊂ TM / TPg / so(n) .

Then, the lift Ze induces the spinorial covariant derivative ∇g on ΣgM. Notice that in terms of a g local representation [e,e σ] of ϕ ∈ Γ(Σ M), where U ⊂ M is an open subset, σ : U ⊂ M → ∆n 40 IOANNIS CHRYSIKOS is a smooth function and e ∈ ΓU (Peg) is a local section which covers a local orthonormal frame 4 e : U → Pg, we have that h 1 X i ∇g ϕ = e,˜ dσ(X) + g(∇g e , e )e · e · σ . X 2 X i j i j i

Lemma 5.16. (see for example [Hij01]) Fix an orthonormal basis {σ1, . . . , σN } of the spinor module N [ n ] g ∆n ' C (N = 2 2 ), such that {ϕq}1≤q≤N with ϕq := [e,e σq] , are local sections of Σ M (this is called a local spinorial frame). Then, the spinorial covariant derivative is locally expressed by n 1 X ∇gϕ = g(∇ge , e )e · e · ϕ , (5.1) q 4 i j i j q i,j=1 where {ej}1≤j≤n denotes a local positively oriented orthonormal basis of TM. Proposition 5.17. For any X,Y ∈ Γ(TM), ω ∈ ΩpM and ϕ, ψ ∈ Γ(ΣgM), the spinorial Levi- Civita connection (or any metric connection on ΣgM) satisﬁes the following relations: g g (α) X(ϕ, ψ) = (∇X ϕ, ψ) + (ϕ, ∇X ψ),

g g g (β) ∇X (Y · ϕ) = (∇X Y ) · ϕ + Y · ∇X ϕ,

g g s (γ) ∇X (ω · ϕ) = (∇X ω) · ϕ + ω · ∇X ϕ. Proof. We shall prove only (α), and (β), (γ) are left to the reader. We follow the method of [Hij01] and apply Lemma 5.16. Consider a local section e = (e1, . . . , en) ∈ ΓU (Pg) and let e ∈ ΓU (Peg) the corresponding local section of the Spin(n)-principal bundle Peg. Let also {ϕq}1≤q≤N be the corresponding local spinorial frame. To prove (α), it is sufficient to set ϕ = ϕq and ψ = ϕr for some [ n ] 1 ≤ q, r ≤ N = 2 2 . Indeed, for any X ∈ Γ(TM) the following holds: (5.1) 1 X (∇g ϕ , ϕ ) = g(∇g e , e )(e · e · ϕ , ϕ ) X q r 4 X i j i j q r i,j (4.5) 1 X = g(∇g e , e )(ϕ , e · e · ϕ ) 4 X i j q j i r i,j 1 X = g(∇g e , e )(ϕ , e · e · ϕ ) 4 X j i q i j r i,j 1 X = − g(∇g e , e )(ϕ , e · e · ϕ ), 4 X i j q i j r i,j which easily yields (α). For arbitrary sections, one proves g g X(fϕ, ψ) = h∇X (fϕ), ψi + hfϕ, ∇X ϕi, for some smooth function f on M, and concludes by bilinearity. 5.3. Spinorial Curvature. Given a Riemannian manifold (M n, g), recall that the Riemannian curvature tensor Rg associated to the Levi-Civita connection on TM is the tensor field of type (1, 3), defined by g g g g g g R (X,Y )Z := ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z,

4 This means Λg(e) = e = (e1, . . . , en) ∈ ΓU (Pg). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 41 for any X,Y,Z ∈ Γ(TM). We can also set Rg(X,Y,Z,W ) := g(Rg(X,Y )Z,W ) and view Rg as a tensor field of type (0, 4). The curvature tensor Rg can be also viewed as a bundle morphism 2 2 ^ ^ Rg : (T ∗M) → (T ∗M) on the vector bundle of 2-forms on M, defined by g(Rg(X ∧ Y ),Z ∧ W ) = g(Rg(X,Y )Z,W ) , for any X,Y,Z,W ∈ Γ(TM). Hence, in terms of some local orthonormal basis {ei}1≤i≤n of TM, one has g X g R (ei ∧ ej) := Rijkèk ∧ e` (5.2) k<` g g g n where Rijk` := g(R (ei, ej)ek, e`). The Ricci tensor Ric of (M , g) is defined to be the contraction of Rg, i.e. Ricg(X,Y ) = tr(Z 7→ Rg(Z,X)Y ) . g g Pn g Hence, Ric is a (symmetric) tensor of type (0, 2), in particular Ric (X,Y ) = i=1 g(R (ei,X)Y, ei). The associated endomorphism Ricg : TM → TM is a tensor field of type (1, 1), defined by g(Ricg(X),Y ) = Ricg(X,Y ), for any X,Y ∈ Γ(TM). This is the so-called Ricci endomorphism and locally we have the expression n g X g Ric (X) := Ric (X, ei)ei . i=1 g Finally we shall denote by Scal : M → R the Riemannian scalar curvature, i.e the trace of the Ricci tensor n n g g X g X g Scal = tr Ric = Ric (ei, ei) = g(Ric (ei), ei) . i=1 i=1 Assume now that (M n, g) is a spin manifold and let Rg be the curvature tensor associated to the spinorial covariant derivative ∇g on the corresponding spinor bundle ΣgM. This is the so-called spinorial curvature tensor, given by g g g g g g R (X,Y )ϕ := ∇X ∇Y ϕ − ∇Y ∇X ϕ − ∇[X,Y ]ϕ for any X,Y ∈ Γ(TM) and ϕ ∈ Γ(ΣgM). Based on the previous discussion (see for example Remark 5.14), one can show that Proposition 5.18. ([LM89, BFGK89, Fr00, BHMMS]) On a Riemannian spin manifold the curvature operators Rg and Rg are related by 1 Rg(X,Y )ϕ = Rg(X ∧ Y ) · ϕ , (5.3) 2 g for any X,Y ∈ Γ(TM) and ϕ ∈ Γ(Σ M). Moreover, in terms of some local orthonormal frame {ei} of TM, the following relation makes sense: n n 1 X 1 X Rg(X,Y )ϕ = g(Rg(X,Y )e , e )e · e · ϕ = g(Rg(X,Y )e , e )e · e · ϕ . (5.4) 2 i j i j 4 i j i j i

Proof. Recall that the Riemannian curvature tensor Rg satisﬁes the properties (see for example [KN69]) Rg(X,Y,Z,W ) = −Rg(Y,X,Z,W ) = −Rg(X, Y, W, Z) .

Moreover, the ﬁrst Bianchi identity states that (below SX,Y,Z denotes the cyclic sum over X,Y,Z) g SX,Y,Z R (X,Y )Z = 0 for any X,Y,Z ∈ Γ(TM), which is equivalent to write g g g Rijk` + Rjki` + Rkij` = 0 , g g where as before we set Rijk` := g(R (ei, ej)ek, e`) for some local orthonormal basis {ei} of TM. This implies that g g g g(R (X, ei)ej, ek) = −g(R (ei, ej)X, ek) − g(R (ej,X)ei, ek) . (5.6) Thus, a combination of (4.4) and (5.4) yields the relation n X 1 X e ·Rg(X, e )ϕ = g(Rg(X, e )e , e )e · e · e · ϕ i i 4 i j k i j k i=1 i,j,k (5.6) 1 X 1 X = − g(Rg(e , e )X, e )e · e · e · ϕ − g(Rg(e ,X)e , e )e · e · e · ϕ 4 i j k i j k 4 j i k i j k i,j,k i,j,k 1 X 1 X = g(Rg(e , e )X, e )e · e · e · ϕ + g(Rg(e ,X)e , e )(−e · e · e ) · ϕ 4 i j k j i k 4 j i k j k i i,j,k i,j,k (5.6) 1 X = − g(Rg(X, e )e , e )(e · e · e − e · e · e ) · ϕ . 4 i j k j i k j k i i,j,k However, by (4.3) it is easy to see that

ej · ei · ek − ej · ek · ei = −ei · ej · ek − 2δijek + ej · ei · ek + 2δikej

= −ei · ej · ek − 2δijek − ei · ej · ek − 2δijek + 2δikej

= −2ei · ej · ek − 4δijek + 2δikej . Therefore, after recalling that

g X g g X g Ric (X,Y ) = R (ei, X, Y, ei), Ric (X) = R (X, ei)ei i i a combination of the previous conclusions, gives the result: n X X 1 X 3 e ·Rg(X, e )ϕ = − Rg(X, e , e , e )e · ϕ + Rg(X, e , e , e )e · ϕ i i i i k k 2 i j i j i=1 i=j,k i=k,j X 1 X = − Rg(X, e , e , e )e · ϕ − Rg(X, e , e , e )e · ϕ i i k k 2 i i j j i=j,k i=k,j X 1 X = − g(Ricg(X), e )e · ϕ − g(Ricg(X), e )e · ϕ k k 2 j j k j 3 = − Ricg(X) · ϕ. 2 DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 43

6. The Dirac operator on a Riemannian spin manifold In this section our main goal is to introduce the Dirac operator (or Atiyah-Singer operator) on a Riemannian spin manifold and examine some of its most basic properties. Next we shall introduce the twistor operator (or Penrose operator), which roughly speaking is the complementary operator on the spinor bundle, relatively with the Dirac operator. 6.1. Definition and basic properties of the Dirac operator. Consider a Riemannian spin manifold (M n, g). Definition 6.1. The Dirac operator or Atiyah-Singer operator D/ g is the first-order differential operator defined by the composition of the Clifford multiplication with the covariant derivative ∇g, g ∇g µ D/ := µ ◦ ∇g : Γ(ΣgM) → Γ(T ∗M ⊗ ΣgM) =∼ Γ(TM ⊗ ΣgM) → Γ(ΣgM) .

In terms of a local orthonormal frame {e1, . . . , en} of TM, the Dirac operator is given by n / g X g D ϕ = ei · ∇ei ϕ . i=1 Exercise 6.2. Prove that the local expression of D/ g is independent of the used local orthonormal basis of TM. The original motivation for constructing a differential operator of this type, was the need of a first-order differential operator whose square is the Laplacian. The first such example was described by P. Dirar and thats why the operator bears its name nowadays. But let us present some details of the original work of Dirac [Dir28]. 3 3 Example 6.3. Consider the 3-dimensional flat space R . A free particle on R is described by a 3 state function ψ(t, x) with t ∈ R and x ∈ R . The properties of the state function are depending on the square root of the wave operator , which is defined by ∂2 ∂2 ∂2 ∂2 := 2 − 2 − 2 − 2 . ∂x0 ∂x1 ∂x2 ∂x3 Dirac [Dir28] found the square root D/ of , in terms of the Pauli matrices 1 0 1 0 0 −i 0 1 σ := , σ := , σ := , σ := . 0 0 1 1 0 −1 2 i 0 3 1 0 Indeed, a short computation shows that 2 σi = −I, σjσk = −σkσj = −iσl , for any j < k with 1 ≤ j, k, l ≤ 3. Define now the 4 × 4 matrices 0 σ0 0 σj γ0 = , γj = , j = 1, 2, 3 . σ0 0 −σj 0 They satisfy the following properties: 2 2 2 2 γ0 = I, γ1 = γ2 = γ3 = −I , γjγk = −γkγj, (j 6= k) . (note that γj :C`1,3 → M4(C) are representations of the Clifford algebra C`1,3, relative with the 1,3 Minkowski space R ). Set now 4 X ∂ ∂ ∂ ∂ ∂ D/ := γ = γ + γ + γ + γ . i ∂x 0 ∂x 1 ∂x 2 ∂x 3 ∂x i=0 i 0 1 2 3 2 Then, from the properties of γj it follows that D/ = I. 44 IOANNIS CHRYSIKOS

n Exercise 6.4. Consider the Euclidean space M = R with the Euclidean metric g = h, i and its g n n n [ n ] canonical spin structure. The spinor bundle Σ := Σ R is given by Σ = R × ∆n = R × C 2 . n [ n ] Hence, any spinor field ϕ ∈ Γ(Σ) is a map ϕ : R → C 2 . What is the Dirac operator in this case? Prove that D/ g is a constant coefficient first-order operator whose square satisfies the relation 2 g X ∂ (D/ )2 = ∆g = − , ∂x2 i i where ∆ ≡ ∆g is the ordinary Laplacian on functions (see [Hij01]). For the even-dimensional case and based on the decomposition of the spinor bundle we obtain the following simple result. Lemma 6.5. Let (M 2k, g) be an even-dimensional Riemannian spin manifold. Then the Dirac operator D/ g splits g g g D/ = D/ + ⊕ D/ − g g g g with D/ ± : Γ(Σ±M) → Γ(Σ∓M). Moreover, the non-zero eigenvalues of D/ are symmetric with respect to the origin.

C Proof. Consider the complex Cliﬀord algebra C`n and its complex volume element, n+1 C [ ] C ω = i 2 e1 · ... · en ∈ C`n . It is straightforward to check that (ωC)2 = 1 x · ωC = (−1)(n−1)ωC · x ,

n C C where x ∈ R . It follows that ω belongs to the centre of C`n if n is odd, while if n is even and E is a complex Cliﬀord module, then we can deﬁne a Z2-grading on E by C E = E+ ⊕ E−,E± = {u ∈ E : ω (u) = ±u} .

For the spinor module ∆n and for n = 2k, it follows that 1 ∆± = (1 ± ωC) · ∆ . n 2 n C ± C Hence, the complex volume form ω acts on ∆n as ± Id. Since n = 2k is even, ω anti-commutes with any X ∈ Γ(TM), X · ωC = −ωC · X. C g + g Moreover, ω is ∇ -parallel. Consider now some ϕ ∈ Γ(Σ+M). Thus we see that C / g + C X g + X C g + ω · D (ϕ ) = ω · ei · ∇ei ϕ = − ei · ω · ∇ei ϕ i i X g C + X g + = − ei · ∇ei (ω · ϕ ) = − ei · ∇ei ϕ i i = −D/ g(ϕ+) . g g C By recalling that Σ± = {ϕ ∈ Σ M : ω · ϕ = ±ϕ}, one easily ﬁnishes with the ﬁrst claim. Consider now some eigenspinor ψ ∈ Γ(ΣgM) of D/ g, i.e. g D/ (ψ) = f · ψ , for some f ∈ R . g g g + − Decompose ψ with respect to the orthogonal splitting Σ M = Σ+M ⊕ Σ−M, i.e. ψ = ψ ⊕ ψ ± g with ψ ∈ Γ(Σ±M). Then D/ g(ψ+) + D/ g(ψ−) = fψ− + fψ+ DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 45 which yields D/ gψ± = fψ∓. Hence it is easy to see that the spinor ϕ := ψ+ − ψ− is an eigenspinor g of D/ with the eigenvalue −f.

n Consider a Riemannian manifold (M , g) and let {ei} be some local orthonormal frame of TM. Recall that the differential and co-differential of a p-form ω on M, i.e. the differential operators

d ≡ dg :Ωp(M) → Ωp+1(M) , δ ≡ δg :Ωp(M) → Ωp−1(M) can be deﬁned in terms of the Levi-Civita connection ∇g (see [LM89]). In particular, the following expressions make sense

n n g X g g X g d ω := ei ∧ ∇ei ω , δ ω := − eiy∇ei ω . i=1 i=1

In these terms one can prove that

Proposition 6.6. Let (M n, g) be a Riemannian spin manifold and D/ g the Riemannian Dirac op- ∞ p g erator. Then, for any f ∈ C (M; R), X ∈ Γ(TM), ω ∈ Ω (M) and ϕ ∈ Γ(Σ M), the following hold:

(1) D/ g(fϕ) = grad(f) · ϕ + fD/ g(ϕ) .

/ g P g / g g (2) D (X · ϕ) = j ej · (∇ej X) · ϕ − X · D (ϕ) − 2∇X ϕ .

/ g p / g g g P g (3) D (w · ϕ) = (−1) ω · D (ϕ) + (d ω + δ ω) · ϕ − 2 j(ejyω) · ∇ej ϕ .

g Proof. (1) By deﬁnition of D/ and in terms of a local orthonormal basis of {e1, . . . , en} of TM, we have that

/ g X g X g D (fϕ) = ej · ∇ej (fϕ) = ej · (ej(f)ϕ + f∇ej ϕ) j j X X g = ej · ej(f)ϕ + f ej · ∇ej ϕ j j = grad(f) · ϕ + fD/ g(ϕ) .

(2) Based on (4.4) and the Leibniz rule (see Proposition 5.17) we obtain that

h i / g X g X g g D (X · ϕ) = ej · ∇ej (X · ϕ) = ej · (∇ei X) · ϕ + X · ∇ej ϕ j j X g X g = ej · (∇ej X) · ϕ + ej · X · ∇ej ϕ j j X g X g X g = ej · (∇ej X) · ϕ − X · ej · ∇ej ϕ − 2 g(ej,X)∇ej ϕ j j j X g / g g = ej · (∇ej X) · ϕ − X · D (ϕ) − 2∇X ϕ. j 46 IOANNIS CHRYSIKOS

(3) Using again the Leibzin rule we get

/ g X g D (ω · ϕ) = ej · ∇ej (ω · ϕ) j X g g = ej · (∇ej ω) · ϕ + ω · ∇ej ϕ j X g X g = ej · (∇ej ω) · ϕ + ej · ω · ∇ej ϕ j j

(4.4) X g X g X g = (ej ∧ ∇ej ω) · ϕ − (ejy∇ej ω) · ϕ + ej · ω · ∇ej ϕ j j j g g X g = (d ω + δ ω) · ϕ + ej · ω · ∇ej ϕ . j

Now, since any vector ﬁeld X and diﬀerential p-form ω satisfy

p X · ω = X ∧ ω − Xyω, ω · X = (−1) (X ∧ ω + Xyω) , where, as usual, we identify X ' X[ with its dual 1-form X[ via g, we get that

p ej · ω − (−1) ω · ej = −2(ejyω) .

Thus the last sum gives rises to

X g p X g X g ej · ω · ∇ej ϕ = (−1) ω · ej · ∇ej ϕ − 2 (ejyω) · ∇ej ϕ j j j p / g X g = (−1) ω · D (ϕ) − 2 (ejyω) · ∇ej ϕ . j

Let us now examine a few analytical properties of the Dirac operator D/ g on a Riemannian spin manifold (M n, g). We begin with its symbol with aim to show tha D/ g is an elliptic diﬀerential operator.

Proposition 6.7. The principal symbol ς(D/ g)(ξ):ΣgM → ΣgM of D/ g is given by

ς(D/ g)(ξ)(ϕ) = ξ] · ϕ , where ξ ∈ T ∗M.

Proof. The principal symbol of a linear differential operator is defined in Section 11.1 of our Appen- dix (so we refer to Section 11.1 for a more detailed description). In particular, here we will apply Proposition 11.6 for the Riemannian Dirac operator D/ g. So, replace E with the spinor bundle ΣgM ∞ and smooth sections by spinors fields. Consider some x ∈ U ⊂ M and f ∈ C (U; R), such that ∗ ] ] ] dfx = ξx ∈ Tx M. Recall that grad(f)(x) = (df) (x) = ξx, where ξ is the vector field defined by DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 47 g(ξ],X) = ξ(X), for any X ∈ Γ(TM). Hence, by using Proposition 6.6, (1), we get that

g 1 −tf(x) g tf ς(D/ )x(ξx)ϕ(x) = lim e D/ (e ϕ)(x) t→∞ t 1 h i = lim e−tf(x) grad(etf ) · ϕ + etf D/ g(ϕ) (x) t→∞ t 1 h i = lim e−tf(x) grad(etf ) · ϕ (x) t→∞ t 1 h i = lim e−tf(x) (detf )] · ϕ (x) t→∞ t h i = (detf )] · ϕ (x)

] = ξx · ϕ(x) . Consequently g ] σ(D/ )p(ξp)ϕ(x) = ξp · ϕ(x) for any x ∈ M, which finishes the proof. Corollary 6.8. The Dirac operator D/ g is a first order elliptic differential operator.

g g g Proof. To see that D/ is elliptic, we have to check that the principal symbol ς(D/ )x(ξ):Σ M → g ] ∗ ] Σ M at x ∈ M is an isomorphism, for any ξ ∈ T M, ξx 6= 0. Indeed, recall that X ·Y ·ϕ+Y ·X ·ϕ = −2g(X,Y )ϕ, for any X,Y ∈ Γ(TM) and ϕ ∈ Γ(ΣgM). Thus, for any ξ] ∈ T ∗M and spinor ﬁeld ϕ we get that ξ] · ξ] · ϕ = −g(ξ, ξ)ϕ = −|ξ|2ϕ . Therefore, ξ] · ϕ = 0 =⇒ ξ] · ξ] · ϕ = 0 ⇐⇒ −|ξ|2ϕ = 0 ⇐⇒ ϕ = 0 and we conclude by Proposition 6.7. Corollary 6.9. The operator −(D/ g)2 is strongly elliptic, i.e. hς(−(D/ g)2)(ξ)ϕ, ϕi = |ξ|2|ϕ|2.

Proof. By Proposition 6.7 we see that h i ς((−D/ g)2)(ξ)(ϕ) = −ς(D/ g)(ξ) ς(D/ g)(ξ)(ϕ) = −ξ] · ξ] · ϕ = |ξ|2 · ϕ.

Hence, the statement easily follows.

g g Consider now the space Γc(Σ M) = {ϕ ∈ Γ(Σ M) : supp(ϕ) is compact} of compactly supported 2 g g spinor ﬁelds. The Hilbert space L (Σ M) is the completion of Γc(Σ M) with respect to the inner product Z g hϕ, ψi0 := (ϕ, ψ)dv , M or equivalently the norm

Z 1/2 2 g 2 kϕk0 := |ϕ| dv , where |ϕ| = (ϕ, ϕ) . M Obviously, when M is compact, then the completion of Γ(ΣgM) with respect to this L2-norm, is exactly the Hilbert space L2(ΣgM). 48 IOANNIS CHRYSIKOS

Proposition 6.10. The Dirac operator D/ g is a formally self-adjoint linear operator with respect 2 g to the L -product. In particular, if ϕ, ψ ∈ Γc(Σ M) are spinor fields with compact support (which vanish on the boundary of M), then Z Z (D/ g(ϕ), ψ)dvg = (ϕ, D/ g(ψ))dvg. M M Proof. By the definition of D/ g, the properties of the Clifford multiplication and the metric compatibility of ∇g (see Proposition 5.17), we compute: / g X g X g (D (ϕ), ψ) = (ej · ∇ej ϕ, ψ) = − (∇ej ϕ, ej · ψ) j j X h g i = − ej(ϕ, ej · ψ) − (ϕ, ∇ej (ej · ψ)) j X X g X g = − ej(ϕ, ej · ψ) + (ϕ, ej · ∇ej ψ) + (ϕ, (∇ej ej) · ψ)) j j j X X g / g = − ej(ϕ, ej · ψ) + (ϕ, (∇ej ej) · ψ)) + (ϕ, D (ψ)) j j g = div(Xϕ,ψ) + (ϕ, D/ (ψ)) .

Here, Xϕ,ψ ∈ Γ(TM ⊗ C) is the complex vector ﬁeld on M deﬁned by (see also Exercise 6.11)

(g ⊗ IdC)(Xϕ,ψ,Y ) = (ϕ, Y · ψ) . Thus, by using spinors with compact support and applying the Green formula (so by integrating), our assertion follows. Exercise 6.11. The divergence of a vector ﬁeld Y on (M, g) (with respect to ∇g) is given by

g X g div (Y ) := g(∇ei Y, ei) . i

Consider two vector ﬁelds X1,X2 ∈ Γ(TM) satisfying g(X1,X) + ig(X2,X) = (ϕ, X · ψ) for any X ∈ Γ(TM) and ϕ, ψ ∈ Γ(ΣgM). Prove that

g g X div (X1) + idiv (X2) = ei(ϕ, ei · ψ) . i g g By Proposition 6.10, in even dimensions one deduces that the formal adjoint of D/ ± is D/ ∓. However, be aware that Proposition 6.10 does not state that D/ g is self-adjoint. This is because D/ g and its adjoint, say (D/ g)∗, do not necessarily share the same domain, neither it is clear if these operators can be extended to the whole Hilbert space L2(ΣgM). To overcome these diﬃculties one should discuss for essential self-adjointness. In particular, whenever (M n, g) is complete, the Dirac g operator D/ is essentially self-adjoint with respect to h , i0, which means that the closure of the Dirac operator D/ g in L2(ΣgM) is self-adjoint. Proposition 6.12. Assume that (M n, g) is a complete Riemannian spin manifold. Then, the Dirac g 2 g operator D/ is essentially self-adjoint with respect to the inner product h , i0 in L (Σ M). A detailed proof of Proposition 6.12 can be found for example in [Fr00, p. 94-96] or [Gi09, Prop. 1.3.5]. We conclude this section with the following corollary Corollary 6.13. ([LM89, Fr00]) On a complete Riemannian spin manifold (M n, g) the kernels of the operators D/ g and (D/ g)2 coincide over L2(ΣgM). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 49

6.2. The twistor operator. On a Riemannian spin manifold (M n, g), complementary to the Dirac operator is the so-called twistor operator. This operator is also known as Penrose operator, since it was ﬁrst introduced in the context of general relativity by R. Penrose during 80’s. n n Fix a Riemannian spin manifold (M , g) and let µ : R ⊗∆n → ∆n be the Cliﬀord multiplication. n Since µ is Spin(n)-equivariant, the Spin(n)-representation R ⊗ ∆n splits into two irreducible parts,

n R ⊗ ∆n = ker(µ) ⊕ ∆n .

By extending the Cliﬀord product to a bundle morphism, we obtain a bundle decomposition

TM ⊗ ΣgM = ker(µ) ⊕ ΣgM, Γ(TM ⊗ ΣgM) = Γ(ker(µ)) ⊕ Γ(ΣgM) .

n Now, there exists a unique universal projection p : R ⊗ ∆n → ker(µ), locally deﬁned by 1 X X ⊗ ϕ 7→ X ⊗ ϕ + e ⊗ e · X · ϕ . n i i i

This allows us to introduce a second diﬀerential operator acting on spinors, namely

Definition 6.14. The Penrose, or twistor operator Pg is the first-order differential operator defined by

∇g p Pg := p ◦ ∇g : Γ(ΣgM) → Γ(T ∗M ⊗ ΣgM) =∼ Γ(TM ⊗ ΣgM) → Γ(ker µ).

g In particular, in terms of a local orthonormal frame {e1, . . . , en}, the twistor operator P is expressed by n X 1 g Pgϕ = e ⊗ {∇g ϕ + e · D/ ϕ} i ei n i i=1 It is easy to prove that

X g ei ·Pei ϕ = 0, (6.1) i for any spinor ﬁeld ϕ ∈ Γ(ΣgM), where for simplicity, one often sets

1 Pg ϕ := ∇g ϕ + X · D/ gϕ , X X n with X ∈ Γ(TM) and ϕ ∈ Γ(ΣgM). Moreover, in a line with Remark 4.24, we extend the Hermitian inner product ( , ) of ΣgM on sections of Γ(T ∗M ⊗ ΣgM) and write

n g 2 g g X g g |P ϕ| = (P ϕ, P ϕ) := (Pei ϕ, Pei ϕ) . i=1 Exercise 6.15. Prove the relation (6.1).

Lemma 6.16. Any spinor ﬁeld ϕ ∈ Γ(ΣgM) satisﬁes the following Pythagorean identity

1 |Pgϕ|2 + |D/ gϕ|2 = |∇gϕ|2 . n 50 IOANNIS CHRYSIKOS

Proof. Based on (6.1) and using the fact that the Clifford multiplication with vectors is skew- symmetric with respect to ( , ), we obtain that n g 2 X g g |P ϕ| = (Pei ϕ, Pei ϕ) i=1 n X 1 g = (Pg ϕ, ∇g ϕ + e · D/ ϕ) ei ei n i i=1 n n X g g 1 X g g = (P ϕ, ∇ ϕ) − (ei·P ϕ, D/ ϕ) ei ei n ei i=1 i=1 n X 1 g = (∇g ϕ + e · D/ ϕ, ∇g ϕ) ei n i ei i=1 n X 1 g = (∇g ϕ, ∇g ϕ) + (e · D/ ϕ, ∇g ϕ) ei ei n i ei i=1 n n 1 X g 1 g X = |∇gϕ|2 − (D/ ϕ, e · ∇g ϕ) = |∇gϕ|2 − (D/ ϕ, e · ∇g ϕ) n i ei n i ei i=1 i=1 1 = |∇gϕ|2 − |D/ gϕ|2. n Elements belonging to the kernel of twistor operator, are of course of special interest. Definition 6.17. A spinor field ϕ ∈ Γ(ΣgM) is said to be a twistor spinor if it belongs to the kernel of the twistor operator, i.e. Pg(ϕ) = 0 .

g g Obviously, a spinor field ϕ ∈ Γ(Σ M) is a twistor spinor if and only if PX ϕ = 0 for any vector field X ∈ Γ(TM), which is equivalent to the relation 1 ∇ ϕ + X · D/ gϕ = 0 , ∀ X ∈ Γ(TM) . (6.2) X n Equation (6.2) is the so-called twistor equation. Some equivalent characterizations of twistor spinors (or equivalently of the twistor equation) are described below. Proposition 6.18. ([BFGK89]) (1) A spinor field ϕ ∈ Γ(ΣgM) is a twistor spinor if and only if 2 X · ∇g ϕ + Y · ∇g ϕ = g(X,Y )D/ gϕ . Y X n for any X,Y ∈ Γ(TM). (2) A spinor field ϕ ∈ Γ(ΣgM) is a twistor spinor if and only if the spinor field g X · ∇X ϕ does not depend on the unit vector field X. Proof. (1) We first multiply the twistor equation (6.2) from the left by some vector fields, say 1 1 Y · ∇g ϕ + Y · X · D/ gϕ = 0,X · ∇g ϕ + X · Y · D/ gϕ = 0 . X n Y n DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 51

Using now the fundamental property of the Cliﬀord multiplication X ·Y +Y ·X = −2g(X,Y ) IdΣgM and adding the last two formulas, we obtain the given condition, 1 X · ∇g ϕ + Y · ∇g ϕ = − (X · Y + Y · X) · D/ gϕ Y X n 2 = g(X,Y )D/ gϕ . (6.3) n For the converse, assume that (6.3) is valid and set Y = ej. We multiply the resulting formula by ej from the left, and summing over j = 1, . . . , n. This yields that 2 X g 2 g X g(X, e )e · D/ ϕ = X · D/ ϕ = e · X · ∇g ϕ + e · e · ∇g ϕ n j j n j ej j j X j j X g g = ej · X · ∇ej ϕ − n∇X ϕ . j

Now, for the first sum, and since ej · X = −X · ej − 2g(X, ej) IdΣgM , it follows that X g X g X g ej · X · ∇ej ϕ = − X · ej · ∇ej ϕ − 2 g(X, ej)∇ej ϕ j j j g g = −X · D/ ϕ − 2∇X ϕ . Therefore one finally gets that 2 X · D/ gϕ = −X · D/ gϕ − 2∇g ϕ − n∇g ϕ , n X X which reduces to the twistor equation (6.2). This proves the first assertion. (2) For the second claim we replace in (6.3) X = Y and assume that X is of unit length. We get that 1 X · ∇g ϕ = D/ gϕ . X n g for all unit vector fields X on M, hence we conclude. Conversely, assume that X·∇X ϕ is independent g of the unit vector field X and set ψ := X · ∇X ϕ. Then, g g g D/ ϕ = nψ, and ∇X ϕ = −X · X · ∇X ϕ = −X · ψ g 1 / g and it is easy to see that the twistor equation ∇X ϕ + n X · D ϕ = 0 is trivially satisfied, 1 −X · ψ + nX · ψ = 0 . n This shows that ϕ is a twistor spinor and finishes the proof. n g n n Example 6.19. Consider the Euclidean space (R , g := h , i)(n ≥ 2) and let Σ R ' R × ∆n be the spinor bundle. The general solution of the twistor equation is given by ϕx = ϕ0 + x · ϕ1 for n some x ∈ R , where ϕ0, ϕ1 are constant spinors (see [BFGK89, p. 28] or [Gi09]). In fact, in this case we obtain the maximal dimension of the kernel of the twistor operator g [ n ]+1 dim ker(P ) = 2 2 .

7. The Schrödinger-Lichnerowicz formula With aim to proceed with further properties of the Dirac operator on a Riemannian spin manifold, but also of Killing and twistor spinors, one needs to develop some further tools, as for example the Schrödinger-Lichnerowicz formula. This identity relates the square of the Dirac operator with the Laplace operator acting on spinors (spinorial Laplacian) and the scalar curvature of the Riemannian manifold (M n, g). In particular, for the ﬂat case it provides the identiﬁcation of the square of D/ g with the spinorial Laplacian, so it gives rise to an answer of Dirac’s physical problem on the 52 IOANNIS CHRYSIKOS

Minkowski spacetime. Our description starts by recalling a few details about the Laplace operator on the spinor bundle.

7.1. The spinorial Laplace operator. Fix a Riemannian spin manifold (M n, g) and denote by ΣgM its spinor bundle. The formal adjoint of the spinorial covariant derivative ∇g, is the operator (∇g)∗ : Γ(T ∗M ⊗ ΣgM) → Γ(ΣgM) , such that ((∇g)∗Θ, ϕ) = (Θ, ∇gϕ) for any compactly supported section Θ ∈ Γ(T ∗M ⊗ ΣgM) and compact supported spinor field ϕ ∈ Γ(ΣgM), where here we have extended ( , ) on sections of T ∗M ⊗ ΣgM as in Remark 4.24. Definition 7.1. The spinorial Laplace operator is the second-order differential operator acting on spinors, defined by ∆g(ϕ) := (∇g)∗∇gϕ, for any spinor field ϕ ∈ Γ(ΣgM).

In terms of a local orthonormal frame (e1, . . . , en) on an open set U ⊂ M, one has n n g X g g g X h g g g g i g ∆ (ϕ) = − ∇ei ∇ei ϕ + ∇ ϕ = − ∇ei ∇ei + div (ei)∇ei ϕ , ∇ei ei i=1 i=1 g P g where div (ei) = j g(∇ej ei, ej). Moreover, it is not hard to prove that g Lemma 7.2. In local normal coordinates, i.e. (∇ ei)x = 0 for any 1 ≤ i ≤ n with x ∈ M, the spinorial Laplacian operator is given by

g g ∗ g X g g ∆ (ϕ) = (∇ ) ∇ ϕ = − ∇ei ∇ei ϕ . i Observer that in the interior of a Riemannian spin manifold, Stoke’s theorem implies that any g two spinor ﬁelds ϕ, ψ ∈ Γc(Σ M) with compact support are such that Z Z Z (∆g(ϕ), ψ)dvg = (∇gϕ, ∇gψ)dvg = (ϕ, ∆g(ψ))dvg , M M M g g Pn g g where (∇ ϕ, ∇ ψ) := i=1(∇ei ϕ, ∇ei ψ).

1 7.2. The 2 -Ricci type formula and a proof of the SL-formula. Our aim now is to discuss a proof of the Schrödinger-Lichnerowicz formula. Theorem 7.3. (The Schrödinger-Lichnerowicz formula) On a Riemannian spin manifold (M n, g) the square of the Dirac operator D/ g satisﬁes the identity Scalg (D/ g)2(ϕ) = ∆g(ϕ) + ϕ , ∀ ϕ ∈ Γ(ΣgM) . (7.1) 4 The traditional proof of the Schrödinger-Lichnerowicz formula (SL-formula in short), is based on a direct computation of (D/ g)2, combining properties of the curvature Rg and the Cliﬀord product, see for example some of the most classical references for the topic [LM89, Fr00, Hij01]. Here we shall 1 1 g present an alternative proof which is based on a 2 -Ricci type formula, the so-called 2 -Ric -formula, introduced by Friedrich-Kim [FrK00]. This is a powerful spinorial formula, which describes the action of the Ricci endomorphism Ricg on the spinor bundle ΣgM in terms of the Dirac operator D/ g and the spinorial covariant derivative ∇g. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 53

1 g n Lemma 7.4. (The 2 -Ric -formula) ([FrK00]) On a Riemannian spin manifold (M , g), any vector field X ∈ Γ(TM) and spinor field ϕ ∈ Γ(ΣgM) satisfies

1 n g / g g g / g X g Ric (X) · ϕ = D (∇X ϕ) − ∇X (D (ϕ)) − ej · ∇∇g X ϕ . 2 ej j=1 Proof. Of course, all our computations can be done locally, at some point x ∈ M. Hence, let us ﬁx a local orthonormal frame {e1, . . . , en} of TxM. We use the relation (5.5), i.e.

n X 1 e ·Rg(X, e )ϕ = − Ricg(X) · ϕ , i i 2 i=1 and we introduce in the one side the expression of the spinorial curvature tensor. Thus,

n 1 g X g g g g g Ric (X) · ϕ = − ej · ∇ ∇e ϕ − ∇e ∇ ϕ − ∇ ϕ 2 X j j X [X,ej ] j=1 X g g X g g X g = ej · ∇ (∇ ϕ) − ej · ∇ (∇ ϕ) + ej · ∇ ϕ ej X X ej [X,ej ] j j j g g X g g X g g X g = D/ (∇ ϕ) − ∇ (ej · ∇ ϕ) + (∇ ej) · (∇ ϕ) + ej · ∇ ϕ X X ej X ej [X,ej ] j j j g g g g X g X g = D/ (∇ ϕ) − ∇ (D/ (ϕ)) + (∇X ej) · (∇ ϕ) + ej · ∇ ϕ , X X ej [X,ej ] j j where for the third equality we applied the Leibniz rule (see Proposition 5.17), i.e.

g g g g g ej · ∇X (∇ej ϕ) = ∇X (ej · ∇ej ϕ) − (∇X ei) · (∇ej ϕ) . Now, since the Levi-Civita connection is torsion-free the above formula reduces to

1 g g g g g X g X g X g Ric (X) · ϕ = D/ (∇ ϕ) − ∇ (D/ (ϕ)) + (∇ e ) · (∇ ϕ) + e · ∇ g ϕ − e · ∇ g ϕ X X X i ej j ∇ e j ∇ X 2 X j ej j j j g g g g X g = D/ (∇X ϕ) − ∇ (D/ (ϕ)) − ej · ∇ g ϕ + A(ϕ) , ∇ej X j P g P g where A(ϕ) := j(∇X ej)·(∇ej ϕ)+ j ej ·∇ g ϕ. However, assuming (without loss of generality) ∇X ej n g g that our orthonormal frame {ej}j=1 is ∇ -parallel, i.e. (∇ ej)x = 0 at x ∈ M, we get A(ϕ) ≡ 0 identically.

1 It turns out that the 2 -Ricci type formula is stronger than the Schrödinger-Lichnerowicz formula, in the sense that the second formula occurs from the ﬁrst one after a contraction. In fact, in Section 1 g 8 we will discuss further powerful applications of the 2 -Ric -identity. But let us proceed now with the proof of theorem Theorem 7.3.

Proof. (Proof of Theorem 7.3) Based on Lemma 7.4, we apply a contraction with respect a g ∇ -parallel local orthonormal frame {ei}. For the left hand side, note that

X g X g X g g ei · Ric (ei) · ϕ = Ric (ei, ek) · ei · ek · ϕ = − Ric (ei, ei) · ϕ = − Scal · ϕ . i i,k i 54 IOANNIS CHRYSIKOS

Let us focus now on the right-hand side. Working locally, we write all together:

1 g X g g X g g X g − Scal · ϕ = e · D/ (∇ ϕ) − e · ∇ (D/ ϕ) − e · e · ∇ g ϕ i ei i ei i j ∇ e 2 ej i i i i,j X g g g 2 X g / / g = ei · D (∇ei ϕ) − (D ) (ϕ) − ei · ej · ∇ ϕ ∇ej ei i i,j X / g g / g 2 = ei · D (∇ei ϕ) − (D ) (ϕ) , i P g / g / g 2 g P / g g since − i ei · ∇ei (D ϕ) = −(D ) (ϕ) and (∇ ei)x = 0. To compute i ei · D (∇ei ϕ) one can use Lemma 6.6, namely the identity / g X g / g g D (X · ϕ) = ej · (∇ej X) · ϕ − X · D (ϕ) − 2∇X ϕ . j g g Replace X by ei and ϕ by ∇ei ϕ. Since the basis {ei} is ∇ -parallel, we conclude that / g g X g g / g g g g / g g g g D (ei · ∇ei ϕ) = ei · (∇ei ei) · ∇ei ϕ − ei · D (∇ei ϕ) − 2∇ei ∇ei ϕ = −ei · D (∇ei ϕ) − 2∇ei ∇ei ϕ . i Consequently, a combination with Lemma 7.2 yields that X / g g X / g g X g g / g 2 g ei · D (∇ei ϕ) = − D (ei · ∇ei ϕ) − 2 ∇ei ∇ei ϕ = −(D ) (ϕ) + 2∆ (ϕ), i i i Thus, it follows that 1 h i − Scalg · ϕ = −2 (D/ g)2(ϕ) − ∆g(ϕ) , 2 which is obviously the Schrödinger-Lichnerowicz formula.

8. Special spinor fields In this section we shall use the SL-formula and other tools described above, in order to examine in a systematic way some special spinor ﬁelds, like harmonic spinors, parallel spinors and Killing spinors.

8.1. Harmonic spinors. Definition 8.1. A harmonic spinor is a non-trivial spinor field ϕ ∈ Γ(ΣgM) belonging in the kernel of the Dirac operator D/ g, i.e. D/ g(ϕ) = 0 . It is obvious that ∇g-parallel spinors are harmonic (details on parallel spinors are given below). However, the converse does not hold in general. Exercise 8.2. Provide an example of a Riemannian spin manifold admitting harmonic spinors which are not parallel. We also mention that the dimension of the kernel of D/ g, ker D/ g = {ϕ ∈ Γ(ΣgM): D/ g(ϕ) = 0} depends on the metric g and the fixed spin structure (a result of Hitchin). However, the dimension of this kernel is in fact, a conformal invariant of (M n, g) (see [Hit74]). Let us explain now how the Schrödinger-Lichnerowicz formula yields naturally vanishing theorems for harmonic spinors. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 55

Theorem 8.3. (Vanishing theorem) On a compact Riemannian spin manifold (M, g) with positive scalar curvature, there are no harmonic spinors, i.e. the kernel of the Dirac operator is trivial, ker D/ g = {0}. Proof. Suppose that ϕ ∈ Γ(ΣgM) is a spinor field such that D/ gϕ = 0. By integrating the Schrödinger-Lichnerowicz formula, we deduce that Z g 2 g g ∗ g g g g 2 Scal |ϕ| dv = −h(∇ ) ∇ ϕ, ϕi0 = −h∇ ϕ, ∇ ϕi0 = −k∇ ϕk0 , M R g where by extending the inner product hϕ, ψi0 = M (ϕ, ψ)dv , we have defined Z g g g g g h∇ ϕ, ∇ ψi0 := (∇ ϕ, ∇ ψ)dv M g g P g g g R g 2 g with (∇ ϕ, ∇ ψ) = i(∇ei ϕ, ∇ei ψ). If Scal (x) > 0 at any x ∈ M, then M Scal |ϕ| dv > 0, a g 2 contradiction, since −k∇ ϕk0 ≤ 0. Exercise 8.4. Prove that on compact Riemannian spin manifold (M n, g) which has zero scalar curvature, every harmonic spinor is ∇g-parallel. Hint: the assertion is treated in an analogous way with Theorem 8.3. 8.2. Atiyah-Singer index theorem. Recall that the analytic index of an elliptic operator D is defined as ind(D) := dim ker(D) − dim coker(D) = dim ker(D) − dim ker(D∗) where both quantities in the right hand side are finite-dimensional and D∗ is the adjoint operator of D. Lemma 8.5. On a Riemannian spin manifold the analytic index of the Dirac operator D/ g is zero. g Proof. This follows since the Dirac operator D/ is elliptic and formally self-adjoint. However, on an even-dimensional Riemannian manifolds one can consider the index of the (half) Dirac operator, g g g g g ind(D/ +) := dim ker(D/ +) − dim coker(D/ +) = dim ker(D/ +) − dim ker(D/ −), where the last equality follows since D/ g is formally self-adjoint and according to Lemma 6.5 the g g formal adjoint of D/ ± is D/ ∓. Due to the Atiyah-Singer Index Theorem [AS63], one can relate the g ˆ index ind(D/ +) with a certain characteristic class with rational coefficients, the so-called A-genus, which we shortly review below. First notice that in the case of the Dirac operator, the index theorem reads as follows. Theorem 8.6. (Atiyah-Singer [AS63]) Let (M n, g) be an even dimensional closed Riemannian spin g manifold and let ind(D/ +) be the analytical index of the positive part of its Dirac operator. Then g ˆ ind(D/ +) = A(M), ˆ R ˆ g ˆ ˆ g n where A(M) = M A(R ) ∈ Z is the integral of the A-genus A(R ) of (M , g). We now remind the definition of the Aˆ-genus Aˆ(Rg) on a closed Riemannian manifold (M n, g), see also [LM89, BHMMS]. Consider the Riemannian curvature tensor Rg ∈ Γ(∧2M ⊗ End(TM)) g g g and define the trace (over End(TM)) of the composition Rk := R ◦ ... ◦ R of k curvature endomorphisms, with values in the even exterior algebra, i.e. g 2k tr(Rk) ∈ Γ(∧ M) . (8.1) This is a closed differential form of degree 2k and one can show that its cohomology class is independent of the choice of the metric g on M n (notice that for k = odd the trace defined by (8.1) is 56 IOANNIS CHRYSIKOS zero). This allows us to define the trace tr(F (Rg)) for any polynomial (or even formal series) F in one variable. Such a series is given for example by X/2 F (X) = sinh(X/2) and then one usually sets 1 1 Aˆ(Rg) := exp tr log(F ( Rg)) . 2 2πi Example 8.7. ([LM89, p. 138]) Consider a four-dimensional compact manifold M 4. Then 1 Aˆ(M 4) = − p (M) ∈/ , 24 1 Z 2 where by p1(M) we denote the first Pontryagin class of M. For CP for example, one computes 2 2 2 H (CP ; Z) ' Z, which yields finally the relation Aˆ(CP ) = −1/8. Formally, we may interpret the differential form Aˆ(Rg) as the square root of the determinant of 1 g ˆ g F ( 2πi R ) (one can also show that A(R ) reads in terms of the Pontryagin classes of M with rational g coefficients). The Atiyah-Singer theorem states that both the analytical index ind(D/ +) and the ˆ R ˆ g ˆ ˆ integral A(M) = M A(R ) are finite. In particular for compact spin manifolds, the A-genus A(M) must be an integer, which it is not the case for non-spin manifolds, according to Example 8.7. Theorem 8.8. On a compact Riemannian spin manifold with non-vanishing Aˆ-genus, there is no metric with positive scalar curvature. g g Proof. By Theorem 8.3, if the scalar curvature is positive, then ker(D/ +) = ker(D/ −) = {0} and g hence ind(D/ +) = 0. The results now follows by the Atiyah-Singer index theorem. Hence, in this case a weak geometrical assumption like Scalg > 0 imposes restrictions on the topology of the manifold (see also Section 10.1, Lemma 10.1). We finally mention that Corollary 8.9. Let M be a compact spin manifold of dimension 4m. If M admits a metric of positive scalar curvature, then Aˆ(M) = 0. 8.3. Parallel spinors. Let (M n, g) be a Riemannian spin manifold. Definition 8.10. A non-trivial spinor field ϕ ∈ Γ(ΣgM) is called ∇g-parallel (or in short parallel) if it satisfies the equation g ∇X ϕ = 0, for any X ∈ Γ(TM). n Example 8.11. Obviously, on the Euclidean space (R , g) endowed with its canonical spin struc- n [ n ] ture, parallel spinor fields are exhausted by constant maps ϕ : → 2 , i.e. constant sections of n R C gn n n n [ ] Σ R ' R × ∆n. Therefore, (R , gn) admits a 2 2 -dimensional space of parallel spinors. g In the “generic case” the equation ∇X ϕ = 0 has no solutions. In particular, existence of parallel fields can always be characterized by a reduction of the holonomy group Hol(g) ≡ Hol(M, g) (see [Joy00, Agr06] for an introduction to holonomy theory). Corollary 8.12. A Riemannian spin manifold (M n, g) admitting a ∇g-parallel spinor is Ricg-flat, i.e. Ricg(X) ≡ 0 for any X ∈ Γ(TM). g g Proof. Assume that 0 6= ϕ0 ∈ Γ(Σ M) is a spinor field satisfying the equation ∇X ϕ0 = 0, for any / g 1 g vector field X. Then D (ϕ0) = 0 identically, and the claim immediately follows by the 2 -Ric - formula, i.e. Lemma 7.4: 1 Ricg(X) · ϕ = 0 2 0 DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 57 and we conclude since ϕ0 has no zeros. For a different proof see for example [Fr00, p. 67]. Consider an oriented Riemannian manifold (M n, g), endowed with a spin structure, and as above write Hol(g) for the holonomy group of the Levi-Civita connection on the tangent bundle level. Next we will mainly be concerned with irreducible Riemannian manifolds, i.e. manifolds whose holonomy representation is irreducible. By the de Rham decomposition theorem, we known that Theorem 8.13. (de Rham, see [KN69, Joy00]) A manifold is irreducible, if and only if its universal cover is not a Riemannian product. It turns out that simply connected irreducible spin manifolds carrying parallel spinors are classified by their (Riemannian) holonomy group in the following way. Let us write Hol(∇g) for the holonomy group of the spinorial Levi-Civita connection. Obviously, one has Hol(g) ⊆ SO(n) and Hol(∇g) ⊆ Spin(n). In particular, the image of Hol(∇g) via the double covering Ad : Spin(n) → SO(n) coincides with Hol(g). When M is simply connected both Hol(g) and Hol(∇g) are connected, in particular, Hol(∇g) is the identity component of Ad−1(Hol(g)) in Spin(n). Consequently, for simply connected Riemannian spin manifolds, the classification of spinorial holonomy groups follows from the classification of Riemannian holonomy groups. Now, by the general holonomy principle g (see [Joy00]) a ∇ -parallel spinor must correspond to an element in ∆n which is invariant under the action of Hol(∇g), via the holonomy representation. A generic Riemannian spin manifold has holonomy group Hol(g) = SO(n) and no parallel spinors. Since the Riemannian holonomy groups have been classified by M. Berger, one can use these results to obtain the classification of holonomy groups of simply connected Riemannian spin manifolds with parallel spinors. In particular, by Corollary 8.12 one deduces that the appearing holonomy groups must be exactly the Ricci flat holonomy groups in Berger’s list, which are given in the following theorem. Theorem 8.14. (Mc Wang [McW89]) Let (M n, g) be a complete, simply connected de Rham irreducible Riemannian spin manifold. Assume that g is not flat and let N be the dimension of the space of ∇g-parallel spinors. Assume also that N ≥ 1. Then, one of the following holds: dim M Hol(g) N terminology n = 2m Hol(g) = SU(n) N = 2 Calabi-Yau n = 4m Hol(g) = Sp(n) N = 2m + 1 Hyper-Kähler n = 7 Hol(g) = G2 N = 1 parallel G2-structures n = 8 Hol(g) = Spin(7) N = 1 parallel Spin(7)-structures Some details related with the geometric structures appearing in this table, are given in the Appendix. Now, in the opposite direction, one can pose the question whether any Riemannian manifold with one of the Ricg-flat holonomy groups is a spin manifold which admits constant spinors under the corresponding holonomy representation, and thus parallel. This admits an affirmative answer, which is based on the following observation. Proposition 8.15. Let (M n, g)(n ≥ 3) an oriented Riemannian manifold with a G-structure R ⊂ Pg, i.e. a reduction of the structure group of its orthonormal frame bundle to G, where G ⊂ SO(n) is a connected and simply connected subgroup. Then M is a spin manifold, with a spin structure Peg induced by the G-structure R. Proof. Consider the embedding ι : G → SO(n). Since n ≥ 3 and G is simply connected, ι lifts to n a homomorphism eι : G → Spin(n), such that Ad ◦eι = ι. Therefore, the G-structure on (M , g) induces a spin structure P := R × Spin(n) = R × Spin(n). eg eι G

Because any of the Lie groups SU(2m), Sp(4m), G2, and Spin(7) are connected and simply connected, we deduce that 58 IOANNIS CHRYSIKOS

Corollary 8.16. Let (M n, g) be a Riemannian manifold with holonomy group G = Hol(g) being one of the Lie groups appearing in Theorem 8.14. Then, M admits a spin structure Peg (induced by the corresponding G-structure) and with respect to this spin structure it admits ∇g-parallel spinors. In particular, the dimension of the kernel ker(∇g) coincides with the number N prescribed by Theorem 8.14.

Hence, focusing in the de Rham irreducible case, we ﬁnally get

Theorem 8.17. An irreducible Riemannian metric g has holonomy group Hol(g) one of the Ricg- ﬂat holonomy groups, if and only if it admits ∇g-parallel spinors.

It worths to mention that parallel spinors, in particular G2-compactiﬁcations and Calabi-Yau manifolds, are extremely useful in supersymmetric string theories and in supergravity (see [Joy00, AW01, A02, AG04, Agr06] and the references therein).

8.4. Killing spinors. We now turn our attention to Killing spinors, which form a special class of twistor spinors.

Deﬁnition 8.18. Let (M n, g) be a Riemannian spin manifold. A non-trivial spinor ﬁeld ϕ ∈ ΣgM solving the equation g ∇X ϕ = κX · ϕ, for any X ∈ Γ(TM) and some complex number κ 6= 0, is called a Killing spinor with Killing number κ. Next we shall denote by n g g K(M , g)κ = {ϕ ∈ Γ(Σ M): ∇X ϕ = κX · ϕ for all X ∈ Γ(TM)} n the space of Killing spinors on (M , g), with Killing number κ 6= 0. Observe that we do not view ∇g-parallel spinors as special cases of Killing spinors.

Remark 8.19. Killing spinors were ﬁrst appeared in the context of supergravity and in superstring theories (see [CR84, CRW84, DNP86]). In fact, nowadays, and due to their strong geometric consequences, Killing spinors are apparent in mathematical physics, especially in supergravity and string theory (see for example [AG04] and [Ert18] for a more recent survey). However, in our context and especially in Chapter 10, such spinors will mainly appear as eigenspinors associated with the minimal eigenvalues of the Dirac operator. Hence, we mention that not every Riemannian spin manifold admits Killing spinors, neither every complex number κ ∈ C can be a Killing number. In particular, by a result of A. Lichnerowicz it follows that Killing spinors fall into two classes: ∗ • real Killing spinors, and then κ ∈ R , ∗ • imaginary Killing spinors, and then κ ∈ iR .

As one can expect, the name given to this special kind of spinor ﬁelds ϕ ∈ K(M, g)κ originates from the following fact.

Lemma 8.20. Let (M n, g) be a connected Riemannian spin manifold admitting a non-trivial Killing ∗ spinor ϕ ∈ K(M, g)κ with Killing number κ ∈ R . Then, the rule n X Vϕ := i(ei · ϕ, ϕ)ei i=1 deﬁnes a Killing vector ﬁeld of (M n, g). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 59

g g Proof. It is suﬃcient to prove that the Lie derivative (LVϕ g)(X,Y ) = g(∇X Vϕ,Y ) + g(X, ∇Y Vϕ) g vanishes, for any X,Y ∈ Γ(TM). Initially, we shall compute the covariant derivative ∇X Vϕ using g a local orthonormal frame {ei} of TxM at some ﬁxed point x ∈ M, with (∇ ei)x = 0. We obtain

g X n g g o ∇X Vϕ = i (ei · ∇X ϕ, ϕ)ei + (ei · ϕ, ∇X ϕ)ei i X = iκ (ei · X − X · ei) · ϕ, ϕ ei. i Hence, g g (LVϕ g)(X,Y ) = g(∇X Vϕ,Y ) + g(X, ∇Y Vϕ) n X X o = iκ g (ei · X − X · ei) · ϕ, ϕ ei,Y + g (ei · Y − Y · ei) · ϕ, ϕ ei,X i i n o = iκ (Y · X − X · Y ) · ϕ, ϕ + (X · Y − Y · X) · ϕ, ϕ = 0 .

n Exercise 8.21. On a Riemannian spin manifold (M , g) with a non-trivial Killing spinor ϕ0 with Killing number κ(6= 0), prove the following properties: g (α) The Killing spinor ϕ0 is a D/ -eigenspinor with eigenvalue −nκ, n g ϕ0 ∈ K(M , g)κ =⇒ D/ (ϕ0) = −nκϕ0 . (8.2) (β) If the dimension is even, then the Killing spinor equation takes the form g + − g − + ∇X ϕ0 = κX · ϕ0 , ∇X ϕ0 = κX · ϕ0 , + − where ϕ0 = ϕ0 ⊕ ϕ0 is the splitting of ϕ with respect to the orthogonal decomposition g g g g Σ M = Σ+M ⊕ Σ−M of the associated spinor bundle Σ M. (γ) Show that any Killing spinor is a non-trivial twistor spinor. In particular, show that a section of ΣgM is a Killing spinor if and only if it is a twistor spinor which is an eigenvector of D/ g.

Remark 8.22. Obviously, a spinor ﬁeld ϕ is a Killing spinor, i.e. ϕ ∈ K(M, g)κ, if and only if it is a parallel section with respect to the (compatible) covariant derivative on ΣgM given by g X 7→ ∇X − κX. n n [ 2 ] Hence ϕ cannot have zeros, and moreover for n ≥ 2 it follows that dim K(M , g)κ ≤ 2 (the equality is realized by the n-sphere Sn, see for example [BFGK89, Gi09]). Killing spinors impose very strong geometric constraints. In particular, on a Riemannian spin manifold (M n, g) the existence of a Killing spinor implies that g is an Einstein metric, i.e. g Ric (X,Y ) = cg(X,Y ), 0 6= c ∈ R . Proposition 8.23. Let (M n, g) be a connected Riemannian spin manifold admitting a non-trivial Killing spinor ϕ ∈ K(M, g)κ with Killing number κ ∈ C. Then g is an Einstein metric. In particular, in the complete case the following is true: ∗ n (α) If κ ∈ R , then (M , g) is a compact Einstein manifold with positive constant scalar curvature g 2 1 g Scal such that κ = 4n(n−1) Scal . ∗ n (β) If κ ∈ iR , then (M , g) is a non-compact Einstein manifold of negative constant scalar curva- g 2 1 g trure Scal such that κ = 4n(n−1) Scal . 60 IOANNIS CHRYSIKOS

n Proof. First we shall prove that such a manifold (M , g) is Einstein, independently of κ ∈ C. For 1 g this claim, we present an alternative proof based on the powerful 2 -Ric -identity. For a traditional proof we refer to [BFGK89, p. 30], [Fr00, p. 118], or [Hij01, p. 56], while a proof based on the theory of twistor spinors can be found in [Gi09, p. 142]. So, consider some element ϕ0 ∈ K(M, g)κ, i.e. g assume that ∇X ϕ0 = κX · ϕ0 for any X ∈ Γ(TM) and some non-zero κ. Then we have g g g ∇ ϕ0 = κ(∇ej X) · ϕ0 ∇ej X 1 g for any local orthonormal frame {ej}. Thus, by applying the 2 -Ric -formula and keeping in mind (8.2) and Proposition 6.6 (β), we easily deduce that

1 g X Ricg(X) · ϕ = D/ (X · ϕ ) + n ∇g ϕ − e · (∇g X) · ϕ 2 0 κ 0 κ X 0 κ j ej 0 j X g / g g = κ ej · (∇ej X) · ϕ0 − κX · D (ϕ0) − 2κ∇X ϕ0 j 2 X g +nκ X · ϕ0 − κ ej · (∇ej X) · ϕ0 j 2 2 2 = nκ X · ϕ0 − 2κ X · ϕ0 + nκ X · ϕ0 2 = 2(n − 1)κ X · ϕ0 . So, g 2 Ric (X) · ϕ0 = 4(n − 1)κ X · ϕ0 g and the result follows since by definition κ is non-zero and ϕ0 cannot have zeros, i.e. Ric = 2 n 4(n − 1)κ IdTM and (M , g) is an Einstein manifold. The formula for the scalar curvature follows easily by a contraction. If κ is a non-zero real, i.e. ϕ0 is a real Killing spinor and the metric g is complete, then, our manifold (M n, g) is a (complete) Einstein manifold with positive scalar curvature. Thus, based on the Theorem of Myers (cf. [Bes86]) we obtain the compactness of M. Suppose now that 0 6= κ ∈ iR, which is equivalent to say that ϕ0 is an imaginary Killing spinor. We can write κ = iα for some g 2 2 2 non-zero real α ∈ R, and then it follows that (D/ ) (ϕ0) = −n α ϕ0. Assume now that M is compact. Because D/ g is formally self-adjoint (see Proposition 6.10), an integration over M gives rise to the following inequality Z Z Z g g g g 2 g 2 2 g 0 ≤ (D/ (ϕ0), D/ (ϕ0))dv = ((D/ ) (ϕ0), ϕ0)dv = −n α (ϕ0, ϕ0)dv , M M M from which we deduce that ϕ0 ≡ 0. This is a contradiction, so M must be non-compact. Let us discuss now some further basic properties of manifolds carrying Killing spinors. Proposition 8.24. A Riemannian spin manifold admitting a non-trivial Killing spinor ϕ with Killing number κ is locally irreducible. g Proof. Assume that ϕ 6= 0 is a Killing spinor, i.e. ∇X ϕ = κX · ϕ, for some κ 6= 0 and for any X ∈ Γ(TM). Then, it is easy to see that g 2 R (X,Y )ϕ = κ (Y · X − X · Y ) · ϕ , (8.3) P for any X,Y ∈ Γ(TM). Now, since i ei · X · ei = (n − 2)X for any orthonormal frame {ei} of TM, we compute that n X g 2 ei ·R (X, ei)ϕ = −2(n − 1)κ X · ϕ, i=1 DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 61 as it should be according to Propositions 5.19 and 8.23. In particular, by Proposition 8.23 we know 2 1 g that κ = 4n(n−1) Scal . However, recall that (see also [Fr00]) n 1 X Rg(X,Y )ϕ = e · Rg(X,Y )e · ϕ , 4 i i i=1 for any X,Y ∈ Γ(TM). Hence, relation (8.3) can be expressed by n X Scalg e · Rg(X,Y )e · ϕ + (X · Y − Y · X) · ϕ = 0.. (8.4) i i n(n − 1) i=1 n n p n−p Suppose now that (M , g) is locally the Riemannian product M = M1 ×M2 of two Riemannian p n−p manifolds M1 and M2 , of dimensions p and n − p, respectively. Consider some vector fields g X ∈ Γ(TM1) and Y ∈ Γ(TM2). Then, R (X,Y )Z = 0 for any Z ∈ Γ(TM), and by (8.4) we obtain that Scalg X · Y · ϕ = 0 . g By assumption, κ 6= 0, so Scal 6= 0. Since X,Y are orthogonal vectors, from the last equation we obtain ϕ = 0, a contradiction according to Remark 8.22. 9. Properties of twistor spinors and their relation with Killing spinors In this section we present elementary properties of twistor spinors, described fro example by Lichnerowicz, Friedrich, Baum and Habermann, see also [Lich87, Fr89, BFGK89, Hab94]. We further discuss the relation of twistor spinors with Killing spinors. 9.1. Structural properties of twistor spinors. Let us start with the following fundamental result of A. Lichnerowicz. Theorem 9.1. ([Lich87]) Let (M n, g)(n ≥ 3) be a Riemannian spin manifold and let ϕ ∈ Γ(ΣgM) be a non-trivial twistor spinor. Then the following conditions are satisfied: n Scalg (D/ g)2ϕ = ϕ, (9.1) 4(n − 1) n h Scalg i n ∇g D/ g(ϕ) = − Ricg(X) · ϕ + X · ϕ = Schog(X) · ϕ, (9.2) X 2(n − 2) 2(n − 1) 2 where Schog : TM → TM denotes the Schouten endomorphism associated to ∇g, i.e. 1 Scalg g(Schog(X),Y ) = Schog(X,Y ), Schog(X) := − Ricg(X) + X , n − 2 2(n − 1) for any X,Y ∈ Γ(TM), where Schog(X,Y ) is the Schouten tensor of (M n, g). 1 Next we present an alternative proof of Theorem 9.1, based on the 2 -Ricci formula (for other 1 g methods see [Lich87, BFGK89, Gi09]). In particular, first we apply Lemma 7.4, i.e. the 2 -Ric - identity 1 n g / g g g / g X g Ric (X) · ϕ = D (∇X ϕ) − ∇X (D (ϕ)) − ej · ∇∇g X ϕ, (9.3) 2 ej j=1 to a non-trivial twistor spinor ϕ ∈ ker(Pg). This yields the following: 1 Lemma 9.2. (The twistorial 2 -Ricci type formula) Any non-trivial twistor spinor ϕ satisfies the following relation: 1 1 n − 2 Ricg(X) · ϕ = X · (D/ g)2(ϕ) − ∇g (D/ g(ϕ)) , (9.4) 2 n n X for any X ∈ Γ(TM). 62 IOANNIS CHRYSIKOS

g g 1 / g Proof. Consider a non-trivial twistor spinor ϕ ∈ ker(P ), i.e. ∇X ϕ = − n X · D (ϕ) for any X ∈ Γ(TM). Then, based on Proposition 6.6, (2), for the ﬁrst term of (9.3) we see that 1 1 D/ g(∇g ϕ) = D/ g(− X · D/ g(ϕ)) = − D/ g(X · D/ g(ϕ)) X n n 1 h X g g g g i = − e · (∇g X) · D/ (ϕ) − X · D/ (D/ (ϕ)) − 2∇g (D/ (ϕ)) n j ej X j 1 X g 1 g 2 g = − e · (∇g X) · D/ (ϕ) + X · (D/ )2(ϕ) + ∇g (D/ (ϕ)) . n j ej n n X j Therefore, for the ﬁrst two terms of (9.3), ones obtains

g g 1 g n − 2 g 1 X g D/ (∇g ϕ) − ∇g (D/ (ϕ)) = X · (D/ )2(ϕ) − ∇g D/ (ϕ) − e · (∇g X) · D/ (ϕ). (9.5) X X n n X n j ej j Let us proceed now with the third term. Because

g 1 g g ∇ g ϕ = − (∇ X) · D/ (ϕ) , ∇ X ej ej n for any X ∈ Γ(TM), it follows that n n X g 1 X g g − e · ∇ g ϕ = e · (∇ X) · D/ (ϕ) , j ∇ X j ej ej n j=1 j=1 and this is canceled by the third term in the right-hand side of (9.5). Thus our claim follows. We present now the proof of Theorem 9.1. Proof of Theorem 9.1. The ﬁrst relation (9.1) follows by using (9.4) and applying a contraction, with respect to some orthonormal frame {ei} of TM. As we already know, this means X g X g X g g ei · Ric (ei) · ϕ = Ric (ei, ek) · ei · ek · ϕ = − Ric (ei, ei) · ϕ = − Scal · ϕ i i,k i and hence, for the left hand side part of (9.4) we obtain that 1 X 1 e · Ricg(e ) · ϕ = − Scalg · ϕ . 2 i i 2 i We proceed with the right hand side and write all together:

1 1 X g n − 2 X g − Scalg · ϕ = e · e · (D/ )2(ϕ) − e · ∇g (D/ (ϕ)) 2 n i i n i ei i i n − 2 = −(D/ g)2(ϕ) − (D/ g)2(ϕ) n 2(n − 1) = − (D/ g)2(ϕ) . n This proves the ﬁrst assertion. 1 Now, for the second relation (9.2), notice that the twistorial 2 -Ricci formula (9.4) can be re-written as follows: n h 1 1 i ∇g (D/ g(ϕ)) = X · (D/ g)2(ϕ) − Ricg(X) · ϕ . X n − 2 n 2 Hence, the result follows after a simple combination with the relation (9.1). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 63

Next we will show that the twistor equation can be viewed as a parallelism condition, with respect to a suitable covariant derivative on the bundle E := ΣgM ⊕ ΣgM. In particular, motivated by Theorem 9.1, one can consider a map ∇g,E : Γ(E) → Γ(T ∗M ⊗ E), defined by 1 n ∇g,E(ϕ ⊕ ϕ ) = ∇g ϕ + X · ϕ ⊕ − Schog(X) · ϕ + ∇g ϕ X 1 2 X 1 n 2 2 1 X 2 g ! ∇X ϕ1 (1/n)X · ϕ2 = , n g g − 2 Scho (X) · ϕ1 ∇X ϕ2 g g for any X ∈ Γ(TM) and ϕ1 ⊕ ϕ2 ∈ Γ(Σ M ⊕ Σ M). Lemma 9.3. ∇g,E defines a covariant derivative on the vector bundle E. Proof. The linearity of ∇g,E is obvious. We need only to check the rule g,E g,E ∇ (f(ϕ1 ⊕ ϕ2)) = df ⊗ (ϕ1 ⊕ ϕ2) + f∇ (ϕ1 ⊕ ϕ2), ∞ for some smooth function f ∈ C (M; R). But this is a simple consequence of the relation g g g ∇X (fϕi) = X(f)ϕi + f∇X ϕi = df(X) ⊗ ϕi + f∇X ϕi . Theorem 9.4. ([Fr89]) Any twistor spinor ϕ ∈ ker(Pg) satisfies the equation g,E g ∇X ϕ ⊕ D/ (ϕ) = 0 . Conversely, if (ϕ ⊕ ψ) ∈ Γ(E) is ∇g,E-parallel, then ϕ is a twistor spinor such that D/ g(ϕ) = ψ. g g 1 / g Proof. Consider a twistor spinor ϕ ∈ ker(P ). Then, ∇X ϕ + n X · D (ϕ) = 0, hence it is obvious that the section ϕ ⊕ D/ g(ϕ) ∈ Γ(E) is parallel under the connection ∇g,E. Conversely, assume that (ϕ ⊕ ψ) ∈ Γ(E) is a ∇g,E-parallel section of E. Then, by the definition of ∇g,E one concludes that the following needs to hold, 1 ∇g ϕ + X · ψ = 0, ∀ X ∈ Γ(TM) . X n g Replacing X by {ei}, multiplying with ei and summing, we get D/ (ϕ) = ψ, see also [BFGK89, p. 25] or [Gi09, p. 136]. Thus, a section ϕ ∈ Γ(ΣgM) is a twistor spinor, if and only if the section ϕ ⊕ D/ g(ϕ) of E = g g g,E g Σ M ⊕ Σ M is ∇ -parallel. Consequently, any element ϕ ∈ ker(P ) is defined by its values ϕx g and (D/ (ϕ))x at some point x ∈ M. Hence Corollary 9.5. Let (M n, g)(n ≥ 3) a connected Riemannian spin manifold. Then, n g [ 2 ]+1 a) The kernel of the twistor operator is a finite dimensional space, i.e. dimC ker(P ) ≤ 2 . b) If ϕ and D/ g(ϕ) vanish at some point x ∈ M and ϕ ∈ ker(Pg), then ϕ ≡ 0.

n g [ 2 ] Proof. Both are consequences of Theorem 9.4. Recall that rnkC(Σ M) = 2 and that parallel sections on vector bundles over connected manifolds are uniquely determined by their value at a single point. In fact, one can show that Corollary 9.5, b) makes sense even for a two-dimensional closed 2 2 manifold. Notice however that the space of twistor-spinors on R (or H ), is inﬁnite-dimensional (see [Gi09, Prop. A.2.3]). Proposition 9.6. ([Fr89, Prop. 2]) Let (M n, g)(n ≥ 2) be a connected Riemannian spin manifold. Then any zero point of a twistor spinor 0 6= ϕ ∈ ker(Pg) is isolated, i.e. the zero-set of ϕ is discrete. 64 IOANNIS CHRYSIKOS

9.2. On the relation of twistor spinors with Killing spinors. Based on the tools described above, one can establish a remarkable relation between twistor spinors and real Killing spinors. Proposition 9.7. ([Hab94]) Assume that ϕ ∈ Γ(ΣgM) is a non-trivial twistor spinor on a connected Riemannian spin manifold (M n, g) with n ≥ 3. If (M n, g) is Einstein with Scalg 6= 0, then ϕ is the sum of two Killing spinors. n g g Scalg Proof. Assume that (M , g) is Einstein with Scal 6= 0, i.e. Ric (X) = n X, for any vector ﬁeld X ∈ Γ(TM). In this case, the relation (9.2) becomes n h Scalg Scalg i Scalg ∇g D/ g(ϕ) = − X · ϕ + X · ϕ = − X · ϕ . (9.6) X 2(n − 2) n 2(n − 1) 4(n − 1) g On the other side, since Scal 6= 0, one may decompose the twistor spinor ϕ as ϕ = ϕ+ + ϕ−, with r 1 4(n − 1) 1 1 ϕ := ϕ ± D/ g(ϕ) = ϕ ± D/ g(ϕ) , ± 2 n Scalg 2 λ where s n Scalg λ := . 4(n − 1) n Based now on (9.6), it is not hard to see that ϕ± are Killing spinors of (M , g) with Killing number λ κ = ∓ n . Indeed, for any X ∈ Γ(TM) we have that 1 1 ∇g ϕ = ∇g ϕ ± ∇gD/ g(ϕ) X ± 2 X λ g (9.6) 1 1 1 Scal = − X · D/ g(ϕ) ± − X · ϕ 2 n λ 4(n − 1) 1 1 1 λ2 = − X · D/ g(ϕ) ± − X · ϕ 2 n λ n λ 1 = ∓ X · ϕ ± D/ g(ϕ) 2n λ λ = ∓ X · ϕ , n ± which proves the claim. Therefore, one can further show that Proposition 9.8. Let ϕ ∈ Γ(ΣgM) be a non-trivial twistor spinor on a connected Riemannian spin manifold (M n, g) with n ≥ 3. If |ϕ| is a non-zero constant, then (M n, g) is Einstein. Moreover, either Scalg = 0 and ϕ is parallel, or Scalg > 0 and ϕ is the sum of two Killing spinors. Proof. A direct calculation shows that locally we have that 2 2 g g |ϕ| |ϕ| Scal 2 Hess∇ (|ϕ|2)(X,Y ) = − R Ricg (X,Y ) + g (X,Y ) + (|D/ g(ϕ)|2) g (X,Y ), x n − 2 x 2(n − 1)(n − 2) x n2 x x (9.7) for any X,Y ∈ Γ(TM) and x ∈ M. If |ϕ| is a non-zero constant, then the left hand side of (9.7) vanishes and we deduce that n Scalg 2(n − 2)|D/ g(ϕ)|2 o Ricg = + g . 2(n − 1) n2|ϕ|2 Consequently, (M n, g) is Einstein. Solving the equation Scalg Scalg 2(n − 2)|D/ g(ϕ)|2 = + , n 2(n − 1) n2|ϕ|2 DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 65 one computes 4(n − 1)|D/ g(ϕ)|2 Scalg = ≥ 0 . n|ϕ|2 Hence, for Scalg > 0, our claim follows by Proposition 9.7, where the corresponding Killing spinors ϕ± have Killing number s 1 Scalg = ∓ ∈ . (9.8) κ 2 n(n − 1) R If Scalg = 0 identically, then the previous expression for Scalg requires D/ g(ϕ) = 0, and hence by g Exercise 8.4, one deduces that ∇ ϕ = 0. This ﬁnishes the proof. In the following section we will see that the Killing number κ given by (9.8), plays an important role in the theory of real Killing spinors.

10. Eigenvalues of the Dirac operator on compact manifolds In Riemannian geometry there are many results related with eigenvalue estimates for the Laplace operator, depending however on different geometric data. For example, by a result of A. Lichnerow- icz ([Lich58]) any non-zero eigenvalue of the Laplace operator ∆g on a closed Riemannian manifold (M n, g), satisfies nk λ ≥ , (10.1) n − 1 g n g where k := Ricmin > 0 denotes the infimum of the Ricci tensor on (M , g), i.e. Ric ≥ k. Here, nk g n the quantity n−1 should be viewed as the first nonzero eigenvalue of ∆ on the round sphere S , of constant curvature k/(n − 1). Moreover, by a result of M. Obata ([Ob62]), the equality case in (10.1) is obtained if and only if (M n, g) is isometric to this sphere. The situation for eigenvalue estimates for the Dirac operator D/ g : Γ(ΣgM) → Γ(ΣgM) on a closed Riemannian spin manifold (M n, g) is in general very different. The most general sharp lower bound for the Dirac spectrum has been proved by Th. Friedrich [Fr80] and depends on the scalar curvature of (M n, g) (a much weaker curvature invariant of (M n, g), than the Ricci tensor). This estimate is also related with the very important notion of Killing spinors. Another important estimate is the so-called “Hijazi’s inequality”, which is an improvement of Friedrich’s estimate, when one uses a conformal equivalent metric. In this chapter we present both these eigenvalue estimates and report some results concerning mainly with the equality case of Friedrich’s estimates and so with manifolds carrying real Killing spinors. More details on the topic can be found for instance in [BFGK89, Fr00, Gi09].

10.1. Lower eigenvalue estimates. First we present a lower estimate which occurs directly via the Schrödinger-Lichnerowicz formula and it was first described by A. Lichnerowicz in 1963. Although it depends on the scalar curvature of (M n, g), Thomas Friedrich proved in 1980 that it is not sharp. Lemma 10.1. ([Lich63]) On a compact Riemannian spin manifold (M n, g) the eigenvalue λ of any non-trivial eigenspinor ϕ ∈ Γ(ΣgM) of D/ g, satisfies 1 λ2 > Scalg , 4 min g g g where Scalmin := minx∈M Scal (x) is the infimum of the scalar curvature Scal : M → R. Hence, D/ g has no eigenvalues in the close interval 1q 1q [− Scalg , Scalg ] . 2 min 2 min 66 IOANNIS CHRYSIKOS

Proof. Integrating the Schrödinger-Lichnerowicz we see that any spinor field ϕ ∈ Γ(ΣgM) satisfies Z Z Z g 2 g g 2 g 1 2 g |D/ (ϕ)| dv = |∇ ϕ| dv + Scalg |ϕ| dv . M M 4 M If ϕ ∈ Γ(ΣgM) is such that D/ g(ϕ) = λϕ, then it follows that 1 λ2 − Scalg ≥ 0 4 min R g 2 g g g g (since M |∇ (ϕ)| dv ≥ 0). If equality holds, then Scal is constant (so Scalmin = Scal ) and it g / g 2 1 g must be ∇ ϕ = 0. Hence, D (ϕ) = 0 and it follows that λ = 4 Scal = 0, a contradiction. Lichnerowicz’s estimate combined with the Aityah-Singer index theorem (see Theorem 8.6), shows that any even-dimensional Riemannian spin manifold with positive scalar curvature must have vanishing topological index (see Corollary 8.9). Hence, as we have already mentioned in Section 8.2, on closed spin manifolds of even dimensions one can impose topological obstructions to the existence of metrics with positive scalar curvature. 10.2. Friedrich’s inequality. Let us pass now to the main result of this section, i.e. Friedrich’s estimate. Theorem 10.2. ([Fr80]) Let (M n, g)(n ≥ 2) be a compact Riemannian spin manifold (without boundary). Then, any eigenvalue λ of D/ g satisfies n λ2 ≥ Scalg . (10.2) 4(n − 1) min Hence, the first positive and negative eigenvalue of the Dirac operator D/ g satisfy s 1 n Scalg |λ | ≥ min . ± 2 n − 1

q n Scalg g 1 min / Moreover, if λ = ± 2 n−1 is an eigenvalue of D and ψ is the corresponding eigenspinor, then s 1 Scalg ∇g ψ = ∓ min X · ψ , X 2 n(n − 1) for any X ∈ Γ(TM). In this case the scalar curvature Scalg is constant. Of course, Friedrich’s inequality (10.2) is trivial if Scalg ≤ 0. However, it is a sharp estimate for Riemannian spin manifolds with Scalg > 0. The main idea of the original proof of Theorem 10.2, is based on a deformed covariant derivative on the spinor bundle ΣgM over M. This is given by f g ∇X ϕ := ∇X ϕ + fX · ϕ, ∞ for some real-valued function f ∈ C (M; R). Due to the properties of Cliﬀord multiplication and by Proposition 5.17, it follows that ∇f is metric, i.e. f f g g (∇X ϕ, ψ) + (ϕ, ∇X ψ) = (∇X ϕ, ψ) + (ϕ, ∇X ψ) + f (X · ϕ, ψ) + (ϕ, X · ψ) (4.5) g g = (∇X ϕ, ψ) + (ϕ, ∇X ψ) = X(ϕ, ψ) , for any X ∈ Γ(TM) and ϕ, ψ ∈ Γ(ΣgM). Let us consider the Laplace operator associated to ∇f , n f X f f X f ∆ := − ∇ei ∇ei − div(ei)∇ei . i=1 i We shall prove that DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 67

Lemma 10.3. On a Riemannian spin manifold (M n, g), the (spinorial) Laplace operators ∆f and ∆g are related by f g g 2 ∆ = ∆ − 2fD/ − gradg(f) + nf , where D/ g : Γ(ΣgM) → Γ(ΣgM) is the corresponding Dirac operator.

Proof. First note that given some orthonormal frame {ei} of TM, one has X g X g X g X g ∇ei ei = g(∇ei ei, ej)ej = − g(∇ei ej, ei)ej = − div (ej)ej. (10.3) i i,j ij j g g Hence, by combining ∇ei (fϕ) = ei(f)ϕ + f∇ei ϕ with Proposition 5.17, we have that X f f X g g ∇ei ∇ei ϕ = (∇ei + fei)(∇ei + fei)ϕ i i X g g g g 2 = ∇ei ∇ei ϕ + ∇ei (fei · ϕ) + fei · (∇ei ϕ) − f ϕ i X g g g g 2 = ∇ei ∇ei ϕ + ei · ei(f)ϕ + f(∇ei ei) · ϕ + 2fei · (∇ei ϕ) − f ϕ i X g g X X g X g 2 = ∇ei ∇ei ϕ + ei · ei(f)ϕ + f ∇ei ei · ϕ + 2f ei · ∇ei ϕ − nf ϕ i i i i (10.3) X g g X g / g 2 = ∇ei ∇ei ϕ + gradg(f) · ϕ − f div (ei)ei · ϕ + 2fD ϕ − nf ϕ. i i Moreover, for the second term of ∆f and by the deﬁnition of ∇f , one can write

X f X g X div(ei)∇ei ϕ = div(ei)∇ei ϕ + f div(ei)ei · ϕ. i i i f Hence, a combination of the last two equations with the deﬁnition of ∆ yields the result. Let us compute now the square of the operator (D/ g − f) (or in other words, the Weitzenböck type formula for the connection ∇f ). Proposition 10.4. ([Fr80]) For the deformed covariant derivative ∇f we have the relation 1 (D/ g − f)2 = ∆f + Scalg +(1 − n)f 2. (10.4) 4 Proof. By Proposition 6.6 (1), it follows that g 2 g g g 2 g 2 (D/ − f) = (D/ − f)(D/ − f) = (D/ ) − 2fD/ − gradg(f) + f , and by the Schrödinger-Lichnerowicz formula, this reduces to 1 (D/ g − f)2 = ∆g + Scalg −2fD/ g − grad (f) + f 2 . 4 g Thus, the claim follows after a combination with Lemma 10.3. We are ready now to prove Friedrich’s inequality. For this, let us also denote by n n f 2 X f 2 X g 2 |∇ ϕ| = |∇ei ϕ| = |∇ei ϕ + feiϕ| i=1 i=1 the (square) length of the 1-form ∇f ϕ. 68 IOANNIS CHRYSIKOS

Proof. (Proof of Theorem 10.2) Take ϕ ∈ Γ(ΣgM) such that D/ gϕ = λϕ. Making the deformation λ / g 2 λ(n−1) 2 parameter f equal to n , and since (D − f) ϕ = ( n ) ϕ, a direct computation shows that the equation (10.4) reduces to 2n − 1 λ 1 g λ ϕ = ∆ n ϕ + Scal ϕ. n 4 Thus, by considering the inner product and integrating over M we obtain that Z Z Z 2n − 1 g λ λ g 1 g g λ (ϕ, ϕ)dv = (∇ n ϕ, ∇ n ϕ)dv + Scal (ϕ, ϕ)dv n M M 4 M Z Z λ 2 g 1 g 2 g = |∇ n ϕ| dv + Scal |ϕ| dv . M 4 M which we can rewrite as Z 2n − 1 2 λ 2 1 g 2 g λ kϕk0 = k∇ n ϕk0 + Scal |ϕ| dv . n 4 M λ 2 g g Since k∇ n ϕk0 ≥ 0 and Scal ≥ Scalmin, we conclude that n − 1 1 λ2 kϕk2 ≥ Scalg kϕk2, n 0 4 min 0 i.e. n λ2 ≥ Scalg , 4(n − 1) min as required. If now this inequality holds as equality, i.e. there exists a spinor ﬁeld ϕ with D/ gϕ = λϕ g 2 n Scalmin g g for λ = 4(n−1) , then we must have Scal = Scalmin and

λ ∇ n ϕ = 0, which is equivalent to say that λ ∇g ϕ = − X · ϕ, X n λ for any X ∈ Γ(TM). This shows that ϕ is a Killing spinor with Killing number κ = − n , where q g 1 n Scalmin λ = ± 2 n−1 . Remark 10.5. (Alternative approaches to Friedrich’s inequality) For pedagogical reasons mainly, let us brieﬂy describe some alternative methods which yield Theo- rem 10.2. 1st alternative method. By integrating the Schrödinger-Lichnerowicz formula over a closed Riemannian spin manifold (M n, g), one has Z Z 1 Z |D/ g(ϕ)|2dvg = |∇gϕ|2 + Scalg |ϕ|2dvg. (10.5) M M 4 M g 2 1 / g 2 g 2 g 2 However, by Lemma 6.16, we know that |P ϕ| + n |D ϕ| = |∇ ϕ| . Hence, replacing |∇ ϕ| in the last equation, it follows that Z n n Z |D/ g(ϕ)|2 − Scalg |ϕ|2 dvg = |Pgϕ|2dvg. (10.6) M 4(n − 1) n − 1 M Assume now that ϕ ∈ Γ(ΣgM) is such that D/ gϕ = λϕ. Then, since |Pgϕ|2 ≥ 0, Friedrich’s inequality (10.2) follows easily by (10.6). Moreover, if (10.2) holds as equality, then the relation (10.6) implies Pg(ϕ) = 0. In this case, according to Exercise 8.21, we conclude that ϕ is a Killing spinor. The converse is also easy and based on the properties of real Killing spinors, see Proposition 8.23. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 69

2nd alternative method. Although almost the same idea, note that the Cauchy-Schwarz inequality yields that √ |D/ gϕ|2 ≤ n|∇gϕ|2, for any ϕ ∈ Γ(ΣgM). Then, one can use this inequality and (10.5) to obtain Theorem 10.2, see for example [Hij01]. n g For Riemannian spin manifolds (M , g) with Scalmin ≤ 0, Friedrich’s inequality can be improved in several different ways based on various techniques, see [Gi09, Chapter 3] for a detailed exposition. Another qualitative improvement of Theorem 10.2 occurs due to the conformal invariance of the Dirac operator (the reader can find details on the conformal invariance in [Hit74, Gi09], for instance). Theorem 10.6. (Hijazi’s inequality) On a compact Riemannian spin manifold (M n, g) any eigenvalue of the Dirac operator D/ g satisfies n λ2 ≥ inf (e2f Scalg) , (10.7) 4(n − 1) M where Scalg is the scalar curvature of (M, g = e2f g). Moreover, (10.7) holds as an equality, if and only if f is constant and the corresponding eigenspinor ϕ ∈ Γ(ΣgM) is a Killing spinor of (M n, g). 10.3. Compact manifolds with real Killing spinors. A combination of Theorem 10.2 and Proposition 8.23 shows that the compact manifolds admitting non-trivial Killing spinors, are just these compact manifolds of positive scalar curvature Scalg which have the smallest possible first g eigenvalue λ+ or λ− of the Dirac operator D/ . Hence, Corollary 10.7. On a compact Riemannian spin manifold with positive scalar curvature, a spinor g q n Scalg g / 1 min field ϕ ∈ Γ(Σ M) is an eigenspinor of D with eigenvalue λ = ± 2 n−1 , if and only if ϕ is a q g λ 1 Scalmin non-trivial Killing spinor with Killing number κ = − n = ∓ 2 n(n−1) . The most basic (and simple) example of compact manifolds carrying non-trivial Killing spinors n are the spheres (S , gcan), (n ≥ 1) endowed with their canonical structure. The following proposition declares that also the topology of manifolds with Killing spinors is somehow constrained. Proposition 10.8. ([Lich87, Hij86-1]) Let (M n, g) be a connected Riemannian spin manifold admitting a non-trivial Killing spinor with Killing number κ 6= 0. Then, the following hold: (1) There are no non-trivial parallel k-forms on M n, for any k 6= 0, n. (2) In the compact case any harmonic form η satisfies η · ϕ = 0. Thus for example, one results with the following important conclusion. Corollary 10.9. A connected Riemannian spin manifold manifold endowed with a Killing spinor cannot be a Kähler manifold. Classification of Riemannian manifolds admitting real Killing spinors. In dimensions n ≤ 4, only spaces of constant (sectional) curvature admit real Killing spinors (see [BFGK89, p. 35]). For example, in dimension four a classical result of Th. Friedrich states that (see also [Fr00, p. 120] or [BFGK89, pp. 34-35]). Theorem 10.10. ([Fr81]) A four-dimensional connected Riemannian spin manifold (M 4, g) admitting a non-trivial real Killing spinor with Killing number κ(6= 0), is isometric to a space of constant 2 sectional curvature 4κ . 4 4 In fact, since RP is not orientable, one concludes that (M , g) must be isometric with the sphere 4 (S , gcan) of radius 1/2|κ|. Notice that there is an analogous result in dimension n = 8 by Hijazi, see [Hij86-2]. In dimension n = 6 the first examples of compact manifolds admitting real Killing 70 IOANNIS CHRYSIKOS spinors were explained by Friedrich and Grunewald in [FrGr85]. In this article it was shown the flag manifolds 3 CP = SO(5)/ U(2) ' Sp(2)/ Sp(1) × U(1), and F := SU(3)/Tmax, which are both classical examples of homogeneous nearly Kähler manifolds, are manifolds with non-trivial real Killing spinors. Recall that (see Appendix, Section 11.2 for more details on almost Hermitian manifolds). Definition 10.11. A nearly Kähler manifold is an almost Hermitian manifold (M 2n, g, J) such that g (∇X J)X = 0, ∀ X ∈ Γ(TM). A nearly Kähler manifold which is not Kähler is usually referred to as a strict nearly Kähler manifold. Any such manifold is spin since the first Chern class of (M,J) vanishes. Moreover, in dimension 6 the metric g must be Einstein (an old result by Alfred Gray, see for example [Agr06]). Hence, for this dimension R. Grunewald improved the result above as folows. Theorem 10.12. ([Gr90]) Let (M 6, g) be a six-dimensional simply connected Riemannian spin manifold. Then M 6 admits a non-trivial Killing spinor if and only if it has an almost Hermitian structure which is strict nearly-Kähler. Let us recall now the notion of the so-called metric cone corresponding to a Riemannian manifold (M n, g).

n Deﬁnition 10.13. Let (M , g) be a Riemannian manifold and set Mc := M × R+. Then, the warped product metric on Mc is the Riemannian metric deﬁned by gˆ := dr2 + f 2(r)g, where r ∈ R+ and f = f(r) is a smooth function called the warping map. If f(r) = r, then the pair (M,c gˆ = dr2 + r2g) is called the Riemannian cone (or metric cone) of M. Basic results related to the curvature properties of the metric cone are summarized as follows:

Lemma 10.14. Consider a Riemannian manifold (M n, g) and let (M,c gˆ) be the corresponding metric cone. Then, gˆ is a Ricci flat metric if and only if g is an Einstein metric with Einstein constant n − 1. Exercise 10.15. Prove Lemma 10.14. In these terms, C. Bär proved that Theorem 10.16. ([Bär93]) Let (M 2n, g) be a compact, even-dimensional Riemannian spin manifold with a non-trivial Killing spinor. Assume that 2n 6= 6. Then, (M 2n, g) is isometric to the standard even-dimensional sphere S2n. This result is based on the so-called cone construction, where one of the main observations is given as follows. Proposition 10.17. Assume that (M, g) is a Riemannian spin manifold. Then, a spinor field g ψ ∈ Γ(Σ M) belongs to K(M, g)1/2, i.e. ψ is a Killing spinor field with Killing number κ = 1/2, if and only if the induced spinor field on the cone (M,c gˆ) is parallel with respect to the associated spinorial Levi-Civita connection. After this proposition, most of Bär’s conclusions occur in combination with Wang’s classification of all simply connected Riemannian spin manifolds admitting parallel spinors (see Theorem 8.14). Exercise 10.18. Prove Proposition 10.17. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 71

Real Killing spinors in odd dimensions are related with Sasakian structures, which form a special class of contact structures (see Appendix, Section 11.3 for a short introduction to contact metric structures). Recall that (see e.g. [B76, BG00]) Definition 10.19. 1) A Riemannian manifold (M, g) is called Sasakian if any of the following (equivalent) conditions hold: (α) There exists a Killing vector field ξ of unit length on M such that the tensor field Φ of type g (1, 1), defined by Φ(X) = −∇X ξ satisfies g (∇X Φ)(Y ) = g(X,Y )ξ − g(ξ, Y )X, ∀ X,Y ∈ Γ(TM) . (β) There exists a Killing vector field ξ of unit length on M so that the Riemannian curvature Rg satisfies Rg(X, ξ)Y = g(ξ, Y )X − g(X,Y )ξ , ∀ X,Y ∈ Γ(TM) . 2 2 (γ) The metric cone (Mc = M × R+, gˆ = dr + r g) is Kähler. 2) An Einstein-Sasakian manifold is a Sasakian manifold (M, g) such that Ricg = cg, for some 0 6= c ∈ R. The vector field ξ is the so-called Reeb vector field, and we denote by η its dual 1-form, which is a contact 1-form, η ∧ (dη)n 6= 0. Proposition 10.20. Any Einstein-Sasakian manifold (M 2n+1, ξ, η, Φ, g) has positive scalar curva- g ture (since Ric (X, ξ) = 2nη(X)) and thus when it is complete, M is compact and π1(M) must be finite. Moreover, in this case the cone (M,c gˆ) on M is a Kähler Ricci-flat (Calabi-Yau). In odd dimensions one of the very first established examples of a manifold carrying real Killing spinors was the 5-dimensional Stiefel manifold SO(4)/ SO(2) (see [Fr80]), which is in fact a homogeneous Einstein-Sasakian manifold. This result was generalized in the early of 90’s in [FrK90]. Theorem 10.21. ([FrK90]) Let (M 2n+1, g) be a simply connected Einstein-Sasakian manifold with spin structure and positive scalar curvature. Then (M 2n+1, g) admits real Killing spinors. A few years later C. Bär proved the converse of this theorem, based again on his correspondence and some holonomy arguments. Theorem 10.22. ([Bär93]) Let (M 2n+1, g) be an odd-dimensional Riemannian manifold admitting a real Killing spinor. Then, M 2n+1 admits an Einstein-Sasakian structure. Definition 10.23. A Riemannian manifold (M, g) is called 3-Sasakian, when its metric cone (M,c gˆ) is hyperkähler. Equivalently, a 3-Sasakian manifold is a Riemannian manifold (M, g) endowed with three Sasakian structures (ξi, ηi, Φi)(i = 1, 2, 3), which are compatible each other in the sense that

g(ξi, ξj) = δij , [ξi, ξj] = 2cijkξk i, j, k{1, 2, 3} . It is well-known that

Proposition 10.24. (see [BGM93, BG05]) Any 3-Sasakian manifold (M, gξi, ηi, Φi) has dimension 4n + 3, is spin and g is an Einstein metric of positive scalar curvature. Moreover, M admits a second Einstein metric which is not homothetic to g. 4n+3 n+1 Exercise 10.25. Prove that the sphere S , viewed as a hypersurface of H is a 3-Sasakian manifold. An odd-dimensional Einstein-Sasakian manifold admitting at least 3 real Killing spinors,5 can be endowed with a 3-Sasakian structure. In particular, by the work of Bär [Bär93] it follows that

5This may happen when the dimension is 4n + 3, n ≥ 1, see for example [Bär93, Gi09]. 72 IOANNIS CHRYSIKOS

Proposition 10.26. Let (M 4n+3, g)(n > 1) be a Riemannian manifold which admits Killing spinors. Then, either M is Einstein-Sasakian but not 3-Sasakian, and in this case there are two linearly independent 1/2-Killing spinors, or M is 3-Sasakian and in this case there are n+2 linearly independent 1/2-Killing spinors. The classiﬁcation of compact Riemannian manifolds with real Killing spinors ﬁnishes within the examination of the 7-dimensional case, which turns out to be the most complicated one. Indeed, in this dimension, and according to [FKMS97], beyond the Einstein-Sasakian manifolds (which are not 3-Sasakian), and the 3-Sasakian manifolds, there is also a third class of manifolds solving the Killing spinor equation, which consists of the so-called weak G2-manifolds or nearly-parallel G2-manifolds (we refer to our Appendix, Section 11.4 for a short summary of G2-structures).

Deﬁnition 10.27. A weak G2-manifold is a 7-dimensional manifold admitting a G2-structure ω ∈ 3 7 6 Ω+(M ) such that dω = c ? ω for some c ∈ R\{0} (and thus d ? ω = 0). The existence of real Killing spinors on such manifolds was proved in [FKMS97].

Proposition 10.28. ([FKMS97]) 1) The existence of a weak G2-structure on a compact 7-manifold (M 7, g) is equivalent to the existence of a spin structure carrying a real Killing spinor. g 3c2 2) Compact weak G2-manifolds are Einstein manifolds, Ric = 8 g, with positive scalar curvature. 7 Compact weak G2-manifolds (M , ω, g) admit also an equivalent description in terms of the metric ˆ 7 7 7 ˆ cone (M = M × R+, gˆ) on M . Since (M , ϕ, g) admits Killing spinors, (M, gˆ) admits parallel spinors and hence has holonomy group Hol(Mˆ ) ⊂ Spin(7). In particular, if (M 7, ϕ, g) is simply- connected and not isometric to the standard sphere, then the inclusions Sp(2) ⊂ SU(4) ⊂ Spin(7) yield the following three natural classes of weak G2-manifolds (for more we refer to [FKMS97, Bär93]) • If Hol(Mˆ ) = Sp(2), then M 7 is 3-Sasakian and it has a 3-dimensional space of Killing spinors. • If Hol(Mˆ ) = SU(4), then M 7 is Einstein-Sasakian and it has a 2-dimensional space of Killing spinors. 7 • If Hol(Mˆ ) = Spin(7), then M is a so-called proper weak G2-manifold, with 1-dimensional space of Killing spinors.

This class of G2-structures, completes the classiﬁcation of compact Riemannian manifolds inducing solutions of the Killing spinor equation.

11. Appendix 11.1. Differential operators. Here we collect a few basic facts of differential operators on vector bundles. Let π : E → M be vector bundle over a Riemannian manifold (M n, g). We consider a linear connection ∇E ≡ ∇ : Γ(E) → Γ(T ∗M ⊗ E) on E which is compatible with g, where as usual Γ(E) is the space of smooth sections of E. Definition 11.1. A (smooth) linear differential operator of order r on E is a linear operator L : Γ(E) → Γ(E) acting on sections of E, such that for any s ∈ Γ(E) and x ∈ M r L(s)(x) = F(x, s(x), ∇s(x),..., ∇ s(x)) ∈ Ex , −1 for some smooth function F, where Ex = π (x) is the fibre over x. n In some local coordinate system of (M , g), say (x1, . . . , xn), by trivializing E we get X ∂r L = La1...ar (x) + lower order terms, ∂xa . . . ∂xa |a|=r 1 r

6 Here ? is the Hodge operator corresponding to the metric g induced by the G2-structure ω, see also Appendix. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 73

a ...ar where the sum is over all multi-incides a = (a1, . . . , ar) of length |a| = r and each L 1 (x) is an endomorphism in End(Ex). By “lower order terms” we mean all the summands appearing in the local expression of L and involving partial derivatives of maximum degree r − 1. Definition 11.2. A differential operator P of order r on a Riemannian manifold (M n, g) is an operator taking real functions on M to real functions on M and depends on f : M → R and its first r derivatives ∇f, . . . , ∇rf (for example with respect to the Levi-Civita connection ∇ ≡ ∇g or any other linear connection on M). Thus P (f) ≡ P f is a real function on M given by P (f)(x) := F(x, f(x), ∇f(x),..., ∇rf(x)), where F is some function on M. Usually one requires that F is at least continuous in its arguments. If F is smooth then P is a smooth differential operator. If we have the relation P (af + bg) = aP (f) + bP (g) for any real number a, b and real functions f, g on M, then P is a linear operator.

Example 11.3. Given some local coordinate system (x1, . . . , xn) of x ∈ U ⊂ M, a general expression of a second-order differential operator on M reads by n n X ∂2f X ∂f P (f)(x) := c (x) + d (x) + b(x)f(x), i,j ∂x ∂x i ∂x i,j=1 i j i=1 i where ci,j, di, b are real functions given in terms of the local chart, with ci,j = cj,i. The algebra of differential operators on E, denoted by D(E) is the subalgebra of End(Γ(E)) generated by elements of Γ(M, End(E)) acting by multiplication on Γ(E) and the covariant derivative ∇X , where ∇ is any connection on E and X ranges over all vector fields of M. Here by Γ(M, End(E)) ' V(M; End(E)) we denote the space of End(E)-valued differential forms on M. The algebra D(E) has a natural filtration defined by ∞ D(E) = ∪i=0Di(E), Di(E) := Γ(M, End(E)) · span{∇X1 · · · ∇Xr : r ≤ i}.

Thus a differential operator of order i on a vector bundle E → M is an element in Di(E). Recall ∞ now that whenever A = ∪i=0Ai is a filtered algebra, i.e. Ai ⊂ Ai+1 with Ai ·Aj ⊂ Ai+j, one may define the associated graded algebra gr(A) ∞ X gr(A) = gri(A), gri(A) := Ai/Ai−1. i=0 P∞ P∞ The projection of the graded algebra i=0 Ai to gr(A), say ς : i=0 Ai → gr(A), is called the symbol map. Its components in degree i is the projection Ai → gri(A) and has kernel Ai−1. Let us apply this formalism on D(E) and denote by Sym•(TM) the bundle of symmetric tensors over M. Proposition 11.4. The associated graded algebra ∞ X gr(D(E)) = gri(D(E)), gri(D(E)) := Di(E)/Di−1(E) i=0 of the filtered algebra D(E) of differential operators is isomorphic to Sym•(TM) ⊗ End(E). The isomorphism i ςi : gri(D(E)) → Γ(M; Sym (TM) ⊗ End(E)) (11.1) is the so-called (principal) symbol of a differential operator L over E → M of order i, where ∗ 7 i := sup{m : ςm(L)(ξ) < ∞} for some covector ξ ∈ T M. Locally we get the expression

7One may identify Γ(M; Symk(TM) ⊗ End(E)) with the subspace of sections Γ(T ∗M; π∗ End(E)) which are polynomials along the ﬁbres of T ∗M. 74 IOANNIS CHRYSIKOS

Definition 11.5. The principal symbol of a linear differential operator L of order r is a bundle map ς(L): T ∗M → End(E) defined by

X a1...ar ς(L)x(ξx) = L (x)ξa1 (x) ◦ ... ◦ ξar (x) ∈ End(Ex) , |a|=r ∗ Pn j where ξx ∈ Tx M is a non-zero linear functional having the local expression ξ = j=1 ξjdx .

Note that ς(L)x is independent of the coordinate system on M, so it is well-deﬁned. In fact, given a linear diﬀerential operator L : Γ(E) → Γ(E) on E → M, its principal symbol ς(L): T ∗M ×E → E admits a coordinate-free expressions given as follows: Proposition 11.6. 1 −tf(x) tf ς(L)x(ξx)s(x) = lim e L(e s)(x) , t→∞ tr ∞ ∗ for any smooth section s ∈ Γ(E), where f ∈ C (U; R) is such that dfx = ξx ∈ Tx M, with U ⊂ M being an open neighbourhood of x ∈ M.

Exercise 11.7. If L1,L2 are two diﬀerential operators whose composition L1 ◦ L2 makes sense, prove that ς(L1 ◦ L2)(ξ) = ς(L1)(ξ) ◦ ς(L2)(ξ), for any non-zero ξ ∈ T ∗M. ∗ ∗ A diﬀerential operator of order r is called elliptic if the section ςr(L) ∈ Γ(T M; π End(E)) is invertible over the open set {(x, ξ), ξ 6= 0}. In other words

Definition 11.8. (1) If the linear map ς(L)x(ξx) is invertible at any point x ∈ M and for any ∗ non-zero linear functional ξx ∈ Tx M, then the differential operator L is said to be elliptic. (2) Whenever the order r is even, L is called strongly elliptic if there exists a real constant c > 0 such that g(ς(L)(ξ)(s), s) ≥ c|ξ|r|s|2 for any non-zero ξ ∈ T ∗M and s ∈ Γ(E). Exercise 11.9. For dim M > 1 prove that an elliptic operator has necessarily even degree. Finally, be aware that if the coefficients of an elliptic operator L are smooth, and L(s) is smooth, then also s is smooth (this is the so-called elliptic regularity). Therefore, if the order of L is greater than zero, then the kernel and all other eigenspaces of L consist of smooth sections. Example 11.10. Consider the 2nd order operator P described in Example 11.3. Then, at each point x ∈ M, the leading coefficients of P are the n × n symmetric matrices ci,j(x). The condition Pn i for P to be elliptic is expressed as follows: ci,jξiξj 6= 0 whenever ξ = i=1 ξidx 6= 0 is a non-trivial linear functional on TM. Thus, P is elliptic if and only if the eigenvalues of the matrix ci,j are either positive or negative. 11.2. Symplectic, complex and hypercomplex manifolds. In this paragraph we recall definitions and a few basic results related with symplectic, (almost) complex and (almost) hypercomplex manifolds. Details for this part, can be found in [KN69, Joy00], for instance. Definition 11.11. A symplectic manifold is a smooth manifold endowed with a globally defined non-degenerate closed 2-form ω ∈ Ω2(M). Such a 2-form ω is called a symplectic form. Because a symplectic form ω is a smooth section of V2 T ∗M, it defines a skew-symmetric bilinear form ωx : TxM × TxM → R on the tangent space TxM of any point x ∈ M. Thus, a symplectic manifold must be even-dimensional. Moreover, on symplectic manifold (M 2n, ω), ω is of maximal rank which means ωn 6= 0. This induces a volume form on M 2n and thus M 2n is orientable. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 75

2n P Obviously, R = {(x1, . . . , xn, y1, . . . , yn): xi, yi ∈ R} endowed with the 2-form ω0 = i dxi ∧ dyi is the simplest example of a symplectic manifold. The 2-sphere S2 and the 2-torus T 2 = S1 × S1 are also symplectic, in particular any oriented surface is a 2-dimensional symplectic manifold, with symplectic form the volume form. Exercise 11.12. Prove that on a symplectic manifold (M 2n, ω), there is a reduction of the frame bundle F(M) to the Lie group Sp(2n; R) = Aut(ω) ⊂ GL2n R. Which is the corresponding Sp(2n; R)- principal bundle over M in this case? (Note that Sp(2n; R) is a non-compact Lie group, e.g. Sp(2; R) = SL2R. Thus it must not be confused with the compact symplectic group Sp(2n)). In the symplectic case, the theorem of Darboux states that, locally, any symplectic structure 2n looks like the standard symplectic structure ω0 on R . 2n Lemma 11.13. (Darboux) On a symplectic manifold (M , ω) there exist coordinates (xi, yi : i = 1, . . . , n) in the neighbourhood of each point x ∈ M, with respect to which ω takes the following canonical form 2n X ω = dxi ∧ dyi . i=1 Remark 11.14. A pair (M 2n, ω) of a 2n-dimensional smooth manifold and a non-degenerate global 2-form ω ∈ Ω2(M) is called an almost symplectic manifold. Definition 11.15. An almost complex structure on a smooth manifold M is a tensor field J : TM → TM of type (1, 1), such that 2 J = − IdTM . Then, the pair (M,J) is referred to as an almost complex manifold. It is easy to see that an almost complex manifold (M,J) is again of even dimension and admits a natural orientation induced by J. In particular, the tangent space of M at x ∈ M is a 2n- dimensional real vector space and the almost complex structure induces at any x ∈ M a linear 2 endomorphism Jx ∈ End(TxM), such that Jx = − IdTxM . Then, one can always pick n vectors {u1, . . . , un} in TxM, such that {u1, . . . , un,Jx(u1),...,Jx(un)} is a basis of TxM and the 2n-form u1 ∧ Jx(u1) ∧ ... ∧ un ∧ Jx(un) induces a volume form. n Example√ 11.16. Consider the complex vector space C = {(z1, . . . , zn): zi ∈ C} and let us set n 2n zi = xi + −1yi for i = 1, . . . , n. Then C is identified with R via the map (z1, . . . , zn) 7→ (x1, . . . , xn, y1, . . . , yn). Moreover, the endomorphism 2n 2n J0 : R → R ,J0(x1, . . . , xn, y1, . . . , yn) = (y1, . . . , yn, −x1,..., −xn) 2n 0 In defines an (almost) complex structure on R . We can identify J0 with the matrix −In 0 and then, the complex general linear group GLn C can be identified with the subgroup of GL2n R consisting of matrices which commute with J0. Exercise 11.17. Prove that there is a bijective correspondence between the set of (almost) complex 2n structures on R and the homogeneous space GL2n R/ GLn C. Exercise 11.18. Let M 2n be a 2n-dimensional smooth manifold. Prove that there is a natural bijective correspondence between the set of almost complex structures on M and reductions of the structure group of the frame bundle F(M) over M, to the Lie group GLn C. Proposition 11.19. An even-dimensional manifold M 2n has an almost symplectic structure if and only if has an almost complex structure. 76 IOANNIS CHRYSIKOS

This yields that Corollary 11.20. Any symplectic manifold is an almost complex manifold. Of course not every even-dimensional smooth manifold admits a symplectic structure. Example 11.21. Consider the 4-sphere S4 and let us assume that ω is a symplectic form on S4. 2 4 Because the second cohomology group H (S , R) is trivial, every closed 2-form must be exact, hence there is some 1-form α such that ω = dα. For the volume form Ω = ω ∧ ω we compute d(ω ∧ α) = dω ∧ α + ω ∧ dα = ω ∧ ω = Ω , so Ω is also exact. But then, Stoke’s theorem yields that Z Z Z Ω = d(ω ∧ α) = ω ∧ α = 0 , S4 S4 ∂ S4 which gives rise to a contradiction, since Ω is a volume form. Thus S4 does not admit a symplectic 2 2n structure. In fact, because H (S , R) = 0 for any n > 1, it follows that the unique sphere admitting a symplectic structure is S2.

Let (M,J) be an almost complex manifold and for any x ∈ M let us extend Jx ∈ End(TxM) C linearly to the complexified tangent space Tx M := TxM ⊗R C. Then, there is a decomposition of the form C 1,0 0,1 Tx M = Tx M ⊕ Tx M where 1,0 C 0,1 C Tx M = {X ∈ Tx M : JxX = iX} ,Tx M = {X ∈ Tx M : JxX = −iX} , are the eigenspaces of Jx corresponding to the eigenvalues ±i. In terms of vector bundles, this yields a decomposition of T CM into two subbundles, whose fibres (over x ∈ M) are the previous eigenspaces, T CM = T 1,0M ⊕ T 0,1M. Example 11.22. Any complex manifold M 2n is equipped with a natural almost complex structure. Indeed, we can consider a local holomorphic coordinate√ system (z1, . . . , zn) in an open set U ⊂ M containing a given point x ∈ M and write zi = xi + −1yi (note that zi are the composition of n the holomorphic charts with the standard coordinates on C ). Then, (x1, . . . , xn, y1, . . . , yn) is a 1,0 0,1 smooth chart in U and moreover, the subbundles TM and TM of TM ⊗ C are generated by the vectors { ∂ } and { ∂ } , respectively, where ∂zi 1≤i≤n ∂zi 1≤i≤n ∂ 1 ∂ √ ∂ ∂ 1 ∂ √ ∂ := − −1 , := + −1 . ∂zi 2 ∂xi ∂yi ∂zi 2 ∂xi ∂yi Define now a linear map J by

J(∂/∂xi) := ∂/∂yi ,J(∂/∂yi) := −∂/∂xi, i = 1, . . . , n . It is easy to see that the definition of J is independent of the choice of local coordinates and hence 2 J induces a bundle map from TM to TM (which we denote again by J), satisfying J = − IdTM . Therefore, J is an almost complex structure on M. Moreover, the fibres of the subbundles TM 1,0 0,1 and TM are just the eigenspaces of the linear extension of J on TM ⊗ C. This is given by ∂ √ ∂ ∂ √ ∂ J( ) = −1 ,J( ) = − −1 , ∂zi ∂zi ∂zi ∂zi 2n for each 1 ≤ i ≤ n. Note also that√ for any (real) vector field X ∈ Γ(TM ) on the underlying real manifold M 2n, the vectors X ∓ −1JX are sections of TM 1,0 and TM 0,1, respectively. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 77

Definition 11.23. The Nijehnius tensor of an almost complex structure J : TM → TM is defined to be the following tensor field of type (1, 2),

NJ (X,Y ) := 2([JX,JY ] − [X,Y ] − J[X,JY ] − J[JX,Y ]) , with X,Y ∈ Γ(TM). An almost complex structure on M 2n is said to be integrable, if it is the one induced by a complex structure. The famous theorem of Newlander and Nirenberg states that Theorem 11.24. (Newlander and Nirenberg) An almost complex structure J is integrable, if and only if NJ ≡ 0 identically. n Exercise 11.25. Show that the complex projective space CP is a complex manifold of even dimension 2n. As one can expect by the relation of symplectic manifolds with almost complex manifolds, not every even-dimensional real manifold admits an almost complex structure; such an example is the 4-dimensional sphere S4. On the other hand, for the 6-dimensional sphere S6 (which is also a non-symplectic manifold), is still unknown if it admits an integrable almost complex structure. Deﬁnition 11.26. A Riemannian metric g on an almost complex manifold (M 2n,J) is called compatible with J if it is invariant by J, i.e., g(JX,JY ) = g(X,Y ) , (11.2) for any X,Y ∈ Γ(TM). An almost Hermitian manifold (M 2n, g, J) consists of an almost complex manifold (M 2n,J) endowed with a compatible Riemannian metric g with J.

The condition (11.2) means that Jx is an isometry of TxM, for any x ∈ M. Moreover, it implies that the structure group SO(2n) of the corresponding principal bundle of positively oriented orthonormal frames Pg = SO(M, g) is reduced to the unitary group U(n) ⊂ SO(2n). For example, it 6 is well-known that S = G2 / SU(3) admits a unique almost Hermitian structure, which is actually G2- invariant and hence its structure group reduces to U(3). Notice however that this almost Hermitian structure is not integrable. Given an almost Hermitian manifold (M 2n, g, J), it is easy to see that g(X,JY ) = −g(JX,Y ), for any X,Y ∈ Γ(TM). Therefore, one can deﬁne a non-degenerate 2-form ω on M, the so called fundamental 2-form ω, via the rule ω(X,Y ) = g(JX,Y ) , for any X,Y ∈ Γ(TM). Then, by (11.2) it follows that ω(JX,JY ) = ω(X,Y ), for any X,Y ∈ Γ(TM). Deﬁnition 11.27. 1) A Hermitian manifold is an almost Hermitian manifold (M 2n, g, J) with integrable J, i.e. NJ ≡ 0. 2) A Kähler manifold is a Hermitian manifold (M 2n, g, J) whose fundamental 2-form ω is closed. In this case g is called a Kahler metric. Thus, Kähler manifolds induce a large category of examples of symplectic manifolds. Proposition 11.28. Given an almost Hermitian manifold (M 2n, g, J) the following conditions are equivalent: 1) (M 2n, g, J) is Kähler, 2) ∇gJ = 0, 3) ∇gω = 0, 4) Hol(g) is contained in U(n) and J is associated to the corresponding U(n)-structure. 78 IOANNIS CHRYSIKOS

Consider an almost complex manifold (M 2n,J) and let us denote by r,s r,0 0,s r+s ∗ Λ (M) := Λ (M) ⊗ Λ (M) ⊂ Λ (T M) ⊗ C the bundle of complex valued forms of type (r, s), where Λr,0(M) := Λr(T ∗M 1,0), Λ0,s(M) := Λs(T ∗M 0,1) are the bundles of holomorphic and anti-holomorphic forms, respectively. Deﬁnition 11.29. The complex line bundle n,0 n ∗ 1,0 KM := Λ (M) = Λ (T M ) is referred to as the canonical line bundle.A holomorphic volume form θ is a non-vanishing holomorphic section of of KM . By the theory described in Section 3.2 it follows that a holomorphic volume form θ exists if and only if KM is trivial. Now, for spin structures the following characterization given by Atiyah ([At71]) makes sense. Proposition 11.30. ([At71]) An almost complex manifold (M 2n,J) admits a spin structure if and only if KM admits a square root, i.e. there exists a complex line bundle L such that ⊗2 L = KM . In order to examine spin structures on a Kähler manifold of course one can apply Propositions 11.30 and 3.42. For example

Example 11.31. Let Σh be a compact Riemannian surface of genus h. Then Σg is a Kähler manifold, with ﬁrst Chern class given by

c1(Σh) = χ(Σh) = 2 − 2h . Here, χ denotes the Euler characteristic. Hence, any Riemann surface is a spin manifold. In particular, since 1 2h H (Σh; Z2) ' Hom(π1(Σh), Z2) ' Z2 , we conclude that there are 22h spin structures. For a paremetrization of these spin structures in terms of holomorphic square roots of KΣh , we refer to [LM89]. Deﬁnition 11.32. A Calabi-Yau manifold is a compact Kähler manifold (M 2n, g, J)(n ≥ 2) with Riemannian holonomy group Hol(g) = SU(2n). As we explained in Section 8.3, Calabi-Yau manifolds are compact Ricci-ﬂat Kähler manifolds, i.e. the Ricci tensor associated to the Kähler metric vanishes. Moreover, it is easy to see that Proposition 11.33. Any Calabi-Yau manifold (M 2n, g, J) is spin.

2n Proof. Since (M , g, J) is Kähler and Hol(g) = SU(2n), the canonical line bundle KM is trivial. Thus, the ﬁrst Chern class vanishes, c1(M,J) = 0 and we conclude by Proposition 3.42. Hence, equivalently a Calabi-Yau manifold is a compact Kähler manifold (M 2n, g, J) (2n ≥ 4) with trivial canonical bundle. Examples and more details of this special type of Kähler manifolds can be found, for instance, in [Joy00]. Let us ﬁnally discuss hypercomplex structures on 4n-dimensional manifolds (for more details see [AM, Joy00]). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 79

Deﬁnition 11.34. Let M be a smooth manifold. An almost hypercomplex structure on M consists of three globally deﬁned almost complex structures I, J, K ∈ End(TM) which satisfy the quaternion relations, I2 = J 2 = K2 = IJK = −1 . We usually denote such a structure by H = (I, J, K) and call the pair (M, H ) an almost hypercomplex manifold. If I, J, K are all integrable complex structures, then (M, H ) is called a hypercomplex manifold.

n Any almost hypercomplex structure H on M establishes an isomorphism TxM ' H for any x ∈ M. It follows that M must be a 4n-dimensional (real) manifold. Equivalently, an almost hypercomplex structure on a 4n-dimensional smooth manifold M can be viewed as reduction of its frame bundle to the group G = GLn H = Aut(H ) ⊂ GL4n R. For the integrable case we mention the following important result of Obata.

Proposition 11.35. ([Ob56]) On a hypercomplex manifold (M 4n, H ) there exists a unique affine torsion-free connection ∇ such that ∇I = ∇J = ∇K = 0 . Definition 11.36. An almost hyper-Hermitian structure on M 4n consists of an almost hypercomplex structure H = (I, J, K) and a Riemannian metric g which is compatible with any of the three almost complex structures, i.e. g(IX,IY ) = g(JX,JY ) = g(KX,KY ) = g(X,Y ) , ∀ X,Y ∈ Γ(TM) . Then the triple (M 4n, g, H ) is referred to as an almost hyper-Hermitian manifold. When H is a hypercomplex structure then the pair (g, H ) is said to be a hyper-Hermitian structure and similarly (M 4n, g, H ) is called a hyper-Hermitian manifold. Exercise 11.37. Prove that an almost hyper-Hermitian structure induces a Sp(n)-structure on M 4n, i.e. a reduction of the structure group of its frame bundle F(M) to Sp(n) = Aut(H , g) ⊂ GL4n R (and this is an equivalent definition of such structures). Given an almost hyper-Hermitian manifold, there are three globally defined non-degenerate 2- 2 forms ωI , ωI , ωJ , ωK ∈ Ω (M) on M, explicitly given by

ωI (X,Y ) = g(X,IY ) , ωJ (X,Y ) = g(X,JY ) , ωK (X,Y ) = g(X,KY ) . Deﬁnition 11.38. A hyper-Kähler manifold is an almost hyper-Hermitian manifold (M 4n, g, H ) whose almost hypercomplex structure H is parallel with respect to ∇g, i.e. ∇gI = ∇gJ = ∇gK = 0 . It follows that on a hyper-Kähler manifold the three almost complex structures are integrable, in particular the Obata connection coincides with the Levi-Civita connection.

n Example 11.39. The simplest example of a hyper-Kähler manifold is the ﬂat space H . Proposition 11.40. An almost hyper-Hermitian manifold (M 4n, g, H ) is hyper-Kähler, if and only if one of the following equivalent conditions is satisﬁed: g g g 1) ∇ ωI = ∇ ωJ = ∇ ωK = 0, 2) dωI = dωJ = dωK = 0, 3) Hol(g) ⊆ Sp(n) and I, J, K are the induced complex structures.

Corollary 11.41. A hyper-Kähler manifold (M 4n, g, H ) is Ricg-ﬂat. 80 IOANNIS CHRYSIKOS

11.3. Contact and Sasakian manifolds. In Section 10.3 we introduced Sasakian manifolds as special examples of contact metric manifolds. Here our aim is to present preliminaries of almost contact metric structures. Basic references on this topic are the books [B76, B10], see also [BG00, BG05]. Definition 11.42. A (2n + 1)-dimensional manifold M 2n+1 is called a contact manifold if there exists a 1-form η such that η ∧ (dη)n 6= 0. Any contact manifold is orientable since the non-zero form η ∧ dη . . . ∧ dη defines a volume form. Let us denote by D := ker η ⊂ TM the contact distribution defined by the subspaces

Dx := ker ηx = {X ∈ TxM : η(X) = 0} . The contact condition means that D is maximally non integrable. Hence one deduces that on any contact manifold there exists a unique (global) vector ﬁeld ξ, called the Reeb vector ﬁeld, such that

ξyη = η(ξ) = 1, dη(ξ, X) = 0 , for any X ∈ Γ(TM).

2n+1 Exercise 11.43. Let R be the Euclidean space with local coordinates (x1, y1, . . . , xn, yn, z). P 2n+1 Then, the 1-form η = dz− i xidyi induces a contact structure on R , which we call the standard ∂ one. In this case, the Reeb vector ﬁeld coincides with ξ = ∂z . Which is the the contact distribution D in this case? In the contact case, the theorem of Darboux states that locally any contact structure looks like 2n+1 the standard one on R . Lemma 11.44. (Darboux) Let (M 2n+1, η) be a contact manifold and x ∈ M a point of M. Then, there exist local coordinates (U, x1, . . . , xn, y1, . . . , yn, z) centered at x ∈ M, with respect to which X η = dz − xidyi. i

3 Figure 5. The standard contact structure on R DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 81

2n+1 2n+2 Exercise 11.45. Prove that the sphere S ⊂ R admits a contact structure given by the contact form 2n+1 X η = (xjdyj − yjdxj) . j=1 Definition 11.46. An almost contact structure on a (2n + 1)-dimensional manifold M 2n+1 consists of a (1, 1)-tensor field Φ and a vector field ξ with dual 1-form η (η(ξ) = 1), such that 2 Φ = − IdTM +η ⊗ ξ. The geometric meaning of an almost contact structure is that the ξ defines a preferred direction on M 2n+1, where Φ2 vanishes, while Φ behaves as an almost complex structure on any linear complement of ξ. It is easy to see that Φ(ξ) = 0. Definition 11.47. An almost contact metric manifold (M 2n+1, g, ξ, η, Φ) consists of an almost contact manifold (M 2n+1, ξ, η, Φ) endowed with a Riemannian metric g which is compatible with Φ, in the sense that g(Φ(X), Φ(Y )) = g(X,Y ) − η(X)η(Y ), for any X,Y ∈ Γ(TM). On an almost contact metric manifold (M 2n+1, g, ξ, η, Φ) the Reeb vector field ξ is of unit length with respect to g and moreover the 1-form η is such that η(X) = g(X, ξ). Exercise 11.48. Prove that the structure group of an almost contact metric manifold (M 2n+1, g, ξ, η, Φ) is U(n) × 1. Example 11.49. Consider a 2n-dimensional almost Hermitian manifold (B2n, g,ˆ J). Then, the 2n+1 2n 2 product manifold M = B × R endowed with the product metric g =g ˆ + d t , admits an almost contact metric structure given by ∂ ∂ ξ = (0, ), η = d t, φ(X, f ) = JX, ∂t ∂t 2n ∂ for some smooth function f : B × R → R and vector field X ∈ Γ(TB). Here, we write (X, f ∂t ) for a vector field on M 2n+1. Note that a similar construction carries out over B2n × S1. In a line with almost Hermitian manifolds, on an almost contact metric manifold (M 2n+1, g, ξ, η, Φ) the metric g satisfies the relation g(Φ(X),Y ) = −g(X, Φ(Y )), for any X,Y ∈ Γ(TM). Thus, one defines a global 2-form F on M via the rule F (X,Y ) = g(X, Φ(Y )) . This again is called the fundamental 2-form of (M 2n+1, g, ξ, η, Φ). Moreover, we see that Theorem 11.50. Let (M 2n+1, η) be a contact manifold with Reeb vector field ξ. Then, there exists an almost contact metric structure (Φ, ξ, η, g) such that F (X,Y ) = g(X, Φ(Y )) = dη(X,Y ). Let (M 2n+1, g, ξ, η, Φ) be an almost contact metric manifold. The Nijenhuis tensor associated to the endomorphism Φ: TM → TM is given by 2 NΦ(X,Y ) := [Φ(X), Φ(Y )] + Φ ([X,Y ]) − Φ([Φ(X),Y ]) − Φ([X, Φ(Y )]) + dη(X,Y )ξ . Definition 11.51. An almost contact metric manifold (M 2n+1, g, ξ, η, Φ) is said to be 1) normal, if NΦ = 0, identically. 2) contact metric, if F = dη. 3) Sasakian, if it is normal and contact metric. 82 IOANNIS CHRYSIKOS

It follows that dF = 0 for a Sasakian manifold, and hence such manifolds should be viewed as the odd-dimensional analogue of Kähler manifolds. Exercise 11.52. Show that the two Definitions 10.19 and 11.51 given for a Sasakian manifold are equivalent. 2n+1 P Example 11.53. Consider R endowed with the contact form η = dz − i yidxi and Reeb vector field ξ = ∂/∂z. Define a metric g by the rule n 1 X g = η ⊗ η + (dx2 + dy2) , 2 i i i=1 and an endomorphism Φ by ∂ ∂ ∂ ∂ ∂ ∂ Φ( ) = + xi , Φ( ) = − , Φ( ) = 0 . ∂xi ∂yi ∂z ∂yi ∂xi ∂z 2n+1 Then (R , g, ξ, η, Φ) is a Sasakian manifold.

11.4. G2-structures. Let us discuss now G2-structures ([Br87, FKMS97, Joy00, Br06] are our main references for this part). Consider the 8-dimensional algebra of octonions O, also called the Cayley numbers, and let {e0, . . . , e7} be the canonical basis {1, i, j, k, , i, j, k}. Using this basis, we may identify O with 8 V2 8 ∗ R . The Lie algebra so(8) is identiﬁed with so(8) = (R ) , and the automorphism group of O is the 14-dimensional compact simple Lie group G2. The orthogonal complement of the unit 7 1 coincides with R and this yields a 7-dimensional irreducible representation of G2, which is the i standard representation. Let us denote by e the dual 1-forms corresponding to the vectors ei and abbreviate ei1 ∧ ... ∧ eip by ei1···ip . Consider the following 3-form 123 145 167 246 257 356 347 ω0 := e + e − e + e + e − e + e . (11.3)

Then one can prove that ω0 has stabilizer Gω0 ⊂ GL7(R) the Lie group G2. Note that G2 is a subgroup of SO(7). 7 Deﬁnition 11.54. A G2-structure on a 7-manifold M is a reduction of the structure group of its frame bundle from GL7 R to G2.

As G2 is the stabilizer of the 3-form ω0, such a reduction is characterized by the existence of 3 a globally defined 3-form ω ∈ Ω (M) which can be pointwise identified with ω0 by means of an 7 isomorphism u : TxM → R . Note that ω0 is a positive stable 3-form à la Hitchin [Hit01], which V3 7 ∗ means a 3-form belonging to an open orbit of the natural action of GL7(R) on (R ) and which 7 V3 7 ∗ ∗ induces an Euclidean metric on R . We denote this open orbit by +(R ) = {A ω0 : A ∈ GL7 R}, 7 3 V3 and for a manifold M we shall write Ω+(M) = Γ +(M) for the corresponding set of positive stable differential 3-forms. It is well-known that Proposition 11.55. On a 7-dimensional manifold M 7 there is a bijective correspondence between 3 elements in Ω+(M) and G2-structures. 3 For any ω ∈ Ω+(M) we shall denote by g = gω the induced Riemannian metric and by ? : Vp ∗ V7−p ∗ Tx M → Tx M the associated Hodge-star operator. Because the Lie group G2 is both connected and simply connected, one has the following topological obstruction (see for example [FKMS97, Joy00]).

Proposition 11.56. On a connected 7-manifold M the existence of a G2-structure is equivalent with the existence of a spin structure and hence the vanishing of the ﬁrst and the second Stiefel-Whitney class of M. DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 83

G2-structures can be classified in algebraic terms by considering the so-called intrinsic torsion. g g This is identified with ∇ ω, where ∇ is the Levi Civita connection of g = gω, and it is completely determined by the exterior derivatives dω and d ? ω (see [Br06, Joy00, Agr06] for more details). Proposition 11.57. When both ω and ?ω are closed, the intrinsic torsion vanishes identically 7 and the Riemannian holonomy group Hol(g) is a subgroup of G2. In this case (M , g, ω) admits a g ∇ -parallel spinor and hence g = gω is Ricci-flat.

Whenever we are in the situation of Proposition 11.57, the G2-structure is said to be parallel, or g torsion-free, or integrable, since it follows that ∇ ω = 0. Otherwise, a G2-structure is called non- integrable. Non-integrable G2-structures fall into four basic classes X1, X2, X3, X4, which correspond 7 ∗ ⊥ to the irreducible submodules of the space X := (R ) ⊗g2 under the G2-action, where one identifies ⊥ g ⊂ so(7) with the standard 7-dimensional representation of G2. We list these classes as follows: • Class X1: nearly parallel or weak G2-structure, dω = c ? ω for some 0 6= c ∈ R, • Class X2: closed or calibrated G2-structure, dω = 0, • Class X3: balanced G2-structure, d ? ω = 0 and dω ∧ ω = 0, • Class X4: locally conformally parallel G2-structure, dω = (3/4)θ ∧ ω and d ? ω = θ ∧ ?ω for some 1-form θ, which is the so-called Lee form of the G2-structure. 11.5. Spin(7)-structures. We close this appendix by devoting a very short section to 8-dimensional manifolds, whose frame bundle admits a reduction to the Lie group Spin(7) ⊂ SO(8). For more details we refer to [Fer82, Br87, LM89, Joy00]. 8 7 Fix again the vector space R = R ⊕ R , denote by {e0, . . . , e7} its canonical basis and consider the 4-form 0 Φ0 = e ∧ ω0 + ?ω0 = e0123 + e0145 − e0167 + e0246 + e0257 + e0347 − e0356 +e4567 + e2367 − e2345 + e1357 + e1346 − e1247 + e1256 , 7 where ω0 is the 3-form on R defined by (11.3). Φ0 has stabilizer GΦ0 ⊂ GL8 R the 21-dimensional compact Lie group Spin(7), i.e. the double covering of SO(7). Spin(7) is a subgroup of SO(8), since 8 1 01234567 it preserves both the Euclidean inner product h , i on R and the volume form 14 Φ0 ∧Φ0 = e (and so the induced orientation). Definition 11.58. A Spin(7)-structure on a 8-dimensional manifold M is a reduction of the structure group of its frame bundle from GL8 R to Spin(7).

As Spin(7) is the stabilizer of the 4-form Φ0, such a reduction is characterized by the existence 4 of a globally deﬁned 4-form Φ ∈ Ω (M) which can be pointwise identiﬁed with Φ0 by means of an 8 isomorphism u : TxM → R . Let us denote the set of such 4-forms by Ax(M) with x ∈ M, and F by A(M) = x∈M AxM the corresponding bundle. A section of A(M) is called admissible and one can prove that Proposition 11.59. On a 8-dimensional manifold M there is a bijective correspondence between admissible 4-forms Φ ∈ ΓA(M) and Spin(7)-structures. In fact by the inclusion Spin(7) ⊂ SO(8) we see that any admissible form Φ gives rise to a Riemannian metric gΦ and to an orientation on M (and so it induces a Hodge operator, which we denote by ∗. Notice also that Φ is self-dual, ∗Φ = Φ). 8 Example 11.60. The Lie group G2 is the stabiliser in Spin(7) of a non-zero vector e0 ∈ R . For 8 7 1 8 7 7 this reason, products of the form M = N × S , or M = N × R, where N is endowed with a 3 G2-structure ω ∈ Ω+(N), admit a natural Spin(7)-structure induced by the 4-form Φ := η ∧ ω + ?ω , 84 IOANNIS CHRYSIKOS

1 where η is a non-trivial 1-form on S , or R. V4 8 ∗ Remark 11.61. By dimension counting, the GL8 R-orbit of Φ0 is not open in (R ) . Conse- quently, an admissible 4-form Φ is not stable in the sense of Hitchin, and this differs significantly from the case of G2-structures on 7-manifolds, which recall that are defined by stable 3-forms satisfying a suitable positivity condition. In eight dimensions, stability occurs for 3- and 5-forms, and the corresponding geometric structures are related to the group PSU(3). Since Spin(7) is both connected and simply connected, a connected 8-manifold M admitting a Spin(7)-structure must be orientable and spin (with a preferred spin structure and orientation). As we know, these conditions are equivalent to the vanishing of the first and second Stiefel-Whitney classes of M. However, notice that not every 8-dimensional Riemannian spin manifold admits a g g g Spin(7)-structure. For example, let us denote by Σ M = Σ+M ⊕ Σ−M the decomposition of the g associated spinor bundle (with respect to g = gΦ) into positive and negative part and by χ(Σ±M) the corresponding Euler characteristics. Let us also denote by 4i pi(M) ∈ H (M; Z) the i-th Pontryagin class of M (see [MilS74, Hus96]). Proposition 11.62. ([LM89]) A 8-dimensional Riemannian manifold (M 8, g) admits a Spin(7)- g g 8 structure if and only if is spin and either χ(Σ+M) = 0, or χ(Σ−M) = 0. Equivalently, M admits a Spin(7)-structure, if and only if, w1(M) = w2(M) = 0 and for an appropriate choice of orientation we have that 2 p1(M) − 4p2(M) + 8χ(M) = 0 . (11.4) Remark 11.63. For products of the type M 8 = N 7 × S1, where N 7 is a smooth 7-manifold, equation (11.4) is automatically satisfied and M 8 is spin if and only if N 7 is spin. Remark 11.64. Consider a closed oriented connected smooth manifold M 4k of even dimension 4k and denote by [M] ∈ H4k(M; Z) the fundamental homology class defined by the orientation. By using the cup product one can define a non-degenerate symmetric bilinear map, the so-called intersection form Z 2k 2k Q : H (M; R) × H (M; R) → R, (α, β) 7→ hα ∪ β, [M]i = α ∧ β . M 4k The signature of M , denoted by σ, is defined to be the signature of the quadratic form qM associated to Q, i.e. σ(M) := sign(qM ). For a manifold M whose dimension is not divided by 4, we set σ(M) = 0. The Pontryagin classes can be expressed in terms of the signature σ, in particular for k = 2 we have 1 σ(M 8) = h7p (M 8) − p2(M 8), [M 8]i . (11.5) 45 2 1 Moreover, in terms of Pontryagin classes the Aˆ-genus of M is given by 2k Y xi/2 1 1 Aˆ(M 4k) = = 1 − p + (7p2 − 4p ) + ··· sinh(x /2) 24 1 5760 1 2 i=1 i Hence for example in dimension 8 we get the relation 1 Aˆ(M 8) = (7p2(M 8) − 4p (M 8)) . (11.6) 5760 1 2 Thus the knowledge of the signature σ and of the Aˆ-genus of M can be combined via equations (11.5) and (11.6), to produce the Pontryagin classes involving in (11.4). This can be a suitable methodology for answering the topological existence of a Spin(7)-structure on a 8-dimensional Riemannian spin manifold (M 8, g). DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 85

Similarly with the approach of G2-structures and based again on the notion of intrinsic torsion, one can algebraically divide the Spin(7)-structures into different types. In this case, the intrinsic torsion is determined by the covariant derivative ∇gΦ of the defining 4-form Φ, with respect to g the Levi-Civita connection ∇ associated to the induced metric g = gΦ, or equivalently by the differential of Φ, i.e. the 5-form dΦ. This is a result of M. Fernández [Fer82], who proved also that there are four classes of Spin(7)-structures, appearing as irreducible Spin(7)-representations of the 8 ∗ ⊥ g space W := (R ) ⊗ spin7 of all possible covariant derivatives ∇ Φ of Φ. Let us examine first the case when the intrinsic torsion vanishes. Proposition 11.65. When Φ is closed, dΦ = 0, then the intrinsic torsion vanishes identically and the Riemannian holonomy group Hol(g) is a subgroup of Spin(7). In this case (M 8, g, Φ) admits a g ∇ -parallel spinor and hence g = gΦ is Ricci-flat. Whenever we are in the situation of Proposition 11.65, the Spin(7)-structure Φ is said to be parallel, or torsion-free, or integrable, since it follows that ∇gΦ = 0. On the other hand, non- integrable Spin(7)-structures fall into just two basic classes (and a mixed one)

• class W1: balanced Spin(7)-structures: ϑ = 0 ⇒ ∗dΦ ∧ Φ = 0. • class W2: locally conformal parallel Spin(7)-structures,: dΦ = ϑ ∧ Φ ⇒ dϑ = 0. • class W1 ⊕ W2: Spin(7)-structures of mixed type. Here, the 1-form ϑ is given by 1 1 ϑ = − ∗ (∗dΦ ∧ Φ) = ∗ (δΦ ∧ Φ) , 7 7 and is the so-called Lee form of the Spin(7)-structure.

References

[A02] B. S. Acharya, M-theory, G2-manifolds and four dimensional physics, Class. Quant. Grav. 19 (2002), 1–56. (cited on p. 58) [AG04] B. S. Acharya, S. Gukov, M-theory and singularities of exceptional holonomy manifolds, Phys. Rep., 392, (3), (2004), 121-189. (cited on p. 58) [Agr06] I. Agricola, The Srní lectures on non-integrable geometries with torsion, Arch. Math. 42 (2006), 5–84. With an appendix by M. Kassuba. (cited on p. 56, 58, 70, 83) [AM] D. V. Alekseevsky, S. Marchiafava, Quaternionic structures on a manifold and subordinated structures, Annali di Matematic pura ed applicata (IV), Vol CLXXI (1996), 205–273. (cited on p. 78) [AVL91] D. V. Alekseevsky, A. M. Vinogradov, V. V. Lychagin, Geometry I - Basic Ideas and Concepts of Differential Geometry, Encyclopaedia of Mathematical Sciences,Vol. 28, Springer-Verlag, Berlin, 1991. (cited on p. 16, 21) [AS63] M. F. Atiyah, I. Singer, The index of elliptic operators: III, Ann. Math., 87 (1963), 546–604. (cited on p.2, 55) [ABS64] M. F. Atiyah, R. Bott, A. Shapiro, Clifford modules, Topology 3, Suppl. 1 (1964) 3–38. (cited on p. 11) [At71] M. F. Atiyah, Riemannian surfaces and spin structures, Annales scientifiques de L’. É. N. S, 4e seríe, (4) 1 (1971), 47–62. (cited on p. 30, 78) [AW01] M. Atiyah, E. Witten, M-theory dynamics on a manifold of G2 holonomy, Adv. Theor. Math. Phys. 6 (2001), 1–106. (cited on p. 58) [Bär93] C. Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (3) (1993), 509–521. (cited on p. 70, 71, 72) [B81] H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, Teubner-Texte zur Mathematik, Band 41, 1981. (cited on p. 29) [BFGK89] H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistors And Killing Spinors On Riemannian Manifolds, Sektion Mathematik der Humboldt-Uiversität zu Berlin, 1989. (cited on p.8, 41, 50, 51, 59, 60, 61, 63, 65, 69) [Bes86] A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1986. (cited on p. 60) [B76] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., vol. 509, Springer, 1976. (cited on p. 71, 80) 86 IOANNIS CHRYSIKOS

[B10] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, 2n Ed., Progress in Mathematics, Vol. 203, Birkhaüser 2010. (cited on p. 80) [BHMMS] J-P. Bourguignon, O. Hijazi, J-L. Milhorat, A. Moroianu, S. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry, EMS Monographs in Mathematics, European Mathematical Society, 2015. (cited on p. 41, 55) [BGM93] C. P. Boyer, K. Galicki, B. M. Mann 3-Sasakian manifolds Proc. Japan Acad., 69, Ser. A (1993) 335–340. (cited on p. 71) [BG00] C. P. Boyer, K. Galicki, Sasakian Geometry, Holonomy, and Supersymmetry, Handbook of pseudo- Riemannian geometry and supersymmetry, Intern. J. Math., 11, (2000), 873–909. (cited on p. 71, 80) [BG05] C. P. Boyer, K. Galicki, Sasakian geometry, hypersurface singularities, and Einstein metrics, Proceedings of the 24th Winter School "Geometry and Physics". Circolo Matematico di Palermo, Palermo, 2005. Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 75, pp. 57–87. (cited on p. 71, 80) [Br87] R. L. Bryant, Metrics with exceptional holonomy, Ann. Math. 126, (1987), 525–576. (cited on p. 82, 83) [Br06] R. L. Bryant, Some remarks on G2-structures, Proceedings of Gokova Geometry-Topology Conference 2005, 75–109, Gokova Geometry/Topology Conference (GGT), Gokova, 2006. (cited on p. 82, 83) [B04] D. Bump, Lie groups, Graduate Text in Mathematics, Vol. 225, Springer New York„ 2004. (cited on p. 13, 15, 16, 17, 18, 33) [CR84] L. Castellani, L. G. Romans, N = 3 and N = 1 supersymmetry in a new class of solutions for D = 11 supergravity, Nucl. Phys. B, 241, (1984), 683–701. (cited on p. 58) [CRW84] L. Castellani, L. G. Romans, N. P. Warner, A classification of compactifying solutions for d = 11 supergravity, Nuclear Phys. B, 241 (1984), 429–462. (cited on p. 58) [Dir28] P. A. M. Dirac, The Quantum Theory of the Electron, Proc. R. Soc. Lond. A117 (1928), 610–624. (cited on p.2, 43) [DNP86] M. J. Duff, B. E. W. Nilsson, C. N. Pope, Kaluza-Klein supergravity, Phys. Rep. 130 (1986), 1–142. (cited on p. 58) [Ert18] U. Ertem, Spin geometry and some applications, arXiv: 1801.06988v1. (cited on p. 58) [Fer82] M. Fernández, A classification of Riemannian manifolds with structure group Spin(7), Ann. Mat. Pura Appl., 143 (1982), 101–122. (cited on p. 83, 85) [Fr80] Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., 97, (1980), 117–146. (cited on p.2, 65, 66, 67, 71) [Fr81] Th. Friedrich, A remark of the first eigenvalue of the Dirac operator on 4-dimensional manifolds, Math. Nachr., 102 (1981), 53–56 (cited on p. 69) [Fr89] Th. Friedrich, On the conformal relation between twistors and Killing spinors, Rend. Circ. Mat. Palermo Ser II. 22, (1989), 59–75. (cited on p. 61, 63) [FrGr85] Th. Friedrich, R. Grunewald, On the first eigenvalue of the Dirac operator on 6-dimensional manifolds, Ann. Global. Anal. Geom. 3 (3), (1985), 265–273. (cited on p. 70) [FrK90] Th. Friedrich, I. Kath, 7-dimensional compact Riemannian manifolds with Killing spinors, Comm. Math. Phys. 133, (1990), 543–561. (cited on p. 71) [FKMS97] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly parallel G2-structures, J. Geom. Phys. 23, (1997), 259–286. (cited on p. 72, 82) [Fr00] Th. Friedrich, Dirac Operators in Riemannian Geometry, Amer. Math. Soc, Graduate Studies in Mathe- matics, Vol. 35, 2000. (cited on p.8, 12, 29, 31, 33, 37, 38, 41, 48, 52, 57, 60, 61, 65, 69) [FrK00] Th. Friedrich, E. C. Kim, The Einstein-Dirac equation on Riemannian spin manifolds, J. Geom. Phys. 33 (2000), 128–172. (cited on p. 52, 53) [Gi09] N. Ginoux, The Dirac Spectrum, Lecture Notes in Mathematics (LNM 1976), Springer-Verlag Berlin Heidel- berg, 2009. (cited on p. 33, 48, 51, 59, 60, 61, 63, 65, 69, 71) [GOV94] V. V. Gorbatzevich, A. L. Onishchik, E. B. Vinberg, Structure of Lie Groups and Lie Algebras, Encycl. of Math. Sci. v41, Lie Groups and Lie Algebras–3, Springer–Verlag 1994. (cited on p. 13, 16) [Gr90] R. Grunewald, Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. Global. Anal. Geom. 8, (1990), 43–59. (cited on p. 70) [Hab94] K. Habermann, Twistor spinors and their zeroes, J. Geom. Phys., 14 (1) (1994), 1–24. (cited on p. 61, 64) [Hij86-1] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys., 104 (1986), 151–162. (cited on p. 69) [Hij86-2] O. Hijazi, Caractérisation de la sphére par les premiéres valeurs propres de l’ opérateur de Dirac en dimension 3, 4, 7 et 8., C.R. Acad. Sci. Paris 303, (1986), 417–419. (cited on p. 69) [Hij01] O. Hijazi, Spectral properties of the Dirac operator and geometrical structures, Geometric Methods for Quan- tum Field Theory: Proceedings of the Summer School on Geometric Methods for Quantum Field Theory, Editors: H. Ocampo, S. Paycha, A. Reys, 116–169, World Scientific 2001. (cited on p. 40, 44, 52, 60, 69) DIRAC OPERATORS IN GEOMETRY - LECTURE NOTES 87

[Hit74] N. Hitchin, Harmonic spinors, Advances in Mathematics, 14, (1974), 1–55. (cited on p. 54, 69) [Hit01] N. J. Hitchin, Stable forms and special metrics, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemp. Math., vol. 288, Amer. Math. Soc., Providence, RI, 2001, 70–89. (cited on p. 82) [Hus96] D. Husemöller, Fibre Bundles, McGraw-Hill, Series in Higher Mathematics, 1996. (cited on p. 10, 13, 17, 21, 24, 25, 26, 27, 29, 30, 31, 38, 84) [Joy00] D. Joyce, it Compact manifolds with special holonomy, Oxford Science Publ., 2000. (cited on p. 56, 57, 58, 74, 78, 82, 83) [Kar68] M. Karoubi, Algébres de Clifford at K-théorie, Ann. Sci. Éc. Norm. Sup. Paris 1 (1968), 161–270. (cited on p. 29) [Kn96] A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, Vol. 140, Birkhäuser, Boston, 1996. (cited on p. 16, 17, 18, 33, 35) [KN69] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry Vol I, II, Wiley - Interscience, New York, 1969. (cited on p. 37, 38, 42, 57, 74) [LM89] H. B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. (cited on p.6,8, 11, 12, 13, 20, 21, 26, 27, 29, 30, 32, 33, 38, 41, 45, 48, 52, 55, 56, 78, 83, 84) [Lich58] A. Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques III, Dunod, Paris 1958. (cited on p. 65) [Lich63] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257, (1963), 7–9. (cited on p. 65) [Lich87] A. Lichnerowicz, Spin manifolds, Killing spinors and universality of the Hijazi inequality, Lett. Math. Phys. 13 (1987), 331–344. (cited on p. 61, 69) [MilS74] J. W. Milnor, J. D. Stasheff, Characteristic classes, Princeton University Press, 1974. (cited on p. 13, 26, 27, 29, 84) [MimT91] M. Mimura, H. Toda, Topology of Lie Groups, I and II, Amer. Math. Soc, Translations of Mathematical Monographs, Vol. 91, 1991. (cited on p. 13, 15, 16, 17, 21) [Mor07] S. Morrison, Classifying Spinor Structures, PhD thesis, arXiv:math-ph/0106007. (cited on p. 29) [Ob56] M. Obata, Affine connections on manifolds with almost complex, quaternion or Hermitian structure, Jap. J. Math., 26 (1956), 43–79. (cited on p. 79) [Ob62] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340. (cited on p. 65) [Ons93] A. L. Onishchik, Foundations of Lie theory, Lie transformation groups, Encycl. of Math. Sci. v20, Lie Groups and Lie Algebras–I, Springer–Verlag, 1993. (cited on p. 16, 17) [McW89] M. Y. Wang, Parallel spinors and parallel forms, Ann. Global Anal. Geom., 7 (1989), 59–68. (cited on p. 57)

University of Hradec Králové, Faculty of Science, Rokitanského 62, 500 03 Hradec Králové, Czech Republic E-mail address: [email protected]