Solitons, Instantons, and Twistors (Oxford Graduate Texts in Mathematics (19))

"Most nonlinear differential equations arising in natural sciences admit chaotic behaviour and cannot be solved analytically. Integrable systems lie on the other extreme. They possess regular, stable, and well behaved solutions known as solitons and instantons. These solutions play important roles in pure and applied mathematics as well as in theoretical physics where they describe configurations topologically different from vacuum. While integrable equations in lower space-time dimensions can be solved using the inverse scattering transform, the higher-dimensional examples of anti-self-dual Yang-Mills and Einstein equations require twistor theory. Both techniques rely on an ability to represent nonlinear equations as compatibility conditions for overdetermined systems of linear differential equations." "The book provides a self-contained and accessible introduction to the subject. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system."--Jacket Contents......Page 8 List of Figures......Page 13 List of Abbreviations......Page 14 1.1 Hamiltonian formalism......Page 16 1.2 Integrability and action–angle variables......Page 19 1.3 Poisson structures......Page 29 2.1 The history of two examples......Page 35 2.1.1 A physical derivation of KdV......Page 36 2.1.2 Bäcklund transformations for the Sine-Gordon equation......Page 39 2.2 Inverse scattering transform for KdV......Page 40 2.2.1 Direct scattering......Page 43 2.2.2 Properties of the scattering data......Page 44 2.2.3 Inverse scattering......Page 45 2.2.4 Lax formulation......Page 46 2.2.5 Evolution of the scattering data......Page 47 2.3 Reflectionless potentials and solitons......Page 48 2.3.1 One-soliton solution......Page 49 2.3.2 N-soliton solution......Page 50 2.3.3 Two-soliton asymptotics......Page 51 3.1 First integrals......Page 58 3.2.1 Bi-Hamiltonian systems......Page 61 3.3 Zero-curvature representation......Page 63 3.3.1 Riemann–Hilbert problem......Page 65 3.3.2 Dressing method......Page 67 3.3.3 From Lax representation to zero curvature......Page 69 3.4 Hierarchies and finite-gap solutions......Page 71 4.1 Lie groups and Lie algebras......Page 79 4.2 Vector fields and one-parameter groups of transformations......Page 82 4.3 Symmetries of differential equations......Page 86 4.3.1 How to find symmetries......Page 89 4.3.2 Prolongation formulae......Page 90 4.4 Painlevé equations......Page 93 4.4.1 Painlevé test......Page 97 5.1 A variational principle......Page 100 5.1.1 Legendre transform......Page 102 5.1.2 Symplectic structures......Page 103 5.1.3 Solution space......Page 104 5.2 Field theory......Page 105 5.2.1 Solution space and the geodesic approximation......Page 107 5.3 Scalar kinks......Page 108 5.3.1 Topology and Bogomolny equations......Page 111 5.3.2 Higher dimensions and a scaling argument......Page 113 5.3.3 Homotopy in field theory......Page 114 5.4 Sigma model lumps......Page 115 6 Gauge field theory......Page 120 6.1 Gauge potential and Higgs field......Page 121 6.1.1 Scaling argument......Page 123 6.1.2 Principal bundles......Page 124 6.2 Dirac monopole and flux quantization......Page 125 6.2.1 Hopf fibration......Page 127 6.3 Non-abelian monopoles......Page 129 6.3.1 Topology of monopoles......Page 130 6.3.2 Bogomolny–Prasad–Sommerfeld (BPS) limit......Page 131 6.4 Yang–Mills equations and instantons......Page 134 6.4.1 Chem and Chem–Simons forms......Page 135 6.4.2 Minimal action solutions and the anti-self-duality condition......Page 137 6.4.3 Ansatz for ASD fields......Page 138 6.4.4 Gradient flow and classical mechanics......Page 139 7.1 Lax pair......Page 144 7.1.1 Geometric interpretation......Page 147 7.2.1 History and motivation......Page 148 7.2.2 Spinor notation......Page 152 7.2.3 Twistor space......Page 154 7.2.4 Penrose–Ward correspondence......Page 156 8.1 Reductions to integrable equations......Page 164 8.2 Integrable chiral model......Page 169 8.2.1 Soliton solutions......Page 172 8.2.2 Lagrangian formulation......Page 180 8.2.3 Energy quantization of time-dependent unitons......Page 183 8.2.4 Moduli space dynamics......Page 188 8.2.5 Mini-twistors......Page 196 9.1 Examples of gravitational instantons......Page 206 9.2 Anti-self-duality in Riemannian geometry......Page 210 9.2.1 Two-component spinors in Riemannian signature......Page 213 9.3 Hyper-Kähler metrics......Page 217 9.4 Multi-centred gravitational instantons......Page 221 9.4.1 Belinskii–Gibbons–Page–Pope class......Page 225 9.5 Other gravitational instantons......Page 227 9.5.1 Compact gravitational instantons and K3......Page 230 9.6 Einstein–Maxwell gravitational instantons......Page 231 9.7 Kaluza–Klein monopoles......Page 236 9.7.1 Kaluza–Klein solitons from Einstein–Maxwell instantons......Page 237 9.7.2 Solitons in higher dimensions......Page 241 10 Anti-self-dual conformal structures......Page 244 10.1 α-surfaces and anti-self-duality......Page 245 10.2 Curvature restrictions and their Lax pairs......Page 246 10.2.1 Hyper-Hermitian structures......Page 247 10.2.2 ASD Kähler structures......Page 249 10.2.3 Null-Kähler structures......Page 251 10.2.4 ASD Einstein structures......Page 252 10.2.5 Hyper-Kähler structures and heavenly equations......Page 253 10.3.1 Einstein–Weyl geometry......Page 261 10.3.2 Null symmetries and projective structures......Page 268 10.3.3 Dispersionless integrable systems......Page 271 10.4 ASD conformal structures in neutral signature......Page 277 10.4.2 Curved examples......Page 278 10.5 Twistor theory......Page 280 10.5.1 Curvature restrictions......Page 285 10.5.2 ASD Ricci-flat metrics......Page 287 10.5.3 Twistor theory and symmetries......Page 298 Appendix A: Manifolds and topology......Page 302 A.1 Lie groups......Page 305 A.2 Degree of a map and homotopy......Page 309 A.2.1 Homotopy......Page 311 A.2.2 Hermitian projectors......Page 313 Appendix B: Complex analysis......Page 315 B.1 Complex manifolds......Page 316 B.2 Holomorphic vector bundles and their sections......Page 318 B.3 Čech cohomology......Page 322 B.3.1 Deformation theory......Page 323 C.1 Introduction......Page 325 C.2 Exterior differential system and Frobenius theorem......Page 329 C.3 Involutivity......Page 335 C.4 Prolongation......Page 339 C.4.1 Differential invariants......Page 341 C.5 Method of characteristics......Page 347 C.6 Cartan–Kähler theorem......Page 350 References......Page 359 C......Page 370 H......Page 371 M......Page 372 S......Page 373 Z......Page 374 Contents 8 List of Figures 13 List of Abbreviations 14 1 Integrability in classical mechanics 16 1.1 Hamiltonian formalism 16 1.2 Integrability and action–angle variables 19 1.3 Poisson structures 29 2 Soliton equations and the inverse scattering transform 35 2.1 The history of two examples 35 2.1.1 A physical derivation of KdV 36 2.1.2 Bäcklund transformations for the Sine-Gordon equation 39 2.2 Inverse scattering transform for KdV 40 2.2.1 Direct scattering 43 2.2.2 Properties of the scattering data 44 2.2.3 Inverse scattering 45 2.2.4 Lax formulation 46 2.2.5 Evolution of the scattering data 47 2.3 Reflectionless potentials and solitons 48 2.3.1 One-soliton solution 49 2.3.2 N-soliton solution 50 2.3.3 Two-soliton asymptotics 51 3 Hamiltonian formalism and zero-curvature representation 58 3.1 First integrals 58 3.2 Hamiltonian formalism 61 3.2.1 Bi-Hamiltonian systems 61 3.3 Zero-curvature representation 63 3.3.1 Riemann–Hilbert problem 65 3.3.2 Dressing method 67 3.3.3 From Lax representation to zero curvature 69 3.4 Hierarchies and finite-gap solutions 71 4 Lie symmetries and reductions 79 4.1 Lie groups and Lie algebras 79 4.2 Vector fields and one-parameter groups of transformations 82 4.3 Symmetries of differential equations 86 4.3.1 How to find symmetries 89 4.3.2 Prolongation formulae 90 4.4 Painlevé equations 93 4.4.1 Painlevé test 97 5 Lagrangian formalism and field theory 100 5.1 A variational principle 100 5.1.1 Legendre transform 102 5.1.2 Symplectic structures 103 5.1.3 Solution space 104 5.2 Field theory 105 5.2.1 Solution space and the geodesic approximation 107 5.3 Scalar kinks 108 5.3.1 Topology and Bogomolny equations 111 5.3.2 Higher dimensions and a scaling argument 113 5.3.3 Homotopy in field theory 114 5.4 Sigma model lumps 115 6 Gauge field theory 120 6.1 Gauge potential and Higgs field 121 6.1.1 Scaling argument 123 6.1.2 Principal bundles 124 6.2 Dirac monopole and flux quantization 125 6.2.1 Hopf fibration 127 6.3 Non-abelian monopoles 129 6.3.1 Topology of monopoles 130 6.3.2 Bogomolny–Prasad–Sommerfeld (BPS) limit 131 6.4 Yang–Mills equations and instantons 134 6.4.1 Chem and Chem–Simons forms 135 6.4.2 Minimal action solutions and the anti-self-duality condition 137 6.4.3 Ansatz for ASD fields 138 6.4.4 Gradient flow and classical mechanics 139 7 Integrability of ASDYM and twistor theory 144 7.1 Lax pair 144 7.1.1 Geometric interpretation 147 7.2 Twistor correspondence 148 7.2.1 History and motivation 148 7.2.2 Spinor notation 152 7.2.3 Twistor space 154 7.2.4 Penrose–Ward correspondence 156 8 Symmetry reductions and the integrable chiral model 164 8.1 Reductions to integrable equations 164 8.2 Integrable chiral model 169 8.2.1 Soliton solutions 172 8.2.2 Lagrangian formulation 180 8.2.3 Energy quantization of time-dependent unitons 183 8.2.4 Moduli space dynamics 188 8.2.5 Mini-twistors 196 9 Gravitational instantons 206 9.1 Examples of gravitational instantons 206 9.2 Anti-self-duality in Riemannian geometry 210 9.2.1 Two-component spinors in Riemannian signature 213 9.3 Hyper-Kähler metrics 217 9.4 Multi-centred gravitational instantons 221 9.4.1 Belinskii–Gibbons–Page–Pope class 225 9.5 Other gravitational instantons 227 9.5.1 Compact gravitational instantons and K3 230 9.6 Einstein–Maxwell gravitational instantons 231 9.7 Kaluza–Klein monopoles 236 9.7.1 Kaluza–Klein solitons from Einstein–Maxwell instantons 237 9.7.2 Solitons in higher dimensions 241 10 Anti-self-dual conformal structures 244 10.1 α-surfaces and anti-self-duality 245 10.2 Curvature restrictions and their Lax pairs 246 10.2.1 Hyper-Hermitian structures 247 10.2.2 ASD Kähler structures 249 10.2.3 Null-Kähler structures 251 10.2.4 ASD Einstein structures 252 10.2.5 Hyper-Kähler structures and heavenly equations 253 10.3 Symmetries 261 10.3.1 Einstein–Weyl geometry 261 10.3.2 Null symmetries and projective structures 268 10.3.3 Dispersionless integrable systems 271 10.4 ASD conformal structures in neutral signature 277 10.4.1 Conformal compactification 278 10.4.2 Curved examples 278 10.5 Twistor theory 280 10.5.1 Curvature restrictions 285 10.5.2 ASD Ricci-flat metrics 287 10.5.3 Twistor theory and symmetries 298 Appendix A: Manifolds and topology 302 A.1 Lie groups 305 A.2 Degree of a map and homotopy 309 A.2.1 Homotopy 311 A.2.2 Hermitian projectors 313 Appendix B: Complex analysis 315 B.1 Complex manifolds 316 B.2 Holomorphic vector bundles and their sections 318 B.3 Čech cohomology 322 B.3.1 Deformation theory 323 Appendix C: Overdetermined PDEs 325 C.1 Introduction 325 C.2 Exterior differential system and Frobenius theorem 329 C.3 Involutivity 335 C.4 Prolongation 339 C.4.1 Differential invariants 341 C.5 Method of characteristics 347 C.6 Cartan–Kähler theorem 350 References 359 Index 370 A 370 B 370 C 370 D 371 E 371 F 371 G 371 H 371 I 372 J 372 K 372 L 372 M 372 N 373 P 373 Q 373 R 373 S 373 T 374 U 374 V 374 W 374 Y 374 Z 374

Most nonlinear differential equations arising in natural sciences admit chaotic behavior and cannot be solved analytically. Integrable systems lie on the other extreme. They possess regular, stable, and well behaved solutions known as solitons and instantons. These solutions play important roles in pure and applied mathematics as well as in theoretical physics where they describe configurations topologically different from vacuum. While integrable equations in lower space-time dimensions can be solved using the inverse scattering transform, the higher-dimensional examples of anti-self-dual Yang-Mills and Einstein equations require twistor theory. Both techniques rely on an ability to represent nonlinear equations as compatibility conditions for overdetermined systems of linear differential equations.

The book provides a self-contained and accessible introduction to the subject. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system.