Mathematical Gauge Theory

1. Front Matter

   Pages i-xviii

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2. Mathematical Foundations

   1. Front Matter

      Pages 1-1

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   2. Chapter 1 Lie Groups and Lie Algebras: Basic Concepts

      • Mark J. D. Hamilton

      Pages 3-82

   3. Chapter 2 Lie Groups and Lie Algebras: Representations and Structure Theory

      • Mark J. D. Hamilton

      Pages 83-126

   4. Chapter 3 Group Actions

      • Mark J. D. Hamilton

      Pages 127-192

   5. Chapter 4 Fibre Bundles

      • Mark J. D. Hamilton

      Pages 193-255

   6. Chapter 5 Connections and Curvature

      • Mark J. D. Hamilton

      Pages 257-318

   7. Chapter 6 Spinors

      • Mark J. D. Hamilton

      Pages 319-398

3. The Standard Model of Elementary Particle Physics

   1. Front Matter

      Pages 399-399

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   2. Chapter 7 The Classical Lagrangians of Gauge Theories

      • Mark J. D. Hamilton

      Pages 401-444

   3. Chapter 8 The Higgs Mechanism and the Standard Model

      • Mark J. D. Hamilton

      Pages 445-526

   4. Chapter 9 Modern Developments and Topics Beyond the Standard Model

      • Mark J. D. Hamilton

      Pages 527-599

4. Appendix

   1. Front Matter

      Pages 601-601

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   2. Appendix A: Background on Differentiable Manifolds

      • Mark J. D. Hamilton

      Pages 603-623

   3. Appendix B: Background on Special Relativity and Quantum Field Theory

      • Mark J. D. Hamilton

      Pages 625-640

5. Back Matter

   Pages 641-657

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The Standard Model is the foundation of modern particle and high energy physics. This book explains the mathematical background behind the Standard Model, translating ideas from physics into a mathematical language and vice versa.

The first part of the book covers the mathematical theory of Lie groups and Lie algebras, fibre bundles, connections, curvature and spinors. The second part then gives a detailed exposition of how these concepts are applied in physics, concerning topics such as the Lagrangians of gauge and matter fields, spontaneous symmetry breaking, the Higgs boson and mass generation of gauge bosons and fermions. The book also contains a chapter on advanced and modern topics in particle physics, such as neutrino masses, CP violation and Grand Unification.

This carefully written textbook is aimed at graduate students of mathematics and physics. It contains numerous examples and more than 150 exercises, making it suitable for self-study and use alongside lecture courses. Only a basic knowledge of differentiable manifolds and special relativity is required, summarized in the appendix.

• gauge theory
• gauge theory mathematics
• standard model of elementary particle physics
• spontaneous symmetry breaking
• HIggs boson standard model
• Higss boson
• electroweak interaction
• quantum chromodynamics qcd theory
• grand unified theory
• Lagrangian
• principal bundle
• vector bundle
• connections and curvature
• spinor
• gauge theory of the Standard Model
• gauge theory and Lagrangians
• MSC (2010) 55R10, 53C05, 22E70, 15A66, 53C27, 57S15, 22E60
• 81T13, 81R40, 81V19, 81V05, 81V10, 81V15

“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)

• Department of Mathematics, Ludwig-Maximilian University of Munich, Munich, Germany

  Mark J.D. Hamilton

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.

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