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Principal Bundles, Vector Bundles and Connections

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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/. E is a topological space called the total space of the bundle, W E ! M is a continuous surjective map, called the canonical projection and F is the typical fiber. The following conditions must be satisfied: (a) 1.x/, the fiber over x, is homeomorphic to F. © Springer International Publishing Switzerland 2016 545 W.A. Rodrigues, E. Capelas de Oliveira, The Many Faces of Maxwell, Dirac and Einstein Equations, Lecture Notes in Physics 922, DOI 10.1007/978-3-319-27637-3 546 A Principal Bundles, Vector Bundles and Connections (b) Let fUi; i 2 Ig,whereI is an index set, be a covering of M, such that: • Locally a fiber bundle E is trivial, i.e., it is diffeomorphic to a product bundle, 1 i.e., .Ui/ ' Ui F for all i 2 I. 1 • The diffeomorphisms ˆi W .Ui/ ! Ui F have the form ˆi. p/ D .. p/; i;x. p//; (A.1) 1. / ij1.x/ Á i;x W x ! F is onto. (A.2) The collection f.Ui;ˆi/g, i 2 I, are said to be a family of local trivializations for E. • The group G acts on the typical fiber. Let x 2 Ui \ Uj. Then, 1 j;x ı i;x W F ! F (A.3) must coincide with the action of an element of G for all x 2 Ui \ Uj and i; j 2 I. • We call transition functions of the bundle the continuous induced mappings . / 1: gij W Ui \ Uj ! G,wheregij x D i;x ı j;x (A.4) For consistence of the theory the transition functions must satisfy the cocycle condition (Fig. A.1) gij.x/gjk.x/ D gik.x/: (A.5) Fig. A.1 Transition functions on a fiber bundle A Principal Bundles, Vector Bundles and Connections 547 Transition functions on a fiber bundle Definition A.2 .P; M; ; G; F Á G/ Á .P; M; ; G/ is called a principal fiber bundle (PFB) if all conditions in Definition A.1 are fulfilled and moreover, if there is a right action of G on elements p 2 P, such that: (a) the mapping (defining the right action) PG 3 . p; g/ 7! pg 2 P is continuous. (b) given g; g0 2 G and 8p 2 P, .pg/g0 D p.gg0/: (c) 8x 2 M; 1.x/ is invariant under the action of G, i.e., each element of p 2 1.x/ is mapped into pg 2 1.x/, i.e., it is mapped into an element of the same fiber. (d) G acts free and transitively on each fiber 1.x/, which means that all elements within 1.x/ are obtained by the action of all the elements of G on any given element of the fiber 1.x/. This condition is, of course necessary for the identification of the typical fiber with G. Definition A.3 A bundle .E; M; 1; G D Gl.m; F/; F D V/,whereF D R or C (respectively the real and complex fields), Gl.m; F/ is the linear group, and V is an m-dimensional vector space over F is called a vector bundle. Definition A.4 A vector bundle .E; M; ; G; F/ denoted E D P F is said to be associated to a PFB bundle .P; M; ; G/ by the linear representation of G in F D V (a linear space of finite dimension over an appropriate field, which is called the carrier space of the representation) if its transition functions are the images under of the corresponding transition functions of the PFB .P; M; ; G/. This means the following: consider the following local trivializations of P and E respectively 1 ˆi W .Ui/ ! Ui G; (A.6) 1 „i W 1 .Ui/ ! Ui F; (A.7) „i.q/ D .1.q/; i.q// D .x;i.q//; (A.8) 1 j 1 Á ; W .x/ ! F; (A.9) i 1 .x/ i x 1 where 1 W P F ! M is the projection of the bundle associated to .P; M; ; G/. Then, for all x 2 Ui \ Uj, i; j 2 I,wehave 1 . 1/: j;x ı i;x D j;x ı i;x (A.10) In addition, the fibers 1.x/ are vector spaces isomorphic to the representation space V. Definition A.5 Let E D .E; M; ; G; F/ be a fiber bundle and U M an open set. A local (cross) section of the fiber bundle .E; M; ; G; F/ on U is a mapping s W U ! E such that ı s D IdU: (A.11) 548 A Principal Bundles, Vector Bundles and Connections A global section s is one for which U D M. Not all bundles admit global sections (see below). E Notation A.6 Let U M. We will say that 2 sec jU (or 2 sec EjU)ifthere exists a local section sj W U ! E, x 7! .x;.x//, with W U ! F. Remark A.7 There is a relation between sections and local trivializations for principal bundles. Indeed, each local section s; (on Ui M) for a principal bundle 1 .P; M; ; G/ determines a local trivialization ˆi W .U/ ! U G; of P by setting ˆ1. ; / . / : i x g D s x g D pg D Rgp (A.12) Conversely, ˆi determines s since . / ˆ1. ; / s x D i x e . (A.13) Proposition A.8 A principal bundle is trivial, if and only if, it has a global cross section. Proposition A.9 A vector bundle is trivial, if and only if, its associated principal bundle is trivial. Proposition A.10 Any fiber bundle .E; M; ; G; F/ such that M is contractible to a point is trivial. Proposition A.11 Any fiber bundle .E; M; ; G; F/ such that M is paracompact and the fiber F is a vector space admits a local section. Remark A.12 Take notice that any vector bundle admits a global section, namely the zero section. However, it admits a global nonvanishing global section if its Euler class is zero. Definition A.13 The structure group G of a fiber bundle .E; M; ; G; F/ is said to be reducible to G0 if the bundle admits an equivalent structure defined with a subgroup G0 of the structure group G. More precisely, this means that the fiber bundle admits a family of local trivializations such that the transition functions takes 0 0 values in G , i.e., gij W Ui \ Uj ! G . A.1.1 Frame Bundle The tangent bundle TM to a differentiable n-dimensional manifoldSM is an . / associated bundle to a principal bundle called the frame bundle F M D x2M FxM, i where FxM is the set of frames at x 2 M.Letfx g be coordinates associated to aˇ local .U ;'/ M T M f @ ˇ g chart i i of the maximal atlas of . Then, x has a natural basis @xi x on Ui M. A Principal Bundles, Vector Bundles and Connections 549 † ;:::; Definition A.14 A frame at TxM is a set x Dfe1jx enjxg of linearly independent vectors such that ˇ @ ˇ e j D Fj ˇ ; (A.14) i x i @ j ˇ x x . j/ j R and where the matrix Fi with entries Ai 2 , belongs to the real general linear . ; R/ . j/ . ; R/ group in n dimensions Gl n . We write Fi 2 Gl n . 1 A local trivialization i W .Ui/!Ui Gl.n; R/ is defined by i. f / D .x; †x/, .f / D x: (A.15) . j/ . ; R/ . / . ; / The action of a D ai 2 Gl n on a frame f 2 F U is given by f a ! fa, . / . / . ; †0 / . / where the new frame fa 2 F U is defined by i fa D x x , f D x,and ˇ ˇ †0 0 ˇ ;:::; 0 ˇ ; x Dfe1 x en xg ˇ ˇ 0ˇ ˇ j: ei x D ej x ai (A.16) † †0 .
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