MASARYKOVA UNIVERZITA ÚSTAV TEORETICKÉ FYZIKY A ASTROFYZIKY. Diplomová práce BRNO 2017 TOMÁŠ MICHALÍK

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1 MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV TEORETICKÉ FYZIKY A ASTROFYZIKY Diplomová práce BRNO 2017 TOMÁŠ MICHALÍK

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3 MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV TEORETICKÉ FYZIKY A ASTROFYZIKY Conformal Geometry and its applications in Physics Diplomová práce Tomáš Michalík Vedoucí práce: prof. Rikard von Unge, Ph.D. Brno 2017

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5 Bibliografický záznam Autor: Název práce: Studijní program: Studijní obor: Vedoucí práce: Bc. Tomáš Michalík Přírodovědecká fakulta, Masarykova univerzita Ústav teoretické fyziky a astrofyziky Conformal Geometry and its applications in Physics Navazující magisterský program Teoretická fyzika a astrofyzika prof. Rikard von Unge, Ph.D. Akademický rok: 2016/2017 Počet stran: xvii + 62 Klíčová slova: konformná geometria, ambientná metrika, traktorový kalkulus, holografický princíp, Penroseova transformácia

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7 Bibliographic Entry Author: Title of Thesis: Degree Programme: Field of Study: Supervisor: Bc. Tomáš Michalík Faculty of Science, Masaryk University Department of Theoretical Physics and Astrophysics Conformal Geometry and its applications in Physics Master s degree programme Theoretical Physics and Astrophysics prof. Rikard von Unge, Ph.D. Academic Year: 2016/2017 Number of Pages: xvii + 62 Keywords: conformal geometry, ambient metric, tractor calculus, holographic principle, Penrose transform

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9 Abstrakt Hlavným ciel om tejto diplomovej práce je znázornit spojitosti medzi nástrojmi konformnej geometrie a fyzikou, ako napríklad traktorový kalkulus, holografický princíp a twistorová korešpondencia na abstraktnej a taktiež na názornej úrovni. Abstract The main focus of this thesis is to present connections between tools of conformal geometry and physics, such as tractor calculus, holographic principle and twistor correspondence on both abstract and explicit level.

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13 Acknowledgement I would first like to thank my thesis advisor prof. Rikard von Unge, Ph.D. The door to Prof. von Unge s office was always open whenever I ran into a trouble spot or had a question about writing. He consistently allowed this thesis to be my own work, but steered me in the right direction whenever he thought I needed it. I would also like to acknowledge the grant " Specifický výzkum 2016" for partially supporting my stay at the University of Auckland in August Finally, I must express my very profound gratitude to my family for providing me with unfailing support and continuous encouragement throughout my years of study. This accomplishment would not have been possible without them. Thank you. Prohlášení Prohlašuji, že jsem svoji diplomovou práci vypracoval samostatně s využitím informačních zdrojů, které jsou v práci citovány. Brno 10. května Tomáš Michalík

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15 Contents Introduction xv Notations and conventions xvii 1. Flat model of conformal geometry The ambient and the Poincaré metrics of the conformal class The metric bundle The ambient space A formal solution to the Ricci flatness condition Examples with explicit ambient metrics The Poincaré metric and holography Conformal curvature tensors Tractor calculus Tractor actions Conformal Killing vectors and spinors Conformal Killing vectors Properties under conformal rescaling The Weyl connection Conformal Killing spinors Twistor correspondence Summary Appendix A. Algebraic preliminaries A.1 Properties of a vector space with inner product A.2 The Mobius Lie algebra of V A.3 The Clifford algebras of V and ˆV A.4 Spinor representations of the Mobius algebra Appendix B. Geometric preliminaries Bibliography xiii

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17 Introduction One of the most efficient techniques both in physics and in mathematics is the discovery and exploitation of symmetries. Equations of both relativity and quantum field theory exhibit natural invariances. For example, special relativity is the result of a Poincare invariant mechanics, which was inspired by Poincare invariance of the theory of electromagnetism. In this thesis we examine the generalization of the Poincare symmetry, called confomal symmetry. In conformally symmetric theories, there is no intrinsic length scale, since conformal symmetry rescales the metric tensor by a positive function instead of keeping it invariant. An example of such a theory is the theory of electromagnetism or, more generally, any Yang-Mills theory in 3+1 dimensions. These theories are of great physical importance and it is very useful to try to exploit the symmetry to the fullest extent. The twistor theory together with the Penrose transformation are examples of such exploitation. They allow us to explicitly solve certain natural differential equations, such as the linear zero rest mass field equations, by simple contour integration. There is even a non-linear generalization of the Penrose transform, the Penrose-Ward transform, which allows us to explicitly solve the full nonlinear Yang-Mills equations in 3+1 dimensions. But this is not all that conformal geometry has to offer. The theory of Poincare metrics allows us to relate the conformal class of metrics on a boundary of a Ricci flat manifold with the metric in the bulk of this manifold, which is the mathematical foundation of the famous holographic principle called AdS/CFT correspondence. The fact that the conformal geometry is an example of so called parabolic geometry allows us to utilize the so called tractor calculus which is very useful in the formulation of manifestly conformally invariant theories. The sturcture of the thesis as follows. We start with the geometric motivation behind the ambient metric - the construction that will lead us to Poincare metrics and holography. Then we will introduce the tractor calculus, which is closely related to the ambient construction and we demonstrate its power in the creation of manifestly conformally invariant field theories. In the fourth chapter we lay the geometric foundation to twistor theory by examining the conformal Killing vectors and spinors. In the fifth and final chapter we take a look at the basics of twistor theory via explicit examples. Unfortunately, we will not be able to explore the twistor theory deeply since it would require at least one more thesis. At the end of the thesis are appendices which summarize geometric and algebraic preliminaries. xv

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19 Notations and conventions, g g + Rm(g), R i j kl Ric(g), R i j Scal(g), R P(g), P i j P(X ) P B(g) L X musical isomorphisms Ambient metric of g Poincare metric of g Riemann curvature tensor of g Ricci tensor of g Scalar curvature of g Schouten tensor of g Schouten operator of g acting of the vector field X Trace of Schouten tensor of g Bach tensor of g Lie derivative in the direction of the vector field X Indices I, J,... ambient, tractor indices (in ambient case we use instead of n + 1) a,b,... spinor indices µ,ν,... indices on Poincaré space i, j,... indices on the conformal manifold xvii

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21 Chapter 1 Flat model of conformal geometry We begin this thesis with a descrption of the flat model of conformal geometry. is Let us assume R n+2 with Lorentzian signature. The scalar product in this space (x, y) = x 0 y 0 n+1 + i=1 x i y i Let b(x) = (x 0 ) 2 n+1 + and let O(1,n + 1) be the orthogonal group i=1 (x i ) 2 O(1,n + 1) = {g GL(n + 2,R) b(g x) = b(x), x R n+2 } We will use basis constructed from the standard basis {e i }, given by where {f 0, f 1,..., f n, f n+1 } = {f 0,e 1,...,e n, f n+1 } f 0 = e n+1 e 0 and f n+1 = e n+1 + e The metric in these coordinates is given by (f I, f J ) = 0 1 n

22 2 Chapter 1. Flat model of conformal geometry Next, we define important subgroups of O(1,n + 1) with respect to this basis a A = 0 1 0, a R, a /= a M = 0 T 0, T O(n) v 1 N 2 b (ṽ) = 0 1 v, v R n N + = v 1 0, v R n 1 2b (ṽ) v 1 (1.1) where ṽ = (0, v,0). We can immediately see that A and N ± are abelian and A and M commute. The group M A normalizes N ±, hence P ± = M AN ± are well defined groups. Real Lie algebras of these groups are denoted by a, m, n ± and p ±. Let us denote b 1 = n, b 1 = n +, b 0 = m a The root decomposition of g with respect to the adjoint action of a is g = b 1 b 0 b 1 with [b i,b j ] b (i+j)mod 3 where the Z 3 has elements { 1,0,1} instead of the usual {0,1,2}. Therefore g is a 1 graded Lie algebra. Also, the adjoint action of M O(n) on n ± R n coincides with the standard action. The action of O(1,n + 1) preserves the light-cone C = {x R 1,n+1 b(x) = 0} Since the action of O(1,n + 1) commutes with dilations x λx, λ R, we can induce an action on the real projective space RP n+1 whose points correspond to lines going through the origin of R n+2. Let Q RP n+1 be the space of lines in the cone C and let π C Q be the natural projection. From now on we will use canonical coordinate system again. Through the identification, illustrated in figure (1.1), we have S n x π((1, x)) Q (1.2) where S n = {x R n+1 n+1 i=1 (xi ) 2 = 1} the orthogonal group acts by

23 Chapter 1. Flat model of conformal geometry 3 C x π π(x) π(c) S n Figure 1.1: The conformal sphere g (1, x) g (1, x) x π((1, x)) π(g (1, x)) = π((1, )) 0 g (1, x) g (1, x) 0 which in coordinates can be written as x i g(1, x)i g(1, x) 0 Let g = ( d c ). We can use the fact that g O(1,n + 1) to calculate the inverse, b A which is g 1 = ( d b c A ). We also find identities between the elements d 2 (c,c) = 1 AA bb = I Ac db = 0 (1.3) Then g(1, x) = (d +(c, x),b + Ax) and g induces the map x b + Ax d +(c, x) We can view the space R 1.n+1 as a Lorentzian manifold with the metric g = d(x 0 ) 2 n+1 + i=1 d(x i ) 2 The restriction of g to C degenerates, since g (X,Y ) = 0 for X = n+1 i=0 xi x i and all Y. The cone C is given by the equation b(x) = (x 0 ) 2 n+1 + i=1 (x i ) 2 = 0

24 4 Chapter 1. Flat model of conformal geometry Hence, on C db = 2x 0 dx 0 n x i dx i = 0 The contraction with the arbitrary vector field Y X(C) is i=1 0 = db(y ) = 2x 0 y 0 n x i y i = 2 g(x,y ) But, for any x C, we have an inner product on T π(x) (Q) by i=1 g π(x) (Y, Z) = g x (π 1 (Y ),π 1 (Z) ), Y, Z T π(x) (Q) Moreover, we get g π(λx) = λ 2 g π(x) since π(x) = π(λx). Thus any section η Q C of π C Q induces a metric η ( g) on Q such that two such sections induce metrics which differ by a positive smooth function, i.e. are conformally equivalent. This is a conformal class on Q. Pulling back the metric to S n using mapping (1.2) leads to the conformal class of the round metric g c. Now O(1,n + 1) operates on Q by conformal diffeomorphisms. This means that for every g O(1,n + 1), there is a non-vanishing function Φ g C (S n ) so that g (g c ) = Φ 2 g g c We can view the pushforward as the pullback via the inverse map, hence and hence where J = g (g c ) = (g 1 ) (g c ) g (g c ) = J g c J 1 g x. This, together with (1.3), allows us to find the scaling function Φ g = 1 d (b, x) O(1,n + 1) acts transitively on Q with isotropy group G π(f0 ) = P. This allows us to write Q O(1,n + 1)/P. In terms of S n, the subgroup P fixes the point (0,0,..., 1). Now we demostrate the flat model of the Poincare metric, which will be introduced in the full generality in next chapter. Let us consider the n ball B n with the metric 4 g + = (1 x 2 ) 2 g E n where g En is a metric on n dimensional Euclidean space.

25 Chapter 1. Flat model of conformal geometry 5 Let M + be a manifold with M + = M, r be a defining function for M +. A defining function fulfills the following conditions r C (M + ), r M 0 + > 0, r M+ = 0, dr M+ /= 0 We say that a smooth metric g + on M + is conformally compact if r 2 g + extends smoothly to M + and r 2 g + M+ is non-degenerate. A conformally compact metric is said to have a conformal infinity (M,[g]) if r 2 g + T M+ [g]. There is a difference between the conformal infinity and general boundary of a manifold. The conformal infinity is in infinite distance, as can be shown by calculation of a length s of a line from the center of a ball to any point of the boundary s = t dt = [ln(1+ 1 t 2 1 t )] = 0 We can apply this to the aforementioned case of B n. First, we will write the metric in hyperspherical coordinates (R,θ 1,...,θ n 1 ) 4 g + = (1 R 2 ) 2 g E n (R,θ 1,...,θ n 1 ) In these coordinates we have M + = {R 2 = 1}. We can choose a defining function of the form r = 1 2 eυ(θ 1,...,θ n 1 ) (1 R 2 ) where Υ is an arbitrary function, which gives us r 2 g + T M+ = r 2 g + = 4e2Υ (1 R 2 ) 2 R 2 =1,dR=0 4(1 R 2 ) 2 g En (1,θ 1,...,θ n 1 ) = e 2Υ g S n 1 [g S n 1] In next chapter we will approach this problem from the opposite direction. We will show how to construct the conformally compact manifold (M +, g + ) for any conformal manifold (M,[g]).

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27 Chapter 2 The ambient and the Poincaré metrics of the conformal class This chapter is based on the ambient construction of Fefferman and Graham, which is summarized in their monograph [1]. 2.1 The metric bundle The conformal class is defined as an equivalence class of the metric on a manifold. Two metrics are equivalent if one is a smooth multiple of the other, the equivalence class therefore looks like [g] = e 2Υ(x) g,υ C (M) For a manifold with a conformal class we can define the R + metric bundle G (h, x), x M,h = s 2 g x, s R + which is equipped with the projection: π G M (h, x) x and the dilation operation: δ s G G δ s (h, x) = (s 2 h, x) On this bundle, we can also naturally define the dilation vector field T = d ds δ s s=1 7

28 8 Chapter 2. The ambient and the Poincaré metrics of the conformal class and a natural symmetric 2-tensor g 0 π T G T M g 0 (X,Y ) = h (π X,π Y ),δ s g 0 = s 2 g 0 G, δ s, π and g 0 depends only on [g]. Once we fix g in [g], we obtain a trivialization of G. We identify In these coordinates we have (t, x) R + M with (t 2 g x, x) G δ s (t, x) (st, x),π (t, x) x,t = t t,g g 0 = t 2 π g The metric g can be regarded as a section of G. The image of this section is a submanifold of G given by t = 1. The choice of g can also be used to determine a horizontal subspace H z T z G, namely H z = ker(dt) z. Now, let us change the representative metric to ĝ = e 2Υ g. The trivialization in this case is (ˆt, x) (ˆt 2 gˆ x, x) G. The R + coordinates change as ˆt = e Υ(x) t. In coordinates, we have g 0 = t 2 g i j dx i dx j Now consider G R with coordinates (z,ρ). We can define the action of a dilation as δ s (z,ρ) = (δ s (z),ρ) Using this action, we can define the dilation vector field in the same manner as before. We can embed G into G R by ι z (z,0) We can easily convince ourselves that the dilation commutes with this embedding. 2.2 The ambient space The pre-ambient space A pre-ambient space for (M,[g]), where [g] is a conformal class of signature (p, q) on M is a pair ( G, g) where 1. G is a dilation-invariant open neighborhood of G {0} in G R 2. g is a smooth metric of signature (p + 1, q + 1) on G R 3. g is homogeneous of degree 2 on G, which means that δ s g = s2 g

29 Chapter 2. The ambient and the Poincaré metrics of the conformal class 9 4. ι g = g 0 A pre-ambient space is called an ambient space, if the following condition is fulfilled in the case when p + q is odd 5. Ric ( g) vanishes to infinite order at every point of G {0} In the case when p + q is even, we need some more preparation on to define the ambient space. Let S I J be a smooth symmetric 2-tensor field on an open neighborhood of G {0}. For an integer m 0, we write S I J = O + I J (ρ m ) if S I J = O (ρ m ) For each z G, the tensor (ι (ρ m S))(z) is of the form π s for some symmetric 2-tensor s at x = π(z) M satisfying tr gx s = 0, where s is allowed to depend on z, not only on x. In coordinates, we have the following conditions on the components of S I J S I J = O (ρ m ) S 00,S 0i, g i j S i j = O (ρ m+1 ) Using this, we are able to define the condition on a pre-ambient space to be ambient in any even dimensions n 4 5. Ric ( g) = O + I J (ρ n/2 1 ) With these conditions, it is impossible to find a unique ambient metric for a given conformal class. We have to further restrict the form of the ambient metric, which leads us to the definition of a straight pre-ambient metric and the normal form The straight pre-ambient space Let ( G, g) be a pre-ambient space for (M,[g]). There is a dilation-invariant open set U G containing G {0}, such that on this subset the following conditions are equivalent 1. T = Id 2. 2T g = d(t 2 ) 3. s δ s p is a geodesic for every p U If U = G the pre-ambient space is called straight.

30 10 Chapter 2. The ambient and the Poincaré metrics of the conformal class The normal form with respect to the representative metric g ( G, g) is said to be in normal form relative to g in (M,[g]) if 1. For each fixed z G, the set of all ρ R such that (z,ρ) G is an open interval I z containing 0 2. For each z G, the parametrized curve I z ρ (z,ρ) is a geodesic of g 3. Let us write (t, x,ρ) for a point of R + M R G R under the coordinatization induced by g. Then, at each point (t, x,0) G {0}, ambient metric has the following form g = g 0 + 2tdtdρ = t 2 g i j dx i dx j + 2tdtdρ Let us take a look at the form of the ambient metric in normal form. First, we use the condition that the ρ-lines are geodesics of the ambient metric. Using the geodesic equation, we find ρ g = 0 ρ g i = 0 ρ g 0 = 0 Using the third condition in the definition of the normal form, we have g = 0 g i = 0 g 0 = t (2.1) Let us examine the homogeneity condition. We write out the ambient metric in the following form g = a (x,ρ) f a (t)dt 2 + 2b i (x,ρ) f b (t)dtdρ + 2tdtdρ + t 2 g i j (x,ρ)dx i dx j The coordinate change induced by the dilation is δ s (t, x,ρ) = (st, x,ρ) If we pullback the ambient metric, we have δ s g = s2 a (x,ρ) f a (st)dt 2 + 2sb i (x,ρ) f b (st)dtdρ + 2s 2 tdtdρ + s 2 t 2 g i j (x,ρ)dx i dx j This gives us the homogeneity degree of the t-functions f a (t) = f a (st) f b (t) = s f b (st)

31 Chapter 2. The ambient and the Poincaré metrics of the conformal class 11 Finally, we ve arrived at the final form of the ambient metric in the normal form g = a (x,ρ)dt 2 + 2tb i (x,ρ)dtdρ + 2tdtdρ + t 2 g i j (x,ρ)dx i dx j where the initial conditions for a, b i and g i j are obtained from the third condition in the definition of the normal form a (x,0) = 0 b i (x,0) = 0 g i j (x,0) = g i j (x) To express the Ricci tensor, we have to calculate the inverse metric and the Christoffel symbols. By straightforward, but extremely tedious calculation we find the following results 0 0 t 1 g I J = 0 t 2 g i j t 2 b i t 1 t 2 b j t 2 (b k b k a) g I J ρ=0 = 0 0 t 1 0 t 2 g i j 0 t j a ρ a 2 Γ I J0 = i a t ( j b i + i b j 2g i j ) t ρ b i ρ a t ρ b j 0 2b k k a t ( j b k k b j + 2g j k ) t ρ b k 2 Γ I Jk = t ( i b k k b i + 2g ik ) 2t 2 Γ i j k t 2 ρ g ik t ρ b k t 2 ρ g j k 0 2 ρ a t ρ b j 0 2 Γ I J = t ρ b i t 2 ρ g i j A formal solution to the Ricci flatness condition In this section, we will seek the unknown functions a (x,ρ),b i (x,ρ) and g i j (x,ρ) such that the ambient metric is Ricci flat to infinite order in ρ. The ambient Ricci tensor is given by R I J = 1 2 g K L ( 2 I L g JK + 2 JK g I L 2 K L g I J 2 I J g K L) + g K L g PQ ( Γ I LP Γ JKQ Γ I JP Γ K LQ ) (2.2)

32 12 Chapter 2. The ambient and the Poincaré metrics of the conformal class Using the previous results, we are able to find the Ricci tensor. R 00 = n 2t 2 (2 ρa) R i0 = 1 2t ( 2 iρ a n ρb i ) R i j = 1 2 [( ρa n) ρ g i j g kl ( ρ g kl ) g i j + i ρ b j + j ρ b i ( ρ b i )( ρ b j )]+R i j where R i j is the Ricci tensor of g i j. By setting the Ricci tensor to zero at ρ = 0 and solving for the unknown functions, we find the following results a = 2ρ +O (ρ 2 ) b i = O (ρ 2 ) g i j (x,ρ) = g i j (x)+2p i j (x)ρ +O (ρ 2 ) (2.3) where P i j is the Schouten tensor P i j = 1 n 2 (R R i j 2(n 1) g i j ) Higher-order flatness We carry out an inductive perturbation for higher orders. Suppose for some m 2 that the metric g (m 1) I J satisfies (2.1) and (2.3). Set g (m) I J = g (m 1) I J + Φ I J, where From (2.2) we have that φ 00 tφ 0j 0 Φ I J = ρ m tφ i0 t 2 φ i j R (m) = R (m 1) I J g K L ( 2 I L Φ JK + 2 JK Φ I L 2 K L Φ I J 2 I J Φ K L)+ + g K L g PQ ( Γ I LP Γ Φ JKQ + Γ JKQ Γ Φ I LP Γ I JP Γ Φ K LQ Γ K LQ Γ Φ I JP )+O (ρm )

33 Chapter 2. The ambient and the Poincaré metrics of the conformal class 13 In this case, g I J and Γ I JK refer to the g (m) I J and 2Γ Φ I JK = JΦ I K + I Φ JK K Φ I J. These are given by (up to the terms of order O (ρ m )) 0 0 ρ Φ 00 2Γ Φ I J0 = 0 0 ρ Φ i0 ρ Φ 00 ρ Φ 0j ρ Φ 0k 2Γ Φ I Jk = 0 0 ρ Φ ik ρ Φ 0k ρ Φ j k 0 ρ Φ 00 ρ Φ 0j 0 2Γ Φ I J = ρ Φ i0 ρ Φ i j This allows us to find the equations for the Φ I J t 2 R (m) 00 = t 2 R (m 1) 00 + m (m 1 n 2 )ρm 1 φ 00 t R (m) 0i R (m) i j = t R (m) 0i + m (m 1 n 2 )ρm 1 φ 0i + m 2 ρm 1 i φ 00 = R (m 1) i j + mρ m 1 [(m n 2 )φ i j 1 2 g kl φ kl g i j ( j φ 0i + i φ 0j )+P i j φ 00 ] t R (m) 0 = t R (m 1) m (m 1)ρm 2 φ 00 R (m) i = R (m 1) i m (m 1)ρm 2 φ 0i R (m) = R (m 1) 1 2 m (m 1)ρm 2 g kl φ kl (2.4) Tensor components in the direction are O (ρ m 1 ) while those which are not are O (ρ m ). Now, let us analyze the case when we are able to determine Φ I J uniquely. First, let us start with I, J /=. We can assume that g (m 1) has been determined so that R (m 1) I J = O (ρ m 1 ) and that g (m 1) is determined uniquely up to the order of O (ρ m ). We can define g (m) as above. If n is odd or if n is even and m n/2, then the coefficient m 1 n/2 appearing in (2.4) will not vanish, so we can choose φ 00 and φ 0i at ρ = 0 to ensure that R (m) 00 and R (m) 0i are O (ρ m ). Let us examine the following mapping appearing in the i j equations. φ i j (m n/2)φ i j 1 2 g kl φ kl g i j (2.5) We can clearly see that (2.5) is not bijective if m = n/2. However, by taking the trace of (2.5) we find out that this mapping is not bijective also if m = n. Thus, the induction can proceed without problems up to the order m < n/2 if n is even and up to the order m < n if n is odd. Now, let us consider the next value of m if n is even. We can uniquely determine φ 00 and φ 0i at ρ = 0 so that R (m) 00 and R (m) 0i are

34 14 Chapter 2. The ambient and the Poincaré metrics of the conformal class O (ρ n/2 ). In addition, we can also determine g kl φ kl so that g kl R kl is O (ρ n/2 ). Thus, for n even, we see that we can guarantee the uniqueness of g I J up to the order of O + I J (ρ n/2 ) by the condition R I J = O + I J (ρ n/2 1 ) for I, J /=. This problem can be solved by introducing logarithmic terms. This is referred to as the generazlized ambient metric [1] and we will not discuss its existence and uniqueness in this thesis. For n odd and m = n, we can uniquely determine φ 00 and φ 0i at ρ = 0 to make R 00, R 0i = O (ρ n ). Now we can uniquely determine the trace-free part of φ i j, so we have R i j = λg i j ρ n 1 up to the order of O (ρ n ) for some unknown function λ. In order to analyze the remaining components R I and to complete the analysis above in the case m = n, we will consider the differential Bianchi identity g JK I R JK = 2 g JK J R I K. We can write this down in terms of the coordinate derivatives 2 g JK J R I K g JK I R JK 2 g JK g PQ Γ JK P R QI = 0 Suppose for some m 2 that R I J = O (ρ m 1 ) for I, J /= and R I = O (ρ m 2 ). We can write out the Bianchi identity for I = 0,i,. Calculating up to the order of O (ρ m 1 ) we get (n 2 2ρ ρ ) R 0 + t ρ R 00 = O (ρ m 1 ) (n 2 2ρ ρ ) R i t i R 0 + t ρ R i0 = O (ρ m 1 ) (n 2 2ρ ρ ) R + g j k j R k + tp k k R g j k r R j k = O (ρ m 1 ) (2.6) Let us finish the case when n is even. Let g I J be the metric determined above. We can show by induction that R I = O (ρ m 1 ) for 1 m n/2. This is clearly true for m = 1. We suppose that it holds for m 1 and write R I γ I ρ m 2. We can substitute this into (2.6) and we find following estimates of γ I (n + 2 2m)γ 0 = O (ρ) (n + 2 2m)γ i = O (ρ) (n m)γ = O (ρ) (2.7) This finally gives us R 0 = O (ρ m 1 ) R i = O (ρ m 1 ) R = O (ρ m 1 ) This completes our induction. If n is odd, let g I J be the metric such that R 00 = O (ρ n ) and R i j = λ(x) g i j ρ n O (ρ n ). Then we can uniquely determine g I J up to the O (ρ n+1 ), except the indeterminacy of the form cg i j ρ n where c is an arbitrary function. Now let us consider the induction based on (2.6). Terms n 2(m 1) in (2.7) will never vanish for any odd n and integral m and we can proceed with the induction to conclude

35 Chapter 2. The ambient and the Poincaré metrics of the conformal class 15 that R I = O (ρ n 2 ). This induction scheme also holds for m = n. The first two equations give us R 0, R i = O (ρ n 1 ). The first term in the third equation vanishes, which gives us λρ n 2 = O (ρ n 1 ). This tells us that λ = O (ρ) and therefore R i j = = O (ρ n ). However, there still remains an indeterminacy and we do not know whether or not R = O (ρ n 1 ). These problems can be solved simultaneously, by choosing c at ρ = 0 such that R = O (ρ n 1 ). Namely set g I J = g I J + Φ I J, with φ 00 = φ 0i = 0 and φ i j = cg i j. This gives us R = R 1 2 n2 (n 1)cρ n 2 +O (ρ 1 ) Therefore, g I J up to the order of O (ρ n+1 ) is uniquely determined by R I J = O (ρ n ) and R I = O (ρ n 1 ). After this step we can continue without problems up to abritrary order. To sum up, the components g I are determined by the requirement that ρ-lines are geodesics. The derivatives of the remaining components are uniquely determined by the requirement that Ric ( g) = O + I J (ρ n/2 1 ) if n is even. If n is odd, we are able to determine the derivatives to arbitrary order by the condition that Ric ( g) vanishes to arbitrary order at ρ = 0. By Borel s theorem, we can find a unique homogeneous symmetric 2-tensor field on a neighborhood of ρ = 0 with the prescribed Taylor expansion. Then, we can choose a dilation-invariant neighborhood G on which g I J has signature (p + 1, q + 1) Ricci-flatness of the straight ambient metric If we consider a special form of the ambient metric, calculations will significantly simplify. For a form of the metric 2ρ 0 t g I J = 0 t 2 g i j (x,ρ) 0 t 0 0 (2.8) one can calculate the Ricci tensor. This calculation is straightforward but again extremely tedious. The result however is that R 0I = 0. In this form, the only unknown components of ambient metric are in the g i j (x,ρ). We can show that the ambient metric of this form is not only in normal form, but also straight. The following conditions are equivalent. 1. g 00 = 2ρ and g 0i = 0 2. p G, s δ s p is a geodesic of g 3. 2T g = d(t 2 ) 4. T = Id Using the straight geodesic condition we find out that Γ 00I = 0. Initial conditions give us g 00 = 2ρ and g 0i = 0. This shows that (1) (2). We can rewrite the third condition in coordinates as 2t g I 0 = I (t 2 g 00 ) This is equivalent to (2) and therefore

36 16 Chapter 2. The ambient and the Poincaré metrics of the conformal class we have (2) (3). The last condition is equivalent to T = T. The geodesic equation in arbitrary parametrization is T = kt for an arbitrary function k. Therefore we finally have (2) (4) and all conditions are equivalent. The straight ambient metric in normal form takes the form (2.8). 2.4 Examples with explicit ambient metrics An Einstein metric Let us consider the conformal class which contains the metric g on the Einstein manifold M λ with the Ricci curvature R i j = 2λ(n 1) g i j In this case, the ambient metric turns out to be extremely simple. Its expansion in ρ is finite. Its i j components take the form g i j (x,ρ) = (1+λρ) 2 g i j (x) Simple manipulations will tell us more about the topology of the ambient space. The ambient metric can be written as g = 2d(ρt)dt +(t + tλρ) 2 g We can introduce the new coordinate u = tρ. The ambient metric in these coordinates is g = 2dudt +(t λu) 2 g If λ /= 0, we can introduce the coordinates r = t λu and s = t + λu. It can be easily shown that the metric in these coordinates has the form g = ds2 2λ + dr 2 2λ + r 2 g Therefore, the ambient space of an Einstein manifold with non-zero Ricci curvature is a direct product of R (0,1) and a cone with the Einstein space as the base space, G = R (0,1) M λ. In the case of vanishing λ, there is no coordinate transformation which would allow us to simplify the ambient metric. The final form of the ambient metric in this case is g = 2dudt + t 2 g

37 Chapter 2. The ambient and the Poincaré metrics of the conformal class pp-wave spacetimes In these spacetimes, the metric tensor has the form n 2 g = i=1 (dx i ) 2 + 2du (dr + h (x i,u)du) Curvature tensors of this spacetime are quite simple, namely R = 0 Ric (g) = h du 2 P (g) = h n 2 du2 B (g) = 2 h n 2 du2 (2.9) where = n 2 i=1 2 i and B(g) is the Bach tensor defined in (B.6). The only non-zero component of Ric (g) is the du 2 component, and this leads us to the following ansatz for the ambient metric g = 2d(ρt)dt + t 2 (g + 2H (ρ, x i,u)du 2 ) = = 2d(ρt)dt + t 2 [2dudr + 2(h + H)du 2 n 2 + i=1 (dx i ) 2 ] (2.10) From the initial condition we see that H (0, x i,u) = 0. The Ricci tensor of the ambient metric again takes a simple form Ric ( g) = [(2 n) ρ H + 2ρ 2 ρ H H h]du2 (2.11) If the ambient metric is Ricci flat, the unknown function H fulfills the following equation (2 n) ρ H + 2ρ 2 ρh H = h We can now Taylor expand H in the ρ variable H = k=1 f k (x i,u)ρ k

38 18 Chapter 2. The ambient and the Poincaré metrics of the conformal class We proceed to calculate terms of the equation ρ H = k f k ρ k 1 k=1 ρ 2 ρ H = ρ k (k 1) f k ρ k 2 = k (k + 1) f k+1 ρ k k=2 H = ( f k )ρ k k=1 k=1 (2.12) Inserting these expressions into the equation (2.11) we obtain k=1 [(2 n)k f k ρ k 1 + 2k (k + 1) f k+1 ρ k f k ρ k ] = h We can see that on the right hand side we have a term without ρ, therefore it is convenient to split the left hand side into two parts (2 n) f 1 + [[(2 n)k + 2k (k 1)] f k f k 1 ]ρ k 1 = h k=2 From this, we can immediately see that f 1 = h 2 n The second part of the equations gives us a recursion relation for the higher order terms (2k n)k f k = f k 1 After inserting the first order term, we get f k = k h k! k i=1 (2i n) We observe that the recursion relation breaks down if k = n/2, exactly as expected by the formal treatment. Another interesting thing is that the expansion stops at the finite order m if m 1 h = 0. For example, if h = 0, the pp-wave spacetime is an Einstein manifold with λ = 0 and the result agrees with the result on Einstein metrics. To sum up, the i j component of the pp-wave ambient metric is g i j (ρ, x i,u) = g i j (x i,u)+ k=1 k h k! k i=1 (2i n)ρk More general pp-wave-like spacetimes with explicit ambient metric are studied in [2]

39 Chapter 2. The ambient and the Poincaré metrics of the conformal class The Poincaré metric and holography The main goal of this section is to show that there is a one to one correspondence between even Poincaré metrics and straight ambient spaces in normal form. We identify M + with a neighbourhood of M 0 in M [0, ). As a coordinate on the 2 nd factor we choose the value of the defining function r. Components on M + will be denoted by lowercase greek indices. We say that a symmetric 2 tensor field on the neighborhood of M [0, ) S αβ is O + αβ (r m ) if S = O (r m ) and tr g (ι (r m S)) = 0, where ι M M [0, ) and ι(x) = (x,0). We say that g + is a Poincaré metric for (M,[g]), where [g] is the conformal class of signature (p, q) on M if g + is a conformally compact metric of signature (p + 1, q) on M 0 + such that 1. g + has conformal infinity (M,[g]) 2. (a) Ric (g + )+ng + vanishes to infinite order along M if n is odd (b) Ric (g + )+ng + O + αβ (r n 2 ) if n 4 is even We could also consider metrics g on (p, q + 1) such that Ric (g )+ng vanishes. If g + is a conformally compact metric, then dr r g + = dr r 2 g +, which can be extended smoothly to M +. We say that a conformally compact metric g + is asymptotically hyperbolic if and only if dr r g + = 1. We say that an asymptotically hyperbolic metric g + is in a normal form relative to g in [g] if g + = 1 r 2 (dr 2 + g r ) where g r is a 1 parameter family of metrics on M with signature (p, q) and g 0 = g. We say that an asymptotically hyperbolic metric g + on M 0 + is even if r 2 g + is the restriction to M + of a smooth metric h on an open set V M (, ) containing M +, such that V and h are invariant under the sign reversal of r. Now we make a few remarks about the geometry of the ambient space which will be useful in the construction of the Poincaré metric. First, we can see that T 2 vanishes to first order on G 0 G, therefore κ = G {T 2 = 1} lies on one side of G 0. Also, each dilation orbit intersects κ only once due to the fact that T 2 is homogeneous of degree 2 with respect to δ s. We can extend the projection Π G M to Π G G R M R by Π(t, x,ρ) = (Π(t, x),ρ). We can also define ξ M R M [0, ) by χ(x,ρ) = (x, 2 ρ ). There is an open set M + in M [0, ) containing M {0} so that χ κ κ M 0 + is a diffeomorphism. [1] ( G, g) is a straight pre-ambient space for (M,[g]) and κ and M + are defined as above. Then g + = ((χ Π) 1 κ ) g is an even asymptotically hyperbolic metric with conformal infinity (M,[g]). If g is in normal form, then also g + is in normal form. We will now prove this statement.

40 20 Chapter 2. The ambient and the Poincaré metrics of the conformal class We can identify G G R R + M R. The pre-ambient metric has the form g = adt 2 + tdtda + t 2 h (x,ρ,dx,dρ) where a (x,0) = 0 and h (x,0,dx,0) = g (x,dx). Also ρ a ρ=0 /= 0 due to the regularity. This gives us T 2 = a (x,ρ) t 2, and therefore κ = {at 2 = 1}. On {a < 0} we introduce new coordinates u > 0 and s > 0 by a = u 2 s = ut (2.13) If we rewrite our pre-ambient metric in new coordinates, we have g = s 2 u 2 (du 2 + h) ds 2 This is a cone with the fibre u 2 (du 2 + h). In our new coordinates κ = {s = 1} which gives g T κ = u 2 (du 2 + h) If we consider the ambient metric, we have ρ a = 2 and h = g ρ (x,dx), where g 0 = g. For the ambient metric we also have u = 2ρ. By the definition of χ we see that u = r is the second coordinate on M [0, ). Hence g + = r 2 (dr 2 + g 1 2 r 2 ) g + is an even asymptotically hyperbolic metric with conformal infinity (M,[g]) in normal form. The ambient metric can be rewritten in the form g = s 2 g + ds 2 Using this form of the metric, it is easy to find a relation between the Riemann tensor of the ambient metric and the Poincaré metric. After a coordinate change s = e y, the ambient metric transforms to g = e 2y (g + dy 2 ). This is a conformal multiple of the product metric h = g + dy 2. Now we use the conformal transformation rule of the Riemann tensor Rm ( g) = e 2y [Rm (h)+2λ h] where Λ = 2 h y + dy2 1 2 dy 2 h h and is defined in (B.4). If we apply this to our

41 Chapter 2. The ambient and the Poincaré metrics of the conformal class 21 case, we find that Rm (h) = Rm (g + ) 2 h y = 0 dy 2 h = 1 (2.14) Λ = 1 2 (g + +dy 2 ) Using these results we find a relation between Rm ( g) and Rm (g + ). By contraction we also find relations between the Ricci tensor and the scalar curvature Rm ( g) = s 2 [Rm (g + )+ g + g + ] Ric ( g) = Ric (g + )+ng + R ( g) = s 2 [R (g + )+n (n + 1)] (2.15) We can see that if g is an ambient metric in normal form with respect to g then g + is the Poincare metric of g. This construction is of a great use in physics. It is the mathematical background behind the famous AdS/CFT correspondence, stating that a conformal field theory on a boundary is dual to a geometry of the Einstein manifold in the bulk. The antide Sitter space, or its deformation, often containing black holes, is often taken as the manifold in the bulk and we are trying to match the boundary geometry with a geometry of some conformal field theory, for example a conformal hydrodynamics. This relationship is explored in [3]. Application of this construction in supergravity is explored in, for example, [4]. In the next section we take a look at some of the examples discussed in [5] Conformal anomaly In this section we will explore a phenomenon which occurs if the conformal manifold is even-dimensional. This phenomenon was studied in [4]. In this section we use different form of the Poincare metric, namely g +µν = l 2 4 ρ 2 dρ 2 + ρ 1 g i j dx i dx j where l is a length scale related to the cosmological constant Λ via and n(n 1) Λ = 2l 2 Ric(g) = Λg

42 22 Chapter 2. The ambient and the Poincaré metrics of the conformal class These coordinates are related to the previously used coordinates in a following way. First, we change r coordinate according to r = ρ Then we rescale the Poincare metric by a constant factor l 2. This factor is absorbed in g as well. In this coordinate system, the Einstein equations for the Poincare metric are ρ (2g 2g g 1 g +Tr(g 1 g ) g )+l 2 Ric(g) (n 2)g Tr(g 1 g ) g = 0 g j k ( i g j k kg i j ) = 0 Tr(g 1 g ) 1 2 Tr(g 1 g g 1 g ) = 0 As we know from the previous section, in the odd-dimensional case we can solve these equations perturbatively to an arbitrary order of ρ. In the even-dimensional case, we can get g = g 0 + ρg 2 + +ρ n 2 gn + ρ n 2 logρhn +O(ρ n 2 +1 ) where the subscript denotes the number of ρ derivatives. We also know that g k for k = 0,2,,n 2 are covariant, Tr(g 0 1g n) is covariant as well and Tr(g 0 1h n) vanishes identically. Now we would like to evaluate the Einstein-Hilbert action, together with Gibbons Hawking [6] and an additional [7] boundary term for the Poincare metric, in the domain ρ > ɛ for some cutoff ɛ > 0. The boundary term is evaluated at ρ = ɛ. where S E H [g + ] = 1 16πG n+1 N d n xl L = n l dρρ d 1 2 detg + ρ n 2 ( 2d ɛ l det g + 4 l ρ ρ det g + α det g) ρ=ɛ We used the fact that g + is an Einstein metric and hence If n is odd, this can be expanded as Scal(g + )+2Λ = 4 n 1 Λ = 2d l 2 while for even n we get L = det g 0 (ɛ n 2 a0 + ɛ n 2 +1 a 2 + +ɛ 1 2 an 1 )+L f in L = det g 0 (ɛ n 2 a0 + ɛ n 2 +1 a 2 + +ɛ 1 a n 2 logɛa n )+L f in

43 Chapter 2. The ambient and the Poincaré metrics of the conformal class 23 where L f in is finite in ɛ 0 limit and a k are terms in the expansion of det g. Since all a k are covariant, we can cancel the divergent terms by subtracting (covariant) counterterms. After the subtraction, we are left with a renormalized effective 1 action 16πG N n+1 d d xl f in which is finite in the ɛ 0 limit. However, this action is not conformally covariant in general. Let us consider the following variation δg 0 = 2δσg 0 δɛ = 2δσɛ for an infinitesimal function δσ. The variation of the finite Lagrangian density is of the form δl f in = M d n x det g 0 δσa where A is the so called conformal anomaly. It vanishes if n is odd, while for n even it is A = 2 16πG N n+1a d This is due to the fact that the logarithm transforms under the variation with a shift, while the rest of the terms are invariant on their own. Since δl f in is a global conformal invariant, a d is of the form [8] nl n 1 (E n + I n + i J i d 1 ) where E n is proportional to the n dimensional Gauss-Bonet term, I n is a local conformal invariant and J i is a vector field. We proceed with a calculation of d 1 the conformal anomaly for M = AdS 5 S 5. Since the determinant g 10 g = 5 Vol(S 5 ) with Vol(S 5 ) = l 5 π 3, we are interested in a 4. After a long and tedious calculation we find a 4 = l 3 ( 1 8 Ri j R i j R2 ) and E 4 = 1 64 (Ri j kl R i j kl 4R i j R i j + R 2 ) I 4 = 1 64 (Ri j kl R i j kl 2R i j R i j R2 ) The ten-dimensional Newton constant GN 10 = 8π6 gstr 2 and l is related to the number N of so-called D3 branes [9] via l = (4πg str ) 1 4 [9] which leads us to A = N 2 π 2 (E 4 + I 4 ) This agrees with a conformal anomaly of the large N limit of the d = 4 N = 4

44 24 Chapter 2. The ambient and the Poincaré metrics of the conformal class superconformal SU(N) Yang-Mills theory and indicates a possible duality between this theory and the gravity on the AdS 5 S 5 spacetime. 2.6 Conformal curvature tensors In this section we examine relations between curvature tensors appearing in conformal geometry and the curvature tensor of the ambient metric. As we will see, conformal rescaling of the metric will lead to a transformation of ambient curvature under parabolic subgroup of O(p + 1, q + 1). It is straightforward to calculate curvature tensor of ambient metric in a normal form. We can easily find that R I JK 0 = 0. The other components are R i j kl = t 2 [R i j kl g il g j k + ρ 2 (g ik g j l g il g j k)] (2.16) R j kl = t 2 2 [ l g j k kg j l ] (2.17) R j k = t 2 [g j k 2 2 g pq g j p g kq ] (2.18) where denotes ρ and R i j kl is a curvature tensor of g i j (x,ρ) with ρ fixed, and is the Levi-Civita connection of this metric. We can also calculate components of covariant derivatives of curvature using these formulae and the formulae for the Levi-Civita connection of the ambient metric. From now on we fix a straight ambient metric g in normal form with respect to g. We can construct tensors on M from covariant derivatives of ambient curvature in the following way. First, we choose the order of covariant derivative r 0. We divide the set of symbols I JK LM 1...M r into three disjoint subsets labeled S 0, S M and S. Indices in S 0 are set to 0, indices in S are set to and those in S M correspond to our manifold in the decomposition R + M R. We evaluate the resulting component at ρ = 0 and t = 1. This description is invariant, since the submanifold {ρ = 0, t = 1} can be described as the image of g viewed as a section of G. For example, for r = 0 we get R i j kl ρ=0,t=1 = W i j kl (2.19) R j kl ρ=0,t=1 = C j kl (2.20) R j k ρ=0,t=1 = B j k n 4 (2.21) where W i j kl is the Weyl tensor defined in (B.3) and C j kl is the Cottont tensor defined in (B.5). Suppose we choose a conformally related metric ĝ = e 2Υ g. All conformally transformed quantities will be denoted by a hat. An ambient metric in normal form with respect to g can be put into normal form with respect to ĝ by unique homogeneous diffeomorphism ψ which restricts to identity on G such that ˆ g = ψ g

45 Chapter 2. The ambient and the Poincaré metrics of the conformal class 25 Hence on G we have ψ(ˆt, ˆx,0) = (ˆte Υ, ˆx,0) This allows us to calculate the derivatives of the diffeomorphism in the ˆt and ˆx directions. Derivatives in the ˆρ directions are calculated from the requirement that at ˆρ = 0. This gives This can be factored ψ g = 2ˆtdˆtd ˆρ + ˆt 2 ĝ i j d ˆx i d ˆx j e Υ ˆte Υ Υ i 1 (ψ ) A I = 2 ˆte Υ Υ k Υ k 0 δ a ˆρ=0 i e 2Υ Υ a 0 0 e 2Υ where ψ ˆρ=0 = d 1 pd 2 1 (gradυ) 1 2 Υ kυ k ˆte Υ 0 0 ˆt p = 0 Id gradυ, d 1 = 0 Id 0, d 2 = 0 Id e 2Υ Each curvature component is homogeneous with respect to dilations and since δ s g = s2 g, it follows that δ s R r = s 2 R r where r denotes how many times we covariantly differentiated the curvature tensor. Since t is homogeneous of degree 1 and ρ and i are of degree 0, the component R I JK L,M1 M r is homogeneous of degree 2 s 0. Hence ˆ R I JK L,M1 M r ˆρ=0,ˆt=1 = e(2 s 0)Υ ˆ R I JK L,M1 M r ˆρ=0,ˆt=e Υ R (r ) is also a tensor and hence ˆ R I JK L,M1 M r = ( R ABCD,F1 F r ψ)(ψ ) A I (ψ ) Fr M r Evaluations of this expression at ˆρ = 0, ˆt = e Υ gives us ˆ R I JK L,M1 M r ˆρ=0,ˆt=e Υ = e(s 0 2s )Υ R ABCD,F1 F r ρ=0,t=1 p A I pfr M r Combining this with the homogeneity of R we finally get ˆ R I JK L,M1 M r ˆρ=0,ˆt=1 = e2(1 s )Υ R ABCD,F1 F r ρ=0,t=1 p A I pfr M r

46 26 Chapter 2. The ambient and the Poincaré metrics of the conformal class Using this formula we are able to calculate transformation rules for (2.19) Ŵ i j kl = ˆ R i j kl ˆρ=0,ˆt=1 = e2υ W i j kl Ĉ j kl = ˆ R j kl ˆρ=0,ˆt=1 = C j kl +(n 2)Υ i W ˆB j k = (n 4) ˆ R j k ρ=0,t=1 = e 2Υ B j k i kl j Calculation of the conformal curvature tensors, together with their derivatives generalizes the jet isomorphism theorem [10] from pseudo-riemannian geometry to conformal geometry. This theorem in the pseudo-riemannian case states the equivalence of a Taylor expansions of metrics in normal coordinates up to order k + 2, abstract curvature k jets 1 and their symmetrizations as vector bundles. In conformal geometry this theorem states that the map from the Taylor coefficients of metrics in so called conformal normal form [1] to the space of all conformal curvature tensors is an isomorphism. The space of all conformal curvature tensors can be realized in terms of covariant derivatives of the ambient curvature. 1 The abstract curvature k jet is a vector space of tensors which has 4 to 4+k indices. A tensor with 4+r indices satisfies algebraic identities resulting from the first and second Bianchi identities.

47 Chapter 3 Tractor calculus In this chapter we will consider conformal densities as sections of a line bundle E[w] with the corresponding weight w and not as components of such sections. This allows us to omit the scaling factor. In this chapter we follow the article [11]. A conformal scale is a nowhere vanishing local section τ of E[1]. This scale uniquely defines a metric τ 2 g i j in the conformal class and, conversely, the choice of a metric in the conformal class uniquely defines the scale. In this chapter we work with the Levi-Civita connection of the representative metric. After rescaling ĝ = e 2Υ g the new connection is related to the old one by ˆ i f = f + wυ i f ˆ i U j = i U j +(w + 1)Υ i U j U i Υ j +U k Υ k δ j i ˆ i ω j = i ω j +(w 1)Υ i ω j ω i Υ j + ω k Υ k δ j i (3.1) where Υ i = i Υ and f, U i and ω i are sections of E[w], E i [w] and E i [w] respectively. Suppose that we try to chose a new scale σ for which σ 2 g i j is Einstein. This can be written as σ = e Υ τ and the metric is rescaled by e 2Υ. The Schouten tensor of the rescaled metric is ˆP i j = P i j i Υ j + Υ i Υ j The new metric is Einstein if and only if the Schouten tensor is pure trace. In τ scale we have σ = e Υ, whence i j σ = σ(υ i Υ j i Υ j ) This shows us that σ 2 g i j is Einstein if and only if σ is a solution of the conformally 27

48 28 Chapter 3. Tractor calculus invariant equation Tf( i j + P i j )σ = 0 (3.2) where Tf denotes the trace-free part. In general, this equation does not have any solutions, not even locally. But it defines a conformally invariant subspace of the space of 2 jets of sections E[1], denoted by J 2 E[1], at each point. This invariant subspace is called the tractor bundle. We can rewrite the equation (3.2) as a pair of equations for µ i and ρ, sections of E i [ 1] and E[ 1] respectively We can check that these quantities scales as follows This can be rewritten as ˆσ ˆµ i ˆρ j σ µ j =0 j µ i + δ i j ρ + P i j σ = 0 (3.3) = σ µ i + Υ i σ ρ Υ j µ j Υ i Υ i σ 2 ˆσ ˆµ i ˆρ = σ Υ i 1 0 µ i 2 Υ i 1 ρ Υj Υ j (3.4) Thus, for any choice of conformal scale, E I is identified with the direct sum E I = E[1] E i [ 1] E[ 1] and the components transform according to (3.4). We are also able to find the connection. We differentiate the second equation in (3.3), which gives us j ρ P j k µ k = 0 When we combine this together with (3.3), we have σ j σ µ j j µ i = j µ i + δj i ρ + P j i σ ρ j ρ P j k µ k We finally arrive at the first result of tractor calculus, which is that the conformal class contains an Einstein metric if and only if there exists a section of the tractor bundle parallel with respect to the tractor connection. The tractor bundle also carries a natural, nondegenerate, symmetric form g I J, called the tractor metric, defined by g I J U I V J = µ i β i + σγ+ρα

49 Chapter 3. Tractor calculus 29 where σ α U I = µ i, V J = β j ρ γ If the underlying conformal manifold has signature (p, q), then the tractor metric has signature (p + 1, q + 1) and is covariantly constant. We are able to split the tractor U I into a primary, secondary and tertiary parts which are σ, µ i and ρ respectively. From the transformation law it is clear that the primary part is conformally covariant. We can project this part out using the canonical tractor X I, given by 0 X I = 0 1 This is the tractor equivalent of the natural vector field T of the ambient metric construction. However, if the primary part vanishes, the secondary part is covariant. If that vanishes as well, the tertiary part becomes conformally covariant. In the notation for the composition series [11], we have E I = E[1]+E i [ 1]+E[ 1] The first non-zero part will be refered to as the projecting part. Similar composition series can be written for tractors with several indices as well. The deviation of a conformal structure from the flat one can be measured by the tractor curvature Ω of the tractor connection. This is defined as On tractors with lower indices we have ( i j j i )U K = Ωi j K L U L ( i j j i )U L = Ωi j K L U K By applying this to a tractor metric, we get 0 = ( i j j i ) g K L = Ω i j(k L) [K L] since i g K L = 0 and thus Ωi j K L = Ω We can apply a similar analysis as in the [i j] previous case to find the composition series. We get E kl [ 2] E [K L] = E k + +E k [ 2] E We do not change the series by tensoring with E [i j], so for the composition of the

50 30 Chapter 3. Tractor calculus tractor curvature we get E [K L] [i j] = E k [i j] + E [kl] [i j] E [i j] [ 2] +E k [i j] [ 2] After performing some calculations, we get the components of the tractor curvature σ σ ( i j j i ) µ i = 2 [i P k W k 0 j] i j l µ i ρ 0 2 [i P j]l 0 ρ The projecting part is the conformally invariant Weyl tensor. If n = 3, the Weyl tensor vanishes and the projecting part is [i P j]l, also known as the Cotton-York tensor. Using the second Bianchi identity, we get the relation between Cotton-York and Weyl tensor k k Wi j l = 2(n 3) [i P j]l Therefore, if the Weyl tensor vanishes so does the Cotton-York tensor (B.5). This leads us to a conclusion that the tractor connection is flat if and only if W i j kl = 0 for n 4 or [i P j]l = 0 when n = 3. These conditions are the necessary and sufficient conditions for a conformal manifold to be locally equivalent to the flat model [12]. However, tractor calculus is equipped with another differential operator, the Thomas D-operator D I E[w] E I [w 1], defined by w (n + 2w 2) f D I f = (n + 2w 2) i f ( + wp) f If 2w = 2 n, then the projecting part is [ +(1 n 2 )P] f, which is the conformally invariant Laplacian. In fact, the whole D-operator can be rewritten using the language of the ambient metric D I = (n + 2 T ) I T I 2 Clearly, the Thomas D-operator and tractor connections are useful tools for generating conformal invariants. One can construct the invariants not only as complete contractions, but also using that the projecting part is conformally invariant from the construction. For example, for a conformal density f with weight w = 1 n 2, the projecting part of the D-operator is the conformally invariant Laplacian. Applications of the tractor calculus to general relativity can be found in [13].