Tau Functions and their Applications [1 ed.] 1108492681, 9781108492683

Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introductio

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Table of contents :
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
List of Symbols
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Tau Functions and their Applications john harnad and ferenc balogh

cambridge monographs on mathematical 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. J o h n H a r n a d is Director of the Mathematical Physics Laboratory at the Centre de Recherches Mathématiques and Professor of Mathematics at Concordia University in Montréal. Over his career he has made numerous contributions to a variety of fields of mathematical physics, including gauge field theory, integrable systems, random matrices, isomonodromic deformations and generating functions for graphical enumeration. He was the recipient of the 2006 Canadian Association of Physicists Prize in Theoretical and Mathematical Physics. F e r e n c B a l o g h is Professor of Mathematics at John Abbott College, Montréal. He completed his Ph.D. in mathematics under the supervision of John Harnad, with whom he has since collaborated on a number of research projects. His doctoral thesis was awarded the 2011 Distinguished Doctoral Dissertation Prize in Engineering and Natural Sciences.

CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS S. J. Aarseth Gravitational N-Body Simulations: Tools and Algorithms † D. Ahluwalia Mass Dimension One Fermions J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach † A. M. Anile Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics J. A. de Azcárraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics † O. Babelon, D. Bernard and M. Talon Introduction to Classical Integrable Systems † F. Bastianelli and P. van Nieuwenhuizen Path Integrals and Anomalies in Curved Space † D. Baumann and L. McAllister Inflation and String Theory V. Belinski and M. Henneaux The Cosmological Singularity † V. Belinski and E. Verdaguer Gravitational Solitons † J. Bernstein Kinetic Theory in the Expanding Universe † G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems † N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Space † K. Bolejko, A. Krasiński, C. Hellaby and M-N. Célérier Structures in the Universe by Exact Methods: Formation, Evolution, Interactions D. M. Brink Semi-Classical Methods for Nucleus-Nucleus Scattering † M. Burgess Classical Covariant Fields † E. A. Calzetta and B.-L. B. Hu Nonequilibrium Quantum Field Theory S. Carlip Quantum Gravity in 2+1 Dimensions † P. Cartier and C. DeWitt-Morette Functional Integration: Action and Symmetries † J. C. Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion † P. D. B. Collins An Introduction to Regge Theory and High Energy Physics † M. Creutz Quarks, Gluons and Lattices † P. D. D’Eath Supersymmetric Quantum Cosmology † J. Dereziński and C. Gérard Mathematics of Quantization and Quantum Fields F. de Felice and D. Bini Classical Measurements in Curved Space-Times F. de Felice and C. J. S Clarke Relativity on Curved Manifolds † B. DeWitt Supermanifolds, 2nd edition† P. G. O. Freund Introduction to Supersymmetry † F. G. Friedlander The Wave Equation on a Curved Space-Time † J. L. Friedman and N. Stergioulas Rotating Relativistic Stars Y. Frishman and J. Sonnenschein Non-Perturbative Field Theory: From Two Dimensional Conformal Field Theory to QCD in Four Dimensions J. A. Fuchs Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory † J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists † Y. Fujii and K. Maeda The Scalar-Tensor Theory of Gravitation † J. A. H. Futterman, F. A. Handler, R. A. Matzner Scattering from Black Holes † A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev Harmonic Superspace † R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity † T. Gannon Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics † A. García-Díaz Exact Solutions in Three-Dimensional Gravity M. Göckeler and T. Schücker Differential Geometry, Gauge Theories, and Gravity † C. Gómez, M. Ruiz-Altaba and G. Sierra Quantum Groups in Two-Dimensional Physics † M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 1: Introduction M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 2: Loop Amplitudes, Anomalies and Phenomenology V. N. Gribov The Theory of Complex Angular Momenta: Gribov Lectures on Theoretical Physics † J. B. Griffiths and J. Podolský Exact Space-Times in Einstein’s General Relativity † T. Harko and F. Lobo Extensions of f(R) Gravity: Curvature-Matter Couplings and Hybrid Metric-Palatini Gravity S. W. Hawking and G. F. R. Ellis The Large Scale Structure of Space-Time † J. Harnad and F. Balogh Tau Functions and their Applications B. B. Hu and E. Verdaguer Semiclassical and Stochastic Gravity F. Iachello and A. Arima The Interacting Boson Model † F. Iachello and P. van Isacker The Interacting Boson-Fermion Model † C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory † C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems †

G. Jaroszkiewicz Principles of Discrete Time Mechanics G. Jaroszkiewicz Quantized Detector Networks C. V. Johnson D-Branes † P. S. Joshi Gravitational Collapse and Spacetime Singularities † J. I. Kapusta and C. Gale Finite-Temperature Field Theory: Principles and Applications, 2nd edition† V. E. Korepin, N. M. Bogoliubov and A. G. Izergin Quantum Inverse Scattering Method and Correlation Functions † K. Krasnov Formulations of General Relativity J. Kroon Conformal Methods in General Relativity M. Le Bellac Thermal Field Theory † L. Lusanna Non-Inertial Frames and Dirac Observables in Relativity Y. Makeenko Methods of Contemporary Gauge Theory † S. Mallik and S. Sarkar Hadrons at Finite Temperature A. Malyarenko and M. Ostoja-Starzewski Tensor-Valued Random Fields for Continuum Physics N. Manton and P. Sutcliffe Topological Solitons † N. H. March Liquid Metals: Concepts and Theory † I. Montvay and G. Münster Quantum Fields on a Lattice † P. Nath Supersymmetry, Supergravity, and Unification L. O’Raifeartaigh Group Structure of Gauge Theories † T. Ortín Gravity and Strings, 2nd edition A. M. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization † M. Paranjape The Theory and Applications of Instanton Calculations L. Parker and D. Toms Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity R. Penrose and W. Rindler Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields † R. Penrose and W. Rindler Spinors and Space-Time Volume 2: Spinor and Twistor Methods in Space-Time Geometry † S. Pokorski Gauge Field Theories, 2nd edition† J. Polchinski String Theory Volume 1: An Introduction to the Bosonic String † J. Polchinski String Theory Volume 2: Superstring Theory and Beyond † J. C. Polkinghorne Models of High Energy Processes † V. N. Popov Functional Integrals and Collective Excitations † L. V. Prokhorov and S. V. Shabanov Hamiltonian Mechanics of Gauge Systems S. Raychaudhuri and K. Sridhar Particle Physics of Brane Worlds and Extra Dimensions A. Recknagel and V. Schiomerus Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes M. Reuter and F. Saueressig Quantum Gravity and the Functional Renormalization Group R. J. Rivers Path Integral Methods in Quantum Field Theory † R. G. Roberts The Structure of the Proton: Deep Inelastic Scattering † P. Romatschke and U. Romatschke Relativistic Fluid Dynamics In and Out of Equilibrium: And Applications to Relativistic Nuclear Collisions C. Rovelli Quantum Gravity † W. C. Saslaw Gravitational Physics of Stellar and Galactic Systems † R. N. Sen Causality, Measurement Theory and the Differentiable Structure of Space-Time M. Shifman and A. Yung Supersymmetric Solitons Y. M. Shnir Topological and Non-Topological Solitons in Scalar Field Theories H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, 2nd edition† J. Stewart Advanced General Relativity † J. C. Taylor Gauge Theories of Weak Interactions † T. Thiemann Modern Canonical Quantum General Relativity † D. J. Toms The Schwinger Action Principle and Effective Action † A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects † R. S. Ward and R. O. Wells, Jr Twistor Geometry and Field Theory † E. J. Weinberg Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics †

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Tau Functions and Their Applications

JOHN HARNAD Concordia University and CRM (Centre de recherches mathématiques), Université de Montréal

FERENC BALOGH John Abbott College, Montréal

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108492683 DOI: 10.1017/9781108610902 c John Harnad and Ferenc Balogh 2021  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 A catalogue record for this publication is available from the British Library. ISBN 978-1-108-49268-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.


Preface Acknowledgements

1 1.1

page xvii xxv

Examples The pendulum and the KdV equation: elliptic function solutions 1.1.1 The pendulum 1.1.2 Travelling wave solutions of the KdV equation 1.1.3 Degeneration to the trigonometric/hyperbolic case: the separatrix 1.2 Multisoliton solutions of KdV and KP 1.3 Schur functions 1.4 Rational solutions of the KP equation and the Calogero–Moser system 1.5 KP τ -functions associated to algebraic curves 1.6 Matrix model integrals 1.7 The Toda lattice: bilinear equations and multisoliton solutions 1.8 Generating function for intersection indices (Kontsevich integral) 1.9 Generating function for simple Hurwitz numbers 1.10 Common features of the examples

2 2.1 2.2

KP flows and the Sato–Segal–Wilson Grassmannian A brief introduction to the Sato–Segal–Wilson Grassmannian: KP τ -functions as Fredholm determinants The Grassmannian element w ∈ GrH+ (H) for the examples 2.2.1 Multisoliton solutions of the KP and KdV hierarchies 2.2.2 Schur functions 2.2.3 Calogero–Moser solutions 2.2.4 τ -functions associated to algebraic curves 2.2.5 Random matrix integrals 2.2.6 One-dimensional Toda lattice multisolitons ¯ g,N 2.2.7 Generating functions for intersection indices on M 2.2.8

Generating function for simple Hurwitz numbers

1 2 2 6 8 9 11 13 17 18 19 21 22 23

27 27 30 30 31 31 32 32 33 33 34



3 3.1

The KP hierarchy and its standard reductions The algebra of pseudodifferential operators 3.1.1 Inverses and nth roots of pseudodifferential operators 3.2 The KP hierarchy 3.3 The dressing method 3.3.1 Gauge equivalence 3.3.2 KP flows on dressing operators 3.4 The Baker function and dual Baker function 3.5 Hirota bilinear residue relations: Baker function 3.6 The τ -function 3.7 Hirota bilinear residue relations in terms of the τ -function and bilinear operators 3.8 Dressing operators of Wronskian type 3.9 Examples of Wronskian solutions: solitons and continuum limits 3.9.1 KP solitons 3.9.2 Multimeasure moment matrices 3.9.3 Single-measure moment matrices 3.10 Addition formula and the Hirota bilinear equations 3.10.1 Four-parametric Gr2 (C4 ) Plücker form of the Hirota equations 3.10.2 Addition formulae for KP τ -functions 3.11 Operator Schur function form of Hirota relations 3.12 Reductions of KP hierarchy to KdV and Gel’fand–Dickey hierarchies 3.12.1 The KdV hierarchy 3.12.2 The Gel’fand–Dickey or nKdV hierarchies 3.12.3 Soliton solutions of the KdV and Gel’fand–Dickey hierarchies 4 4.1 4.2 4.3 4.4 4.5 4.6

4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

Infinite dimensional Grassmannians A finite dimensional model The Sato–Segal–Wilson Grassmannian Plücker coordinates on GrH+ (H) The big cell of GrH+ (H) and affine coordinates The action of the groups Γ+ and Γ− on the infinite Grassmannian The (dual) determinantal line bundle and the Segal–Wilson τ -function 4.6.1 The (dual) determinantal line bundle 4.6.2 The Segal–Wilson τ -function The Baker function associated to a subspace Schur function expansion Equivalence between the Hirota equations and Plücker relations Hook partitions and the generalized Giambelli identity Finitely truncated subspaces in the Sato–Segal–Wilson Grassmannian Gauge transformations and Γ− symmetries The Borodin–Okounkov formula Time reversal and the dual Grassmannian

35 35 36 38 41 42 43 43 44 48 51 54 58 58 58 59 59 59 62 65 68 68 70 71 73 73 76 81 82 83 87 87 88 89 93 94 95 97 99 101 102

Contents 4.15 Gel’fand–Dickey/nKdV reductions and the Grassmannian Fermionic representation of τ -functions and Baker functions 5.1 Fermionic Fock space and Clifford algebra representation 5.1.1 Fermionic Fock space 5.1.2 The Clifford algebra of fermionic creation and annihilation operators 5.1.3 Fermionic representation of the GL(H) group action 5.1.4 The charge, energy and charge raising operators on the Fock space F 5.2 Plücker coordinates and fermionic bilinear relations 5.2.1 Plücker coordinates and the Plücker map 5.2.2 Fermionic representation of the Plücker relations 5.3 Fermionic representation of the KP and modified KP τ -functions 5.4 Fermionic representation of the Baker function 5.5 The Plücker relations and the Hirota bilinear equations 5.5.1 Fermionic bilinear commutation relations 5.5.2 Equivalence between the Plücker relations and the Hirota equations (second proof) 5.6 Bose–Fermi equivalence 5.6.1 Bosonic Fock space 5.6.2 Vertex operators 5.7 Adapted bases 5.8 Expansion of Baker function in adapted bases 5.9 Multipair correlators and their fermionic representation 5.10 Evaluation of KP τ -functions on finite power sums 5.11 The (dual) Baker function evaluated on power sums

ix 103


6 6.1 6.2

6.3 6.4 6.5

Finite dimensional reductions of the infinite Grassmannian and their associated τ -functions Gekhtman–Kasman finite determinantal formula Examples of finite determinant type τ -functions 6.2.1 Example 1. Rational solutions 6.2.2 Example 2. KP multisolitons 6.2.3 Example 3. Generic case: degeneration of KP solitons 6.2.4 Example 4. Calogero–Moser pole dynamics as rational solutions of KP Finite Grassmannians as sub-quotients of the Sato–Segal–Wilson Grassmannian KP τ -functions as finite determinants Fermionic representation

105 105 105 106 109 112 113 113 115 116 119 120 120 122 123 123 125 127 130 131 135 137

139 140 142 142 143 147 148 153 158 164

x 6.6 6.7 6.8


7 7.1




Contents f Direct proof that τ(A,B,C,D,F ) (t) satisfies the Hirota bilinear equations Subspaces with geometric affine coordinates KP solitons and reductions to KdV 6.8.1 Special KP multisolitons 6.8.2 Reductions to KdV solitons Solutions of the KP hierarchy related to algebraic curves 6.9.1 KND data on an algebraic curve 6.9.2 The τ -function associated to algebraic curve data

Other related integrable hierarchies The small BKP hierarchy 7.1.1 BKP Baker function and τ -function: Sato formula 7.1.2 Examples of BKP τ -functions and the Cartan relations The BKP addition formula and Cartan relations 7.2.1 The addition formula for BKP τ -functions 7.2.2 3-parameter form of the BKP addition formula Isotropic Grassmannians and BKP fermionic representation 7.3.1 Isotropic Grassmannians 7.3.2 Neutral fermions 7.3.3 A Clifford subalgebra of A(H + H∗ , Q) and the associated spin group Spin(Hφ , Qφ ) 7.3.4 The Cartan map and Cartan relations 7.3.5 Bilinear fermionic relations and Cartan relations 7.3.6 The Pfaffian Giambelli identity 7.3.7 Neutral fermion currents and the BKP flow group 7.3.8 Fermionic representation of the BKP τ -function; equivalence of BKP Hirota equations and Cartan relations 7.3.9 Fermionic representation of BKP multisoliton solutions 7.3.10 Fermionic representations of multiple integral BKP τ -functions The large BKP and DKP hierarchies 7.4.1 Quadratic form, Clifford algebra, spin group and fermionic representation 7.4.2 Fermionic representation of the spin groups Spin(J , Q) ˜ and the bilinear relations and Spin(J˜, Q) 7.4.3 Isotropic Grassmannians for large BKP and DKP and the Cartan map 7.4.4 Bilinear relations for large BKP and DKP; Cartan relations 7.4.5 Large BKP and DKP τ -functions and Hirota equations in residue form 7.4.6 The large BKP and DKP Hirota equations (bilinear differential equation form)

167 169 172 172 174 175 176 179 181 181 181 184 188 188 190 191 191 191 193 195 197 200 201 204 206 207 208 209 211 213 215 216 217



7.6 7.7

8 8.1 8.2 8.3 8.4 8.5 8.6

9 9.1


9.3 9.4



7.4.7 Examples of (large) BKP and DKP τ -functions The mKdV and mKP hierarchies 7.5.1 mKdV Hierarchy: Darboux–Bäcklund transformations 7.5.2 mKP Hierarchy 7.5.3 mKP τ -functions and the Hirota bilinear equation 7.5.4 The fermionic representation of mKP τ -functions and Baker functions 2D-Toda lattice hierarchy Discrete KP and BKP hierarchies 7.7.1 Discrete KP and the octahedron recursions 7.7.2 Discrete BKP hierarchy and cube recursion relations

218 220 221 225 228

Convolution symmetries Convolution symmetries of τ -functions Fermionic derivation of the Schur function expansion Convolution action on H and GrH+ (H) Convolution action on Fock space Convolutions and Schur function expansions Applications: KP, mKP and 2D Toda τ -functions of hypergeometric type

236 236 237 237 239 241

230 231 233 233 234


Isomonodromic deformations 246 246 The Painlevé transcendent PII 246 9.1.1 Scaling reduction of KdV to PII 247 9.1.2 PII as an isomonodromic deformation equation 250 9.1.3 Rational solutions of PII Fuchsian differential operators and their deformations 251 9.2.1 Schlesinger equations 251 9.2.2 Hamiltonian structure of the Schlesinger equations 254 9.2.3 The isomonodromic τ -function for Schlesinger systems 255 256 9.2.4 The case r = 2, n = 3: reduction to PV I (α, β, γ, δ) 9.2.5 σ-equation for PV I and bilinear equation for the τ -function 260 9.2.6 Special solutions: Riccati equation and 261 2 F1 hypergeometric functions 9.2.7 Picard–Fuchs solutions 262 The reduced r = 2 Schlesinger system for arbitrary n: the Garnier system 263 Non-Fuchsian systems 268 9.4.1 Irregular singularities at λ = ∞ 268 9.4.2 Schlesinger-like systems with second order poles at infinity 270 9.4.3 Hamiltonian structure and isomonodromic τ -function 271 9.4.4 Symplectic lift of the Hamiltonian system 272 Classical R-matrix, isospectral and isomonodromic deformations 273




Dual isomonodromic deformations 9.6.1 Dual pairs of isomonodromic systems 9.6.2 The case r=2, n=2: reduction to PV


Integrable integral operators and dual isomonodromic deformations Integrable Fredholm kernels and the Riemann–Hilbert problem Special integrable kernels and isomonodromic deformations Isomonodromic families of operators supported on a union of intervals 10.3.1 Deformation of the endpoints 10.3.2 Deformations of exponents The Fredholm determinant Monodromy data and the Fredholm determinant as a τ -function Example: sine kernel solution for PV and Garnier-like systems Grassmannian interpretation Duality Duality theorem Discrete symplectic reductions

10.1 10.2 10.3

10.4 10.5 10.6 10.7 10.8 10.9 10.10 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 12 12.1 12.2 12.3 12.4 12.5 12.6

Random matrix models I. Partitions functions and correlators Hermitian matrix models: conjugation invariant measures, partition function and orthogonal polynomials Polynomial potentials and orthogonal polynomials supported on curves in the complex plane The partition function as a KP τ -function The Grassmannian subspace associated to a matrix model partition function Multi-point correlators associated to matrix models 2 × 2 differential system associated to one-matrix models and the spectral curve Partition functions as isomonodromic τ -functions Proofs of Theorems 11.6.1 and 11.6.2 Random matrix models II. Level spacings Level spacings for random matrices Correlation functions (marginal distributions) and gap probabilities Fredholm determinant representation of gap probabilities Dynamics of distribution functions Hamiltonian Structure of the JMMS equations The Airy Kernel System and distributions at the edge of the spectrum: the Tracy–Widom distribution

277 277 281

286 286 291 292 292 296 297 299 300 302 305 309 310

314 314 318 321 325 326 329 335 338 341 341 342 343 346 347 348

Contents 12.7 Example: m= 1 ; Hastings–McLeod solution of PII Generating functions for characters, intersection indices and Brézin–Hikami matrix models 13.1 The Frobenius character formula 13.2 Intersection indices and Kontsevich matrix integral 13.3 Convolution symmetries and Brézin–Hikami externally coupled Hermitian matrix models

xiii 353


14 14.1 14.2 14.3 14.4


Generating functions for weighted Hurwitz numbers: enumeration of branched coverings Hurwitz numbers: pure and weighted Hypergeometric τ -functions as generating functions for weighted Hurwitz numbers Fermionic representation Examples: classical and quantum 14.4.1 Classical weighted Hurwitz numbers 14.4.2 Quantum Hurwitz numbers 14.4.3 Generating functions for Hurwitz numbers: classical counting of branched covers  14.4.4 The τ -functions τ (E(q),β) , τ (E (q),β) as generating functions for quantum Hurwitz numbers 14.4.5 Classical limits of examples E(q), E  (q) Quantum Hurwitz numbers 14.5.1 Symmetrized monomial sums and q-weighted Hurwitz sums 14.5.2 Quantum Hurwitz numbers: the case E(q) 14.5.3 Bose gas model

356 356 357 366

369 369 373 376 377 377 380 382 383 383 384 384 385 385

Appendix A: Integer partitions A.1 Partitions and Young diagrams A.2 Particle positions and Maya diagrams A.3 Frobenius notation for partitions A.4 Symmetric and strict partitions A.5 Young tableaux and Young’s lattice

388 388 389 390 391 391

Appendix B: Determinantal and Pfaffian identities B.1 Vandermonde and Cauchy determinants B.2 The von Koch formula B.3 The Desnanot–Jacobi and Sylvester identities B.4 The Cauchy–Binet identity B.5 The Schur complement B.6 The Weinstein–Aronszajn identity B.7 Gessel’s identity

393 393 393 394 395 395 396 397

xiv B.8 B.9 B.10 B.11

Contents Pfaffians The Andréev identity The de Bruijn identities Vandermonde-like Pfaffians

398 399 399 400

Appendix C: Grassmann manifolds and flag manifolds C.1 The Grassmann manifold Grn (FN ) C.2 The Grassmannian as a homogeneous space C.3 Particle positions and coordinate charts labelled by partitions C.4 The Plücker embedding and Plücker coordinates C.5 The Plücker image and Plücker relations C.6 Plücker relations in terms of maximal minors of the frame matrix C.7 The dual determinantal line bundle C.8 Hook partitions and affine coordinates: the generalized Giambelli identity C.9 Short Plücker relations and the Desnanot–Jacobi identity C.10 Dual Grassmannian C.11 Flag manifolds C.12 Schubert cell decomposition of the Grassmannian C.13 Formal Taylor series and jet spaces C.14 Finite and infinite dimensional Grassmannians C.14.1 Subquotients and embeddings C.14.2 Intertwining of embeddings i(n,N ) , ˆi(n,N ) by the Plücker map

401 401 402 403 405 406 409 411

Appendix D: Symmetric functions D.1 Schur polynomials sλ (x) as Plücker coordinates D.1.1 Cauchy–Littlewood identity and generating functions D.1.2 Jacobi–Trudi formula D.1.3 Hook Schur functions in terms of hk and ek D.1.4 Giambelli identity D.2 Dimensions of irreducible representations of U (n) D.2.1 Plücker coordinate formula for Dλ,n D.2.2 Giambelli identity for Dλ,n D.3 Dimensions of irreducible representations of Sn dλ as a Plücker coordinate: determinantal formula D.3.1 |λ|! D.3.2 Product formula for dλ D.3.3 The Giambelli formula for dλ D.4 Symmetric function bases D.4.1 The algebra Λ of symmetric functions D.4.2 The symmetric functions sλ , mλ , eλ , hλ and pλ as Q-bases for Λ D.4.3 The Z-bases {˜ sλ }, {mλ }, {eλ } , {hλ } and {fλ } for Λ

429 429 430 433 435 437 437 438 439 440 440 442 443 443 443

412 413 419 421 422 424 425 426 427

445 446

Contents D.4.4 D.4.5


Pieri formula and proof that sλ = s˜λ Orthogonality and variants on the Cauchy–Littlewood relations Q-Schur functions

Appendix E: Finite dimensional fermions: Clifford and Grassmann algebras, spinors, isotropic Grassmannians E.1 Clifford algebra E.1.1 Irreducible Clifford module: even dimensions E.1.2 Clifford module: odd dimensions E.1.3 The Spin group E.1.4 Weyl spinors E.1.5 Fermionic expression of the Plücker relations E.2 Isotropic Grassmannians, the Cartan map and pure spinors E.2.1 Totally isotropic subspaces, Grassmannians and flags (even dimensions) E.2.2 Spinor orbits and pure spinors E.2.3 The Cartan map E.2.4 Coordinate atlas on Gr0N (V), isotropic Giambelli identity and an explicit representation of the Cartan map E.2.5 The image of the Cartan map: the Cartan relations Appendix F: Riemann surfaces, holomorphic differentials and θ functions F.1 Compact Riemann surfaces: homology basis, holomorphic differentials, abelian integrals F.2 The Jacobi variety and the Abel map F.2.1 The Jacobi variety F.2.2 Divisors, symmetric powers and the Abel map F.3 Riemann θ functions F.4 Solution of the Jacobi inversion problem

xv 450 454 455

458 458 459 461 462 463 463 465 465 466 466 467 471

474 474 475 475 476 478 479

Appendix G: Orthogonal polynomials G.1 Moments and Heine’s formulae G.2 Orthogonality along contours in the complex plane

482 482 484

Appendix H: Solutions of selected exercises


References List of Symbols Index

494 510 513


The modern theory of integrable systems began with the discovery of solitons [289], which are strongly stable, localized, propagating solutions of nonlinear evolution equations. The occurrence of solitary waves in narrow shallow water channels was observed much earlier, in the mid-19th century [240], and the phenomenon modelled by exact solutions of the Korteweg-de Vries (KdV) equation [176]. But the very strong stability properties of these solitary waves, manifested in the fact that they can collide and pass through each other with no distortion in shape, were first noticed by Kruskal and Zabusky [289] in 1965. This remarkable insight soon led to an important methodological breakthrough with the discovery of the inverse scattering method [95–98] by Gardner, Greene, Kruskal and Miura. This turned out to be applicable to a large class of highly coherent nonlinear evolution equations, such as the cubically nonlinear Schrödinger equation, of importance in nonlinear optics, and many further dynamical evolution equations having a variety of physical applications, including certain well-known lattice systems, both finite and infinite, interacting particle systems on a line and spin chains. This was followed by very substantial further developments in which an array of common characteristics came to be understood within the unifying framework of inverse spectral methods [6, 290–293]. A key feature was the fact that they could be viewed as compatible systems of evolution equations that preserve the spectrum of a linear operator: isospectral deformations [185]. Another was the presence, in nearly all cases, of an underlying Hamiltonian structure, and the existence of a “maximal” number of Poisson commuting conserved quantities, implying that, at least in the finite dimensional cases, these systems were completely integrable in the Liouville–Arnold sense. Equations of this type, with suitable boundary conditions, thus came to be seen as infinite dimensional analogs of classical integrable systems [22,75]. The scattering transform was interpreted as a canonical transformation to linearizing coordinates, similar to action-angle variables, though not necessarily periodic. Solitons, in particular, were shown to correspond to the bound state spectrum. The inverse scattering approach was soon generalized to include other boundary conditions, in which the asymptotically decreasing property of the interaction potential appearing in the linear operator was replaced by periodicity or



quasi-periodicity [70–72,141,142,178]. The analogs of solitons were quasi-periodic solutions of “finite band” type, with their associated Bloch wave functions. The ultimate generalization of inverse spectral methods was eventually recognized to be the matrix Riemann–Hilbert method [213, 290, 293]. Another remarkable feature of these systems was developed by Sato [243–245] and by Segal and Wilson [250] in the light of earlier work by Hirota [134], who noticed that they could alternatively be expressed as bilinear differential or finite difference equations for an auxiliary quantity, named the τ -function. Generally, this function of the commuting flow variables could be expressed as a finite or infinite determinant in which the entries were elementary functions with linear dependence of their arguments on the flow variables. In the case of multisoliton or rational solutions, the determinants were finite, with linear exponential or quasi-polynomial entries. These authors had the remarkable insight of recognizing a parallel between the bilinear differential or difference equations satisfied by the τ -functions and a classical geometrical problem also characterized by bilinear equations, namely, the Plücker relations, which determine the embedding of a Grassmann manifold into the projectivization of a suitably defined exterior space. This led to a beautiful characterization of the dynamics of such systems as generated by the action of an abelian group of flows on an infinite dimensional Grassmannian, with the latter viewed as a sort of universal phase space. The τ -function is interpreted in this setting as the determinant of the orthogonal projection of an element of the Grassmannian in the group orbit onto a standard element, the “origin”. In the exterior space setting, this is just the Plücker coordinate, relative to the “vacuum state”, of the point in the Grassmannian moving under the action of the commuting group flows. The bilinear equations arising in the classical integrable systems context can also appear in a discretized form, and interpreted as addition formulae that generalize the Fay identities satisfied by Riemann θ functions, or as systems of recursion relations defining dynamical systems on a lattice. This approach is not aimed so much at providing new techniques for computing solutions but rather a universal phase space, in which the various solutions may be viewed as arising from different initial points under the same abelian group action. In general, both the phase space and the group are understood as infinite dimensional, with the dynamics determining not just one but an infinite family of compatible evolution equations and their commuting flows. Together, these form infinite families, or hierarchies of evolution equations, such as the Kadomtsev–Petviashvili (KP) or two dimensional Toda hierarchies, for which any given τ -function provides solutions of all equations of the hierarchy simultaneously. At a conceptual level, it may be seen as a dynamical generating function, somewhat like Hamilton’s characteristic function in the context of classical integrable Hamiltonian systems, evaluated on the simultaneous level sets of the conserved quantities.



A closely related development concerns systems obtained from the standard integrable hierarchies through multiscaling reductions. The reduced systems are also “integrable”, in a sense, but they typically involve equations of nonautonomous type, like the Painlevé transcendents [143, 150]. Such equations have numerous applications, both in statistical physics and in random matrix theory [83, 116, 120, 197]. They may also be understood as deformation equations which, rather than preserving the spectrum of an associated linear operator, preserve the monodromy of a rational covariant derivative operator with respect to the associated spectral parameter [81, 120, 143, 152, 153, 156], when the solution of the linear system is analytically continued around singular points in the complex plane: thus, isomonodromic deformation equations. There is an associated isomonodromic τ -function characterizing such systems, which is closely related to the underlying Hamiltonian structure. A variant of the Riemann–Hilbert method is applicable here as well, particularly in finding asymptotic limits when some small parameter tends to zero or the matrix size tends to infinity [55,57,58]. This has led to important advances in the asymptotic analysis of spectral distributions of random matrices, as well as the dispersionless asymptotics of integrable nonlinear evolution equations. Amongst the many applications of such isomonodromic deformation systems are: the computation of partition functions, correlators and gap probabilities in statistical mechanical models and in the spectral theory of random matrices [83, 116, 197, 270], as well as further applications to random processes of integrable type [38, 116], conformal field theory [61, 92, 138] and supersymmetric Yang–Mills theory [17, 101, 207]. However, these still do not exhaust the wide array of applications that τ functions may have. The discrete bilinear equations may also be related to the so-called Bethe Ansatz equations that underlie the quantum inverse scattering approach to quantum integrable systems and solvable lattice models [175]. New applications and interpretations of τ -functions were found in the context of quantum systems, as partition functions or correlation functions for various integrable quantum mechanical or field theoretical systems. The exterior space, which is fundamental in the Grassmannian approach, may be viewed as a fermionic Fock space, on which various group actions are realized in terms of Fermi creation and annihilation operators, providing a suggestive link to quantum applications. Yet another range of applications of τ -functions arose from a completely different direction: the computation of enumerative geometric and topological invariants associated to problems in algebraic geometry and combinatorics, such as graphical enumeration on Riemann surfaces [148], Gromov–Witten invariants [219], intersection indices [149, 174] and Hurwitz numbers [112, 218, 230]. In this setting, the τ -function is not interpreted analytically, but rather as a formal power series in a finite or infinite number of auxiliary variables, in which the coefficients are the enumerative invariants of interest. It thus serves as a generating function in the combinatorial sense for various enumerative geometrical and topological invariants.



The various applications of τ -functions mentioned here are developed in this volume, with primary focus on their use in the theory of classical integrable systems, their appearance as partition functions or correlation functions in the spectral theory of random matrices and as combinatorial generating functions in various problems of enumerative geometry.

Readership and prerequisites This monograph is aimed at graduate students, postdoctoral fellows and researchers working in various domains of mathematical physics who require a self-contained introduction to the rapidly expanding subject of τ -functions. Its intention is to develop, in detail, a representative set of key results and techniques that currently are accessible only in scattered form throughout the research literature. The reader is assumed to have reasonable familiarity with the standard topics comprising an undergraduate curriculum in mathematics, including complex variables, functional analysis, operator theory, differential equations and the rudiments of group theory, as well as the standard concepts and methods of classical and quantum mechanics. The bibliography lists numerous supplementary sources, including textbooks, research papers and surveys that provide further details and technical background on the material presented in the text.

Summary of contents The main topics covered are the following. - Chapter 1 consists of a “bird’s eye view”, listing several specific examples of τ -functions that will appear throughout the book. - Chapter 2 interprets these very briefly within the infinite Grassmann manifold setting of Sato and Segal–Wilson. - Chapter 3 introduces the KP hierarchy and gives its standard representation as commuting isospectral flows in the algebra of formal pseudodifferential operators. The Baker function and τ -function are introduced and related through the Sato formula. It is then shown how the equations of the KP hierarchy are equivalently expressed through the infinite set of Hirota bilinear equations. - In Chapter 4, the infinite dimensional Grassmannians of Sato and Segal– Wilson are formally introduced. The τ -function is defined in terms of flows of holomorphic sections of the (dual) determinantal line bundle induced by an abelian group action. The Schur function expansion of the τ -function is derived, and used to prove the equivalence between the Hirota bilinear equations and the Plücker relations determining the image, under the Plücker map, of the Grassmannian in the projectivization of the semi-infinite exterior







product space, i.e., the fermionic Fock space. The addition formulae for KP τ -functions are derived and also shown equivalent to the Hirota relations. Chapter 5 provides representations of the τ -function and associated Baker function as fermionic vacuum state expectation values (VEV’s), and this is used to rederive the expansion of τ -functions in the basis of Schur functions. The Plücker coordinates and Plücker relations are represented fermionically, and the equivalence between the Hirota bilinear equations and the Plücker relations is rederived using the fermionic operator formalism. VEV representations are also provided for all further relevant quantities, such as the basis of the Hilbert space adapted to a given element of the infinite Grassmannian, pair correlators (integral kernels) and multipair correlators. The Bose-Fermi correspondence is explained and fermionic VEV representations of the Baker function and multipair correlators are derived. The notion of vertex operators is introduced, expressed both bosonically and fermionically, and it is shown how the KP τ -function, when evaluated on power sums of a finite number of bosonic variables, can be expressed as a finite determinant. In Chapter 6 a finite dimensional microcosm of the Sato–Segal–Wilson Grassmannian is constructed using a subquotient procedure. The resulting τ -functions are all expressed as finite dimensional determinants with quasi-polynomial entries. These include the multisoliton solutions of the KP hierarchy, and their various degenerations, such as rational solutions and solutions with pole dynamics satisfying the Calogero–Moser particle system. This chapter also introduces the algebro-geometric solutions of the KP hierarchy related to algebraic curves, as developed by Matveev and Its, using inverse spectral methods, and by Krichever, Dubrovin and Novikov using direct constructions. These τ -functions are associated to divisors on algebraic curves of arbitrary genus and expressed explicitly in terms of Riemann θ-functions using the Abel map to linearize the flows. Chapter 7 concerns other integrable hierarchies that are closely related to the KP hierarchy: the BKP and DKP hierarchies, providing analogs of the Plücker relations and the embedding of isotropic Grassmannians into the projectivized fermionic Fock space, as well as the modified KP (mKP) hierarchy. Discrete analogs of the KP and BKP hierarchies are also derived, in which the τ functions are defined over suitable infinite lattices and the Plücker relations and their analogs interpreted as finite recursion relations. The 2D-Toda lattice hierarchy, which extends the modified KP lattice systems, is introduced and the fermionic VEV representation is given for the associated τ -function. Chapter 8 deals with applications of another type of infinite abelian group action on the Grassmannian: generalized convolution symmetries, which are diagonal in the standard Fourier basis and hence also act diagonally on the Plücker coordinate basis. This leads, in particular, to certain constructions that appear in later chapters: the relation between externally coupled matrix









models and self-coupled ones, detailed in Chapter 13, Section 13.3, and the τ -functions of hypergeometric type, which appear as generating functions for weighted Hurwitz numbers in Chapter 14. Chapter 9 gives a brief introduction to the Hamiltonian theory of isomonodromic deformations, and the corresponding isomonodromic τ -functions. As examples, it includes the Schlesinger equations of arbitrary rank and pole number, their reduction to the Garnier system, non-Fuchsian systems with a second order pole at ∞ and the Painlevé transcendents. It also develops the notion of dual isomonodromic systems. Chapter 10 develops further the notion of dual pairs of isomonodromic systems, and shows how special isomonodromic τ -functions can be realized as Fredholm determinants of integral operators of integrable type, whose resolvents may be computed through the solution of an associated matrix Riemann–Hilbert problem. Chapters 11 and 12 are devoted to the spectral theory of random matrices with conjugation invariant measures, studied from the viewpoint of integrable systems, isomonodromic deformations and τ -functions. In Chapter 11 the standard self-coupled Hermitian matrix model partition function with deformed measures defined by exponential series in the trace invariants is shown to be a KP τ -function, with the relevant Grassmannian element spanned by the associated orthogonal polynomials, as well as an isomonodromic τ -function. This is further extended to measures defining orthogonal polynomials supported on more general contours in the complex plane. In Chapter 12, the equations of Tracy and Widom determining generalized gap probabilities are derived and shown to be isomonodromic deformation equations for associated first order linear systems in the spectral parameter, in which the end points of the gaps are the pole locus. The spacing distributions near the edge of the spectrum are expressed as Fredholm determinants of the integral operator with Airy kernel, supported on the gaps, and shown to be the associated isomonodromic τ -functions. The same is done with respect to the sine kernel, similarly providing solutions to the equations of generalized Garnier type that determine gap probabilities in the bulk region. Chapter 13 deals with various examples of τ -functions that serve as generating series for enumerative geometric and topological invariants relating to Riemann surfaces, such as the Kontsevich integral, which generates intersection indices on the moduli space of marked Riemann surfaces and the Brézin–Hikami externally coupled matrix model. Chapter 14 develops the use of KP and 2D-Toda τ -functions as generating functions for enumerative invariants of another type: weighted Hurwitz numbers, which give weighted enumerations of N -sheeted branched coverings of the Riemann sphere or, equivalently, factorizations of elements in the symmetric group SN into products of elements of specified conjugation classes, or paths in its Cayley graph. The various weightings considered include: those



that enumerate the classical case of simple branched covers; the enumeration of weakly and strongly monotonic paths in the Cayley graph, and a new class of weightings, quantum weighted Hurwitz numbers, whose weighted enumeration is closely related to the energy distribution of a Bosonic gas.

Appendices With a view to making the volume as self-contained as possible, several appendices have been included, providing introductions to some of the more specialized mathematical tools required in various parts of the book. These are intended to serve as a helpful summary of material that may not be familiar to all readers, including: A) A survey of the basics of integer partitions. B) A list of various useful determinantal and Pfaffian identities, such as the Cauchy–Binet identity, the Schur complement and the de Bruijn identities. C) A summary of the algebraic geometry of Grassmann manifolds and the Plücker embedding. D) An introduction to the theory of symmetric functions, with all results proved in the spirit of τ -functions, i.e., by making use of Grassmannians and Plücker coordinates in the interpretation of Schur functions, and in the computation of dimensions and identities. E) A brief introduction to Clifford algebras, spinors and Grassmann algebras, with the exterior space viewed as a finite dimensional fermionic Fock space, and Cartan’s analog of the Plücker embedding for the case of maximal isotropic Grassmannians. F) A brief review of compact Riemann surfaces, abelian differentials and Riemann θ functions. G) A brief survey of basic results on orthogonal polynomials. H) Detailed solutions of a number of examples and problems appearing as exercises throughout the text.

Exercises The exercises left for the reader supplement the material treated, including details of proofs of certain results stated in the text, as well as new examples that illustrate or enhance the topic at hand. In some cases, they provide elements essential to the continuity of the text. Following many of the exercises, hints are provided, which should be sufficient to guide the reader in carrying out the proof or computation suggested in detail. Appendix H provides complete solutions of some of the exercises (indicated (S)) and sketches of solutions of others.


Preface What is not covered

It would be unrealistic to try to include more than a representative sample of the many advances in the modern theory of integrable systems of the past half-century in a single volume of moderate size. Only results central to the theory of τ -functions and their applications have been retained, and even here, choices had to be made on what was suitable for inclusion or not, and the degree of detail that was appropriate. Most of the topics omitted are, however, dealt with in the many excellent monographs and reviews listed in the bibliography. The main themes that are not treated in this work in any depth, but nevertheless represent important aspects of the current theory of integrable systems, are here listed, with references to where they may be found developed in detail. • Inverse spectral methods and the Riemann–Hilbert problem, the dressing method, Bäcklund–Darboux transformations [5, 75, 82, 140, 194, 213, 239]. • Hamiltonian structure of integrable systems and the spectral transform, classical R-matrices, separation of variables [8–11, 22, 75]. • Asymptotics of isomonodromic systems and dispersionless limits of isospectral evolution systems [56–58, 82, 143]. • Discrete integrable dynamics as recursion systems [36, 65, 259]. • Quantum integrable systems and lattice models: quantum R-matrices, YangBaxter equations, Bethe Ansatz methods [28, 100, 155, 175, 182]. • Integrable random processes [38, 116].


The authors have greatly benefited, over many years, from stimulating discussions and collaborative interactions with colleagues, co-workers and friends. It is impossible to adequately express our gratitude to all these for their valuable contributions and challenges to our understanding. Particular mention deserves to be made of those who have actively collaborated with the authors on results specifically related to the content of this work. These include: Alexander Alexandrov, Malcolm Adams, Marco Bertola, Guillaume Chapuy, Tiago Dinis da Fonseca, Victor Enolski, Bertrand Eynard, Mathieu Guay-Paquet, Jacques Hurtubise, Alexander Its, Alexander Orlov, Emma Previato, Steve Shnider, Craig Tracy, Johan Van de Leur and Harold Widom, to whom we wish to express our hearty thanks. The bibliography contains several references to joint works whose content is essential to the development of the text. This gives some indication of the degree to which we are indebted to our collaborators for shared insights regarding the various topics covered in this volume. We have kindly received permission from all those who have contributed to joint publications to include material drawn from them, adapted to the context and purpose of the present work. A detailed list of jointly authored works whose content is drawn upon follows, with chapters and sections of the relevant passages specified. Wherever the copyright for original publications has been assigned to a journal publisher, this is indicated, as well as when the copyright remains with the authors and collaborators. In all cases, permission has been kindly accorded for the use of such material. Chapter 5. Sections 5.7–5.9. Reproduced in part from: [18] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad, Fermionic approach to weighted Hurwitz numbers and topological recursion, Commun. Math. Phys., 360(2), 777–826, 2018, with permission c 2018 Springer-Verlag). from Springer-Nature ( Chapter 6. Sections 6.1, 6.2.1–6.2.2, 6.3–6.7. Reproduced in part from: [25] F. Balogh, T. Fonseca and J. Harnad, Finite dimensional Kadomtsev-Petviashvili τ functions. I. Finite Grassmannians, J. Math. Phys., 55(8):083517, 32 2014, with the c 2014 AIP publishing). permission of AIP publishing ( Chapter 8. Reproduced in part from: [123] J. Harnad and A. Yu. Orlov, Convolution symmetries of integrable hierarchies, matrix models and τ -functions. Chapter of Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Vol.



65 of MSRI publications, pp. 247–275 (2014). Cambridge University Press, Eds. P. c 2014 J. Harnad and A. Yu. Orlov). Deift and P. Forrester edition, 2014. ( Chapter 10. Sections 10.1–10.5, 10.7–10.10. Reproduced in part from: [119] J. Harnad and A. R. Its, Integrable Fredholm operators and dual isomonodromic deformations, Commun. Math. Phys., 226, 497–530, (2002), with permission from c 2002 Springer-Verlag). Springer-Nature ( Chapter 11. Sections 11.3, 11.6–11.8. Reproduced in part from: [31] M. Bertola, B. Eynard and J. Harnad, Partition functions for matrix models and isomonodromic τ -functions. J. Phys. A. Math. Gen., 36, 3067–3983 (2003). Reproduced c 1993 IOP Publishing Ltd). with permission from IOP Publishing ( Chapter 12. Sections 12.4–12.6. Reproduced in part from: [129] J. Harnad, C. Tracy and H. Widom. Hamiltonian structure of equations appearing in random matric 2016 J. Harnad, C. Tracy and ces NATO Sci. Ser. B, 315:231–246 (1993). ( H. Widom). Chapter 13. Section 13.3. Reproduced in part from: [123] J. Harnad and A. Yu. Orlov, Convolution symmetries of integrable hierarchies, matrix models and τ -functions. Chapter of Random Matrix Theory, Interacting Particle Systems, and Integrable Systems, Vol. 65 of MSRI publications, pp. 247–275 (2014). Cambridge c 2014 J. Harnad University Press, Eds. P. Deift and P. Forrester edition, 2014. ( and A. Yu. Orlov). Chapter 14. Reproduced in part from: [117] J. Harnad, Weighted Hurwitz numbers and hypergeometric τ -functions: an overview, Proc. Symp. Pure Math. (AMS), 93, c 2016 J. Harnad) and [112] M. Guay-Paquet and J. Harnad, 289–332 (2016) ( Generating functions for weighted Hurwitz numbers, J. Math. Phys., 58, 083503 c 2017 M. Guay-Paquet and J. Harnad). (2017) (

1 Examples

In this introductory chapter, we list a number of concrete examples of τ -functions, noting the elements they have in common, but postponing a formal definition to subsequent chapters. The first case is the simplest nonlinear periodic Hamiltonian system with one degree of freedom: the pendulum. In the Hamilton–Jacobi approach, Hamilton’s characteristic function, evaluated on the energy level sets, is the logarithmic derivative of the Weierstrass σ-function. This is our first example of a τ -function. The equations of motion are expressible as a bilinear equation for the σ-function, providing the first instance of an equation of Hirota type. Turning to nonlinear integrable evolution equations that are PDE’s in one spatial and one time dimension, such as the KdV equation, the simplest reduction is to travelling wave solutions with constant velocity. These again satisfy a Weierstrass-type equation, just like the pendulum. The separatrix, where the τ -function is simply a hyperbolic cosine, corresponds to 1-soliton solutions. The τ -functions corresponding to multisoliton solutions are expressed as the determinant of a matrix whose entries are linear exponential functions of the flow variables, which again satisfies a bilinear system of Hirota type. Multisoliton solutions of the more general integrable KP (Kadomtsev–Petviashvili) hierarchy are similarly given in terms of τ -functions having determinantal exponential form, also satisfying the Hirota bilinear equations. We next consider the basic building blocks from which all KP τ -functions are constructed: the Schur functions, which are polynomials in the flow parameters, whose logarithmic derivatives provide rational solutions of the hierarchy. Other examples include the Toda lattice, an integrable multiparticle system on the line with exponential nearest-neighbour interactions and the Calogero–Moser system, another integrable multiparticle system on the line whose dynamics coincide with the pole dynamics of rational solutions of the KP hierarchy. The “ultimate” generalization of the pendulum then follows: the so-called finite gap or multi-quasi-periodic solutions of the KP hierarchy, where the τ -function is



simply expressible in term of multivariable Riemann θ functions associated to the period lattice of an algebraic curve of arbitrary genus. Further specific examples of KP τ -functions are provided by the partition functions for various types of random matrix models. These include cases where the matrix integrals do not necessarily converge, nor does the expansion in the basis of Schur functions. They may, however, be viewed as formal expansions that serve as generating functions for various combinatorial invariants such as: intersection indices on the moduli space of marked Riemann surfaces, or Hurwitz numbers, which enumerate branched covers of the Riemann sphere. The characteristic features shared by all these examples are listed at the end of the chapter. A preliminary interpretation of these is given in Chapter 2, in terms of abelian group actions on a Grassmann manifold. This anticipates the Sato– Segal–Wilson approach to KP τ -functions, whose detailed development begins in Chapters 3 and 4 and continues throughout the remainder of the book.

1.1 The pendulum and the KdV equation: elliptic function solutions 1.1.1 The pendulum Consider the motion of a simple pendulum, consisting of a point mass m suspended on a massless rigid rod of length L, subject to the force of gravity (Fig. 1.1). The Lagrangian of the system, expressed in terms of the angle φ from the vertical, is the difference between kinetic and potential energies: ˙ = L(φ, φ)

φ 1 mL2 φ˙ 2 − 2mgL sin2 , 2 2


where g is the acceleration due to gravity.

φ L

2L sin2

φ 2


Fig. 1.1. The pendulum

The total energy, which is conserved, is the sum E=

1 φ mL2 φ˙ 2 + 2mgL sin2 . 2 2


1.1 The pendulum and the KdV equation: elliptic function solutions


Introducing the coordinate q =: sin we have q˙ =

φ , 2


1 φ 1 ˙ cos φ˙ = 1 − q 2 φ, 2 2 2


and the Lagrangian takes the form L(q, q) ˙ = 2mL2

q˙2 − 2mgLq 2 . 1 − q2


The momentum conjugate to q is p :=

q˙ ∂L = 4mL2 , ∂ q˙ 1 − q2


and the Legendre transformation gives the Hamiltonian as the sum of kinetic and potential energies: H(q, p) =

1 (1 − q 2 )p2 + 2mgLq 2 . 8mL2


Since the system is autonomous, the total energy is constant 1 (1 − q 2 )p2 + 2mgLq 2 = E 8mL2


and a first integral of the equations of motion is given by its level curves. Substituting (1.1.6), this can be integrated directly, giving q(t) implicitly in terms of an elliptic integral. It is worthwhile, however, to also consider the problem using the Hamilton– Jacobi method. For this, we define a new momentum variable P , which is a constant of motion, by 2mgLP 2 := H(q, p),


and seek a generating function S(q, P ), Hamilton’s principal function, for the transformation from (q, p) to new canonical coordinates (Q, P ) in which the equations of motion are trivial. The transformation is defined by p=

∂S , ∂q


∂S , ∂P

and the Hamilton–Jacobi equation is   ∂S H q, =E ∂q or, more explicitly,

∂S ∂q


= 16m2 gL3

P 2 − q2 . 1 − q2






The solution is given by an elliptic integral of the second kind:  q 2  P − x2 S(q, P ) = 4m gL3 dx, 1 − x2 q0


where the constant of integration is absorbed into the choice of initial point q0 . The coordinate canonically conjugate to P is thus given by an elliptic integral of the first kind  q  dx  , (1.1.14) Q = 4m gL3 P (1 − x2 )(P 2 − x2 ) q0 defined on the curve z 2 = (1 − x2 )(P 2 − x2 ).


In the canonical coordinates (Q, P ), the equations of motion have the trivial form ∂H dP =− = 0, dt ∂Q dQ ∂H = = 4mgLP, dt ∂P

(1.1.16) (1.1.17)

which, when integrated, give a linear flow in time Q(t) = Q0 + 4mgLP t,

P (t) = P0 .


Viewing Hamilton’s characteristic function S(q, P ) as a function of time, evaluated on the energy level sets, we have    q(t) P 2 − x2 dx, (1.1.19) S(q(t), P ) = 4mL gL 1 − x2 q(0) Changing the integration variable in eq. (1.1.13) from x to y := x2 gives   v(t) dy g  t, (1.1.20) =2 L y(y − 1)(y − e) v0 where v(t) := q 2 (t), and

v0 := v(0),

  S(q(t), P ) = 2mL gL



e := P 2


e−y dy. y(1 − y)


Introducing the rescaled, translated function  L e+1 u(t) = v t − , g 3


1.1 The pendulum and the KdV equation: elliptic function solutions


the inverse of the elliptic integral in (1.1.20) becomes a first order differential equation in standard Weierstrass form: (u )2 = 4u3 − g2 u − g3 , with coefficients 4 g2 = (e2 − e + 1), 3

g3 =

4 (e + 1)(e − 2)(2e − 1). 27



The general solution to (1.1.24) is given by the Weierstrass ℘-function u(t) = ℘(t − t0 )


for these parameter values, and any initial value constant t0 ∈ C. In general, ℘ is defined by

1 1 1 − , ℘(z) := 2 + z (z − w)2 w2



where the sum is over the integer lattice L in the complex plane L = {2mω1 + 2nω2 : m, n ∈ Z}


generated by any non-collinear pair of elliptic periods (2ω1 , 2ω2 ∈ C+ ). This satisfies the Weierstrass equation [280] (℘ )2 = 4℘3 − g2 ℘ − g3 ,


for modular constants (g2 , g3 ) determined from the lattice periods by the Eisenstein series

1 1 , g3 = 140 . (1.1.30) g2 = 60 4 w w6 w∈L\{0}


It can also be expressed as a second logarithmic derivative: ℘(z) = −

d2 ln σ(z) dz 2

in terms of the Weierstrass σ-function  

 z z2 z + σ(z) := z . 1− exp w w 2w2




Taking the first derivative of (1.1.29) gives ℘ (t) = 6℘2 (t) −

g2 . 2


Substituting (1.1.31) in (1.1.33) gives the equation of motion in bilinear form in terms of σ g2 σσ  − 4σ  σ  + 3(σ  )2 − σ 2 = 0, (1.1.34) 2



where  := way [74] as

d dt .

Equivalently, (1.1.34) may be expressed in a more symmetrical (Δ4 − g2 ) (σ(t − t0 )σ(t − t0 )) |t=t = 0,


where Δ :=

d d − . dt dt


Differentiating Hamilton’s characteristic function (1.1.19) with respect to t gives ∂S(q(t), P ) = 4mgL2 ℘(t − t0 ) + E ∂t


where E :=

4 (mgL2 − E). 3


So, within an integration constant, we have S(q(t), P ) = −

∂(ln σ) + Et. ∂t


Remark 1.1.1. The logarithmic derivative formula (1.1.31) expressing the general solution u(t) in terms of the Weierstrass σ-function will reappear in subsequent examples, as will the bilinear form (1.1.35) of the equation it satisfies. This is the first example of a τ -function generating the solution of an integrable nonlinear equation. It is seen here as closely related to Hamilton’s characteristic function S(q(t), P ); i.e., the complete solution of the Hamilton–Jacobi equation evaluated on the level sets of the conserved quantities.

1.1.2 Travelling wave solutions of the KdV equation The Weierstrass ℘-function also appears in another context relating to integrable systems: travelling wave solutions of the nonlinear partial differential equation 4ut = 6uux + uxxx ,


known as the Korteweg–de Vries (KdV) equation, which describes nondissipative shallow water waves in a narrow channel∗ . Choosing u(x, t) to have the form of a travelling wave u(x, t) = U (x + ct), ∗


The renewed study of the KdV equation, started in the mid 1960’s, led to the discovery of solitons, the inverse scattering method [95–98] and the subsequent flood of interest in completely integrable systems with infinite degrees of freedom.

1.1 The pendulum and the KdV equation: elliptic function solutions


where U is a function of a single variable z := x + ct, and c is the velocity, the KdV equation reduces to the ODE 4cU  = 6U U  + U  .


Integration and multiplication by U  gives 4cU U  = 3U 2 U  + U  U  + αU  ,


where α is an integration constant, which can again be integrated to give the first order equation 2cU 2 = U 3 +

1  2 (U ) + αU + β, 2


where β is a second constant of integration. This can now be reduced to the Weierstrass standard form by the substitution U (z) = −2℘(z + z0 ) +

2c , 3


where z0 ∈ C is an arbitrary constant and the modular forms (g2 , g3 ) determining ℘(z) are g2 =

α 4c2 − 3 2

The formula u(x, t) = 2

g3 = −


cα β 8c3 + + . 27 6 4


 ∂ 2  c x2 6 ln e σ(x + ct + z ) 0 ∂x2


thus gives the general travelling wave solution to the KdV equation, a simple example of an elliptic function solution to a nonlinear evolution equation. The function c


τ (x, t) := e 6 x σ(x + ct + z0 ) η1

= Ke 6 x e 2ω1 (x+ct+z0 ) θ c



x + ct + z0 ω2 ω2 1 + + ; 2ω1 2 2ω1 ω1

where θ(z; τ ) is the Jacobi θ function

2 θ(z; τ ) := eπiτ n +2πizn , n∈Z

τ :=

ω2 , ω1

(1.1.48) (1.1.49)


K is a nonzero constant and η1 :=

σ  (ω1 ) σ(ω1 )


is another example of a τ -function that, in this case, generates the elliptic function solution of the KdV equation representing generic travelling waves for this case.


Examples 1.1.3 Degeneration to the trigonometric/hyperbolic case: the separatrix

In terms of the pendulum, the elliptic integral (1.1.20) degenerates to a trigonometric one at the critical energy (1.1.52)

Ecrit = 2mgL,

and therefore the solution, either for the pendulum or the travelling wave of the KdV equation, can be written in terms of elementary trigonometric/hyperbolic functions. The corresponding solution of the pendulum problem is known as the separatrix, i.e., the special level curve E = Ecrit of the energy on the phase space ˙ (See Fig. 1.2, where the separatrix of the pendulum in the coordinates (φ, φ). E = Ecrit is indicated.) φ˙ E = Ecrit φ

˙ plane Fig. 1.2. Level curves of the energy E of a pendulum in the (φ, φ)

Similarly, if both integration constants α and β in (1.1.44) are chosen to be zero, the discriminant for the Weierstrass equation vanishes: Δ = g23 − 27g32 = 0,


and the general solution to (1.1.44) can be obtained by using elementary hyperbolic functions: √ U (z) = 2c sech2 ( c(z + z0 )). (1.1.54) Since U (z) can be written as a second logarithmic derivative U (z) = 2

√ d2 ln cosh( c(z + z0 )), dz 2


the corresponding solution to the KdV equation can be represented as u(x, t) = 2

 √ ∂2  ln cosh( c(x + ct + z0 )) , ∂x2


which is known as the one-soliton solution to the KdV equation, associated to the simple exponential type of τ -function  √ √ 1  √c(x+ct+z0 ) e + e− c(x+ct+z0 ) . (1.1.57) τ (x, t) = cosh( c(x + ct + z0 )) = 2

1.2 Multisoliton solutions of KdV and KP


1.2 Multisoliton solutions of KdV and KP For a given positive integer N , choose 2N complex numbers {αk }k=1,...,N and {γk }k=1,...,N


with all αk ’s pairwise distinct and all γk ’s nonzero. Define N functions ∞

yk (t) := e

i i=1 ti αk


+ γk e

i=1 ti (−αk )



k = 1, . . . , N,


where t is an infinite sequence of variables t = (t1 , t2 , . . . ),


referred to as the higher KdV flow variables or times. Note that  ∞  ∞ i i ∂l (l) (1.2.4) yk (t) := l yk (t) = (αk )l e i=1 ti αk + (−1)l γk e i=1 ti (−αk ) ∂t1 ⎧ ∞

⎪ 1 ⎪ 2i+1 ⎪ cosh t2i+1 αk − log γk l even ⎪ ⎨ ∞ 2 l 1/2 t2i α2i i=0 k  i=1 = 2(αk ) γk e ∞

⎪ 1 ⎪ ⎪ sinh t2i+1 αk2i+1 − log γk l odd. ⎪ ⎩ 2 i=0 (1.2.5) (N ) τα1 ,...,αN ,γ1 ,...,γN (t)

Now define the τ -function   y1 (t)   y  (t)  1 (N ) τα1 ,...,αN ,γ1 ,...,γN (t) :=  .. .   (N −1)  y1 (t)

as the Wronskian determinant  y2 (t) ··· yN (t)   y2 (t) ··· yN (t)   (1.2.6) .. .. ..  . . .   (N −1) (N −1) y2 (t) · · · yN (t)    y1 (t0 ) y2 (t0 ) ··· yN (t0 )    y  (t )  y2 (t0 ) ··· yN (t0 )   ∞ N  1 0 2i α t 2i  , = e i=1 k=1 k  .. .. .. ..  . . . .    (N −1)  (N −1) (N −1)  y1 (t0 ) y2 (t0 ) · · · yN (t0 )

where t0 := (t1 , 0, t3 , 0, . . . ), and the derivatives {yi  . . . yi (N −1)} } are taken with (N ) respect to x = t1 . The function τα1 ,...,αN ,γ1 ,...,γN (t) has the remarkable property that twice its second logarithmic derivative, evaluated at the parameter values (t1 = x, t2 = 0, t3 = t, ti = 0, i > 3) ∂2 ) log τα(N (x, 0, t, 0, 0 . . . ) (1.2.7) 1 ,...,αN ,γ1 ,...,γN ∂x2 satisfies the KdV equation (1.1.40). Solutions of this form are called standard N -soliton solutions to the KdV equation. More generally, if we choose 3N complex constants u(x, t) := 2

{αk , βk , γk }k=1,...,N




with αk , βk ’s all distinct, γk = 0, and define the functions ∞

i i=1 ti αk

yk (t) := e


+ γk e

i i=1 ti βk


k = 1, . . . , N,

we arrive at the more general Wronskian determinant    y1 (t) y2 (t) ··· yN (t)    y  (t)  y2 (t) ··· yN (t)   1 (N ) . τ (t) :=  .. .. .. ..  α ,β, γ . . . .     (N −1) (N −1) (N −1)  y1 (t) y2 (t) · · · yN (t) The function u(x, y, t) := 2

  ∂2 (N ) log τ (x, y, t, t , . . . ) 4 γ α ,β, ∂x2




can then be shown to satisfy the 2 + 1 dimensional nonlinear partial differential equation 3uyy = (4ut − 6uux − uxxx )x ,


known as the Kadomtsev–Petviashvili (KP) equation (which plays a prominent rôle in plasma physics and in the study of shallow water ocean waves), together with an infinite set of further nonlinear autonomous PDEs, each involving partial derivatives of finite order with respect to a finite number of the KP flow parameters t = (t1 , t2 , . . . ). These are collectively known as the KP hierarchy. They may all be deduced from a single family of bilinear relations known as the Hirota bilinear equations, (see Section 1.10 below), satisfied by the τ -function (N ) τ (t), and by all solutions of the KP hierarchy. Solutions of the form (1.2.10) α ,β, γ are referred to as standard N -soliton solutions of the KP hierarchy in Wronskian form. If βj = −αj for all j, the standard KP-solitons are independent of y = t2 and all further even flow parameters {t2i } and reduce to KdV N -solitons, since   2 3 2 3 2 3 3 exαk +yαk +tαk + γk e−xαk +yαk −tαk = eyαk exαk +tαk + γk e−xαk −tαk , (1.2.13) and the second logarithmic derivative in x eliminates the y-dependence. (N ) The Wronskian formula (1.2.10) for the τ -function τ (t) can also be rewritα ,β, γ ten in a more general determinantal form [102, 104] (detailed in Section 6.1) as τ where, for (1.2.10), A is ⎡ 1 1 ⎢ α1 α2 ⎢ A=⎢ . .. . ⎣ . . α1N −1

α2N −1

(N ) γ (t) α ,β,


= det(Ae

i=1 ti B


C T ),


the N × 2N double Vandermonde-type matrix ⎤ ··· 1 1 1 ··· 1 ··· αN β1 β2 ··· βN ⎥ ⎥ (1.2.15) . . . .. ⎥ , .. .. .. .. .. . . . ⎦ ···

N −1 αN

β1N −1

β2N −1


N −1 βN

1.3 Schur functions


B is the diagonal 2N × 2N matrix B = diag(α1 , α2 , . . . , αN , β1 , β2 , . . . , βN ) and C is also N × 2N , with ⎡ 1 ⎢0 ⎢ C = ⎢. ⎣ ..


the special form 0 1 .. .

··· ··· .. .

0 0


0 γ1 0 0 .. .. . . 1 0

0 γ2 .. .

··· ··· .. .

0 0 .. .




⎤ ⎥ ⎥ ⎥. ⎦


General N -soliton solutions of the KP hierarchy are of the form ∞

(N )

τA,B,C (t) = det(Ae

i=1 ti B


C T ),


where A and B are as above, but C can be an arbitrary N × 2N complex matrix of full rank. Such solutions were introduced and systematically studied by Matveev, Zakharov and others [102, 104, 142, 290, 293]. The necessary and sufficient conditions that they be nonsingular for arbitrary real values of the flow parameters (t1 , t2 , · · · ) were derived by Kodama and Williams [46,170–173].

1.3 Schur functions Recall that, for a set of indeterminates x = (x1 , . . . , xn ), the n × n Vandermonde matrix V (x1 , . . . , xn ) is defined as , Vij (x1 , . . . , xn ) := xn−i j

1 ≤ i, j ≤ n.

Its determinant, denoted )= Δ(x1 , . . . , xn ) := det(xn−j i

(xi − xj ),