DOCSLIB.ORG

Connections on Smooth Lie Algebra Bundles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Connections on Smooth Lie Algebra Bundles Indian J. Pure Appl. Math., 50(4): 891-901, December 2019 °c Indian National Science Academy DOI: 10.1007/s13226-019-0362-3 CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES K. Ajaykumar and B. S. Kiranagi Department of Studies in Mathematics, University of Mysore, Mysore 570 006, India e-mails: [email protected]; [email protected] (Received 25 April 2018; accepted 4 September 2018) We define the notion of Lie Ehresmann connection on Lie algebra bundles and show that a Lie connection on a Lie algebra bundle induces a Lie Ehresmann connection. The converse is proved for normed Lie algebra bundles. We then show that the connection on adjoint bundle corre- sponding to the connection on principal G¡bundle to which it is associated is a Lie Ehresmann connection. Further it is shown that the Lie Ehresmann connection on adjoint bundle induced by a universal G-connection is universal over the family of adjoint bundles associated to G-bundles. Key words : Adjoint bundle; Lie algebra bundle; Lie connection; Lie Ehresmann connection; parallel transport, universal connection. 1. INTRODUCTION Transporting a certain motion in the base manifold of a Lie algebra bundle to the motion in its fibres, needs an additional structure. This is provided by the notion of connection on Lie algebra bundle. Mackenzie [9], defines Lie connection on a Lie algebra bundle in terms of covariant derivative by defining Lie connection on Lie algebroids and proves the existence. A proof of the existence of Lie connection on Lie algebra bundles is also in Hasan Gu¨ndog˘an’s diploma thesis [5]. We define the notion of connection on smooth Lie algebra bundles in a more geometric way in terms of horizontal fields, which we call Lie Ehresmann connection. We prove that our notion is equivalent to the notion of Lie connection defined in [9] for normed Lie algebra bundles. Further we define connection on adjoint bundle adP associated to a smooth principal bundle P in terms of a connection on P and show that it is Lie Ehresmann connection in a natural way. 892 K. AJAYKUMAR AND B. S. KIRANAGI Parallel transport on adjoint bundles is defined in terms of parallel transport on principal bun- dles. We then show that the connection recovered by such a parallel transport is a Lie Ehresmann connection. A universal G¡connection has been constructed by Narasimhan and Ramanan [10, 11] and Schlafly [12]. Later in 2012, Biswas et al. [1] have given a simple and transparent construction of universal principal G¡connection. Here we study the Lie Ehresmann connection on adjoint bun- dle induced by universal G¡connection and show that such an induced Lie Ehresmann connection is G¡universal over the family of adjoint bundles. Notations and Terminologies Unless specified otherwise, all bundles are smooth bundles over smooth manifolds that are haus- dorff, second countable and paracompact. All principal bundles have a structure group that is a finite dimensional Lie group. All Lie algebras are real. 2. PRELIMINARY DEFINITIONS Definition 2.1 — A real normed vector space (L; j ¤ j) is said to be a normed Lie algebra [14] w.r.t a Lie product [; ] on L if it has the property j[x; y]j · 2jxjjyj Definition 2.2 — A smooth weak (normed) Lie algebra bundle (»; ¼; M) is a smooth vector bundle together with a smooth morphism Θ: » © » ! » inducing a (normed) Lie algebra structure on each fibre »x [2]. Definition 2.3 — A locally trivial smooth (normed) Lie algebra bundle [4, 7], for short a smooth Lie algebra bundle, is a smooth vector bundle » = (»; ¼; M) whose standard fibre is a (normed) Lie algebra say L, in which each fibre is a (normed) Lie algebra such that for each x in M there is an open set U in M containing x and a diffeomorphism ' : U £ L ! ¼¡1(U) such that for each x in ¡1 U; 'x : fxg £ L ! ¼ (x) is a (normed) Lie algebra isomorphism. Remark 2.1 : Every locally trivial smooth Lie algebra bundle is a weak Lie algebra bundle. But the converse is not true in general [8]. Definition 2.4 — A connection on a principal bundle ¼ : P ! M is a collection H = fHp j p 2 P g of subspaces Hp ½ TpP called horizontal subspaces, such that 1. the assignment p 7! Hp depends smoothly on p 2 P CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 893 2. TpP = Vp © Hp, where Vp = ker((¼¤)p : TpP ! T¼(p)M) called vertical subspace 3. HRg(p) = ((Rg)¤)pHp for all g 2 G and p 2 P , where (Rg)¤ is the differential of Rg, the right action on P Definition 2.5 — For a vector bundle ¼ : E ! M, let a : E ! E be the diffeomorphism defined by a(e) = a ¢ e for every non-zero a 2 R.A linear Ehresmann connection on a vector bundle ¼ : E ! M is a collection H = fHe j e 2 Eg of subspaces He ½ TeE called horizontal subspaces, such that 1. the assignment e 7! He depends smoothly on e 2 E 2. TeE = Ve © He, where Ve = ker((¼¤)e : TeE ! T¼(e)M) called vertical subspace 3. Ha¢e = ((a)¤)eHe, for all e 2 E and non-zero a 2 R, where (a)¤ is the differential of a 3. CONNECTION ON LIE ALGEBRA BUNDLES Definition 3.1 — (Lie Ehresmann connection). Let » = (»; ¼; M; Θ) be a smooth Lie algebra bundle. A Lie Ehresmann connection on » is a collection H = fHl j l 2 »g, such that 1. H is a linear Ehresmann connection on » 2. Θ¤(H £ H) = H, where H = [l2»Hl and H £ H is the corresponding horizontal subbundle on » © ». That is, Θ¤(Hl1 £ Hl2 ) = (Θ¤)(l1;l2)(H(l1;l2)) = HΘ(l1;l2) = H[l1;l2], where Θ¤ is the differ- ential of the smooth morphism Θ: » © » ! ». In otherwords, a linear Ehresmann connection on » for which the differential of the morphism Θ carries horizontal tangent vectors of » ©» to the corresponding horizontal tangent vector of » is called a Lie Ehresmann connection. Definition 3.2 — (Lie Connection). Let » = (»; ¼; M; Θ) be a smooth Lie algebra bundle. A Lie connection [9] is a linear map r : Γ(TM) ! End(Γ(»));X 7! rX satisfying the following conditions: 1 1. rX (aA) = (X:a)A + arX A for all X 2 Γ(TM); a 2 C (M; R);A 2 Γ(») 894 K. AJAYKUMAR AND B. S. KIRANAGI 1 2. r(aX+bY )A = arX A + brY A for all X; Y 2 Γ(TM); a; b 2 C (M; R);A 2 Γ(») 3. rX [A; B] = [rX A; B] + [A; rX B] for all X 2 Γ(TM); A; B 2 Γ(») In short a Lie connection on a Lie algebra bundle » is a covariant derivative rX on vector bundle » such that it is a derivation on Γ(»), the Lie algebra of smooth sections of ». We now establish the inter-relation between Lie connection and Lie Ehresmann connection on Lie algebra bundles. Theorem 3.1 — Let » = (»; ¼; M; Θ) be a smooth Lie algebra bundle. Then a Lie connection r on » induces a Lie Ehresmann connection on » and the converse holds if » is a normed Lie algebra bundle. PROOF : Let r be a Lie connection on ». Define, Hl = fA¤(Xp): p = ¼(l) and rXp (A) = 0g for each l 2 ». That is Hl = A¤(TpM) where ¼(l) = p and A 2 Γ(») is such that rA(p) = 0. Then Hl does not depend on the choice of A 2 Γ(») and H = fHl : l 2 »g forms a linear Ehresmann connection on the vector bundle » [13, Addendum 3]. Now to show H is a Lie Ehresmann connection, it suffices to show Θ¤H(l1;l2) = HΘ(l1;l2) for any (l1; l2) 2 »p £ »p ½ » © ». The horizontal subspace 0 H(l1;l2) = S¤(TpM) where S = (A1;A2) 2 Γ(» © ») such that r S(p) = (rA1(p); rA2(p)) = 0. Consider Θ¤(H(l1;l2)) = Θ¤(S¤(TpM)) = (Θ ± S)¤(TpM) = [A1;A2]¤(TpM) = H[l1;l2] = HΘ(l1;l2) (3.1) since, r[A1;A2](p) = [rA1(p);A2(p)] + [A1(p); rA2(p)] = 0 and ¼([l1; l2]) = p. Thus, H = fHl : l 2 »g is a Lie Ehresmann connection on ». Now suppose » is a normed Lie algebra bundle. We show that every Lie Ehresmann connection H on » is induced by some Lie connection r on ». Suppose H is a Lie Ehresmann connection on ». Then clearly H is a vector bundle connection on ». Let hX and vX be the horizontal and vertical component of any tangent vector X 2 »l induced by the direct sum decomposition »l = Vl © Hl. Then rXp A = vA¤(Xp), for Xp 2 TpM, p 2 M and A 2 Γ(») defines a connection r on the vector bundle » [13, Addendum 3]. Then r is a Lie connection on » since rX [A; B] = [rX A; B] + CONNECTIONS ON SMOOTH LIE ALGEBRA BUNDLES 895 [A; rX B] for any A; B 2 Γ(») and X 2 Γ(TM). For, rXp [A; B] = v[A; B]¤(Xp) = v([A¤(Xp);B] + [A; B¤(Xp)]) (as [ , ] is bilinear and » is normed) = v[A¤(Xp);B] + v[A; B¤(Xp)] = [vA¤(Xp);B] + [A; vB¤(Xp)] (by 2 in the Definition 3.1) = [rXp A; B] + [A; rXp B]2 (3.2) 4. CONNECTION ON ADJOINT BUNDLE We define connection on adjoint bundle adP associated to a principal bundle P in terms of connection on P . Let ¼ : P ! M be a principal G¡bundle and ¼Ad : adP ! M be the adjoint bundle [6, 8] corresponding to P . Explicitly, the elements of the adjoint bundle are equivalence classes of pairs f(p; x)g, p 2 P and x 2 g, the Lie algebra of G, such that f(p:g; x)g = f(p; Adg(x))g; for all g 2 G For w 2 adP , choose (p; x) 2 P £ g such that w = f(p; x)g.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Complete Connections on Fiber Bundles
    Complete connections on fiber bundles Matias del Hoyo IMPA, Rio de Janeiro, Brazil. Abstract Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples. We also establish a version of the theorem involving Riemannian submersions. 1 Introduction: A rather tricky exercise An (Ehresmann) connection on a submersion p : E → B is a smooth distribution H ⊂ T E that is complementary to the kernel of the differential, namely T E = H ⊕ ker dp. The distributions H and ker dp are called horizontal and vertical, respectively, and a curve on E is called horizontal (resp. vertical) if its speed only takes values in H (resp. ker dp). Every submersion admits a connection: we can take for instance a Riemannian metric ηE on E and set H as the distribution orthogonal to the fibers. Given p : E → B a submersion and H ⊂ T E a connection, a smooth curve γ : I → B, t0 ∈ I, locally defines a horizontal lift γ˜e : J → E, t0 ∈ J ⊂ I,γ ˜e(t0)= e, for e an arbitrary point in the fiber. This lift is unique if we require J to be maximal, and depends smoothly on e. The connection H is said to be complete if for every γ its horizontal lifts can be defined in the whole domain. In that case, a curve γ induces diffeomorphisms between the fibers by parallel transport.
    [Show full text]
  • Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections
    SMOOTHING MAPS INTO ALGEBRAIC SETS AND SPACES OF FLAT CONNECTIONS THOMAS BAIRD AND DANIEL A. RAMRAS n Abstract. Let X ⊂ R be a real algebraic set and M a smooth, closed manifold. We show that all continuous maps M ! X are homotopic (in X) to C1 maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds. 1. Introduction The first goal of this paper is to prove the following result about the differential topology of algebraic sets. Theorem 1.1 (Section2) . Let X ⊂ Rn be a (possibly singular) real algebraic set, and let f : M ! X be a continuous map from a smooth, closed manifold M. Then there exists a map g : M ! X, and a homotopy H : M × I ! X connecting f and g g, such that the composite M ! X,! Rn is C1. The problem of smoothing maps into algebraic sets seems natural, but we have not found mention of it in the literature. We consulted several experts in real algebraic geometry; some expected our result to hold, and some did not. Our proof proceeds by embedding X as the singular set of an irreducible, quasi- projective variety Y and using a resolution of singularities Ye ! Y for which the inverse image of X is a divisor with normal crossing singularities. Basic facts about neighborhoods of algebraic sets then reduce the problem to the case of normal crossing divisors, which can be handled by differential-geometric means.
    [Show full text]
  • WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
    WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5].
    [Show full text]
  • Unstable Bundles in Quantum Field Theory
    Contemporary Mathematics Volume 17, 1998 Unstable Bundles in Quantum Field Theory M. Asorey†, F. Falceto† and G. Luz´on‡ Abstract. The relation between connections on 2-dimensional manifolds and holomorphic bundles provides a new perspective on the role of classical gauge fields in quantum field theory in two, three and four dimensions. In particular we show that there is a close relation between unstable bundles and monopoles, sphalerons and instantons. Some of these classical configurations emerge as nodes of quantum vacuum states in non-confining phases of quantum field theory which suggests a relevant role for those configurations in the mechanism of quark confinement in QCD. 1. Introduction The role of principal bundles in the theory of gauge fields is now well established as the kinematical geometric background where classical and quantum fluctuations of gauge fields evolve. However, it is much less known that holomorphic bundles also play a fundamental role in the dynamics of the theory. There are two cases where the theory of holomorphic bundles proved to be very useful in the analysis of the classical dynamics of Yang-Mills fields. The first applica- tion arises in the construction of self-dual solutions of four-dimensional Yang-Mills equations (instantons), i.e. connections which minimize the Yang-Mills functional in non-trivial bundles. U(N) instantons in S4 are in one–to–one correspondence with holomorphic bundles of rank N on CP 3 which are trivial along the canonical U(1) fibres [29]. The second application occurs in 2-D Yang-Mills theory, where arXiv:hep-th/9812173v1 18 Dec 1998 the Yang-Mills functional becomes an almost equivariantly perfect Morse function in the space of 2-D connections [8].
    [Show full text]
  • Automorphisms of Ehresmann Connections
    Acta Math. Hungar., 123 (4) (2009), 379395. DOI: 10.1007/s10474-008-8139-x First published online December 12, 2008 AUTOMORPHISMS OF EHRESMANN CONNECTIONS J. PÉK and J. SZILASI¤ Institute of Mathematics, University of Debrecen, H-4010 Debrecen, P.O.B. 12, Hungary e-mail: [email protected] (Received July 14, 2008; accepted July 28, 2008) Abstract. We prove that a dieomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesics, considered as parametrized curves, to geodesics) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative). 1. Introduction It is well-known (and almost trivial) that two covariant derivative oper- ators on a manifold are equal, if and only if, they have the same geodesics and equal torsion tensors. Heuristically, this implies that a dieomorphism of a manifold with a covariant derivative is an automorphism of the covari- ant derivative, if and only if, it is a totally geodesic map (i.e. sends geodesics, considered as parametrized curves, to geodesics) and preserves the torsion. ¤The second author is supported by the Hungarian Nat. Sci. Found. (OTKA), Grant No. NK68040. Key words and phrases: Ehresmann connection, automorphism, strong torsion, totally geodesic map. 2000 Mathematics Subject Classication: 53C05, 53C22. 02365294/$ 20.00 °c 2008 Akadémiai Kiadó, Budapest 380 J. PÉK and J. SZILASI A formal proof of this conceptually important result may be found e.g.
    [Show full text]
  • Projective Connections and the Algebra of Densities
    , Projective Connections and the Algebra of Densities Jacob George School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK Abstract. Projective connections first appeared in Cartan’s papers in the 1920’s. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall the notion of projective connection and describe its relation with the algebra of densities on a manifold. In particular, we construct a Laplace-type operator on functions using a Thomas projective connection and a symmetric contravariant tensor of rank 2 (‘upper metric’). Keywords: Projective connections, densities PACS: PACS 02.40.Dr Euclidean and projective geometries, PACS 02.10.De Algebraic structures and number theory INTRODUCTION This paper is concerned with the geometry of differential operators on a manifold and their relation with projective connections. The notion of projective connection is an old one, first appearing in Cartan’s papers of the 1920s and then in various modified forms throughout the 20th century. They have surfaced periodically in mathematical physics, in particular in integrable systems and more recently in projectively equivariant quantisation. This paper establishes that the algebra of densities introduced in [1] and projective connections on a given manifold are fundamentally linked. We recall the notion of a projective connection; in fact there are two distinct but related notions which will be detailed here. We define the manifold M (cf. Mˆ in [1]) for which the algebra of densities V(M) may be interpreted as a subalgebra of the algebra of smooth ∞ b functions C (M).
    [Show full text]
  • On Fibre Bundles and Differential Geometry
    Lectures On Fibre Bundles and Differential Geometry By J.L. Koszul Notes by S. Ramanan No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Insti- tute of Fundamental Research, Apollo Pier Road, Bombay-1 Tata Institute of Fundamental Research, Bombay 1960 Contents 1 Differential Calculus 1 1.1 .............................. 1 1.2 Derivationlaws ...................... 2 1.3 Derivation laws in associated modules . 4 1.4 TheLiederivative... ..... ...... ..... .. 6 1.5 Lie derivation and exterior product . 8 1.6 Exterior differentiation . 8 1.7 Explicit formula for exterior differentiation . 10 1.8 Exterior differentiation and exterior product . 11 1.9 The curvature form . 12 1.10 Relations between different derivation laws . 16 1.11 Derivation law in C .................... 17 1.12 Connections in pseudo-Riemannian manifolds . 18 1.13 Formulae in local coordinates . 19 2 Differentiable Bundles 21 2.1 .............................. 21 2.2 Homomorphisms of bundles . 22 2.3 Trivial bundles . 23 2.4 Induced bundles . 24 2.5 Examples ......................... 25 2.6 Associated bundles . 26 2.7 Vector fields on manifolds . 29 2.8 Vector fields on differentiable principal bundles . 30 2.9 Projections vector fields . 31 iii iv Contents 3 Connections on Principal Bundles 35 3.1 .............................. 35 3.2 Horizontal vector fields . 37 3.3 Connection form . 39 3.4 Connection on Induced bundles . 39 3.5 Maurer-Cartan equations . 40 3.6 Curvatureforms. 43 3.7 Examples ......................... 46 4 Holonomy Groups 49 4.1 Integral paths . 49 4.2 Displacement along paths . 50 4.3 Holonomygroup ....................
    [Show full text]
  • Group-Valued Momentum Maps for Actions of Automorphism Groups
    Group-valued momentum maps for actions of automorphism groups Tobias Dieza and Tudor S. Ratiub February 5, 2020 The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more general class of group-valued momentum maps which is inspired by the Poisson Lie setting. In this approach, the group-valued momentum map assigns to every section of the symplectic fiber bundle a principal circle-bundle with connection. The power of this general framework is illustrated in many examples: we construct generalized Clebsch variables for fluids with integral helicity; the anti-canonical bundle turns out to be the momentum map for the action of the group of symplectomorphisms on the space of compatible complex structures; the Teichmüller moduli space is realized as a symplectic orbit reduced space associated to a coadjoint orbit of SL 2; R and spaces related to the other coadjoint orbits are identified and studied.¹ º Moreover, we show that the momentum map for the group of bundle automorphisms on the space of connections over a Riemann surface encodes, besides the curvature, also topological information of the bundle. Keywords: symplectic structure on moduli spaces of geometric structures, infinite- dimensional symplectic geometry, momentum maps, symplectic fiber bundles, differential characters, Kähler geometry, Teichmüller space MSC 2010: 53D20, (58D27, 53C08, 53C10, 58B99, 32G15) a Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany and Institut für Theoretische Physik, Universität Leipzig, 04009 Leipzig, Germany and Institute of Ap- arXiv:2002.01273v1 [math.DG] 4 Feb 2020 plied Mathematics, Delft University of Technology, 2628 XE Delft, Netherlands.
    [Show full text]
  • Field Theory from a Bundle Point of View
    Field theory from a bundle point of view Laurent Claessens December 8, 2008 2 Contents 1 Differential geometry 9 1.1 Differentiable manifolds ................................ 9 1.1.1 Definition and examples ............................ 9 Example: the sphere .............................. 10 Example: projective space ........................... 10 1.1.2 Topology on manifold and submanifold ................... 10 1.1.3 Tangent vector ................................. 12 1.1.4 Differential of a map .............................. 14 1.1.5 Tangent and cotangent bundle ........................ 15 Tangent bundle ................................. 15 Commutator of vector fields .......................... 16 Some Leibnitz formulas ............................ 16 Cotangent bundle ............................... 17 Exterior algebra ................................ 17 Pull-back and push-forward .......................... 18 Differential of k-forms ............................. 19 Hodge operator ................................. 19 Volume form and orientation ......................... 20 1.1.6 Musical isomorphism .............................. 20 1.1.7 Lie derivative .................................. 20 1.2 Example: Lie groups .................................. 22 1.2.1 Connected component of Lie groups ..................... 22 Õ 1.2.2 The Lie algebra of SUÔ2 ........................... 23 1.2.3 What is g ¡1dg ? ................................ 24 1.2.4 Exponential map ................................ 24 1.2.5 Invariant vector fields ............................
    [Show full text]
  • Connections on Principal Bundles
    LECTURE 5: CONNECTIONS 1. INTRODUCTION:THE MATHEMATICAL FRAMEWORK OF GAUGE THEORIES Gauge theories such are extremely important in physics. Examples are given by Maxwell’s theory of electromagnetism but also the Yang–Mills theories describing the strong and weak forces in the standard model. Central ingredient in this is the choice of a compact Lie group G that physicist call the gauge group. For Maxwell theory this group is abelian, namely G = U(1), but for other important examples this group must be chosen to be nonabelian, e.g. G = SU(N). Mathematically, the framework for such gauge theories is given by the theory of principal bundles. Each ingredient of the phys- ical theory has a solid mathematical meaning in the theory of principal bundles. The table below gives a “dictionary” to translate from physics to mathematics, and back: Physics Mathematics Gauge group Structure group G of a principal bundle P ! M Gauge field Connection A on P Field strength Curvature F(A) Gauge transformation Bundle isomorphism j : P ! P Matter fields in a representation V of G Sections of the associated vector bundle (P × V)/G The most important ingredient of gauge theories is of course the gauge field itself, which, mathematically, corresponds to connections on a principal bundle. The goal of this lecture is to explain what a connection precisely is. We explain this concept on a principal bundle, as well as on a vector bundle, in view of the last item when we want to introduce matter fields. 2. CONNECTIONS In the previous lecture we have discussed the fundamental dichotomy Associated bundle / Principal bundles o Vector bundles Frame bundle Let us briefly recall the associated bundle construction: suppose p : P ! M is a prin- cipal G-bundle, and let V be a representation of G.
    [Show full text]
  • INTRODUCTION to GAUGE THEORY 1. Bundles, Connections
    INTRODUCTION TO GAUGE THEORY LORENZO FOSCOLO 1. Bundles, connections and curvature M smooth manifold + G Lie group with Lie algebra g • principal G–bundle P over M • – smooth manifold P – free right G–action R: P G P × → – smooth projection to orbit space π : P M = P/G → – local G-equivariant trivialisations π 1(U) U G − ≃ × gauge group = Aut(P ) • G principal bundles and vector bundles • – representation ρ: G GL(V ) ❀ vector bundle E = P V → ×ρ e.g. ρ = Ad ❀ adjoint bundle ad P – G = stabiliser of φ r V s V , (E, φ) modelled on (V, φ ) ❀ principal G–bundle 0 ∈ ⊗ ∗ 0 of φ–adapted frames � � Examples: Hopf circle bundle S3 S1, quaternionic Hopf bundle S7 S4, frame bundle • → → Exercise 5.1 A connection on π : P M is a G–invariant splitting T P = ker π H of • → ∗ ⊕ 0 ker π T P π∗T M 0 → ∗ → → → H is called the horizontal subspace 1 Since ker π P g, a connection is a 1-form A: T P g such that Rg∗A = Ad(g− )A • ∗ ≃ × → – Product/trivial connection on M G induced from identification T G G g × ≃ × – loc. trivialisation P U G: A = trivial + a for a g-valued 1-form a on U |U ≃ × – change trivialisation given by u: U G ❀ A = trivial + u 1du + u 1au → − − infinite-dimensional affine space of connections on P modelled on Ω1(M; ad P ) • A Note: action of gauge group on by pull-back G A Exercise 5.2 horizontal lift XH of a vector field X on M: the unique G–invariant vector field on P such • that XH H T P at every point of P and π XH = X ∈ ⊂ ∗ parallel transport of A • – hor.
    [Show full text]