Tau Functions and their Applications

A thorough introduction to tau functions, from the basics through to the most recent results, with applications in mathematical physics.

John Harnad (Author), Ferenc Balogh (Author)

9781108492683, Cambridge University Press

Hardback, published 4 February 2021

520 pages, 17 b/w illus.
25.1 x 17.6 x 3.4 cm, 1.114 kg

'This book is a magnificent handbook on the applications of the ? -functions.' Dimitar A. Kolev, zbMATH

Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introduction, starting from the basics and extending to recent research results. It covers a wide range of applications, including generating functions for solutions of integrable hierarchies, correlation functions in the spectral theory of random matrices and combinatorial generating functions for enumerative geometrical and topological invariants. A self-contained summary of more advanced topics needed to understand the material is provided, as are solutions and hints for the various exercises and problems that are included throughout the text to enrich the subject matter and engage the reader. Building on knowledge of standard topics in undergraduate mathematics and basic concepts and methods of classical and quantum mechanics, this monograph is ideal for graduate students and researchers who wish to become acquainted with the full range of applications of the theory of tau functions.

Preface
List of symbols
1. Examples
2. KP flows and the Sato-Segal-Wilson Grassmannian
3. The KP hierarchy and its standard reductions
4. Infinite dimensional Grassmannians
5. Fermionic representation of tau functions and Baker functions
6. Finite dimensional reductions of the infinite Grassmannian and their associated tau functions
7. Other related integrable hierarchies
8. Convolution symmetries
9. Isomonodromic deformations
10. Integrable integral operators and dual isomonodromic deformations
11. Random matrix models I. Partition functions and correlators
12. Random matrix models II. Level spacings
13. Generating functions for characters, intersection indices and Brézin-Hikami matrix models
14. Generating functions for weighted Hurwitz numbers: enumeration of branched coverings
Appendix A. Integer partitions
Appendix B. Determinantal and Pfaffian identities
Appendix C. Grassmann manifolds and flag manifolds
Appendix D. Symmetric functions
Appendix E. Finite dimensional fermions: Clifford and Grassmann algebras, spinors, isotropic Grassmannians
Appendix F. Riemann surfaces, holomorphic differentials and theta functions
Appendix G. Orthogonal polynomials
Appendix H. Solutions of selected exercises
References
Alphabetical Index.

Subject Areas: Mathematical physics [PHU], Relativity physics [PHR], Quantum physics [quantum mechanics & quantum field theory PHQ]