Tau Functions and their Applications

John Harnad, Ferenc Balogh

• Publisher: Cambridge University Press
• Year: 2021
• Print ISBN: 9781108492683
• eBook ISBN: 9781316999929
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

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.

Table of contents

• Cover
• Half-title page
• Series page
• Title page
• Copyright page
• Contents
• Preface
• Acknowledgements
• 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 τ-functions and Baker functions
• 6 Finite dimensional reductions of the infinite Grassmannian and their associated τ-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. Partitions 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 θ functions
• Appendix G: Orthogonal polynomials
• Appendix H: Solutions of selected exercises
• References
• List of Symbols
• Index