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 stud- ied 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 sec- tions 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 Hodge2 star operator :Λp Λ2l−p as: ∗ ! α β = α; β ! ^ ∗ h i where ! denotes the volume form2, α; β Λp and ; the induced inner product on p-forms by the metric on the manifold.2 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: ⊕ 0 1 s B W+ + Id Z C B 12 C B C B C (2.1) B C B t s C @ Z W− + 12 Id A 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: 2 ^ (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 ! = g dx dx . j j 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 uni- versal 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 ex- pressed 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 decom- poses using Clebsch-Gordan2 formula (A.3.3) : Dk=2 Dl=2 = Dk+l=2 Dk+l−2=2 Djk−lj=2 (2.2) ⊗ ⊕ ⊕ · · · ⊕ So the representation ring4 of Spin(4) is based on the two fundamental repre- sentations D± , where we use to distinguish the representation of each of the 1=2 ± SU(2), Spin(4) = SU(2)+ SU(2)−.