Bundles of Irreducible Clifford Modules and the Existence of Spin
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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 . -
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. -
THE DIRAC OPERATOR 1. First Properties 1.1. Definition. Let X Be A
THE DIRAC OPERATOR AKHIL MATHEW 1. First properties 1.1. Definition. Let X be a Riemannian manifold. Then the tangent bundle TX is a bundle of real inner product spaces, and we can form the corresponding Clifford bundle Cliff(TX). By definition, the bundle Cliff(TX) is a vector bundle of Z=2-graded R-algebras such that the fiber Cliff(TX)x is just the Clifford algebra Cliff(TxX) (for the inner product structure on TxX). Definition 1. A Clifford module on X is a vector bundle V over X together with an action Cliff(TX) ⊗ V ! V which makes each fiber Vx into a module over the Clifford algebra Cliff(TxX). Suppose now that V is given a connection r. Definition 2. The Dirac operator D on V is defined in local coordinates by sending a section s of V to X Ds = ei:rei s; for feig a local orthonormal frame for TX and the multiplication being Clifford multiplication. It is easy to check that this definition is independent of the coordinate system. A more invariant way of defining it is to consider the composite of differential operators (1) V !r T ∗X ⊗ V ' TX ⊗ V,! Cliff(TX) ⊗ V ! V; where the first map is the connection, the second map is the isomorphism T ∗X ' TX determined by the Riemannian structure, the third map is the natural inclusion, and the fourth map is Clifford multiplication. The Dirac operator is clearly a first-order differential operator. We can work out its symbol fairly easily. The symbol1 of the connection r is ∗ Sym(r)(v; t) = iv ⊗ t; v 2 Vx; t 2 Tx X: All the other differential operators in (1) are just morphisms of vector bundles. -
Differential Forms As Spinors Annales De L’I
ANNALES DE L’I. H. P., SECTION A WOLFGANG GRAF Differential forms as spinors Annales de l’I. H. P., section A, tome 29, no 1 (1978), p. 85-109 <http://www.numdam.org/item?id=AIHPA_1978__29_1_85_0> © Gauthier-Villars, 1978, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincare, Section A : Vol. XXIX. n° L 1978. p. 85 Physique’ théorique.’ Differential forms as spinors Wolfgang GRAF Fachbereich Physik der Universitat, D-7750Konstanz. Germany ABSTRACT. 2014 An alternative notion of spinor fields for spin 1/2 on a pseudoriemannian manifold is proposed. Use is made of an algebra which allows the interpretation of spinors as elements of a global minimal Clifford- ideal of differential forms. The minimal coupling to an electromagnetic field is introduced by means of an « U(1 )-gauging ». Although local Lorentz transformations play only a secondary role and the usual two-valuedness is completely absent, all results of Dirac’s equation in flat space-time with electromagnetic coupling can be regained. 1. INTRODUCTION It is well known that in a riemannian manifold the laplacian D := - (d~ + ~d) operating on differential forms admits as « square root » the first-order operator ~ 2014 ~ ~ being the exterior derivative and e) the generalized divergence. -
Principal Bundles, Vector Bundles and Connections
Appendix A Principal Bundles, Vector Bundles and Connections Abstract The appendix defines fiber bundles, principal bundles and their associate vector bundles, recall the definitions of frame bundles, the orthonormal frame bun- dle, jet bundles, product bundles and the Whitney sums of bundles. Next, equivalent definitions of connections in principal bundles and in their associate vector bundles are presented and it is shown how these concepts are related to the concept of a covariant derivative in the base manifold of the bundle. Also, the concept of exterior covariant derivatives (crucial for the formulation of gauge theories) and the meaning of a curvature and torsion of a linear connection in a manifold is recalled. The concept of covariant derivative in vector bundles is also analyzed in details in a way which, in particular, is necessary for the presentation of the theory in Chap. 12. Propositions are in general presented without proofs, which can be found, e.g., in Choquet-Bruhat et al. (Analysis, Manifolds and Physics. North-Holland, Amsterdam, 1982), Frankel (The Geometry of Physics. Cambridge University Press, Cambridge, 1997), Kobayashi and Nomizu (Foundations of Differential Geometry. Interscience Publishers, New York, 1963), Naber (Topology, Geometry and Gauge Fields. Interactions. Applied Mathematical Sciences. Springer, New York, 2000), Nash and Sen (Topology and Geometry for Physicists. Academic, London, 1983), Nicolescu (Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000), Osborn (Vector Bundles. Academic, New York, 1982), and Palais (The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, 1981). A.1 Fiber Bundles Definition A.1 A fiber bundle over M with Lie group G will be denoted by E D .E; M;;G; F/. -
Two-Spinors and Symmetry Operators
Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools Master’s thesis in Physics Simon Stefanus Jacobsson DEPARTMENT OF MATHEMATICAL SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se Master’s thesis 2021 Two-spinors and Symmetry Operators ¦ An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology Gothenburg, Sweden 2021 Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON © SIMON STEFANUS JACOBSSON, 2021. Supervisor: Thomas Bäckdahl, Mathematical Sciences Examiner: Simone Calogero, Mathematical Sciences Master’s Thesis 2021 Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2021 iii Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Chalmers University of Technology Abstract This thesis employs spinor techniques to find what conditions a curved spacetime must satisfy for there to exist a second order symmetry operator for the massive Dirac equation. Conditions are of the form of the existence of a set of Killing spinors satisfying some set of covariant differential equations. -
Arxiv:Math/9802033V1 [Math.DG] 6 Feb 1998
Cartan Spinor Bundles on Manifolds. ∗ Thomas Friedrich, Berlin July 26, 2021 1 Introduction. Spinor fields and Dirac operators on Riemannian manifolds M n can be introduced by means of a reduction of the frame bundle from the structure group O(n) to one of the groups Spin(n), P in(n) or SpinC(n). In any case the existence of a corre- sponding reduction imposes topological restrictions on the manifold M n. The aim of the present paper is the construction of spinor bundles of Cartan type over certain non-orientable manifolds. The bundles under consideration do not split into sub- bundles invariant under the action of the Clifford algebra and therefore they are not induced from P inC structure of the manifold. Moreover, we study the case of the real projective space RPn and its spinor bundle in more detail. These bundles over RPn admit a suitable metric connection ∇S and the corresponding Dirac operator. In particular it turns out that there are Killing spinors in the constructed twisted spinor bundle over RPn for all dimensions. C ∼ Let us fix some notations. In case n ≡ 0 mod 2 we denote by γ : Cliff (n) → End(∆n) the isomorphism between the Clifford algebra and the algebra of endomorphisms of the space ∆n of all Dirac spinor. In this way we obtain the so called Dirac representation γ of the Clifford algebra. In case n ≡ 1 mod 2 we have the Pauli C C C representation γ : Cliff (n) → End(∆n). Denote by α : Cliff (n) → Cliff (n) the canonical involution. -
Something on Spin Structures Sven-S. Porst
something on spin structures sven-s. porst∗ spring abstract This will give a brief introduction to spin structures on vector bundles to pave the way for the definition and introduction of Dirac operators. The presentation is mainly a brief summary of that given in [1]. principal G-bundles definition Let M be a manifold, π : P → M be a bundle over M and G be a Lie group. If there is a right action of G on P preserving fibres and acting freely on each fibre, π : P → M is called a principal G-bundle. In particular, a principal G-bundle is locally of the form U × G → U. equivalence Given two principal G-bundles π : P → M and π0 : P 0 → M over the same base space M, we say that these ϕ P / P 0 bundles are called equivalent if there exists a diffeo- AA | 0 0 AA || morphism ϕ : P → P such that π = π ϕ (a bundle AA || π AA || π0 isomorphism) that is compatible with the group ac- A |~ | tion, i.e. ϕ(x.g) = ϕ(x).g for all x ∈ M and g ∈ G. M This gives an equivalence relation on the set of principal G-bundles over M and the set of equivalence classes will be denoted PrinG(M). example: covering spaces Let π : M˜ → M be a normal covering space, i.e. the group G(M) of deck transformations acts transitively on each π−1(x). This is a principal G(M)-bundle where G(M) is equipped with the discrete topology. -
Bilinear Covariants and Spinor Field Classification in Quantum Clifford Algebras
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT BILINEAR COVARIANTS AND SPINOR FIELD CLASSIFICATION IN QUANTUM CLIFFORD ALGEBRAS R. ABLAMOWICZ, ´ICARO GONC¸ALVES AND ROLDAO˜ DA ROCHA June 2013 No. 2013-2 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505 Bilinear Covariants and Spinor Field Classification in Quantum Clifford Algebras Rafa l Ab lamowicz†, ´Icaro Gon¸calves‡ and Rold˜ao da Rocha‡ † Department of Mathematics, Box 5054, Tennessee Technological University, Cookeville, TN 38505, USA ‡ Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC, 09210-170, Santo Andr´e, SP, Brazil E-mail: [email protected], [email protected], [email protected] Abstract. In this letter, the Lounesto spinor field classification is extended to the spacetime quantum Clifford algebra and the associated quantum algebraic spinor fields are constructed. In order to accomplish this extension, the spin-Clifford bundle formalism is employed, where the algebraic and geometric objects of interest may be considered. In particular, we describe the bilinear covariant fields in the context of algebraic spinor fields. By endowing the underlying spacetime structure with a bilinear form that contains an antisymmetric part, which extends the spacetime metric, the quantum algebraic spinor fields are exhibited and compared to the standard case, together with the bilinear covariants that they induce. Quantum spinor field classes are hence introduced and a correspondence between them and the Lounesto spinor field classification is provided. A physical interpretation regarding the deformed parts and the underlying Zn-grading is also given. The existence of an arbitrary bilinear form endowing the spacetime, already explored in the literature in the context of quantum gravity [1], plays a prominent role in the structure of the Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flag-dipoles, which are shown to be capable to probe interesting features associated to the spacetime. -
Notes on Cli Ord Algebras and Spin Groups
Notes on Cliord Algebras and Spin Groups Thomas Hjortgaard Danielsen Contents 1 Cliord Algebras 5 1.1 Elementary Properties . 5 1.2 Classication of Cliord Algebras . 11 1.3 Representation Theory . 15 2 Spin Groups 19 2.1 The Cliord Group . 19 2.2 Pin and Spin Groups . 22 2.3 Double Coverings . 25 2.4 Spin Group Representations . 28 2.5 Spin Structures . 31 2.6 The Dirac Operator . 38 3 4 Chapter 1 Cliord Algebras 1.1 Elementary Properties In this chapter we will introduce the Cliord algebra and discuss some of its elementary properties. The setting is the following: Let V be a nite-dimensional vector space over the eld K (predominantly R or C) and ϕ : V × V −→ K a symmetric bilinear form on V . ϕ is said to be positive (resp. negative) denite if for all 0 6= v ∈ V we have ϕ(v, v) > 0 (resp. ϕ(v, v) < 0). ϕ is called non-degenerate if ϕ(v, w) = 0 for all v ∈ V implies w = 0. From a bilinear form we construct a quadratic form Φ: V −→ K given by Φ(v) := ϕ(v, v). We can recover the original bilinear form by the polarization identity: Φ(u + v) = ϕ(u + v, u + v) = Φ(u) + Φ(v) + 2ϕ(u, v), hence 1 ϕ(u, v) = (Φ(u + v) − Φ(u) − Φ(v)). (1.1) 2 Thus we have a 1-1 correspondence between symmetric bilinear forms and quadratic forms. Thus a quadratic form Φ is called positive denite/negative denite/non-degenerate if ϕ is. -
GENERAL RELATIVITY and SPINORS Jean Claude Dutailly
GENERAL RELATIVITY AND SPINORS Jean Claude Dutailly To cite this version: Jean Claude Dutailly. GENERAL RELATIVITY AND SPINORS. 2015. hal-01171507v3 HAL Id: hal-01171507 https://hal.archives-ouvertes.fr/hal-01171507v3 Preprint submitted on 16 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License GENERAL RELATIVITY AND SPINORS Jean Claude Dutailly 15th of November 2016 Abstract This paper exposes, in a comprehensive and consistent way, the Geometry of General Relativity through fiber bundles and tetrads. A review of the concept of motion in Galilean Geometry leads, for its extension to the relativist context, to a representation based on Clifford Algebras. It provides a strong frame-work, and many tools which enable to deal easily with the most general problems, including the extension of the concepts of deformable and rigid solids in RG, at any scale. It is then natural to represent the moments of material bodies, translational and rotational, by spinors. The results hold at any scale, and their implementation for elementary particles gives the known results with anti-particles, chirality and spin.