Harmonic Maps, Loop Groups, and Integrable Systems

Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

Preface; Acknowledgements; Part I. One-Dimensional Integrable Systems: 1. Lie groups; 2. Lie algebras; 3. Factorizations and homogeneous spaces; 4. Hamilton's equations and Hamiltonian systems; 5. Lax equations; 6. Adler-Kostant-Symes; 7. Adler-Kostant-Symes (continued); 8. Concluding remarks on one-dimensional Lax equations; Part II. Two-Dimensional Integrable Systems: 9. Zero-curvature equations; 10. Some solutions of zero-curvature equations; 11. Loop groups and loop algebras; 12. Factorizations and homogeneous spaces; 13. The two-dimensional Toda lattice; 14. T-functions and the Bruhat decomposition; 15. Solutions of the two-dimensional Toda lattice; 16. Harmonic maps from C to a Lie group G; 17. Harmonic maps from C to a Lie group (continued); 18. Harmonic maps from C to a symmetric space; 19. Harmonic maps from C to a symmetric space (continued); 20. Application: harmonic maps from S2 to CPn; 21. Primitive maps; 22. Weierstrass formulae for harmonic maps; Part III. One-Dimensional and Two-Dimensional Integrable Systems: 23. From 2 Lax equations to 1 zero-curvature equation; 24. Harmonic maps of finite type; 25. Application: harmonic maps from T2 to S2; 26. Epilogue; References; Index.

This book is an introduction to an important area of mathematics and mathematical physics. It is accessible to university students, but leads to topics of current research in the theory of harmonic maps. It is the first book on this subject at this level.