(london mathematical society lecture note series )lionel mason, yavuz nutku-geometry and...

http://www.cambridge.org/9780521529990

GEOMETRY AND INTEGRABILITY

Many integrable systems owe their origin to problems in geometry and allare perhaps best understood in a geometrical context. This is especiallytrue today when the heroic early days of study of KdV-type integrabilityare over. The problems that can be solved using the inverse scatteringtransformation are now well studied and there are diminishing returnsin this direction. Two major techniques have emerged more recently fordealing with multi-dimensional integrable systems: Twistor theory andthe d-bar method, both of which form the subject of this book. It isintended to be an introduction, though by no means an elementary one,to current research on integrable systems in the framework of differentialgeometry and algebraic geometry.

This book arose from a semester, held at the Feza Gursey Institute, tointroduce advanced graduate students to this area of research. The arti-cles are all written by leading researchers and are designed to introducethe reader to contemporary research topics.

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES

Managing Editor: Professor N.J. Hitchin, Mathematical Institute,University of Oxford, 24–29 St Giles, Oxford OX1 3LB, United Kingdom

The titles below are available from booksellers, or, in case of difficulty, from CambridgeUniversity Press at www.cambridge.org.

170 Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK171 Squares, A.R. RAJWADE172 Algebraic varieties, GEORGE R. KEMPF173 Discrete groups and geometry, W.J. HARVEY & C. MACLACHLAN (eds)174 Lectures on mechanics, J.E. MARSDEN175 Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds)176 Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds)177 Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE &

A. PITTS (eds)178 Lower K-and L-theory, A. RANICKI179 Complex projective geometry, G. ELLINGSRUD et al180 Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT181 Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds)182 Geometric group theory II, G.A. NIBLO & M.A. ROLLER (eds)183 Shintani zeta functions, A. YUKIE184 Arithmetical functions, W. SCHWARZ & J. SPILKER185 Representations of solvable groups, O. MANZ & T.R. WOLF186 Complexity: knots, colourings and counting, D.J.A. WELSH187 Surveys in combinatorics, 1993, K. WALKER (ed)188 Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN189 Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY190 Polynomial invariants of finite groups, D.J. BENSON191 Finite geometry and combinatorics, F. DE CLERCK et al192 Symplectic geometry, D. SALAMON (ed)194 Independent random variables and rearrangement invariant spaces, M. BRAVERMAN195 Arithmetic of blowup algebras, WOLMER VASCONCELOS196 Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND197 Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI et al198 The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN199 Invariant potential theory in the unit ball of Cn, MANFRED STOLL200 The Grothendieck theory of dessins d’enfant, L. SCHNEPS (ed)201 Singularities, JEAN-PAUL BRASSELET (ed)202 The technique of pseudodifferential operators, H.O. CORDES203 Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH204 Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT &

J. HOWIE (eds)205 Ergodic theory and its connections with harmonic analysis, K. PETERSEN &

I. SALAMA (eds)207 Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds)208 Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD &

W.M. OXBURY (eds)209 Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA & N. YUI210 Hilbert C∗-modules, E.C. LANCE211 Groups 93 Galway/St Andrews I, C.M. CAMPBELL et al (eds)212 Groups 93 Galway/St Andrews II, C.M. CAMPBELL et al (eds)214 Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders,

V. KOWALENKO et al215 Number theory 1992–93, S. DAVID (ed)216 Stochastic partial differential equation, A. ETHERIDGE (ed)217 Quadratic forms with applications to algebraic geometry and topology, A. PFISTER218 Surveys in combinatorics, 1995, PETER ROWLINSON (ed)220 Algebraic set theory, A. JOYAL & I. MOERDIJK221 Harmonic approximation, S.J. GARDINER222 Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds)223 Analytic semigroups and semilinear initial boundary value problems, KAZUAKITAIRA224 Computability, enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN &

S.S. WAINER (eds)225 A mathematical introduction to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA,

S. SCARLATTI226 Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI &

J. ROSENBERG (eds)227 Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI &

J. ROSENBERG (eds)

228 Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds)229 Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK230 Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS &

E.V. FLYNN231 Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds)232 The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS233 Finite fields and applications, S. COHEN & H. NIEDERREITER (eds)234 Introduction to subfactors, V. JONES & V.S. SUNDER235 Number theory 1993–94, S. DAVID (ed)236 The James forest, H. FETTER & B. GAMBOA DE BUEN237 Sieve methods, exponential sums, and their applications in number theory,

G.R.H. GREAVES et al

238 Representation theory and algebraic geometry, A. MARTSINKOVSKY &G. TODOROV (eds)

239 Clifford algebras and spinors, P. LOUNESTO240 Stable groups, FRANK O. WAGNER241 Surveys in combinatorics, 1997, R.A. BAILEY (ed)242 Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds)243 Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds)244 Model theory of groups and automorphism groups, D. EVANS (ed)245 Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al246 p-Automorphisms of finite p-groups, E.I. KHUKHRO247 Analytic number theory, Y. MOTOHASHI (ed)248 Tame topology and o-minimal structures, LOU VAN DEN DRIES249 The atlas of finite groups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds)250 Characters and blocks of finite groups, G. NAVARRO251 Grobner bases and applications, B. BUCHBERGER & F. WINKLER (eds)252 Geometry and cohomology in group theory, P.KROPHOLLER, G.NIBLO,

R. STOHR (eds)253 The q-Schur algebra, S. DONKIN254 Galois representations in arithmetic algebraic geometry, A.J. SCHOLL &

R.L. TAYLOR (eds)255 Symmetries and integrability of difference equations, P.A. CLARKSON &

F.W. NIJHOFF (eds)

256 Aspects of Galois theory, HELMUT VOLKLEIN et al257 An introduction to noncommutative differential geometry and its physical applications 2ed,

J. MADORE258 Sets and proofs, S.B. COOPER & J. TRUSS (eds)259 Models and computability, S.B. COOPER & J. TRUSS (eds)260 Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al261 Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al263 Singularity theory, BILL BRUCE & DAVID MOND (eds)264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS &

M.REID (eds)265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds)

268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJOSTRAND269 Ergodic theory and topological dynamics, M.B. BEKKA & M. MAYER270 Analysis on Lie Groups, N.T. VAROPOULOS & S. MUSTAPHA271 Singular perturbations of differential operators, S. ALBEVERIO & P. KURASOV272 Character theory for the odd order function, T. PETERFALVI273 Spectral theory and geometry, E.B, DAVIES & Y. SAFAROV (eds)274 The Mandelbrot set, theme and variations, TAN LEI (ed)275 Computational and geometric aspects of modern algebra, M.D. ATKINSON et al (eds)276 Singularities of plane curves, E. CASAS-ALVERO277 Descriptive set theory and dynamical systems, M. FOREMAN et al (eds)278 Global attractors in abstract parabolic problems, J.W. CHOLEWA & T. DLOTKO279 Topics in symbolic dynamics and applications, F. BLANCHARD, A. MAASS &

A. NOGUEIRA (eds)280 Characters and Automorphism Groups of Compact Riemann Surfaces, T. BREUER281 Explicit birational geometry of 3-folds, ALESSIO CORTI & MILES REID (eds)282 Auslander-Buchweitz approximations of equivariant modules, M. HASHIMOTO283 Nonlinear elasticity, R. OGDEN & Y. FU (eds)284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SULI (eds)285 Rational points on curves over finite fields: Theory and Applications,

H. NIEDERREITER & C. XING286 Clifford algebras and spinors 2nd edn, P. LOUNESTO287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA &

E. MARTINEZ (eds)288 Surveys in combinatorics, 2001, J.W.P. HIRSCHFELD (ed)289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE290 Quantum groups and Lie theory, A. PRESSLEY291 Tits buildings and the model theory of groups, K. TENT292 A quantum groups primer, S. MAJID293 Second order partial differential equations in Hilbert spaces, G. DA PRATO &

J. ZABCZYK294 Introduction to operator space theory, G. PISIER295 Geometry and integrability, L. MASON & Y. NUTKU (eds)296 Lectures on invariant theory, I. DOLGACHEV297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES298 Higher Operads, higher categories, T. LEINSTER299 Kleinian groups and hyperbolic 3-manifolds, Y.KOMORI, V.MARKOVIC &

C. SERIES (eds)300 Introduction to Mobius differential geometry, U. HERTRICH-JEROMIN301 Stable modules and the D(2)-problem, F. E. A. JOHNSON302 Discrete and continous nonlinear Schrodinger systems, M. ABLOWITZ, B. PRINARI &

D. TRUBATCH304 Groups St Andrews 2001 in Oxford v1, C. M. CAMPBELL, E. F. ROBERTSON & G. C.

SMITH (eds)305 Groups St Andrews 2001 in Oxford v2, C. M. CAMPBELL, E. F. ROBERTSON & G. C.

SMITH (eds)306 Peyresq lectures on geometric mechanics and symmetry, J.MONTALDI & T.RATIU (eds)307 Surveys in combinatorics 2003, C. D. WENSLEY (ed)308 Topology, geometry and quantum field theory, U. L. TILLMANN (ed)309 Corings and comodules, T. BRZEZINSKI & R. WISBAUER310 Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds)

311 Groups: topological, combinatorial and arithmetic aspects, T. W. MULLER (ed)

London Mathematical Society Lecture Note Series. 295

Geometry and Integrability

Edited by

Lionel MasonUniversity of Oxford

and

Yavuz NutkuFeza Gurzey Institute, Instanbul

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University PressThe Edinburgh Building, Cambridge , United Kingdom

First published in print format

ISBN-13 978-0-521-52999-0 paperback

ISBN-13 978-0-511-06555-2 eBook (NetLibrary)

© Cambridge University Press 2003

2003

Information on this title: www.cambridge.org/9780521529990

This book is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press.

ISBN-10 0-511-06555-8 eBook (NetLibrary)

ISBN-10 0-521-52999-9 paperback

Cambridge University Press has no responsibility for the persistence or accuracy ofs for external or third-party internet websites referred to in this book, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate.

Published in the United States by Cambridge University Press, New York

www.cambridge.org

http://www.cambridge.org

http://www.cambridge.org/9780521529990

To our families

Contents

List of contributors page xPreface xi1 Introduction 12 Differential equations featuring many periodic solutions 93 Geometry and integrability 214 The anti-self-dual Yang–Mills equations and their reductions 605 Curvature and integrability for Bianchi-type IX metrics 896 Twistor theory for integrable systems 977 Nonlinear equations and the ∂-problem 135

ix

Contributors

F. CalogeroUniversita di Roma ‘La Sapienza’R. Y. DonagiUniversity of PennsylvaniaL. J. MasonThe Mathematical Institute, OxfordP. M. SantiniUniversita di Roma ‘La Sapienza’K. P. TodThe Mathematical Institute, OxfordN. M. J. WoodhouseThe Mathematical Institute, Oxford

x

Preface

Integrable systems continue to fascinate because they are examples ofsystems with nontrivial nonlinearities that one can nevertheless system-atically analyse and often solve exactly analytically. However, there is noroyal road to complete integrability, or even a precise all-encompassingdefinition and so, instead, one must resort to patterns and themes. Thisvolume is concerned with a theme that emerges time and again of thedeep links that integrability has with geometry. The motivation for hold-ing a research semester devoted to ‘Geometry and Integrability’ at theFeza Gursey Institute was precisely for the purpose of exposing studentsand post-docs to modern geometrical structures that form the naturalsetting for completely integrable systems.

xi

1

IntroductionLionel Mason

The Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK

1.1 Background

Integrable systems are systems of partial or ordinary differential equa-tions that combine nontrivial nonlinearity with unexpected tractability.Often one can find large families of exact solutions, and general methodsfor generic solutions. This volume is concerned with the deep links thatintegrability has with geometry. There are two rather different waysthat geometry emerges in the study of integrable systems.

1.1.1 Geometrical context for integrable equations

The first is from the context of the differential equations themselves:even those integrable equations whose origins, perhaps in the theory ofwater waves or plasma physics, seem a long way from geometry canusually be expressed in the context of symplectic geometry as possiblyinfinite dimensional Hamiltonian systems with many conserved quanti-ties and often with much more further structure. But geometry is itselfalso a rich source of integrable systems; one of the first examples ofa completely integrable nonlinear partial differential equation, the sine-Gordon equation first appeared in the 19th century theory of surfaces, asa formulation of the constant mean curvature condition on a 2-surfaceembedded in Euclidean 3-space. Now there are many more examplesfrom geometry in many dimensions, from the two-dimensional systemsgiven by harmonic maps from Riemann surfaces to symmetric spaces,to the anti-self-duality equations in 4-dimensions and more generallyquaternionic structures in 4k-dimensions.

The contributions of Tod, Mason and Woodhouse focus on the anti-self-duality equations either on a Yang–Mills connection on a vector

1

2

bundle over R4, or on a 4-dimensional conformal structure. The systems

discussed by Santini also have a geometric origin, in their discrete form asquadrilateral lattices, and in their continuous limits as conjugate nets.The reductions and specializations of these systems then form manymore geometrical examples of integrable systems: although the systemsdiscussed by Donagi are presented as arising from complex algebraicgeometry rather than Riemannian geometry, they have their origin inreductions of the real anti-self-dual Yang-Mills equations.

1.1.2 Geometrical transforms and solution methods for

differential equations

The second way that geometry appears in the theory of integrable sys-tems is in the transforms and solution methods that are brought to bearon integrable systems. There are many different strands here. The sym-plectic framework for integrable equations leads to the first definitionof an integrable system, that due to Arnol’d and Liouville, in termsthe existence of sufficiently many constants of motion satisfying variousrequirements. The Arnol’d–Liouville theorem leads to a transform ofthe system to action-angle variables by quadratures in which the actionvariables are constant and the motion is linear in the angle variables. Infact many interesting integrable systems admit further structures thatimply Arnol’d–Liouville integrability. Those considered by Donagi arealgebraically completely integrable so that the structures in question arecomplexified and required furthermore to be algebraic. Another struc-ture that guarantees complete integrability is a bi-Hamiltonian struc-ture.

These structures in finite dimensions lead, at least in principle, to thegeneral solution by quadratures. Integrable partial differential equationscan often be expressed as infinite dimensional examples of systems sat-isfying the Arnol’d–Liouville requirements often by virtue of admittinga bi-Hamiltonian structure. However, the infinite number of degrees offreedom mean that one can no longer solve the system in a finite numberof quadratures. Nevertheless, new techniques become available. On theone hand there are hidden symmetries, both discrete, such as Backlundtransforms, and continuous, such as those generated by flows associatedto the Arnol’d–Liouville constants of motion, and these can help gen-erate new exact solutions. But also there are transforms that apply togeneral solutions; historically, the inverse scattering transform was thefirst important example of this and was used to provide the transform to

Introduction 3

action angle variables for solutions subject to rapidly decreasing bound-ary conditions in precise analogy with the transform provided by thefinite dimensional Arnol’d–Liouville theorem.

There are now a number of such transforms such as the inverse spectraltransform, the Penrose and Ward transforms and so on. A remarkablefeature of many of these transforms is the appearance of sophisticatedcomplex holomorphic, and often even algebraic geometry. This complexanalysis often plays a deep role in the finite dimensional case also. Inthe contribution of Woodhouse we see twistor theory as providing a sim-ilar transform between solutions to integrable equations and geometricstructures, holomorphic vector bundles, that can be described in termsof free functions. This construction has the additional benefit that itapplies to the general local analytic solution. A related method is basedon the non-local ∂-problem, so called ∂-dressing. In the local case this isoften simply an independent formulation of the twistor correspondence,but in the non-local case, such constructions go beyond standard twistortheory.

1.2 The contributions

The following is intended to provide some introduction to, and contextfor, the various contributions. I should make a disclaimer here that thecontext and background I give are perhaps rather one-sided and reflectmy own point of view; there are a number of different points of viewthat might be taken on this material that are not presented here!

1.2.1 Notes on reductions of the anti-self-dual Yang-Mills

equations and integrable systems, L. J. Mason;

Curvature and integrability for Bianchi-type IX metrics,

K. P. Tod;

Twistor theory and integrability, N. M. J. Woodhouse

These contributions are connected by an overview on the theory of in-tegrable systems based on reductions of the anti-self-dual Yang-Mills(ASDYM) equations and anti-self-dual conformal structures.

The ASDYM equations can be thought of as integrable by virtue of theexistence of the Ward correspondence between solutions to these equa-tions and holomorphic vector bundles on an auxilliary complex manifold,twistor space. For ASDYM fields on Minkowski space, twistor space is aportion of CP

3, complex projective 3-space. If one allows the transform

4

between solutions to the ASDYM equations and twistor data, this con-struction amounts to providing, in a geometric form, the general solutionto the ASDYM equations. There is a similar construction due to Penrosegiving a correspondence between anti-self-dual conformal structures anddeformations of the complex structure on twistor space.

A key observation of Richard Ward’s is that many of the most famousintegrable equations are symmetry reductions of the ASDYM equations.The various aspects of the integrability of such reductions of the ASDYMequations can then be understood by reduction of the correspondingtheory for the full ASDYM equations.

The contribution of Mason concerns the integrability of the ASDYMequations that can be understood without using twistor theory. Thusits Lax pair, Backlund transformations, Hamiltonian formulation andrecursion operator and hierarchy are presented. Some of the more sig-nificant reductions are reviewed also.

Paul Tod’s lectures on spinor calculus and conformal invariance weretaken from his book with Huggett, Introduction to Twistor Theory (sec-ond edition), published by CUP as LMS Student-Text 5, and so are notincluded here. The book gives useful details of space-time geometry thatprovide a background for the twistor correspondence and the interestedreader can refer to it for full details.

Nick Woodhouse’s contribution is an introduction to twistor methodsand explains how the Ward transform applies to ASDYM fields anddescends to provide correspondences for reductions of ASDYM fields.In particular it is shown how twistor methods can give new insight intothe KdV equations and the isomonodromy problem that arises in thestudy of Painleve equations. One aspect of integrability that emergesparticularly clearly is a ‘geometric’ explanation of the Painleve test forintegrability. The lectures build on those of Tod and Mason.

There is a further contribution from Paul Tod which concerns vari-ous equations on metrics in 4-dimensions that admit an SU(2) symmetry.The metric may be required to be Kahler, Einstein or have anti-self-dualWeyl tensor. The latter equation is usually thought to imply integrabil-ity because of Penrose’s twistor correspondence. With this symmetry,the equations reduce to ODE’s. If the metric is Einstein, it is no re-striction to assume it is diagonal (although it is a nontrivial restrictionfor general anti-self-dual conformal structures). When Ricci flat, oneobtains (with a further assumption) the Chazy equation. This is some-what of a novelty for integrable systems theory as this equation admits

Introduction 5

solutions with movable natural boundaries, contradicting the Painleveproperty. An explanation of this paradox is proposed.

1.2.2 Geometry and integrability, R.Y. Donagi

The contribution of Donagi is concerned with the theorem that the Mod-uli space of ‘meromorphic Higgs bundles’ over a Riemann surface Σ hasthe structure of an algebraically completely integrable system. Thiscombines the symplectic geometry underlying the Arnol’d–Liouville def-inition of an integrable system with algebraic geometry. The Arnol’d–Liouville definition of a completely integrable system as above can beabstracted by taking an integrable system to mean a Poisson manifold,M , with sufficiently many commuting Hamiltonians, the collection be-ing thought of as a map from H : M → R

n, satisfying certain technicalrequirements to guarantee satisfactory global properties. This definitioncan be complexified so that M is a complex manifold and the Pois-son structure is a complex holomorphic bivector and H : M → C

n areholomorphic. The algebraic condition is then that M be an algebraicmanifold with all the structures being expressible in terms of algebraicfunctions of algebraic coordinates on M . Although this definition mightseem somewhat special, there are a remarkable number of interestingsystems that turn out to be integrable in this way.

A Higgs bundle E is a holomorphic vector bundle equipped with aglobal holomorphic section, the Higgs field Φ, of the associated bundleof 1-forms with values in the endomorphisms of E, End(E) ⊗ Ω1(Σ).These first arose in the context of Hitchin’s study of reductions of theanti-self-dual Yang-Mills equations on a connection on a bundle over Eu-clidean R

4 by two translational symmetries. Remarkably, the reducedsytem acquires 2-dimensional conformal invariance and so makes senseon an arbitrary Riemann surface. The anti-self-duality condition reducesto equations on a connection and Higgs field on the Riemann surface; theHiggs field should be holomorphic and the curvature of the connectionbe given in terms of the Higgs fields. According to the philosophy ofthe contributions by Woodhouse, Tod and Mason, this Hitchin systemis an integrable system. Since it is a system of elliptic partial differen-tial equations, it doesn’t naturally fall into a Hamiltonian framework.However, the space of solutions on a compact Riemann surface is finitedimensional and one might expect this moduli space to inherit somevestige of integrability.

Hitchin proves that, for a compact Riemann surface and certain bun-

6

dles, a solution is determined just by the holomorphic data of the holo-morphic vector bundle and Higgs field. Thus the study of the modulispace can be reduced to a problem in complex geometry and this is theapproach that is adopted in this article. Naively the moduli space can bethought of as the cotangent bundle of the moduli space of holomorphicvector bundles on Σ as the Higgs fields are Serre-dual to deformations ofthe complex structure on a holomorphic vector bundle. Thus the Higgsbundle moduli space is a complex phase space. Furthermore, the coeffi-cients of the characteristic polynomial of the Higgs field can be thoughtof as defining a system of commuting Hamiltonians and so one has acomplex (holomorphic) integrable system which turns out to be alge-braic. However, there are a number of technicalities concerning stabilityand semi-stability that need to be addressed to make these ideas precise,and render the above discussion heuristic.

In keeping with the expository aim of the lectures, the bulk of thesenotes concern not the theorem and its applications, but the many ingre-dients which go into its proof. Students with a fairly modest backgroundin geometry should be able to work through these notes, learning a fairamount of algebraic geometry and symplectic geometry along the way,and may be motivated to follow some of the leads in the last sectiontowards open problems and further development of the subject.

1.2.3 The ∂ dressing method and integrable geometries,

P. Santini

In the previous contributions, it can be seen that a prominent role isplayed by complex structures. One way of formulating a complex struc-ture is in the form of a ∂-operator and, in the case of the Ward transform,the inverse transform from twistor data to the solution on space-timerequires the solution of a linear ∂-equation. Dressing can be understoodas a process by which one takes the transform for a well understood,perhaps trivial, solution where all the ingredients of the tansform areknown, and then change the ∂-data that appears in the ∂-equation togive a more general solution (perhaps the general solution). Over the lastfew decades such methods have been developed (independently of twistortheory) and extended to include a non-local element in the ∂-equation,so that the source term in the ∂-equation is given by integrating againsta kernel. These non-local terms seem to be essential for certain systemsin 2+1 dimensions such as the KP equations etc..

In this contribution the ∂-dressing method is shown to apply to cer-

Introduction 7

tain integrable geometric structures: quadrilateral lattices, a discretesystem consisting of lattices in which each elementary quadrilateral isplanar, and its continuous limit, the conjugate net, a system studied byDarboux.

The connection between the ∂-dressing method and these integrablegeometries relies upon the following facts:

(1) the simple, linear dependence of the ∂ data on the coordinates,described by the given linear differential and/or difference equations,defines some basic elementary singularities in the complex plane of thespectral parameter λ (the complex parameter with respect to which the∂-problem is defined): essential singularities, poles and branch points, inwhich the coordinates appear as parameters of the essential singularities,positions of the poles and strength of the branch points.

(2) These elementary singularities and their defining equations have of-ten an elementary and basic geometric meaning. For instance, (a) thematrix equation ψ0x = iλσ3ψ0 and its solution ψ0(x, λ) = exp(iλxσ3)define the Frenet frame of a straight line in R3, parallel to the third axiswith constant torsion λ and arclength x; (b) the vector difference equa-tions: ∆iψ0j = 0, i = 1, .., N, j = 1, ..,M define the tangent vectorsψ0j = (0, .., λθj , .., 0)T of an N - dimensional regular lattice in RM .

(3) Through the ∂ dressing method the above basic elementary func-tions ψ0 get dressed into new functions ψ which satisfy dressed linearequations in configuration space, whose integrability conditions are theintegrable nonlinear systems. In this dressing procedure, the originalgeometric meaning is usually preserved and suitably deformed. Forinstance, the linear equation of example (a) is dressed up into ψx =(iλσ3 + Q)ψ and describes an arbitrary curve in R3; while the lin-ear equations of example (b) are dressed up into the linear equations∆iψj = qjiψi, i = 1, .., N, j = 1, ..,M which describe the planarityof the elementary quadrilaterals of the N -dimensional lattice (what wecall: a quadrilateral, or planar lattice).

(4) The associated ∂ problem provides at the same time:

(i) large classes of solutions of the above geometries, which can thereforebe called “integrable”;

(ii) geometrically distinguished symmetry transformations and symme-try reductions of the above geometries.

8

1.2.4 Differential equations featuring many periodic

solutions, F. Calogero

The contribution of Francesco Calogero, who is the originator of mod-ern super-integrable systems amongst many other things, shows a way toobtain evolutionary PDEs which possess many periodic solutions. Thisdevelopment has obvious potential in the context of applications (espe-cially in the modelling of periodic phenomena), but it also sheds light(as more fully shown in other papers by Calogero and others) on a ratherfundamental question: the connection between the integrability of evo-lution equations and the analyticity in complex time of the solutions ofsuch equations, an issue related to the ‘Painleve property’.

1.3 Conclusion

There are many areas of interaction between geometry and integrabilitythat have not been touched on here — the infinite-dimensional grass-manians of Segal & Wilson, the theory of quaternion-Kahler manifolds,the various special integrable classes of two-surfaces embedded into sym-metric spaces and so on, but it is to be hoped that these articles willstimulate the reader into further study.Acknowledgements: I am grateful to Professor Nutku and TUBITAK forthe invitation to the Feza Gursey institute and for their generous hospi-tality. I should also like to thank Professor Nutku and the contributorsfor helpful paragraphs in writing this introduction.

2

Differential equations featuring manyperiodic solutions

F. CalogeroDipartimento di Fisica, Universit di Roma “La Sapienza”,Istituto Nazionale di Fisica Nucleare, Sezione di Roma

[email protected], [email protected]

Mathematics Subject Classification 2000: 34C25, 35B10Physics and Astronomy Classification Scheme: 02.30.Hq, 02.30Jr

Abstract

A simple trick is reviewed, which yields differential equations (bothODEs and PDEs) of evolution type featuring lots of periodic solutions.Several examples (PDEs) are exhibited.

2.1 Introduction

Recently a simple trick has been introduced that allows us to manu-facture evolution equations (both ODEs and PDEs) which possess lotsof periodic solutions – in particular, completely periodic solutions cor-responding, in the context of the initial-value problem, to an open setof initial data of nonvanishing measure in the space of initial data [1]-[5]. The purpose and scope of this presentation is to review this trick –most completely introduced and described in [5] – and to display, andtersely discuss, certain new (classes of) evolution PDEs yielded by it;the alert reader, after having grasped the main idea, can easily manu-facture many more examples, possibly also featuring several dependentand independent variables – here for simplicity we restrict attention tojust one (complex) dependent variable and to just two (real) independentvariables (the standard 1 + 1 case: one ‘time’ and one ‘space’ variablesonly).

The trick is described tersely in Section 2.2. Some examples of evo-lution equations – different from those reported in [5] – are displayed

9

10

in Section 2.3, which should be immediately seen by the browser whowishes to decide whether to invest time in reading the rest of this paper.Justification for these examples – namely, arguments that justify the ex-pectation that these evolution equations indeed feature lots of periodicsolutions – are given in Section 2.4, and in some cases they are backedby the display there of some periodic solutions.

2.2 The trick

Suppose that the function ϕ of the two complex variables ξ, τ , ϕ ≡ϕ(ξ, τ), satisfies an evolution equation in the (time-like) variable τ , andthat the structure of this evolution equation guarantees that there exista lot of solutions ϕ(ξ, τ) which are holomorphic in τ in an open disk ofradius 1/ω centered at τ = i/ω in the complex τ -plane (where ω is apositive constant), and that are as well holomorphic in ξ in an open diskof radius ρ (where ρ is another positive constant, possibly arbitrarilylarge) centered at ξ = 0 in the complex ξ-plane. Then introduce a(complex) function w ≡ w(x, t) of the two real variables x, t by setting

w(x, t) = exp(iλωt)ϕ(ξ, τ) (2.1)

with

τ =[exp(iωt)− 1

]/(iω), (2.2)

so that

τ ≡ dτ/dt = exp(iωt), (2.3)

τ(0) = 0, τ(0) = 1, (2.4)

and

ξ = x exp(iµωt). (2.5)

It is then clear that, if λ and µ are two rational numbers, all the nonsingu-lar functions w(x, t) defined by (2.1) are, at least for |x| < ρ, completelyperiodic functions of the real independent variable t, with a period whichis an integer multiple of 2π/ω.

On the other hand, if ϕ ≡ ϕ(ξ, τ) is determined by the requirementto satisfy an evolution equation of analytic type, say

ϕτ = F (ϕ,ϕξ, ϕξ, xi, . . . , ξ, τ) (a) (2.6)

Differential equations featuring many periodic solutions 11

or

ϕττ = F (ϕ,ϕξ, ϕξ,ξ, . . . , ξ, τ) (b) (2.7)

with F an analytic function of all its arguments, then it is indeed clear,from the standard existence/uniqueness/analyticity theorem for the ini-tial value-problem of analytic evolution PDEs, that there exist a set ofsolutions, of nonvanishing measure in the functional space of all solu-tions, that satisfy the requirements specified above, namely are holo-morphic in the variables and sectors specified above. This is clear if oneimagines obtaining these solutions ϕ ≡ ϕ(ξ, τ) of (2.6), (2.7) by solv-ing an initial-value problem, with the initial datum assigned at τ = 0and such that the right-hand side of (2.6), (2.7), evaluated at τ = 0,is sufficiently small (and, in the case of (2.7), the additional conditionthat ϕτ (ξ, 0) also be sufficiently small, say for all values of the complexvariable ξ such that |ξ| < ρ).

The trick consists now in inserting the ansatz (2.1) with (2.2), (2.3),(2.4) and (2.5) in the evolution equation, say of type (2.6) and (2.7),satisfied by ϕ ≡ ϕ(ξ, τ), and thereby to obtain an evolution equation forw ≡ w(x, t) that clearly then has a lot of solutions completely periodic int – at least in an appropriately restricted space region (say, |x| < ρ ; wewill not keep repeating this condition below, but the reader should notforget it). What makes this development interesting is the possibilitythat the evolution equations for w ≡ w(x, t) so manufactured have aneat structure – a possibility already demonstrated elsewhere [1]-[5] andalso displayed immediately below.

2.3 Evolution equations featuring lots of periodic solutions

In this section we display, with no comments other than those needed toexplain the notation, examples of evolution PDEs which possess lots ofperiodic solutions – in the sense explained above. These equations areobtained via the trick described in the preceding Section 2.2, as demon-strated in the following Section 2.4. Two such equations (or rather,classes of such equations) read as follows:

wt − iΩw =∑

p0,p1,p2,...=0∑n=0

pn=p

ap0p1p2···(w)p0(wx)p1(wxx)p2···, (2.8)

12

wtt − i[(p+ 3)/2]Ωw1 − [(p+ 1)/2]Ω2w =p∑

p0,p1,p2,...=0∑n=0

pn=p

ap0p1p2···(w)p0(wx)p1(wxx)p2 · · · . (2.9)

Here, and below, w ≡ w(x, t) is the (complex) dependent variable, xand t are the (‘space’ and ‘time’, hence real) independent variables,subscripted independent variables denote (partial) derivatives, Ω is anarbitrary real and nonvanishing (hereafter, without loss of generality,positive, Ω > 0) constant, p is an arbitrary positive integer (p > 1 so thatthese evolution equations are indeed nonlinear), the summation indicespn are of course nonnegative integers not larger than p, and the constantsap0p1p2... are arbitrary (possibly complex). These two evolution PDEs,of first-order, respectively second-order, have lots of solutions which arecompletely periodic with periods T = 2π/Ω, respectively T = 2π/Ω (ifp is odd) or T = 4π/Ω (if p is even).

Two other (classes of) evolution PDEs read as follows:

w1 − iλωw − iµωxwx

=p∑

p0,p1,p2,...=0∑n=0

pn=p,∑

n=1npn=P

ap0p1p2···(w)p0(wx)p1(wxx)p2 · · ·

+q∑

q0,q1,q2,...=0∑n=0

qn=q,∑

n=1nqn=Q

bq0q1q2...(w)q0(wx)q1(wxx)q2 . . .

+∑j=0

r(j)∑r(j)0 ,r

(j)1 ,r

(j)2 ,···=0∑

n=0r(j)n =r(j)∑

n=1nr

(j)n =R(j)

c(j)

r0(j)r(j)1 r

(j)2 ···(w)r

(j)0 (wx)r

(j)1 (wxx)r

(j)2 · · · (2.10)

and

wtt − 2iµωxwxt − i(2λ+ 1)ωwt

= µ2ω2x2wxx + (λ+ 1)µω2xwx + λ(λ+ 1)ω2w

+p∑

p0,p1,p2,...=0∑n=0

pn=p,∑

n=1npn=P

ap0p1p2···(w)p0(wx)p1(wxx)p2 · · ·


+q∑

q0,q1,q2,...=0∑n=0

qn=q,∑

n=1nqn=Q

bq0q1q2...(w)q0(wx)q1(wxx)q2 · · ·

+∑j=0

r(j)∑r(j)0 ,r

(j)1 ,r

(j)2 ,···=0∑

n=0r(j)n =r(j)∑

n=1nr

(j)n =R(j)

c(j)

r(j)0 r

(j)1 r

(j)2 ...

(w)r(j)0 (wx)r

(j)1 (wxx)r

(j)2 · · · . (2.11)

In these evolution PDEs, (2.10) and (2.11), the arbitrary constant ω isreal and nonvanishing (hence without loss of generality we hereafter as-sume it is positive, ω > 0); the constants ap0,p1,p2,..., bq0,q1,q2,..., c

(j)

r(j)0 ,r

(j)1 ,...

are arbitrary (possibly complex); pn, qn, j, r(j)n are nonnegative integers

over which the sums run, subject to the constraints characterized by theparameters p, P, q,Q, r(j); the limit of the sum over j is determined bythe limits over the integers r(j) and R(j), see below (this might entail thissum is altogether missing); the positive integers p, q and the nonnegativeintegers P,Q are arbitrary, except for the condition

(p− 1)Q− (q − 1)P = 0, (2.12)

but we assume (without loss of generality) p = q, p = r(j), q = r(j) andr(j) = r(k) if j = k, hence (again, without loss of generality) we set

p < q < r(0) < r(1) < r(2) < · · · . (2.13)

The nonnegative integers R(j) are given by the formula

R(j) = (p− q)−1[pQ− qP + (P −Q)r(j)

], (2.14)

and the corresponding positive integers r(j) are arbitrary except for thecondition that R(j) , as given by this formula, (2.14), be itself a non-negative integer (note that the relations r(j) =

∑n=0

r(j)n , R(j) =

∑n=1

nr(j)n

also hold, see (2.10) and (2.11)). Finally the rational numbers λ and µ

in (2.10) are given by the formulas

λ = (Q− P )/[(p− 1)Q− (q − 1)P ], (2.15)

µ = (p− q)/[(p− 1)Q− (q − 1)P ], (2.16)

14

while the rational numbers λ and µ in (2.11) are given by the formulas

λ = 2(Q− P )/[(p− 1)Q− (q − 1)P ], (2.17)

µ = 2(p− q)/[(p− 1)Q− (q − 1)P ]; (2.18)

note the consistency of these formulas, (2.15), (2.16) and (2.17), (2.18),with the condition (2.12).

These evolution PDEs, (2.10), (2.11) and (2.17), (2.18), have lotsof solutions multiply periodic with the 3 periods T1 = 2π/ω, T2 =T1/|λ| and T3 = T1/|µ| (of course with λ and µ given by (2.15), (2.16)respectively (2.17), (2.18), hence completely periodic (possibly only fora restricted set of values of the real “space” variable x , say |x| < ρ)with a period T that is the smallest common integer multiple of these3 periods T1, T2, T3. (In making this argument we implicitly assumethat neither µ nor λ vanish; the first condition is indeed guaranteed bythe condition (2.13), see (2.16) and (2.18); on the other hand λ couldvanish, in which case the reference made above to the period T2 shouldbe ignored, namely in this case there would be a lot of solutions multiplyperiodic with the 2 periods T1 and T2, hence completely periodic with aperiod T that is the smallest common integer multiple of these 2 periodsT1, T3.)

An example of an evolution PDE belonging to the class (2.8) (withp = 4) reads as follows:

w1 − iΩw = a4w4 + a13ww

3x + a3001w

3wxxx + a2110w2wxwxx. (2.19)

For a4 = a13 = 0, a2110 = 3a3001, or a4 = a13 = 0, a2110 = (3/2)a3001,this evolution PDE, (2.19), is C-integrable, while for a4 = a13 = a2110 =0 it is S-integrable. [6], [7]. This evolution PDE, (2.19), possesses ofcourse many solutions completely periodic with period 2π/Ω.

An example of an evolution PDE belonging to the class (2.10) (withp = 1, P = 3, q = 3, Q = 2, r(0) = 5 hence R(0) = 1, see (2.15), (2.16),and λ = 1/6, µ = 1/3, see (3.3 (b), (c)) reads

w1−i(ω/6)w−i(ω/3)xwx = a0001wxxx+b201w2wxx+b12ww

2x+c

(0)41 w

4wx.

(2.20)This evolution PDE, (2.20), is C-integrable if b12 = 3b201 and c

(0)41 =

−a0001(b201)2/3. [6]. For arbitrary values of all the constants it fea-tures (with the only restriction that ω be real and nonvanishing, indeed,without loss of generality, positive, ω > 0) this evolution PDE, (2.20),possesses many completely periodic solutions, with period T = 12π/ω.


Another example of an evolution PDE belonging to the class (2.10)(with p = 1, P = 3, q = 2, Q = 2, r(0) = 3 hence R(0) = 1, λ = µ = 1/3)reads

wt − i(ω/3)w− i(ω/3)xwx = a0001wxxx + b101wwxx + b02w2x − c21w

2wx.

(2.21)For b1001 = b02 = 0 it is S-integrable (related by a change of variables tothe modified Korteweg–de Vries equation). Again, it generally featuresmany completely periodic solutions, with period T = 6π/ω.

An example of an evolution PDE belonging to the class (3.2) (withp = 2) reads

wtt − i(5/2)Ωwt − (3/2)Ω2w

= a2w2 + a11wwx + a02w

2x + a101wwxx + a011wxwxx + a002w

2xx.

(2.22)

Hence it features a lot of completely periodic solutions with period T =4π/Ω.

2.4 Proofs

To obtain (2.8), we start from the evolution PDE

ϕτ = F (ϕ,ϕx, ϕxx, . . . ) (2.23)

with

F (ϕ,ϕx, ϕxx, . . . ) =p∑

p0,p1,p2,...=0,∑

n=0pn=p

ap0p1p2...(ϕ)p0(ϕx)p1(ϕxx)p2 · · ·

(2.24)so that F satisfy the scaling property

F (αϕ, αϕx, αϕxx, . . . ) = αpF (ϕ,ϕx, ϕxx, . . . ). (2.25)

In (2.23) ϕ ≡ ϕ(x, τ) is of course the dependent variable, while x and τ

are the independent variables.Now we use the change of dependent variables (2.1) with (2.2), (2.3)

(2.4) (but with µ = 0, see (2.5)), namely

w(x, t) = exp(iλωt)ϕ(x, τ) (2.26)

with

τ [exp(IωT )− 1]/(Iω), (2.27)

16

and we note that it entails (see (2.3))

wt = iλωw + exp[i(λ+ 1)ωt]ϕτ , (2.28)

hence, via (2.23), (2.25) and (2.26),

wt − iλωw = expi[λ(1− p) + 1]ωtF (w,Wx, wxx, . . . ). (2.29)

We now set

λ = 1/(p− 1), (2.30)

Ω = λω = ω.(p− 1), (2.31)

and via (2.24) we get (2.8), and we thereby justify the assertions madeabove (after (2.9)) about the evolution PDE (2.8).

The proof of (2.9) is analogous: by differentiating (2.28) we get

wtt = iλωwt + i(λ+ 1)ω exp[i(λ+ 1)ωt]ϕτ + exp[i(λ+ 2)ωt]ϕtt, (2.32)

and via (2.28) this yields

wtt = iλωwt + i(λ+ 1)ω[wt − iλωw] + exp[i(λ+ 2)ωt]ϕtt. (2.33)

We now assume that ϕ ≡ ϕ(x, τ) satisfy the second-order evolution PDE

ϕtt = F (ϕ,ϕx, ϕtt, . . . ) (2.34)

rather than the first-order evolution PDE (4.1), with F (ϕ,ϕx, ϕxx, . . . )defined as above, see (2.24), and therefore satisfying the scaling property(2.25). Hence we get

wtt−i(2λ+1)ωwt−λ(λ+1)ω2 = expi[λ(1−p)+2]ωtF (w,wx, wxx, . . . ).(2.35)

We now set

λ2/(p− 1), (2.36)

Ω = λω = 2ω/(p− 1), (2.37)

and via (2.24) we get (2.9) and we thereby justify the assertions madeabove (after (2.9)) about the evolution PDE (2.9).

The derivations of (2.10) and (2.11) are analogous, except that wenow use the change of variables (2.1) with nonvanishing µ, see (2.5).The relevant relations implied by this change of variables read then asfollows:

wt = iλωw + iµω exp(iµωt)xϕξ + exp[i(λ+ 1)ωt]ϕt, (2.38)


as well as

wx = exp(iµωt)ϕξ (2.39)

wxx = exp(2iµωt)ϕξξ (2.40)

and so on, hence

wt − iλωw − iµωxwx = exp[i(λ+ 1)ωt]ϕτ (2.41)

We now assume that ϕ ≡ ϕ(ξ, τ) satisfies the evolution PDE

ϕτ = F (p)(ϕ,ϕx, ϕxx, . . . ) + F (q)(ϕ,ϕx, ϕxx, . . . )

+∑j=0

F r(j)(ϕ,ϕx, ϕxx, . . . ) (2.42)

with

F (p)(ϕ,ϕξ, ϕξξ), . . . )

=p∑

p0,p1,p2,...=0∑n=0

pn=p,∑

n=1npn=P

ap0p1p2···(ϕ)p0(ϕξ)p1(ϕξξ)p2··· (2.43)

and F (q) respectively F (r(j)) defined by analogous formulas except forthe systematic replacement of the letters p, P and a by q,Q and b respec-tively by r(j), R(j) and c(j) (preserving of course the integer subscriptswherever they appear). Note that (2.44) and its analogs entail the scal-ing properties

F (p)(αϕ, αβϕξ, αβ2ϕξξ) = αpβPF (ϕ,ϕξ, ϕξ,ξ, . . . ), (2.44)

F (q)(αϕ, αβϕξ, αβ2ϕξξ) = αqβQF (ϕ,ϕξ, ϕξ,ξ, . . . ), (2.45)

F (r(j))(αϕ, αβϕξ, αβ2ϕξξ) = αr(j)

βR(j)F (ϕ,ϕξ, ϕξ,ξ, . . . ).(2.46)

Hence, from (2.41), (2.42), (2.1), (2.39), (2.40), (2.41), (2.44), (2.45),(2.46), (2.44) and the analogous equations to (2.44), we get (2.10), pro-vided there hold the following relations:

λ(p− 1) + µP = 1, (2.47)

λ(q − 1) + µQ = 1, (2.48)

λ(r(j) − 1) + µR(j) = 1. (2.49)

It is then easily seen that, provided the condition (2.12) holds, the twoequations (2.48), (2.49) yield (2.15), (2.16); and by then inserting these

18

expressions, (2.15), (2.16), of λ and µ in (2.49) one obtains (2.14). Thederivation of (2.10) is thereby completed, and via this derivation thestatements made above (after (2.11)) about the evolution PDEs (2.10)are validated as well.

The proof of (2.11) is entirely analogous, except for the replacement ofthe first-order evolution PDE (2.34) that served as starting point for thederivation of (2.10), (2.11) with the second-order evolution PDE thatobtains from (2.34) by replacing, in its left-hand side, ϕ, with ϕττ . Thedetailed treatment is left as an easy exercise for the diligent reader.

Let us end this section by displaying, for some of the above evolu-tion PDEs, certain simple solutions which indeed feature the periodicityproperties mentioned above (but are of course not the only ones to pos-sess this property).

Consider the evolution PDE

wt − iΩw =p∑

p0,p1,p2,...=0∑n=0

pn=p,∑

n=1npn=p

ap0p1p2···(w)p0(wx)p1(wxx)p2 · · · (2.50)

which is clearly a subcase of (2.8) (due to the additional restriction∑n=1

npn = P on the summations indices pn). Here p is an arbitrary

positive integer larger than unity, p > 1, P is an arbitrary nonnegativeinteger different from unity, P = 1, the constants ap0,p1,p2 are arbitrary(possible complex), and Ω > 0.

It is then easily seen that this evolution PDE, (2.50), possesses thesolution

w(x, t) = exp(iΩt)Ax+B exp[i(p− 1)Ωt] + CP , (2.51)

β = (P − 1)/(p− 1), (2.52)

B = −i[(P − 1)Ω]−1AP ×p∑

p0,p1,p2,...=0∑n=0

pn=p∑n=1

npn=P

ap0p1p2···∏n=1

[β(β − 1) · · · (β − n)]p, (2.53)

with A and C arbitrary (complex) constants. This solution is periodicin t with period 2π/Ω provided x = x±, while it becomes singular at the


real times t = t± mod 2π/[(p− 1)Ω] for x = x±, with

x± = −Re(AC∗)±[

Re(AC∗)]2 + |AB|2]1/2

(2.54)

exp[i(p− 1)Ωt±

]= −(Ax± + C)/B. (2.55)

Here we are of course assuming that B, see (2.53), does not vanish,B = 0.

Likewise, consider the evolution PDE

wtt − i[(p+ 3)/2

]Ωwt −

[(p+ 1)/2

]Ω2w

=p∑

p0,p1,p2,...=0∑n=0

pn=p,∑

n=1npn=P

ap0p1p2···(w)p0(wx)p1(wxx)p2 · · · (2.56)

which is clearly a subcase of (2.9) (due to the additional restriction∑n=1

npn = P on the summations indices pn). Here p is again an ar-

bitrary positive integer different from unity, p > 1, P is an arbitrarynonnegative integer different from 2, P = 2, the constants ap0,p1p2··· arearbitrary (possibly complex), and Ω > 0.

It is then easily seen that this evolution PDE, (2.56), also possesses asolution analogous to (2.54), namely

w(x, t) = exp(iΩt)Ax+B exp

[i(p− 1)Ωt

]+ C

β, (2.57)

β = (P − 2)/(p− 1), (2.58)

B =± 2i[(P − 2)(P − p− 1)Ω

]−1Ap/2

× p∑

p0,p1,p2,...=0∑n=0

pn=p,∑

n=1npn=P

ap0p1p2···∏n=1

[β(β − 1) · · · (β − n)]p1/2

,

(2.59)

with A and C again arbitrary (complex) constants. This solution has ofcourse the same periodicity properties described above, see after (2.54).

It is clearly easy to identify subclasses of (2.10) and (2.11) that possesssolutions analogous to (2.54), (2.57), namely

w(x, t) = exp(iλwtAx exp(iµωt) +B exp(iωt) + C

β. (2.60)

We leave this task, and the discussion of the properties of such solutions,as an exercise for the diligent reader.

20

References[1] F. Calogero, A class of integrable hamiltonian systems whose solutions are

(perhaps) all completely periodic, J. Math. Phys. 38 (1997), 5711-5719.[2] F. Calogero and J.-P. Francoise, Solution of certain integrable dynamical

systems of Ruijsenaars–Schneider type with completely periodictrajectories, Ann. Henri Poincare 1 (2000), 173-191.

[3] F. Calogero, Classical many-body problems amenable to exact treatments,Lecture Notes in Physics Monograph m 66, Springer, 2001.

[4] F. Calogero and J.-P. Francoise, Periodic solutions of a many-rotatorproblem in the plane, Inverse Problems, (in press).

[5] F. Calogero and J.-P. Francoise, Periodic motions galore: how to modifynonlinear evolution equations so that they feature a lot of periodicsolutions, (in preparation).

[6] F. Calogero, The evolution PDE ut = uxxx + 3(uxxu2+3ux2u) + 3uxu4,J. Math. Phys. 28 (1987), 538-555.

[7] F. Calogero, Why are certain nonlinear PDEs both widely applicable andintegrable?, in: What is integrability? (V. E. Zakharov, editor),Springer, 1990, pp.1-62.

3

Geometry and integrabilityRon Y. Donagi

University of Pennsylvania

3.1 Introduction

These lectures are centered around the following result and its variousspecial cases, applications, and extensions:

Theorem. There is an algebraically integrable system on the modulispace of meromorphic Higgs bundles on a curve.

This was proved independently by Markman [M] and Bottacin [Bo],and is closely related to results of Mukai [Mu] and Tyurin [T]. It in-corporates and generalizes earlier work of Hitchin [H] and many others.The theorem combines ideas from algebraic geometry and symplectic ge-ometry. In keeping with the expository aim of the lectures, the bulk ofthese notes concerns not the theorem and its applications, but the manyingredients which go into its proof. It is my hope that students with afairly modest background in geometry will be able to work through thesenotes, learning a fair amount of algebraic geometry and symplectic ge-ometry along the way. They may also be motivated to follow some of theleads in the last section towards open problems and further developmentof the subject.

The symplectic geometry needed for the statement and proof of thetheorem is covered in Sections 3.2, 3.3, and 3.7, while the algebraic ge-ometry is in Sections 3.4, 3.5, 3.6. Section 3.2 introduces the basics ofsymplectic and Poisson manifolds, while Section 3.3 discusses integrablesystems. The notions of moment map and symplectic reduction, whichare used in the proof, are explained in Section 3.7. The main algebraicgeometry input is the study, in Section 3.5, of various aspects of the mod-uli of vector bundles and related objects such as principal G-bundles andHiggs bundles. Since vector bundles exhibit quite a range of complicatedbehavior, I preceded this section with a review of the much simpler story

21

22

for line bundles, in Section 3.4. Finally, Section 3.6 contains the specificsabout spectral and cameral covers which are needed for exhibiting theCasimirs and Hamiltonians of the integrable system of the theorem.

In some ways, these notes are an elementary introduction to the morecomplete earlier version [DM]. But I have also taken this opportunityto update some of the results of [DM] and to point out their recentvariations and applications. This is done mostly in Section 3.8. Manyexamples and special cases of the theorem are discussed there, togetherwith various applications in mathematics and physics, further develop-ments and some open problems. However, since these notes were gettingto be too long, this discussion is not as leisurely as most of the rest ofthe text. Instead, only the main points of each special case, application,or open problem are explained, and the interested reader is referred tothe literature for more details.

It is a pleasure to thank Yavuz Nutku and the Gursey Institute for theinvitation to deliver these lectures and to write these notes. I would alsolike to acknowledge partial support from NSF Grant # DMS-9802456.

3.2 Symplectic geometry

3.2.1 Symplectic manifolds

A Symplectic manifold (M,ω) consists of:M : a C∞ manifold,ω ∈ A2(M) := Γ(M,Λ2T ∗(M): a closed, nondegenerate 2-form on

M .

Note: any 2-form ω determines an interior (or: contraction) mapiω : TM → T ∗M . We say ω is nondegenerate if iω is an isomorphism.This is possible only when dimM = 2n is even. Darboux’s theoremguarantees the existence of local coordinates pi, q

i(i = 1, ..., n) on M

such that locally

ω = Σni=1dpi ∧ dqi.

A Holomorphically symplectic manifold (M,ω) consists of:M : a complex (analytic) manifoldω: a closed, non-degenerate holomorphic two form on M .

Note: saying that ω is holomorphic means that in terms of local holo-morphic coordinates zi it can be written as ω = Σfij(z)dzi ∧ dzj

Geometry and integrability 23

(fij is holomorphic and there are no dzi.) The holomorphic version ofDarboux’s theorem says that for a holomorphically symplectic ω thereare local holomorphic coordinates pi, qi such that ω = Σn

i=1dpi∧dqi. Thecomplex dimension of X must be even, the real dimension is divisibleby 4.

3.2.2 Examples

There are two basic examples:

(i) For any C∞ manifold X, the cotangent bundle M = T ∗X issymplectic. A choice of local coordinates qi, i = 1, . . . , n on X

determines pullback functions, still denoted qi, on M , as wellas fiber coordinates pi along the cotangent spaces. The locallydefined two form ω = Σdpi ∧ dqi is easily seen to be independentof the choice of local coordinates, so it is a (global) symplecticform on M . In fact the 1-form α := Σpidqi is already globallydefined, so ω = dα is actually exact.

If we start instead with a complex manifold X, the same con-struction produces a holomorphically symplectic manifold (M,ω)where M := T ∗X is the cotangent bundle and ω is as above.

(ii) Let G be a Lie group and g its Lie algebra. The action of G onitself, by conjugation, induces the adjoint action of G on g andthe coadjoint action on the dual vector space g∗:

Ad : G× g→ g, Ad∗ : G× g∗ → g∗.

For each ξ ∈ g∗ consider its orbit O = Oξ := G · ξ ⊂ g∗ underthe coadjoint action. There is a natural symplectic form ω on O(discovered by Kirillov and Kostant), making O into a symplecticmanifold for real G and a holomorphically symplectic manifoldfor complex G.

Explicitly, we use the map G → O sending g → gξ in orderto identify the tangent space TξO to O at each ξ ∈ O with thequotient g/gξ, where Gξ := g ∈ G| g ξ = ξ is the stabilizerof ξ in G, and gξ = X ∈ g| (ξ, [X,Y ]) = 0 ,∀Y ∈ g is the Liealgebra of Gξ, a subalgebra of g. With this identification, letX, Y ∈ TξO ≈ g/G∗ be the images of X,Y ∈ g. The symplecticform is then defined by

ω : X, Y → (ξ, [X,Y ]),

24

which of course is independent of the representatives X,Y used.Note: when G is semisimple, the Killing form gives an isomor-

phism of g→g∗ which is G-equivariant. So coadjoint orbits canbe naturally identified with adjoint orbits, in g.

Example. The Lie algebra of G = SO(3, R) is

g = so(3, R) ≈ R3 ≈

0 a c

−a 0 b

−c −b 0

∣∣∣ a, b, c ∈ R

.

Here co-adjoint orbit = adjoint orbit = sphere a2+b2+c2 = r2in R3.

3.2.3 Poisson manifolds

A Poisson manifold (M,ψ) consists of a manifold M with a 2-vectorψ ∈ Γ(Λ2TM) such that the (Poisson) bracket

f, g ∈ C∞(M) → f, g := (df ∧ dg, ψ)

is a Lie algebra bracket on C∞(M), i.e it satisfies the Jacobi identity

f, g, h+ g, h, f+ h, f, g = 0.

In this case the map

v : C∞(M)→ V F (M) := Γ(M,TM),

f → (df, ψ) = idfψ

is automatically a homomorphism of Lie algebras:

v(f, g) = [v(f), v(g)].

The vector field v(f) is called the Hamiltonian vector field of the functionf , while f is referred to as the Hamiltonian function of v = v(f).

Examples

• A symplectic manifold (M,ω) is Poisson: the 2-vector ψ is deter-mined by the isomorphism iψ : T ∗M → TM which is the inverse ofiω : TM→T ∗M . The Jacobi identity for ψ turns out to be equivalentto the closedness of ω. So a Poisson manifold is symplectic if and onlyif ψ is non-degenerate.


• The dual g∗ of a Lie algebra g is a Poisson manifold. For F,G ∈C∞(g∗) and ξ ∈ G∗, the bracket is defined by

F,Gξ :=(ξ, [dξF, dξG]

).

Note that in this case the 2-vector ψ is degenerate. Its restriction toeach coadjoint orbit O ⊂ g∗ is non-degenerate and corresponds pre-cisely to the standard symplectic form on O described earlier. In fact,the coadjoint orbits are the largest loci on which ψ is non-degenerate:the conormal space to the orbit Oξ at ξ is precisely the nullspace ofψ at ξ.

3.2.4 Symplectic leaves

A general result of Weinstein asserts that any Poisson manifold (M,ψ) isthe disjoint union of the submanifolds ofM on which ψ is non-degenerateand which are maximal with respect to this property. These subman-ifolds inherit a symplectic structure and they are called the symplecticleaves of (M,ψ). Their conormal space at each point is the nullspace ofψ there. The symplectic leaves of g∗ with the Kirillov–Kostant Poissonstructure are, of course, the coadjoint orbits. For so(3, R) the picture isvery simple: the symplectic leaves are the spheres centered at the origin,as well as the origin itself. Another extreme is the case where ψ is alge-braic and non-degenerate somewhere: there is then an open symplectic“leaf”, and possibly others, of lower dimension, in its closure. In general,the rank of the alternating 2-vector ψ is not constant but only semicon-tinuous, i.e. its value at a point is less than or equal to its values atnearby points; this rank equals the dimension of the symplectic leaves,which can be lower for a special leaf than for nearby ones.

3.3 Integrable systems

3.3.1 Definitions

The notion of a Hamiltonian map gives a coordinate-free way to dis-cuss a collection of commuting Hamiltonians. A map H : (M,ψ) → B

from a Poisson manifold (M,ψ) to another manifold B is called Hamil-tonian if for every two functions f, g ∈ C∞(B), the pullback functionsH∗f := f H and H∗g := g H Poisson commute:

H∗f,H∗g = 0.

26

A Casimir on the Poisson manifold (M,ψ) is a function f ∈ C∞M

whose Hamiltonian vector field vanishes: v(f) = 0. Equivalently, f

Poisson-commutes with every function on M . More generally, a mapM → C is called Casimir if the pullback of any function on C is aCasimir on (M,ψ).

An integrable system is a Hamiltonian map of maximal rank. WhenM is symplectic of dimension 2n, this amounts to a Hamiltonian mapH : M → B which is onto an n-dimensional base B. More generally, let(M,ψ) be Poisson with ψ of constant rank 2n, where dim(M) = 2n+ c.Then we want B = H(M) to have dimension c + n. Locally, then, Hcan be expressed in terms of n + c independent functions on M . Ofthese, c will be Casimirs and the remaining n Hamiltonians need toPoisson-commute.

3.3.2 Liouville’s Theorem

Let H : (M,ψ)→ B be an integrable system where

dim(M) = 2n+ c,

rank(ψ) = 2n,

dimB = n+ c,

H is a proper, submersive, Hamiltonian map.

Then:

• The connected components of the fibers of H are tori, i.e. they arediffeomorphic to (S1)n = Rn/Zn.

• The Hamiltonian vector fields v(H∗f) for f ∈ C∞(B) are translationinvariant vector fields on these tori, so the corresponding flows arelinear.

• The symplectic foliation of M is (locally) pulled back from a foliationof B.

Note: saying that H is submersive means that its differential

dH : TmM → TbB, b := H(m)

is surjective for all m ∈ M . This guarantees that the fibers H−1(b) are


non-singular manifolds. In practice, an integrable system is often sub-mersive only over some dense open subset of the base B, and Liouville’stheorem applies to the fibers over this subset.

3.3.3 Algebraically integrable systems

The most natural way to complexify the notion of an integrable sys-tem is to turn Liouville’s theorem into a definition. Thus an ana-lytically integrable system consists of a complex analytic manifold M ,an analytic Poisson structure ψ, (i.e. a holomorphic two-vector onM satisfying the Jacobi identity), a proper, holomorphic Hamiltonianmap H : (M,ψ)→B whose generic fibers are complex tori Cn/Λ, whereΛ ≈ Z2n is a maximal lattice in Cn. An algebraically integrable systeminvolves data (M,ψ,H,B, ...) which are algebraic. In particular, thecomplex tori now become abelian varieties. This restricts the lattices Λwhich may arise: they must satisfy Riemann’s first and second bilinearrelations, cf. [GH]. Note that we do not require H to be submersive.Accordingly, we expect the generic fiber (i.e. the fiber over b in a Zariskiopen subset of B) to be a complex torus or abelian variety, but we allowsome of the fibers to degenerate. Once a system is shown to be algebraic,it can be ‘solved’ explicitly: the solutions are flows which are tangent toan abelian variety fiber. On the universal cover Cn this flow is linear; onthe abelian variety itself it can therefore be expressed in terms of thetafunctions.

3.4 Line bundles

Before discussing moduli spaces of vector bundles and more compli-cated objects, it may make sense to review the much simpler case of linebundles. In this section we work over a fixed (compact, non-singular)Riemann surface X. Since we are switching here from symplectic toalgebraic language, we think of X as a 1-dimensional algebraic varietyover C, or a “curve” for short. In fact much of what we say will workover a (projective, non-singular) complex variety X of any dimension.

A vector bundle on X is another variety V , together with a mapπ : V → X which is locally isomorphic to the product X × Cn (withthe projection map): every point x ∈ X has a neighborhood U and atrivialization π−1(U)→U × Cn which commutes with projection to U(i.e. it sends fibers to fibers), and on the intersection U1 ∩ U2 of twosuch neighborhoods, the difference between these trivializations is linearin the fibers and algebraic (or holomorphic) along the base, i.e it is

28

given by an n× n transition matrix g(x) whose entries are algebraic (orholomorphic) functions of x ∈ U1 ∩ U2. The vector bundles we definedhere are algebraic (or holomorphic; it turns out not to matter). There isan analogous, and more familiar, notion of a C∞ complex vector bundlewhere the matrices g(x) are allowed to be complex-valued C∞ functionsof x.

3.4.1 Pic and Jac

A line bundle is a vector bundle of rank n = 1. The moduli space ofline bundles is usually called the Picard variety, and denoted Pic(X).It is an algebraic variety whose points are in one to one correspondencewith the isomorphism classes of line bundles on X. It is also a group,the operation being the tensor product of line bundles. As a variety,Pic(X) is disconnected. The connected component of the trivial bun-dle is called the Jacobian, and is denoted by Jac(X). The map whichsends a line bundle to its first Chern class, or degree, is continuous (i.e.constant on connected components) and is a surjecive group homomor-phism Pic(X) → Z. Its kernel is Jac(X). Algebraic line bundles havemore structure than C∞ bundles: two algebraic line bundles are iso-morphic as C∞ bundles if and only if their Chern classes agree. TheJacobian therefore parametrizes all possible algebraic (or holomorphic)structures on the trivial C∞ bundle. An analogous description still holdsfor the Picard variety parametrizing line bundles on higher dimensional(non-singular, projective) X, the main difference being that the group ofconnected components of Pic(X) (or the group of isomorphism classesof the underlying C∞ line bundles) can be bigger that Z.

3.4.2 Cohomological description

The description of vector bundles in terms of their transition matricescan be refined to show that an isomorphism class of vector bundles isuniquely determined by a cohomology class in H1(X,GLn(OX)). Forline bundles, the sheaf GL1(OX) = O∗

X is just the sheaf of nowherevanishing holomorphic (or algebraic) functions on X, so we can identifyPic(X) with H1(X,O∗

X). This can be described via the exponentialsequence of sheaves on X:

0→ Z→ OXexp−→ O∗

X → 1.


Indeed, the corresponding long exact sequence of cohomology gives

0→ H1(X,OX)/H1(X,Z)→ H1(X,O∗X) c1−→ H2(X,Z)→ 0

which can be identified with the previous sequence

0→ Jac(X)→ Pic(X) c1−→ Z→ 0.

(For higher dimensional X, the group H2(X,Z) can be bigger than Zand the image of the Chern class map c1 may be a proper subgroup: itis the kernel of H2(X,Z)→ H2(X,OX).)

Let X be a (smooth, compact) curve of genus g. Then H1(X,Z) isa free abelian group of rank 2g and H1(X,C) is its complexification,a 2g-dimensional complex vector space. The Hodge theorem says thatH1(X,C) decomposes as the sum of two g-dimensional complex sub-spaces, H1,0 ⊕H0,1, where

H1,0 = H0(X,KX)

H0,1 = H1(X,OX),

namely, the spaces of holomorphic and anti-holomorphic 1-forms on X.(We use KX as another notation for the canonical bundle of X, KX =T ∗X , and we denote the trivial bundle by OX .) The map H1(X,Z) →

H1(X,OX) is the composition of the inclusion H1(X,Z) → H1(X,C)with the projection H1(X,C) → H0,1 = H1(X,OX). Since complexconjugation is an automorphism of H1(X,C) which fixes H1(X,Z) butinterchanges H1,0 with its orthogonal complement H0,1, it follows thatthe image of H1(X,Z) in H1(X,C) does not intersect H1,0. The com-position H1(X,Z) → H1(X,OX) is therefore injective, i.e. H1(X,Z)is a maximal lattice in the g-dimensional vector space H1(X,OX). Weconclude that Jac(X) = H1(X,OX)/H1(X,Z) is a g-dimensional com-plex torus. (In fact, it is an abelian variety.) The same is true for theith component Pici(X) of Pic(X), i.e. for the component mapping byc1 to i ∈ Z: it is (non-canonically) isomorphic to Pic0(X) = Jac(X).Such an isomorphism is given by tensoring with any fixed line bundle ofdegree i.

3.4.3 Flat bundles

A flat bundle is given by transition matrices g(x) which are constant, i.e.independent of x. Such a bundle admits a natural flat connection: theflat sections are those which are locally constant in terms of any of the

30

local trivializations, and this condition does not depend on the trivializa-tion used since any two trivializations differ by a constant matrix. Con-versely, specifying a bundle with a flat connection uniquely determinesa flat bundle. Cohomologically, the moduli space of flat bundles on X isgiven by H1(X,C∗). It is the product of 2g copies of C∗, when X is acurve of genus g. Note that sections of the sheaf C∗ are locally constant,non-zero functions, compared to sections of O∗

X which are holomorphicor algebraic non-zero functions. The inclusion i : C∗ → O∗

X induces thehomomorphism i∗ : H1(X,C∗) → H1(X,O∗

X) = Pic(X) which sends aflat bundle to the bundle with the same algebraic transition matrices,forgetting that they actually happen to be constant.

Topologically, this map can be described as follows. Its image isPic0(X) = Jac(X) ≈ (S1)2g, and the map (C∗)2g ≈ H1(X,C∗) →Jac(X) ≈ (S1)2g is the product of 2g copies of the argument map

C∗ → S1

z → z/|z|.

Thus, a line bundle L ∈ Pic(X) admits a flat structure if and only if itsdegree is 0, and in this case the family of flat structures on L, or moresimply the family of flat connections on L, is an affine space modelledon H0(X,KX) = H1,0 ≈ Cg ≈ R2g.

The general flat bundle has structure group (or holonomy group) C∗.There is the special class of unitary flat bundles, for which the locallyconstant transition matrices take values in the unitary group U(1) ≈ S1.The moduli space of these unitary flat bundles is given by

H1(X,U(1)) ≈ (U(1))2g ≈ (S1)2g.

It is thus diffeomorphic to the Jacobian. The theorem of Narasimhan-Seshadri says (in this case) that any degree 0 holomorphic line bundleon X admits a unique flat unitary connection, i.e. the restriction of i∗to the moduli space of unitary flat bundles is an isomorphism to theJacobian.

3.4.4 Abel–Jacobi

Any point p of the curve X determines a line bundle Lp as follows.We cover X by two open sets: a small disc Up containing p, and thecomplement X − p. Let z = z(x) be a local coordinate for x ∈ Up,vanishing at p. The line bundle Lp is trivialized on each of the open


sets; on their intersection the 1× 1 transition “matrix” g is taken to bethe coordinate z. It is straightforward to check that this Lp dependsonly on p ∈ X and not on the coordinate z. The fact that z vanishes ata single point implies that degree(Lp) = 1. We get this way a map

AJ : X → Pic1(X)

p → Lp

called the Abel-Jacobi map. It turns out to be an algebraic map, andmuch can be learned about the geometry of a curve from the behaviorof its Abel-Jacobi map. But we will not pursue this here except for thefollowing remarks:

• The choice of a base point p0 ∈ X allows us to view AJ as goingto Jac(X) = Pic0(X) instead of Pic1(X): simply replace Lp byLp ⊗ (Lp0)

−1, where (Lp0)−1 is the inverse or dual line bundle of

Lp0 , obtained using the same open cover and the inverse transitionfunction g−1(x).

• When g = 0 we have Pic(X) ≈ Z, Jac(X) is a point and the Abel-Jacobi map is a constant. When g = 1, the Abel-Jacobi map is anisomorphism of X with Jac(X) (or, more naturally, with Pic1(X)).For g ≥ 2 it is injective, but of course not surjective: Only some specialline bundles can be described algebraically via a cover involving onlytwo open sets as above.

3.5 Vector bundles

3.5.1 Complications

The set of isomorphism classes of rank n vector bundles on a curve X

is given by the cohomology group H1(X,GLn(OX)). Algebraically thisis more complicated than the case of line bundles because GLn(OX) isnon-abelian for n > 1. But the main cause of complications in the theoryis geometric: it is the existence of the jump phenomenon. There existvector bundles V on the product X ×∆ of X with a parameter space ∆(which can be taken to be “a small disc”) and two points 0, 1 ∈ ∆ suchthat the restriction Vδ of V to X × δ, for δ ∈ ∆, is isomorphic to thevector bundle V1 for δ = 0, but V0 is not isomorphic to V1. If V0 andV1 represent points of a moduli space of vector bundles, this examplepresents us with an (algebraic) map v from the parameter space ∆ tothe moduli space, which jumps: v(∆ − 0) is one point, while v(0) is

32

another point. In other words, the topology of the moduli space wouldbe non-separated. (We will see an explicit example of such a jump atthe end of this section.)

This forces us to accept some compromise. We could disallow somebundles and thus settle for a moduli space which parametrizes only somesubset of all bundles, and thereby avoids the jump phenomenon. Or, wecould allow certain non-isomorphic vector bundles to be represented bythe same point of the moduli space: instead of excluding either V0 orV1, we allow both but declare them equivalent. A third possibility is toaccept non-separatedness of the moduli space and to develop a languagefor studying it. All three approaches can be carried out: they lead tothe moduli space of stable bundles, the moduli space of S-equivalenceclasses of semi-stable bundles, and the moduli stack of all bundles. Wewill describe some of their features below.

3.5.2 Stability

Just as for line bundles, a vector bundle has a well-defined degree orfirst Chern class. (One way to define this is to set c1(V ) := c1(detV )where detV is the line bundle on X whose transition functions are thedeterminants of the transition matrices of V .) If V has rank n anddegree d, we define its slope to be d/n: if V happens to be the directsum ⊕Li of line bundles, then µ is the average degree of the Li.

A bundle V is called stable if for every subbundle V ′ ⊂ V (other thanV itself and the zero subbundle),

µ(V ′) < µ(V ).

Similarly V is semistable if for every subbundle V ′ ⊂ V ,

µ(V ′) ≤ µ(V ).

S-equivalence is the equivalence relation on the set of isomorphismclasses of semistable bundles generated by setting V0 equivalent to V1whenever there is a jump from V1 to V0, as above. More concretely,consider a short exact sequence

0→ V ′ → V1 → V ′′ → 0

where V ′, V ′′ are semistable bundles of the same slope. With V ′, V ′′

fixed, the extension is specified by an extension class ε ∈ H1(X, (V ′′)−1⊗V ′). But since V ′, V ′′ themselves have a C∗ of scalar automorphisms, itfollows that the isomorphism class of the bundle V1 depends on ε only


up to multiplication by a non-zero scalar. By rescaling the extensionclass, we therefore obtain a jump from V1, for all non-zero values of thescalar, to V0 := V ′⊕V ′′ when the scalar becomes 0. In fact, it turns outthat any two S-equivalent bundles can be linked by a sequence of movesof this particular type.

With these definitions, we can describe two of the three types of mod-uli spaces mentioned above. There exists a projective variety M =MX(n, d) parametrizing S-equivalence classes of semistable bundles ofrank n and degree d on X. There is also a quasi-projective varietyMs =Ms

X(n, d), identifiable with an open subset ofM, and parametriz-ing isomorphism classes of stable bundles on X. The latter is “almost”smooth: its singularities are quotient singularities, corresponding to bun-dles which have non-scalar automorphisms.

3.5.3 Tangent spaces

We can describe the tangent space to Ms at a non-singular point V ∈Ms as follows. The bundle V is given in terms of some open cover Uiof X by a 1-cocycle gij, where gij is an n× n matrix of holomorphicfunctions on Ui ∩ Uj . A tangent vector to Ms at V is determined bymapping a small disc ∆ ⊂ C to Ms so that 0 ∈ ∆ goes to V . This isachieved by deforming the transition matrices:

gεij(x) = gij(x) + εg′ij(x) + · · · , ε ∈ ∆.

The tangent vector itself is then encoded in the 1-cocycle g′ij(x) givingthe leading term of the deformation. The multiplicative cocycle condi-tion for the bundle V :

gij · gjk = gik

then translates into an additive cocycle condition for hij := g−1ij · g′ij:

g−1jk · hij · gjk + hjk = hik.

The conjugation by gjk means that the class of the deformation doesnot live in H1(X, gln(O)); rather, it lives in H1(X, adV ). Here

adV = V ∗ ⊗ V = gl(V )

is the bundle of endomorphisms of V . Its fiber at each point x ∈ X isisomorphic to the Lie algebra gln, but the isomorphism is not naturaland cannot be chosen globally over X. Anyway, the conclusion is that

34

there is a natural identification

TVMs ≈ H1(X, ad(V )),

or by Serre duality and the self-duality of ad(V ) (via the Killing form):

T ∗VMs ≈ H0(X, ad(V )⊗KX).

Note that in the abelian case n = 1, and only then, ad(V ) is trivial:

ad(V ) = V ⊗ V ∗ ≈ OX for n = 1.

The result for vector bundles therefore specializes to:

TLPic(X) ≈ H1(X,OX),

which also follows trivially from the exponential sequence of Section3.4.2.

The Riemann–Roch formula for a vector bundle W on a curve X ofgenus g says that:

χ(W ) := h0(X,W )− h1(X,W ) = degree(W )− (g − 1) · rank(W ).

We apply this to W := ad(V ). This has rank n2 and degree 0, so wefind

χ(ad(V )) = n2(1− g).

Now the only endomorphisms of a stable bundle V are scalar, thuswe have h0(ad(V )) = 1. This determines the dimension of TVMs ≈H1(X, ad(V )) and hence also of Ms:

Corollary

dimMsX(n, d) = (g − 1)n2 + 1.

3.5.4 The moduli stack

We have considered the moduli space of stable bundles, in which somebundles have to be excluded for not being stable, as well as the modulispace of semistable bundles, in which fewer bundles are excluded, butsome of these have to be identified with each other. The third approachis to insist on including all bundles. The resulting structure is no longerthat of an algebraic variety or even a scheme, but an algebraic stack M =MX(n, d). We cannot discuss these here, so we only point out one oftheir features: each point [V ] of the stack (representing the isomorphism


class of some vector bundle V ) can be assigned a dimension, in such away that whenever V1 jumps to V0 we have

dim[V1] > dim[V0].

In our case this assignment is straightforward: dim[V ] is defined to be

dim[V ] := −dim (End(V )),

where End(V ) is the vector space of global endomorphisms of V , i.e.global sections of ad(V ). In general, the main difference between astack and a scheme is that points of the stack “remember” the presenceof some automorphisms. Locally, an algebraic stack can be describedby an “equivalence relation” which may involve a continuous family ofidentifications of a family with itself.

The stack itself can be assigned a dimension. If it contains a Zariskiopen subset U which looks like an N -dimensional variety except thatall its points have stack dimension = −r, then the dimension of thestack is N − r. For example, the moduli stack of vector bundles onX has dimension (g − 1)n2, since stable bundles have a 1-dimensionalfamily of automorphisms. This notion behaves well in fibrations: givena morphism of stacks π : M → M whose fibers have dimension N , wehave

dim(M) = N + dim(M).

This is useful when M is the moduli stack of some objects V (e.g. bun-dles) while M is the moduli space of some enriched objects (V, δ) con-sisting of an object V plus some additional structure δ on V which isnot preserved by any automorphism of V , so the pair (V, δ) has no au-tomorphisms. Then dim(M), as a variety, equals dim(M), as a stack,plus the number of parameters required to specify δ.

3.5.5 Examples

First we consider vector bundles on X = P1. It is well known that everyvector bundle on P1 is a direct sum of line bundles. Further, we haveseen that Jac(P1) = (0) and Pic(P1) = Z. The line bundle of degree d

on P1 is denoted OP1(d), or O(d) for short. We can therefore enumerateall vector bundles on P1. For example, when n = 2 and the degree isd = 0, the possibilities are:

O ⊕O, O(1)⊕O(−1), O(2)⊕O(−2), . . . .

36

We see immediately that none of these is stable, O⊕O is semistable, andthe others are unstable because they contain a line bundle of positiveslope (=degree). Therefore:

MsP1(2, 0) = ∅, MP1(2, 0) = (point).

The moduli stack MP1(2, 0) contains one point xk for each k ≥ 0, rep-resented by the bundle O(k)⊕O(−k). An endomorphism of this bundle

consists of a matrix(

a b

c d

)where a, d are sections of OP1 ( i.e. com-

plex numbers), b is a section of OP1(2k) and c is a section of OP1(−2k),hence c ≡ 0 unless k = 0. We find that:

dim(xk) =

−4 k = 0−(2k + 3) k ≥ 1.

So the semistable point O ⊕O has the largest dimension (namely, −4),and all others are smaller. In fact, the other points are in its closure.A similar picture holds for any even degree d. For odd d, MP1(2, d) isempty while the stack MP1(2, d) still involves an infinite nested sequenceof points O(k)⊕O(d− k).

Let V be a vector bundle of rank n and degree d on a curve X whichis a branched cover π : X → Y of another curve Y , and let l be thedegree of the cover π. There is a natural way to construct a direct imagebundle π∗V on Y . This is defined in such a way that for any open setU ⊂ Y , sections of π∗V on U correspond precisely to sections of V onπ−1(U) ⊂ X. Thus if y ∈ Y is a regular value (= not a branch point) ofπ and π−1(y) = x1, ..., xl, then

(π∗V )y = ⊕li=1Vxi

.

There are three relations among the invariants of V and π∗V :

rank(π∗V ) = l · rank(V ) = l · n

h0(π∗V ) = h0(V )

d− l · n · (g − 1) = d− n · (g − 1).

In the last formula, g and g are the genera of Y and X respectively, andd, d are the degrees of V, π∗V . (This formula can best be rememberedas stating that the holomorphic Euler characteristic is preserved underdirect image.)

For example, if X ≈ Y ≈ P1 and π : P1 → P1 is the standard doublecover z → w := z2, branched at 0 and ∞, the above formulas imply


that π∗O(1) is a vector bundle on P1 of rank 2 and degree 0 with a2-dimensional space of sections. In fact π∗O is O ⊕ O and a section ofO(1) upstairs corresponds downstairs to the pair of sections representingits even and odd parts:

f(z) = f+(w) + z f−(w) → (f+(w), f−(w))

where

f+(z2) :=12(f(z) + f(−z))

f−(z2) :=12z

(f(z)− f(−z)).

A more interesting example arises when Y is P1 and X is the ellipticcurve

X : y2 = x(x− 1)(x− λ)

for some λ ∈ C − 0, 1, and π : X → Y sends (x, y) → x. ForL ∈ Pic0(X) we find that π∗L is a rank 2 vector bundle of degree −2 onP1, and h0(π∗L) = h0(L). For all L other than the trivial bundle OX ,we get h0(π∗L) = 0, so the only possibility is π∗L ≈ O(−1) ⊕ O(−1).However, for L = OX , π∗L must have a non-zero section, and we seeeasily that in fact π∗OX ≈ O⊕O(−2). This gives us an explicit exampleof the jump phenomenon: as L varies continuously in Pic2(L), its directimage π∗L varies continuously in the stack MP1(2, 0). It equals thesemistable (or: generic) point O(−1)⊕O(−1) for most L, but jumps tothe “smaller” point O ⊕O(−2) at L = OX .

In order to see an example of S-equivalence, consider rank 2 bundleson an elliptic curve X. Any pair L1, L2 ∈ Jac(X) gives the semistablebundle

L1 ⊕ L2 ∈MX(2, 0).

This gives a map from the second symmetric product of X ≈ Jac(X) toMX(2, 0), a map which turns out to be an isomorphism.

Nevertheless, there are bundles on X which are not direct sums of linebundles. The group parametrizing extensions

0→ L1 → V → L2 → 0

is

Ext1(L2, L1) ≈ H1(L∗2 ⊗ L1) ≈ H0(L2 ⊗ L∗

1)∗.

It is non-zero if and only if L1 ≈ L2. So for every L ∈ Jac(X) there

38

is a non-trivial extension V of L by itself. This V is not isomorphic toL1 ⊕ L2, but they are S-equivalent, so they occupy the same point inMX(2, 0).

Note that MX(2, 0) is 2-dimensional, while the dimension computedin Section 3.5.3 for the open subsetMs

X(2, 0) was (g− 1)n2 +1 = 1. Infact, there are no stable bundles, so the open subset is empty and thereis no contradiction. Each semistable bundle has a 2-dimensional familyof endomorphisms, so the dimension of the stackMX(2, 0) is 2− 2 = 0,in accordance with the general formula (g − 1)n2.

On the other hand, MX(2, 1) is 1-dimensional as it “should” be, andevery semistable bundle of degree 1 is automatically stable. The sit-uation for all ranks and degrees was worked out by Atiyah [At]: thedimension of MX(n, d) is the greatest common divisor (n, d) of n andd; every S-equivalence class contains a unique representative which isthe direct sum of (n, d) stable bundles, each of rank n/(n, d) and degreed/(n, d); and stable bundles exist if and only if (n, d) = 1, in which caseall semistable bundles are stable and MX(n, d) is 1-dimensional, as itshould be. In general, the dimension of MX(n, d) is (n, d) while thestack is always 0-dimensional. Finally, when the genus of X is g ≥ 2,the open subsetMs

X(n, d) is always non-empty and dense inMX(n, d),which has the predicted dimension (g − 1)n2 + 1.

3.5.6 Higgs bundles

A Higgs bundle is a pair (V, ϕ) where V is a vector bundle on X and

ϕ : V → V ⊗KX

is a 1-form valued endomorphism of V . A Higgs bundle (V, ϕ) is stable ifthe slope of every proper subbundle W which is ϕ-invariant is less thanthe slope of V ; one defines semistable Higgs bundles similarly. Modulispaces Higgs := HiggsX(n, d) and Higgs′ ⊂ Higgs exist, with proper-ties analogous to those of M and Ms. Note from our identification ofcotangent vectors toMs in Section 3.5.3 that there is a natural inclusion

T ∗Ms ⊂ Higgss.

In fact for a stable V , the set of stable Higgs bundles (V, ϕ) can beidentified with T ∗

VMs. However, the stability condition for a Higgsbundle (V, ϕ) is weaker than that for the underlying vector bundle V ,so Higgss is strictly bigger than T ∗Ms. Still, it can be checked that


Higgss is holomorphically symplectic: the symplectic form on T ∗Ms

extends to it and remains (closed and) non-degenerate.Fix an effective divisor D on X, i.e. D = Σmipi with pi ∈ X, mi ≥ 0.

A meromorphic Higgs bundle with poles on D is a pair (V, ϕ) where V

is a vector bundle of rank n and degree d on X, and

ϕ : V → V ⊗KX(D)

is an endomorphism of V taking values in meromorphic differentials onX

with pole on D. These have a moduli space HiggsD := HiggsX,D(n, d)which is a key ingredient in the Theorem in section 3.6.1.

3.5.7 Other groups

The moduli spaces we have been considering have analogues for everyreductive group G, the cases we have already seen corresponding toG = GL(n). We go very briefly through some of the main points of thegeneral case.

Instead of vector bundles, we consider principal G-bundles V on X.Given such a V, each representation

ρ : G→ GL(n)

of G associates to V a vector bundle ρ(V). The notions of stabilityand semistability can be defined directly for the principal bundles, interms of the reductions of V to various parabolic subgroups of G. Givenany representation ρ, we can also consider the (semi)stability of ρ(V).Unfortunately, the (semi)stability of ρ(V) may depend on the represen-tation ρ. Fortunately, (semi)stability of ρ(V) when ρ is in the adjointrepresentation turns out to be equivalent to the (semi)stability of theprincipal bundle. One obtains a projective moduli space M = MG

X

parametrizing S-equivalence classes of semistable bundles, and an opensubset Ms = Ms,G

X parametrizing isomorphism classes of stable G-bundles, just as in the case G = GL(n). There is also a moduli stackMG

X parametrizing all G-bundles.The tangent space to Ms,G

X at a non-singular point V is given, as inthe case of G = GL(n), by H1(X, ad(V)). For semisimple G,

dim(MGX) = (g − 1) · dim(G) (G semisimple)

since the generic G-bundle has no non-trivial automorphisms. For re-ductive groups there is a correction term equal to the dimension of the

40

center of G, accounting for the automorphisms of stable bundles:

dim(MGX) = (g − 1) · dim(G) + dim(Z(G)) (G reductive).

For the stack, the correction term drops out, so

dim(MGX ) = (g − 1) · dim(G) (G reductive).

A G-Higgs bundle is a pair (V, ϕ) where V is a principal G-bundle and

ϕ ∈ Γ(X, ad(V)⊗KX),

and a meromorphic G-Higgs bundle with poles on an effective divisor Dis a pair (V, ϕ) where now

ϕ ∈ Γ(ad(V)⊗KX(D)).

The moduli spaces HiggsGX,D,Higgss,GX,D,HiggsGX and Higgss,GX existand have the expected properties.

Finally note that, since we allow the group G to be non-semisimple,the resulting moduli spaces may be disconnected. The the componentsMG

X,d and HiggsGX,D,d are indexed by the “degree” d of the G-bundleV. This degree is a cocharacter of the center Z of G, i.e. d is anelement of the lattice Hom(C∗, Z). When G is semisimple, its centeris finite so there are no non-trivial cocharacters or components. ForG = GL(n) the cocharacter lattice is just the integers, so the “degree”is the usual degree, d, of the rank-n vector bundle V associated to theGL(n)-bundle V. In this case we retrieve the moduli spaces MX(n, d)of §5.2 and HiggsX,D(n, d) of Section 3.5.6.

3.5.8 Other constructions

As for line bundles, we can ask which rank n vector bundles over acurve X admit a flat connection with structure group either GL(n,C)or U(n). A necessary condition is that the Chern class c1 must vanish.The theorem of Narasimhan and Seshadri states that a rank n vectorbundle with c1 = 0 admits a flat connection if and only if it is polystable,i.e. the direct sum of stable bundles, each with c1 = 0. In particular,each S-equivalence class of semistable bundles contains a unique repre-sentative which admits a flat U(n) connection, soMX(n, 0) can also beconsidered as the moduli space of flat U(n) bundles on X. (This resulthas analogues, due to Donaldson, Uhlenbeck and Yau, valid over anycompact Kahler X. There are also analogues involving Gcompact flat


connections on polystable G-bundles, where G is the complexification ofa compact group Gcompact.)

The analogue of the Abel-Jacobi map involves the Hecke correspon-dences. A correspondence between two varieties is any subvariety oftheir product, considered as the graph of a

“multivalued map” between them. Given a point x ∈ X, the ith Heckecorrespondences T i ⊂M×M assigns to a vector bundle V the family ofall vector bundles V ′ such that the sheaf of sections of V ′ contains thatof V , with index i, and is contained in that of V ⊗OX(x). (This familyis parametrized by the Grassmannian of i-dimensional subspaces of thefiber Vx of V at x.) Starting with a given vector bundle V0, any othervector bundle V can be reached by a sequence of Hecke correspondences(going both “up” and “down” if necessary). We will discuss neither flatbundles nor the Hecke correspondences any further in these notes.

3.6 Algebraic geometry of Higgs bundles

3.6.1 The theorem

Fix a curve X, an effective divisor D on X and a reductive group G. Asin Section 3.5.7, we have a moduli space MG

X parametrizing G-bundlesV on X and a moduli space HiggsGX,D parametrizing meromorphic G−Higgs bundles (V, ϕ) with Higgs field ϕ ∈ Γ(X, ad(V )⊗KX(D)). As inSection 3.5.2, each of these moduli spaces comes in three flavors: spacesparametrizing stable objects or S-equivalence classes of semistable ob-jects, and a stack parametrizing all objects. We are finally ready to statethe central result:

TheoremHiggsGX,D is the total space of an algebraically integrable system.

This holds in full generality for the stack version. Some minor restric-tions (excluding curves X of low genus, or divisors D of low degree) arerequired in order to make this work for the moduli spaces of stable orsemistable Higgs bundles, due to special features such as the existence ofunexpected automorphisms, cf. Theorem (4.8) of [DM]. In these noteswe will ignore these technical complications.

In order to prove the theorem, we need to exhibit on HiggsGX,D analgebraic Poisson structure ψ and an algebraic map H : HiggsGX,D →BG

X,D which is Hamiltonian with respect to ψ and whose generic fibers,or ‘Liouville tori’, are abelian varieties. The symplectic foliation of

42

(HiggsGX,D, ψ) should be pulled back from an algebraic foliation of BGX,D.

In fact, we will find another space CGX,D and an algebraic map

BGX,D → CG

X,D

whose fibers give the generic leaves in BGX,D: the Casimirs on HiggsGX,D

are the pullbacks of functions on CGX,D, while the larger collection of all

Hamiltonians consists of the pullbacks of functions on BGX,D.

In the remainder of Section 3.6 we discuss the algebro-geometric as-pects of the theorem: we construct the Hamiltonians and Casimirs, andidentify the Liouville tori as abelian varieties. The symplectic aspect,i.e. the construction of ψ, will be taken up in Section 3.7.

3.6.2 Spectral data and cameral covers

In this section we present a heuristic description of the Hamiltonianmap and the Liouville tori. We start with the case G = GL(n). Wewant to think of the Higgs bundle (V, ϕ) as consisting of two distinctpieces: its eigenvalues and eigenspaces. Very roughly, the Hamiltoniansfix the eigenvalues while a point in the Liouville torus determines theeigenspaces.

An n × n matrix is regular semisimple if it is diagonalizable withdistinct eigenvalues. Specifying such a matrix is equivalent to specifyingn distinct eigenvalues plus the eigenline associated to each.

For a pair (V, ϕ) where V is a rank n vector bundle on X and ϕ : V →V is an everywhere regular semisimple endomorphism, the n movingeigenvalues determine an n-sheeted cover π : X → X contained inX×C,while the moving eigenlines form a line bundle L ∈ Pic(X). For an(everywhere regular semisimple) Higgs bundle (V, ϕ : V → V ⊗KX(D)),the only change is that X sits in the total space Tot(KX(D)), ratherthan in X ×C which is the total space of the trivial bundle OX .

But the requirement that ϕ be everywhere regular semisimple is un-realistic. For generic (V, ϕ) we expect ϕ to be regular semisimple atall but a finite number of points of X. In this situation, the cover X

becomes a branched cover π : X → X, still contained in Tot(KX(D)).In general, X could be any subscheme of Tot(KX(D)) whose projectionto X is finite of degree n. When X is singular or non-reduced, the linebundle L can degenerate to various other sheaves on X: the eigenspacesneed not be eigenlines. But we can still recover the underlying vectorbundle V via the direct image construction encountered in Section 3.5.5:V = π∗L.


So in conclusion, in case G = GL(n), we can let BGL(n)X,D be the space

of all such “spectral covers” X. The Hamiltonian map

H : HiggsGL(n)X,D → B

GL(n)X,D

then sends (V, ϕ) to its spectral cover X. The Liouville torus thenparametrizes all the allowable sheaves L on X. In the generic situationX will be non-singular, the sheaves L are just line bundles, and theLiouville torus is a component of Pic(X). (As noted in Section 5.5, thedegree of L is not the same as the degree of V . Rather, there is a shift:it is the Euler characteristic which is preserved.)

The generalization to arbitrary reductive G is very important and verypretty, but it requires a recasting of the simple eigenvalue/eigenvectorpicture. In the case G = GL(n) this amounts to replacing the degree n

spectral cover π : X → X by a degree n! cameral cover X → X. Overa point x ∈ X where ϕ(x) is regular semisimple, so π−1(x) consistsof n distinct points x1, ..., xn, the points of X correspond to the n!ways of ordering the xi. X has the advantage that it is a Galois coverof X, with Galois group Sn, the symmetric group. The n projectionsπi : X → X give the n line bundles Li := π∗

i L on X. Let T ≈ (C∗)n bethe maximal torus of GL(n), i.e the subgroup of diagonal matrices. Thedecomposable rank n vector bundle ⊕n

i=1Li determines a principal T -bundle T and we can work out how T transforms under the action of Sn.Our recasting amounts to replacing the pair (X, L) by (X, T ), where X

is Sn-Galois over X and T is a T -bundle with the correct transformationproperty.

This seems more complicated, but it has the advantage that it extendsto all groups G. The symmetric group Sn is replaced by the Weyl groupW of G, W = NG(T )/T where NG(T ) is the normalizer of T in G. Then! points in X over a point x ∈ X for which ϕ(x) is regular semisimpleare replaced by the collection of chambers in the Cartan subalgebracontaining ϕ. The cover X is therefore called the cameral cover. Moregenerally, as long as ϕ(x) is regular (but not necessarily semisimple), thefiber of X over x ∈ X can be identified with the set of Borel subalgebrasin the Lie algebra ad(VX) containing the element ϕ(x) ⊗ α−1, for anynon-zero element α in the fiber (KX(D))|x of the line bundle KX(D)at the point x. Thus a G-Higgs bundle (V, ϕ) determines a cameralcover X → X together with a principal B-bundle on X, where B is aBorel subgroup of G. Since the maximal torus T is recovered from B asT = B/[B,B], there is an associated T -bundle on X which we denoteT . The bundles T which arise this way transform under the Weyl group

44

W according to an affine transformation law which is worked out ingeneral in [D1] and [DG]. It is a shifted form of the natural action ofW on T -bundles, which is induced from the action of W on the latticeΛ of characters of G, but the affine shift is quite delicate. The bottomline is that there is a categorical equivalence between regular G-Higgsbundles (V, ϕ) and pairs (X, T ) consisting of a cameral cover and aproperly-transforming T -bundle on it. We let BG

X,D be a parameterspace for cameral covers. The Hamiltonian map H : HiggsGX,D → BG

X,D

sends a G-Higgs bundle (V, ϕ) to the point of BGX,D parametrizing the

corresponding cameral cover X. The family of all T -bundles on X isthe product of n copies of Pic(X) where n is the rank of G, i.e. thedimension of T or the rank of the lattice Λ. Or, more intrinsically, thisfamily is given by Hom(Λ, P ic(X)). This is an abelian variety, as isits subgroup HomW (Λ, P ic(X)) of W -equivariant T -bundles, called thegeneralized Prym variety. The Liouville torus is a nontrivial coset of thissubgroup (reflecting the fact that the action is affine rather than linear),so it is also isomorphic to an abelian variety.

It is worth noting that each representation

ρ : G→ GL(N)

converts a G-Higgs bundle to its associated GL(n)-Higgs bundle. Thelatter has an N -sheeted spectral cover Xρ, depending on ρ. Hitchin’soriginal approach in [H], in case D = 0 and G a classical group, wasbased on these spectral covers for the “classical representations” of G.The Xρ for all representations ρ are recovered as associated covers ofthese cameral covers X.

3.6.3 Hamiltonians and Casimirs

We want to write down explicit expressions for the Hamiltonian baseBG

X,D, the Casimir base CGX,D, and the maps to them.

For G = GL(n), recall that the spectral cover X is an arbitrary sub-scheme of Tot(KX(D)) which is finite of degree n over X. In terms ofa local coordinate x on X and a linear coordinate y along the fibers ofKX(D), such an X is given by a polynomial equation

f(x, y) = yn + b1(x)yn−1 + · · ·+ bn(x) = 0.

This still works globally, but the bi are not functions – they are sectionsof line bundles (KX(D))⊗i. In fact if X is the spectral cover of (V, ϕ),


then f(x, y) is just the characteristic polynomial of ϕ, so

bi(x) = si(ϕ)

where si is the ith symmetric function in the roots of ϕ. So BGL(n)X,D ,

which is the space parametrizing all equations f(x, y) = 0, is the sum ofthe spaces parametrizing the individual coefficients:

BGL(n)X,D =

n⊕i=1

Γ(X, (KX(D))⊗i

),

and

H : HiggsGL(n)X,D → B

GL(n)X,D

sends

(V, ϕ) → (s1(ϕ), ..., sn(ϕ)).

We can also describe the quotient CGL(n)X,D , i.e. specify the subset of the

Hamiltonian functions which will turn out (once we defined ψ) to be theCasimirs. It is

CGL(n)X,D =

n⊕i=1

Γ(D, ((KX(D))⊗i)|D

),

and the map

Res : BGL(n)X,D −→ C

GL(n)X,D

simply sends the n-tuple (bi) of sections of line bundles on X to then-tuple of their restrictions to the divisor D. The composition

HiggsGL(n)X,D −→ B

GL(n)X,D −→ C

GL(n)X,D

then sends a Higgs bundle (V, ϕ) to the n-tuple of symmetric functionsof the residue resD(ϕ) which is a “Higgs field” on the 0-dimensionalscheme D, with values in the canonical line bundle (KX(D))|D ≈ KD

of D itself:

HiggsGL(n)X,D

HGX,D−→ B

GL(n)X,D

res =↓ residue Res =↓ Restriction

HiggsGL(n)D

HGD−→ C

GL(n)X,D .

For a reductive group G of rank n there are n polynomial functionssGi , i = 1, ..., n on the Cartan subalgebra t of G ( i.e. the Lie algebraof the maximal torus T ), which generate the algebra of W -invariantpolynomials on t. For G = GL(n), these are the elementary symmetric

46

functions, so the degree of sGL(n)i is i. In general, the degrees di =

deg(sGi ) are invariants of the group G, e.g. for G = SO(2n + 1) wehave di = 2i, while for G = SO(2n) we have di = 2i for i < n, anddn = n. (The corresponding invariant function s

SO(2n)n is the Pfaffian

– the square root of the determinant, which is a well defined SO(2n)-invariant function of a skew symmetric matrix.)

It turns out in general that fixing a cameral cover is equivalent tofixing the values of these n invariant polynomials. The Hamiltonianbase is therefore:

BGX,D =

n⊕i=1

Γ(X, (KX(D))⊗di

),

and the Hamiltonian

H : HiggsGX,D −→ BGX,D

sends

(V, ϕ) →(sGi (ϕ)

)ni=1

.

The Casimir base CGX,D is obtained similarly, by restricting from X to

D.

3.7 Symplectic geometry of Higgs bundles

3.7.1 Moment maps

Recall from Section 3.2.4 that a Poisson structure ψ on a manifold M

determines a Lie algebra homomorphism v : C∞(M)→ V F (M). Vectorfields in the image of v, i.e. of the form v(f) for a C∞ function f on M ,are called Hamiltonian. A vector field X is locally Hamiltonian if thereis an open covering Ui ofM such that the restriction of X to each Uiis Hamiltonian.

An action ρ : G ×M → M of a connected Lie Group G (with Liealgebra g) on a manifold M determines an infinitesimal action dρ whichis a Lie algebra homomorphism:

dρ : g→ V F (M).

Conversely, ρ can be recovered by exponentiating dρ.When (M,ϕ) is a Poisson manifold we say that the action ρ is Poisson

(or: liftable) if there is a Lie algebra homomorphism K : g → C∞(M)which lifts dρ, i.e. such that v K = dρ. In particular, the image of dρ


consists of Hamiltonian vector fields. If the action ρ is Poisson, then forevery g ∈ G the diffeomorphism ρ(g) : M → M preserves the Poissonstructure ψ. In fact, the latter condition is equivalent to the image ofdρ consisting of locally Hamiltonian vector fields.

The lift K determines (and is equivalent to) a map

µ : M → g∗

defined by

< µ(m), A >:= K(A)(m).

This map µ is called the moment map for the Poisson action ρ. It is aPoisson map, i.e. it sends the Poisson structure ψ on M to the Kirillov-Kostant Poisson structure on g∗. It is also G-equivariant. We illustratethis with the two examples from Section 3.2:

(i) Any action ρ of G on a manifold Y induces an action T ∗ρ of G onthe symplectic M := T ∗Y . This action T ∗ρ is Poisson. Indeedthe infinitesimal action of ρ

dρ : g→ V F (Y ) = Γ(Y, TY )

composes with the evaluation map

eval : Γ(Y, TY ) −→ C∞(T ∗Y )

to yield the desired lift

K : g −→ C∞(T ∗Y ).

The corresponding moment map

µ : T ∗Y −→ g∗

is the fiber by fiber dual of dρ. It can also be interpreted as thepullback, via ρ, of differential forms from Y to G.

(ii) The coadjoint action of G on the Poisson manifold g∗ is also aPoisson action. Its moment map is the identity

g∗ → g∗.

3.7.2 Symplectic reduction

Let ρ be a Poisson action of a connected Lie group G on a symplecticmanifold M,ω. Assume the action is nice enough that the quotient M/G

48

is a manifold. We then have the quotient map

M →M/G

and the moment map

µ : M → g∗.

If G is abelian, the G-equivariance of µ means that it factors throughM/G:

µ : M →M/G→ g∗

and for each ξ ∈ g∗, the subquotient

Mξ := µ−1(ξ)/G

is again a symplectic manifold, called the symplectic reduction of M atξ.

When G is non-abelian, µ does not factor through M/G. Instead weconsider the coadjoint orbit Oξ ⊂ g∗ which is the symplectic leaf in g∗

containing ξ, and set

Mξ := µ−1(Oξ)/G ≈ µ−1(ξ)/G(ξ),

where Gξ is the stabilizer of ξ in G. Again, Mξ inherits a symplecticstructure from M , and is called the symplectic reduction of M at ξ.

In fact, all these symplectic reductions fit together. The invertiblePoisson structure ω−1 on M is immediately seen to descend to a Pois-son structure on the quotient M/G. This is no longer symplectic; itssymplectic leaves are precisely the symplectic reductions Mξ, so they areparametrized by the coadjoint orbits Oξ in g∗.

3.7.3 The Poisson structure on HiggsGX,D

We want to obtain the Poisson structure on HiggsGX,D by symplecticreduction. The idea is as follows. Let a group G act on a manifoldY with a nice quotient Z = Y/G. The induced action ρ of G on thecotangent bundle M := T ∗Y is automatically Poisson, as we saw inExample (1) of Section 3.7.1. So, as in Section 3.7.2, the symplecticstructure on T ∗Y descends to a Poisson structure on W := T ∗Y/G. IfG acts freely on Y , this W is a vector bundle over Z: if y ∈ Y maps toz ∈ Z then the fiber of W over z can be identified with T ∗

y Y .We want to rig things so that Z =MG

X , W = HiggsGX,D, and the mapW → Z sends a G-Higgs bundle (V, ϕ) to the underlying G-bundle V.


There are various complications and technicalities. For instance, thereare (semi)stable Higgs bundles (V, ϕ) involving an unstable V, so the“map” W → Z is not everywhere defined (unless we work with stacks).A related problem is that some of the objects have extra automorphisms,causing the action of G not to be very nice. These details are handledin [M], [DM] and elsewhere. So here I will ignore them. This produces aPoisson structure only on a Zariski open subset of HiggsGX,D and extrawork is needed in order to extend it, cf. [DM], Section 3.5.4.

So we are searching for a space Y with an action of a group G suchthat Y/G is (generically) MG

X , while a typical cotangent space can beidentified with the fiber of HiggsGX,D over MG

X :

T ∗y Y ≈ Γ(X, ad(V)⊗KX(D))

for y ∈ Y above V ∈ MGX .

Recall the deformation theory worked out in Section 3.5.3 for GL(n)-bundles and in Section 3.5.7 for arbitrary G-bundles. We have identifi-cations

TVMGX ≈ H1(X, ad(V))

and dually

T ∗VMG

X ≈ Γ(X, ad(V)⊗KX).

Compare with what we want of Y :

T ∗y Y ≈ Γ(X, ad(V)⊗KX(D)),

or dually:

TyY ≈ H1(X, ad(V)⊗OX(−D)).

So while a deformation of a G-bundle V ∈ MGX is given by an ad(V)-

valued 1-cocycle hij, a deformation of a point of Y should be givenby a cocycle hij which vanishes on D.

This suggests the correct choice: for Y we take the moduli spaceMG

X,D of G-bundles on X together with a level D structure. By defini-tion a level-D structure on a G-bundle V is a trivialization of V|D, i.e.an isomorphism

η : V|D ∼→ G(OD)

where G(OD) is the trivial G-bundle on D. A deformation of the pair(V, η) is given by a 1-cocycle gij which is compatible with the trivial-ization η; this means precisely that the additive cocycle hij = g−1ij g′ijvanishes on D, as required.

50

For the group acting on MGX,D with quotient MG

X we can take

G′D := Aut(G(OD)).

When D consists of d distinct points, an element of G′D is a d-tuple

of elements of G. An element g ∈ G′D acts on MG

X,D sending (V, η)to (V, g η). On the other hand, if α is an automorphism of V, thenthe pairs (V, η) and (V, η α) represent the same point of MG

X,D. Inparticular, every element z of the center Z(G) of G determines suchan automorphism, α(z). The action of G′

D on MGX,D therefore factors

through an action of the quotient group:

GD := Aut(G(OD))/Z(G).

This quotient group acts generically freely onMGX,D, with quotientMG

X .As usual, this action lifts to T ∗MG

X,D, and the quotient T ∗MGX,D/GD

can be identified (away from a bad locus) with HiggsGX,D, which thushas a natural Poisson structure. The symplectic leaves are precisely thefibers over the Casimir base, which has a natural interpretation via thecoadjoint action of GD:

CGX,D ≈ (gD)∗/GD.

3.8 Examples, further developments, open problems

3.8.1 Some examples

The abelian caseThe case G = GL(1) is quite trivial: the moduli space of bundlesMGX

is Pic(X), the moduli space of Higgs bundles HiggsGX,D is the productPic(X)×Γ(X,KX(D)), the Hamiltonian base is BG

X,D = Γ(X,KX(D)),and the Hamiltonian map is the projection to the second factor. TheCasimir base is CG

X,D = Γ(D,KX(D)|D), and each symplectic leaf inHiggsGX,D is isomorphic to the cotangent bundle T ∗Pic(X) = Pic(X)×Γ(X,KX).

Hitchin’s systemThe case D = 0 of holomorphic Higgs bundles was studied by Hitchin[H]. As we saw in Section 3.5, the total space HiggsGX is a partialcompactification of the cotangent bundle T ∗MGX to moduli. Hitchinwrote down the Hamiltonians (i.e. the coefficients of the characteristicpolynomial) in this case, and proved the complete integrability whenthe group G is one of the classical groups GL(n), SL(n), SO(n), Sp(n).Proofs of several versions of the integrability for general G can be foundin [BK, D1, DG, F, S].


Rational baseIn case X = P1 and G = GL(n), the system of meromorphic Higgs bun-dles was described explicitly in [AHH, Be]. As we saw in Section 3.5.5,the stack MP1(n, 0) is stratified: it has one dense point, correspondingto the trivial bundle O⊕n

X , plus an infinite nesting of other points in itsclosure. The forgetful map HiggsGP1,D →MP1(n, 0) induces a stratifica-

tion ofHiggsGP1,D, with a dense open subset (HiggsGP1,D)0 parametrizingHiggs bundles whose underlying vector bundle is the trivial bundle. IfD happens to be d times the point p∞ where a coordinate t on P1 be-comes infinite, a Higgs bundle in the open subset (HiggsGP1,d·p∞)0 can

be given by a polynomial∑d−2

i=0 Aiti dt whose coefficients Ai are n × n

matrices. If D is the sum of d distinct points pi where t = ti, the Higgsbundle in the open subset (HiggsGP1,d·p∞)0 can be given instead by a

rational function∑d

i=1Bi/(t− ti) dt, with n× n matrices Bi satisfying∑di=1Bi = 0. Markman’s Poisson structure can be written explicitly in

terms of the A’s or B’s. The special case of Theorem 3.6.1 proved in[AHH, Be] says that the set of such polynomial (or rational) matriceswith fixed characteristic polynomial (equivalently, spectral cover C) isan open subset of the Jacobian of C. The complement, coming fromHiggs bundles whose underlying vector bundle is not trivial, can also bedescribed explicitly in terms of the theta divisor of Jac(C).

The particular case of G = SL(2) was studied earlier by Mumford[Mum], and used in his solution of the Schottky problem for hyperellipticcurves. Particular choices of the residues lead to very classical systemssuch as the geodesic flow on an ellipsoid and Neumann’s system, cf. [Be,DM].

Elliptic baseA similarly explicit description is available when the base X has genusg = 1, using the description of the moduli space given in Section 3.5.5.A particularly important case arises when D is one point p∞ and theresidue there is in the conjugacy class of the diagonal matrix O :=diag(1, . . . , 1, 1 − n). The spectral curves which arise are the ellipticsolitons studied by Krichever [Kr] and Treibich–Verdier [TV]. These aren-sheeted covers π : C → X, where C has genus n and at least n−1 of then sheets come together above p∞. Equivalently, Jac(C) contains a copyof the elliptic curve X, and the image AJ(C) of the Abel-Jacobi map(cf. Section 3.4.4) is tangent to X at the ramification point. (Accordingto Krichever, the Jacobian Jac(C) together with the linear flows on itgiven by the derivatives of AJ constitute a KP soliton, i.e. a finite-

52

dimensional orbit of the infinite-dimensional KP hierarchy. An ellipticsoliton is such a solution in which the first KP flow is tangent to anelliptic curve, hence is doubly periodic.)

The same system arises, from very different considerations, in su-persymmetric Yang-Mills theory, cf. [DW]. The relation of integrablesystems to Seiberg–Witten theory and supersymmetric Yang-Mills isdiscussed in [D2], where various related spectral curves are worked outexplicitly. This elliptic soliton system is also the SL(n) case of the ellip-tic Calogero–Moser system, which suggests an analogue for other groups,cf. [DHP].

3.8.2 Further developments

Bundles on elliptic fibrationsWe noted in Section 3.6.2 that there is a categorical equivalence be-tween regular G-Higgs bundles (V, ϕ) and pairs (X, T ) consisting of acameral cover and a properly-transforming T -bundle on it. The ver-sion we needed there involved, on the one hand, KX(D)-valued G-Higgsbundles, and on the other hand, cameral covers X mapped to the to-tal space Tot(KX(D)): this allowed us to have parameter spaces BG

X,D,for the cameral covers, and HiggsGX,D, for the Higgs bundles, with amap HiggsGX,D → BG

X,D whose fibers are the generalized Prym varietiesHomW (Λ, P ic(X)). However, the result of [D1] and [DG] is stronger: itis a categorical equivalence between abstract G-Higgs bundles (with nospecification of the bundle KX(D) in which they take their values), andabstract spectral data (X, T ), consisting of an abstract cameral coverX (with no specification of a map to the total space of a bundle) plus aproperly-transforming T -bundle on it. (The definition of an abstract G-Higgs bundle is somewhat technical: it is defined [D1, Def. 7] to be a pair(V, c), where V is, as usual, a principal G-bundle on X, and c ⊂ ad(V)is a subbundle which fiber by fiber consists of regular centralizers, i.e.centralizers in g of regular elements of G.) The “valued” result followsimmediately from the “abstract” one by adding to the Higgs bundle(V, c) a section ϕ ∈ Γ(X, c ⊗KX(D)), and to the spectral data (X, T )an appropriate map X → Tot(KX(D)). (Actually, this should be a W -equivariant collection of maps, cf. [D1, Def. 6].)

But given the notions of abstract Higgs bundles and spectral data,we can clearly consider Higgs bundles and spectral data with valuesin any family Y of groups over X, not necessarily in a line bundle:to a an abstract Higgs bundle add a section ϕ ∈ Γ(X, c ⊗ Y ), and


to the abstract spectra data (X, T ) add an appropriate collection ofmaps to Tot(Y ). With the obvious notation, we get parameter spacesHiggsGY/X and BG

Y/X and a map HiggsGY/X → BGY/X whose fibers are

the generalized Prym varieties HomW (Λ, P ic(X)).An important case involves “values” in an elliptic fibration, i.e. a

variety Y fibered over X with the generic fiber being an elliptic curve.This is important due to the observation [D3] that the moduli spaceMG

Y of G-bundles on Y (satisfying an appropriate version of stability)can be identified with the moduli space HiggsGY/X of G-Higgs bundleson X with values in the fibration Y → X. Combining this with theequivalence of Higgs bundles with spectral data, we see thatMG

Y can befibered over a base BG

Y/X , the fibers being the same generalized Prym

varieties HomW (Λ, P ic(X)) which occur for line-bundle-valued Higgsbundles.

In case Y is a 2-dimensional algebraically integrable system, fiberedover a curve X with elliptic fibers, the resultingMG

Y is an algebraicallyintegrable system with base BG

Y/X and generalized Prym varieties forfibers. (The case that G = GL(n) is also a special case of Mukai’ssystem, see below.) When dim(Y ) ≥ 3 we lose the symplectic aspectsof an algebraically integrable system, and retain only the fibration byabelian varieties. The point is that the moduli space of bundles on Y

can be analyzed in terms of objects on the lower dimensional base X.Another approach to the study ofMG

Y is in [FMW]. This is based onLooijenga’s result [Lo] which says that the moduli of G bundles on anelliptic curve, for any semisimple group G, is a weighted projective space,and on the use of minimally unstable bundles and their deformations.

Non-linear deformationsMukai [Mu] showed that on an algebraically symplectic surface Y , eachcomponent of the moduli space of coherent sheaves is itself algebraicallysymplectic (away from its possible singularities). We have already en-countered the case of vector bundles on an elliptically fibered symplecticsurface. In order to obtain another important case, choose a curve C

with an embedding i : C → Y and a line bundle on it, L ∈ Pic(C), andconsider the moduli space MukY,C,L of coherent sheaves on Y whoseHilbert polynomial equals that of the direct image i∗(L). The supportof such a sheaf is a curve algebraically equivalent to C, and there isa well-defined support map H : MukY,C,L → |C| to the moduli space(often, the linear system) |C| of curves in Y algebraically equivalent toC. This support map is Lagrangian with respect to the symplectic struc-

54

ture on MukY,C,L, and the fiber over a non-singular C is a componentof Pic(C).

In the case when Y = T ∗X and C = n ·X is n times the zero section,MukT∗X,n·X,L is just Hitchin’s system. Note that although Y in thiscase is not compact, the linear system |C = n·X| is finite dimensional, infact it is precisely the system of Hitchin spectral curves. (More generally,Tyurin [T] extended Mukai’s result to Poisson surfaces Y . Theorem 6.1is obtained by taking Y := Tot(KX(D)).)

On the other hand, we can take X to be any non-singular curve con-tained in a K3 surface Y . It is shown in [DEL] that MukY,n·X,L is anon-linear deformation of Higgs

GL(n)X . The fiber over the non-reduced

divisor n ·X is seen to be a certain affine twist of the nilpotent cone inHitchin’s system, analyzed by Laumon [La]. In particular, this fiber isreducible: one component parametrizes vector bundles on the reducedX, while other components parametrize sheaves on Y whose support isthe full non-reduced C = n ·X, for example line bundles in Pic(n ·X).This analysis seems to be useful for understanding the D-branes of non-perturbative string theory, cf. [GS] for a recent example.

3.9 Open problems

Seiberg-Witten integrable systemsQuantum field theoretic considerations [SW1,2] predict the existence ofa collection of Seiberg-Witten integrable systems. These should dependon the choice of an elliptic curve X plus a complex semisimple groupG and a collection of representations of G satisfying certain numericalconditions, cf. Section 1 of [D2]. The Liouville tori for these systemsshould be r-dimensional, where r is the rank of G, and they should beJacobians or Prym varieties of the Seiberg-Witten curves, which playthe role of spectral curves for these systems. In particular, there shouldexist a Seiberg-Witten integrable system (SWIS) for each elliptic curveX and complex semisimple group G taken with its adjoint representa-tion. The case G = SL(n) was solved in [DW]. The system obtainedthere is equivalent to the elliptic solitons system. It is tempting to guessthat the SWIS for any group G (with its adjoint representation) is simi-larly obtained from the meromorphic Higgs bundles on X with structuregroup G. The problem is that the dimension of the Liouville tori for sucha system equals half the dimension of the coadjoint orbit to which the


residue is confined; for groups other than SL(n), a small enough orbit(i.e. of dimension 2r) does not seem to exist.

D’Hoker and Phong have shown [DHP] that integrable systems sat-isfying the SW requirements can be obtained by choosing appropriateparameters in the elliptic Calogero-Moser system for the group G. How-ever, these choices are made on a case-by-case basis. It would be veryinteresting to understand a priori which choices work, and to derive theentire collection of SWIS simultaneously from some uniform geometricconstruction, presumably some variant of the moduli space of meromor-phic Higgs bundles.

Mathematics of string dualityThe central development in string theory in the last few years has beenthe discovery of string dualities. The various individual perturbativestring theories are now believed to be equivalent to each other and torepresent different limits of a single M-theory. The two central dualitiesare the one between types IIA and IIB strings, known as mirror sym-metry, and the duality between the heterotic string and F-theory, cf.[V, MV]. Among other things, this conjectured duality predicts a nat-ural isomorphism between the moduli spaces of the two theories. Theheterotic moduli space parametrizes collections of data which include,among other fields, a Calabi–Yau n-fold Z together with a G-bundle onZ, where G is E8 × E8 (where E8 is the 248-dimensional exceptionalLie group) or a subgroup in it. The corresponding F-theory modulispace parametrizes data which includes, among other fields, an ellipti-cally fibered Calabi-Yau n+ 1-fold Y → X, and a “Ramond–Ramond”field, which amounts to a point in the Deligne cohomology group of Y , acertain extension of the intermediate Jacobian Jac(Y ) of Clemens andGriffiths [CG]. The best understood case [MV] is when Z is itself el-liptically fibered over a base B. X is then a P1-bundle over the sameB, and Y is fibered over B with K3 fibers (and these K3’s in turn areelliptically fibered over the P1’s). In the limit when one of the auxiliaryfields on the heterotic side goes to zero, the F-theory varieties degener-ate into reducible varieties X = X1 ∪ X2 and Y = Y1 ∪ Y 2, with eachYi fibered elliptically over Xi and fibered with rational elliptic surfacefibers over B. The intersection Y1 ∩ Y2 is then the heterotic Calabi-YauZ. In this limit the birational isomorphism of the moduli spaces can beproved geometrically [CD, D5]. This combines the description ofMG

Z interms of data on B, as outlined in Section 3.8.2, with an analysis of theintermediate Jacobian and Deligne cohomology group of rational ellip-

56

tic surface fibrations, extending earlier work of Kanev [K]. The picturewhich emerges is of an equivalence of two integrable systems: one is theversion of the meromorphic Higgs bundles applicable to moduli of bun-dles on elliptic fibrations, the other a system of intermediate Jacobiansover the moduli of Calabi-Yaus [DM2]. Some information on the exten-sion of this birational isomorphism to various geometric strata of themoduli was obtained in [AD] using the non-linear deformation pictureof [DEL], and a computation of the vanishing cycles as we degeneratefrom a general point of the F-theory moduli space into the geometriclimit was done in [A]. Still, the conjectured isomorphism away from thisgeometric limit remains elusive, and even understanding the precise pic-ture along all the strata inside the geometric limit should be very usefuland pretty.

Interpretation of the trigonometric caseThere are three types of connected, 1-dimensional groups: an ellipticcurve E, the multiplicative group Gm := C∗, and the additive groupGa := C. The latter two can be considered as the groups of non-singularpoints in degenerations of the elliptic curve E with 1 or 2 vanishingcycles, respectively, i.e. degenerations of E to a curve with a nodeor a cusp. The natural functions on these three types of groups are,respectively, elliptic (doubly periodic), trigonometric (singly periodic),and rational.

Hitchin’s system on the moduli of bundles on an arbitrary curve X, aswell as Markman’s meromorphic extension, involve Higgs bundles witharbitrary structure group G but with values in a line bundle, which is agroup variety over X with fiber Ga. The moduli space of G-bundles onan elliptic fibration involves Higgs bundles with structure group G andwith values in the elliptic fibration. Is there an interesting geometricinterpretation of the remaining “trigonometric” case, where the valuesare taken in Gm?

Abelian solitonsAn abelian soliton is a k-dimensional abelian subvariety A ⊂ Jac(C) ofa g-dimensional Jacobian, having the property that the kth osculatingsubspace to the Abel-Jacobi image AJ(C) at some point p ∈ C equalsthe tangent space TAJ(p)A. This is equivalent to saying that the firstk flows of Krichever’s KP hierarchy solution corresponding to C, whichordinarily would evolve on the entire Jac(C), happen to be confined toA. The case k = 1 is that of elliptic solitons, and the complementarycase k = g − 1 of coelliptic solitons was studied in [DP]. In both cases,


the Jacobians or Pryms fit together to form an integrable system: asymplectic leaf of Markman’s system for the elliptic solitons, and a cer-tain non-linear reduction for the coelliptic solitons. Are there abeliansolitons in other dimensions? Do they form integrable systems?

Polygonal systemsThere are a number of other algebraically integrable systems which areprobably related to meromorphic Higgs bundles, although this relationhas not been worked out yet. One particularly beautiful example is thesystem of closed space polygons [Kl, KM]. An (n + 3)-gon in R3 (or,for that matter, in C3) with a vertex at the origin can be specified bylisting an ordered (n + 3)-tuple of vectors vi, i = 1, . . . , n + 3 satisfying∑n+3

i=1 vi = 0. The moduli space of these polygons (modulo the actionof SO(3)) is a 3n+ 3-dimensional Poisson variety, with 2n-dimensionalsymplectic leaves obtained by fixing the lengths of the n+3 sides. Thisis easiest to see by identifying affine 3-space with the Lie algebra so(3) aswe did in Section 3.2.2. The configuration space of polygons is then thereduction (so(3))n+3//SO(3), and its symplectic leaves are obtained byfixing the Casimirs of the individual so(3) factors, i.e. the side lengths.What is a little less obvious is that you get an algebraically integrablesystem by taking as Hamiltonians the lengths of the n diagonals

dj :=j∑

i=1

vi, j = 2, . . . , n+ 1.

The flow on the space of polygons corresponding to the jth Hamiltonianis simply the rotation of the first j vertices and edges around the jth di-agonal. It is not known whether this system can be obtained by choosingthe residues appropriately in some version of Markman’s system.

References[A] Aspinwall: Aspects of the hypermultiplet moduli space in string duality,

JHEP 9804 (1998) 019, hep-th/9802194.[AD] Aspinwall, Donagi: The heterotic string, the tangent bundle, and

derived categories, Adv. Theor. Math. Phys. 2 (1998) 1041-1074,hep-th/9806094.

[AHH] Adams, Harnad, Hurtubise: Isospectral Hamiltonian flows in finiteand infinite dimensions, II: Integration of flows, Comm. Math. Phys. 134(1990), 555-585.

[At] Atiyah: Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. 7(1957), 414-452.

[Be] Beauville: Jacobiennes des courbes spectrales et systemes hamiltonienscompletement integrables, Acta Math. 164 (1990), 211-235.

58

[Bo] Bottacin: Thesis, Orsay, 1992.[BK] Beilinson, Kazhdan: Flat projective connections, unpublished (1990).[CG] Clemens, Griffiths: The intermediate Jacobian of the cubic threefold,

Ann. of Math. 95 (1972), 281-356.[CD] Curio, Donagi: Moduli in N=1 heterotic/F-theory duality, Nucl. Phys.

B518 (1998), 603-631. hep-th/9801057.[D1] Donagi: Spectral covers, in: Current topics in complex algebraic

geometry, MSRI 28(1992), 65-86. (alg-geom/9505009).[D2] Donagi: Seiberg-Witten integrable systems, in: Proceedings, Santa Cruz

Alg. Geom. Conf.; reprinted in: Surveys in Differential Geometry, ed.S.T.Yau

[D3] Donagi: Principal bundles on elliptic fibrations, Asian J. Math. Vol. 1(June 1997), 214-223. alg-geom/9702002.

[D4] Donagi: Taniguchi lecture on principal bundles on elliptic fibrations, in:Integrable Systems and Algebraic Geometry; Saito, Shimizu and Ueno,eds. World Sci. (1998). hep-th/9802094.

[D5] Donagi: ICMP lecture on Heterotic/F-theory duality. hep-th/9802093.[DEL] Donagi, Ein, Lazarsfeld: Nilpotent cones and sheaves on K3 surfaces.

alg-geom/9504017.[DG] Donagi, Gaitsgory: The gerbe of Higgs bundles. math. AG/0005132[DHP] D’Hoker, Phong: Spectral curves for super-Yang-Mills with adjoint

hypermultiplet for general Lie algebras, Nucl. Phys. B534 (1998),697-719, hep-th/9804126.

[DM] Donagi, Markman: Spectral curves, algebraically completely integrableHamiltonian systems, and moduli of bundles, in: Integrable Systemsand Quantum Groups, LNM 1620 (1996), 1-119. alg-geom/9507017.

[DM2] Donagi, Markman: Cubics, integrable systems, and Calabi-Yauthreefolds, in: Proceedings of the Algebraic Geometry Workshop on theOccasion of the 65th Birthday of F. Hirzebruch, 1993.alg-geom/9408004.

[DP] Donagi, Previato: Abelian solitons. nlin.SI/0009004.[DW] Donagi, Witten: Supersymmetric Yang-Mills systems And integrable

systems, Nucl. Phys. B460 (1996) 299-334. hep-th/9510101.[F] Faltings: Stable G-bundles and projective connections, J. Algebr. Geom. 2

(1993), 507-568.[FMW] Friedman, Morgan, Witten: Vector bundles and F theory, Commun.

Math. Phys. 187 (1997) 679-743. hep-th/9701162.[GH] Griffiths, Harris: Principles of algebraic geometry, Pure and Applied

Mathematics. Wiley-Interscience (1978).[GS] Gomez, Sharpe: D-branes and scheme theory. hep-th/0008150[H] Hitchin: Stable bundles and integrable systems, Duke 54 (1987), 91-114.[K] Kanev: Spectral curves, simple Lie algebras and Prym–Tjurin varieties,

Proc. Symp. Pure Math. 49 (1989), Part I, 627-645.[Kl] Klyachko: Spatial polygons and stable configurations of points on the

projective line, Alg. Geom. and its Applns, Yaroslavl (1992), 67-84.[KM] Kapovich, Millson: The symplectic geometry of polygons in Euclidean

space, J. Differ. Geom. 44 (1996), 479-513.[Kr] Krichever: Elliptic solutions of the Kadomtsev–Petviashvili equation and

integrable systems of particles, Functional Anal. Appl. 14 (1980),282-290.

[La] Laumon: Un analogue global du cone nilpotent, Duke 57 (1988), 647-671.


[Lo] Looijenga: Root systems and elliptic curves, Inv. Math. 38 (1976), 17-32.[M] Markman: Spectral curves and integrable systems, Comp. Math. 93

(1994), 255-290.[Mu] Mukai: Symplectic structure of the moduli space of sheaves on an

abelian or K3 surface, Inv. Math. 77 (1984), 101-116.[Mum] Mumford: Tata Lectures on Theta II, Birkhaeuser-Verlag, Basel,

Switzerland and Cambridge, MA (1984).[MV] Morrison, Vafa: Compactifications of F-theory on Calabi–Yau

threefolds, I: Nucl. Phys. B473 (1996) 74-92. hep-th/9602114; and II:Nucl. Phys. B476 (1996) 437-469. hep-th/9603161.

[S] Scognamillo: An elementary approach to the abelianization of the Hitchinsystem for arbitrary reductive groups. alg-geom/9412020.

[SW1] Seiberg, Witten: Monopole condensation, and confinement In N=2supersymmetric Yang-Mills theory, Nucl. Phys. B426 (1994) 19-52;Erratum-ibid. B430 (1994) 485-486. hep-th/9407087.

[SW2] Seiberg, Witten: Monopoles, duality and Chiral symmetry breaking inN=2 supersymmetric QCD, Nucl. Phys. B431 (1994), 484-550.hep-th/9408099.

[T] Tyurin: Symplectic structures on the varieties of moduli of vector bundleson algebraic surfaces with pg > 0, Math. USSR Izvestiya 33 (1989).

[TV] Treibich, Verdier: Solitons elliptiques. The Grothendieck Festschrift, Vol.III, 437-480, Birkhauser Boston (1990).

[V] Vafa: Evidence for F-Theory, Nucl. Phys. B469 (1996), 403-418.hep-th/9602022.

4

The anti-self-dual Yang–Mills equations andtheir reductions

Lionel MasonThe Mathematical Institute, Oxford

[email protected]

Abstract

These notes provide an introduction to an approach to the theory ofintegrable systems that arises from the observation that many integrablesystems are reductions of the anti-self-duality equations so that the the-ory of these equations can be understood as a reduction of the corre-sponding theory for the anti-self-duality equations.

We start with a general discussion of integrable systems and relationsbetween them arising from symmetry reductions and give some standardexamples. We then give an introduction to gauge theory and the self-dual Yang–Mills equations.

The anti-self-dual Yang–Mills equations can be seen to be an inte-grable system; it has a Lax pair, admits Backlund transformations, thereare ansatze for solutions, and it has topological solutions, instantons.We go on to discuss its reductions to three dimensions, monopoles andChiral models.

Another way to see the self-dual Yang–Mills equations as an integrablesystem is to present Hamiltonian and Lagrangian formulations and arecursion operator. This leads to the generalised self-dual Yang–MillsHierarchies.

We then discuss general principals of reduction. Translation reduc-tions lead to the KdV and nonlinear Schrodinger equations. Non transla-tional reductions give rise to the Ernst equations and Painleve equations.Finally we discuss further developments of the overview.

60

The anti-self-dual Yang–Mills equations and their reductions 61

4.1 Introduction

These notes are intended to provide an introduction to an overview onthe theory of integrable systems based on reductions of the anti-self-dual Yang–Mills equations and its twistor construction. This overviewwas originally proposed by Ward (1985) and developed in Mason &Woodhouse (1996) and this latter book contains full details of most ofthe material presented here and much more. These notes are intendedto be accessible to graduate students and so introductory material onconnections and the Yang–Mills equations is provided, together withsome exercises at the end.

4.1.1 Standard aspects of integrability

Integrable systems are differential equations that, despite their nontriv-ial nonlinearity, are surprisingly tractable. The following more precisedefinitions do not always apply:

• Theorem 1 (Arnol’d-Liouville) Suppose that a Hamiltonian sys-tem in 2n-dimensions has n constants of the motion Hi in involutionsuch that the map to R

n determined by the Hi is proper and regu-lar, then there exists a coordinate system of ‘action-angle’ variables,obtainable by quadratures, in which the flows are linear.

• Theorem 2 (Magri) If a Hamiltonian system admits a recursionoperator satisfying certain conditions, then the system is integrable inthe Arnol’d-Liouville sense.

• The system should admit a twistor correspondence.• The system is the consistency condition for a ‘Lax pair’, i.e. an aux-

illiary system of overdetermined linear equations.

4.1.2 Properties of solutions to integrable systems

Integrability usually means that explicit formulae for solutions are read-ily obtainable. Such solutions have characteristic properties

• The Painleve property: for ODE’s, solutions to linear equationsare always regular (except at the fixed singularities of the equations).For complex nonlinear equations, solutions generically have branchingand essential singularities or worse. The Painleve property requires

62

that all moveable singularities be rational, i.e. any singularity whoselocation depends on the initial conditions must be rational.

It is conjectured, but not yet proved, that is a defining property.• Solitons: integrable equations often admit ‘particle-like’ lump solu-

tions, solitons. These can be superposed and the lumps often simplypass through each other, perhaps with some simple scattering, butmaintaining their integrity.

• IST: the inverse scattering transform expresses the general solutionas a nonlinear superposition of radiative or dispersive modes, andsolitons.

4.1.3 Key examples

Because of the imprecise nature of the definition of an integrable system,it is important to familiarize oneself with the key examples in order toget a feel for the subject.

The Euler top

The configuration space is SO(3) =frames fixed in the top, and thephase space is T ∗SO(3). The moving frame ei, i = 1, 2, 3 satisfies

deidt

= ω ∧ ei , ω = ω1e1 + ω2e2 + ω3e3 ,

and, for the Euler top with principal moments of inertia Ii (constants)

I1dω1dt

= (I2 − I3)ω2ω3 , + cyclic.

Constants of motion: the motion is generated by the Hamiltonian12

∑i Iiω

2i and there are two other constants of the motion in involution;

the total angular momentum L ·L =∑

i I2i ω

2i and L · k, L =

∑i Iiωiei.

The equations for ωi can be rescaled to give

ω1 = 2ω2ω3 , ω2 = −2ω3ω1 , ω3 = 2ω1ω2.

Lax pair: these equations are equivalent to the integrability, [L0, L1] =0 of the Lax pair

L0 =ddt− i(Ω3 + λ(Ω1 + iΩ2))

L1 = λ(Ω1 + iΩ2) + 2Ω3 − (Ω1 − iΩ2)/λ


where

Ω1 =(

0 ω1ω1 0

), Ω2 =

(0 −ω2ω2 0

), Ω3 =

(ω3 00 ω3

).

Note that L1 evolves by conjugation, so that tr(L1)p is constant. Thisyields the conserved quantities.

The Korteweg–de Vries equations

This is an equations for the evolution of the height u(x, t) of shallowwater waves in a channel

u = u(x, t) , 4ut − uxxx + 6uux = 0 .

This equation is equivalent to the integrability conditions [L0, L1] = 0for the Lax pair

L0 = ∂x +(q −1p −q

)− λ

(0 01 0

), L1 = ∂t +B − λ∂x

where p and B are determined in terms of q by the consistency conditionsand u = −2qx.

Solitons: this equation admits ‘soliton’ solutions, in particular the onesoliton is

u = 2c cosh−2(c(x− ct)) , c = velocity .

There exist n-soliton solutions with n lumps moving at different speedsthat keep their identity under interaction.

The Kadomtsev–Petviashvilii equations

These equations control the evolution of shallow water waves in 2-dimensions

u := u(x, y, t) , ∂x(4ut − uxxx + 6uux) = uyy .

The Lax pair is

L0 = ∂y − ∂2x − 2u , L1 = ∂t − ∂3x + 3u∂x + v

where the consistency conditions determine’s v in terms of u in additionto forcing the KP equation.

64

4.1.4 Reductions of integrable systems

Reduction can mean the imposition of a symmetry or the specializa-tion of a parameter in a system of equations. For example imposing asymmetry along ∂y reduces the KP equation to the KdV equation.

Generally, reduction is compatible with the structures such as theLax Pair associated with integrability and so yield an integrable system.Hence it gives a partial ordering on the set of integrable equations.

Is there a universal integrable system? The answer is almost certainlyno; one can construct integrable systems in arbitrarily high dimensionsfor example. However, the anti-self-dual Yang–Mills (ASDYM) equa-tions reduce to many of the most popular integrable systems. Its flexi-bility lies in the fact that it is really a family of equations, one for eachchoice of gauge group.

This leads to two programmes:

(i) classify those integrable systems that can be obtained by reduc-tion from the ASDYM equations;

(ii) unify the theory of these equations by reduction of the corres-ponding theory for the ASDYM equations.

This leads to a self contained theory restricted to these systems, butalso highlights the distinctions from systems that are not reductions ofASDYM. In particular the KP equations do not appear to be a reductionof ASDYM with finite dimensional gauge group, although in Ablowitz &Clarkson (1991) it is shown to be a reduction if an infinite-dimensionalgauge group is allowed. (This may not, however, mean so much asit is not so clear that the ASDYM equations are integrable in such ameaningful way with an infinite-dimensional gauge group.)

4.1.5 Twistor theory

The basic aim of Penrose’s twistor theory is to find 1–1 correspondencesbetweenSolutions to physical equa-tions; Yang–Mills, Einstein

←→

Deformed complex struc-tures on twistor space

The hope has been that twistor space provides the correct geometricarena for the correct formulation of quantum gravity. Evidence for theexistence of such constructions comes from the twistor correspondencesfor the self-duality equations and these suggest the larger (unfulfilled)programme for the full Yang–Mills and Einstein equations.


These twistor correspondences for the self-duality equations provide aparadigm for ‘complete integrability’. They give a geometric construc-tion for the general local solution to the equations. The fact that theself-duality equations yield ‘most’ integrable systems under symmetry re-duction implies that these reductions also inherit reduced twistor corres-pondences. This suggests that it is the existence of a twistor constructionthat underlies integrability. This can be more general than merely look-ing at reductions of self-dual Yang–Mills as many more equations, inarbitrarily high dimension for example, admit a twistor correspondence.

These notes are too brief to give an introduction to Twistor theoryand the interested reader is referred to Mason & Woodhouse (1996) partII for a full treatment of these ideas. However, it is the twistor theorythat is the real impetus for many of these ideas.

4.2 The self-dual Yang–Mills equations

We need some background geometry first. The Yang–Mills equationsdepend on the choice of a Lie group, G. When G = U(1) they reduce toMaxwell’s equations. The anti-self-duality condition picks out circularlypolarized solutions. The full equations make good sense for space-timesof any signature and dimension. However, the self-duality condition onlymakes sense in dimension four, and only admits real solutions when thesignature is Euclidean or ultrahyperbolic, (+ +−−).

4.2.1 Bundles, connections and curvature

The Yang–Mills equations are equations on connections on vector-bundles.A vector bundle on a manifold M is an attachment of a vector space Ex

to each x ∈M . More formally:

Definition 1 A vector bundle of rank n over a manifold M is a manifoldE fibred over M , p : E → M such that the fibres p−1(x), x ∈ M havethe structure of n-dimensional real or complex vector spaces dependingsmoothly on x ∈M (sometimes holomorphically if the fibres and M arecomplex in which case the bundle will be said to be holomorphic).

A complex vector bundle is unitary if each fibre has a Hermitean met-ric (depending smoothly on x ∈ M) and special unitary if there is aholomorphic volume form on each fibre.

66

A bundle is usually described by means of local trivializations associatedto a cover by topologically trivial open sets Ui ⊂ M , i.e. isomorphismsρi : E|Ui

∼= Ui×Cn. On overlaps, U1 ∩U2, there will be an n×n matrix

function ρ12 = ρ1ρ−12 satisfying, for consistency ρ12ρ23 = ρ13 on triple

overlaps. A vector bundle is said to have structure group G if the ρij canbe chosen so as to take values in G. One can take direct sums, E1⊕E2,duals E∗ and tensor products E1 ⊗ E2 of vector bundles fibrewise overeach x ∈ M following the definitions for vector spaces. The bundleE⊗E∗ is of particular note as the bundle End(E) of endomorphisms ofE.

Example. A simple nontrivial example is TS2, the tangent bundle ofthe sphere; TxS

2 is the space of tangent vectors at x ∈ S2. This isalso a holomorphic complex line bundle and the structure group can bereduced to SO(2) = U(1) (exercise).

A gauge transformation g is an automorphism, i.e. a diffeomorphismg : E → E that sends each fibre to itself by a linear transformation(in G if the structure group is G). In a local trivialization it will berepresented by a matrix function g(x), x ∈ M on Ui with values in G.Even when the bundle is topologically trivial, E = M ×C

n, the bundleconcept is nontrivial in the sense that the given trivialization is not partof the structure and any gauge transformation maps this to a differentbut equivalent trivialization.

A section s ∈ Γ(E,M) is a smooth map s : M → E. These canbe added together and multiplied by functions on M . We would like adefinition of differentiation of sections that does not depend on a choiceof (local) trivialization.

Definition 2 A connection on a bundle E is a linear map D fromsections of E to sections of T ∗M ⊗ E such that D(fs) = (df)s + fDs

where f is a function on M and d is the exterior derivative.

In a local trivialization ρi, put v = ρi(s), here v is an n-componentcolumn vector, then

ρi(Ds) = dxa(

∂

∂xa+Ai

a

)ρi(s) = (d+A)v , A = dxaAa

where xa are coordinates on M and Aia = ρiDaρ

−1i are matrices. Usu-

ally one misses out the ρi in the above formula working instead with afunction v with values in C

n and writes Dv = dv+Av. Under a change


of local trivialization, we must have

Aia = ρijA

jaρ

−1ij + ρijdρ−1ij .

Gauge transformations send D to g D g−1 = D+ (gDg−1), the sameformula as above if expressed in a frame. When the bundle has structuregroup G, then the Aa are usually taken to take values in Lie(G), the Liealgebra TeG of G.

The connection naturally extends to E∗ and E ⊗E∗ etc by requiringthe Leibnitz (product) rule over contractions and tensor products. Fora section γ of E ⊗ E∗ we will have, in a local trivialization ρi(Dγ) =dρi(γ) + [A, ρi(γ)].

The curvature is a 2-form with values in E ⊗ E∗ defined to be F =D ∧D or in indices in a local trivialization

Fab = ∂[aAb] + [Aa, Ab] .

The curvature transforms homogeneously under gauge transformations(and changes of trivialization) F → g−1Fg and satisfies the Bianchiidentity

D ∧ F = 0 , or in indices D[aFbc] = 0

which follows from the Jacobi identity [Da, [Db,Dc]] = 0.

4.2.2 The full Yang–Mills equations

The Yang–Mills equations arise from the Lagrangian density L =−tr(FabF

ab) where M is now assumed to be endowed with a metricwhich is used to raise and lower indices and tr is shorthand for a neg-ative definite invariant inner product on Lie(G) (which is indeed thetrace for SU(n)). The Euler-Lagrange equations are DaFab = 0.

When G = U(1) and M is Minkowski space, this gives Maxwell’sequations by identifying F0j = iEj the electric field, and Fjk = iεjklBl

the magnetic field, j, k, l = 1, 2, 3.Note that the equations are gauge invariant so that if D is a solution

to the Yang–Mills equations, then so is g D g−1. This connectionis regarded as equivalent to D and one is in general interested only ingauge equivalence classes of solutions.

Unlike Maxwell’s equations, the Yang–Mills equations only acquirefull relevance for physics as a quantum field theory in which they de-scribe the weak and the strong nuclear interactions. However, they makeperfectly good sense as a set of differential equations. Nevertheless this

68

fact means that solutions to the equations in Euclidean signature havemore physical significance than one might expect on account of the Eu-clidean path integral approach to quantum field theory. In particular,solutions that are absolute minima of the action in Euclidean signatureplay a significant role as ‘instantons’, vacua about which the theory canbe expanded.

4.2.3 Anti-self-duality and the different signatures

Minkowski space M is R4 together with a metric ηab = diag(1,−1,−1,−1)

and volume form εabcd = ε[abcd], ε0123 = 1. Euclidean space E andultrahyperbolic space U have the volume form and metrics of signature(+ + ++) and (+ +−−) respectively.

On 2-forms, Fab we can define the duality operation

F ∗ab =

12εabcdF

cd

and it can be seen (exercise) that F ∗∗ = −F in Minkowski signature,whereas F ∗∗ = F in Euclidean and ultra hyperbolic signature. Thusthe eigenvalues are ±i in Minkowski signature, but ±1 in Euclidean andultrahyperbolic signature. The duality operation is also conformallyinvariant on 2-forms. In terms of two component spinors (cf. Huggett& Tod 1994) the decomposition into eigenspaces of ∗ corresponds to thespinor decomposition of 2-forms

FAA′BB′ = εABφA′B′ + εA′B′φAB ,

2φAB = FAA′BB′

, 2φA′B′ = −FAA′BB′ ,

where the first term is self-dual and the second anti-self-dual. We oftenwrite F = F+ + F− for the decomposition of a 2-form into its SD andASD parts.

The anti-self-dual Yang–Mills equations are the conditions that F ∗ =−F in Euclidean or ultra-hyperbolic signature, orF ∗ = −iF inMinkowskisignature, i.e. F = F−.

Solutions to the ASDYM equations are also solutions to the full YMequations as the full YM equations can be written as D ∧ F ∗ = 0, butD ∧ F ∗ = −D ∧ F by anti-self-duality and D ∧ F = 0 by the Bianchiidentity. Note also that the equations are conformally invariant (like thefull Yang–Mills equations) as the ∗ operator is conformally invariant on2-forms in four dimensions.

If one changes the sign of the volume form, a self-dual 2-form becomesanti-self-dual and vice-versa. Thus there is not much to choose betweenthe self-dual and anti-self-dual Yang–Mills equations. However, when


one has also made a choice of complex structure on E or U, there isa natural choice of orientation and the anti-self-dual Yang–Mills con-nections are automatically also holomorphic. For this reason it is moreusual to work with the anti-self-dual Yang Mills equations.

4.2.4 Anti-instantons

Definition 3 Instantons are defined to be the absoluted minima of theaction S =

∫R4(F, F )ν where (F, F ) = −tr(FabF

ab) is a positive definitemetric on the curvature and the signature is taken to be Euclidean.

These relate to the ASDYM equations on S4 since, firstly a theorem ofUhlenbeck’s shows that if the action is finite, then the gauge field extendsto S4 = R

4 ∪ point (the identification here is given by stereographicprojection), Uhlenbeck (1982). Further the action

S =∫S4

(F, F )ν =∫S4

((F+, F+) + (F−, F−))ν ,

whereas

8π2k =∫S4

((F+, F+)− (F−, F−))ν

is a topological invariant, the second Chern class, or instanton number.Assuming this to be negative (we wish to work with ASD fields)

S = −8π2k + 2∫S4

(F+, F+)ν

so that F+ = 0 clearly gives absolute minima for a bundle in a giventopological class (with k negative).

Any solution of the ASDYM equations on R4 with finite action is

therefore an anti-instanton. For a full discussion of instantons and theirtwistor theory, see Atiyah (1979).

Reduction by 1 symmetry

The simplest way to impose a symmetry is to drop the dependence onx0 = t. In this case one must also restrict the gauge transformations sothat they too do not depend on t and this implies that the componentΦ = A0 transforms homogeneously under gauge transformations Φ →gΦg−1. Then the data reduces to (Di,Φ), where Di is a connection

70

on the bundle E over R3 and Φ is a section of E ⊗ E∗. The ASDYM

equations reduce to

Fij = εijkDkΦ , or F =∗ DΦ

where now ∗ is the 3-dimensional ∗-operator. This makes sense both fora reduction from E and from U, and we get equations on E

3 and R2+1

respectively.The natural boundary condition on E

3 is that the ‘energy’

E =∫

E3(Fij , F

ij) + (DiΦ,DiΦ)dvol3

should be finite. The solutions will be trivial unless one also requires that−tr(Φ2) → 1 as |x| → ∞. This characterizes the monopole solutions.These also admit topological invariants of a more subtle kind; considerthe n = 2 case. Then the eigenspaces of Φ for large enough |x| arecomplex line bundles on S2 and these have integral Chern class

c1 = limr→∞

14π

∫|x|=r

tr(ΦF ) ,

the monopole number.On R

2+1, the equations are evolution equations and the finite energycondition is a condition that one would like to impose on data on aninitial 2-dimensional hypersurface. There are lump solutions here, butthe lump number is not understood as a topological invariant.

4.3 ASDYM as an integrable system

One feature of integrable systems is the ease, despite their nonlinearity,with which one can write down exact solutions. The simplest of these forthe ASDYM equations is the t’Hooft ansatz. This has a neat geometricinterpretation in terms of spinors in curved space-time. Consider thecurved metric ds2 = φ2δ where δ is the flat Euclidean metric. Theconnection on the spin bundle for ds2 can be represented in terms of theflat metric by

∇AA′αB′ = ∂AA′αB′ − αA′∂AB′ log φ+12αB′∂AA′ log φ

where the flat space spinor αA′ has been identified with the spinorφ1/2αA′ for ds2. The gauge potential can be alternatively written as

AAA′B′C′

= −εA′ (C′∂A

B′) log φ ,


or

A = iσ ·∇ log φdt+ i(σ ∧∇ log φ+ σ∂t log φ) · dx ,

where σ = (σ1.σ2, σ3) are the Pauli matrices.The Weyl curvature for ∇AA′ is zero as it is conformally flat, so the

curvature is

[∇a∇b]αC′ = −εA′B′ΦABC′D′αD′ − 2ΛεABα(A′εB′)C′

and one can see that if 4Λ = ∂a∂a log φ + (∂a log φ)(∂a log φ) = 0 theconnection is ASD. However, this will be the case if φ = 0. Thus, anynon vanishing solution of the Laplacian on E gives rise to a solution ofthe ASDYM equation (locally) so long as φ = 0.

The solutions φ = 1 +∑k

i=1 λi/|x − ai|2, λi > 0 are particularlysignificant as the singularities in φ do not lead to singularities in theASDYM field and they are nowhere vanishing and lead to anti-instantonsolutions on S4 with instanton number k as described above. These arenot all the instanton solutions except for k = 1, 2. In general there is an8k − 3-dimensional moduli space but the other solutions are harder todescribe, although they do have a complete description in terms of somenonlinear algebraic equations via the ‘Atiyah–Drinfeld–Hitchin–Manin(ADHM)’ construction, see Atiyah (1979).

4.3.1 The ASDYM Lax pair

The spinor correspondence leads to coordinates in which the anti-self-duality condition can be seen to be a flatness condition on a certainfamily of 2-planes called α-planes.

xAA′=

1√2

(x0 + x1 x2 − ix3

x2 + ix3 x0 − x1

)=

1√2(tI + r · σ) =

(z w

w z

)where the xa are rectilinear Minkowski space coordinates. In the(z, w, z, w) coordinates, the metric and volume form are

ds2 = 2(dzdz − dwdw) , ν = dw ∧ dw ∧ dz ∧ dz .

We will usually consider either Euclidean or ultra hyperbolic signa-ture which are obtained with z = z and w = ∓w respectively. Withthese reality conditions this choice of coordinates expresses the ds2 as a(pseudo-) Kahler metric with (pseudo-)Kahler form

iω = dw ∧ dw − dz ∧ dz

72

where the reality conditions z = z, w = ∓w are imposed for E, U

respectively. Ultrahyperbolic signature is also obtained with (z, w, z, w)all real.

Define the vector fields l and m by

l = ∂w − λ∂z , m = ∂z − λ∂w, .

Then the bivector l∧m is SD for all λ and varies over all such as λ variesover the Riemann sphere. Such two-planes are called α-planes. Theseare generally complex 2-planes, except in U where they can be real when(z, w, z, w) and λ are all real.

Self-dual bivectors are orthogonal to ASD 2-forms and indeed theASD 2-forms F are characterized by the condition that F (l,m) = 0.This gives:

Proposition 1 The ASD Yang–Mills equations on a connection areequivalent to the condition that it is flat on α-planes, i.e. that

[Dw − λDz ,Dz − λDw] = F (l,m) = 0.

Note that this integrability condition is equivalent to the existence ofsolutions to the equations

(Dw − λDz)ψ = (Dz − λDw)ψ = 0.

where ψ = ψ(xa, λ) is a basis of E at each point.

4.3.2 Backlund transformations

One of the first applications of a Lax pair is to perform Backlund trans-formations. One approach to these is to perform a gauge transformationto the Lax pair,

(Dw − λDz ,Dz − λDw)→ (D′w − λD′

z ,D′z − λD′

w)

= g(Dw − λDz ,Dz − λDw)g−1

where g depends on λ. This is well defined and will lead to a new solutionto the ASDYM equations if g (Dw −λDz ,Dz −λDw) g−1 is linear inλ; it must necessarily still commute and so will determine an ASDYMsolution since, if a Lax pair with its linear dependence on λ is given, theconnection coefficients can be reconstructed from it.

An example is

g =

(1

λ−µA 00 A

)


for some 2× 2 block decomposition relative to k+ k = n, A being k× k

and A being k × k and µ being a complex constant. In general it isnecessary to solve linear differential equations in order to find a gaugeand A for which this works. These are already solved if we are given asolution ψ(xa, λ) to the Lax pair.

We illustrate the technique in the case k = k = 1, n = 2, and g =diag(1, λ). If we put

Φ =(a b

c d

), then gΦg−1 =

(a λ−1bλc d

).

We need to ensure that there are no terms of order λ−1 or λ2 in thetransformed Lax pair. This will be the case if we can find a gaugein which Aw and Az are lower triangular and Aw and Az are uppertriangular. We can do this by performing a gauge transformation to anew frame (e1, e2) such that Dze1 ∝ Dwe1 ∝ e1 and Dze2 ∝ Dwe2 ∝ e2.The existence of such e1 and e2 follows from the integrability conditions[Dz,Dw] = 0 = [Dz,Dw], and indeed, if we have a solution ψ to the Laxpair that is regular both near λ = 0 and λ = ∞, then we can take onecolumn at λ = 0 for e1 and the other column for e2 near λ = ∞. Inthis gauge one can now perform the ‘singular gauge transformation’ andobtain a new solution. Moreover, one can continue the process since,in the new gauge, ψ is lower triangular by construction at λ = 0 andupper triangular at λ = ∞ so that ψ′ = gψg−1 is regular at λ = 0,∞and satisfies the Lax pair with the new potentials. Thus one has thenecessary ingredients to perform the procedure again and again.

These Backlund transformations are often thought of as ‘hidden sym-metries’, they are transformations from one solution to another that donot simply arise from point transformations of space-time. However, onecan understand them as an (infinite dimensional) group of symmetriesand investigate their linearized action.

4.3.3 Solution generation

The above works even if we start with the trivial solution and one can,instead of iterating, work harder to construct a more general g. Thiscan be presented somewhat differently as follows. Note that if one isgiven ψ, then the connection is determined by

((∂w − λ∂z)ψ)ψ−1 = Aw − λAz

((∂z − λ∂w)ψ)ψ−1 = Az − λAw . (4.1)

74

One approach to constructing exact solutions is to make an ansatz forψ so that the RHS of the equations above are linear in λ thus allowingone to construct a connection A for which the Lax pair commutes. Afruitful ansatze is

ψ = I +k∑

i=1

Mi(x)λ− µi

.

If we wish the connection to be unitary, we require that ψ(x, λ)ψ∗

(x,∓1/λ) = αI in E or U respectively where α is a scalar indepen-dent of λ (when the coordinates are all real on U one can also useψ(x, λ)ψ∗(x, λ) = α1). The condition that the RHS of equation (4.1) hasno poles gives further conditions: theMi should have rank 1, Mi = ui⊗viwith the ui determined in terms of vi (or vice versa) and the vi dependon the coordinates only through (z + µiw, w + µiz).

4.3.4 Recursion operators

The Arnol’d-Liouville definition of integrability can be obtained when asystem has the additional structure of a recursion operator.

Definition 4 Let (M2n,Ω) be a finite dimensional phase space withHamiltonian H and corresponding Hamiltonian vector field X, X Ω+dH = 0 and Poisson bracket h1, h2 = Πab(∂ah1)(∂bh2), Π = Ω−1. Arecursion operator on M is a section Ra

b ∈ Γ(End(TM)) such that(1) a, + bR, , where Rh1, h2 = ΠabRc

a(∂bh1, ∂ch2), is a Poissonstructure ∀ constants a and b, and(2) X is also Hamiltonian with respect to R, .

Proposition 2 If a (simply connected) Hamiltonian system admits asuitably non-degenerate recursion operator, then the system is Arnol’d-Liouville integrable.

Proof: note first that since , is non-degenerate then so is , + tR, for small t and corresponds to the symplectic form

Ωt = (1− tR)−1Ω =∞∑i=0

tiRiΩ :=∞∑i=0

tiΩi

for all t. Thus each Ωi = RiΩ is closed and non-degenerate and istherefore a symplectic form. Since X is Hamiltonian wrt , and R, ,LXΠ = LXRΠ = 0 so that LXR = 0 and hence LXΩi = 0. Hence


X is Hamiltonian with respect to each Ωi with Hamiltonian hi, dhi =−Ωi(X, ) = −Ω(RiX, ). Furthermore, hi, hj = 0 since

hi, hj = Ω(RiX,RjX) = Ωj+k(X,X) = 0

by skew symmetry of Ω and the property RacΩab = −Ra

bΩac that followsfrom skew symmetry of RΩ etc..

Thus, if n of the hi are independent, then we have an Arnol’d-Liouvilleintegrable system (and this is what is meant by the ‘suitable’ in theassumption of non-degeneracy of R).

Example. In the case of the KdV equations, M = S∞(R) the space ofsmooth functions on the line with rapid decrease at ∞.

We have two Poisson structures

F [u], G[u]1 =∫

R

δF

δu(x)∂x

δG

δu(x)dx

F [u], G[u]2 =∫

R

δF

δu(x)(14∂3x + u∂x +

12ux)

δG

δu(x)dx .

The Hamiltonian h0 =∫

12u

2dx generates translations in x wrt , 1and the KdV flow wrt , 2. Thus the system is bi-Hamiltonian withrecursion operator

R = (14∂3x + u∂x +

12ux) ∂−1

x .

Proposition 3 (Magri, 1978) R is a recursion operator for the KdVflow.

We obtain higher Hamiltonians

h1 =∫

18(2u3 − u2x)dx

h2 =∫

132

(5u4 − 10uu2x + uxx2)dx

h3 =∫

1128

(14u5 − 70u2u2x + 14uuxx2 − uxxx2)dx

and flows

∂1u = Rux =14(∂3xu+ 6uux)

∂2u = R2ux =116

(∂5xu+ 20uxuxx + 10uuxxx + 30u2ux)

∂3u = R3ux =164

(∂7xu+ 42uxuxxxx + · · · )

76

We will not prove this, but instead show that ASDYM has a recursionoperator and that the KdV equation is a symmetry reduction of theASDYM equations. The ASDYM recursion operator will then reduce tothe above for KdV. Other reductions of ASDYM will similarly inheritthis structure. In order to do this we will need to introduce two potentialformulations for the ASDYM equations.

Potential forms of the ASDYM equations

The Lax pair is a family of integrability conditions and these can beused in order to simplify the dependent variables from the four con-nection matrices down to one potential matrix. Such potentials areimportant both for making contact with hermitian geometry and withLagrangian/Hamiltonian formulations. There are two prominent suchpotentials, which we will refer to as K-matrices and J-matrices respec-tively. They are both obtained by use of the ASDYM equations in theabove coordinates:

[Dw,Dz] = 0 , [Dw,Dw]− [Dz,Dz] = 0 , [Dz,Dw] = 0 . (4.2)

The derivation of both forms starts with the observation that the firstof these implies that there exists a frame for the bundle (a gauge) whosebasis vectors are constant along Dw and Dz so that Dw = ∂w andDz = ∂z, i.e. Az = Aw = 0. When the coordinates are complex, thisshows that the connection is compatible with the complex structure andadmits holomorphic sections which yield a holomorphic frame.The J-matrix: the last of equations (4.2) is the integrability conditionfor the existence of a second frame that is constant along Dz and Dw,and so it will represented relative to the first by a matrix function J

such that

(∂z+Az)J = (∂w+Aw)J = 0 , so that Az = −JzJ−1 , Aw = −JwJ−1 .

This determines J up to J → h(z, w)Jh(z, w), the function h arisingfrom the freedom in the choice of holomorphic frame. The final equationyields

∂z(JzJ−1)− ∂w(JwJ−1) = 0 . (4.3)

If we choose a reality structure in which the above complex coordinatesyield a Kahler metric, and choose unitary gauge group, then J can bechosen to be the hermitian metric expressed in a holomorphic frameand there is a standard expression for the curvature in this frame, F =∂((∂J)J−1) where d = ∂ + ∂ is the usual decomposition of d into its


holomorphic and anti-holomorphic parts. Then (4.3) is just ω ∧ F = 0.This is a natural equation for a hermitian metric on a holomorphic vectorbundle over a Kahler manifold in 4 dimensions.

The K-matrix: to obtain the the K-matrix potential we start with thegauge in which Aw = Az = 0 and use the second equation of (4.2) todeduce that

∂wAw − ∂zAz = 0

which implies that there exists a potential K taking values in the (com-plexified) Lie algebra of the gauge group such that

Aw = ∂zK , and Az = ∂wK .

K is unique up to

K → hKh−1 + zh∂wh−1 + wh∂zh

−1 + f(z, w) , where h = h(z, w).

The last ASDYM equation then implies that

(∂z∂z − ∂w∂w)K + [∂wK, ∂zK] = 0 .

Lagrangians

Although there is a Lagrangian and corresponding phase space frame-work for the full Yang–Mills equations, that Lagrangian does not leadto a phase space framework (i.e. Hamiltonian and symplectic form) forthe ASDYM equations. However, there are Lagrangians for each of thepotential forms of the equations given above.

The K-matrix Lagrangian is

L[K] =∫

R4tr((∂aK)(∂aK) +

13K[∂zK, ∂wK]) dwdzdwdz .

The J-matrix Lagrangian doesnt have such a straightforward formula.The simplest, due to Donaldson, is

L[J ] =∫

xaxatr(F ∧ F ) .

It can also be written using a ‘Wess–Zumino–Witten (WZW)’ term, orby expressing J = UL−1 where U is upper triangular with 1’s down thediagonal and L is lower triangular. Then

S[J ] =12

∫tr(2U−1∂U ∧ L−1∂L− L−1∂L ∧ L−1∂L) ∧ ω .

78

In order to develop the theory, we only need to know that

δS = −∫

tr(J−1δJ)∂(J−1∂J) ∧ ω .

The Hamiltonian formulation

Neither E or U signature gives rise to an evolution equation equationalthough one cast the equations as evolution equations in the complex.Nevertheless, one can use the two Lagrangians to obtain two formalexpressions for the symplectic structures and Hamiltonians. These giverise to different symplectic structures which are compatible in the senserequired of a recursion operator.

Formally, these Lagrangians endow the phase space

M = Space of solutions to ASDYM eqs./ gauge

with a symplectic structure (here we are being sloppy about the residualgauge freedom in the potential forms of the equations which must befactored out to obtain the space of solutions to the ASDYM equations— see Mason & Woodhouse (1996) for a more careful treatment).

The tangent space to M at a given solution A ∈ M is the space ofsolutions δA to DδA = −∗DδA. It can be identified with perturbationsJ−1δJ to the J- or K-matrix equations. These satisfy

DaDaδK = 0, DaDaJ−1δJ = 0

and so both potential forms identify TAM with

W = space of solutions φ to DaDaφ = 0 .

The symplectic structure in both cases is

Ω(φ1, φ2) =∫φ1∗Dφ2 − φ2

∗Dφ1 .

One can check that it is closed etc.However, J and K give two different maps j, k :W → TAM with

j(φ) = Dwφdw +Dzdz , k(φ) = Dzφdw +Dwφdz .

These maps can be used to push forward Ω to give two different sym-plectic forms on TAM . Alternatively it defines the symplectic form anda recursion operator R = k j−1.

Either Lagrangian can be used to obtain the Hamiltonians that gen-erate space-time translations, but the K-matrix version is easiest to use


and gives

HV =∫

V btr∂aK∂bK −12δab ∂

cK∂cK +43αacK[∂bK, ∂cK]

−43αcdδabK∂cK∂dK)d3xa ,

where αabdxadxb = dw∧dz, for the Hamiltonian generating flows alongthe vector V = V a∂a,

Proposition 4 The symplectic structure above and R together endowthe space of solutions to ASDYM with a recursion operator for which theflows along the coordinate axes are biHamiltonian.

Sketch proof: note that identifying W with TAM using k gives therecursion operator (also denoted R) R = j−1k on W

φ′ = Rφ ⇔ DA0φ = DA1φ′ .

To prove the result we need to show first that Ω(Rφ, φ′) = Ω(φ,Rφ′).This follows from an integration by parts (exercise). From this it followsthat the forms Ωk(φ, φ′) = Ω(Rkφ, φ′), k ∈ Z are skew. One must checka further condition to show that these forms are closed, see Mason andWoodhouse (1996) for full details.

ASDYM hierarchies and generalizations

As before for the KdV equation, ∂aK defines a linearized solution onwhich R can act and so we can use Ri∂a(K) to define flows of a hierarchy.Write, for a = 0, 1, i = 0, 1,

xAi =(z w

w z

).

We discover that the field equations are equivalent to

∂K

∂xA1= R

(∂K

∂xA0

)(exercise). We can now define higher flows by setting

∂K

∂xA,i+1= R

(∂K

∂xAi

).

From the definition of the recursion operator this can be recast as theequations on K

∂K

∂xA,i+1= R

(∂K

∂xAi

)which implies DB0

∂K

∂xA,i+1= DB1

(∂K

∂xAi

).

80

These equations have the Lax system

LAi = ∂Ai +AAi − λ∂A,i−1 where AAi = ∂A,i−1K .

These equations are a hierarchy for the ASDYM equations.The above formulation of the ASDYM hierarchy arose from a special

gauge choice. To remove this, we can consider the Lax system

LAi = ∂Ai +AAi − λ(∂A,i−1 + AA,i−1)

which can be reduced to the first by using the commutativity conditions.Note that i can be continued to negative values of i also. The conditionthat this Lax system commutes, then, gives the field equations of theASDYM hierarchy.

A final generalization is to let the index A extend from 1, . . . , N also.This gives the generalised ASDYM hierarchy. This is particularly nat-ural when i ranges over just 2 values and N = 2k as the Lax systemarises from a connection on a 4k-dimensional space and with appropri-ate reality conditions determines a hypercomplex connection on bundleson H

k, i.e. a connection that is holomorphic with respect to the threecomplex structures of a quaternionic Kahler manifold. Thus this systemdefines a hierarchy for the equations on a connection compatible witha hypercomplex structure (a specification of three complex structures(I, J,K) that satisfy the quaternion relations).

4.4 Reductions of the ASDYM equations

The easiest way to spot reductions is simply by comparison of the LaxPair of the desired system with that of the ASDYM equations; oneneeds only to express the Lax pair in a form that is linear in λ withappropriate derivatives and matrices. (As an exercise the reader mightlike to reformulate the Lax pair for the Euler top so that it has thisproperty.) This is effective but one would often like to do more. Onewould like to understand the precise natural conditions on the ASDYMfield so that it reduces to the required sytem. One would also like tobe able to express the dependent variables of the reduced system asgauge invariants that determine the full connection so that there is a1-1 correspondence between gauge equivalence classes of solutions to theASDYM equations and the space of solutions to the integrable system inquestion. Also, there are some surprises upon symmetry reduction. Onediscovers that the reduced equations often have additional symmetries,


sometimes infinite-dimensional. Sometimes two quite different routesgive the same equation.

Reductions are classified by the ingredients involved, i.e.

• A choice of gauge group.• A choice of subgroup H of the conformal group and action on the

bundle.• A choice of reality structure.• A choice of potential formulation or gauge.• A choice of certain constants of integration that arise in the reduction

process.

For each generator h of H one must have an action Lh on the bundlesatisfying Lh(fs) = h(f)s+fLhs. This gives rise to a natural endomor-phism of the bundle, Φh = Dh − Lh ∈ Γ(End(E)). We refer to Φh as a‘Higgs field’ since this is precisely how Higgs fields arise in Kaluza–Kleintheory. An invariant gauge is one in which the frame is Lie derived alongh, so Lh = h and Φh = A(h) (cf. the discussion of monopoles).

Many examples are described in Mason & Woodhouse (1996). Herewe will examine the reduction by a null translation, and then a furthertranslation to give the KdV and nonlinear Schrodinger (NLS) equations.Finally we consider the Ernst equations. These all illustrate interestingfeatures. The case of the Painleve equations will be the subject of thenotes in this volume by N. M. J. Woodhouse. We will consider onlygauge groups that are real forms of SL(2,C).

Reduction by a null translation

Let us choose gauge group SL(2,R), the totally real coordinates on U

and symmetry in the ∂z direction with associated Higgs field Φ. Notethat in general one of the ASDYM equations reduces to the conditionDwΦ = 0 and this implies that the invariants tr(Φr) are functions of(z, w) alone. These are constants of integrations; the remaining equa-tions are deterministic on the remaining variables.

Remark. The K-matrix equation is

∂w∂wK + [∂zK, ∂wK] = 0

and this equation exhibits a new larger coordinate freedom, (w, z, w)→(w′, z′, w′) = (f(w, z), z, w) for arbitrary f , (use (∂w′ , ∂z′ , ∂w′) =

82

(f−1w ∂w, ∂z − fzf

−1w ∂w, ∂w)). Under this transformation Φ = Kw →

Φ/fw.

To reduce the equations, choose an invariant gauge such that Aw = 0also. The residual gauge freedom is now given by g = g(z, w). We thenfind the reduced equations

∂wΦ = 0 , DzΦ + [∂w,Dw] = 0 , [Dz,Dw] = 0 .

The first equation implies that Φ is independent of w. The residualgauge freedom can be used to reduce Φ to normal form, and there aretwo nontrivial cases.

(i) trΦ2 = 0 The normal form here is Φ =(0 01 0

)with residual

gauge freedom g =(

1 0h(w, z) 1

). If we set Aw =

(q p

r −p

), we

dicover that ∂wp = 0. Thus p is another constant of integrationand can be chosen to be −1 (or alternatively we can, when p = 0reduce p to −1 by means of the coordinate freedom). We thenobtain ∂w(r + ∂wq + q2) = 0 and we can use the residual gaugefreedom to set r+ ∂wq+ q2 = 0. With this, and further work, wediscover that all entries of Aw and Az are determined in terms ofq, and

4qwz − qwwww − 8qwqww − 4qwqww = 0. (4.4)

The residual gauge freedom now yields q → q + a(z).(ii) For tr(Φ2) = 0, as above, we can either consider tr(Φ2) as a con-

stant of integration and choose it to be −2 in order to obtainautonomous equations, or we can perform the coordinate trans-formation above to reduce it to that value. Then the gauge canbe used to reduce Φ to Φ = diag(i,−i) with residual gauge free-dom consisting of diagonal matrices. This can be used to reduce

the diagonal entries of Aw to 0, so that Aw =(0 q

p 0

)for some

p amd q. The equations imply that Az = 14

(−V 2qw−2pw V

)for

some V and

Vw = −2(pq)w , 2qz = qww + qV , 2pz = −pww − pV .

This has real form, starting from an SU(2) connection on U, and


replacing (z, w, w) by (t, x, y),

iψt = ψxy + V ψ , Vx = 2(|ψ|2)y . (4.5)

These are degenerate versions of the KP and Davey–Stewartson equa-tions respectively. (The full forms of the KP and Davey–Stewartsonequations are unlikely to be reductions of the Yang–Mills equations witha finite dimensional gauge group, see Mason (1995) for a more extendeddiscussion, although they do appear in Ablowitz & Clarkson (1991) asreductions from ASDYM with an infinite dimensional gauge group.)

With a further symmetry along ∂w − ∂w it can be readily seen thatthese reductions give the non-linear Schrodinger and KdV equations re-spectively. Putting x = y (= w = w), then for KdV, we put u = qx andthen (4.4) reduces to the KdV equation, 4ut−uxxx−12uux = 0. For thenonlinear Schrodinger equation, it is clear from (4.5) that with x = y,V = |ψ|2 + f(t) for some f(t) and this f can be reduced to zero by aresidual gauge transformation yielding the non-linear Schrodinger equa-tion (NLS), iψt = ψxx + |ψ|2ψ. These reductions extend to reductionsof the hierarchy of the ASDYM equations to that of KdV and NLS.

Alternatively, with different gauge choices, and a further symmetryalong ∂z, we obtain the Sine–Gordon equation and nonlinear σ modelswith values in SU(2) and Hitchin’s Higgs bundle equation. This lattersystem now has conformal invariance in 2-dim, (w, w) → (f(w), f(w))whose origins can already be seen in the infinite dimensional symmetrythat arises after the null translation reduction above. Higher rank gaugegroups yield systems such as the Toda field theory and harmonic mapsinto symmetric spaces. Reduction by a further translation yields theEuler spinning top equations and large families of generalizations.

Non-autonomous reductions

If we impose symmetries that are not translations, the reduced equationswill generally no longer be translation invariant. With one symmetryaround a rotation in a 2-plane in E, one obtains the Bogomolny equationson hyperbolic 3-space, i.e. the equations FA =∗ DAΦ on a connection A

and Higgs field Φ where now the ∗ is that of hyperbolic 3-space. Witha further symmetry and gauge group SL(2,C) one obtains the Ernstequations, the Einstein vacuum equations from general relativity withtwo commuting symmetries and a 2-surface orthogonality condition, andwith a 3rd symmetry (still with SL(2,C) gauge group, we obtain thePainleve equations.

84

The Ernst equations

We first discuss the Ernst equations in a form due to Ward (1983). Weconsider a metric on a space-time with two commuting symmetries suchthat the 2-planes orthogonal to the symmetries are integrable. If weintroduce coordinates (x, r, yi) such that ∂/∂yi are the symmetries, themetric can be put in the form

ds2 = Jijdyidyj − Ω2(dx2 + dr2) where det J = −r2 .

The Einstein vacuum equations then reduce to

∂x(rJ−1∂xJ) + ∂r(rJ−1∂rJ) = 0 ,

2∂ξ(log rΩ2) = irtr((J−1∂ξJ)2) , ξ = x+ ir .

The equation for Ω is linear and integrable (it is overdetermined) as aconsequence of the equation for J , and so one usually focuses on the firstequation.

This can be reduced from Yang’s J-matrix equation version of theASDYM equations by setting z = x + it and w = reiθ and imposingsymmetries along ∂t and ∂θ. It is easily seen that the equation thenreduces to the above.

It can be shown that a version of the Backlund transformation intro-duced earlier gives rise again to this same reduction of Yang’s equation,but now in terms of the twist and magnitude of one of the Killing vectors,and this is in fact Ernst’s original form of these equations. The interplaybetween these two representations is nontrivial and can be used to buildthe Geroch group of transformations of solutions to these equations (agauge transformation of one does not lead to a gauge transformation ofthe other).

The Painleve equations

The Painleve equations arose from Painleve’s classification of all secondorder ODEs that satisfy the Painleve property that solutions admit nomovable branching or essential singularities. This led to a list of sixnew equations that could not be reduced to known equations and whosesolutions define new transcendental functions. It turns out that thereare 5 distinct abelian 3-dimensional subgroups of the conformal groupthat satisfy a certain non-degeneracy property. Reduction of SL(2,C)ASDYM by the first of these yields Painleve’s first and second equations,according to a choice of normal form of a null Higgs field. Each of the


remaining four Painleve equations correspond to each of the remaining4 groups. For a full discussion see Mason & Woodhouse (1996).

4.5 Further topics

Clearly many more equations can be obtained by considering otherchoices of symmetries and gauge groups. The aim in Mason & Wood-house (1996) is to show how familiar examples arise and are related toeach other and to explore some related equations that are obtained whenmore general choices are obtained rather than develop a full classifica-tion. A full classification is likely to be rather unwieldy as, for example,there are more than 50 distinct (families of) 2-dimensional subgroups ofthe conformal group, and, for higher dimensional Lie groups, the classi-fication of conjugacy classes of elements of the Lie algebra, as requiredfor the choices of normal forms of Higgs fields, is already complicated.

Another related programme is to consider reductions of the anti-self-dual vacuum Einstein equations (ASDVE equations) on a metric. Suchmetrics have vanishing Ricci tensor and anti-self-dual Wey tensor. In Eu-clidean signature this condition is equivalent to the conditions that thereare three Kahler structures, (I, J,K), satisfying the quaternion relationsand so such metrics are known as hyperkahler. These equations are alsointegrable in the sense that there is a Lax pair and twistor construction.The equations can be expressed as a reduction of ASDYM, but onlyif infinite-dimensional gauge groups (diffeomorphism groups) are used,see Mason & Newman (1989), Ward (1990). The scheme of reductionsof these equations with one and two symmetries is now reasonably wellunderstood.

A spinoff from understanding the various chains of reductions is thatby knowing solutions to a reduction, one can deduce solutions to theoriginal equation, so for example, by knowing that the Painleve equa-tions are reductions of the Ernst equations, one can find solutions to thefull Einstein vacuum equations in terms of the Painleve transcendents,Calvert & Woodhouse (1997).

The most important corollary of the the realization of integrable sys-tems as reductions of the ASDYM or ASDVE equations is the existenceof twistor correspondences for integrable systems for all these equations.Indeed, many equations that are unlikely to be reductions of the AS-DYM or ASDVE equations nevertheless admit twistor correspondencesin some form.

The second part of Mason & Woodhouse (1996) is concerned with

86

twistor theory. As mentioned above there are twistor constuctions forthe ASDYM equations and the ASDVE equations. In the first case, solu-tions to the ASDYM equations are transformed via the Ward transformto holomorphic vector bundles over regions in CP

3 (complex projective3-space which in this context is twistor space), and in the second, solu-tions correspond to deformations of the regions in twistor space them-selves. These constructions amount to a geometric general solution ofthe equations in the sense that the data on twistor space is freely pre-scribable (although the reconstruction of the solution on space-time isfar from trivial). It emerges that one can see that much of the standardmachinery of integrable systems theory arises from these constructions.In particular methods based on the Riemann–Hilbert problem such asdressing procedures and the inverse scattering transform are specializa-tions of the twistor constructions.

It is difficult for twistor theory to improve on existing methods for dif-ferential equations with a well developed theory such as the KdV equa-tions. The real benefit of the interactions has been in the other direction;well developed theory for equations such as the KdV equation can beapplied in certain cases to the full ASDYM equations or other reductionsusing twistor theory. For example, the spectral curve construction formonopoles, Hitchin (1982), extends a technique for what are in effect in-tegrable ordinary differential equations to equations in three dimensionssubject to boundary conditions. Another example, is proposition 10.5.1in Mason & Woodhouse (1996) which gives a construction for global so-lutions to the ASDYM equations in split signature based on the inversescattering tranform for global solutions to the KdV equations on theline. One might hope that ideas from the theory of integrable systemswill eventually provide some of the techniques that twistor theory needsto realize its more fundamental aspirations in mathematical physics.

4.6 Exercises

(i) Check that the KdV and KP Lax pairs do indeed give rise to thecorresponding equations and identify the various entries in thematrices or operators in terms of the KdV potential q or the KPu.

(ii) Show that F ∗∗ = ±F depending on the signature.(iii) Show that F → F ∗ interchanges the electric and magnetic fields

with factors of ±1 or ±i and determine the different cases in thecorresponding signatures.


(iv) Find an expression for the t’Hooft ansatze solutions using ordi-nary space-time derivatives and Pauli matrices. Check directlythat the ASDYM equations follow from the Laplacian on φ.

(v) Show in the case of the k = 1 instanton solutions using the t’Hooftansatz that it is regular at a. Use the gauge transformationxAA′

/|x| to show that there exists a gauge in which A → 0 as|x| → ∞. (If you’re feeling brave, check by direct integrationthat the instanton number is −1.)

(vi) Make the ansatz Φ = f(r)x · σ and A = g(r)x ∧ σ where σ =(σ1, σ2, σ3) are the Pauli sigma matrices and solve the monopoleequation. This should give the 1 monopole solution.

(vii) Work through the conditions arising from the solution generatingansatze on U with real coordinates in the case n = 2, k = 1,and with v depending only on ω = µ(µz + w) + µw + z = µ2z +µ(w+w)+ z to determine a ψ and a corresponding solution to theASDYM equations. Put 2x = w+ w and t+ y = z, t− y = z andshow that following the ansatze through will lead to a solutionto the R

2+1 solution to the Bogomolny equation. Note that thissolution only depends on (x, y), i.e. is static when µ = i. [Hint:first try to show that

ψ = 1 +µ− µ

λ− µ

(v∗ ⊗ v

v · v∗

)satisfies the required conditions, where v = v(µz + w, µw + z).For a simple special example, put v = (1, ω).]

References

Ablowitz, M. & Clarkson, P. A. (1991). Solitons, nonlinear evolutionequations and inverse scattering, LMS lecture notes, 149, CUP.

Atiyah, M. F. (1979). Geometry of Yang–Mills fields, Academia NazionaleDei Lincei, Scuola Normale Superiore, Pisa.

Calvert, G. & Woodhouse, N. M. J. (1997). Painleve transcendents andEinstein’s equations, Class. & Quant. Grav., 13, No 4, L33-9.

Hitchin, N. J. (1982). Monopoles and geodesics, Comm. Math. Phys.,83, 589-602.

Huggett, S. A. & Tod, K. P. (1994). An introduction to twistor theory,LMS student texts, 4, CUP.

88

Magri, F. (1978). A simple model of the integrable Hamiltonian equa-tion, J. Math. Phys., 19, 1156-62.

Mason, L. J. (1995). Generalized twistor correspondences, d-bar prob-lems and the KP equations, in Twistor Theory, ed. S. Huggett, LectureNotes in Pure and Applied Mathematics, vol. 169, Marcel Dekker.

Mason, L. J. & Newmam, E. T. (1989). A connection between theEinstein and Yang–Mills equations, Comm. Math. Phys., 121, 659-68.

Mason, L. J. & Woodhouse,, N. M. J. (1996). Integrability, self-dualityand twistor theory, LMS Monographs, OUP.

Uhlenbeck, K. (1982). Removable singularities in Yang–Mills fields,Comm. Math. Phys., 83, 11-29.

Ward, R. S. (1983). Stationary axisymmetric space-times: a new ap-proach, Gen. Rel. Grav., 15, 105-9.

Ward, R. S. (1985). Integrable and solvable systems and relations amongstthem, Phil. Trans. Royal Soc., 315, 451-7.

Ward, R. S. (1990). The SU(∞) chiral model and self-dual vacuumspaces, Class. & Quant. Grav., 7, L217-22.

Ward, R. S. & Wells, R. (1990). Twistor geometry and Field theory,CUP.

5

Curvature and integrability for Bianchi-typeIX metrics

K. P. TodMathematical Institute, St Giles, Oxford OX1 3LB

Abstract

In this seminar, I review the various curvature conditions that one mightwish to impose on a Bianchi-type IX metric, and the direct route to theself-dual Einstein metrics obtained from solutions of the Painleve VIequation.

In four-dimensions there are several ‘nice’ conditions which one may im-pose on a Riemannian metric. For example, one may require it to beKahler, or Einstein, or to have anti-self-dual (ASD) Weyl tensor. Thesepossibilities are set out in figure 5.1, which I have used before (Tod 1995,1997). The overlapping regions in the figure also correspond to interest-ing conditions: a Kahler metric with zero-scalar curvature (‘scalar-flat’in the terminology of (LeBrun 1991)) has ASD Weyl tensor; ASD Ein-stein is known as quaternionic Kahler; ASD Ricci-flat is hyper-Kahler.Elsewhere in the top circle of the diagram are hyper-complex metrics,which have three integrable complex structures with the algebra of the(unit, complex) quaternions but are not Kahler.

Conditions on the curvature are most readily imposed using Cartan cal-culus, so suppose that (e0, e1, e2, e3) is a normalised basis of 1-forms insome Riemannian 4-manifold. Define a basis of SD 2-forms by

φ1 = e0 ∧ e1 + e2 ∧ e3

φ2 = e0 ∧ e2 + e3 ∧ e1

φ3 = e0 ∧ e3 + e1 ∧ e2

89

90

Einstein Kahler

Kahler Kahler

Kahler

Hyper

KahlerEinstein

ASD Weyl

Scalar-flatQuaternionic

Fig. 5.1. Different field equations in 4-dimensions

so that Qφi = φi, in terms of the Hodge star or dual, and a basis of ASD2-forms by

ψ1 = e0 ∧ e1 − e2 ∧ e3

ψ2 = e0 ∧ e2 − e3 ∧ e1

ψ3 = e0 ∧ e3 − e1 ∧ e2

so that Qψi = −ψi. Now define the connection 1-forms for the SD partof the Levi–Civita connection, αij = −αji, by

dφi = αij ∧ φj

and the corresponding curvature 2-forms Ωij by

Ωij = dαij − αik ∧ αkj .

Since the Ωij are 2-forms, we may expand them in terms of φi and ψi as

Ωij = Wijkφk + Φijkψk

in terms of two sets of coefficients, Wijk and Φijk. The first Bianchi

Curvature and integrability for Bianchi-type IX metrics 91

identity is

Ωij ∧ φj = 0

which implies that Wijj = 0. It follows that Wijk has 6 independentcomponents, 5 corresponding to the SD Weyl tensor and one, the totally-anti-symmetric part, corresponding to the Ricci scalar, while Φijk has 9components corresponding to the trace-free Ricci tensor.

Various field equations can now be imposed as, for example:

(i) ASD Weyl tensor iff Wijk = Λεijk where Λ is a multiple of the Ricciscalar (since this condition on the Weyl tensor is conformally-invariant,it is often convenient to choose the conformal scale so that Λ vanishes).

(ii) ASD Einstein iff Wijk = Λεijk and Φijk = 0.

(iii) Hyper-Kahler iff Wijk = 0 = Φijk.

From now on, we shall be interested just in Bianchi-type IX metrics, thatis metrics with an action of SU(2) transitive on hypersurfaces. Such ametric can be written in terms of the basis (σ1, σ2, σ3) of left-invariant1-forms on SU(2) in the form

ds2 = w1w2w3dt2 +

w2w3

w1dσ21 +

w3w1

w2dσ22 +

w1w2

w3dσ23 (5.1)

where the wi are three functions of t, and the σi satisfy

dσ1 = σ2 ∧ σ3

and cyclic permutations of this.

The virtue of writing the metric in the form of (1) is that the basis ofSD 2-forms can be taken to be

φ1 = w2w3dt ∧ σ1 + w1σ2 ∧ σ3

and cyclic permutations of this. Then it turns out that the connection1-forms αij can be written in terms of three more functions a1, a2, a3 oft as

α12 =a3w3

σ3

and cyclic permutations of this, where the ai are determined by the

92

equations

dw1

dt= w2w3 − w1(a2 + a3)

dw2

dt= w3w1 − w2(a3 + a1)

dw3

dt= w1w2 − w3(a1 + a2)

which we shall call the first system.

We may continue with the Cartan calculus and find the curvature 2-forms in terms of derivatives of the ai. If we now make a choice offield equations, then we obtain a second system of first-order differentialequations on the ai. If we choose something outside the top circle inthe figure then typically we arrive at equations that are not integrable(for Einstein–Kahler, (Dancer and Strachan 1994)) or even chaotic (forEinstein, see e.g.(Barrow 1982)). However field equations from insidethe top circle are integrable (as is to be expected from the ‘self-dualityimplies integrability’ heuristic; see e.g. Mason 1990).

We first consider the case (i) above, the case of vanishing Wijk. Thenthe second system turns out to be

da1dt

= a2a3 − a1(a2 + a3)

da2dt

= a3a1 − a2(a3 + a1) (5.2)

da3dt

= a1a2 − a3(a1 + a2)

This attractive system is widely known as the Chazy system (Chazy1910, Ablowitz and Clarkson 1991), although it was studied earlier byBrioschi (1881). We shall solve it below. Before that, we note a specialsolution: if we insist that all the ai are constant in time then without lossof generality two, say a2 and a3, must be zero. With a1 non-zero, onecan reduce the first system to a special case of the Painleve-III equation(Tod 1991). Also the form φ1 is covariant constant in this case so thatthis is a scalar-flat Kahler solution, the one first found by Pedersen andPoon (1990).


We have found the solution which in figure 5.1 lies in the intersectionon the right. The solutions in the triple intersection, the hyper-Kahlermetrics, are characterised now by vanishing Φijk. There are two classes.In the first, all ai are zero, which certainly satisfies the second system,and then the first system is solved by elliptic functions. These solutionswere found by Belinski et al (1979). In the second, for each i, ai = wi, sothat the first system reduces to the second which we shall solve below.These solutions were found by Atiyah and Hitchin (1985, 1988).

For the general solution of (2) we follow Brioschi and introduce a newdependent variable x by

x =a1 − a2a3 − a2

It is now straightforward to obtain a single third-order equation for x

d3x

dt3=

32(d

2xdt2 )

2

dxdt

− 12

(dxdt

)3( 1x2

+1

x(x− 1)+

1(x− 1)2

)and this equation is solved by x equal to the reciprocal of the ellip-tic modular function. Now the elliptic modular function has a naturalboundary in the t-plane, so that the ai and hence also the wi have a nat-ural boundary in the t-plane, whose location depends on the constantsof integration i.e. a movable natural boundary. This suggests that theseself-duality equations are not integrable despite what was said above(integrability in the context of ODEs is usually regarded as being equiv-alent to the Painleve property, that all movable singularities are poles).We shall see the resolution of this puzzle below. Meanwhile, we returnto the question of solving the first system.

We introduce new dependent variables Ω1,Ω2,Ω3 according to

w1 =1√

x(1− x)dx

dtΩ1

w2 =1

x√

(1− x)dx

dtΩ2

w3 =1

(1− x)√x

dx

dtΩ3

94

and switch independent variables from t to x. The first system becomes

dΩ1

dx=

Ω2Ω3

x(1− x)dΩ2

dx=

Ω3Ω1

x(5.3)

dΩ3

dx=

Ω1Ω2

(1− x)

which has the first integral

γ =12(−Ω2

1 + Ω22 + Ω2

3)

while the metric becomes

ds2 = Θ( dx2

x(1− x)+

σ21Ω21

+(1− x)σ22

Ω22

+xσ21Ω23

)(5.4)

where

Θ =Ω1Ω2Ω3

x(1− x)dx

dt.

(The system (3) with different derivations and motivations, has beenstudied by Fokas et al (1986) and by Dubrovin (1990).)

To solve the new version, (3), of the first system, we seek an equationfor Ω3 alone. Because of the existence of the first-integral γ this will besecond-order. To make it recognisable, we introduce a new independentvariable z by

x =4√z

(1 +√z)2

and a new dependent variable V by

Ω3 =z

V

dV

dz− V

2(z − 1)− 1

2+

z

2V (z − 1).

Now we find that V satisfies the Painleve-VI equation with the parame-ters (α, β, γ, δ) in the notation of Ince (1956) or Ablowitz and Clarkson(1991), equal to (18 ,−

18 , γ,

12 (1 − 2γ)). (The reduction of the system

(3) to Painleve-VI was also found by Chakravarty (1993)). Thus as an-ticipated the equations for the conformal structure have the Painleveproperty (being in fact reducible precisely to one of the Painleve equa-tions!) but it is the conformal factor in (4) which contains the functionx(t), which has the natural boundary. This choice of conformal factor


can be thought of as just a gauge choice, to make the Ricci scalar vanish.

At this point, we have found the general solution of the form (1) insidethe top circle in the figure. (Note that there are ASD Bianchi-type IXmetrics which are not diagonal in the chosen invariant basis of 1-forms;see (Maszczyk et al 1993)). Now we ask: can we change the conformalfactor Θ in order to make these metrics Einstein? This would give thegeneral metric in the left-hand intersection in the figure (which must bediagonal in this basis by a general argument, see e.g. (Tod 1995)).

This question can be answered by a brute-force calculation: simply writedown the desired condition as a set of equations on Θ and try to solve.We find (Tod 1995) that this is only possible if γ = 1/8 and that thenthe solution may be written as

Θ =N

D2

where

N = 2Ω1Ω2Ω3(4xΩ1Ω2Ω3 + P )

P = x(Ω21 + Ω2

2)− (1− 4Ω23)(Ω

22 − (1− x)Ω2

1)

D = xΩ1Ω2 + 2Ω3(Ω22 − (1− x)Ω2

1).

Since the equation for Ω3 was second-order, the metric depends on twoarbitrary constants. Despite the complexity of these expressions, a gooddeal can be said about the metrics (Hitchin 1995). In particular, withappropriate choices, these are ASD Einstein metrics on the 4-ball whichfill-in the general left-invariant metric on the 3-sphere in just the waythat the 4-dimensional hyperbolic metric fills in the round metric on S3.

Acknowledgements. I am grateful to Professor Nutku and TUBITAKfor the invitation to visit the Fesa Gursey Institute and their generoussupport and hospitality during the Research Semester on Geometry andIntegrability.

References

M. J. Ablowitz and P. A. Clarkson 1991 Solitons, nonlinear evolutionequations and inverse scattering LMS Lecture Note Series 149

96

M. F. Atiyah and N. J. Hitchin 1985 Phys. Lett. A107 21M. F. Atiyah and N. J. Hitchin 1988 The geometry and dynamics of

magnetic monopoles Princeton University PressJ. D. Barrow 1982 Gen. Rel. Grav. 14 523-530V. A. Belinski, G. W. Gibbons, D. W. Page and C. N. Pope 1979 Phys.

Lett. B76 433F. Brioschi 1881 C. R. Acad. Sci. tXCII 1389S. Chakravarty 1993 private communicationJ. Chazy 1910 C. R. Acad. Sci. 150 456A. S. Dancer and I. A. B. Strachan 1994 Math. Proc. Camb. Phil.

Soc. 115 513-525B. A. Dubrovin 1990 Funct. Anal. Applns. 24 280A. S. Fokas, R. A. Leo, L. Martina and G. Soliani 1986 Phys. Lett.

A115 329N. J. Hitchin 1995 J. Diff. Geom. 42 30-112E. L. Ince 1956 Ordinary differential equations Dover reprintC. R. LeBrun 1991 J. Diff. Geom. 34 223-253L. J. Mason 1990 in Further Advances in Twistor Theory eds. L. J.

Mason, L. P. Hughston and P. Z. Kobak, PitmanR. Maszczyk, L. J. Mason and N. M. J. Woodhouse 1993 Class. Quant.

Grav. 11 65-71H. Pedersen and Y.-S. Poon 1990 Class. Quant. Grav. 7 1707K. P. Tod 1991 Class. Quant. Grav. 8 1049-1051K. P. Tod 1994 Phys. Lett. A190 221-224K. P. Tod 1995 in Twistor Theory ed S. Huggett, Dekker Lecture notes

in pure and applied mathematics 169K. P. Tod 1997 in Geometry and Physics eds. J. E. Andersen, J.

Dupont, H. Pedersen and A. Swann, Dekker Lecture notes inpure and applied mathematics 184

6

Twistor theory for integrable systemsN. M. J. Woodhouse,

The Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, [email protected]

Abstract

Integrable systems that arise as reductions of the anti-self-dual Yang-Mills equations have a twistor correspondence by reduction of the Wardcorrespondence for the anti-self-dual Yang-Mills equations themselves.These lecture notes begin by reviewing the twistor correspondence, firstof all for linear fields and anti-self-dual Yang-Mills fields. They thenmove onto the reduced twistor correspondences for reductions of theanti-self-dual Yang-Mills equations with particular attention paid to thespecial examples of the Korteweg–de Vries equations and the Painleveequations.

6.1 Lecture 1

6.1.1 Introduction and background

In two dimensions, one can solve Laplace’s equation

uxx + uyy = 0

by introducing a complex coordinate w = x + iy. The equation thenbecomes

∂2u

∂w∂w= 0 , (6.1)

which is satisfied by u = f(w) for any holomorphic function f .How many different ways can one do this? That is, in how many

different ways can one introduce a complex linear coordinates on R2 so

that Laplace’s equation reduces to (6.1)? It is not hard to see that theonly freedom is to replace w by aw for some constant a = 0, or to replacew by x− iy. The first gives nothing new since holomorphic functions of

97

98

w are also holomorphic functions of aw; the second tells us that f(w) isalso a solution.

In four dimensions the situation is more interesting. If we start with

uxx + uyy + uyy + utt = 0,

then we can reduce it to the form

∂2u

∂w1∂ω1+

∂2u

∂w2∂ω2= 0 (6.2)

by putting w1 = x + iy, w2 = z + it. We have solutions of the formu = f(w1, w2) for any holomorphic function of w1 and w2. Again wecan replace the complex coordinates by any complex linear functions ofw1 and w2 (which gives nothing new). But in this case we have otherpossibilities that mix up the complex coordinates with their complexconjugates. For any constant α, β ∈ C, not both zero, the substitution

w1 = α(x+ iy)− β(z − it), w2 = α(z + it) + β(x− iy) (6.3)

also reduces Laplace’s equation to (6.2).Suppose that we choose instead different constants α′, β′. If α/β =

α′/β′, then corresponding ws are related by a complex linear transfor-mation; but if α/β = α′/β′, then this is not true, and we get a new classof solutions from holomorphic functions of the new complex coordinates.To put this more formally, we have a family of complex structures on R

4,labelled by the points of the projective line (the Riemann sphere withcomplex coordinate ζ = α/β).† There is also a second family that oneconstructs in the same way, but starting instead with x+ iy and z − it.

Now let us consider a closed 2-form F = dΦ (signature apart, we thinkof F as an electromagnetic field). We say that F is anti-self-dual if ithas no dw1 ∧ dw2 or dw1 ∧ dw2 components for any choice of ζ (thereis a more geometric definition below).

Suppose that this condition holds and fix for the moment a choice ofζ. If we write

Φ = Φ1 dw1 + Φ2 dw2 + Φ1 dw1 + Φ2 dw

2 ,

then the ASD condition implies that

∂Φ1

∂w2 =∂Φ2

∂w1

† In the notation of Paul Tod’s lectures, w1 and w2 are the components of the spinorωA and α, β are the components of the spinor πA′ . The point of view here is basedon that of Atiyah et al (1978a).

Twistor theory for integrable systems 99

and hence that

Φi +∂f

∂wi= 0 i = 1, 2

for some function f . Written in terms of the original Cartesian coordi-nates, this is

ζ(∂x + i∂y)f − (∂z − i∂t)f + ζ(Φx + iΦy)− (Φz − iΦt) = 0

ζ(∂z + i∂t)f + (∂x − i∂y) + ζ(Φz + iΦt) + β(Φx − iΦy) = 0 . (6.4)

Clearly, as α and β vary, we can find a solution f that depends holo-morphically on ζ = α/β. However, we cannot find a solution that isholomorphic in ζ on the whole Riemann sphere.

What is always possible, however, is to find one solution f which isholomorphic in ζ for ζ =∞ and another f which is holomorphic in ζ forζ = 0 (including the point at infinity). It then follows that h = f − f isholomorphic with respect to the three variables

ζ(x+ iy)− (z − it), ζ(z + it)− (x− iy), ζ

(away from ζ = 0,∞) since (6.4) implies ∂wih = 0. We call h the twistorfunction of F . It encodes F since we can recover f and f , and hence Φ,by splitting the Laurent series of

h(ζ(x+ iy)− (z − it), ζ(z + it)− (x− iy), ζ

)into its positive frequency part (f) and negative frequency part (f), asa function of ζ for fixed values of the Cartesian coordinates x, y, z, t.

The picture is typical of twistor theory: a system of partial differentialequation (the ASD condition on F ) in four independent variables (thespace-time coordinates) has been replaced by a condition that a singlefunction (h) should be holomorphic in the three variables w1, w2, ζ.

A number of interesting linear equations, such as the hypergeometricequation and Bessel’s equation, are symmetry reductions of the ASDcondition; and by imposing corresponding symmetry conditions on h,one can recover classical contour integral formulas for the solutions.†

In these lectures, I shall look at the corresponding nonlinear theory,where the electromagnteic field is replaced by a Yang–Mills field, and

† The idea of looking systematically at integrable systems as reductions of the ASDYang–Mills equations is due to Ward (1985), who also first showed that ASD Yang–Mills equation could be solved by twistor methods (Ward 1977). Ward’s approachto integrability was developed, in particular, by Mason, and is surveyed in Masonand Woodhouse (1996). Twistor methods themselves are due to Penrose, who firstapplied them in a nonlinear context in Penrose (1976). The twistor approach tothe ASD condition was applied and extended in Atiyah and Ward (1977), Atiyah(1979), and Atiyah et al (1978b).

100

the reductions are integrable systems. We shall look in particular at theKdV equation and the sixth Painleve equation. In the nonlinear theory,h is replaced by the transition matrix of a holomorphic vector bundle.

6.1.2 Complex manifolds

We shall make some use of the theory of complex manifolds and holomor-phic vector bundles. Although we shall not go very far into the theory,it is important to get some intuitive feeling for the basic structures, andin particular for the rather subtle way in which ideas from complex anal-ysis and differential geometry come together. In the remainder of thislecture, I shall try to convey this, without giving full definitions andrigorous proofs.

First, an n-dimensional complex manifold M is defined in the sameway as a real manifold, except that the transition maps between localcoordinate patches are required to be holomorphic. Thus in a neigh-bourhood of each point, we can introduce local holomorphic coordinatesza (a = 1, 2, . . . , n). The transition between two coordinate systems za

and wa is must satisfy:

• the functions za(w1, . . . wn) are holomorphic (that is, ∂za/∂wb = 0);• the Jacobian matrix ∂za/∂wb is non singular on the overlap of the

two coordinate patches.

One can think of this as a generalization of the notion of a real mani-fold, and so take over the standard apparatus of differential geometryto define holomorphic functions, vector fields, and differential forms,by simply adding the requirement that the components should dependholomorphically on the coordinates.

Alternatively, we can think of an n-dimensional complex manifold as a2n-dimensional real manifold with some additional structure, by writing

za = xa + iya ,

with xa and ya real, and taking xa, ya as real coordinates. The addi-tional structure is a real tensor field J , with one upper index and onelower index. As a linear operator on tangent vectors, it is given by

J

(∂

∂xa

)=

∂

∂ya, J

(∂

∂ya

)= − ∂

∂xa.

It has the property that J2 = −1.Conversely a given tensor field J on an even-dimensional smooth real


manifold M such that J2 = −1 is called an almost complex structure.Such a tensor has two n-dimensional eigenspaces, corresponding to theeigenvalues ±i. The additional property of J that characterizes M as acomplex manifold is that J should be integrable. That is, if two complexvector fields Z and Z ′ are eigenvector fields with

JZ = iZ, JZ ′ = iZ ′ ,

then [Z,Z ′] is also an eigenvector with J [Z,Z ′] = i[Z,Z ′]. When thisholds, it is possible to giveM the structure of a complex manifold by tak-ing the local holomorphic coordinates to be solutions za to the Cauchy–Riemann equations

(X + iJX)za = 0 for every vector field X.

It is not hard to prove this from the Frobenius theorem when M isanalytic (see Appendix 8 of Kobayashi and Nomizu 1969); but it isa hard theorem (the Newlander–Nirenberg theorem) if one starts withsmooth objects. When n = 1 (so that M has two dimensions as areal manifold), an integrable almost complex structure is the same thingas a conformal metric together with an orientation – since X → JX

determines a rotation through π/2 in a positive sense. In this case, theNewlander–Nirenberg theorem is the assertation that it is possible tofind isothermal coordinates, in which the conformal metric is a multipleof

dx2 + dy2 .

6.1.3 Examples

Two complex manifolds M,M ′ are isomorphic if they are diffeomorphicby a diffeomorphism f : M →M ′ that is biholomorphic – that is, f andf−1 are holomorphic in local holomorphic coordinates. Biholomorphicmappings are much harder to find than smooth ones because their ana-lyticity gives them an extra rigidity: a diffeomorphism can be deformedin all sorts of ways in a neighbourhood of a point, but a biholomor-phic map is determined by analytic continuation once it is known in asmall neighbourhood of a point of M . Consequently it is possible fortwo complex manifolds to be diffeomorphic but not biholomorphicallyequivalent.

Example. The torus. Let α, β be two nonzero complex numbers suchthat their ratio is not real. We can then construct a one-dimennsional

102

complex manifold M by taking the quotient of the Argand plane C bythe equivalence relation z ∼ w whenever

z = w + nα+mβ for some n,m ∈ Z.

As a smooth real manifold, M is diffeomorphic to the 2-dimensionaltorus, irrespective of the choice of α and β. So every choice gives thesame real manifold. If however we construct M ′ by making a differentchoice α′, β′, then M and M ′ are isomorphic as one-dimensional complexmanifolds if and only if we can find a holomorphic function F withnowhere vanishing derivative such that

F (z + α) = F (z) + aα′ + bβ′, F (z + β) = F (z) + cα′ + dβ′

for some integers a, b, c, d such that ad− bc = ±1. But then dF/dz is aholomorphic on M . It must therefore be constant by Liouville’s theorem(see Exercise 1.1). Since we are free to add a constant to F , we havewithout loss of generality F (z) = kz for some constant k. But then

kα = aα′ + bβ′, kβ = cα′ + dβ′ .

We can conclude from this that M and M ′ are isomorphic if and only ifω = α/β and ω′ = α′/β′ are related by

ω =aω′ + b

cω′ + d

for some integers a, b, c, d with ad−bc = ±1. More details of this exampleare given in Gunning’s book Lectures on Riemann surfaces (Gunning1966).

We shall be more exclusively interested in projective spaces, and inmanifolds constucted from projective spaces.

Example. The projective line. This is the familiar Riemann sphere,which is a one-dimensional complex manifold with local coordinates de-fined by stereographic projection. If a point on the sphere has imagez ∈ C under projection from the North pole and image w under pro-jection from the South pole, then w = 1/z, and we have a holomorphiccoordinate transformation. The domain of z is the whole sphere less theNorth pole; the domain of w is the whole sphere less the South pole.Unlike the torus, the sphere has a unique complex structure.

Example. Projective spaces. More generally, the n-dimensional projec-tive space CPn is the quotient of C

n+1 by the equivalence relation

(Z0, Z1, . . . , Zn) ∼ (λZ0, λZ1, . . . , λZn) for some λ = 0 ∈ C .


The homogeneous coordinates Zα label the points uniquely, up to anoverall complex scaling factor. We make CPn into a complex manifoldby using the corresponding inhomogeneous coordinates. For example onthe open set in which Z0 = 0, we define za (a = 1, . . . , n) by

z1 = Z1/Z0, z2 = Z2/Z0, . . . , zn = Zn/Z0 .

When n = 1, the projective space is the same as the Riemann spheresince we can take z = Z1/Z0 and w = Z0/Z1 as the two local coordinates(in the southern and northern hemispheres).

6.1.4 Holomorphic vector bundles

In the same way, we can extend the notion of a smooth bundle E →M tothat of a holomorphic bundle by requiring that E (the ‘total space’) andM (the ‘base space’) should be complex manifolds and that the maps inthe usual definition should be holomorphic. Thus a holomorphic vectorbundle E of rank k has local trivializations in which we identify therestriction of E to a suitable open subset U ⊂M with U × C

k. On theoverlap of two local trivializations we have

(m, v′) ∼ (m, v) =(m,F (m)v′

),

where F : U∩U ′ → GL(k,C) is holomorphic (that is, F is a non-singularholomorphic matrix-valued function of the local coordinates). We callF the patching matrix or transition function.

Of course, all the usual operations (E ⊕E′, E ⊗E′, . . . ) from differ-ential geometry make sense in for holomorphic bundles. But there aretwo key ways in which holomorphic bundles behave differently: these weshall illustrate below.

Two holomorphic vector bundles E,E′ are isomorphic if there is abiholormphic map ρ : E → E′ which maps Em linearly to E′

m for eachm ∈M .

One way that we can specify a holomorphic vector bundle is by givingits transition maps Fij between the open sets of some open cover Ui (i insome indexing set): these are holomorphic maps matrix-valued functionson the intersections with the cocyle property

FijFjkFki = 1

on each triple intersection Ui ∩ Uj ∩ Uk. It is not hard to see thatevery holomorphic vector bundle can be represented in this way. Whatis rather less obvious, but nonetheless true, is that every holomorphic

104

bundle can be represented in this way for some fixed special choice ofopen cover.

A holomorphic section s is given by holomorphic maps si : Ui → Ck,

with the transition rule si = Fijsj . We denote the space of holomorphicsections over V ⊂M by Γ(V,E). A fundamental fact is that:

• if M is compact, then Γ(M,E) is finite dimensional.

In the case of the trivial bundle over the Riemann sphere, this is adirect consequence of Liouville’s theorem (since all sections in that caseare constant).

The bundle is trivial, that is isomorphic to M × Ck, if we can find

holomorphic splitting matrices fi : Ui → GL(k,C) such that

Fij = f−1i fj .

In this case, we can specify global sections by putting si = f−1i c for

some constant vector c.We shall give an example below to illustrate that

• E can be trivial as a smooth bundle, but non-trivial as a holomorphicbundle.

Thus a holomorphic bundle can carry information beyond that containedin its toplogical structure. Indeed it is possible to deform a holomorphicvector bundle holomorphically into an inequivalent bundle. This fact iscentral to the twistor constructions.

Example. Bundles over projective space. If we represent the Riemannsphere as the projective line, as above, then we have the projectionC2 → CP 1. This gives us a line bundle (i.e. a rank-one vector bundle)

with total space L = C2. The fibre above the point with coordinate

z = Z1/Z0 is the one-dimensional subspace of C2 spanned by (Z0, Z1).

We define holomorphic sections e and e over the z and w coordinatepatches by taking

e = (1, Z1), e = (Z0, 1) .

These are related by e = ze; thus if s and s are two functions respesentingthe same section in the z and w coordinate patches, respectively, thens = s/z and so the transition function is F = 1/z = w on the annularintersection of the two patches.

A local section of L is the same thing as a function f on C2 homo-

geneous of degree -1, since we can put s(z) = f × (Z0, Z1) to get awell-defined point of L which depends only on the ratio z = Z0/Z1.


Since there are no global holomorphic homogeneous functions of degree-1 on C

2−0, there are no global sections of L, a fact that we can alsosee in another way: a global section would be represented by an entirefunction s(z) with the property that zs(z) = s(w−1)/w is holomorphicin w = 1/z at w = 0. By expanding zs in a Laurent series, it is easy tosee that this is impossible.

Because its sections are functions of degree −1, it is customary todenote L by O(−1). We can similarly define O(1) to be the dual bundle,and O(m) by taking the |m|th tensor product of O(1) or O(−1), asappropriate. The transition function for O(m) is F = zm, and its globalsections are homogeneous functions of degree m in (Z0, Z1). So we notethat

Γ(CP 1,O(m)

)= 0 (m < 0), dimΓ

(CP 1,O(m)

)= m+1 (m ≥ 0) .

The second equality can be seen in two ways: first a global section ofO(m) is an entire function g(z) such that z−mg(z) is holomorphic atinfinity – that is, g is a polynomial of degree at most m; second, a globalhomogeneous function of degree m is of the form φAB···CZAZB · · ·ZC

for some symmetric constant ‘spinor’ φAB···C .The same definitions of the line bundles O(m) can be given over

higher-dimensional projective spaces.

6.1.5 Exercises

(1.1) Prove the following variants of Liouville’s theorem.

(i) If M is a compact complex manifold, then every holomorphicfunction φ : M → C is constant. [Hint: suppose that φ is notconstant and consider a point where |φ| achieves its maximumand apply the maximum principle to φ as a function of each ofthe local coordinates to get a contradiction.]

(ii) If φ(z) is an entire holomorphic function and φ/zn is bounded forsome integer n ≥ 0, then φ is a polynomial of degree at most n.

(iii) If φ(z) is a meromorphic function on CP 1 (so its only singular-ities are a finite collection of poles), then φ = s1/s2 for someholomorphic sections s1, s2 of O(n) for some n.

(1.2) Show that if (6.4) holds for some f which is holomorphic in ζ on thewhole Riemann sphere, then F = 0. [Hint: apply the previous exerciseto deduce that f is independent of ζ; then show that Φ is a gradient.]

106

(1.3) Show that the holomorphic tangent and cotangent bundles of CP 1

are, respectively, O(2) and O(−2).

6.2 Lecture 2

6.2.1 Holomorphic bundles

In the case of CP 1, the line bundles O(m) are in some sense the wholestory, since we have the following.

Theorem 1 (Grothendieck) A rank-k holomorphic vector bundleE → CP 1 is isomorphic to a direct sum O(m1) ⊕ · · · ⊕ O(mk) forsome integers mi.

It is also true that any vector bundle E must have trivial restriction tothe z and w coordinate patches. Thus the theorem is equivalent to thestatement that the transition matrix

F : C− 0 → GL(k,C)

can be written in the form

F = f−1

zm1 0 . . . 00 zm2 0

. . .0 zmk

f (6.5)

where f : U → GL(k,C) and f : U → GL(k,C) are holomorphic.Thus Gothendieck’s theorem is closely related to Birkhoff’s factorizationtheorem, which states that any smooth map F from the unit circle inthe complex plane to GL(k,C) can be written in this form, where f, f

extend holomorphically to the inside and outside of the circle on theRiemann sphere. (It is important to note that the theorem does nothold if ‘smooth’ is replaced by ‘continuous’. Birkhoff’s theorem is thelink between the twistor view in which solutions to integrable systems arerepresented by holomorphic vector bundles, and the older approaches inwhich equations are solved by solving Riemann–Hilbert problem – thatis, factorization problems for matrix-valued functions on the circle. Aproof is given in Pressley and Segal 1986.)

Example. Put

F =(z 0t z−1

).


For t = 0, F is already in the form given by Birkhoff’s theorem, withf = f = 1, m1 = 1, m2 = −1. But for t = 0, we have

F =(1 t−1z0 1

)(0 −t−1t z−1

),

so that m1 = m2 = 0. If we think of F as a patching matrix for abundle Et, then E0 2 O(1) ⊕ O(−1), but Et is the trivial bundle fort = 0. This is an example of ‘jumping’: as t changes through 0, theholomorphic structure of the bundle changes discontinuously, in spite ofthe fact that the bundles Et are all the same (and all trivial) from thetopological point of view.

6.2.2 Complex space-time

With these preliminaries, we turn to Ward’s theorem, which gives a cor-respondence between anti-self-dual Yang–Mills fields and holomorphicbundles over (open subsets of) of CP 3. It leads to a correspondencebetween equivariant bundles and solutions to integrable systems that beregarded as symmetry reductions of the anti-self-dual Yang–Mills equa-tions, and thence to more general twistor constructions. To understandits statement and proof, we need some basic facts about conformal ge-ometry in four dimensions (these were covered in more detail by PaulTod). A much fuller account is given in Ward and Wells (1990).

We shall work almost exclusively with complex solutions, the generalphilosophy being to understand the twistor transforms as entirely holo-morphic constructions first, and then to impose reality conditions at alater stage.

We begin, therefore, by representing complex space-time as C4 with

coordinates z, w, z, w, and metric and volume element

ds2 = 2(dz dz − dw dw) ν = dw ∧ dw ∧ dz ∧ dz .

(It should be noted that the apparent signature of the metric is notrelevant: in the complex one can change the distribution of plus andminus signs in a diagonal metric by complex coordinates transforma-tion. Conversely the analytic continuations to C

4 of the real metrics onMinkowski space and Euclidean space are the same: one recovers theEuclidean metric by putting z = z, w = −w and the Minkowski metricby putting w = w, and taking z and z to be real.)

These determine a metric tensor gab and an alternating tensor εabcd.

108

For a 2-form α, we put

∗αab = 12εab

cdαcd ,

and say that α is self-dual or anti self-dual as ∗α = ±α. For example

dw ∧ dz, dw ∧ dz, dw ∧ dw − dz ∧ dz

are self-dual, while

dw ∧ dz, dw ∧ dz, dw ∧ dw + dz ∧ dz

are anti-self-dual.

Lemma 1 The duality operation α → ∗α does not depend on the scaleof the metric.

A conformal transformation ρ : U ⊂ C4 → ρ(U) is a biholomorphic

mapping such ρ∗(g) ∝ g. If ρ is conformal, then either

ρ∗(∗α) = ∗ρ∗(α) or ρ∗(∗α) = −∗ρ∗(α)

for every 2-form α. We shall consider only proper transformations, forwhich the first is true. A central result underlying the twistor con-structions, which has been explained from other starting points in otherlectures, is that the proper conformal transformations form a groupisomorphic to PGL(4,C) 2 GL(4,C)/C×. There are two points to es-tablish here: first, that we do indeed have a group structure (which isawkward because we cannot compose transformations with disjoint do-mains, while if we insist that U = C

4, then we exclude all but isometriesand dilatations). Second, that the transformations can be identified withprojective linear transformations.

Both points are dealt with by labelling the points of space-time by4× 4 matrices by putting

X = λ

0 s −w z

−s 0 −z w

w z 0 1−z −w −1 0

for some λ = 0, where s = zz−ww. This gives a correspondence between

• points of space-time, and• skew 4× 4 matrices X with detX = 0 and X34 = 0, up to scale.

We then have


Lemma 2 ds2 =√

det(dX).

Note that the right-hand side is in fact quadratic in dX because a skew-symmetric matrix has a natural square root (the Pfaffian). We haveto make a choice for λ: different choices give different metrics in theconformal class.

The isomorphism is then given by M ∈ GL(4,C) → ρ, where

ρ(X) = MXM t . (6.6)

Since X is determined only up to scale, M and any nonzero scalar multi-ple of M give the same transformation, which is why the isomorphism isbetween the conformal transformations and the projective linear group,rather than GL(4,C) itself.

One now sees how to address the first point: the action of ρ is notdefined by (6.6) on the whole of C

4 because it does not preserve thecondition X34 = 0. In fact, this condition looks rather unnatural; if wedrop it, and admit matrices that do not satisfy it, then what we have isnot space-time itself, but its conformal compactification, which we canrepresent as the quadric hypersurface det X = 0 in the projective spaceCP 5 (the entries in X are the homogeneous coordinates on CP 5). Theoriginal space-time C

4 is the dense open subset X34 = 0, while the extrapoints are the points of a light-cone at infinity. What we have shownis that the group PGL(4,C) acts on the compactification as a group ofconformal transformations, in a similar way to that in which SL(2,C)acts on the Riemann sphere, which is a compactification of the complexplane. What is a little harder to show is that this action gives all theproper conformal transformations.

6.2.3 The anti-self-dual Yang–Mills equations

We now turn to the anti-self-dual Yang–Mills equations themselves.These are nonlinear differential constraints on a connection D. We shallconsider only connections D = d + Φ on the trivial bundle V × Ck,where V ⊂ C

4 is some open set. We shall thus ignore any topologicalcomplications in space-time, and concentrate on the local behaviour ofYang–Mills fields. We shall not, however, assume that the bundle hasa fixed trivialization, so we are free to make gauge transformations byg : V → GL(k,C), under which

Φ → g−1Φg + g−1d g .

110

The curvature 2-form

Fab = ∂aΦb − ∂bΦa + [Φa,Φb]

then transforms homogeneously by F → g−1Fg. Consequently it makessense to make the gauge-independent definition:

Definition 1 A connection D is a solution to the anti-self-dual (ASD)equations if F = −∗F .

Equivalently, the operators

Dw − ζDz and Dz − ζDw

commute for every fixed value of the spectral parameter ζ. This is theLax pair or zero-curvature formulation of the Yang–Mills equation.

If D is ASD, then DF = 0 (the Bianchi identity), and consequentlyD∗F = 0; thus, as the terminology implies, an ASD connection is asolution to the Yang–Mills equations. We also have

Proposition 1 If d + Φ is ASD, then so is d + ρ∗Φ for any properconformal transformation ρ.

6.2.4 Exercise

(2.1) Show that√

det(dX) = ds2.

6.3 Lecture 3

6.3.1 Null 2-planes

Ward’s theorem is based on the observation that the ASD condition isthe integrability condition for D over a special family of 2-planes in C

4.A 2-plane Z ⊂ C

4 is null if g(X,Y ) = 0 for every X,Y tangent to Z.We define the tangent 2-form ω to such a plane by

ω = ν(X,Y, ·, ·)

and note that ω is determined by Z up to scale.

Proposition 2 The tangent 2-form of a null 2-plane is either self-dualor anti-self-dual.

Proof. The tangent 2-form is characterized (again up to scale) byω(X, ·) = 0 for every X tangent to Z. If Z is null, then ∗ω also has


this property. Consequently ∗ω = λω for some λ. But the eigenvaluesof ∗ as a linear operator on 2-forms are ±1.

Definition 2 A null 2-plane with a self-dual tangent 2-form is called anα-plane.

The α-planes are given by linear equations of the form

Z0 − zZ2 − wZ3 = 0 = Z1 − wZ2 − zZ3 (6.7)

for some constant Zα, with Z2, Z3 not both zero. Since we obtain thesame α-plane by rescaling Zα, we see that the space of α planes arelabelled by the points of the projective space CP 3 on which the Zαsare homogeneous coordinates, excluding the projective line I on whichZ2 = Z3 = 0: this is the twistor space described in Paul Tod’s lectures.

6.3.2 Integrability conditions

Proposition 3 A connection D is ASD if and only if the equation Ds =0 is integrable over every α-plane.

Proof. If we put ζ = Z3/Z2, then we can rewrite (6.7) as

z + ζw = const., w + ζz = const. .

So the tangent space to an α-plane is spanned by the vectors

L = ∂w − ζ∂z, M = ∂z − ζ∂z ,

for some constant ζ. Thus the proposition follows by writing the ASDcondition as the condition that the operators

Dw − ζDz and Dz − ζDw

should commute.

The key to Ward’s theorem is the observation that the solutions tothe equation Ds = 0 on the α-planes form the fibres of a holomorphicvector bundle over an open subset of the twistor space CP 3.

Before we can prove this, we need one other piece of geometry, whichwill also be familiar from Paul Tod’s lectures.

112

6.3.3 The Klein correspondence

Consider all the α-planes through a given point x ∈ C4. These are given

by (6.7), but where we now fix w, z, z, w (the coordinates of x) and allowZα to vary. We thus have two homogeneous linear equations in the fourvariables Zα: they determine a 2-dimensional subspace of C

4 and hencea projective line x ⊂ CP 3.

We thus have a correspondence (the Klein correspondence) betweenpoints x of C

4 and lines x ⊂ CP 3. Every line that does not intersectI determines a point of C

4 (the lines that do intersect I are the pointsat infinity in the compactified space-time). The conformal action ofPGL(4,C) on compactified space-time that we looked at above is simplythat induced by the natural action of the projective linear group onCP 3.

6.3.4 Ward’s theorem

Ward’s theorem is an example of a Penrose transform, that is, a corre-spondence between solutions of a differential equation in space-time andholomorphic objects on the corresponding twistor space.

If U ⊂ C4 is some subset, then the twistor space of U is the set ZU

of α-planes Z such that Z ∩ U = ∅. The twistor space of C4 itself is

CP 3 − I; the twistor space of a single point x is the projective linex ⊂ CP 3. For a general U , we have the incidence condition: if x ∈ U ,then x ⊂ ZU .

With this notation, we have the following.

Theorem 2 (Ward) Let U ⊂ C4 be an open set such that U ∩ Z is

connected and simply connected for every Z ∈ ZU . Then there is a one-to-one correspondence between solutions of the ASD Yang–Mills equationon U with gauge group GL(k,C) and holomorphic vector bundles E →ZU such that E|x is trivial for every x ∈ U .

We have seen from the general discussion of holomorphic vector bundlesthat it is possible for Ex to jump from a trivial bundle to a non-trivialone as x varies. However, it can be shown that its behaviour is semi-continuous in the sense that, if Ex is trivial, then so is E|y for all y someopen neighbourhood of x.

Proof. I shall not give the full proof, but simply show how to constructE from D and then indicate in outline how to recover D from E.

Given an ASD connection D with gauge group GL(k,C), we construct


the fibre EZ of the corresponding bundle E → ZU at Z ∈ ZU by takingEZ to be the space of solutions to the linear equation Ds = 0 on Z ∩U .This space is k-dimensional since the linear equation is integrable (by theASD condition) and because Z ∩ U is connected and simply connected,by hypothesis.

It is not immediately obvious that the fibres EZ fit together to form aholomorphic vector bundle E → ZU . To show this, we have to constructholomorphic local trivializations.

Given x ∈ U , we can identify all the spaces EZ , Z ∈ x, with Ck by

evaluating the solutions s of the linear equation at x. If we could finda two-dimensional holomorphic surface S ⊂ U which intersected each Z

in exactly one point, then we could go further and identify every spaceEZ with C

k by evaluating s at S ∩ Z. This is not in fact possible, butit is almost possible: if we take S itself to be an α-plane through somex ∈ U , then it intersects all other α-planes in exactly one point, withthe exception of those with the same (or proportional) tangent 2-form.By making different choices for S, we can identify EZ with C

k for Z

in various open subsets of U . The transition maps Ck → C

k betweenthe identifications determined by S and S′ are given by integrating theequation Ds = 0 from S ∩ Z to S′ ∩ Z: they are holomorphic functionsof the coordinates on ZU .

Thus E → ZU is a rank-k holomorphic vector bundle. Its restrictionto x is trivial for any x ∈ U since EZ = C

k for every Z ∈ x by theconstruction above.

Going in the other direction, suppose that we are given E → ZU withthis property; suppose that ZU is covered by two open sets V , V oneach of which E is trivial and suppose that x ∩ V ∩ V is an annulus foreach x ∈ U . Let the transition matrix between these trivializations beF (λ, µ, ζ), where

λ = Z0/Z2, µ = Z1/Z2, ζ = Z3/Z2.

Then if the point x has coordinates w, z, w, z, the line x is given by

λ = ζw + z, µ = ζz + w . (6.8)

We assume that the coordinates have been set up so that V ∩ x is adisc in the ζ-plane containing ζ = 0, and that V ∩ x is a disc containingζ = ∞, and that x ∩ V ∩ V is an annular neighbourhood of the unitcircle in the ζ-plane.

Since E|x is trivial, we must have a Birkhoff factorization of the form

F (ζw + z, ζz + w, ζ) = f−1f , (6.9)

114

where f and f are functions of the five variables w, z, w, z, ζ with valuesin GL(k,C), with f holomorphic at ζ = 0 and f holomorphic at ζ =∞.A crucial point, however, is that f and f will not in general be expressibleas functions of λ, µ and ζ (this happens only when E itself is trivial).Now

∂wF − ζ∂zF = 0 .

Hence

∂wff−1 − ζ∂zff

−1 = ∂wf f−1 − ζ∂z f f

−1 .

Both sides, therefore, must be equal to a global rational function of ζwith a simple pole at infinity (see Exercise 1.1). Therefore

∂wff−1 − ζ∂zff

−1 = −Φw + ζΦz

where Φw and Φz are matrix-valued functions of the space-time coor-dinates (but not of ζ). By repeating this argument for the operator∂z − ζ∂w, we get that f and f are solutions of a linear system of theform

(∂w + Φw)f − ζ(∂z + Φz)f = 0

(∂z + Φz)f − ζ(∂w + Φw)f = 0 ,

the integrability condition for which (for all constant ζ) is precisely thatd + Φ should be an ASD connection.

The transition matrix can be chosen freely; thus the result reduces thesolution of the ASD Yang–Mills equation to the solution of a Riemann–Hilbert problem – the factorization (6.9). In the proof, we made as-sumptions about the covering V , V ; these can, in fact, be justified.Alternatively, one can work with a more general cover, in which case thefactorization problem is of the form Fij = f−1

i fj .

6.3.5 Exercises

(3.1) Show that the factorization (6.9) is unique up to multiplication off and f on the left by a nonsingular matrix g depending on the space-time coordinates, but not ζ. Show that different choices of factorizationgive gauge-equivalent connections.

(3.2) Show that if, in the proof of Ward’s theorem, f and f can beexpressed as functions of λ, µ and ζ, then D is gauge-equivalent toΦ = 0.


(3.2) In the proof of Ward’s theorem, show that the if D is constructedfrom E by Birkhoff factorization, and that if E′ is constructed from D,then E 2 E′.

(3.3) Suppose that

F =(1 h

0 1

)where h = h(λ, µ, ζ). Find Φ in terms of the electromagnetic field gen-erated by h.

(3.4) Show that the natural action of PGL(4,C) on CP 3 induces anaction on the space of lines in CP 3 and hence an action on space-time.Show that this coincides with the conformal action described in Lecture2. [Hint: the notation of Paul Tod’s lectures helps here.]

6.4 Lecture 4

6.4.1 Equivariant holomorphic bundles

In Lionel Mason’s lectures, it has been shown how various integrable sys-tems of nonlinear differential equations arise as reductions of the ASDYang–Mills equations by subgroups of the conformal group. We shallnow introduce Penrose transforms for these equations by imposing sym-metry on Ward’s construction.

A subgroup of the conformal group is the same thing as a subgroupof the projective linear group PGL(4,C), which acts transitively on theprojective space CP 3, by the linear action of GL(4,C) on the homoge-neous coordinates Zα. The generators are the holomorphic vector fieldsof the form

YA = AαβZ

β ∂

∂Zα,

for some constant 4 × 4 matrix A (note that Y is well defined on CP 3

because the expression on the right-hand side is homogeneous in Z ofdegree zero). Thus to solve some reduced form of the ASD Yang–Millsequation, we are led to consider holomorphic bundles E which are equiv-ariant under under this action on CP 3.

In general, of course, an element of the conformal group does notgive a well-defined mapping ZU → ZU since it may move an α-planethat intersects U to one that does not. For this reason, we concentrateon the equivariance condition at the Lie algebra level. Before makingthis precise, we need some definitions and some general results aboutequivariant bundles.

116

Definition 3 Let E → M be a holomorphic vector bundle and letρ : M → M be a biholomorphic transformation. We say that E isequivariant under ρ if ρ lifts to a biholomorphic map ρ : E → E suchthat ρ maps the fibre Em linearly onto the fibre Eρ(m) for each m ∈M .

Thus an equivariant bundle is one for which ρ∗E 2 E (the pull-backbundle ρ∗E is the bundle with fibres (ρ∗E)m = Eρ(m)). In practicalterms, if E has transition matrices Fij relative to some cover Ui, thenρ∗E has transition matrices Fij ρ relative to the cover ρ−1Ui.

As usual, one should be careful about taking over intuitive ideas fromthe behaviour of smooth bundles: if ρ is close to the identity, then everysmooth bundle is equivariant along ρ (in the sense of the above definition,with holomorphic maps replaced by smooth ones); but this is certainlynot true in general for holomorphic bundles.

At the Lie algebra level, equivariance is characterized by the existenceof a lift of a Lie algebra of vector fields on M to vector fields on E. Moreconveniently, we can think in terms of the existence of ‘Lie derivativeoperators’.

Definition 4 Let E →M be a holomorphic vector bundle over a complexmanifold M and let Y be a holomorphic vector field on M . We say thatE is equivariant along Y if there exists a first-order differential operatorLY on the local sections of E which takes the form

LY = Y + θY

for some holomorphic matrix-valued function θY in a local trivialization.

We note that if LY exists, then θY transforms by

θY → g−1θY g + g−1Y (g) (6.10)

under transformations of the local trivialization. If we introduce localcoordinates za on M and linear coordinates wi on the fibres of E, then

Y = Y a ∂

∂za− θijw

j ∂

∂wi

is a vector field on the total space of E, where the Y as are the com-ponents of Y and the θijs are the entries in the matrix θY . If we canintegrate Y to construct a family of biholomorphic transformations ρ,then by integrating Y , we obtained the lifted transformations ρ. ThusDefinition (4) is indeed an infinitesimal form of Definition (3).


If E is equivariant and is given by a transition matrix F between localtrivializations over the V and V (where M = V ∪ V ), then

Y (F ) = −θY F + F θY ,

where LY = Y + θY in V and LY = Y + θY in V . Conversely, if we canwrite Y (F ) in this form, with θY holomorphic in V and θY holomorphicin V , then E is equivariant.

If Y (m) = 0 for some point m, then we can solve the equation

Y (g) + θY g = 0

for g : V → GL(k,C) in some neighbourhood V of Y to obtain a localtrivialization of E such that θY = 0. Any other local trivialization withthis property is then given by transforming the local frames by someinvertible matrix-valued function F such that Y (F ) = 0. It follows thatif Y has no zeros, then we can choose local trivializations of E such thatthe transition functions are constant along Y . If it is also true that thespace of integral curves of Y is Hausdorff manifold M ′, and that thecurves are simply connected, then a Y -equivariant bundle is the samething as the pull-back of a holomorphic vector bundle E′ → M ′ by theprojection map M →M ′.

Example. The Bogomolny equations Consider the weighted projectivespace given by taking C

3−W 2 = W 3 = 0 and making the identification

(W 1,W 2,W 3) ∼ (λ2W 1, λW 2, λW 3) λ = 0 ∈ C.

This is a two-dimensional complex manifold, which is fibred over CP 1

by the projection (W 1,W 2,W 3) → (W 2,W 3). (It is in fact the totalspace of the tangent bundle TCP 1). We also have a projection

CP 3 − I → TCP 1 : (Z0, Z1, Z2, Z3) → (W 1,W 2,W 3)

= (Z3Z1 + Z2Z0, Z2, Z3) .

The fibres of this are tangent to the vector field

Y = Z3 ∂

∂Z0− Z2 ∂

∂Z1= ζ

∂

∂λ− ∂

∂µ,

which has no zeros on CP 3−I. We thus have a correspondence betweenholomorphic vector bundles E′ → TCP 1 and Y -equivariant holomor-phic vector bundles E → CP 3 − I.

Now Y is a generator of a conformal action on space-time. In fact, wesee from (6.8) that it generates the translation

∂

∂w− ∂

∂w.

118

Thus we expect a correspondence between (i) solutions of the ASD Yang–Mill equations which are invariant under translation by ∂w−∂w and (ii)holomorphic vector bundles over TCP1. The corresponding symmetryreductions of the ASD Yang–Mills equations are the Bogomolny equa-tions

F = ∗Dφ ,

where D is a connection on a vector bundle B over some region in three-dimensional Euclidean space (complexified), φ (the ‘Higgs field’) is asection adj(B), and the ∗ is the three-dimensional duality operator.

This is in fact true, and has been exploited by Hitchin (1982) in theconstruction of monopoles.

6.4.2 Normal forms

The situation is more interesting if Y has zeros (so that its flow hasfixed points). If Y (m) = 0, then g−1Y (g) = 0 at m and so θY (m)transforms by conjugation under change of local trivialization. Thus ifθY (m) = 0 in some local trivialization, then θY (m) = 0 in every localtrivialization. In fact the flow of Y induces a one (complex) parameterfamily of linear transformations of the fibre Em over the fixed point,generated by θY (m). The best that we can do in this case is to reduceθY to normal form in some neighbourhood of m.

The full range of possible behaviours is very complicated; we shalljust look at some simple examples, which give the flavour of the sortof phenomena one can expect, and indicate the connection with theclassification of singular points of ordinary linear differential equations.In the following, we suppose that z and w are two coordinates on acomplex manifold.

Logarithmic fixed points. Suppose that dimM = 1 and that Y =z∂z. This has a zero at the point z = 0. If θ = θY (0) has distincteigenvalues, no two of which differ by an integer, then we can choose alocal trivialization in which θ is a diagonal matrix m, and we can find afundamental matrix solution y to the ordinary differential equation

zdydz

+ θY y = 0

of the form y = h(z)z−m, where h is a power series in z. That is,

h−1Y (h) + h−1θY h = m.

Thus we can use h to reduce θY to m everywhere (in a neighbourhood of


z = 0). Because the solution y is not single-valued in general, we cannotmake θY = 0.

Irregular singular points. If we take instead Y = z2∂z, then theproblem of finding a good local normal form is more involved. We startin this case with the linear ordinary differential equation

z2dydz

+ θY y = 0, ordydz

= Ay

where A = θY /z2 = c−2/z2+c−1/z+c0+· · · is a matrix-valued function

of z with a double pole at the origin (the ci are the coefficients in itsLaurent expansion). Provided that the eigenvalues of c−2 are distinct,we can can choose a local trivialization in which c−2 is diagonal, and wecan find a formal solution of the form

y = h exp(−c−2/z +m log z

)where h = 1 + h1z + h2z

2 + · · · is a formal power series with matrixcoefficients and m is a constant diagonal matrix. If this power seriesconverges, then

h−1Y (h) + h−1θY h = −mz − c−2

and so we can reduce the Lie derivative operator to diagonal form, withθY depending linearly on z. The constant term gives the action of LY

on the fibre above the fixed point. In general, however, the series doesnot converge. By truncating it, however, at some power of z, we canachieve

θY = −mz − c−2 +O(zp) ,

for p arbitrarily large.

Dressing. Another type of behaviour will be important in the KdVexample below. Suppose that M is 2-dimensional and that Y = z2∂w.This vanishes on the z-axis. Let us suppose in this case that

θY = N + z2U N =(0 1z 0

),

where trU = 0 and U is upper triangular at z = 0. In this case, the linearaction on the fibre above the fixed point has the nilpotent generator(

0 10 0

).

The question that arises here is: can we remove U by changing the

120

local trivialization? The answer is that we can, but again only formally,by the ‘dressing procedure’ described by Drinfeld and Sokolov (1985).The idea is to begin by noting that U can be written as a convergentpower series

U =∞∑0

(hi + αi 0

0 hi − αi

)N i

where the coefficients hi, αi depend on w alone. We can reduce thecoefficients αi to zero by an iterative procedure. Suppose that i is theleast index such that αi = 0 and put

g = 1 +(αi 00 0

)N i−1 .

Then, since z2 = N4 and since[(αi 00 0

),

(0 1z 0

)]=(αi 00 −αi

)(0 1z 0

)we have that g−1Y (g) + g−1θY g has an expansion with i increased byone. By this algebraic procedure, therefore, we can arrange that

θY = N + z22p∑0

hiNi +O(zp+2)

for any positive p. A further iteration, where now we take at each stageg to be of the form g = 1 + gi(w)N i for scalar gi(w) reduces the histo zero up to any finite i by solving a differential equation for gi as afunction of w at each stage. Thus for any positive p, we can choose theframe in a neighbourhood of the origin so that

LY = z2∂w +N +O(zp) .

Formally, we can take the procedure to the limit; but the relevant powerseries do not converge in general.

Finally, we remark that we have only considered here equivariancealong a single vector field. More generally, E is said to be equivariantunder a Lie algebra h of holomorphic vector fields if it is equivariantalong each Y ∈ h, and if Y → LY is a representation of h by differentialoperators.


6.4.3 Exercises

(4.1) Prove that every holomorphic line bundle over the Riemann sphereis equivariant under the action of SL(2,C).

(4.2) Show that if E is a smooth bundle on a smooth manifold M , then E

is equivariant along any smooth vector field on M . [Hint: piece togetherlocal Lie derivative operators by using a partition of unity.]

6.5 Lecture 5

6.5.1 The KdV equation

In Lionel Mason’s lectures, he shows that the KdV equation is a sym-metry reduction of the ASD Yang–Mills equation by two orthogonaltranslations, one null and the other non-null (a result due to Mason andSparling 1989). In our notation, these are the two commuting vectorfields

X ′ = ∂z, Y ′ = ∂w − ∂w .

We have already seen that the second corresponds to the vector field

Y = Z3 ∂

∂Z0− Z2 ∂

∂Z1= ζ

∂

∂λ− ∂

∂µ,

on CP 3. The first corresponds to X = Z2∂/∂Z0.In the example of the Bogomolny equation above that we saw that

we can take a quotient along Y to reduce a Y -equivariant bundle to oneover TCP 1. The projection of X onto this space is ∂γ , which we alsodenote by X, in the coordinates

ζ = Z3/Z2, γ = (Z3Z1 + Z2Z0)/(Z2)2 = µζ + λ .

These do not, of course, cover the fibre over the point ζ =∞: there wemust use another coordinate system, for example

ζ = Z3/Z2 = 1/ζ, γ = (Z3Z1 + Z2Z0)/(Z3)2 = γ/ζ2 .

In these, X = ζ2∂γ , so that X vanishes at ζ = ∞. Thus we cannottake a further quotient (which is fortunate, because otherwise the wholeconstruction would be trivial).

Each point in space-time determines a line in CP 3 and hence anembedded copy of CP 1 in TCP 1. From (6.8), this is given in terms ofthe coordinates w, z, w, z by

γ = zζ2 + (w + w)ζ + z .

122

That is, it is a section of the bundle TCP 1 → CP 1

The Yang–Mills theory suggests that there should be a correspondencebetween

• solutions to the KdV equation which are holomorphic in t = z andx = w + w in some region of the t, x-plane and

• holomorphic vector bundles over some corresponding region in TCP 1

which are equivariant along X and trivial on the copies of CP 1.

Not every such bundle will do: it is important that the Lie derivativeoperator LX should have the right normal form at the zeros of X.

We cannot here go into the detailed argument that leads to the correctnormal form. Essentially there are two possibilities (apart from thetrivial one): that θX should be either be semi-simple or nilpotent at ζ =0. The choice is determined by the form of the Higgs field correspondingto X ′: for the KdV equation it is nilpotent; for the nonlinear Schrodingerequation, it is semi-simple. However the story is not quite that simple(a lot is hidden in the qualification ‘essentially’).

What we shall do instead of pursuing the full details of the reduction ofthe ASD Yang–Mills equation is to show directly that the bundles witha particular normal form give rise to solutions of the KdV equation.

6.5.2 Solutions to the KdV equation

Suppose that E is a holomorphic SL(2,C)-vector bundle E over a neigh-bourhood of the zero section in TCP 1 satisfying the triviality conditionon sections of TCP 1 and and equivariant along

X = ∂γ = ζ2∂γ .

The KdV bundles are distinguished by the form of LX in a neighbour-hood of the fibre at infinity (ζ = 0), where X vanishes. They satisfy thecondition that there should exist a frame in a neighbourhood of ζ =∞in which

LX = ∂γ − ζ−1Λ + ζ−2U ; Λ =(0 ζ

1 0

),

where U is upper triangular at ζ = ∞. By the iterative proceduredescribed above, it is then possible to change the frame so that U =O(ζ−p) as ζ →∞ for arbitrary p > 0.

If we use this together with an invariant frame over the complement of


ζ =∞ to construct local trivializations of E → O(2), then the transitionmatrix F (ζ, γ) satisfies

F−1 ∂F∂γ

= −ζ−1Λ +O(ζ−p−2)

as ζ →∞.The solution itself is recovered from F by substituting γ = xζ + tζ2,

and making a Birkhoff factorization F = f−1f in ζ, with f = 1 atinfinity (this fixes the gauge). If this can be found, then

∂xff−1 = ∂xf f

−1 − fΛf−1 +O(ζ−p−1)

ζ∂xff−1 − ∂tff

−1 = ζ∂tf f−1 − ∂xf f

−1 ,

since ζ∂xF = ∂tF = ζ2∂γF . Both sides of the two equations must beglobal rational functions of ζ. By examining their behaviour at ζ =∞,we conclude that the two sides of the two equations are respectively ofthe forms −Λ+A and B, where A and B depend only on x, t, and A isupper triangular. Therefore

∂xf +Af − Λf = 0, ∂tf +Bf − ζ∂xf = 0 ,

and the two operators on the left-hand sides commute. The integrabilityof this linear system implies that

A =(−q r

0 q

), B =

(∗ ∗−qx ∗

).

Now f also satisfies the second linear equation, while instead of the first,it satisfies

∂xf = fΛ− Λf +Af +O(ζ−p−1) (6.11)

But

f = 1 + ζ−1(σ ρ

τ −σ

)+O(ζ−2)

as ζ → ∞, for some σ, ρ, τ depending on x, t. By substituting this into(6.11), we deduce that

r = −2σ, τ = −q, τx + 2σ = τq ,

and hence that r = qx − q2. The commutation relation now gives thatv = 2qx satisfies the KdV equation 4vt − vxxx − 6vvx = 0. In fact everylocal analytic solution of the KdV solution arises in this way.

124

We can generate solutions by taking

F (ζ, γ) = g(ζ)e−γΛ/ζ

where g is a holomorphic SL(2,C)-valued function on an annulus in theζ plane. This is the case in which the iteration converges, since

∂F

∂γ= −FΛ/ζ .

For example, when g is rational, we obtain a multi-soliton solution.Given one patching matrix F (ζ, γ) satisfying the equivariance condi-

tion, we can obtain others by replacing F by

F (ζ, γ) = F(ζ, γ + h(ζ)

),

where h(ζ) is holomorphic in ζ is a neighbourhood of ζ = 0. If we thinkof the coefficients in the Taylor expansion of h as the ‘times’, then theresult is the KdV hierarchy.

6.5.3 The isomonodromy problem

Finally, we turn to the twistor theory of the Painleve equations andthe general isomonodromy deformation equations. The aim is to gainsome geometric insight into why these lie at the heart of the theory ofintegrable systems.

6.5.4 Fuchsian equations

Suppose that we have an ordinary equation of the form

dydζ

= Ay (6.12)

where y takes values in the k × k matrices and A(ζ) is a matrix-valuedfunction of ζ with rational coefficients. In the Fuchsian case, A has onlysimple poles, so that

A =N∑1

Aα

ζ − aα(6.13)

where the residues Aα are constant matrices.There are two simple transformations that preserve the Fuchsian form:

first, gauge transformations (for constant g)

A → g−1Ag, y → g−1y ;


and second Mobius transformations of the coordinate

ζ → pζ + q

rζ + s.

The matrix A has poles at the points aα; if

A0 = −A1 −A2 − · · · −AN

is nonzero, then it has a further simple pole at ζ =∞ with residue A0.We see this by making the transformation ζ → 1/ζ. Thus in generalthere are N + 1 poles.

We also note that if y is invertible at one point, then it is invertibleeverywhere (except at the poles), and that any other solution is of theform yM for some constant matrix M .

The elementary theory of ordinary differential equations tells us thatfor any point ζ0 (other than a pole) there is a unique local holomorphicsolution y such that y(ζ0) = 1. However it is not single-valued in thelarge. If we analytically continue y around a closed path γ in the com-plement of the poles, with endpoints at ζ0, then it it does not returnto its original value at ζ0, but to Mγ (an element of GL(k,C)). Thisdepends only on the homotopy class of γ. The map

π1(C− a1, . . . , aN

)→ GL(N,C) : γ →Mγ

is a representation of the fundamental group π1, called the monodromyrepresentation. The isomonodromy problem is to find all the Fuchsianequations with the same monodromy as the given one. Since we caneasily change the representation to a conjugate one,

γ → g−1Mγg, g constant ,

by changing ζ0 or by making a gauge transformation, we understand‘same’ in this context to mean ‘same up to conjugacy’. (The relatedRiemann–Hilbert problem is to find a Fuchsian equation with given mon-odromy.)

6.5.5 Exercises

(5.1) Show that the vector fields Y and X in the KdV example are globalon CP 3. [Hint: find expressions for them in homogeneous coordinates.]

(5.2) Show that

exp(−γΛ/ζ

)=(

cosh (γ/ζ1/2) −ζ1/2sinh (γ/ζ1/2)−ζ−1/2sinh (γ/ζ1/2) cosh (γ/ζ1/2)

).

126

(5.3) Let α, β ∈ C with |α|, |β| < 1 and let C : C → SL(2,C) be entire.Show that if

F (ζ) =1

ζ − α

(ζ − β 0

0 ζ − α

)C(ζ)−1

and

f(ζ) = 1− 1M(ζ − α)

C(β)(1 00 0

)C(α)−1

then (i) f is holomorphic and invertible for all ζ outside of the unit disc(including ζ = ∞), (ii) det f = (ζ − β)/(ζ − α), and (iii) f = fF−1

is holomorphic and invertible inside the unit disc. Hence find explicitlythe solution to the KdV equation generated by

F (γ, ζ) =1

ζ − α

(ζ − β 0

0 ζ − α

)exp(−γΛ/ζ

).

[Hints: use Exercise (5.2); use the fact that f satisfies the second linearequation to find qx.]

6.6 Lecture 6

6.6.1 Fuchsian equations from equivariant bundles

The twistor solution to the isomonodromy problem comes from a cor-respondence between Fuchsian equations and equivariant holomorphicbundles over (subsets) of CPN .†

Let Z ⊂ CPN be a neighbourhood of a projective line X0 ⊂ CPN

and let E → Z be a rank-k holomorphic vector bundle such that

• E|X0 is trivial;• E is equivariant under the Lie algebra of the diagonal subgroup of

PGL(N + 1,C).

The diagonal subgroup is generated by any N of the N + 1 commutingvector fields

Y0 = Z0 ∂

∂Z0, Y1 = Z1 ∂

∂Z1, . . . , YN = ZN ∂

∂ZN

(their sum is everywhere zero). However, rather than discard one gen-erator, it is simpler to treat all the vector fields on the same footing. So

† The connections between the Painleve equations and twistor methods were firstexplored by Hitchin (1995). Various reductions of the ASD Yang-Mills equationsto the Painleve equations are catalogued in Ablowitz and Clarkson (1991).


if we denote the corresponding Lie derivatives by Lα = LYα= Yα + θα

(in a local trivialization), then for consistency we shall require

N∑0

Lα =N∑0

θα = 0 ,

as well as the commutation relations [Lα,Lβ ] = 0.Except on the coordinate hyperplanes Zα = 0, the vector fields span

the tangent space to CPN . We denote the union of these by Σ anddefine a connection ∇ on the restriction of E to Z − Σ by taking

∇T =∑

tαLα, where T =∑

tαYα .

In fact if T = Tα∂/∂Zα, then tα = Tα/Zα. By the consistency condi-tion, this is independent of the way in which T is written as a combina-tion of the Yαs; and, by the commutation relation, ∇ is flat.

Now suppose that X is a line near X0. Then E|X is also trivial, andso we can write can write E|X = X × C

k. It is claimed that (i) therestriction of ∇ to X is a Fuchsian differential operator of the form

d−Adζ

where A is as above and ζ is a stereographic coordinate on X; and (ii),as X changes, the mononodromy of the Fuchsian equation

dydζ

= Ay

is preserved.To prove the first statement, we write the equation of X in the form

Zα = Bα − ζCα α = 0, 1, . . . , N

for constants Bα and Cα. The tangent vector T = d/dζ can be written

T = −∑α

Cα ∂

∂Zα=∑α

Cα

Bα − ζCαYα .

Thus

∇T =ddζ−∑α

θαζ − aα

where aα = Bα/Cα. If we now work in the global trivialization of E|X ,then the sum on the right-hand side a global rational function on theRiemann sphere with simple poles at the points ζ = aα. So we havean operator of the required form (except that the poles ζ = aα are in

128

general position: to move a0 to ζ = ∞, we have to make a Mobiustransformation).

To prove the second statement, we note that the monodromy of theFuchsian equation is simply the holonomy of ∇ around the singularitieson the coordinate hyperplanes; and this is independent of the choice ofX.

Thus the construction gives us, from an equivariant vector bundle,a family of Fuchsian equations with the same monodromy. The polesof the equation are the intersection points of X with the coordinatehyperplanes. These are N + 1 points in general position. ProvidedN > 2, we cannot in general map the poles for one choice of X to thoseof another by a Mobius transformation (since the cross-ratio of any fourpoles is preserved by a Mobius transformation). Thus the equations ondifferent lines are genuinely distinct.

6.6.2 Explicit calculation

Suppose that E is given by a patching matrix F (Zα), between trivial-izations over a two-set open cover U, U of Z. Since F is a function onCPN , it is homogeneous of degree zero in the coordinates Zα. As inthe Ward construction, we make the simplifying assumption that X ∩Uis a neighbourhood of ζ = 0, X ∩ U is a neighbourhood of ζ = ∞ andX ∩U ∩ U is an annular neighbourhood of the unit circle in the ζ-plane.

There is no loss of generality in restricting to the case Cα = 1, Bα =aα. Since E|X is trivial, we can find f, f such that

F (aα − ζ) = f−1f (6.14)

where f : X∩U → GL(k,C) and f : X∩U → GL(k,C) are holomorphic.Then as X varies, f and f become holomorphic functions of aα and ζ.

By the equivariance condition

YαF = −θαF + F θα

where Yα+θα and Yα+θα are the Lie derivatives in the two trivializationsof E. Also, with Zα = aα − ζ, we have

∂ζF = −N∑0

∂F

∂Zα=

N∑0

θαF − F θαaα − ζ

∂aαF =

∂F

∂Zα=−θαF + F θα

aα − ζ.


By substituting from (6.14) into the first of these

∂ζff−1 +

N∑0

fθαf−1

aα − ζ= ∂ζ f f

−1 +N∑0

f θαf−1

aα − ζ.

Both sides must therefore be equal to a global rational function on ζ,with simple poles at aα. We denote this by

A =N∑0

Aα

ζ − aα.

Since∑Lα = 0, we also have

∑Aα = 0, so the corresponding Fuchsian

equation has no pole at infinity: it is the same Fuchsian equation as theone we constructed above by a more abstract argument.

Now from the construction, we see that

∂f

∂ζ= Af,

∂f

∂aα= − Aαf

ζ − aα.

The compatibility condition for these linear equations is the Schlesingerequation:

∂Aα

∂aβ=

[Aα, Aβ ]aα − aβ

α = β ,

which is well known to determine the isomonodromic deformations of aFuchsian system.

Almost exactly the same argument works if E is given by a family oflocal trivializations with patching matrices Fij .

6.6.3 The inverse construction

We see that an equivariant bundle over U generates a family of Fuchsianequations all with the same monodromy; the equations in the family arelabelled by the positions of the poles aα, and the dependence of thecoefficients Aα is determined by the Schlesinger equations.

Suppose, instead, we start with one Fuchsian equation (6.12), withpoles at ζ = aα, α = 0, 1, . . . N ; for simplicity, we shall assume that thereis no pole at infinity, so

∑Aα = 0. Can we construct the corresponding

equivariant bundle, and hence the isomonodromic deformations? Weshall think of the given equation as being associated with an ‘initialline’ Zα = aα − ζ.

Let y be a invertible matrix solution. Then, of course, y has singu-larities at the poles and is multi-valued: if we continue it around one ofthe poles, then it becomes yM for some constant matrix M .

130

For each α, choose some α′ = α and put

ζα =aα′Zα − aαZ

α′

Zα − Zα′ .

Then ζα = ζ on the initial line. We also pick an open cover Uα of aneighbourhood U of the initial line in CPN so that Uα contains thepoints Zα = 0, but not any of the points Zβ = 0 for β = α. Wethen define a holomorphic vector bundle E → U by taking its transitionmatrices between trivializations over Uα and Uβ to be

Fαβ = y(ζα)y(ζβ)−1

on Uα ∩ Uβ . It does not matter which branch of y we choose since themonodromy matrix M cancels when we replace one branch by another.Clearly, from their form, the F s satisfy the cocycle condition, and sothey determine a holomorphic vector bundle E → U .

It is claimed that E is equivariant, and that it generates the givenequation, together with its isomonodromic deformations. It is not hardto see this, but we look at the argument below only in the special case ofPVI, where there are only four poles and it is a little easier to see whatis going on.

6.6.4 The sixth Painleve equation

In the special case that N = 3 (four poles) and that E has struc-ture group SL(2,C), the Schlesinger equation comes down to the sixthPainleve equation. In this case, there is essentially one independent vari-able since three of the poles can be moved to ζ = 0, 1,∞ by a Mobiustransformation, and so all that is left to vary is the fourth (or, moreinvariantly, we can take the independent variable to be the cross-ratioof the four poles).

The case N = 3 is also the Yang–Mills case. Thus there we see directlythat that the reduction of the Yang–Mills equations by the diagonalsubgroup of the conformal group is the sixth Painleve equation.

To be more explicit, the generators of the diagonal group on space-time are the three conformal Killing vector fields

X ′ = −z∂z − w∂w, Y ′ = −z∂z − w∂w, Z ′ = z∂z + w∂w

(corresponding to the three vector fields

X = ζ∂ζ , Y = −λ∂λ − µ∂µ − ζ∂ζ , Z = µ∂µ


on CP 3). If we put p = − logw, q = − log z, r = log(w/z), t = zz/ww,then these become

X = ∂p, Y = ∂q, Z = ∂r .

We can choose the gauge for the Yang–Mills field so that

Φ = P dp+Qdq +R dr (6.15)

with P constant and Q, R depending on t; then the ASD Yang–Millsequations reduce to

tQ′ = [R,Q], t(1− t)R′ = [tP +Q,R] .

It is straightforward to show that if we take a root x of the quadratic

det[P, xQ−R] = 0 ,

and write x = (y−t)/y(t−1), then y satisfies the sixth Painleve equation

y′′ =12

(1y+

1y − 1

+1

y − t

)y′2 −

(1t+

1t− 1

+1

y − t

)y′

+y(y − 1)(y − t)

t2(t− 1)2

(α+

βt

y2+

γ(t− 1)(y − 1)2

+δt(t− 1)(y − t)2

),

where α, β and γ are constants (that determine invariants of P , Q, andR). Conversely any solution y determines a self-dual Yang–Mills field ofthe form (6.15).

6.6.5 Twistor construction of solutions

Let us return to the initial Fuchsian equation. In the case of the sixthPainleve equation, this is of the form

dydζ

=(A0

ζ+

A1

1 + ζ+

At

ζ + t

)y

where we think of ζ as a coordinate on the line w = z = w = 1, z = t.We pick an initial value of t, and put ξ = ζ/(µ− ζ), ξ = λ− t0 (these

are local holomorphic function on CP 3 with the property that ξ = ξ = ζ

on the initial line). Then ξ is constant along Y,W , and ξ is constantalong X,Z, while

X(ξ) =µζ

(µ− ζ)2, Y (ξ) = −λ.

132

Therefore if y is a solution of the initial Fuchsian equation

X(y(ξ)) =1

µ− ζ

(µA0 + ζA1 +

At

ζ(1− t0) + t0µ

)y

Y (y(ξ)) =(−λ A0

λ− t0+

−λA1

λ+ 1− t0+At

)y .

We follow the general template for the Schlesinger equation, and put

F = y(ξ)y(ξ)−1 ,

and take F to be the transition matrix of a holomorphic bundle betweentwo open sets: U containing ζ = 0,−1 and U containing ζ = −t0,∞.Then because the expressions in brackets above are holomorphic in U

and U , respectively, at least near the initial line, we have an equivariantbundle.

If we take some other line given by w = z = w = 1, z = t, then theextraction of the deformed Fuchsian equation comes down to solving theRiemann–Hilbert problem

y(ζ)y(ζ + t− t0)−1 = f(t, ζ)−1f(t, ζ)

with f , f holomorphic in U and U respectively.

6.6.6 The Painleve test

This construction fits into general pattern: to obtain a family of ODEsfrom an a holomorphic vector bundle E →M , we need two ingredients.First, an equivariance condition under a Lie algebra of holomorphic vec-tor fields of the same dimension as M , with the vector fields spanningthe tangent space at almost every point of M . Second, a family of em-beddings CP 1 → M . The ODE will not generally be Fuchsian, but A

will always be rational. The deformations are always isomonodromic.This fact explains why symmetry reductions to ODEs of the integrable

systems that arise from twistor constructions lead to Painleve and moregeneral isomonodromy deformation equations – in short, why they passthe Painleve test.


6.6.7 Exercise

(6.1) Show that if instead of X,Y, Z, we take the generators of thesubgroup of PGL(4,C) of matrices of the form

a b c d

0 a b c

0 0 a b

0 0 0 a

then the result is a family of ODEs in which A has a single pole oforder 4.

References

Ablowitz, M. J. and Clarkson, P. A. (1991). Solitons, nonlinear evo-lution equations and inverse scattering. London Mathematical SocietyLecture Notes in Mathematics, 149, Cambridge University Press, Cam-bridge.

Atiyah, M. F. (1979). Geometry of Yang–Mills fields. Lezioni Fermiane.Accademia Nazionale dei Lincei and Scuola Normale Superiore, Pisa.

Atiyah, M. F., Hitchin, N. J. and Singer, I. M. (1978a). Self-duality infour-dimensional Riemannian geometry. Proc. Roy. Soc. Lond., A 362,425–61.

Atiyah, M. F., Hitchin, N. J., Drinfeld, V. G. and Manin, Yu.I. (1978b).Construction of instantons. Phys. Lett., A65, 185–7.

Atiyah, M. F. and Ward, R. S. (1977). Instantons and algebraic geom-etry. Commun. Math. Phys., 55, 111–24.

Drinfeld, V. G. and Sokolov, V. V. (1985). Lie algebras and equationsof Korteweg–de Vries type. J. Sov. Math., 30, 1975–2036.

Gunning, R. C. (1966). Lectures on Riemann surfaces. Princeton Math-ematical Notes. Princeton University Press, Princeton, New Jersey.

Hitchin, N. J. (1995). Twistor spaces, Einstein metrics and isomon-odromic deformations. J. Diff. Geom., 42, 30–112.

Kobayashi, S. and Nomizu, K. (1969). Foundations of differential geom-etry, Vol. 2. Wiley, New York.

Mason, L. J. and Sparling, G. A. J. (1989). Nonlinear Schrodinger andKorteweg–de Vries are reductions of self-dual Yang–Mills. Phys. Lett.,A137, 29–33.

134

Mason, L. J. and Woodhouse, N. M. J. (1996). Integrability, self-duality,and twistor theory. Oxford University Press, Oxford.

Penrose, R. (1976). Nonlinear gravitons and curved twistor theory. Gen.Rel. Grav., 7, 31–52.

Pressley, A. and Segal, G. B. (1986). Loop groups. Oxford UniversityPress, Oxford.

Ward, R. S. (1977). On self-dual gauge fields. Phys. Lett., 61A, 81–2.

Ward, R. S. (1985). Integrable and solvable systems and relations amongthem. Phil. Trans. R. Soc., A315, 451–7.

Ward, R. S. and Wells, R. O. (1990). Twistor geometry and field theory.Cambridge University Press, Cambridge.

7

Transformations and reductions of integrablenonlinear equations and the ∂-problem

Paolo Maria Santini

Dipartimento di Fisica, Universita di Roma ”La Sapienza”Piazz.le Aldo Moro 2, I-00185 Roma, Italy

Istituto Nazionale di Fisica Nucleare, Sezione di RomaP.le Aldo Moro 2, I-00185 Roma, Italy

[email protected]

Abstract

The ∂ dressing method is used to construct and solve integrable nonlin-ear equations as well as to describe their transformations and reductions.The theory is illustrated on a distinguished example: the quadrilaterallattice and its continuous limit: the conjugate net.

7.1 The ∂ dressing method

All the nonlinear equations integrable via the Inverse Spectral Trans-form (ISM) are the compatibility condition of two (or more) linear prob-lems and are characterized by the property that they are linearized ina suitable spectral space. The ISM establishes the proper connectionbetween the configuration space and this spectral space. In the spectralspace the underlying mathematical structure is an analyticity problem:a Riemann–Hilbert or, more generally, a ∂ problem on the complex planeC [1], [2], [3], [4], [5].

The ∂ dressing method is a powerful evolution of the ISM; its startingpoint is not the nonlinear equation and its associated linear systems,

135

136

but a linear ∂ problem. A linear (simple) dependence of the ∂ data onthe coordinates implies algebrically that the solution of the ∂ problemsolves a set of linear equations in configuration space, whose integra-bility condition is the integrable nonlinear equation. Therefore the ∂

dressing method allows one to construct, at the same time, integrablenonlinear equations together with a large class of their solutions [6], [7],[8]. As we shall see in the following, the ∂ dressing method provides alsothe convenient setting in which to study symmetry transformations andsymmetry constraints of the integrable nonlinear equation.

These short notes are organized as follows. In Section 7.1.1 we presenta short introduction to the theory of the ∂ problem. In Section 7.1.2we discuss the basic ideas of the ∂ dressing method and in Section 7.2we illustrate the method on a basic example: the quadrilateral latticeand its continuous limit: the conjugate net. In Section 7.3.1 we brieflydiscuss the basic symmetry transformations in the ∂ formalism and wederive the basic bilinear formulas which are used in Section 7.3.2 toconstruct a class of symmetry constraints leading to basic geometricreductions of the quadrilateral lattice and of the conjugate net. Theseexplicit reductions are considered in Sections 7.3.3–5.

7.1.1 The ∂ problem

We first give a short introduction to the ∂ (DBAR) problem; more detailscan be found in [5] and [4].

It is well known that, is φ(λ, λ) is analytic in the domain D, then∂λφ = 0, λ ∈ D, where ∂λ = ∂/∂λ, and the famous Cauchy formulashold:

∫∂D

φ(λ)dλ = 0, (7.1)

φ(λ) =1

2πi

∫∂D

φ(λ′)λ′ − λ

dλ′, λ ∈ D. (7.2)

Consider now a function φ(λ, λ) which is not analytic in D, whose de-parture from analyticity is described by the equation

∂λφ(λ, λ) = h(λ, λ), λ ∈ D (7.3)

Our goal is to generalize the Cauchy equations (7.1) and (7.2) to this

Nonlinear equations and the ∂-problem 137

more general situation. The basic tool is provided by the well-knownGauss–Green formula∫

∂A(Pdy −Qdx) =

∫A(∂xP + ∂yQ)dx ∧ dy, (7.4)

where A is a simply connected domain of the (x, y) plane, and by itsspecial cases

−∫∂A

gdx =∫A∂ygdx ∧ dy, (7.5)

∫∂A

gdy =∫A∂xgdx ∧ dy, (7.6)

obtained setting P = 0, Q = g(x, y) and P = g(x, y), Q = 0 respectively.Through the usual change of variables (x, y)→ (λ, λ):

λ = x+ iy, λ = x− iy, (7.7)

of the plane, which implies

∂λ =12(∂x − i∂y), ∂λ =

12(∂x + i∂y), (7.8)

dλ ∧ dλ = −2idx ∧ dy, (7.9)

equations (7.5) and (7.6) take the form:∫∂D

f(λ, λ)dλ = −∫D∂λf(λ, λ)dλ ∧ dλ, (7.10)

∫∂D

f(λ, λ)dλ =∫D∂λf(λ, λ)dλ ∧ dλ, (7.11)

where A → D and f(λ, λ) = g(x, y). Notice that equation (7.10) pro-vides the wanted generalization of the Cauchy theorem (7.1); to get thegenerlization of the Cauchy formula (7.2), we first prove that 1/πλ isthe localized Green’s function of the ∂λ operator; i.e.:

∂λ(1

λ− λ0) = πδ(λ− λ0), (7.12)

where the delta function is defined in the natural way

δ(λ− λ0) = δ(λR − λ0R)δ(λI − λ0I) (7.13)

(λR = Reλ, λI = Imλ); which implies, from (7.9), that∫Dδ(λ− λ0)f(λ, λ)dλ ∧ λ = −2if(λ0, λ0), λ0 ∈ D. (7.14)

138

Therefore the formal inverse of the ∂λ operator is given by

∂λ−1 =

12πi

∫dλ′ ∧ dλ′

λ′ − λ. (7.15)

To prove equation (7.12) we choose f = a(λ)λ−λ0

in (7.10), where a(λ) isanalytic in D, and we obtain, using the Cauchy theorem (7.1),

2πia(λ) = −∫Da(λ)∂λ(

1λ− λ0

)dλ ∧ dλ, (7.16)

valid ∀a(λ); which implies equation (7.12).If we choose instead f = φ(λ,λ)

λ−λ0in (7.10) and we use (7.12), we obtain

the following generalization of the Cauchy formula (7.2):

φ(λ, λ) =1

2πi

∫∂D

φ(λ′, λ′)λ′ − λ

dλ′ +1

2πi

∫D

dλ′ ∧ dλ′

λ′ − λ∂λφ(λ

′, λ′), λ ∈ D.(7.17)

We conclude these general considerations remarking that, while the ∂

equation (7.3) admits the general solution

φ(λ, λ) = a(λ) +1

2πi

∫D

dλ′ ∧ dλ′

λ′ − λh(λ′, λ′), λ ∈ D, (7.18)

where a(λ) is an arbitrary analytic function in D, the ∂-boundary valueproblem

∂λφ(λ, λ) = h(λ, λ), λ ∈ D (7.19)

φ(λ, λ) = B(λ, λ), λ ∈ ∂D (7.20)

is solvable only if the boundary condition B and the forcing h satisfythe equation

B(λ, λ) =1

2πi

∫∂D

B(λ′, λ′)λ′ − λ

dλ′ +1

2πi

∫D

dλ′ ∧ dλ′

λ′ − λh(λ′, λ′), λ ∈ ∂D,

(7.21)which follows directly from equation (7.17) in the limit λ→ ζ ∈ ∂D.

The two solvable ∂-boundary value (BV) problems which are relevantin the ∂ dressing are matrix M × M ∂-problems and take both thefollowing form:

∂λφ(λ, λ) = ∂λη(λ)+∫Dφ(λ′)R(λ′, λ′, λ, λ)dλ′ ∧ dλ′, λ ∈ D, (7.22)

φ(λ, λ) = B(λ, λ), λ ∈ ∂D, (7.23)


where η(λ) is a given normalization of φ(λ, λ) and R(λ′, λ′, λ, λ) is agiven M ×M matrix ∂-datum.

In the first case, the so-called ‘canonical’ ∂-BV problem, D = C andφ → I, λ → ∞; therefore η = 1. We call hereafter the solution of thecanonical ∂ BV problem χ(λ), omitting for simplicity its dependence onλ.

In the second case, the ‘simple pole normalization’ case, D = Cµ,where Cµ is the complex plane without a small circle around λ = µ andthe boundary condition reads:

φ→ 0, λ→∞; φ ∼ 1λ− µ

, λ ∼ µ. (7.24)

Therefore in this case η = (λ − µ)−1. We call hereafter the solution ofthis ∂ BV problem χ(λ, µ), omitting for simplicity its dependence on λ.

It follows that the solutions of these two ∂ BV problems are expressedrespectively in terms of the following linear integral equations

χ(λ) = I +1

2πi

∫C

dλ′ ∧ dλ′

λ′ − λ

∫Cχ(λ′′)R(λ′′, λ′)dλ′′ ∧ dλ′′ λ ∈ C, (7.25)

χ(λ, µ) =1

λ− µ+

12πi

∫C0

dλ′ ∧ dλ′

λ′ − λ

∫C0

χ(λ′′, µ)R(λ′′, λ′)dλ′′ ∧ dλ′′,

λ ∈ C0, (7.26)

omitting hereafter the dependence of R on λ, λ′.In all our considerations we shall assume that the solution of the ∂

problem (7.22), expressed through the linear integral equation

φ(λ) = η(λ) +1

2πi

∫C

dλ′ ∧ dλ′

λ′ − λ

∫Cφ(λ′′)R(λ′′, λ′)dλ′′ ∧ dλ′′ λ, λ′ ∈ C,

(7.27)be unique; this implies that the solution of the homogeneous ∂ problemsatisfying the homogeneous boundary condition φ → 0, λ → ∞, andcorresponding therefore to the case η = 0, is φ = 0.

It is important to introduce also the “adjoint” ∂ problem:

∂λφ∗(λ) = −∂λη(λ)−

∫CR(λ, λ′)φ∗(λ′)dλ′ ∧ dλ′ , λ, λ′ ∈ C. (7.28)

The ∂ problem (7.22) and its adjoint (7.28) imply the basic bilinearidentity:∫

γ∞φ1(λ)φ∗

2(λ)dλ+∫C[(∂λη1)(λ)φ

∗2(λ)− φ1(λ)∂λη2(λ)]dλ ∧ dλ+

140∫C

∫Cφ1(λ′)[R1(λ′, λ)−R2(λ′, λ)]φ∗

2(λ)dλ ∧ dλ dλ′ ∧ dλ′ = 0, (7.29)

where γ∞ is the circle with center at the origin and arbitrarily largeradius (and the corresponding integration is counter-clockwise). Equa-tion (7.29) involves the solutions φ1 of (7.22), corresponding to the nor-malization η1 and to the ∂ datum R1, and φ∗

2 of (7.28), correspondingto the normalization η2 and to the ∂ datum R2 respectively. To obtainthis identity, multiply equation (7.22) for φ1 from the right by φ∗

2 andequation (7.28) for φ∗

2 from the left by φ1; add the resulting equations,apply the operator

∫C dλ ∧ λ′ and finally use the identity∫

Cdλ ∧ λ′∂λf = −

∫γ∞

dλf. (7.30)

Of particular importance in the following, together with the two basic so-lutions χ(λ) and χ(λ, µ) of equations (7.22), corresponding respectivelyto the ‘canonical normalization’ η = 1 and to the ‘simple pole normal-ization’ η = (λ − µ)−1, will be also the solutions χ∗(λ) and χ∗(λ, µ) ofthe corresponding adjoint problems (7.28) .

The solutions of the ∂ problem (7.22) are expressed in terms of the in-tegral equation (7.27). Explicit solutions can be found for special choicesof the arbitrary ∂ kernel R; in particular, if this kernel is separable, theintegral equation reduces to an algebraic system of equations (see forinstance [4] and [9]).

7.1.2 The ∂-dressing method

The ∂-dressing method is a powerful method to construct, starting fromthe general linear ∂ problem (7.22) in the spectral space (the λ plane),integrable nonlinear equations in the configuration space parametrizedby the, in general complex, parameters x = (x1, .., xN ), together withlarge classes of solutions.

The dependence on the set x = (x1, .., xN ) of (space) parameters isintroduced through the basic function ψ0(x.λ) satisfying the followingdifferential equations:

∂iψ0 = Ki(x, λ)ψ0, i = 1, .., N (7.31)

where ∂i = ∂/∂xi and/or the following difference equations:

Tiψ0 = Ai(x, λ)ψ0, i = 1, .., N, (7.32)


where Ti is the translation operator in the xi variable:

Tif(x) = f(x1, .., xi + 1, .., xN ). (7.33)

The linear equations (7.31) and (7.32) must be compatible; i.e. thematrix functions Ki(x, λ), Ai(x, λ) must satisfy the equations:

∂iKj − ∂jKi + [Kj ,Ki] = 0, (7.34)

(TjAi)Aj − (TiAj)Ai = 0, (7.35)

(TjKi)Aj − ∂iAj −AjKi = 0. (7.36)

The dressing method is based on the assumption that the ∂ datum R

depends on the parameters x through ψ0 in the following form:

R(λ′, λ;x) = ψ0(x, λ′)R0(λ′, λ)(ψ0(x, λ))−1 (7.37)

where R0(λ′, λ) is independent of x. This structure of R and equations(7.34), (7.35), (7.36) imply that, if φ and φ∗ are solutions of the ∂ prob-lem (7.22) and of its adjoint (7.28), then also the ‘covariant’ derivativeand translations of φ:

Diφ(λ) = ∂iφ+ φKi, (7.38)

Tiφ(λ) = (Tiφ)Ai, (7.39)

and their adjoint operations:

D∗i φ

∗(λ) = −∂iφ∗ +Kiφ∗, (7.40)

T +i φ(λ) = A−1

i (Tiφ∗), (7.41)

T −i φ(λ) = T−1

i (Aiφ∗) (7.42)

(T +i and T −

i correspond to forward and backward adjoint translationsrespectively) satisfy the same ∂ problems but with different normaliza-tions:

∂λ(Diφ) = (∂λη)Ki + φ∂λKi +∫C(Diφ)R (7.43)

∂λ(Tiφ) = (∂λη)Ai + (Tiφ)∂λAi +∫C(Tiφ)R (7.44)

∂λ(D∗i φ

∗) = −Ki(∂λη) + (∂λKi)φ∗ −∫CR(D∗

i φ∗) (7.45)

142

∂λ(T +i φ∗) = −A−1

i (∂λη) + (∂λA−1i )(Tiφ∗)−

∫CR(T +

i φ∗) (7.46)

∂λ(T −i φ∗) = −(T−1

i Ai)(∂λη) + T−1i ((∂λAi)φ∗)−

∫CR(T −

i φ∗). (7.47)

Also any polynomial combination L(D1, ..,DN )φ of the operators Di

with matrix coefficients which are functions of x but not of λ solve the ∂problem (7.22), with different normalizations (the same considerationshold for the adjoint operators), forming a ring of operators satisfyingthe ∂ problem (7.22).

In this ring of operators a crucial role is played by those operators Lsuch that Lφ satisfy the homogeneous ∂ problem (7.22), correspondingto η = 0 (i.e. Lφ→ 0, λ→∞); the uniqueness of the ∂ problem impliesindeed that Lφ = 0. The operators L form the left ideal of the ring; abasis Li of such ideal gives rise to a complete set of spectral problems

Liφ = 0 (7.48)

associated to the ∂ problem (7.22) [6]. These matrix differential and/ordifference equations are the nontrivial result of the dressing of the simpleequations (7.31),(7.32); they are of course compatible by constructionand their compatibility is equivalent to a set of (integrable) nonlinearequations, which are therefore constructed and solved simultaneously.

7.2 ∂-formulation of conjugate nets and quadrilateral lattices

In this section we illustrate an important application of the ∂-dressingmethod to the solution of the Darboux equations:

∂iQjk = QjiQik, i = j = k = i (7.49)

which characterize N-dimensional conjugate nets (CNs) inRM , i.e. mani-folds parametrized by conjugate coordinates [10], and of their differenceanalogues, the quadrilateral lattice (QL) equations [11],[12],[13]:

∆iQjk = (TiQji)Qik, ∆i = Ti − 1, i = j = k = i, (7.50)

which characterize the quadrilateral (or planar) lattices, N-dimensionallattices in RM , whose elementary quadrilaterals are planar [12].

To obtain solutions of the Darboux equations (7.49) and of the QLequations (7.50) respectively we choose the basic function ψ0 in thefollowing two ways [14], [9]:


ψ0(x, λ) = exp(λN∑k=1

xkPk) = diag(eλx1 , .., eλxN , 1, .., 1) (7.51)

ψ0(x, λ) =N∏k=1

[I + (λ− 1)Pk]xk = diag(λx1 , .., λxN , 1, .., 1) (7.52)

where Pi is the usual ith projection matrix: (Pi)jk = δijδik. Therefore:

Ki = λPi, Ai = I + (λ− 1)Pi (7.53)

We remark that equation (7.51) can be obtained in a straightforwardway from equation (7.52) through the substitution:

λ→ eελ ∼ 1 + ελ, x→ x

ε. (7.54)

Therefore all the relevant formulas for CNs can be immediately derivedfrom the corresponding formulas for QLs; in particular, in spectral space,one has the following simple recipe [9]:

f(λ)→ f(λ); f(1/λ)→ f(−λ); λf(λ)→ f(λ).

Nevertheless, for the sake of completeness, we shall write down equationsfor both the CN and the QL.

Let us derive now the complete set of spectral problems associatedwith the Darboux equations. Let χ(λ) be the canonical solution of the∂ problem (7.22):

χ ∼ I +Q

λ, λ→∞, (7.55)

then Diχ is also a solution of the homogeneous ∂ problem with theasymptotics Diχ ∼ λPi + QPi, λ → ∞, and so is PjDiχ ∼ PjQPi,λ→∞. Therefore the combination Lχ=PjDiχ− PjQPiχ is a solutionof the homogeneous ∂ problem and Lχ→ 0, λ→∞. Uniqueness implies

Lχ = PjDiχ− PjQPiχ = 0, i = j (7.56)

This is the compatible linear system of spectral problems we were lookingfor; the Darboux equations (7.49) follow from (7.56) in the λ→∞ limit,using (7.55).

In a similar way one derives the compatible linear system of spectralproblems

Pj(Ti − 1)χ− Ti(PjQPi)χ = 0, i = j (7.57)

144

which reduces down to the QL equations (7.50) in the λ → ∞ limit.Furthermore, if we define the matrix function

ψ(λ) := χ(λ)ψ0(λ), (7.58)

then ψ satisfies

∆iψjk(λ) = (TiQji)ψik(λ), i = 1, .., N, j, k = 1, .., N, i = j, (7.59)

[∂iψjk(λ) = Qjiψik(λ)], i = 1, .., N, j, k = 1, .., N, i = j, (7.60)

Hereafter the first equation corresponds to the QL case, while the secondone, in square brackets, corresponds to the CN and is obtainable eitherdirectly or through the straightforward continuous limit (7.54) from thefirst.

Similar considerations hold for the adjoint equations; if χ∗(λ) is thecanonical solution of (7.28) and ψ∗(λ) := (ψ0(λ))−1χ∗(λ), then the fol-lowing adjoint linear systems are satisfied:

((T −i − 1)χ∗)(λ)Pj = −χ∗(λ)T−1

i (PiQPj), i = j, (7.61)

[(D∗i χ

∗)(λ)Pj = −χ∗(λ)PiQPj ], i = j, (7.62)

∆iψ∗kj(λ) = (Tiψ∗

ki(λ))Qij , i = 1, .., N, j, k = 1, .., N, i = j, (7.63)

[∂iψ∗kj(λ) = ψ∗

ki(λ)Qij ], i = 1, .., N, j, k = 1, .., N, i = j, (7.64)

where Qij are the (ij)-components of the matrix Q.Also the solution χ(λ, µ) of the ∂ problem (7.22), corresponding to

the simple pole normalization, plays an important role in the theory.Using exactly the same procedure as before, it is possible to show thatthe matrix function

ψ(λ, µ) := (ψ0(µ))−1χ(λ, µ)ψ0(λ), (7.65)

is connected to the canonical solutions of the ∂ problem through thefollowing equations:

Di((ψ0(µ))−1χ(λ, µ)) = (Tiψ∗(µ))Piχ(λ), (7.66)

[Di((ψ0(µ))−1χ(λ, µ)) = ψ∗(µ)Piχ(λ)],

∆iψjk(λ, µ) = (Tiψ∗ji(µ))ψik(λ), i = 1, .., N, j, k = 1, .., N, i = j;

(7.67)

[∂iψjk(λ, µ) = ψ∗ji(µ)ψik(λ)], i = 1, .., N, j, k = 1, .., N, i = j;


χ∗(µ) = limλ→∞

λχ(λ, µ), χ∗(λ) = − limµ→∞µχ(λ, µ),

χij(0, µ) = −(Tjχ∗ij(µ))χjj(0), χij(λ, 0) = χii(0)T−1

i χij(λ)).

Furthermore, the matrix function

ψ∗(λ, µ) := (ψ0(n, λ))−1χ∗(λ, µ)ψ0(n;µ) (7.68)

is connected to ψ(λ, µ) through

ψ∗(λ, µ) = ψ(µ, λ), j, k = 1, ..,M. (7.69)

From the solutions ψ(λ, µ), ψ(λ) and ψ∗(λ) of the ∂ problem onecan construct a family of parallel N-dimensional quadrilateral latticesr(k), k = 1, ..M in RM , together with the corresponding tangentvectors Xi and Lame coefficients Hi, through the following formulas:

Ω =∫Cdλ ∧ dλ

∫Cdµ ∧ dµM∗(µ)ψ(λ, µ)M(λ), (7.70)

Xi =∫Cdλ ∧ dλψi(λ)M(λ) = (H∗

i(1),H∗i(2), ..,H

∗i(M)), (7.71)

X∗i =

∫Cdµ ∧ dµM∗(µ)ψ∗

i (µ) = (H(1)i,H(2)i, ..,H(M)i), (7.72)

in such a way that

∆ir(k) = (TiH(k)i)Xi,

[∂ir(k) = H(k)iXi],

where r(k) is the kth row of matrix Ω, ψi(λ) is the ith row of matrixψ(λ), ψ∗

i (µ) is the ith column of matrix ψ∗(µ) and M(λ), M∗(µ) arearbitrary M ×M matrices independent of x.

The evaluation of equations (7.56) at the distinguished point λ = 0leads [16] to the spectral formulation of the τ -function of the QL. Indeed,at λ = 0, equations (7.56) read:

∆iχjj(0) = (TiQji)χij(0),

χji(0) + (TiQji)χii(0) = 0, (7.73)

and imply thatTiχjj(0)χjj(0)

= 1− (TiQji)(TjQij). (7.74)

146

Since the RHS of equation (7.74) is symmetric wrt i and j, we canintroduce the potential τ , the τ - function of the QL, in the followingway:

χii(0) =Tiτ

τ. (7.75)

Using equation (7.75), equation (7.74) becomes

τ(TiTjτ)(Tiτ)(Tjτ)

= 1− (TiQji)(TjQij). (7.76)

The introduction of the second potential τ allows one to write the τ

function representation of the QL equations (7.50) [15]:

τTiτjk − (Tiτ)τjk = (Tiτji)τik, i = j = k = i,

τTiTjτ − (Tiτ)(Tjτ) + (Tiτji)(Tjτij) = 0 i = j,

where

τij = τQij , i = j.

Finally equation (7.73) gives:

χij(0) = −Tjτijτ

, i = j. (7.77)

and, analogously:

χ∗ii(0) =

T−1i τ

τ, χ∗

ij(0) =T−1i τijτ

. (7.78)

7.3 Transformations and reductions of conjugate nets andquadrilateral lattices

The theory of the symmetry transformations and of the symmetry con-straints for integrable nonlinear systems takes a particularly simple formin the ∂ dressing context. Hereafter we assume, without loss of general-ity, that the basic matrix function ψ0 be diagonal.

7.3.1 Symmetry transformations

If R(λ′, λ) is the ∂ datum associated with a certain nonlinear systems,then equation

R(λ′, λ) = β(λ′, λ)F (λ′)R(λ′, λ)F−1(λ), (7.79)


is a general symmetry transformation of that integrable nonlinear sys-tem, where β(λ′, λ) is an arbitrary scalar function of λ and λ′, and F (λ)is an arbitrary diagonal matrix function of λ (but independent of x).We call this symmetry, which always exists, of the ‘first kind’.

The particular form of ψ0 as function of λ often implies the existenceof additional symmetries. For instance, for QLs and CNs we have theobvious symmetries:

ψ0(λ−1) = ψ−10 (λ),

[ψ0(−λ) = ψ−10 (λ)], (7.80)

implying that, if R(λ′, λ) is the ∂ kernel of a QL [CN], then

R(λ′, λ) = RT (λ−1, λ′−1) = ψ0(λ′)RT0 (λ

−1, λ′−1)ψ−10 (λ)

[R(λ′, λ) = RT (−λ,−λ′) = ψ0(λ′)RT0 (−λ,−λ′)ψ−1

0 (λ)] (7.81)

are also symmetry transformations [17]. Therefore the symmetry (7.80)implies the existence of the additional symmetry transformation

R(λ′, λ) = β(λ′, λ)F (λ′)RT (λ−1, λ′−1)F−1(λ)

[R(λ′, λ) = α(λ′, λ)F (λ′)RT (−λ,−λ′)F−1(λ)], (7.82)

which we call of ‘second kind’.It is interesting to remark that the ‘square’ of a second kind symmetry

transformation is a symmetry transformation of first kind; indeed:

ˆR(λ′, λ) = β(λ, λ′)F (λ′)R(λ′, λ)F−1(λ),

β(λ′, λ) = β(λ′, λ)β(λ−1, λ′−1), F (λ) = F (λ)F−1(λ−1),

[β(λ′, λ) = β(λ′, λ)β(−λ,−λ′), F (λ) = F (λ)F−1(−λ)]; (7.83)

but it is not true that any symmetry transformation of first kind is thesquare of a symmetry transformation of second kind.

It is possible to show that one of the implications of the symme-try transformation of the second kind (7.82) is that λ−2F (λ−1)φT (λ−1)satisfies the adjoint ∂ problem (7.28), while φ∗T (λ−1)F−1(λ−1) satisfiesthe ∂ problem (7.22):

∂λ(λ−2F (λ−1)φT (λ−1)) = ∂λ(λ

−2F (λ−1))φT (λ−1)

+ λ−2F (λ−1)∂λη(λ−1)−

∫CR(λ, λ′)(λ′−2F (λ′−1)φT (λ′−1))dλ′ ∧ dλ′,

(7.84)

148

∂λ(φ∗T (λ−1)F−1(λ−1)) = φ∗T (λ−1)∂λF

−1(λ−1)

− (∂λη(λ−1))F−1(λ−1) +

∫C(φ∗T (λ′−1)F−1(λ′−1))R(λ′, λ)dλ′ ∧ dλ′,

(7.85)

[λ−1 → −λ]

where φ and φ∗ are the solutions of the ∂ problems (7.22) and (7.28)corresponding to R.

These equations, through the bilinear identity (7.29), imply the fol-lowing nonlocal quadratic relations [17]:∫

γ∞φ(λ)(λ−2F (λ−1))φT (λ−1)dλ+

∫C[φ(λ)∂λ(λ

−2F (λ−1))φT (λ−1)

+φ(λ)F (λ)(∂λη(λ−1)) + (∂λη(λ))F (λ)φT (λ−1)]dλ ∧ dλ = 0, (7.86)∫

γ∞φ∗T (λ−1)F−1(λ−1)φ∗(λ)dλ+

∫C[φ∗T (λ−1)(∂λF

−1(λ−1))φ∗(λ)

−φ∗T (λ−1)F−1(λ−1)(∂λη(λ))

−(∂λη(λ−1))F−1(λ−1)φ∗(λ)]dλ ∧ dλ = 0, (7.87)

[λ−1 → −λ]. (7.88)

Therefore the symmetry transformation of second kind establishes anontrivial quadratic connection, whose nature depends on the particularchoice of F (λ) , between the solutions of the ∂ problem (7.22) for R andof the adjoint ∂ problem (7.28) for R or, equivalently, between QL’sand the dual objects, the quadrilateral hyperplane lattices [16], of thetransformed QL.

7.3.2 Symmetry constraints

The transformation of second kind (7.82) allows to introduce in a naturalway the symmetry constraint R(λ′, λ) = R(λ′, λ). This constraint canbe written down in terms of R in the following way:

R(λ′, λ) = |λ|−4λ′−2F (λ′)RT (λ−1, λ′−1)F−1(λ), (7.89)

[R(λ′, λ) = F (λ′)RT (−λ,−λ′)F−1(λ)]


and is admissible iff

F (λ) = F±(λ) = λ−1(A(λ)±A(λ−1)), (7.90)

[F±(λ) = A(λ)±A(−λ)], (7.91)

where A(λ) is an arbitrary diagonal matrix.The nonlocal quadratic relations (7.86), (7.87) become the following

nonlocal quadratic constraints [16]:∫γ∞

φ(λ)(λ−2F (λ−1))φT (λ−1)dλ+∫C[φ(λ)∂λ(λ

−2F (λ−1))φT (λ−1)

+φ(λ)F (λ)(∂λη(λ−1)) + (∂λη(λ))F (λ)φT (λ−1)]dλ ∧ dλ = 0, (7.92)

∫γ∞

φ∗T (λ−1)F−1(λ−1)φ∗(λ)dλ+∫C[φ∗T (λ−1)(∂λF

−1(λ−1))φ∗(λ)

− φ∗T (λ−1)F−1(λ−1)(∂λη(λ))

− (∂λη(λ−1))F−1(λ−1)φ∗(λ)]dλ ∧ dλ = 0, (7.93)

[λ−1 → −λ].

Therefore the symmetry constraint (7.89) establishes a nontrivial quad-ratic connection, whose nature depends on the particular choice of F (λ),between the solutions of the ∂ problem (7.22) and of its adjoint (7.28)or, equivalently, between QL’s and their dual objects, the quadrilateralhyperplane lattices [16].

In the following we shall identify the matrix functions A(λ) whichcorrespond to the symmetric, circular and Egorov lattices [symmetric,orthogonal and Egorov nets].

7.3.3 ∂-formulation of the symmetric lattice

It is possible to show that, for the following choice [16]:

A(λ) = 1/2 ⇒ F+(λ) = λ−1I, ⇒R(λ′, λ) = |λ|−4λ′−2λλ′−1RT (λ−1, λ′−1)

[F+(λ) = I ⇒ R(λ′, λ) = RT (−λ,−λ′)] (7.94)

the following equations hold:

ψT (λ, µ) = (λµ)−1ψ(µ−1, λ−1), (7.95)

150

[ψT (λ, µ) = ψ(−µ,−λ)], (7.96)

λ−1ψij(λ−1) =Tiτ

τψ∗ji(λ), (7.97)

[ψij(−λ) = ψ∗ji(λ)], (7.98)

χT (0) = χ(0), (7.99)

[QT = Q]. (7.100)

Using equations (7.78) and (7.77) it is easy to identify equation (7.99)with the symmetric constraint of a QL [CN] [16]:

Ti(τQji) = Tj(τQij), i = j, (7.101)

[Qij = Qji],

and equation (7.70) allows to construct a parallel system of symmetricquadrilateral lattices, provided that

M∗(λ) = λ|λ|−4MT (λ), (7.102)

[M∗(λ) = MT (λ)]. (7.103)

7.3.4 ∂ formulation of the circular lattice

The choice [9] [[14],[18]]:

A(λ) = (λ− 1)I ⇒ F−(λ) =λ+ 1

λ(λ− 1)I [F−(λ) = λ−1I] (7.104)

implies the following consequences of the bilinear constraints (7.92) and(7.93):

χ(0) + χT (0) = 2χ(1)χT (1), (7.105)

[χ(0)χT (0) = I], (7.106)

χ∗(0) + χ∗T (0) = 2χ∗T (−1)χ∗(−1), (7.107)

[∂iQji + ∂jQij +∑k =i,j

QkiQkj = 0, i = j], (7.108)

λ+ 1λ(1− λ)

χT (λ−1, µ−1) =µ(µ+ 1)1− µ

χ(µ, λ) + χ(1, λ)χT (1, µ−1), (7.109)


λ− 1λ(1 + λ)

χT (µ−1, λ−1) =µ(µ− 1)1 + µ

χ(λ, µ) + χT (µ−1,−1)χ(λ,−1),

(7.110)

4χ(1,−1)χT (1,−1) = I. (7.111)

Through the identifications:

x(i) = (ψi1(1, µ), .., ψiM (1, µ)), (7.112)

Xi = (ψi1(1), .., ψiM (1)), H(n)i = ψ∗(µ), (7.113)

X∗i = (ψ∗

1i(−1), .., ψ∗Mi(−1))T , (7.114)

one obtains:

χii(0) =Tiτ

τ= |Xi|2, (7.115)

and equation (7.105) coincides with the circularity [orthogonality] con-straint

Xi · TiXj +Xj · TjXi = 0, i = j, (7.116)

[Xi ·Xj +Xj ·Xi = 0, i = j].

We also recognize in equations (7.108) the Lame equations for orthogonalnets.

7.3.5 ∂ formulation of the egorov lattice

The Egorov lattice [19][net [20],[10]] arises when the circular and sym-metric constraints are simultaneously satisfied; i.e., when

|λ′|−4λ−2RT (λ′−1, λ−1) = λR(λ, λ′)λ′−1

=(

λ+ 1λ(λ− 1)

)−1R(λ, λ′)

λ′ + 1λ′(λ′ − 1)

.

This implies the equation:

2λ(λ− λ′)λ′(λ′ − 1)(λ+ 1)

R(λ, λ′) = 0, (7.117)

which admits the (unique) distributional solution

R(λ, λ′) =i

2δ(λ− λ′)R(λ). (7.118)

152

Therefore the ∂ formulation of the Egorov lattice is given in terms ofthe local ∂ problem

∂λφ(λ) = φ(λ)R(λ), (7.119)

RT (λ−1) = |λ|4λ2R(λ), (7.120)

[RT (−λ) = R(λ)].

It is straightforward now to verify that this formulation corresponds tothe Egorov lattice [net] [16].

BibliographyV. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevsky. Theory

of solitons. The inverse problem method. Plenum Press, 1999.M. J. Ablowitz and H. Segur. Solitons and the inverse scattering transform.

SIAM, Philadelphia, 1981.F. Calogero and A. Degasperis. Spectral transform and solitons: tools to

solve and investigate nonlinear evolution equations, vol.1.North-Holland. Amsterdam, 1982.

B. G. Konopelchenko. Solitons in multidimension. World Scientific,Singapore, 1993.

M. J. Ablowitz and P. A. Clarkson. Solitons, nonlinear evolution equationsand inverse scattering. Cambridge Univ. Press, Cambridge, 1991.

V. E. Zakharov and S. V. Manakov. Construction of multidimensionalnonlinear integrable systems and their solution, Funk. Anal. Appl. 19(1985) 89.

L. V. Bogdanov and S. V. Manakov. The nonlocal ∂-problem and(2+1)-dimensional soliton equations, J. Phys. A: Math. Gen. 21 (1988)L537- L544.

V. E. Zakharov. On the dressing method in: Inverse Methods in Action, ed.P. C. Sabatier, Berlin, Springer, 1990, 602-623.

A. Doliwa, S. V. Manakov and P. M. Santini. ∂-reductions of themultidimensional quadrilateral lattice: the multidimensional circularlattice, Comm. Math. Phys. 196 (1998) 1-18.

G. Darboux. Lecons sur le systemes orthogonaux et le coordonnes curvilignes,2-eme ed., Gauthier-Villars, Paris, 1910.

L. V. Bogdanov and B. G. Konopelchenko. Lattice and q-differenceDarboux–Zakharov–Manakov systems via ∂-method, J. Phys. A: Math.Gen. 28 (1995) L173-L178.

A. Doliwa and P. M. Santini. Multidimensional quadrilateral lattices areintegrable, Phys. Lett. A 233 (1997) 365-372.

L. V. Bogdanov and B. G. Konopelchenko. Analytic bilinear approach tointegrable hierarchies II. Multicomponent KP and 2D Toda latticehierarchies, J. Math. Phys. 39 (1998) 4701-4728.

V. E. Zakharov. Description of the N-orthogonal curvilinear coordinatesystems and hamiltonian integrable systems of hidrodynamic type, I:integration of the Lame equations, Duke Math. J. 94 (1998) 103-139.


A. Doliwa, M. Manas, L. Martinez Alonso, E. Medina and P. M. Santini.Charged free fermion, vertex operators and classical transformations ofconjugate nets, J. Phys. A 32 (1999) 1197-1216.

A. Doliwa and P. M. Santini. The symmetric, d-invariant and Egorovreductions of the quadrilateral lattice, Solv-int/9907012. J. Phys. andGeom. (in press).

P. M. Santini. Symmetry transformations and symmetry constraints of thequadrilateral lattice in the ∂-formalism, in preparation.

V. E. Zakharov and S. V. Manakov. Reduction in systems integrated by themethod of the inverse scattering problem, Dokl. Math. 57 (1998)471-474.

W. K. Schief. Talk given at the Workshop: Nonlinear systems, solitons andgeometry, Oberwolfach, 1997.

L. Bianchi. Lezioni di geometria differenziale, 3-a ed. Zanichelli, Bologna,1924.

(london mathematical society lecture note series )lionel mason, yavuz nutku-geometry and...

Documents

area of research

current research

contemporary research

twistor theory

algebraic geometry

schnepsed201 singularities

feza gursey institute

botvinnik171 squares