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Mathematical Gauge Theory


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Table of Contents

Part I Mathematical foundations1 Lie groups and Lie algebras: Basic concepts1.1 Topological groups and Lie groups1.2 Linear groups and symmetry groups of vector spaces1.3 Homomorphisms of Lie groups1.4 Lie algebras1.5 From Lie groups to Lie algebras1.6 From Lie subalgebras to Lie subgroups1.7 The exponential map1.8 Cartan's Theorem on closed subgroups1.9 Exercises for Chapter 12 Lie groups and Lie algebras: Representations and structure theory2.1 Representations2.2 Invariant metrics on Lie groups2.3 The Killing form2.4 Semisimple and compact Lie algebras2.5 Ad-invariant scalar products on compact Lie groups2.6 Homotopy groups of Lie groups2.7 Exercises for Chapter 23 Group actions3.1 Transformation groups3.2 Definition and first properties of group actions3.3 Examples of group actions3.4 Fundamental vector fields3.5 The Maurer-Cartan form and the differential of a smooth group action3.6 Left or right actions?3.7 Quotient spaces3.8 Homogeneous spaces3.9 Stiefel and Grassmann manifolds3.10 The exceptional Lie group G23.11 Godement's Theorem on the manifold structure of quotient spaces3.12 Exercises for Chapter 34 Fibre bundles4.1 General fibre bundles4.2 Principal fibre bundles4.3 Formal bundle atlases4.4 Frame bundles4.5 Vector bundles4.6 The clutching construction4.7 Associated vector bundles4.8 Exercises for Chapter 45 Connections and curvature5.1 Distributions and connections5.2 Connection 1-forms5.3 Gauge transformations5.4 Local connection 1-forms and gauge transformations5.5 Curvature5.6 Local curvature 2-forms5.7 Generalized electric and magnetic fields on Minkowski spacetime of dimension 45.8 Parallel transport5.9 The covariant derivative on associated vector bundles5.10 Parallel transport and path-ordered exponentials5.11 Holonomy and Wilson loops5.12 The exterior covariant derivative5.13 Forms with values in Ad(P)5.14 A second and third version of the Bianchi identity5.15 Exercises for Chapter 56 Spinors6.1 The pseudo-orthogonal group O(s; t) of indefinite scalar products6.2 Clifford algebras6.3 The Clifford algebras for the standard symmetric bilinear forms6.4 The spinor representation6.5 The spin groups6.6 Majorana spinors6.7 Spin invariant scalar products6.8 Explicit formulas for Minkowski spacetime of dimension 46.9 Spin structures and spinor bundles6.10 The spin covariant derivative6.11 Twisted spinor bundles6.12 Twisted chiral spinors6.13 Exercises for Chapter 6Part II The Standard Model of elementary particle physics7 The classical Lagrangians of gauge theories7.1 Restrictions on the set of Lagrangians7.2 The Hodge star and the codifferential7.3 The Yang-Mills Lagrangian7.4 Mathematical and physical conventions for gauge theories7.5 The Klein-Gordon and Higgs Lagrangians7.6 The Dirac Lagrangian7.7 Yukawa couplings7.8 Dirac and Majorana mass terms7.9 Exercises for Chapter 78 The Higgs mechanism and the Standard Model8.1 The Higgs field and symmetry breaking8.2 Mass generation for gauge bosons8.3 Massive gauge bosons in the SU(2)U(1)-theory of the electroweak interaction8.4 The SU(3)-theory of the strong interaction (QCD)8.5 The particle content of the Standard Model8.6 Interactions between fermions and gauge bosons8.7 Interactions between Higgs bosons and gauge bosons8.8 Mass generation for fermions in the Standard Model8.9 The complete Lagrangian of the Standard Model8.10 Lepton and baryon numbers8.11 Exercises for Chapter 89 Modern developments and topics beyond the Standard Model9.1 Flavour and chiral symmetry9.2 Massive neutrinos9.3 C, P and CP violation9.4 Vacuum polarization and running coupling constants9.5 Grand Unified Theories9.6 A short introduction to the Minimal Supersymmetric Standard Model (MSSM)9.7 Exercises for Chapter 9Part III AppendixA Background on differentiable manifoldsA.1 ManifoldsA.2 Tensors and formsB Background on special relativity and quantum field theoryB.1 Basics of special relativityB.2 A short introduction to quantum field theoryReferencesIndex

About the Author

Mark Hamilton has worked as a lecturer and interim professor at the University of Stuttgart and the Ludwig-Maximilian University of Munich. His research focus lies on geometric topology and mathematical physics, in particular, the differential topology of 4-manifolds and Seiberg-Witten theory.


"Assuming an introductory course on differential geometry and some basic knowledge of special relativity, both of which are summarized in the appendices, the book expounds the mathematical background behind the well-established standard model of modern particle and high energy physics... I believe that the book will be a standard textbook on the standard model for mathematics-oriented students." (Hirokazu Nishimura, zbMATH 1390.81005)

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