Two-Spinors and Symmetry Operators
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:Math/0212058V2 [Math.DG] 18 Nov 2003 Nosbrusof Subgroups Into Rdc C.[Oc 00 Et .].Eape O Uhsae a Spaces Such for Examples 3.2])
THE SPINOR BUNDLE OF RIEMANNIAN PRODUCTS FRANK KLINKER Abstract. In this note we compare the spinor bundle of a Riemannian mani- fold (M = M1 ×···×MN ,g) with the spinor bundles of the Riemannian factors (Mi,gi). We show that - without any holonomy conditions - the spinor bundle of (M,g) for a special class of metrics is isomorphic to a bundle obtained by tensoring the spinor bundles of (Mi,gi) in an appropriate way. For N = 2 and an one dimensional factor this construction was developed in [Baum 1989a]. Although the fact for general factors is frequently used in (at least physics) literature, a proof was missing. I would like to thank Shahram Biglari, Mario Listing, Marc Nardmann and Hans-Bert Rademacher for helpful comments. Special thanks go to Helga Baum, who pointed out some difficulties arising in the pseudo-Riemannian case. We consider a Riemannian manifold (M = M MN ,g), which is a product 1 ×···× of Riemannian spin manifolds (Mi,gi) and denote the projections on the respective factors by pi. Furthermore the dimension of Mi is Di such that the dimension of N M is given by D = i=1 Di. The tangent bundle of M is decomposed as P ∗ ∗ (1) T M = p Tx1 M p Tx MN . (x0,...,xN ) 1 1 ⊕···⊕ N N N We omit the projections and write TM = i=1 TMi. The metric g of M need not be the product metric of the metrics g on M , but L i i is assumed to be of the form c d (2) gab(x) = Ai (x)gi (xi)Ai (x), T Mi a cd b for D1 + + Di−1 +1 a,b D1 + + Di, 1 i N ··· ≤ ≤ ··· ≤ ≤ In particular, for those metrics the splitting (1) is orthogonal, i.e. -
Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
Abstract Tensor Systems and Diagrammatic Representations
Abstract tensor systems and diagrammatic representations J¯anisLazovskis September 28, 2012 Abstract The diagrammatic tensor calculus used by Roger Penrose (most notably in [7]) is introduced without a solid mathematical grounding. We will attempt to derive the tools of such a system, but in a broader setting. We show that Penrose's work comes from the diagrammisation of the symmetric algebra. Lie algebra representations and their extensions to knot theory are also discussed. Contents 1 Abstract tensors and derived structures 2 1.1 Abstract tensor notation . 2 1.2 Some basic operations . 3 1.3 Tensor diagrams . 3 2 A diagrammised abstract tensor system 6 2.1 Generation . 6 2.2 Tensor concepts . 9 3 Representations of algebras 11 3.1 The symmetric algebra . 12 3.2 Lie algebras . 13 3.3 The tensor algebra T(g)....................................... 16 3.4 The symmetric Lie algebra S(g)................................... 17 3.5 The universal enveloping algebra U(g) ............................... 18 3.6 The metrized Lie algebra . 20 3.6.1 Diagrammisation with a bilinear form . 20 3.6.2 Diagrammisation with a symmetric bilinear form . 24 3.6.3 Diagrammisation with a symmetric bilinear form and an orthonormal basis . 24 3.6.4 Diagrammisation under ad-invariance . 29 3.7 The universal enveloping algebra U(g) for a metrized Lie algebra g . 30 4 Ensuing connections 32 A Appendix 35 Note: This work relies heavily upon the text of Chapter 12 of a draft of \An Introduction to Quantum and Vassiliev Invariants of Knots," by David M.R. Jackson and Iain Moffatt, a yet-unpublished book at the time of writing. -
Generic Properties of Symmetric Tensors
2006 – 1/48 – P.Comon Generic properties of Symmetric Tensors Pierre COMON I3S - CNRS other contributors: Bernard MOURRAIN INRIA institute Lek-Heng LIM, Stanford University I3S 2006 – 2/48 – P.Comon Tensors & Arrays Definitions Table T = {Tij..k} Order d of T def= # of its ways = # of its indices def Dimension n` = range of the `th index T is Square when all dimensions n` = n are equal T is Symmetric when it is square and when its entries do not change by any permutation of indices I3S 2006 – 3/48 – P.Comon Tensors & Arrays Properties Outer (tensor) product C = A ◦ B: Cij..` ab..c = Aij..` Bab..c Example 1 outer product between 2 vectors: u ◦ v = u vT Multilinearity. An order-3 tensor T is transformed by the multi-linear map {A, B, C} into a tensor T 0: 0 X Tijk = AiaBjbCkcTabc abc Similarly: at any order d. I3S 2006 – 4/48 – P.Comon Tensors & Arrays Example Example 2 Take 1 v = −1 Then 1 −1 −1 1 v◦3 = −1 1 1 −1 This is a “rank-1” symmetric tensor I3S 2006 – 5/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... I3S 2006 – 6/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... PARAFAC cannot be used when: • Lack of diversity I3S 2006 – 7/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... PARAFAC cannot be used when: • Lack of diversity • Proportional slices I3S 2006 – 8/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA . -
Multivector Differentiation and Linear Algebra 0.5Cm 17Th Santaló
Multivector differentiation and Linear Algebra 17th Santalo´ Summer School 2016, Santander Joan Lasenby Signal Processing Group, Engineering Department, Cambridge, UK and Trinity College Cambridge [email protected], www-sigproc.eng.cam.ac.uk/ s jl 23 August 2016 1 / 78 Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. 2 / 78 Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. 3 / 78 Functional Differentiation: very briefly... Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. 4 / 78 Summary Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... 5 / 78 Overview The Multivector Derivative. Examples of differentiation wrt multivectors. Linear Algebra: matrices and tensors as linear functions mapping between elements of the algebra. Functional Differentiation: very briefly... Summary 6 / 78 We now want to generalise this idea to enable us to find the derivative of F(X), in the A ‘direction’ – where X is a general mixed grade multivector (so F(X) is a general multivector valued function of X). Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. Then, provided A has same grades as X, it makes sense to define: F(X + tA) − F(X) A ∗ ¶XF(X) = lim t!0 t The Multivector Derivative Recall our definition of the directional derivative in the a direction F(x + ea) − F(x) a·r F(x) = lim e!0 e 7 / 78 Let us use ∗ to denote taking the scalar part, ie P ∗ Q ≡ hPQi. -
1.2 Topological Tensor Calculus
PH211 Physical Mathematics Fall 2019 1.2 Topological tensor calculus 1.2.1 Tensor fields Finite displacements in Euclidean space can be represented by arrows and have a natural vector space structure, but finite displacements in more general curved spaces, such as on the surface of a sphere, do not. However, an infinitesimal neighborhood of a point in a smooth curved space1 looks like an infinitesimal neighborhood of Euclidean space, and infinitesimal displacements dx~ retain the vector space structure of displacements in Euclidean space. An infinitesimal neighborhood of a point can be infinitely rescaled to generate a finite vector space, called the tangent space, at the point. A vector lives in the tangent space of a point. Note that vectors do not stretch from one point to vector tangent space at p p space Figure 1.2.1: A vector in the tangent space of a point. another, and vectors at different points live in different tangent spaces and so cannot be added. For example, rescaling the infinitesimal displacement dx~ by dividing it by the in- finitesimal scalar dt gives the velocity dx~ ~v = (1.2.1) dt which is a vector. Similarly, we can picture the covector rφ as the infinitesimal contours of φ in a neighborhood of a point, infinitely rescaled to generate a finite covector in the point's cotangent space. More generally, infinitely rescaling the neighborhood of a point generates the tensor space and its algebra at the point. The tensor space contains the tangent and cotangent spaces as a vector subspaces. A tensor field is something that takes tensor values at every point in a space. -
Appendix A: Matrices and Tensors
Appendix A: Matrices and Tensors A.1 Introduction and Rationale The purpose of this appendix is to present the notation and most of the mathematical techniques that will be used in the body of the text. The audience is assumed to have been through several years of college level mathematics that included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in undergraduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles and the vector cross and triple scalar products. The solutions to ordinary differential equations are reviewed in the last two sections. The notation, as far as possible, will be a matrix notation that is easily entered into existing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad etc. The desire to represent the components of three-dimensional fourth order tensors that appear in anisotropic elasticity as the components of six-dimensional second order tensors and thus represent these components in matrices of tensor components in six dimensions leads to the nontraditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in Sect. A.11, along with the rationale for this approach. A.2 Definition of Square, Column, and Row Matrices An r by c matrix M is a rectangular array of numbers consisting of r rows and c columns, S.C. -
Parallel Spinors and Connections with Skew-Symmetric Torsion in String Theory*
ASIAN J. MATH. © 2002 International Press Vol. 6, No. 2, pp. 303-336, June 2002 005 PARALLEL SPINORS AND CONNECTIONS WITH SKEW-SYMMETRIC TORSION IN STRING THEORY* THOMAS FRIEDRICHt AND STEFAN IVANOV* Abstract. We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the V- parallel spinors. In particular, we obtain partial solutions of the type // string equations in dimension n = 5, 6 and 7. 1. Introduction. Linear connections preserving a Riemannian metric with totally skew-symmetric torsion recently became a subject of interest in theoretical and mathematical physics. For example, the target space of supersymmetric sigma models with Wess-Zumino term carries a geometry of a metric connection with skew-symmetric torsion [23, 34, 35] (see also [42] and references therein). In supergravity theories, the geometry of the moduli space of a class of black holes is carried out by a metric connection with skew-symmetric torsion [27]. The geometry of NS-5 brane solutions of type II supergravity theories is generated by a metric connection with skew-symmetric torsion [44, 45, 43]. The existence of parallel spinors with respect to a metric connection with skew-symmetric torsion on a Riemannian spin manifold is of importance in string theory, since they are associated with some string solitons (BPS solitons) [43]. Supergravity solutions that preserve some of the supersymmetry of the underlying theory have found many applications in the exploration of perturbative and non-perturbative properties of string theory. -
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. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
A Some Basic Rules of Tensor Calculus
A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e. -
General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ).