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Geometric and topological methods for quantum field theory : proceedings of the 2009 Villa de Leyva summer school /
"Based on lectures given at the renowed 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...
| Clasificación: | Libro Electrónico |
|---|---|
| Otros Autores: | , , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2013.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Contributors; Introduction; 1 A brief introduction to Dirac manifolds; 1.1 Introduction; 1.1.1 Notation, conventions, terminology; 1.2 Presymplectic and Poisson structures; 1.2.1 Two viewpoints on symplectic geometry; 1.2.2 Going degenerate; 1.3 Dirac structures; 1.4 Properties of Dirac structures; 1.4.1 Lie algebroid; 1.4.2 Presymplectic leaves and null distribution; 1.4.3 Hamiltonian vector fields and Poisson algebra; 1.5 Morphisms of Dirac manifolds; 1.5.1 Pulling back and pushing forward; 1.5.2 Clean intersection and smoothness issues
- 1.6 Submanifolds of Poisson manifolds and constraints1.6.1 The induced Poisson bracket on admissible functions; 1.6.2 A word on coisotropic submanifolds (or first-class constraints); 1.6.3 Poisson-Dirac submanifolds and the Dirac bracket; 1.6.4 Momentum level sets; 1.7 Brief remarks on further developments; Acknowledgments; References; 2 Differential geometry of holomorphic vector bundles on a curve; 2.1 Holomorphic vector bundles on Riemann surfaces; 2.1.1 Vector bundles; 2.1.2 Topological classification; 2.1.3 Dolbeault operators and the space of holomorphic structures; 2.1.4 Exercises
- 2.2 Holomorphic structures and unitary connections2.2.1 Hermitian metrics and unitary connections; 2.2.2 The Atiyah-Bott symplectic form; 2.2.3 Exercises; 2.3 Moduli spaces of semi-stable vector bundles; 2.3.1 Stable and semi-stable vector bundles; 2.3.2 Donaldson's theorem; 2.3.3 Exercises; References; 3 Paths towards an extension of Chern-Weil calculus to a class of infinite dimensional vector bundles; Introduction; Part 1: Some useful infinite dimensional Lie groups; 3.1 The gauge group of a bundle; 3.2 The diffeomorphism group of a bundle
- 3.3 The algebra of zero-order classical pseudodifferential operators3.4 The group of invertible zero-order dos; Part 2: Traces and central extensions; 3.5 Traces on zero-order classical dos; 3.6 Logarithms and central extensions; 3.7 Linear extensions of the L2-trace; Part 3: Singular Chern-Weil classes; 3.8 Chern-Weil calculus in finite dimensions; 3.9 A class of infinite dimensional vector bundles; 3.10 Frame bundles and associated do-algebra bundles; 3.11 Logarithms and closed forms; 3.12 Chern-Weil forms in infinite dimensions; 3.13 Weighted Chern
- Weil forms; discrepancies
- 3.13.1 The Hochschild coboundary of a weighted trace3.13.2 Dependence on the weight; Part 4: Circumventing anomalies; 3.13.3 Exterior differential of a weighted trace; 3.13.4 Weighted traces extended to admissible fibre bundles; 3.13.5 Obstructions to closedness of weighted Chern
- Weil forms; 3.14 Renormalised Chern-Weil forms on do Grassmannians; 3.15 Regular Chern-Weil forms in infinite dimensions; Acknowledgements; References; 4 Introduction to Feynman integrals; 4.1 Introduction; 4.2 Basics of perturbative quantum field theory; 4.3 Dimensional regularisation