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Geometric and Topological Methods for Quantum Field Theory

Book Description

Based on lectures given at the renowned Villa de Leyva summer school, this book provides a unique presentation of modern geometric methods in quantum field theory. Written by experts, it enables readers to enter some of the most fascinating research topics in this subject. Covering a series of topics on geometry, topology, algebra, number theory methods and their applications to quantum field theory, the book covers topics such as Dirac structures, holomorphic bundles and stability, Feynman integrals, geometric aspects of quantum field theory and the standard model, spectral and Riemannian geometry and index theory. This is a valuable guide for graduate students and researchers in physics and mathematics wanting to enter this interesting research field at the borderline between mathematics and physics.

Table of Contents

  1. Cover
  2. Half Title
  3. Title
  4. Copyright
  5. Table of Contents
  6. List of contributors
  7. Introduction
  8. 1 A brief introduction to Dirac manifolds
    1. 1.1 Introduction
    2. 1.2 Presymplectic and Poisson structures
    3. 1.3 Dirac structures
    4. 1.4 Properties of Dirac structures
    5. 1.5 Morphisms of Dirac manifolds
    6. 1.6 Submanifolds of Poisson manifolds and constraints
    7. 1.7 Brief remarks on further developments
    8. References
  9. 2 Differential geometry of holomorphic vector bundles on a curve
    1. 2.1 Holomorphic vector bundles on Riemann surfaces
    2. 2.2 Holomorphic structures and unitary connections
    3. 2.3 Moduli spaces of semi-stable vector bundles
    4. References
  10. 3 Paths towards an extension of Chern–Weil calculus to a class of infinite dimensional vector bundles
    1. 3.1 The gauge group of a bundle
    2. 3.2 The diffeomorphism group of a bundle
    3. 3.3 The algebra of zero-order classical pseudodifferential operators
    4. 3.4 The group of invertible zero-order ψ dos
    5. 3.5 Traces on zero-order classical $ψ dos
    6. 3.6 Logarithms and central extensions
    7. 3.7 Linear extensions of the L2-trace
    8. 3.8 Chern–Weil calculus in finite dimensions
    9. 3.9 A class of infinite dimensional vector bundles
    10. 3.10 Frame bundles and associated ψ do-algebra bundles
    11. 3.11 Logarithms and closed forms
    12. 3.12 Chern–Weil forms in infinite dimensions
    13. 3.13 Weighted Chern–Weil forms; discrepancies
    14. 3.14 Renormalised Chern–Weil forms on ψ do Grassmannians
    15. 3.15 Regular Chern–Weil forms in infinite dimensions
    16. References
  11. 4 Introduction to Feynman integrals
    1. 4.1 Introduction
    2. 4.2 Basics of perturbative quantum field theory
    3. 4.3 Dimensional regularisation
    4. 4.4 Loop integration in D dimensions
    5. 4.5 Multi-loop integrals
    6. 4.6 How to obtain finite results
    7. 4.7 Feynman integrals and periods
    8. 4.8 Shuffle algebras
    9. 4.9 Multiple polylogarithms
    10. 4.10 From Feynman integrals to multiple polylogarithms
    11. 4.11 Conclusions
    12. References
  12. 5 Iterated integrals in quantum field theory
    1. 5.1 Introduction
    2. 5.2 Definition and first properties of iterated integrals
    3. 5.3 The case P1 \{0, 1, ∞} and polylogarithms
    4. 5.4 The KZ equation and the monodromy of polylogarithms
    5. 5.5 A brief overview of multiple zeta values
    6. 5.6 Iterated integrals and homotopy invariance
    7. 5.7 Feynman integrals
    8. References
  13. 6 Geometric issues in quantum field theory and string theory
    1. 6.1 Differential geometry for physicists
    2. 6.2 Holonomy
    3. 6.3 Strings and higher dimensions
    4. 6.4 Some issues on compactification
    5. Exercises
  14. 7 Geometric aspects of the Standard Model and the mysteries of matter
    1. 7.1 Radiation and matter in gauge theories and General Relativity
    2. 7.2 Mass matrices and state mixing
    3. 7.3 The space of connections and the action functional
    4. 7.4 Constructions within noncommutative geometry
    5. 7.5 Further routes to quantization via BRST symmetry
    6. 7.6 Some conclusions and outlook
    7. Exercises
    8. Appendix: Proof of relation (7.11a)
    9. References
  15. 8 Absence of singular continuous spectrum for some geometric Laplacians
    1. 8.1 Meromorphic extension of the resolvent and singular continuous spectrum
    2. 8.2 Analytic dilation on complete manifolds with corners of codimension 2
    3. References
  16. 9 Models for formal groupoids
    1. 9.1 Motivation and plan
    2. 9.2 Definitions and examples
    3. 9.3 Algebraic structure for formal groupoids
    4. 9.4 The symplectic case
    5. References
  17. 10 Elliptic PDEs and smoothness of weakly Einstein metrics of Hölder regularity
    1. 10.1 Introduction
    2. 10.2 Basics on function spaces
    3. 10.3 Elliptic operators and PDEs
    4. 10.4 Riemannian regularity and harmonic coordinates
    5. 10.5 Ricci curvature and the Einstein condition
    6. References
  18. 11 Regularized traces and the index formula for manifolds with boundary
    1. 11.1 General heat kernel expansions and zeta functions
    2. 11.2 Weighted traces, weighted trace anomalies and index terms
    3. 11.3 Eta-invariant and super-traces
    4. Acknowledgements
    5. References
  19. Index