Twistor Theory and Gravitational Instantons

Treball ﬁnal de grau GRAU DE MATEMATIQUES` Facultat de Matem`atiques Universitat de Barcelona

TWISTOR THEORY AND GRAVITATIONAL INSTANTONS

Autor: Alejandro Fern´andezPiqu´e

Director: Dr. Ignasi Mundet Riera Realitzat a: Departament d’Algebra` i Geometria

Barcelona, 27 de Juny de 2016 Abstract

The aim of this work is to understand and introduce a self-contained account of the construction of certain kind of gravitational instantons made by Hitchin in his work “Polygons and Gravitons”. Gravitational instantons are solutions of the Einstein equations which have a complete, non-singular and positive deﬁnite metric. Hitchin use twistor theory to construct these solutions. We present beforehand some concepts of self-duality, diﬀerential operators and connections to understand twistor theory. Afterwards, the construction of the aforementioned instantons is explained in full detail.

Acknowledgements

I would like to thank my advisor Ignasi Mundet Riera for all the answers and detailed explanations he has given me. I also would like to emphasize his patience which has been tested after being asked every week. I also thank my family and friends for being always by my side. Finally, I would like to thank Ainhoa for being so supportive and understanding.

i Contents

1 Preface 1

2 Riemannian and Spin Geometry 2 2.1 Spinors and p-forms ...... 2 2.2 Self-duality ...... 6

3 Twistor non-linear graviton 8 3.1 Frobenius and Newlander-Nirenberg theorems ...... 8 3.2 First order diﬀerential operators and jet bundles ...... 9 3.3 Integrability of the Twistor space ...... 13 3.4 Properties of the twistor space of a self-dual manifold ...... 20

4 Construction of Gravitational Instantons 22 4.1 Gravitational Instantons ...... 22 4.2 Hitchin’s Construction ...... 22

5 Conclusions 33

A Appendix. Basic Material 34 A.1 Lie Groups ...... 34 A.2 Cliﬀord Algebras and Spin Groups ...... 36 A.3 Representation Theory ...... 39 A.4 Fibre Bundles ...... 40 A.5 Connections in Fibre Bundles ...... 42 A.6 Complex Geometry ...... 45

ii 1 Preface

The project

Yang-Mills instantons are non-perturbative solutions of the Yang-Mills equations which helped resolve some of the problems that appeared during the 1970s. Due to their relevance in the path integral quantization, similar solutions for Euclidean quantum gravity were studied. They were called gravitational instantons. The Yang-Mills instantons that had self-dual curvature were classiﬁed and studied using algebraic geometry and also using the ideas introduced by Penrose and Ward. This methodology let the interpretation of a 4-dimensional Minkowski space in terms of a 3-dimensional complex space. This methodology is called twistor theory. A similar approach can be taken with gravitational instantons. If the curvature derived from the metric is self-dual, it is posible to use the “non-linear graviton” ideas of Penrose to convert the problem of ﬁnding such metrics as one of complex geometry. Hitchin used this technique to construct solutions of Einstein equations called ”gravitational multi-instantons”, which were introduced previously by Gib- bons and Hawking [11]. The aim of this “Treball de Fi de Grau” is to give a self-contained explanation of the non-linear graviton introduced by Penrose and to construct the gravitational multi-instanton metric as Hitchin did. The main reason for taking this amazing journey is to give a more mathematical perspective to the previous “Treball de Fi de Grau” of the Physics Degree, which focused also on gravitational instantons. In order to explain all the concepts of this construction in full detail and in a reader-friendly manner, the text is divided into three sections:

1. The first section gives an introduction of Riemann geometry. It gives an elementary view on Clifford algebra, spinors and their representations. It also introduces the notion of self-duality for differential forms, for Riemann manifolds and for connections.

2. The aim of the second section is to understand two very important proposi- tions that will let us define the twistor space of a manifold. For this purpose, an introduction to first order differential equations and jet bundles is given.

3. Finally, on the third section we explain the construction made by Hitchin with the tools of the previous sections. We construct the metric making some assumptions that are proved subsequently.

In the appendix there are some concepts that will be necessary through these sections in order to understand the explanations. It is divided into topics and contain deﬁnitions and statements that will be needed through the construction.

1 2 Riemannian and Spin Geometry

2.1 Spinors and p-forms

Let M be an oriented Riemannian manifold of an even dimension 2l, i.e. a smooth manifold with a smooth inner product defined on the tangent space of each m M. Let Λp denote the bundle of exterior p-forms1 . It is posible to define the Hodge∈ star operator :Λp Λ2l−p as: ∗ → α β = α, β ω ∧ ∗ h i where ω denotes the volume form2, α, β Λp and , the induced inner product on p-forms by the metric on the manifold.∈ This innerh producti is determined by the condition: v1 vp, w1 wp = det ( vi, wj ) h ∧ · · · ∧ ∧ · · · ∧ i h i If we consider the case l = 2, then 2 = ( 1)p(2l−p)s = ( 1)p(4−p)s, where s is the signature of the inner product. For an∗ oriented− Riemannian− manifold, the signature p 2 2 is s = 1 and the vector space Λ for p = 2, Λ , splits into the direct sum of Λ+ and 2 Λ−, the subspaces of the self-dual and anti-self-dual 2-forms respectively. Now we will introduce another notion involving the Hodge star that will let us later define the desired manifolds properly. The curvature of a manifold of dimension n 4 can be decomposed into three irreducible terms. Furthermore, if n = 4 one of the≥ terms decomposes into two terms . Taking the curvature tensor as a 2 2 2 2 2 linear map from Λ Λ and recalling the decomposition of 2-forms, Λ = Λ+ Λ−, we can view the curvature→ tensor as the following matrix: ⊕   s  W+ + Id Z   12      (2.1)    t s   Z W− + 12 Id 

where s is the scalar curvature, Z is the traceless Ricci tensor and W+ and W− together give the Weyl curvature W = W+ + W− . The terms on the upper left- hand side and lower right-hand side are self-adjoint [4]. The 2-forms can be identified, using the metric, with the Lie Algebra of SO(4), so(4), i.e. with skew adjoint transformations of Λ1. This identification can be derived by associating to a pair of vectors v, w Λ1 the skew-symmetric endomor- phism v w defined by: ∈ ∧ (v w)(x) = v, x w w, x v ∧ h i − h i 1 A covariant p-tensor on a real vector space V (in out case is T ∗M) is said to be alternating if any permutation of the arguments causes its value to be multiplied by the sign of the permutation. Alternating covariant p-tensor are called p-forms. The wedge product ( ) acts as follows: a b = a b b a. ∧ ∧ ⊗ − ⊗ p 2In local coordinates it is described by ω = g dx dx . | | 1 ∧ · · · ∧ 2l 2 The resulting map Λ2 so(4) is a vector space isomorphism. The notion of Lie bracket can be applied to→ 2-forms making the previous identification an isomorphism 2 2 2 of Lie algebras. Moreover, the decomposition described before, Λ = Λ+ Λ−, corresponds to the isomorphism of Lie Algebras so(4) = so(3) so(3). We⊕ can 2 ⊕ see thereby that Λ± are 3-dimensional real vector spaces which are isomorphic as vector spaces to so(3), the vector space of skew-symmetric matrices of dimension 3 . Moreover, we can make this isomorphism a Lie algebra isomorphism. The 3- dimensional real vector space have an inner product derived from the metric on M. We can define then the vectorial product as: (a, b) a b = a b, where is the Hodge dual. This also gives the Lie bracket to this→ space× becoming∗ ∧ thereby∗ a Lie algebra. We have obtained then the identification Λ2 3 so(3) due to the + ∼= R ∼= vectorial product. There is also an important identification in 4-dimensional manifolds. The universal cover of SO(4), Spin(4), is not a simple group3 but decomposes as

Spin(4) = SU(2) SU(2) × In order to understand this equality, R4 must be identified by the quaternions H (see Definition A.1.2). Taking the elements p, q of SU(2) = Sp(1) (see Definition A.1.3) as unit quaternions, the map x pxq−1 is an orthogonal transformation of the quaternions with unit determinant→ which has as kernel the element ( 1, 1). Therefore, SU(2) SU(2) is a doble covering of SO(4) and it implies Spin− (4)− = SU(2) SU(2). × × Due to the previous identification, the representations of Spin(4) can be expressed using the representations of SU(2). The fundamental representation D1/2 is SU(2) acting on C2 and all other irreducible representations are symmetric powers k + Dk/2 = S D1/2, with k Z . The tensor product of two representations decomposes using Clebsch-Gordan∈ formula (A.3.3) :

Dk/2 Dl/2 = Dk+l/2 Dk+l−2/2 D|k−l|/2 (2.2) ⊗ ⊕ ⊕ · · · ⊕ So the representation ring4 of Spin(4) is based on the two fundamental representations D± , where we use to distinguish the representation of each of the 1/2 ± SU(2), Spin(4) = SU(2)+ SU(2)−. The representation D+ D− has dimen- × k/2 ⊗ l/2 sion (k + 1)(l + 1) and factors through SO(4) if and only if k + l is even. These are orthogonal irreducible representations of Spin(4) and all others are symplectic 56 .

It is posible to deﬁne, at least locally, the complex vector bundles V+ and V− which are associated to the principal spin bundle through irreducible representations + of Spin(4). The vector bundle V+ is associated with the representation D C 1/2 ⊗ 3A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. 4It is the ring formed from all the ﬁnite-dimensional linear representations of the group. 5If V is an irreducible representation of G and W is an irreducible representation of H, then V W is an irreducible representation of G H, see [6, page 82]. ⊗6A symplectic representation of a Lie Group× G into V is a representation that preserves the symplectic form ω of V , i.e ω(ga, gb) = ω(a, b) for a, b V . ∈

3 − of Spin(4) = SU(2) SU(2) and V− with the representation C D . The total × ⊗ 1/2 spin bundle is defined as V = V+ V−, it is a Z2-graded vector bundle. Spinors are defined, consequently, as sections⊕ of these fibre bundles.

The Z2-graded vector spaces are also called superspaces. If the superspace is the direct sum of two complex vector spaces with Hermitian structures, it is called a Her- mitian superspace. A superalgebra A is an algebra whose underlying vector space i j i+j is a superspace and whose product respects the Z2-grading, i.e. A A A . The Clifford algebra7 C`(Λ1, g) is a superalgebra, C`(Λ1, g) = C`0(Λ1,· g) ⊂C`1(Λ1, g). The algebra of endomorphisms End(E) of a superspace E is a superalgebra,⊕ such that: End+(E) = Hom(E+,E+) Hom(E−,E−) End−(E) = Hom(E+,E−) ⊕ Hom(E−,E+) ⊕ If E is a Hermitian superspace, we say that u End−(E) is odd self-adjoint if it has the form ∈ 0 u− u = u+ 0 where u+ : E+ E− and u− is the adjoint of u+. → A very important proposition concerning modules8 of the Clifford algebra of the tangent space of an 2l-dimensional manifold is the next one: Proposition 2.1. If E is a 2l-dimensional oriented Euclidean vector space , then + − there is a unique Z2-graded Clifford algebra module S = S S , called the spinor bundle , such that ⊕ C`(V ) C = End(S) ⊗ ∼ Since Spin(2l) C`(E), it follows that both S+ and S− are representations of Spin(2l). They⊂ are called half-spinor representations.

See [3, page 109] for the proof. Therefore, the previous representations of Spin(4) can be also understood as restrictions to Spin(4) of the representations of the Clifford bundle. ∗ p At the same time, the complexified Clifford algebra is isomorphic to Λc = pΛc p p ⊕ as a graded vector bundle, where Λc = Λ C [16, page 95][10, page 13]. Hence, we have the following vector bundles isomorphisms:⊗

∗ ∗ 1 1 Λ (T M = Λ ) = C`c(Λ , g) = End(V = V+ V−) c ∼ ∼ ⊕ Therefore, it is posible to associate the p-forms to endomorphisms of the spinors. They act on spinors in the following way: Λ1 Hom(V ,V ) Hom(V ,V ) Λ3 c ∼= + − ∼= − + ∼= c 7An introduction to Cliﬀord algebras and spinors can be found at the Appendix A.2. 8 + We say that E = E E− is a Cliﬀord module if there is a bundle map c : T ∗M End(E) such that: ⊕ → c(α )c(α ) + c(α )c(α ) = 2(α )(α ) • 1 2 2 1 − 1 2 + c(α) swaps the bundle E and E− •

4 1 Moreover, the real bundle of Λc can be identified with Hom(V+,V−) Hom(V−,V+). 2 ⊕ On the other hand, Λ+c End(V+) are the traceless endomorphisms of V+ and the ⊂ 9 2 real bundle are those which are also skew hermitian . In the case of Λ−c it works on the opposite way. As we have seen before, Λ2 so(3) which implies that Λ2 so(3, ). Recalling + ∼= +c ∼= C that V+ is the vector bundle obtained by the fundamental representation of SU(2), 2 we know that there is a volume form ν in V+ which is invariant by the action of ∧2 SU(2). We can represent each element of S (V+), i.e. the symmetric tensor product 2 2 of V+, as a polynomial ax + bxy + cy . Therefore, we can use the discriminant of 2 b2−ac the polynomial to give a Euclidean inner product to S (V+) such that ν C. In this way, we can use this inner product to define the Hodge dual and to define∈ the 2 notion of Lie Algebra by [a, b] = a b for a, b S (V+). This algebra has complex ∗ ∧ ∈ 2 2 dimension 3 and is isomorphic to so(3, C). Hence, Λ+c = S V+. This also makes ∗ 2 ∼ ∗ sense taking into account that V+ = V+,Λ+c End(V+) = V+ V+ = V+ V+, 2 ∼ ⊂ ∼ ⊗ ∼ ⊗ S V+ V+ V+ and have the same dimension. ⊂ ⊗ An important remark that will be used in the next section derives from the previous discussion. The bundle of self-dual Weyl tensors is identified with the 10 2 bundle of self-adjoint traceless endomorphisms of Λ+. The bundle of all endo- 2 2 morphisms of this vector space is Λ+ Λ+ and using the Clebsch-Gordan formula 2 2 2 2 4 ⊗ 2 0 (2.2), Λ+ Λ+ = S V+ S V+ = S V+ S V+ S V+ are obtained. As previously 2⊗ ∼ ⊗ ∼ ⊕ ⊕ stated, S V+ is the bundle of skew hermitian transformations and hence it can not 0 be identified with the Weyl term. S V+ is a one-dimensional bundle of scalar trans- 4 formations which are not clearly traceless. Thus, S V+ is the bundle of self-dual − 4 Weyl tensors. Similarly, W Γ(S V−). ∈ 1 1 Another important point of Λ Λ = Hom(V+,V−) is their relation with the ⊂ c ∼ almost complex structures on M. Fixing a non-zero spinor φ (V+)m at m gives a real isomorphism Λ1 (V ) defined by the Clifford multiplication∈ α α φ and m ∼= − m in turn this identification gives a complex structure on the tangent space→ of· M at m. This complex structure is compatible with the metric and the orientation of M. Multiplying φ by a scalar λ C∗ does not modify this complex structure, hence ∈ P (V+)m parametrizes the set of compatible complex structures. The same can be done using V−, but in this case, the orientation is the inverse. Finally, it is important to realize that the fundamental representation of SU(2), i.e. D1/2, has a symplectic structure, because SU(2) SL(2, C) and a unitary structure because SU(2) U(2). Therefore, ≤ ≤ ¯ V+ = V+ = V+ ∼ ∗ ∼ Hence, the composition is a quaternionic structure:

j : V+ V+ → where j(λv) = λj¯ (v) and j2 = 1. − 9 It is minus its conjugate transpose, S = S†. 10A self-adjoint operator A on a vector space− V with inner product , is that which satisﬁes for all x, y V : Ax, y = x, Ay . h i ∈ h i h i 5 2.2 Self-duality

The Hodge star operator is conformally invariant11 on l-forms, where 2l is the dimension of the manifold. If we multiply the metric by a factor λ, then the result of the product of two vectors is multiplied by λ and for 1-forms by λ−1. Consequently, the result is multiplied by λ−l on l-forms. The volume form is multiplied by λl so (α, β)ω = α β remains invariant. ∧ ∗ Recalling the decomposition of the curvature in (2.1) , we can deﬁne a self-dual manifold as follows:

Deﬁnition 2.1. An oriented Riemannian 4-manifold is self-dual if its Weyl tensor W = W+, i.e if W− = 0.

The Weyl tensor is also conformally invariant, therefore this is a property of the conformal structure. A very important notion in diﬀerential geometry is the connection or the covariant derivative. A brief introduction to connections on ﬁbre bundles can be found at the Appendix (A.5). The notion of equivalence between connections is called gauge equivalence or gauge invariant, i.e the invariance of the connection under a gauge transformation:

Definition 2.2. A gauge field transformation on a principal G-bundle P is a dif- feomorphism f : P P such that (1) f(gp) = gf(p) g G, p P , (2) f preserves each fibre, i.e acts trivially→ on the base space M. ∈ ∈

As we did for manifolds, we can also characterize the self-duality of a connection:

Deﬁnition 2.3. On a 4-manifold M, a connection is said to be self-dual if its curvature Ω is in A2 (g)12 (i.e Ω = Ω) and anti-self-dual if Ω A2 (g) (Ω = Ω). + ∗ ∈ − −∗ Finally, the next proposition will clarify the properties of the 4-dimensional manifolds that will be used and the kind of connections of the ﬁbre bundles on the manifolds.

Proposition 2.2. Let M be a 4-manifold with an Einstein metric13 . Then the induced connections on the bundle of self-dual spinors V+ and the bundle of self-dual 2 2-forms Λ+ are self-dual. The induced connections on the corresponding anti-self- dual bundles are anti-self-dual. Conversely, if the induced connections on V+ and 2 Λ+ are self-dual, the metric is an Einstein metric.

2 Proof. The vector bundle of self-dual forms Λ+ deﬁned over M has an associated principal bundle called the frame bundle which has SO(3) as a Lie group, the group 2 of rotations of Λ+. If we take a metric on M we can induce a connection on this

11It is invariant under conformal transformations, see Deﬁnition A.4.13. 12 k Vk 2 2 A (E) denotes the bundle of sections in Γ(E T ∗M). In this case, A = Γ(g Λ+). 13A metric is Einstein if its Ricci tensor is proporcional⊗ to the metric. However, in this⊗ context we refer to Einstein metrics as those whose Ricci tensor also vanishes.

6 principal bundle from the Levi-Civita connection of M. The adjoint bundle is the associated vector bundle14 obtained due to the adjoint representation15 of the Lie 2 group. The lie algebra of SO(3), so(3), can be identiﬁed with Λ+ as we have seen before. Therefore, the adjoint bundle of this principal bundle is the vector bundle 2 2 Λ+. The curvature then is a section of A (g), which is the part of the Riemann 2 2 curvature tensor which lives in Λ+ Λ . Looking at the matrix representation of the curvature tensor (2.1), we see that⊗ the curvature on the vector bundle is the ﬁrst row: s Ω = W + Id + Z + 12 2 2 Since Z is the only term in A−(Λ+), the connection on the vector bundle is self-dual if and only if Z = 0, i.e. if the manifold is an Einstein manifold. The self-duality property of the connection doesn’t change between associated bundles, hence V+ is also self-dual if M has an Einstein metric and is a spin manifold.

14see Deﬁnition A.4.7 . 15see Deﬁnition A.1.7 .

7 3 Twistor non-linear graviton

Most of the work done by Hitchin at [12] relies on two statements of “Self-Duality in Four-Dimensional Riemannian Geometry” [1]. The first one is the Proposition 3.3 which gives two necessary and sufficient conditions for V (D¯) T ∗E∗ (a vector subspace derived from a differential operator D¯ that acts on the vector⊂ space E) to be involutive. Afterwards, Theorem 3.3 makes use of the aforementioned proposition to define an integrable almost complex structure on P (V−), a bundle defined over M, which is the vanishing of W−. These two statements will let us define the twistor space. Some tools of differential geometry are needed in order to prove them and understand their implications.

3.1 Frobenius and Newlander-Nirenberg theorems

First, it is necessary to introduce the concept of distribution and some properties that can be applied to it. Due to length restrictions, most of the statements will be given without proof. Our aim is to recall some concepts that will be used later on. Definition 3.1. Let c be an integer, 1 c d.A c-dimensional distribution D on ≤ ≤ a d-dimensional manifold M is a choice of a c-dimensional subspace Dm of TmM for each m M. D is smooth if for each m in M there is a neighborhood U of ∈ ∞ m and there are c vector field X1,...Xc of class C on U which span D at each point of U. A vector field X on M is said to belong to the distribution D(X D) ∈ if Xm Dm for each m M. A smooth distribution D is called involutive if [X,Y ] ∈D whenever X and∈ Y are smooth vectors fields lying in D. ∈ Definition 3.2. A submanifold (N, ψ) of M where ψ is an embedding is an integral manifold of a distribution D on M if

dψ(Nn) = D(ψ(n)) for each n N ∈ Definition 3.3. A smooth distribution D is said to be integrable if each point of M is contained in an integral manifold of D Definition 3.4. A smooth distribution D TM is completely integrable if there exist a flat chart for D in a neighborhood of⊂ each point of M. Proposition 3.1. Every integrable smooth distribution is involutive

See [22, page 42] for the proof.

In order to understand the proposition (3.3), it is necessary to deﬁne the distribution in terms of diﬀerential forms. Lema 3.1. Suppose M is a smooth n-manifold and D TM a distribution of rank k. Then D is smooth if and only if each point m of M⊂ has a neighborhood U on which there are smooth 1-forms ω1, . . . , ωn−k such that for each q U: ∈ 1 n−k Dq = kerω q ker ω q | ∩ · · · ∩ | 8 See [17, page 493] for the proof.

On the next proposition we will find equivalent conditions for a distribution to be involutive in terms of differential forms. Proposition 3.2. Let D be a smooth distribution of rank k on a smooth n-manifold M, and let ω1, . . . , ωn−k be a smooth defining forms for D on an open U M. The following statements are equivalent: ⊂

1. D is involutive on U

2. dω1, . . . , dωn−k annihilate D

3. There exists smooth 1-forms αi : i, j = 1, . . . , n k such that j − n−k i X j i dw = w αj j=1 ∧

See [17, page 495] for the proof.

Finally, let us introduce two very important theorems which, with the help of the implications of the aforementioned statements, will enable the construction of a complex manifold. Theorem 3.1. (Frobenius). Every involutive distribution is completely integrable.

See [22, page 43] for the proof. Remark 3.1. Therefore, completely integrable integrable involutive ⇔ ⇔ Theorem 3.2. (Newlander and Nirenberg). Let (M,J) be a 2l-dimensional almost complex manifold. If J is integrable, the manifold M is a complex manifold with the almost complex structure J.

See [19] for the proof.

3.2 First order diﬀerential operators and jet bundles

There are two ways to understand the proposition 3.3 and one of them is in terms of jet bundles. Let us begin by defining the notion of a differential operator: Definition 3.5. A differential operator of order k on M is a linear map P : Γ(E) Γ(F ), where E and F are smooth complex vector bundles over M, with the following→ property. Each point of M has a neighborhood U with local coordinates (x1..., xn) p q and local trivializations: E U U C and F U U C , in which P can be written in the form: | → × | → × X ∂|α| P = Aα(x) ∂xα |α|≤k

9 where each Aα(x) is a q p-matrix of smooth complex-valued functions and where Aα = 0 for some α with×α = k. A real diﬀerential operator of order k is deﬁned 6 | | similarly with C replaced by R.

Another notion related to a differential operator is the symbol map: ∂|α| Definition 3.6. The Symbol map σ of a differential operator P = P Aα(x) |α|≤k ∂xα of order k, is described by: X σ(P ) = Aα(x)ψα |α|=k where ψ = (dx1, . . . , dxr). It can be defined intrinsically: taking a covector v ∗ ∞ ∈ Tx M, a section se of E such se(x) = e Ex and a function f C (M) such f(x) = 0 and df = v for x M , then: ∈ ∈ ∈ k D(f se)(x) = σ(D)(x)(v) Fx · ∈ Remark 3.2. From now on, only first-order differential operators will be considered, i.e k = 1.

Now, let us define a new type of fibre bundle, the jet bundle, that will be very important to understand the first proposition. First of all, we will define an equivalence relation and then we will used it to define the fibre bundle.

Definition 3.7. Let us consider a fibre bundle π : E M, where M has dimension n. Taking m M, denote Γ(E) all the sections of E→that contain m in the domain. Let us define∈ an equivalence relation in Γ(E). Two sections σ and η have the same k-jet at m if |I| α |I| α ∂ σ ∂ η I = I , 0 I k. ∂x m ∂x m ≤ | | ≤ where I is any ordered n-tuple of integers I = (I(1),I(2),...I(n)), such that

n m I(i) X ∂|I| Y ∂ I = I(i) , := . ∂xI ∂xi | | i i=1

k The k-jet with representative σ is denoted by jσ or jk(σ). Remark 3.3. By using the previous definition we can define a k-th jet manifold J k(E) as: J k(E) := jrσ : p M, σ Γ(E) p ∈ ∈ and it defines fibre bundle with the projections:

( r ( r πr : J (E) M πr,0 : J (E) E → , → jrσ p jrσ σ(p) p 7→ p 7→

10 Let us define now the subspace V (D¯) T ∗(E∗ 0) mentioned at the beginning of the section. Let E be a real vector bundle⊂ on M\ . Section s Γ(E) defines by duality a function sv C∞(E∗) on the dual bundle such that: ∈ ∈ v s (m) = s(m), m h i where m is a point of M defined by x1, . . . , xn . Let us see these operators in local { } ∗ terms: taking the basis (e1, . . . , ek) for E and (1, . . . , k) for E , a parametrization of E and E∗ can be done locally by:

k X (f1, . . . , fk, x1, . . . , xn) fi(m)ei(m) → i

k X (λ1, . . . , λk, x1, . . . , xn) λi(m)ei(m) → i v P v The action of s for any s = i fiei and the 1-form obtained by derivation of s has the form: k v X s (λ1, . . . , λk, x1, . . . , xn) = λifi(m) i k k v X X ds = dλifi + λidfi i i The homomorphism from sections of E, Γ(M,E) to covectors of the vector space E∗, i.e to Γ(E∗,T ∗E∗) also factors through the 1-jet bundle. Taking p : E∗ M as the projection, one obtains the following diagram. →

duality ∗ d ∗ ∗ ∗ Γ(M,E) ∼ Γ(E , R) / Γ(E ,T E ) = 6 V

∗ p ∗ Γ(M,J1(E)) / Γ(E ,J1(E))

∗ ∗ ∗ ∗ The homomorphism V : Γ(E ,J1(E)) Γ(E ,T E ) is characterized by the property → ∗ v V (p j1(s)) = ds and is surjective oﬀ the zero section of E∗. This can be interpreted as follows: ∗ the zero section m = 0 of E is determined by λi(m) = 0, i, therefore for every ∗ v ∀ p j1(s) m=0, the image through V is the 1-forms ds , which at this point m = 0 | v Pk ∗ ∗ ∗ ∗ has the form ds = i dλifi. However, the subspace p (T M) T E is generated by dxi, for i = 1, . . . , n. Taking dλ1, . . . dλn, dx1, . . . , dxn as a⊂ basis for T ∗E∗, we ∗ ∗ ∗ { } see that p (T M) V (p (J1(E))) = 0. ∩ ∗ 1 ∗ The homomorphism V restricted to p (E Λ ) p (J1(E)) gives: ⊗ ⊂ ∗ ∗ ∗ V ((em αm) ) = em, m p αm (T E ) (3.1) ⊗ m h i ∈ m

11 ∗ 1 ∗ Let us prove this statement. The ﬁbre of the sub-bundle p (E Λ ) p (J1(E)) 1 ⊗ ⊂ (we can do this identiﬁcation J1(E) = E E Λ ) at m is formed by those 1-jets ∼ ⊕ ⊗ whose representatives vanish at m, i.e. those which vanish at m. In local terms, those sections are determined by the condition fi(m) = 0, i. The image by V of v Pk Pk P∀n dfi r this sub-bundle is then 1-forms ds = i λidfi = i λi r dxr dx . Contracting k s P this 1-form to a vector X = ∂x + i ai∂λi ,

k X dfi dsv(X) = λ i dxs i

1 Pn Pk fj i An element of E Λ will be of the form ( ( i ej) dx ) which applied to V ⊗ i j dx ⊗ ∗ Pn Pk ∂fj ∗ l Pn Pk ∂fj l using (3.1) gives em, m p αm = l j λj ∂xl p (dx ) = l j λj ∂xl dx . Applying this 1-form to Xh: i k X dfj e , p∗α (X) = λ m m m j dxs h i j which proves the statement. Therefore, a linear first-order differential operator D¯ is defined as a homomorphism from J(E) to F , whose kernel R is taken to be a vector bundle. Then the vector bundle V (D¯) defined in the proposition is just V (p∗R). We can also define it ∗ ∗ as the sub-bundle of the cotangent space of E 0 such that at each point m E 0, v \ ∈ \ this vector bundle is formed by 1-forms (ds )m derived from sections s Γ(M,E) ¯ ∈ which D(s)m = 0. Let us introduce an important example that will be of main use for the proof of the first proposition. Exemple 3.1. Let us take D¯ = . Taking as ξ any section of E, the covariant derivative16 applied to this section∇ gives as follows:

k k k k ¯ X X X X Dξ = D(ξαeα) = eα dξα + eβ ωαβξα ⊗ ⊗ α=1 α=1 α=1 β=1 where the property (fs) = df s + f s ∇ ⊗ ∇ is used on the second equality. The covariant derivative of the section is zero iﬀ for any α: β dξα + wβαξ = 0 (3.2) v P P The image through V of this sections are ds = dλiξi + λidξi. Using the condition (3.2) gives as follows:

v X X X X X X X X ds = dλiξi λiwjiξj = dλiξi λjwijξi = (dλi λjwij)ξi i − i j i − i j i − j P Therefore, θi = dλi wijλj span V ( ). − j ∇ 16 e = P w e . ∇ i j ij ⊗ j 12 3.3 Integrability of the Twistor space

The two statements introduced at the beginning of the prior section will be proved in this section. Let us take D¯ as a general diﬀerential operator of the form σ 1 ∇ where σ is the symbol of the operator. Let S1 = R E Λ be the kernel of the 2 1 ∩ ⊗ 17 symbol and S2 E Λ the image of S1 Λ under exterior multiplication . Let us recall the ﬁrst⊂ proposition:⊗ ⊗

¯ ∗ ∗ Proposition 3.3. V (D) T (E 0) is involutive iﬀ (1) D1Γ(S1) Γ(S2), (2) ⊂ 0 \ 2 ⊂ 1 ΩΓ(E) Γ(S2), where Ω: A (E) A (E) is the curvature of and D1 : A (E) A2(E) the⊂ extended covariant derivative.→ (Note that the ﬁrst condition∇ is a “torsion”→ condition on the connection - the vanishing of certain components of the connection matrix - and the second condition is a condition on the curvature)

Proof. As we have seen in Proposition 3.2, the subspace V (D¯) will be involutive ∗ ¯ P and therefore involutive if for any section s Γ(E ,V (D)), then ds = si αi for ∗ ¯ ∗ ∗ ∗ ∈ ∧ vi Γ(E ,V (D)) and αi Γ(E ,T (E 0)). The kernel of the diﬀerential operator is the∈ direct sum of the kernel∈ of the covariant\ derivative, E, and the kernel of the ¯ ∗ symbol, S1. Therefore, V (D) = V (p (E S1)). This subspace is spanned by: ⊕ X θi = dλi wijλj , 1 i k − j ≤ ≤

v X σ = sijkλjdxk , 1 i m i ≤ ≤ j,k P where σi = j,k sijkej dxk is taken as a local basis of S1 and we have used the ⊗ v equation (3.1) to compute σi . Taking the tuple θ1, . . . , θk, dx1, . . . , dxn as the ∗ ∗ 2 18 { 1 } basis of T (E 0), the subspace V2 ΛE∗ (the image of V ΛE∗ under exterior \ ⊂ v ⊗ v multiplication) is spanned by θi θj, θi dxj, σ dxj. The term σ θj is not ∧ ∧ i ∧ i ∧ added because this term is also generated by θi dxj. S2 is generated by σi dxj, v ∧ ∗ 1 ∧ thus the forms σ dxj span V (S2) V2, where V (S2) is V (p S1) ΛM . Taking the exterior derivative∧ of the generators⊂ of V (D¯): ⊗ X X dθi = λjdwij dλj i j = − j − j ∧ X X X X = λjdwij wjk wijλk θj wij − − ∧ − ∧ j j k j X X = λkΩik θj wij − − ∧ k j where the last equality follows from: ! X X X X a b λk dwik + wij wjk = λkΩik = λkΩik,abdx dx (3.3) ∧ ∧ k j k k

17We use the relation a b b a = a b. 18 ⊗ − ⊗ ∧ We express explicitly with this notation that the domain of the sections is E∗.

13 P Since the element θj wij is a linear combination of elements θi dxj, dθi V2 P j ∧ ∧ ∈ if and only if k λkΩik V2. If this 2-form is in V2 it implies that is a linear ∈ v combination of θi θj, θi dxj and σ dxj, i.e. exist αu,v, α¯u,¯ v¯ andα ˜u,˜ v˜ such as: ∧ ∧ i ∧ k k k n n n n n k X X X X X X X X X αuvθu θv+ α¯u¯v¯θu¯ dxv¯+ α˜u˜v˜θu˜ θv˜ = λcΩic,abdxa dxb ∧ ∧ ∧ ∧ u v u¯ v¯ u˜ v˜ a b c

Thus, evaluating (∂λr , ∂λs ) on both sides of the equality implies:

αrs = αsr

Pk Pk Taking any r, s 1, . . . k and recalling that θu θv = θv θu gives αuvθu ∈ ∧ − ∧ u v ∧ θv = 0. Now, evaluating (∂λr , ∂xs ) on both sides again implies:

α¯rs = 0

Taking all r 1, . . . k and s 1, . . . , n we ﬁnally arrive to the point where; if (3.3) ∈ ∈ v is in V2, then it is a linear combination of only σ dxj. Therefore: i ∧ X X dθi Γ(V2) λkΩik Γ(V2) λkΩik Γ(V (S2)) ΩΓ(E) Γ(S2) ∈ ⇔ ∈ ⇔ ∈ ⇔ ⊂ k k On the other hand,

v X X dσ = sijkdλj dxk + λjdsijk dxk = i ∧ ∧ j,k j,k X X X = sijkdθj dxk + sijkλmwjm dxk + λjdsijk dxk = ∧ ∧ ∧ (3.4) j,k j,k,m j,k X v = sijkdθj dxk + (D1σi) ∧ j,k where it is used that X X D1σi = dsijk ej dxk + sijkD1(ej dxk) = ⊗ ⊗ ⊗ jk jk X X = dsijk ej dxk + sijk ej dxk = ⊗ ⊗ ∇ ⊗ jk jk ! X X X = dsijk ej dxk + sijk wjm em dxk = ⊗ ⊗ ⊗ ⊗ jk jk m ! X X X = dsijk ej dxk + simk wmj ej dxk ⊗ ⊗ ⊗ ⊗ jk mk j ! X X = dsijk + simkwmj ej dxk ⊗ ⊗ jk m

14 which combined with (3.1) gives: ! v X X (D1σi) = λj dsijk + simkwmj dxk ∧ jk m X X = sijkλmwjm dxk + λjdsijk dxk ∧ ∧ j,k,m j,k X X = sijkλmwjm,adxa dxk + λjdsijk dxk ∧ ∧ j,k,m,a j,k

v v Similarly to the previous case, (D1σi) could only be spanned by σi dxj. Therefore, the following result is obtained: ∧

v v v dσ Γ(V2) (D1σi) Γ(V2) (D1σi) Γ(V (S2)) D1Γ(S1) Γ(S2) i ∈ ⇔ ∈ ⇔ ∈ ⇔ ⊂ Remarks 3.1. 1. In the case D¯ = the symbol operator is the identity and ∇ therefore S1 = 0. This implies that for V ( ) to be involutive, the curvature must be zero. ∇

2. The proposition still holds for a complex vector space E if the complex structure commutes with the covariant derivative and the differential operator. In this case, V (D¯) is a sub-bundle of the complexified cotangent bundle T ∗(E∗ 0). c \ ∗ ¯ 3. In the complex case, if the distribution V Tc is such that V V = 0 and ¯ ∗ ⊂ ∩ V + V = Tc , an almost complex structure J can be defined: J acts on the elements of V by multiplying by i and by i for the elements of V¯ . It can be also defined on the opposite way. If it− is involutive, then because of the Theorem 3.2, the structure is integrable.

Let us use this proposition to prove the next theorem:

Theorem 3.3. Let M be an oriented 4-manifold . Then a conformal structure on M defines in a natural way an almost complex structure on P (V−), which is integrable iff W− = 0, i.e iff M is self-dual. (Note that the spin representations are well defined projective representations of SO(4), so we need not assume that M is a spin manifold)

Proof. Let us consider the spin bundle V− and a metric within the conformal structure. The Dirac operator is deﬁned by:

∇ 1 σ D : Γ(V−) Γ(V− Λ ) Γ(V+) −→ ⊗ −→ where σ is the Cliﬀord multiplication. On the other hand, the twistor operator D¯ is deﬁned by: ¯ ∇ 1 σ¯ ⊥ D : Γ(V−) Γ(V− Λ ) Γ(V ) −→ ⊗ −→ +

15 ∗ 1 whereσ ¯ is the orthogonal projection 1 σσ onto the kernel of σ in Γ(V− Λ ). This projectionσ ¯ =: (V , Λ1) Ker(σ−) is described locally by [2]: ⊗ ± → 1 X σ¯(t ψ) = t ψ + e e t ψ n i i ⊗ ⊗ i ⊗ · · where n is the dimension of the manifold. Therefore, ! ¯ X X Dψ =σ ¯ ( ψ) =σ ¯ ( iψ dxi) = (¯σ( iψ dxi)) = ∇ i ∇ ⊗ i ∇ ⊗ ! X 1 X = ψ dx + e e ψ e = i i 4 j j i i i ∇ ⊗ j ⊗ · ∇ · 1 X = ψ + ei Dψ ei ∇ 4 · ⊗ P ¯ where it is used that Dψ = i ei ei ψ. Now let us focus on the properties of D. · ∇ 1 The kernel of the symbol, i.e. S1 are those elements of V+ embedded in V− Λ by the Clifford multiplication: ⊗ ! X X 1 X X σ¯( e ψ e ) = e ψ e + e e e e ψ = j j j j 4 i i j j j · ⊗ j · ⊗ i ⊗ j · · ! X 1 X X X X = e ψ e + e e 1 ψ = e ψ e e ψ e = 0 j j 4 i i j j i i j · ⊗ i ⊗ j − · j · ⊗ − i · ⊗ ∗ Using (3.1) at a point φ V , the elements of V (S1) are ∈ − X X V ( ej ψ ej) = ej ψ, φ ej j · ⊗ j h · i for each ψ Γ(V+). This is the subspace of holomorphic 1-forms of the complexified ∈ 1,0 1 ¯ ∗ ∗ cotangent space of M, i.e Λ Λc . Thus, V (D) = V (p S1 p V−) and is a 4- ⊂ ∗ ∗ ¯ ⊕ ¯ dimensional complex sub-bundle of Tc (V− 0) such that V (D) V (D) = 0. As it has been seen above, this defines an almost\ complex structure,∩ that applying the previous proposition, will be integrable. Let us see if this sub-bundle verifies the conditions of the proposition.

P P 1. Taking φ = ei ψ ei S1 where ψ V+, let us see that (φ) = ei · ⊗ ∈ ∈ ∇ · ψ ei: Doing the covariant derivative of φ we obtain: ∇ ⊗ X X X φ = ei ψ ei + ei ψ ei + ei ψ ei ∇ ∇ · ⊗ · ∇ ⊗ · ⊗ ∇ We need to prove that the sum of the ﬁrst item and the third item equals 0. X X ei ψ ei + ei ψ ei = 0 (3.5) ∇ · ⊗ · ⊗ ∇

16 This comes from the compatibility of the connection with the metric, i.e.: ! X X g = 0 ei ei = 0 ( ei ei + ei ei) = 0 (3.6) ∇ ⇔ ∇ i ⊗ ⇔ i ∇ ⊗ ⊗ ∇

Now, the Cliﬀord multiplication by an element ψ V+ is a lineal operation: ∈ ∗ ∗ ∗ ⊗V+ ∗ ∗ ∗ · ∗ ∗ T M T M T M T M V+ T M T M V− T M T M ⊗ ⊗ −−→ ⊗ ⊗ ⊗ −→ ⊗ ⊗ Therefore, multiplying φ to the compatibility condition (3.6) of the metric and realizing that this is a linear operator we prove (3.5) and again: X (φ) = ei ψ ei ∇ · ∇ ⊗ Hence, X D1φ = A( φ) = ei jψ ej ei Γ(S2) ∇ · ∇ ⊗ ∧ ∈ This is also seen by direct computation: ! X X X D1φ =D1 ei ψ ei = D1(ei ψ ei) = ( (ei ψ) ei + ei ψ dei) = i · ⊗ i · ⊗ i ∇ · ∧ · ⊗ X = ( j(ei ψ) ej ei + ei ψ dei) i,j ∇ · ⊗ ∧ · ⊗

Now, using that dei = 0, i and that the connection is torsion free, i.e. iej = ∀ ∇ jei, we have: ∇ X D1φ = (ei jψ ej ei) i,j · ∇ ⊗ ∧

1 2. The elements of Γ(S2) are generated by the image of the map V+ Λ 2 P ⊗ → V− Λ deﬁned by ψ α ei ψ ei α. If this map is injective, then ⊗ 1 2 ⊗ → · ⊗ ∧ V+ Λ V− Λ . Let us see then that the kernel of the map is zero. If P ⊗ ⊂ ⊗ P ψj ej is in the kernel, then ei ψj ei ej = 0. This implies eiψj = ejψi for i =⊗j following that: · ⊗ ∧ 6

eiψj = ejψi eieiψj = eiejψi ψj = eiejψi = ejeiψi ⇒ ⇒ − − ⇒

ejψj = ejejeiψi = ejψj = eiψi ⇒ − This implies that eiψi = 0 which at the same time is equivalent to ψi = 0. Let us prove this by contraposition. If ψi = 0, then ψi = 0 eieiψi = 0 6 − 6 ⇒ 6 and thus eiψi = 0. 6 The map is injective. Therefore

1 2 V+ Λ V− Λ ⊗ ⊂ ⊗ Decomposing these two bundles into irreducible components:

1 2 0 2 V+ Λ = V+ V+ V− = (S V+ S V+) V− = (S V+ V−) V− ⊗ ∼ ⊗ ⊗ ∼ ⊕ ⊗ ∼ ⊗ ⊕ 17 2 2 2 2 2 2 3 V− Λ = V− (Λ Λ ) = (S V+ V−) (S V− V−) = (S V+ V−) V− S V− ⊗ ∼ ⊗ +⊕ − ∼ ⊗ ⊕ ⊗ ∼ ⊗ ⊕ ⊕ it can bee seen that ΩΓ(V−) Γ(S2) if and only if the composition ⊂ Ω 2 Re 3 V− V− Λ S V− −−→ ⊗ −−→ 3 is zero, i.e. the restriction Re to S V−. That would imply that the image 1 is in V+ Λ and therefore, an element of Γ(S2). The homomorphisms from 3⊗ 3 ∗ 3 V− S V− correspond to elements of S V− V− and thus also of V− S V−, → ∗ 4 ⊗ 3 ⊗ because V− = V−. We have that S V− V− S V− and we need this ∼ ⊂ 4⊗ homomorphism to vanish, therefore, the part S V− of the curvature must vanish. This part corresponds tp W− as we have seen in the first section. Therefore, the condition requieres W− = 0. ∗ This complex structure is defined on V− 0. Due to the invariance under scalar x \ ∗ multiplication λ C , this complex structure can be also defined on P (V−) = ∗ ∈ ¯ P (V−). The kernel of the twistor operator D is conformally invariant, hence the complex structure in determined by the conformal class of the metric.

Remarks 3.2. 1. This fibre bundle π : P (V−) M with the complex structure is defined as the twistor space of M, also denoted→ by . It represents the Z fibre bundle where each element Jx is a complex structure on TxM which gives the opposite orientation. Therefore,∈ Z a section of defines an almost complex structure on M. If the complex structure is involutive,Z then it is a 3-dimensional complex manifold. The twistor space is also identified with π : S√ (Λ2 ) M, where S√ is the subspace of Λ2 withZ norm equal to √2. 2 − → 2 − 2. The construction of the complex structure can be explained more geometrically. Fixing a metric in the conformal structure splits the tangent bundle of P (V−) into a vertical and horizontal subspace:

∗ TP (V−) = VP (V−) π TM ⊕

Recalling that V− is a complex vector bundle, it is easy to see that on the vertical subspace VP (V−), the complex structure is defined by the fibres. On the other hand, the complex structure on the horizontal part is defined as follows: Each point φ P (V−)m over m M defines a complex structure ∈ ∈∗ on TmM because of the real isomorphism T M V+ given by α α φ. → → · This complex structure defines the opposite orientation on M. Calling Jm ∗ this complex structure, the complex structure Jφ on π TM is:

∗ JφX = π (Jm)X = Jm(π∗X)

∗ for X π TM φ. ∈ |

3. A similar construction can be done using V+, but in this case, the complex structures in M preserve the orientation. For the almost complex structure on P (V+) to be involutive, it is necessary in this case W+ = 0 [23].

18 4. The fibres of π : P (V−) M are complex submanifolds. Each fibre is a → rational curve, i.e. a copy of the complex projective line CP1 and has as a normal bundle N = (1) (1) (see Definition A.6.7). Let us see this: we ∗ O ⊕ O take fibre `m = (V−)m with λ1, λ2 as linear coordinates. The conormal bundle ∗ v v N at this fibre is spanned by σ1 and σ2 , where:

4 v X σα = ei ψα, φ ei i h · i

∗ ∗ ∗ and ψα (V+)m, φ (V−)m. Both are holomorphic sections of N on (V−)x since, as∈ seen in (3.4)∈ , do not have dλ¯ terms. Thus, they trivialize N ∗ on ∗ ∗ (V−)m. Since they are linear in φ, on P (V−) they trivialize (1)N and ∗ O therefore N = (1) (1) or equivalently N = (V )m (1). O ⊕ O + ⊕ O 0 5. Holomorphic sections H (`m, (1)) are parametrized by (V−)m, hence: O 0 ∗ 1 H (`m,N) = (V V−)m = Hom(V+,V−)m = (Λ )m ∼ + ⊗ ∼ ∼ c as seen in the ﬁrst section.

∗ 0 6. A decomposable element φ ψ (V V−)m deﬁnes a section φlψ of H (`, N) ⊗ ∈ + ⊗ where lψ is the linear form identiﬁed with ψ. This section of N vanishes where ψ, θ = 0 for θ `m. Conversely, those which vanish somewhere in `m are h i ∈ ∗ the set of decomposable elements (V+ V−)m, i.e. elements of rank equal to ⊗ 1 or less than 1 in Hom(V+,V−)m. If α (Λc )m annihilates a spinor ψ, then 0 = α α ψ = (α, α)ψ and α has length∈ 0, and conversely, is α has length 0 then· its· rank is− less than or equal to 1.

7. We have a quaternionic structure on V−, i.e. a function j : V− V− such 2 → that j = Id. Therefore, the restriction of j in P (V−), τ = jP (V−) is an − 2 antiholomorphic involution, τ = Id. On each fibre CP1, i.e. each twistor line `m, τ acts as the antipodal map taking the fibre as S2. P (V−) has no fixed points but leaves the fibres invariant.

Let us ﬁnish this section with a theorem that gives the two main properties that deﬁne the twistor space of a self-dual manifold. The previous remarks prove the following:

Theorem 3.4. Let be the twistor space of a self-dual manifold M, i.e W− = 0. Then [4]: Z

1. The ﬁbres of π : M are holomorphic curves in . Each is a rational Z → Z curve CP1 whose normal bundle in is isomorphic to (1) (1) Z O ⊕ O 2. possesses a free antiholomorphic involution τ : which transforms Zeach ﬁbre to itself. Z → Z

19 Z

T (CP1)= (2) O

1 π− (m)= ℓm ∼= CP1

b m M

Figure 1: Diagram of twistor space

3.4 Properties of the twistor space of a self-dual manifold

Let us study now properties of the 3-complex manifold , the twistor space of Z a self-dual manifold M. For each point m, `m denotes the fibre at this point, the twistor line. A vector v TmM defines a vectorv ˜ of the normal bundle ∈ N = T ` /T `m whose value at a point z `m is given by the inverse of the Z| m ∈ mapping π∗z : Tz TmM evaluated at v. Furthermore, if a conformal structure is defined on M,Z a → horizontal distribution is defined and the sectionv ˜ can be g H g −1 interpreted as a sectionv ˜ T `m , wherev ˜ (z) = (π∗z Hz ) (v). Most of the statements of the next theorem∈ followsZ| from the remarks of| the previous section:

Theorem 3.5. If the conformal structure on M is self-dual, then [23]

0 1. The sections v˜ are in fact holomorphic, i.e v˜ H (`m,N). Furthermore, g ∈ 0 given a metric in the conformal class of M, v H (`m,T ` ). Thus, H` ∈ Z| m m is a holomorphic sub-bundle of T ` isomorphic to N. Z| m 2. The real structure i on induces a real structure (C-antilinear involution) i∗ 0 Z on H (`m,N), which leaves the sections v˜ invariant. Conversely, if a section 0 σ H (`m,N) is real invariant then σ =v ˜ for some v TmM. ∈ ∈ 3. The complexiﬁcation of the linear correspondence v v˜ induces an iso- 0 → morphism (TcM)m = H (`m,N) that preserves the respective real structures. ∼ 0 Thus, TmM is a real form of H (`m,N).

0 4. Every non-zero section σ H (`m,N) vanishes in at most one point of `m. ∈ Sections coming from TmM are nowhere vanishing.

0 5. The metric g on TmM induces a quadratic form g˜ on H (`m,N) whose null 0 cone consist of the sections σ H (`m,N) that vanish at some point of `m. ∈

20 Taking N = (1) (1) as the normal bundle for each twistor line `m, we have 1 O ⊕ O that H (`m,N) = 0 (see [13, page 92]). The twistor lines are compact complex submanifolds. Due to the theorem proved by Kodaira [14], there exists a complex manifold M c that parametrizes the twistor lines and there exists a canonical iso- c 0 c morphism TpM ∼= H (`p,N), where p M and `p is the corresponding twistor line. Due to this isomorphism, M c has a∈ complex dimension 4. The antipodal map c gives, as previously mentioned, a real structure to P (V−) but also to M . The fixed point set of M c is M. We can also derive from the remarks of the previous section that the cone of c 1 null vectors in M is in correspondence with sections of H (lp,N) that vanishes at some point. Taking that N = (1) (1) = (aµ + b, cµ + d), a section vanishes at some point if and only if ad Ocb =⊕ 0. O Therefore, we can say that a vector on the tangent space of M c is null if− the corresponding section obeys the previous quadric form. This null cone is real and defines a conformal structure on M c and also on M using the real structure. A metric within this conformal structure can be fixed afterwards by a volume form. Let us focus now on the case where M is also Ricci flat, thus being an Einstein manifold. The connection on V− induced by the Levi-Civita connection is also flat. If M is simply connected we can use parallel transport to identify all the fibres of M in a way that does not depend on the path. Moreover, if V− is flat, the identification under a closed path is the identity. We have then a fibration π : CP1. If Z → one considers the tangent space at a point z , Tz = Tπ(z)CP1 Nπ(z) = ∈ Z Z ∼ ⊕ ∼ (2)π(z) (1)π(z) (1)π(z), using that T CP1 = (2). Hence, the dual space O ⊕ O ⊕ O ∼ O is: T ∗ ( 2) ( 1) ( 1) . Recalling that K = V3 T ∗ is the z ∼= π(z) π(z) π(z) canonicalZ bundleO − of ⊕, then O −19; ⊕ O − Z Z 3 ^ ∗ Kz = ( ( 2)π(z) ( 1)π(z) ( 1)π(z)) = ( 4)π(z) K = π ( 4) ∼ O − ⊕ O − ⊕ O − ∼ O − ⇒ ∼ O − The bundle Hom(π∗ ( 4),K) is isomorphic to H0(π∗ (4) K) and a non-vanishing section of this vectorO bundle− is identified with an isomorphismO ⊗ in Hom(π∗ ( 4),K). Hence, the twistor space of a Ricci flat manifold M must have a non-vanishingO − section of K π∗ (4). This condition is also sufficient20, and will be used in the next section. ⊗ O

19This is also proved in [13, page 92]. 20This is proved in [4, page 390].

21 4 Construction of Gravitational Instantons

The aim of this section is to construct gravitational instantons using the tools of the previous section. For this purpose we will use the Reverse Penrose construction. First of all, a brief introduction to gravitational instantons is presented. Afterwards, the ideas introduced by Hitchin at [12] are explained.

4.1 Gravitational Instantons

Classical vacuum Einstein equations derive from the Euclidean gravitational action: Z Z 1 1 4 1 1 3 S[g] = (R 2Λ) g 2 d x [K]h 2 d Σ (4.1) −16πG M − − 8πG ∂M where Λ is the cosmological constant, R is the Ricci scalar and the second integral is the boundary term that will be described latter. Gravitational instantons could be defined to be complete non-singular positive definite solutions of the classical vacuum Einstein equations, either with a Λ term or without it. The property of self-duality is sometimes included by some authors into the previous definition (because of the notion of instanton). Some different topological properties can be found among these solutions which let classify them. There are solutions which can be considered the closest analogues to Yang-Mills instantons. These metrics are non-compact solutions with Λ = 0 and are asymptotically flat in all four dimensions. The Euclidean gravitational action (4.1) is not invariant under conformal transformations. As a result, since any asymptotically Euclidean metric must be an ex- tremum, it will have zero action. Due to the Positive Action Theorem [20] (which states that any asymptotically Euclidean self-dual metric has positive action except for the flat space which has zero action), there won’t be any self-dual asymptotically Euclidean metrics. The latter theorem does not exclude working with Asymptotically Locally Eu- clidean (ALE) spaces. These solutions tend to a flat metric modulo identification under a discrete subgroup of SO(4). This solutions are Ricci flat. The objective of this section is obtain these last solutions using twistor theory. We will work for the case when the topology of the manifold at infinity is S3/Γ where Γ is a cyclic subgroup of SU(2) acting on the unit sphere S3 in C2.

4.2 Hitchin’s Construction

The aforementioned cyclic subgroup acting on C2 is represented by the set of matrices: 2πin ! e k 0 − 2πin , 0 n k 0 e k ≤ ≤

22 We can see that the origin of the quotient space C2/Γ contains a singularity. As it is commonly done in algebraic geometry, affine algebraic varieties can be represented 2 in terms of the ring of functions from the variety itself to a field. If (z1, z2) C , k k ∈ the monomials z1 , z2 , z1z2 are invariant under left multiplication by elements of Γ. k k k If we take x = z1 , y = z2 , z = z1z2, they satisfy xy = z . This gives an isomorphism 2 3 k between C /Γ and (x, y, z) C xy = z , i.e. the set of solutions of the complex ∈ | surface xy zk = 0 in C3. − We can use now deformation theory to solve this singularity. One way to make a deformation is by adding lower terms in z and obtaining a family of deformations that also have the topology of S3/Γ at infinity:

k k−1 xy = z + a1z + ak ··· If the discriminant of the right hand side is non-zero for a particular selection of the lower terms , the resulting manifold has no singularity. This occurs because of the regular value21 theorem22. Therefore, we would take M to be the underlying 4-dimensional diﬀerentiable manifold of such a surface. In the previous section the relation between self-dual manifolds and complex geometry has been explained. This reformulation of the self-dual Einstein equations means that in order to ﬁnd these gravitationals instantons it is necessary to look for the following:

(1) A complex 3-manifold , the total space of a holomorphic fibration π : Z Z → CP1 (2) A 4-parameter family of sections, each with normal bundle N = (1) (1) O ⊕ O (3) A non-vanishing holomorphic section θ of K π∗ (4) ⊗ O (4) A real structure on such that π and s are real, and is fibred by the real Z Z sections of the family. Here CP1 is given the real structure of the antipodal map, i.e u u¯−1 → − In order to construct we need a family of complex structures on M which are Z parametrized by u CP1. The fibre of each u CP1 defines a curve in (where each point of the curve∈ is parallel transported from∈ u) which is also a sectionZ from M to . This section defines a complex structure on M as we have seen in the previousZ section. We can say then that we can construct a family of complex structure parametrized by u. We call ¯ the first approximation of and is defined by: Z Z k k−1 xy = z + a1(u)z + ak(u) (4.2) ··· where ai(u) is locally a holomorphic function of u. We emphasize the term locally because CP1 is a compact complex manifold and therefore global functions are 21see Definition A.6.6. 22If the discriminant is not zero, all the roots are different and then the differential of xy zk k 1 k k 1 − − a z − a is not 0 at the roots. Therefore, the value 0 of the function xy z a z − a 1 −· · ·− k − − 1 −· · ·− k is a regular value and the pre-image is a smooth submanifold of C3.

23 constant functions. Vector bundles of dimension 1, i.e line bundles, on CP1 are isomorphic to (k) for k Z. Therefore, we take ai(u) as a line bundle, i.e. ai(u) (n i) and itO implies x ∈ (l), y (m), z (n) where l + m = kn in order for∈ theO equation· to make sense.∈ O Hence,∈ O¯ is a 3-dimensional∈ O hyper-surface solution of the aforementioned equation in the complexZ 4-manifold (l) (m) (n). Each O ⊕ O ⊕ O u CP1 will then define the 2-dimensional complex manifold. ∈ ¯ 2 We need a real structure on . The map on C defined by (z1, z2) ( z¯2, z¯1) Z → − induces a real structure on CP1, which is the aforementioned antipodal map σ. Since Γ acts by left multiplication, it conmutes with the real structure and it induces also a real structure on C2/Γ given by: x ( 1)ky,¯ y x,¯ z z¯. We can use this to define a real structure on (l) →(m)− (n): → → − O ⊕ O ⊕ O τ(x, y, z, u) = ( 1)kσ(y), σ(x), σ(z), σ(u) − y¯ x¯ z¯− = ( 1)k , , , u¯−1 − u¯k u¯k −u¯2 − where σ : (m) (m) is the antipodal map defined before. This real structure O → O ¯ requires that l = m. If the sections are also real, i.e. σ(ai(u)) = ai(σ(u)), then ¯ Z has a real structure and the fibration is also real π : CP1. Z → At this point, we have obtained a real structure for ¯ and a real projection Z to CP1. However, as we have explained before, at the points uk CP1 where the discriminant is zero the resultant is not a manifold. To obtain a∈ non-singular ¯ manifold from we can resolve these singularities by taking a finite covering of CP1 Z branched over those points uk. That would imply that the base Riemann manifold where the new non-singular manifold is fibred is of higher genus23. However, taking the assumption that Z

k k X k−i Y n ai(u)z = (z pj(u)) pj Γ(H ) i=0 j=1 − ∈ we obtain a complex 3-manifold with a real structure and a compatible fibration Z π : CP1. So, is not fibred over another Riemannian surface but over CP1. TheZ reason → for thisZ is found in [5] . In order to determine the values of l, m, n, we use the previous condition l = m, l + m = kn because of the compatibility with (4.2) and finally we use now the aforementioned condition (2). The normal bundle of CP1 can be identified as the tangent space of the fibres, i.e. with the tangent space of each π−1(u). If we take a section which misses the singularity, the tangent space of the fibres is the kernel of this vector bundle homomorphism

(fx, fy, fz): (l) (m) (n) (kn) O ⊕ O ⊕ O → O This is also due to the regular value theorem: all the elements contained in the submanifold have as image through f the regular value mentioned before, hence

23The genus g of a closed Riemann surface can be deﬁned in terms of the Euler characteristic, χ = 2 2g. − 24 the tangent vectors on this manifold are in the kernel of df. Using the ﬁrst Chern class, we know that c1(N) = c1( (1) (1)) = 2 and in our case, c1(N) = l + m + n kn = n. Therefore, n =O 2 and⊕ Ol = m = k. −

b b

TCP1 b NCP1

CP1 b b

b b M

Figure 2: Projection to CP1 We also need a non-vanishing section of the vector bundle K π∗ (4) on because of condition (3) . Let us make the assumption that a constant⊗ O multipleZ of the standard form is non-vanishing on : Z dx dy dz dy dz du dx dz du dx dy du ∧ ∧ = ∧ ∧ = ∧ ∧ = ∧ ∧ fu fx fy fz Once we will have the metric this fact will be noticeable from its existence.

Finally, let us see how the sections of this vector bundle on CP1 are parametrized by a 4-dimensional complex manifold as Kodaira’s theorem implies. Through this identification we define then sections of (k) (k) (2). A section of (m) can be defined by a polynomial of complexO coefficients⊕ O ⊕ of O degree m in u (seeO [13, page 91]) . Hence, we take polynomials x(u), y(u), z(u) such that:

k Y x(u)y(u) = (z(u) pj(u)) j=1 −

2 2 where z(u) = au + 2bu + c and pi = aiu + 2biu + ci. Taking αi and βi the roots of the polynomial z(u) pi(u) and doing a simple factorization we obtain: − k k Y Y x = A (u αi) , y = B (u βi) i=1 − i=1 −

25 Q where AB = (a ai). At this point, the curves are parametrized locally by (a, b, c, A) where the− four of them are complex numbers. Let us see how the real lines are parametrized. To obtain real lines we must impose that they are compatible with the real structure. Therefore, as it is seen before

x(σ(u)) = ( 1)kσ(y(u)), y(σ(u)) = σ(x(u)), z(σu) = σ(z(u)) − − and the third condition implies that c = a,¯ b = ¯b: − z(σu) = au¯−2 2bu¯−1 + c − a¯u¯2 + 2bu¯ + c σ(z(u)) = = cu¯−2 2bu¯−1 a¯ − − u¯2 − − − Therefore, the roots are:

p 2 p 2 2 2(b bi) 4(b bi) 4(a ai)(c ci) (b bi) (b bi) + a ai − − ± − − − − = − − ± − | − | 2(a ai) a ai − − 2 Calling ∆i the real positive square root of (b bi) (a ai)(c ci), then − − − − k k Y (b bi) ∆i Y (b bi) + ∆i x = A u + − − , y = B u + − (4.3) a ai a ai i=1 − i=1 −

Finally, imposing again the reality condition on x(u) and y(u) and that AB = Q (a ai): − ¯ Y AA = ((b bi) + ∆i) (4.4) − Therefore, A 2 = AA¯ can be described in terms of a, b and we only need its angle to define A|completely.| Therefore, with the real variables Re(a), Im(a), b and the angle arg(A) we can parametrize the real lines. We still have the indeterminacy of the polynomials pi, i 1 i k, therefore each metric is determined by fixing the ∈ ≤ ≤ coefficients, which is the same as taking k points in R3. Taking a general point (a, b, c, A) of the manifold which parametrizes the sections in ¯, i.e. the manifold M c, we obtain a section x(u), y(u), z(u) by Kodaira’s theorem.Z This theorem also associated each tangent vector of the point (a, b, c, A) to a section of the normal bundle of the associated section x(u), y(u), z(u). We will call (a0, b0, c0,A0) the tangent vector and x0(u), y0(u), z0(u) the section of the normal bundle. We can now define the null cone on M c as the sections of the normal bundle which vanish at some point. Therefore, we need a common zero for x0(u), y0(u), z0(u): z0(u) = a0u2 + 2b0u + c0 = 0 (4.5) x0(u) A0 X α0 = i = 0 (4.6) x(u) A − u αi − x0(u) 0 Q 0 where the second equation follow from: x(u) = ln(x(u)) = ln (A (u αi)) = 0 0 − P 0 A P αi (ln A + ln(u αi)) = . − A − u−αi

26 c−ci On the other hand, using αiβi = and that z pi = (a ai)(u αi)(u βi) we a−ai obtain: − − − − −1 z pi = (u αi) (a ai)u (c ci)α − − − − − and diﬀerentiating it gives:

0 0 −1 0 0 −1 −2 0 0 = z (u) = α (a ai)u (c ci)α ) + (u αi)(a u c α + (c ci)α α ) = − i − − − i − − i − i i 0 −1 −2 0 0 −1 = α (a ai)u + (c ci)α + (u αi)(c ci)α + (u αi)(a u c α ) i − − − i − − i − − i ⇒ 0 0 0 0 0 α a uαi c αia u c i = − = − ⇒ u αi (a ai)u(αi βi) 2u∆i − − − 2 1/2 where 2∆i = (a ai)(αi βi) = 2(b ac) . Once we obtain− this result,− we substitute− it in (4.6):

0 0 0 0 0 A X αia u c 2A X αia X c = − u 0 + = 0 A 2u∆i ⇒ A − ∆i ∆i

Changing the variables γ = P 1 , δ = P αi : ∆i ∆i 2A γc0 u δa0 + γc0 = 0 u = − A0 − ⇒ 2A δa0 A0 − and substituting in (4.5):

γ2c02 γc0 2A 2A 2 a0 +2b0 − +c0 = 0 γ2a0c0 2γb0 δa0 + δa0 = 0 2A 02 2A 0 0 0 δa 0 δa ⇒ − A − A − A0 − A − 0 2A 02 where we have divided the entire equation by c and multiplied by A0 δa . Expanding now the equation: − 4A0 4A02 4A0 γ2a0c0 γb0 + 2δa0γb0 + + δ2a02 δa0 = 0 − A A2 − A Now, adding and subtracting γ2b02, we obtain:

2A0 2 γ2(a0c0 b02) + δa0 γb0 = 0 − A − − which deﬁnes the null cone. At this point , we have considered a general point curve x(u), y(u), z(u). The computations for real lines made before gives:

0 0 0 2A X (b bi) + ∆ Re = − i = γb0 + Re(δa0) A (b bi) + ∆i − Let us see this equality. From (4.4):

¯ ¯ Y X ln AA = ln A+ln A = 2Re ln A = ln ((b bi) ∆i) = ln ((b bi) ∆i) − − − − ⇒ 0 0 0 0 A 2A X (b bi) + ∆ (2Re ln A)0 = 2Re = Re = − i ⇒ A A (b bi) + ∆i − 27 On the other hand, 0 0 a ai 0 1 0 1 0 a | − | = (ln a ai ) = ln ((a ai)(¯a a¯i)) = (2Re ln(a ai)) = Re a ai | − | 2 − − 2 − a ai | − | − −(b−bi)+∆i Hence, using that αi = a−ai 0 0 X a ∆i (b bi) X a ∆i (b bi) Re(δa0) = Re − − = Re − − a ai ∆i a ai ∆i ⇒ − − 0 X a ai ∆i (b bi) Re(δa0) = | − | − − ⇒ a ai ∆i | − | and therefore we ﬁnally obtain: 0 0 X a ai ∆i (b bi) b Re(δa0) + γb0 = | − | − − + = a ai ∆i ∆i | − |  0  a ai −1 2 0 −1 0 | − | ∆i ∆i (b bi) + b + ∆i (b bi)b X  a ai − − −  =  | − |  =  (b bi) + ∆i  −

0 2 0  a ai a ai (b bi)b  | − | | − | + b0 + − X  a ai ∆i ∆i  =  | − |  =  (b bi) + ∆i  −

 0 0  a ai a ai + (b bi)b 0 | − || − | − + b 0 0 X  ∆i  X (b bi) + ∆i =   = −  (b bi) + ∆i  (b bi) + ∆i − −

Therefore, applying this to the null cone formula and using c0 = a¯0 − 2A0 2 γ2(b02 + a0a¯0) + Im δa0 A − where we have obtained a positive definite metric. We can use the proved equality 0 2 2A0 02 to express the metric in the Hermitian form. Taking (γb ) = Re A δa , we have that − 2A0 2A¯0 γ2a0a¯0 + δa0 δa0 (4.7) A − A¯ − At this point we have expressed the conformal structure in terms of the coordinates that parametrizes the complex sections. We can now express this conformal structure in terms of local coordinates on a non-singular fibre of π : CP1. We take this non-singular fibre as π−1(0) without loss of generality, i.e. Zu →= 0 24. We have then that the resulting surface are those x, y, z such that: Y Y xy = (z ci) = (z +a ¯i) (4.8) − 24We can always do a translation of the coordinates in order to miss the singularity.

28 for ai = aj for i = j. We have that (x, y) are coordinates at those points where 6 6 25 fz = 0 and (y, z) are also coordinates where fx = y = 0 . We begin first by taking (y,6 z) as the coordinates. Therefore, z(0) = c =a ¯ 6and − Y (b bi) + ∆i 1 Y ¯ y = B − = ((b bi) + ∆i) = A (a ai) A − − where the first equality follows from (4.3). Substituting these results on (4.7): 2dy 2dy¯ γ2dzdz¯ + + δdz¯ + δdz¯ (4.9) y y¯ We use now the holomorphic section defined before to fix the metric in the conformal structure, i.e. dy dz du dy dz du s = ∧ ∧ = ∧ ∧ fx y which define the holomorphic 2-form on each fibre and the volume form is thereby dy∧dz∧dy¯∧dz¯ 2 ω = |y|2 . Now, computing the volume form of (4.9) gives 4γ ω. Hence, the metric up to constant multiplication is: 2dy 2dy¯ γdzdz¯ + γ−1 + δdz¯ + δdz¯ y y¯ where

X 1 X X (b bi) ∆i 2 2 − 2 −1 γ = ((b bi) + z¯ + ai ) = ∆i , δ = − − − | | ∆(¯z + ai) and the term b is deﬁned by the equation Y ((b bi) + ∆i) = yy¯ (4.10) −

Taking x1 = Re(¯z), x2 = Im(¯z), x3 = b an after some manipulations, we obtain: − − γd~xd~x + γ−1(dτ + ~wd~x)2 (4.11) where γ = P 1 and ~ γ = ~ ω. This is the metric deﬁned by Gibbons and |~x−~xi| ∇ ∇ × Hawking at [11]. It may appear in (4.9) that there is a singularity at y = 0. However, at this point, (y, z) are not coordinates. At y = 0, due to (4.8), z = ai for some i. Since ai = aj − 6 and fz = 0 in a neighborhood of z = ai we have that (x, y) are coordinates. Let us see if6 it is well deﬁned. −

Due to (4.8), we have that near z = ai: − xy z + ai = Q = hxy j6=i(z + aj)

25This can bee derived also recalling the regular value theorem. The projection to the coordinates x, y near a point p of will be one to one if ∂ is not in T ( ). Recalling that the tangent Z z p Z space are those vectors (a, b, c) that contracted to the forms fx, fy, fz are equal to 0, we see that ∂ = (0, 0, 1) is not in the tangent space if f = 0. z z 6 29 where h is differentiable and non-vanishing . Moreover, due to (4.10), we have that yy¯ (b bi) + ∆i = Q = gyy¯ − ((b bj) + ∆j) j6=i − where g is also differentiable and positive. Using now the definition if ∆i and the previous equations:

2 2 2 2 2 2 2  ∆ = (b bi) + z + ai = (b bi) + h x y i − | | − | | | | | |  g2 y 2 h 2 x 2 b bi = | | − | | | | 2 2 4 2 2 ⇒ − 2g ∆ = g y + (b bi) 2(b bi)g y  i | | − − − | | g2|y|2+|h|2|x|2 2 2 2 2 and therefore, ∆i = . Taking now r = g y + h x : 2g | | | | | | 1 X 1 2g γ = + = 2 + γ0 ∆i ∆j r j6=i where γ0 is diﬀerentiable. On the other hand:

|h|2|x|2 (b bi) ∆i X (b bj) ∆j g hx δ = + = − + δ = 2 + δ − − − − r2 0 2 0 ∆i(¯z + ai) ∆j(¯z + aj) ¯ − r y¯ j6=i 2g hx¯y¯ where δ0 is again diﬀerentiable. Using the previous expressions and dz = h(xdy + ydx) + xydh:

2dy ¯ 2 2 2 ¯ 2 ¯ + δdz = g ydy¯ h xdx¯ h x dh + δ0dz y r2 − | | − | | Now, the metric near y = 0 is well-defined if and only if 2g r2 4 dzdz¯ + (g2ydy¯ h 2xdx¯ h¯ x 2dh)(g2ydy¯ h 2xdx¯ h¯ x 2dh) r2 2g r4 − | | − | | − | | − | | is regular. Making some manipulations, we rewrite it as: 2g−1 h2dxdx¯ + g2dydy¯ + hxdhd¯ x¯ + hxd¯ hdx¯ + x 2dhdh¯ | | which is regular since g is non-vanishing. Let us use the form of the metric introduced by Hawking and Gibbons (4.11) to see the asymptotically locally behavior of the metric. The topology of the manifold at infinity can be computed taking ~xi to be zero for all i = 1, . . . , n. Therefore, n n cos θ 3 γ r and ω = 0 ~ar + 0 ~aθ + r sin θ ~aφ where r, θ, φ are spherical coordinates in R . The' 1-form connection· w· d~r, where· d~r = (dr, rdθ, r sin θdφ), satisfies the condition · of self-duality, i.e ~ γ = ~ ω. Hence, the metric at infinity is: ∇ ∇ × r n ds2 (dτ + n cos θdφ)2 + dr2 + r2dθ2 + r2 sin2 θdφ2 ' n r 2 changing r ρ , → 4n ! ρ2 dτ 2 ds2 dρ2 + + cos θdφ + dθ2 + sin2 θdφ2 ' 4 n

30 The local topology of the manifold at each ~xi is equivalent to taking n to be 1 2 (r would be the distance to the new origin, ~xi). Fixing θ, φ, the metric is ds 2 2 dτ 2 ' dρ + ρ 2 . The canonical singularity at ~xi is avoided if 0 τ 4π. Turning to the topology at infinity and taking 0 τ 4π, the metric≤ (4.11)≤ for any n is equivalent to the flat metric with the Euler≤ angle≤ ψ/n. Hence, the topology of the 4 2 manifold at long distances of the positions ~xi is R /Zn C /Zn. ' Finally , let us see again the passage from ¯ to , where we have resolve the Z Z singularities. For every u in CP1, a section on is defined, which at the same time determines a complex structure on M, dueZ to the holomorphic map from M ¯. In particular, for a singular fibre in ¯ we would have a complex manifold M →mappingZ onto the singular fibres, hence procuringZ a resolution of the singularity. Q The singular surfaces arise when we have values of u for which xy = (x pi(u)) − have singularities. This occurs when we have double points i.e. when pi(u) = pj(u) for some u = u0 and for some i = j. At this point, the singularity is in x = 0, y = 26 6 0, z = p(u0) . Using the metric we will see the transformation of this singular point in ¯ to a sphere in . We take a real curve that goes through the singularity, Z Z hence we have (x(u), y(u), z(u)) that goes through the singularity (0, 0, p(u0)) at u = u0. We have that pi(u0) = pj(u0) and that z, pi, pj also have the same value at −1 u¯0 i.e. the antipodal point, because they are real polynomials. − 2 Let us see this. If f(u0) = au0+2bu0+c = k, then for a real quadratic polynomial, i.e.a ¯ = c and ¯b = b: − 2 2 a + 2bu¯0 cu¯ c¯ + 2bu¯0 +a ¯u¯ f( u¯−1) = a( u¯−1)2 + 2b( u¯−1) + c = − − 0 = 0 = − 0 − 0 − 0 u¯2 u¯2 − 0 − 0 au2 + 2bu + c = 0 0 = ku−2 u¯2 − 0 − 0 Therefore, if the two polynomials have the same value for u0, they do also on the antipodal image. Then, there is a linear dependence of the polynomials, i.e.: z pi = λ(pj pi) 0 − 0 − λ = 1 λ z pj = λ (pi pj) ⇒ − − − for a real λ. The roots of the pj pi are: −

(bj bi) ∆ij − − ± aj ai − therefore, taking the positive roots for x of both polynomials, z pi and z pj, we obtain: − − λ(bj bi) + λ ∆ij (λ 1)(bj bi) + λ 1 ∆ij − − | | − − − | − | λ(aj ai) (λ 1)(aj ai) − − − Assuming that pr(u0) = ps(u0) for any other pair (r, s), then these two roots have 6 two be distinct, in order to x vanish at both roots of pj pi. Let us see what is the − 26 Again, due to the regular value theorem, we need fx = fy = fz = f = 0 at this point.

31 requirement for both roots to be diﬀerent:

λ(bj bi) + λ ∆ij (λ 1)(bj bi) + λ 1 ∆ij − − | | − − − | − | = λ(aj ai) − (λ 1)(aj ai) − − −

( λ (λ 1) λ λ 1 )∆ij λ (λ 1) λ λ 1 = | | − − | − | = 0 | | − − | − | = 0 λ(λ 1)(aj ai) 6 ⇔ λ(λ 1) 6 ⇔ − − − Sign(λ 1) = Sign(λ) ⇔ − − Therefore, Sign(λ 1) = Sign(λ) and it implies λ (0, 1). On the non-singular ﬁbre u = 0, the corresponding− − real lines trace out a 2-sphere:∈

x = pλ(λ 1)f(λ)e−iθ − y = pλ(λ 1)g(λ)eiθ − z = a¯i + λ(¯ai a¯j) − − for 0 λ 1, 0 θ 2π and f, g determined by a change of coordinates from a, b, A≤to x,≤ y, z.≤ This≤ gives in a certain way the form of the resolution of the singularity seeing the intersection of the curves x(u), y(u), z(u) mentioned before in a non-singular ﬁbre. What we have done actually is a blow-up27 of the singularity. Therefore, the process for the resolution involves ﬁrst the branching over the singular points and then a blow-up.u

Figure 3: Representation of a blow-up

27see Deﬁnition A.6.8

32 5 Conclusions

Twistor theory has let us construct solutions of the Einstein equations using algebraic methods. Thanks to the relations between self-dual manifolds and complex geometry, we have been able to use the more powerful tools of complex geometry to find the metric. We have also derived the requirements for a manifold to have a twistor space and as well as how to define a metric on this manifold inside its conformal class. The aim of this “Treball de Fi de Grau” has been to construct gravitational instantons for the specific case when Γ is a cyclic subgroup of SU(2). Hitchin suggested on his work [12] the existence of other families of ALE gravitational instantons with other finite subgroups Γ. As a matter of fact, some solutions were actually found. One example is the work by Cherkis and Hitchin [7], where they constructed gravitational instantons for the case Γ = Dk. One of the two methods they used for this construction follows the twistor approach; but in this case for the polynomial x2 zy2 = a, in contrast to the polynomial xy = zk used in this work. − The conjecture made by Hitchin was later proved by Kronheimer [15]. In this work, Kronheimer proves that every ALE solution to self-dual Einstein equations is diffeomorphic to a minimal resolution of C2/Γ.

33 A Appendix. Basic Material

A.1 Lie Groups

Deﬁnition A.1.1. A Lie group is a diﬀerentiable manifold which is endowed with a group structure such that the group operations

(i)∆: G G G,(g1, g2) g1 g2 × → 7→ · (ii) −1: G G, g g−1 → 7→ are diﬀerentiable. The dimension of a Lie group G is deﬁned to be the dimension of G as a manifold.

Exemple A.1.1. Let V be a finite-dimensional vector space over R or C. The set Aut(V ) of linear automorphisms of V is an open subset of finite-dimensional vector space End(V ) of linear maps V V (we can deduce it using the continuity of the determinant function). Thus Aut(→ V ) has a structure of a differentiable manifold. The group operation of Aut(V ) is matrix multiplication , which is algebraic and hence differentiable. Therefore Aut(V ) has a canonical structure as a Lie group, and we get the groups

n n GL(n, R) = AutR(R ) GL(n, C) = AutC(C ). Definition A.1.2. There is up to isomorphism only one proper finite field exten- sions of R, which is C. There is, however, a skew field containing C of complex dimension 2 and real dimension 4, called the quaternion algebra H, which could be defined as: the R-algebra H is the algebra of (2 2) complex matrices of the form: × a b ¯b a¯ − with matrix addition and multiplication. Every nonzero h H has a multiplicative ∈ inverse. H is a complex vector space, C acting by left multiplication. It standard basis is: 1 0 0 1 1 = and j = 0 1 1 0 − with the rules for multiplication zj = jz¯ for z C and j2 = 1. This gives the standard isomorphism of complex vector spaces ∈ − 2 a b C H, (a, b) a + bj = → → ¯b a¯ − H has a conjugation anti-automorphism:

i : H H, h = a + bj i(h) = h¯ =a ¯ bj, a, b C → → − ∈ The norm on H is deﬁned by N(h) = h h¯ = h¯ h · · 34 As a real vector space H has a standard basis consisting of the four elements 1 0 i 0 0 1 0 i 1 = , i = , j = , k = 0 1 0 i 1 0 i 0 − − − with the rules for multiplication

i2 = j2 = k2 = 1, ij = ji = k, jk = kj = i, ki = ik = j − − − − The quaternion ai + bj + ck with a, b, c R are called pure quaternions. Each ∈ h H has a unique expression h = a + b where a R and b is a pure quaternion. Conjugation∈ in this notation is expressed as ∈

i(a + b) = a b − and therefore N(a + b) = a2 b2. − Deﬁnition A.1.3. The group

Sp(1) = h H N(h) = 1 { ∈ | } is called the quaternion group. In matrix notation Sp(1) consist of the matrices

a b such aa¯ + b¯b = 1 ¯b a¯ − Deﬁnition A.1.4. Let a and g be elements of a Lie group G. The right-translation (right action) Ra : G G and the left-translation (left action) La : G G of g by a are deﬁned by→ → Rag = ga

Lag = ag

By definition, Ra and La are diffeomorphisms from G to G. Hence, the maps Ra and La induce La∗ : TgG TagG and Ra∗ : TgG TgaG. → → Definition A.1.5. Let X be a vector field on a Lie group G. X is said to be left-invariant vector field if La∗ X = X . |g |ag Definition A.1.6. The set of left-invariant vector fields g with the Lie bracket [ , ]: g g g is called the Lie algebra of a Lie group G. × → Theorem A.1.1. Let µ : G M M be an action of G on M on the left. Assume × → that p M is a fixed point, that is, µσ(p) = p for each σ G. Then the map ∈ ∈

ϕ : G Aut(TpM) → deﬁned by ϕ(σ) = d(µσ) = µσ∗ : TpM TpM → is a representation of G.

See [22] for the proof.

35 Deﬁnition A.1.7. A Lie group G acts on itself on the left by inner automorphisms:

−1 φ : G G G, φ(σ, τ) = στσ = φσ(τ) × → The identity is a ﬁxed point of this action. Hence, by the previous theorem, the map σ d(φσ): TeG = g TeG = g 7→ ∼ → ∼ is a representation of G into Aut(g). This is called the adjoint representation for a Lie group and is denoted by

Ad : G Aut(g) → Deﬁnition A.1.8. It is also posible to deﬁne a Lie Algebra representation taking the derivative of Ad at the identity. Therefore, it is obtained:

d(Ad)e = ad : g End(g) → such that, for x, y g ∈ ad(x)(y) = adx(y) = [x, y] This is called the adjoint representation for a Lie algebra.

A.2 Cliﬀord Algebras and Spin Groups

Definition A.2.1. Let V be a finite dimensional K-vector space over a commutative field K and Q : V K a quadratic form. The Clifford Algebra C`(Q) is an → associative K-algebra with unit 1 defined as follows. Let

∞ r X O T (V ) = V r=0 denote the tensor algebra of V , and define IQ(V ) to be the ideal in T (V ) generated by elements of the form v v+Q(V )1 for v V . Then the Clifford algebra is defined to be the quotient ⊗ ∈ C`(Q) T (V )/IQ(V ) ≡ There is a natural embedding

j : V, C`(Q) → which is the image of V under the canonical projection

πQ : T (V ) C`(Q) −→ Corollary A.2.1. The linear map j : V, C`(Q) from the vector space V of the quadratic form into its Cliﬀord algebra is→ injective.

36 Remark A.2.1. The algebra C`(Q) is generated by the vector space V C`(Q) subject to the relations: ⊂ v v = Q(V )1 · − for v V . If the characteristic of K is not 2, then for all v, w V ∈ ∈ v w + w v = 2Q(v, w) · · − where 2Q(v, w) Q(v + w) Q(v) Q(w) is the polarization of Q. ≡ − − Deﬁnition A.2.2. An element x T (V ) is of pure degree s if x Ns V . ∈ ∈ Proposition A.2.1. Let f : V A be a linear map into an associative K-algebra with unit, such that → f(v) f(v) = Q(v)1 · − for all v V . Then f extends uniquely to a K-algebra homomorphism f˜ : C`(Q) ∈ → A . Furthermore, C`(Q) is the unique associative K-algebra with this property.

See [16, page 8] for the proof. Proposition A.2.2. The Cliﬀord algebra C`(Q) of a quadratic form is equipped with an involution β : C`(Q) C`(Q) such that: → a) β is an algebra homomorphism and an involution, i.e. β2 = Id b) Setting C`0(Q) = x C`(Q): β(x) = x , C`1(Q) = x C`(Q): β(x) = x we have the splitting{ ∈ } { ∈ − } C`(Q) = C`0(Q) C`1(Q) ⊕ and the relations C`0(Q) C`0(Q) C`0(Q), C`0(Q) C`1(Q) C`1(Q) as well as C`1(Q) C`1(Q) C`0·(Q). In particular,⊂ C`0(Q) · C`(Q)⊂ is a subalgebra. · ⊂ ⊂ See [10, page 5] for the proof.

Deﬁnition A.2.3. An algebra with this decomposition is called Z2-graded algebra. Proposition A.2.3. Let V be an n-dimensional vector space. Then the vector space C`(Q) has dimension 2n.

See [10, page 8] for the proof. Definition A.2.4. There is a natural filtration F˜0 F˜1 F˜2 T (V ) of the tensor algebra , which is defined by ⊂ ⊂ ⊂ · · · ⊂

s r X O F˜s = V r=0

˜s ˜s0 ˜s+s0 s ˜s and has the property that F F F . If we set F = πQ(F ) we obtain a ⊗ ⊆ ﬁltration F 0 F 1 F 2 C`(Q) which also has the property that ⊂ ⊂ ⊂ · · · ⊂ 0 0 F s F s F s+s · ⊆ 37 for all s, s0. This makes C`(Q) into a ﬁltered algebra. Using this last property, the multiplication map descends to a map (F r/F r−1) (F s/F s−1) (F r+s/F r+s−1) ∗ L∞ r r ×r r−1 → for all r, s. Setting G r=0 G , where G F /F , we obtain the associated graded algebra.≡ ≡ Proposition A.2.4. For any quadratic form Q, the associated graded algebra of C`(Q) is naturally isomorphic to the exterior algebra Λ∗V .

See [16, page 10] for the proof. Proposition A.2.5. There is a canonical vector space isomorphism Λ∗V ≈ C`(Q) −−→ compatible with ﬁltrations.

See [16, page 11] for the proof. Definition A.2.5. Consider the real vector space V Rn and the Clifford algebra ≡ 2 C`n (notation has changed for this particular case) of the quadratic form x1 2 n n − − ... xn. R itself is a linear subspace of C`n. For every vector x R , the equality − ∈ x x = x 2 · −k k −1 holds in C`n and hence the inverse element x is given by x x−1 = − x 2 k k We define then the pin group P in(n) C`n as the group which is multiplicatively generated by all vectors x Sn−1. Therefore⊂ , the elements of P in(n) are the ∈ n−1 products x1 xm with xi S . The spin group, Spin(n), is defined as Spin(n)=Pin(·····n) C`0 . ∈ ∩ n Definition A.2.6. Let X and X˜ be connected topological spaces. The pair (X,˜ p), or simply X˜, is called the covering space of X if there exists a continuous map p : X˜ X such that → (i) p is surjective (ii) for each x X, there exists a connected open set U X containing x, such that p−1(U∈) is a disjoint union of open sets in X˜, each⊂ of which is mapped homeomorphically onto U by p.

In particular, if X˜ is simply connected, (X,˜ p) is called the universal covering space of X. Definition A.2.7. Certain groups are known to be topological spaces, they are called topological groups. If X and X˜ in the previous definition happen to be topological groups and p : X˜ X to be a group homomorphism, the universal covering space is called the (universal→ ) covering group. Definition A.2.8. The universal covering group Spin(n) of SO(n) is the spin group.

38 A.3 Representation Theory

Deﬁnition A.3.1. A representation of the Lie group G on the (ﬁnite dimensional complex) vector space V is a continuous action

ρ : G V V × → of G on V such that for each g G the translation Lg : v ρ(g, v) is a linear map. We call the pair (V, ρ) a complex∈ representation and V →the representation space. V is also called a G-module. The dimension of V (as a complex vector space) is called the dimension of the representation. A representation is called faithful if the associated homomorphism G Aut(v) is injective. We call this representations V as complex G-modules. → Remark A.3.1. Usually gv denotes ρ(g, v). Definition A.3.2. A matrix representation of G is a continuous homomorphism ρ : G Gl(n, C). → Definition A.3.3. If V is a complex G-module, an (Hermitian) inner product V V C,(u, v) u, v is called G-invariant if gu, gv = u, v for all g G and×u, v→ V . A representation→ h i together with a G-invarianth i innerh producti is called∈ unitary∈ representation. Definition A.3.4. Let V be a G-module . A subspace U V which is G-invariant (i.e., gu U for g G and u U) is called a submodule of ⊂V or a subrepresentation. A nonzero∈ representation∈ V is∈ called irreducible if it has no submodules other than 0 and V . A representation which is not irreducible is called reducible. { } Proposition A.3.1. Let G be a compact group. If U is a submodule of the G- module V , then there is a complementary submodule W such that V = U W . Each G-module is a direct sum of irreducible submodules. ⊕

Deﬁnition A.3.5. The representations of SU(n), U(n), and Gl(n, C) on Cn in which elements of the stated Lie groups simply operate by matrix multiplication are called the standard representations. A representation is called trivial if each group element acts as the identity.

Proposition A.3.2. Let Vn be the space of homogeneous polynomials of degree n Z in two variables z1 and z2. The dimension of Vn is n+1. Viewing polynomials ∈ as functions on C2, we obtain a left action of SU(2) on polynomials by letting (gP )z = P (zg) where a b P C[z1, z2], g = , z = (z1, z2) ∈ c d

Since each g acts as a homogeneous linear transformation, the subspaces Vn ⊂ C[z1, z2] are SU(2)-invariant. The representations Vn of SU(2) are irreducible. Furthermore, every irreducible representation of SU(2) is isomorphic to one of the Vn.

39 See [6, page 85-86] for the proof.

Remark A.3.2. Usually Dk/2 also denotes Vk Proposition A.3.3. Clebsch-Gordan formula:

q M Vk Vl = Vk+1−2j q = min k, l ⊗ j=0 { }

See [6, page 87] for the proof.

A.4 Fibre Bundles

Definition A.4.1. A (differentiable) fibre bundle (E, π, M, F, G) consists of the following elements:

(i) A diﬀerentiable manifold E called the total space.

(ii) A diﬀerentiable manifold M called the base space.

(iii) A differentiable manifold F called the fibre (or typical fibre).

(iv) A surjection π : M E called the projection. The inverse image π−1(p) = → Fp ∼= F is called the ﬁbre at p. (v) A Lie group G called the structure group, which acts on F on the left.

−1 (vi) A set of open covering Ui of M with a diﬀeomorphism φi : Ui F π (Ui) { } × → such that π φ(p, f) = p. The map φi is called the local trivialitzation since −1 ◦−1 φ maps π (ui) onto the direct product Ui F . i ×

(vii) If we write φi(p, f) = φi,p(f), the map φi,p : F Fp is a diﬀeomorphism. −1 → On Ui Uj = , we require that tij(p) φ φj,p : F F be an element ∩ 6 ∅ ≡ i,p ◦ → of G. Then φi and φj are related by a smooth map tij : Ui Uj G as ∩ → φj(p, f) = φi(p, tij(p)f). The maps tij are called the transition functions. Remark A.4.1. If all transition functions can be taken to be identity maps, the ﬁbre bundle is called a trivial bundle.

Definition A.4.2. A section (or a cross section) s : M E is a smooth map → which satisfies π s = idM . The set of sections on M is denoted by Γ(M,F ). If U M, we may talk◦ of a local section which is defined only on U. ⊂ Definition A.4.3. A vector bundle E π M is a fibre bundle whose fibre is a vector space. −−→

Definition A.4.4. A principal bundle has a fibre F which is identical to the structure group G. A principal bundle P π M is also denoted by P (M,G) and is often called a G bundle over M. The−−→ transition function acts on the fibre

40 on the left. In addition, we may also deﬁne the action of G on F on the right. −1 −1 Let φi : Ui G π (Ui) be the local trivialization given by φi (u) = (p, gi), ×−1 → −1 where u π (Ui) and p = π(u). The right action of G on π (Ui) is deﬁned by −1 ∈ φi (ua) = (p, gia), that is ua = φi(p, gia) −1 for any a G and u π (Ui). ∈ ∈

Definition A.4.5. Given a section s1(p) over Ui, we define a preferred local trivial- −1 −1 ization φi : Ui G π (Ui) as follows. For u π (Ui), p Ui, there is a unique × → ∈ ∈ −1 element gu G such that u = si(p)gu. Then we define φi by φ (u) = (p, gu). In ∈ i this local trivialization, the section si(p) is expressed as

si(p) = φi(p, e)

This local trivialization is called the canonical local trivialization.

Definition A.4.6. Given a principal fibre bundle P (M,G), we may construct an associated fibre bundle as follows. Let G act on a manifold F on the left. Define an action of g G on P F by ∈ × (u, f) (ug, g−1f) −→ where u P and f F . Then the associated fibre bundle (E, π, M, G, F, P ) is an equivalence∈ class P ∈ F/G in which two points (u, f) and (ug, g−1f) are identified. × Definition A.4.7. Considering the previous manifold F a k-dimensional vector space V . Let ρ be the k-dimensional representation of G. The associated vector −1 bundle P ρ V is defined by identifying the points (u, v) and (ug, ρ(g) v) of P V , where u ×P , g G and v V . × ∈ ∈ ∈ Definition A.4.8. Associated with a tangent bundle TM over an m-dimensional S manifold M is a principal bundle called the frame bundle LM LpM ≡ p∈M where LpM is the set of frames at p. More explicitly, if we introduce coordinates µ µ x on a chart Ui, the bundle TpM has a natural basis ∂/∂x on Ui. A frame { } u = X1,...,Xm at p is expressed as { } µ µ Xα = X ∂/∂x 1 α m α ≤ ≤ µ where (Xα ) is an element GL(m,R) so that Xα are linearly independent. { } Definition A.4.9. Suppose M is an n-dimensional Riemannian manifold with or without boundary with metric g, and S M is a k-dimensional Riemannian submanifold (also with or without boundary).⊂ For any p S we say that a vector ∈ v TpM is normal to S if v is orthogonal to every vector in TpS with respect to ∈ the inner product , . The normal space to S at p is the subspace NpS TpM consisting of all vectorsh i that are normal to S at p, and the normal bundle≤of S is the subset NS TM consisting of the union of all the normal spaces at points of S. The projection≤ π : NS S is defined as the restriction to NS of π : TM M. → →

41 Definition A.4.10. Let M be a smooth manifold and S M be an embedded submanifold. Define the conormal bundle of S to be the≤ subset N ∗S T ∗M defined by ⊂ ∗ N S = (s, α) T M : s S, α T S 0 { ∈ ∗ ∈ | s ≡ } Definition A.4.11. Let π : E M be a real n-dimensional vector bundle over a manifold M. Assume this bundle→ is equipped with a Riemannian structure, that is, a positive definite inner product continuously defined in the fibres. Assume also that the bundle is oriented, i.e., that there is an orientation continuously defined on the fibres. Let PSO(E) be its principal bundle of oriented orthonormal frames. Then a spin structure on E is a principal Spin-bundle PSpin(E) together with a 2-sheeted covering ϕ : PSpin(E) PSO(E) → such that ϕ(pg) = ϕ(p)ϕ0(g) (where ϕ0 is the double covering Spin(n) SO(n)) → for all p PSpin(E) and all g Spin(n). ∈ ∈ Remark A.4.2. A spin manifold is an oriented Riemannian manifolds with a spin structure on its tangent bundle. Definition A.4.12. Let E be an oriented Riemannian vector bundle with spin structure ϕ : PSpin(E) PSO(E). A real spinor bundle of E is a bundle of the form → S(E) = PSpin(E) ρ M × where M is a left module for C`n and where ρ: Spin(n) SO(n) is the representation given by left multiplication by elements of Spin(n). → Definition A.4.13. Let (M, g) be a (pseudo-)Riemannian manifold. A diffeomor- phism F : M M is called a conformal transformation if it preserves the metric up to scale→ ∗ 2σ f gf(p) = e gp 2σ where σ : M R. Namely, gf(p)(f∗X, f∗Y ) = e gp(X,Y ) for X,Y TpM. This let us to create→ an equivalence relation between metrics: the metric∈g and g0 are conformally related if g = e2σg0. The equivalence class is called the conformal structure.

A.5 Connections in Fibre Bundles

Definition A.5.1. Let π : P M be a principal G-bundle and let m M and → ∈ p P . The vertical subspace Vp TpP are those vectors of the tangent space at ∈ ⊂ a point p that are tangent to the fibres at that point, i.e. Vp = kerπ∗ : TpP TmM. → Remark A.5.1. A vector field v is vertical if v(p) Vp for all p P . ∈ ∈ Remark A.5.2. It is posible to identify the vertical space with the Lie algebra g of G. The right action of the group G at p P gives and a map G P . If one ∈ → takes the differential of this map at the identity, a map σp : g TpP is obtained: → d σ (X) = (pexp(tX)) p dt t=0 42 d The image of this map is in the vertical space: π∗(σp(X)) = dt (π(pexp(tX)))t=0 = d dt (π(p))t=0 = 0. Since G acts on P freely, counting the dimensions of the Lie Algebra and the vertical space it is seen that this map is an isomorphism of g and Vp Remark A.5.3. At this point we have a vertical space as a subset of the tangent space. A connection defines a complement of this vertical space called horizontal space. Definition A.5.2. A connection on P is a smooth choice of horizontal subspaces Hp TpP complementary to Vp: ⊂

TpP = Vp Hp ⊕ with the condition that (Rg)∗Hp = Hpg. Deﬁnition A.5.3. The connection one-form of a connection H TP is the g-valued one-form ω A1(g) such that ⊂ ∈ X if ξ = σ(X) ω(ξ) = 0 if ξ is horizontal In other words, the action of this one-form on the vertical vectors gives the corresponding element of the Lie algebra by the isomorphism σ. Remark A.5.4. A one-form is said to be horizontal if it annihilates the vertical vectors. Proposition A.5.1. The connection one-form obeys

∗ −1 Rgω = g ωg

See [9, page 37] for the proof. Definition A.5.4. The horizontal projection h : TP TP is the projection onto the horizontal distribution along the vertical distribution.→ At a point p P is defined by: ∈ ξ if ξ Hp hp(ξ) = ∈ 0 if ξ Vp ∈ Remark A.5.5. We will denote h∗ : T ∗P T ∗P the dual map, such that for a k- ∗ → form α,(h α)(ξ1, . . . , ξk) = α(hξ1, . . . , hξk). It does not commute with the exterior derivative. Definition A.5.5. Let ω A1(g) be the connection 1-form for a connection H TP . The 2-form Ω = h∗dω∈ A2(g) is called the curvature of the connection. By⊂ definition ∈ Ω(u, v) = dω(hu, hv) = ω([hu, hv]) − Remark A.5.6. The curvature is 0 if and only if [hu, hv] is horizontal. The curvature of the connection measures the failure of the integrability of the horizontal distribution H TP . ⊂ 43 Remark A.5.7. The map d∇ = D(φ) = (h∗d)(φ) applied to a k-form φ is called the exterior covariant derivative. Therefore, the curvature is the covariant derivative of the connection one-form. Definition A.5.6. On a vector bundle π : E M, the notion of a connection (Koszul connection) is given by the linear map→ : A0(E) A1(E) , called covariant derivative, which satisfies the following∇ property: → (fs) = df s + f s ∇ ⊗ ∇ 0 0 If X is a vector field, then X : A (E) A (E) and satisfies the following properties: ∇ → fX+gY = f X + g Y ∇ ∇ ∇ X (fs) = X(s) + f X s ∇ ∇ Remark A.5.8. The covariant derivative on a vector bundle E has a natural ex- 1 2 tension D1 : A (E) A (E), defined by → D1(e α) = e α + e dα ⊗ ∇ ∧ ⊗ Remark A.5.9. The curvature on a vector bundle E is defined as the composition 2 D1 A (EndE). ∇ ∈ Definition A.5.7. A F -valued k-form αAk(E) is basic if it is horizontal (i.e h∗α = ∗ −1 α) and G-equivariant (i.e Rgα = ρ(g )α, where ρ : G GL(F ) is a representation of G). → Proposition A.5.2. Let C∞(P,E)G denote the space of equivariant maps from P to E, that is, those maps s : P E that satisfy s(p g) = ρ(g−1)s(p). There is → ∞· G a natural isomorphism between Γ(M,P G E) and C (P,E) , given by sending ∞ G × s C (P,E) to sM by ∈ sM (x) = [p, s(p)]; −1 here p is any element of π (x) and [p, s(p)] is the element of P G E corresponding to (p, s(p)) P E. × ∈ × See [3, page 19] for the proof. Corollary A.5.1. There is an isomorphism of C∞-modules k k A (P,F ) = A (P G F ) G ∼ × See [9, page 41] for the proof. Remark A.5.10. This corollary lets identify the connections on a vector bundle with the connection on the corresponding frame bundle. This can be seen in the following way: Lets apply the exterior covariant derivative to a section α of P G F . k × Because of the corollary, there is an equivariant functionα ¯ AG(P,F ) such that ∗ −1 ∈ ∗ Rgα¯ = ρ(g )¯α and whose covariant derivative is given by Dα¯ = h dα¯. Doing some manipulations it can be obtained that Dα¯ = dα¯ + ρ(w)¯α, where ρ it also denote the representation of the Lie Algebra. This 1-form is basic (by definition) and equivariant [9, page 41] , therefore it can be associated with a 1-form α 1 ∇ ∈ A (P G F ) on the associated vector bundle. Hence,× a connection on the principal bundle defines the covariant derivative on the associated vector bundle and the other way around.

44 A.6 Complex Geometry

Definition A.6.1. A complex valued function f : Cn C is holomorphic if → µ µ µ f = f1 + if2 satisfies the Cauchy-Riemann relations for each z = x + iy , ∂f ∂f ∂f ∂f 1 = 2 , 2 = 1 ∂xµ ∂yµ ∂xµ −∂yµ Definition A.6.2. An almost complex structure on a connected manifold M is a smooth field of automorphism J of the tangent bundle TM satisfying

J 2 = Id, m M m − ∈ Equivalently, an almost complex structure on M is an action of the field C of the complex numbers on TM giving to it a structure of complex vector bundle over M. If a manifold admits an almost complex structure is called an almost complex manifold. Remark A.6.1. The dimension n of an almost complex manifold M is even. We set n = 2m, where m is the complex dimension of M, and the complex operator J induces a preferred orientation on M, for which adapted orthonormal frames e1, Je1, . . . , em, Jem are positively oriented { } Remark A.6.2. The almost complex structure J induces a splitting of the complexified tangent bundle TM C into two complementary complex sub-bundles, conjugate to each other: ⊗ ¯ TcM = TM C = V V ⊗ ⊕ ¯ where at each point m of M, the fibre Vm (resp. Vm) is the eigenspace of Jm relative to the eigenvalue i (resp. i). Vectors inside V are called (1, 0) or holomorphic and those in V¯ are called (0, 1)− or antiholomorphic. Definition A.6.3. A complex manifold of complex dimension m is a paracom- pact, Hausdorff topological space admitting a covering by open subsets, homeomor- phic with open subsets of Cm and with holomorphic transition functions. Definition A.6.4. A Hermitian metric on M is a Riemannian metric g such that g(JX,JY ) = g(X,Y ), X,Y TmM, m M ∀ ∈ ∀ ∈ Definition A.6.5. A symplectic manifold (M, ω) is a manifold M equipped with a non-degenerate closed 2-form ω. Such a form is called a symplectic form . In local coordinates xµ on M, µ ν ω = ωµνdx dx , dω = 0

The condition of being non-degenerate means that ωµν is invertible. An invertible antisymetric matrix has an even number of rows and columns, so symplectic manifolds are therefore of even real dimension. Proposition A.6.1. Symplectic manifolds are almost complex

45 See [21, page 16] for the proof.

Deﬁnition A.6.6. Let U Cm be an open subset and let f : U C be a holomorphic map. The complex⊂ Jacobian of f at a point z U is the matrix→ ∈ ∂f J(f)(z) := i (z) ∂zj 1≤i≤n 1≤j≤m

A point z U is called regular point if J(f)(z) is surjective. If every point z f −1(w)∈ is regular then w is called a regular value. ∈ n+1 Proposition A.6.2. The set ( 1) CPn C that consists of all pairs (l, z) n+1 O − ⊂ × ∈ CPn C with z l forms in a natural way a holomorphic line bundle over CPn. × ∈ See [13, page 68] for the proof.

Definition A.6.7. The line bundle (1) is the dual (1)∗ of ( 1). For k > 0, let (k) be the line bundle (1) O (1) (k-times).O Analogously,O − for k < 0 one definesO (k) := ( k)∗. O ⊗ · ⊗ O O O − n Definition A.6.8. Let (z1, . . . , zn) be Euclidean coordinates in a disc ∆ in C ˜ and l = [l1, . . . , ln] corresponding homogeneous coordinates on CPn−1. Let ∆ ⊂ ∆ CPn−1 be the submanifold of ∆ CPn−1 given by the quadratic relations × × ˜ ∆ = (z, l): zilj = zjli, i, j { ∀ } n If we consider the points l CPn−1 as lines in C , then writing these equations as z l = 0 we see that this is just∈ the incidence correspondence defined as (z, l): z l∧. Now ∆˜ maps onto ∆ via projection π on the first factor ; from{ the geometric∈ } interpretation it follows that the map is an isomorphism away from the origin in ∆, and π−1(0) is just the projective space of lines in ∆. In effect, ∆˜ consist of all the lines through the origin in ∆ made disjoint. ∆, together with its projection map πto∆, is called the blow-up of ∆ at 0 (the blow-up of the origin in a disc in Cn). Now let M be a complex manifold of dimension n, m M any point, and z : U ∆ a coordinate polydisc centered around m M. The∈ restriction of the projection→ map ∈ π : ∆˜ E U m M \ → − { } ⊂ gives and isomoprhism between a neighborhood of E = π−1x in ∆ and a neigh- ˜ borhood of m in M; we define the blow-up Mm of M at m to be the complex manifold ˜ ˜ Mm = M m π ∆ − { } ∪ obtained by replacing ∆ M with ∆,˜ together with the natural projection map ˜ ⊂ π : Mm M. →

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