Twistor and Killing Spinors in Lorentzian Geometry
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The Simplicial Ricci Tensor 2
The Simplicial Ricci Tensor Paul M. Alsing1, Jonathan R. McDonald 1,2 & Warner A. Miller3 1Information Directorate, Air Force Research Laboratory, Rome, New York 13441 2Insitut f¨ur Angewandte Mathematik, Friedrich-Schiller-Universit¨at-Jena, 07743 Jena, Germany 3Department of Physics, Florida Atlantic University, Boca Raton, FL 33431 E-mail: [email protected] Abstract. The Ricci tensor (Ric) is fundamental to Einstein’s geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman’s proof of the Poincar`e conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher-dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area – an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimension. arXiv:1107.2458v1 [gr-qc] 13 Jul 2011 The Simplicial Ricci Tensor 2 1. -
Killing Spinor-Valued Forms and the Cone Construction
ARCHIVUM MATHEMATICUM (BRNO) Tomus 52 (2016), 341–355 KILLING SPINOR-VALUED FORMS AND THE CONE CONSTRUCTION Petr Somberg and Petr Zima Abstract. On a pseudo-Riemannian manifold M we introduce a system of partial differential Killing type equations for spinor-valued differential forms, and study their basic properties. We discuss the relationship between solutions of Killing equations on M and parallel fields on the metric cone over M for spinor-valued forms. 1. Introduction The subject of the present article are the systems of over-determined partial differential equations for spinor-valued differential forms, classified as atypeof Killing equations. The solution spaces of these systems of PDE’s are termed Killing spinor-valued differential forms. A central question in geometry asks for pseudo-Riemannian manifolds admitting non-trivial solutions of Killing type equa- tions, namely how the properties of Killing spinor-valued forms relate to the underlying geometric structure for which they can occur. Killing spinor-valued forms are closely related to Killing spinors and Killing forms with Killing vectors as a special example. Killing spinors are both twistor spinors and eigenspinors for the Dirac operator, and real Killing spinors realize the limit case in the eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds of positive scalar curvature. There is a classification of complete simply connected Riemannian manifolds equipped with real Killing spinors, leading to the construction of manifolds with the exceptional holonomy groups G2 and Spin(7), see [8], [1]. Killing vector fields on a pseudo-Riemannian manifold are the infinitesimal generators of isometries, hence they influence its geometrical properties. -
Gsm025-Endmatter.Pdf
http://dx.doi.org/10.1090/gsm/025 Selected Titles in This Series 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. -
Arxiv:1910.04634V1 [Math.DG] 10 Oct 2019 ˆ That E Sdnt by Denote Us Let N a En H Subbundle the Define Can One of Points Bundle
SPIN FRAME TRANSFORMATIONS AND DIRAC EQUATIONS R.NORIS(1)(2), L.FATIBENE(2)(3) (1) DISAT, Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129 Torino, Italy (2)INFN Sezione di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy (3) Dipartimento di Matematica – University of Torino, via Carlo Alberto 10, I-10123 Torino, Italy Abstract. We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer re- quirements, transformations allowed by spin frames are more general than usual spin transformations and they usually do not preserve the induced metric structures. We study how these new transformations affect connections both on the spin bundle and on the frame bundle and how this reflects on the Dirac equations. 1. Introduction Dirac equations provide an important tool to study the geometric structure of manifolds, as well as to model the behaviour of a class of physical particles, namely fermions, which includes electrons. The aim of this paper is to generalise a key item needed to formulate Dirac equations, the spin structures, in order to extend the range of allowed transformations. Let us start by first reviewing the usual approach to Dirac equations. Let (M,g) be an orientable pseudo-Riemannian manifold with signature η = (r, s), such that r + s = m = dim(M). R arXiv:1910.04634v1 [math.DG] 10 Oct 2019 Let us denote by L(M) the (general) frame bundle of M, which is a GL(m, )-principal fibre bundle. -
Spinors of Real Type As Polyforms and the Generalized Killing Equation 11
SPINORS OF REAL TYPE AS POLYFORMS AND THE GENERALIZED KILLING EQUATION VICENTE CORTÉS, CALIN LAZAROIU, AND C. S. SHAHBAZI Abstract. We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler- Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module Σ of real type as a real algebraic variety in the Kähler-Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of AdS4 space-time. Contents 1. Introduction 1 2. Representing real vectors as endomorphisms in a paired vectorspace 6 3. From real spinors to polyforms 14 4. Constrained Generalized Killing spinors of real type 28 5. Real Killing spinors on Lorentzian four-manifolds 39 6. Supersymmetric heterotic configurations 47 Appendix A. -
Math 865, Topics in Riemannian Geometry
Math 865, Topics in Riemannian Geometry Jeff A. Viaclovsky Fall 2007 Contents 1 Introduction 3 2 Lecture 1: September 4, 2007 4 2.1 Metrics, vectors, and one-forms . 4 2.2 The musical isomorphisms . 4 2.3 Inner product on tensor bundles . 5 2.4 Connections on vector bundles . 6 2.5 Covariant derivatives of tensor fields . 7 2.6 Gradient and Hessian . 9 3 Lecture 2: September 6, 2007 9 3.1 Curvature in vector bundles . 9 3.2 Curvature in the tangent bundle . 10 3.3 Sectional curvature, Ricci tensor, and scalar curvature . 13 4 Lecture 3: September 11, 2007 14 4.1 Differential Bianchi Identity . 14 4.2 Algebraic study of the curvature tensor . 15 5 Lecture 4: September 13, 2007 19 5.1 Orthogonal decomposition of the curvature tensor . 19 5.2 The curvature operator . 20 5.3 Curvature in dimension three . 21 6 Lecture 5: September 18, 2007 22 6.1 Covariant derivatives redux . 22 6.2 Commuting covariant derivatives . 24 6.3 Rough Laplacian and gradient . 25 7 Lecture 6: September 20, 2007 26 7.1 Commuting Laplacian and Hessian . 26 7.2 An application to PDE . 28 1 8 Lecture 7: Tuesday, September 25. 29 8.1 Integration and adjoints . 29 9 Lecture 8: September 23, 2007 34 9.1 Bochner and Weitzenb¨ock formulas . 34 10 Lecture 9: October 2, 2007 38 10.1 Manifolds with positive curvature operator . 38 11 Lecture 10: October 4, 2007 41 11.1 Killing vector fields . 41 11.2 Isometries . 44 12 Lecture 11: October 9, 2007 45 12.1 Linearization of Ricci tensor . -
Elementary Differential Geometry
ELEMENTARY DIFFERENTIAL GEOMETRY YONG-GEUN OH { Based on the lecture note of Math 621-2020 in POSTECH { Contents Part 1. Riemannian Geometry 2 1. Parallelism and Ehresman connection 2 2. Affine connections on vector bundles 4 2.1. Local expression of covariant derivatives 6 2.2. Affine connection recovers Ehresmann connection 7 2.3. Curvature 9 2.4. Metrics and Euclidean connections 9 3. Riemannian metrics and Levi-Civita connection 10 3.1. Examples of Riemannian manifolds 12 3.2. Covariant derivative along the curve 13 4. Riemann curvature tensor 15 5. Raising and lowering indices and contractions 17 6. Geodesics and exponential maps 19 7. First variation of arc-length 22 8. Geodesic normal coordinates and geodesic balls 25 9. Hopf-Rinow Theorem 31 10. Classification of constant curvature surfaces 33 11. Second variation of energy 34 Part 2. Symplectic Geometry 39 12. Geometry of cotangent bundles 39 13. Poisson manifolds and Schouten-Nijenhuis bracket 42 13.1. Poisson tensor and Jacobi identity 43 13.2. Lie-Poisson space 44 14. Symplectic forms and the Jacobi identity 45 15. Proof of Darboux' Theorem 47 15.1. Symplectic linear algebra 47 15.2. Moser's deformation method 48 16. Hamiltonian vector fields and diffeomorhpisms 50 17. Autonomous Hamiltonians and conservation law 53 18. Completely integrable systems and action-angle variables 55 18.1. Construction of angle coordinates 56 18.2. Construction of action coordinates 57 18.3. Underlying geometry of the Hamilton-Jacobi method 61 19. Lie groups and Lie algebras 62 1 2 YONG-GEUN OH 20. Group actions and adjoint representations 67 21. -
Chapter 13 Curvature in Riemannian Manifolds
Chapter 13 Curvature in Riemannian Manifolds 13.1 The Curvature Tensor If (M, , )isaRiemannianmanifoldand is a connection on M (that is, a connection on TM−), we− saw in Section 11.2 (Proposition 11.8)∇ that the curvature induced by is given by ∇ R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] for all X, Y X(M), with R(X, Y ) Γ( om(TM,TM)) = Hom (Γ(TM), Γ(TM)). ∈ ∈ H ∼ C∞(M) Since sections of the tangent bundle are vector fields (Γ(TM)=X(M)), R defines a map R: X(M) X(M) X(M) X(M), × × −→ and, as we observed just after stating Proposition 11.8, R(X, Y )Z is C∞(M)-linear in X, Y, Z and skew-symmetric in X and Y .ItfollowsthatR defines a (1, 3)-tensor, also denoted R, with R : T M T M T M T M. p p × p × p −→ p Experience shows that it is useful to consider the (0, 4)-tensor, also denoted R,givenby R (x, y, z, w)= R (x, y)z,w p p p as well as the expression R(x, y, y, x), which, for an orthonormal pair, of vectors (x, y), is known as the sectional curvature, K(x, y). This last expression brings up a dilemma regarding the choice for the sign of R. With our present choice, the sectional curvature, K(x, y), is given by K(x, y)=R(x, y, y, x)but many authors define K as K(x, y)=R(x, y, x, y). Since R(x, y)isskew-symmetricinx, y, the latter choice corresponds to using R(x, y)insteadofR(x, y), that is, to define R(X, Y ) by − R(X, Y )= + . -
The Ricci Curvature of Rotationally Symmetric Metrics
The Ricci curvature of rotationally symmetric metrics Anna Kervison Supervisor: Dr Artem Pulemotov The University of Queensland 1 1 Introduction Riemannian geometry is a branch of differential non-Euclidean geometry developed by Bernhard Riemann, used to describe curved space. In Riemannian geometry, a manifold is a topological space that locally resembles Euclidean space. This means that at any point on the manifold, there exists a neighbourhood around that point that appears ‘flat’and could be mapped into the Euclidean plane. For example, circles are one-dimensional manifolds but a figure eight is not as it cannot be pro- jected into the Euclidean plane at the intersection. Surfaces such as the sphere and the torus are examples of two-dimensional manifolds. The shape of a manifold is defined by the Riemannian metric, which is a measure of the length of tangent vectors and curves in the manifold. It can be thought of as locally a matrix valued function. The Ricci curvature is one of the most sig- nificant geometric characteristics of a Riemannian metric. It provides a measure of the curvature of the manifold in much the same way the second derivative of a single valued function provides a measure of the curvature of a graph. Determining the Ricci curvature of a metric is difficult, as it is computed from a lengthy ex- pression involving the derivatives of components of the metric up to order two. In fact, without additional simplifications, the formula for the Ricci curvature given by this definition is essentially unmanageable. Rn is one of the simplest examples of a manifold. -
Operator-Valued Tensors on Manifolds: a Framework for Field
Operator-Valued Tensors on Manifolds: A Framework for Field Quantization 1 2 H. Feizabadi , N. Boroojerdian ∗ 1,2Department of pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran. Abstract In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commu- tative C⋆-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered. Keywords: Operator-valued tensors, Operator-Valued Semi-Riemannian Met- rics, Levi-Civita Connection, Curvature, Hodge star operator MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. 0 Introduction arXiv:1501.05065v2 [math-ph] 27 Jan 2015 The aim of the present paper is to extend the theory of semi-Riemannian metrics to operator-valued semi-Riemannian metrics, which may be used as a framework for field quantization. Spaces of linear operators are directly related to C∗-algebras, so C∗- algebras are considered as a basic concept in this article. This paper has its roots in Hilbert C⋆-modules, which are frequently used in the theory of operator algebras, allowing one to obtain information about C⋆-algebras by studying Hilbert C⋆-modules over them. Hilbert C⋆-modules provide a natural generalization of Hilbert spaces arising when the field of scalars C is replaced by an arbitrary C⋆-algebra. -
Geometric Algebra Techniques for General Relativity
Geometric Algebra Techniques for General Relativity Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor. PACS numbers: 02.40.-k; 04.20.Cv Keywords: General relativity; Clifford algebras; solution techniques I. INTRODUCTION Geometric (or Clifford) algebra provides a simple and natural language for describing geometric concepts, a point which has been argued persuasively by Hestenes [1] and Lounesto [2] among many others. Geometric algebra (GA) unifies many other mathematical formalisms describing specific aspects of geometry, including complex variables, matrix algebra, projective geometry, and differential geometry. Gravitation, which is usually viewed as a geometric theory, is a natural candidate for translation into the language of geometric algebra. This has been done for some aspects of gravitational theory; notably, Hestenes and Sobczyk have shown how geometric algebra greatly simplifies certain calculations involving the curvature tensor and provides techniques for classifying the Weyl tensor [3, 4]. Lasenby, Doran, and Gull [5] have also discussed gravitation using geometric algebra via a reformulation in terms of a gauge principle. In this paper, we formulate standard general relativity in terms of geometric algebra. A comprehensive overview like the one presented here has not previously appeared in the literature, although unpublished works of Hestenes and of Doran take significant steps in this direction. -
Clifford Algebras
A geometric construction of exceptional Lie algebras José Figueroa-O’Farrill Maxwell Institute & School of Mathematics Leeds, 13 February 2008 2007 will be known as the year where E8 (and Lie groups) went mainstream... Introduction Hamilton Cayley Lie Killing É. Cartan Hurwitz Hopf J.F. Adams This talk is about a relation between exceptional objects: • Hopf bundles • exceptional Lie algebras using a geometric construction familiar from supergravity: the Killing (super)algebra. Real division algebras R C H O ≥ ab = ba ab = ba ! (ab)c = a(bc) (ab)c = a(bc) ! These are all the euclidean normed real division algebras. [Hurwitz] Hopf fibrations S1 S3 S7 S15 S0 S1 S3 S7 S1 S2 S4 S8 " " " " S0 R S1 C S3 H S7 O ⊂ ⊂ ⊂ ⊂ 1 2 3 2 7 2 15 2 S R S C S H S O ⊂ ⊂ ⊂ ⊂ 1 2 4 8 S ∼= RP1 S ∼= CP1 S ∼= HP1 S ∼= OP1 These are the only examples of fibre bundles where all three spaces are spheres. [Adams] Simple Lie algebras (over C) 4 classical series: 5 exceptions: An 1 SU(n + 1) G2 14 ≥ F4 52 Bn 2 SO(2n + 1) ≥ E6 78 Cn 3 Sp(n) ≥ E7 133 Dn 4 SO(2n) E8 248 ≥ [Lie] [Killing, Cartan] Supergravity Supergravity is a nontrivial generalisation of Einstein’s theory of General Relativity. The supergravity universe consists of a lorentzian spin manifold with additional geometric data, together with a notion of Killing spinor. These spinors generate the Killing superalgebra. This is a useful invariant of the universe. Applying the Killing superalgebra construction to the exceptional Hopf fibration, oneS 15 obtains a triple of exceptional Lie algebras: S7 S8 " E8 “Killing superalgebra” B4 F4 Spinors Clifford Clifford algebras V n , real euclidean vector space !− −# V C!(V ) = filtered associative algebra v v + v 21 ! ⊗ ! | | # C!(V ) ∼= ΛV (as vector spaces) C!(V ) = C!(V ) C!(V ) 0 ⊕ 1 C!(V ) = ΛevenV odd 0 ∼ C!(V )1 ∼= Λ V orthonormal frame e1, .