Geometric, algebraic and topological methods for quantum field theory : proceedings of the 2011 Villa de Leyva Summer School, Villa de Leyva, Colombia, 4-22 July 2011 / editors, Alexander Cardona, Universidad de los Andes, Colombia, Carolina Neira-Jimenez, Universitat Regensburg, Germany, Hernan Ocampo, Universidad del Valle, Colombia, Sylvie Paycha, Universitat Potsdam, Germany, Andres F. Reyes-Lega, Universidad de los Andes, Colombia.

Format:

Book

Conference/Event

Author/Creator:

Villa de Leyva Summer School, Corporate Author.

Contributor:

Cardona, Alexander, editor.

Neira-Jimenez, Carolina, editor.

Ocampo, Hernan, editor.

Paycha, Sylvie, editor.

Reyes-Lega, Andrés F., 1973- editor.

Conference Name:

Villa de Leyva Summer School (7th : 2011 : Leiva, Boyacá, Colombia)

Villa de Leyva Summer School

Series:

Gale eBooks

Language:

English

Subjects (All):

Geometric quantization--Congresses.

Geometric quantization.

Quantum field theory--Mathematics--Congresses.

Quantum field theory.

Topology--Congresses.

Topology.

Physical Description:

1 online resource (x, 367 pages) : illustrations

Place of Publication:

New Jersey : World Scientific, [2014]

Language Note:

English

Summary:

Based on lectures held at the 7th Villa de Leyva summer school, this book presents an introduction to topics of current interest in the interface of geometry, topology and physics. It is aimed at graduate students in physics or mathematics with interests in geometric, algebraic as well as topological methods and their applications to quantum field theory. This volume contains the written notes corresponding to lectures given by experts in the field. They cover current topics of research in a way that is suitable for graduate students of mathematics or physics interested in the recent developme

Contents:

Introduction; CONTENTS; Part A LECTURES; Spectral Geometry B. Iochum; 1. Motivations; 2. Wodzicki residue and kernel near the diagonal; 2.1. A quick overview on pseudodifferential operators; 2.2. Case of manifolds; 2.3. Singularities of the kernel near the diagonal; 2.4. Wodzicki residue; 3. Dixmier trace; 3.1. Singular values of compact operators; 3.2. Dixmier trace; 4. Dirac operator; 4.1. Definition and main properties; 4.2. Dirac operators and change of metrics; 5. Heat kernel expansion; 5.1. The asymptotics of heat kernel; 5.2. Wodzicki residue and heat expansion

6. Noncommutative integration6.1. Notion of spectral triple; 6.2. Notion of pseudodifferential operators; 6.3. Zeta-functions and dimension spectrum; 6.4. One-forms and fluctuations of D One-forms and fluctuations of D; 6.5. Tadpole; 6.6. Commutative geometry; 6.7. Scalar curvature; 6.8. Tensor product of spectral triples; 7. Spectral action; 7.1. On the search for a good action functional; 7.1.1. Einstein-Hilbert action; 7.1.2. Quantum approach and spectral action; 7.1.3. Yang-Mills action; 7.2. Asymptotic expansion for Asymptotic expansion for Lambda going to infinity

7.3. Remark on the use of Laplace transform7.4. About convergence and divergence, local and global aspects of the asymptotic expansion; 7.5. On the physical meaning of the asymptotics of spectral action; 8. The noncommutative torus; 8.1. Definition of the nc-torus; 8.2. Kernels and dimension spectrum; 8.3. The spectral action; Acknowledgments; References; Index Theory for Non-compact G-manifolds M. Braverman and L. Cano; 1. The Fredholm Index; 1.1. Finite dimensional case; 1.2. The Fredholm index; 1.6. An example; 1.10. Application of the index

1.11. Connected components of the set of Fredholm operators1.12. The group action; 1.17. The ring of characters; 1.22. The equivariant index; 2. Differential operators; 2.1. Differential operators; 2.9. Matrix-valued differential operators; 2.10. Vector bundles; 2.12. Differential operators on manifolds; 2.14. Sobolev spaces of sections; 2.16. The symbol of a differential operator; 2.20. The leading symbol as a section of the pullback bundle; 2.21. Elliptic differential operators; 3. The Atiyah-Singer index theorem; 3.1. Index as a topological invariant; 3.5. K-theory

3.10. The pushforward map in K-theory3.12. The topological index; 3.14. Equivariant vector bundles; 3.17. Equivariant K-theory; 3.18. The Atiyah-Singer index theorem; 3.20. The case of an open manifold; Question 1; 4. Transversal elliptic operators; 4.1. A motivating example; 4.3. The transversal cotangent bundle; 4.5. The analytical index of transversally elliptic operators; 4.9. Transversal K-theory and the topological index; Question 2; 5. Dirac-type operators; 5.1. Clifford action; 5.6. A Clifford bundles; 5.10. A Clifford connection; 5.12. A generalized Dirac operator; 5.19. A grading

5.26. The group action

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on online resource; title from PDF (ebrary, viewed December 30, 2013).

ISBN:

9789814460057

9814460052

OCLC:

864899698