Group-Valued Momentum Maps for Actions of Automorphism Groups
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Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
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]. -
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/. -
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 .................... -
Foliated Manifolds, Algebraic K-Theory, and a Secondary Invariant
Foliated manifolds, algebraic K-theory, and a secondary invariant Ulrich Bunke∗ June 25, 2018 Abstract We introduce a C=Z-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential K-theory, or alternatively, in terms of η-invariants of Dirac operators and local correction terms. Initially, the construction of the element in C=Z involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' e- invariant, the ρ-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic K-theory of smooth functions on a manifold to its connective K-theory with C=Z coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological K-theory. arXiv:1507.06404v4 [math.KT] 22 Jun 2018 Contents 1 Introduction 2 2 Basic notions 4 2.1 Foliated manifolds . .4 2.2 Filtrations on the de Rham complex . .6 2.3 Vector bundles with flat partial connections . .8 2.4 Connections and characteristic forms . 10 2.5 Characteristic forms of real foliations . 12 2.6 Transgression . 14 ∗NWF I - Mathematik, Universit¨at Regensburg, 93040 Regensburg, GERMANY, [email protected] 1 3 Differential K-theory 18 3.1 Basic structures . -
CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie Groups a Lie Group Is A
CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie groups A Lie group is a group G which is also a smooth manifold such that the mul- tiplication map G × G → G, (a, b) 7→ ab, and the inverse G → G, a 7→ a−1, are smooth. Actually, one can prove (but this is not easy) that it is sufficient to assume continuety, smoothness comes free. (This was one of the famous problems listed by David Hilbert in his address to the international congress of mathematicians in 1900. The result was proven by D. Montgomery and L. Zippin in 1952.) Examples The vector space Rn is a Lie group. The group multiplication is just the addition of vectors. The set GL(n, R) of invertible real n × n matrices is a Lie group with respect to the usual matrix multiplication. Theorem. Any closed subgroup of a Lie group is a Lie group. The proof is complicated. See for example S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces, section II.2. The theorem gives an additional set of examples of Lie groups: The group of real orthogonal matrices, the group of complex unitary matrices, the group of invertible upper triangular matrices .... For a fixed a ∈ G in a Lie group we can define a pair of smooth maps, the left translation la : G → G, la(g) = ag, and the right translation ra : G → G, ra(g) = ga. 1 We say that a vector field X ∈ D (G) is left (resp. right) invariant if (la)∗X = X (resp. (ra)∗X = X) for all a ∈ G. -
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 ............................ -
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. -
A Foliated Seiberg-Witten Theory a Dissertation Presented to The
A Foliated Seiberg-Witten Theory A Dissertation presented to the Faculty of the Graduate School University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by ANDREW RENNER Dr. Shuguang Wang, Dissertation Supervisor MAY 2016 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled A Foliated Seiberg-Witten Theory presented by Andrew Renner, a candidate for the degree of Doctor of Philosophy of Mathematics, and hereby certify that in their opinion it is worthy of acceptance. Professor Shuguang Wang Professor Zhenbo Qin Professor Jan Segert Professor Jianguo Sun ACKNOWLEDGEMENTS I'd like to thank my advisor Dr. Shuguang Wang for his support and guidance throughout my graduate studies here at the University of Missouri. I would further like to thank my committee members Dr. Jan Segert, Dr. Zhenbo Qin, and Dr. Jianguo Sun for their support and encouragement. Finally, I'd like to thank all of the other members of the MU mathematics department faculty, staff and students, they have all made my time here memorable and enjoyable. ii TABLE OF CONTENTS Acknowledgements ii Abstract v 1 Introduction 1 2 Foliations 7 2.1 Transverse Geometry . 10 2.2 The Transverse Riemannian Connection . 16 3 Foliated Dirac Bundles 24 3.1 Foliated Principal Bundles . 24 3.2 Dirac Bundles . 27 3.3 Associated Connections . 29 3.4 Second Order Differential Operators . 33 3.5 Global Analysis . 36 3.6 Weitzenb¨ock Identities . 40 4 Elliptic Theory 47 4.1 Sobolev Spaces . 47 4.1.1 Classical Theory . 47 iii 4.1.2 Basic Sobolev Theory . -
Orientability of Vector Bundles Over Real Flag Manifolds
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 159 (2012) 2774–2786 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Orientability of vector bundles over real flag manifolds ✩ ∗ Mauro Patrão a, Luiz A.B. San Martin b, ,1, Laércio J. dos Santos c, Lucas Seco a a Department of Mathematics, Universidade de Brasília, Brasília – DF, Brazil b Institute of Mathematics, Universidade Estadual de Campinas, Campinas – SP, Brazil c Department of Mathematics, Universidade Federal de Juiz de Fora – MG, Brazil article info abstract Article history: We investigate the orientability of a class of vector bundles over flag manifolds of real Received 27 June 2011 semi-simple Lie groups, which include the tangent bundle and also stable bundles of Accepted 5 March 2012 certain gradient flows. Closed formulas, in terms of roots, are provided. © 2012 Elsevier B.V. All rights reserved. MSC: 57R22 14M15 Keywords: Vector bundles Orientability Flag manifold 1. Introduction We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, the so- called (generalized) real flag manifolds. These include the tangent bundle and also stable bundles of some gradient flows on these manifolds which were considered elsewhere (see Section 3 of Duistermat, Kolk and Varadarajan [7] and Section 5 of the present article). We get closed formulas, in terms of roots associated to the real flag manifolds, to decide when they are orientable. As far as we know, our results and methods of proof are not known. -
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. -
Introduction to Quantum Principal Bundles
INTRODUCTION TO QUANTUM PRINCIPAL BUNDLES MICHO DURDEVICH 1. Introduction In diversity of mathematical concepts and theories a fundamental role is played by those giving a unified treatment of different and at a first sight mutually inde- pendent circles of problems. As far as classical differential geometry is concerned, such a fundamental role is given to the theory of principal bundles. Various basic concepts of theoretical physics are also naturally expressible in the language of principal bundles. Classical gauge theory and general relativity theory are paradigmic examples. But classical geometry is just a very special case of a much deeper quantum geometry. So it is natural to ask what would be the analogs of principal bundles in quantum geometry. And it is reasonable to expect that such quantum principal bundles would play a similar fundamental role in quantum geometry, as is the role of classical principal bundles in classical geometry. A general theory of quantum principal bundles, where quantum groups play the role of structure groups and general quantum spaces play the role of base manifolds, has been developed in [3, 4]. A very brief exposition (without proofs) can be found in [2]. Here we shall discuss basic ideas of the theory of quantum principal bundles, trying to speak informally and paying a special attention to interesting purely quantum phenomenas appearing in the game. It is not difficult to incorporate the basic geometrical idea of a principal bundle, into the noncommutative context. Let G be a compact matrix quantum group [20], represented by a Hopf *-algebra A (here, A plays the role of polynomial functions over G).