Topological Solitons

by Nicholas Manton , Paul Sutcliffe

• ISBN13: 9780521838368
• ISBN10: 0521838363
• Format: Hardcover
• Copyright: 2004-07-05
• Publisher: Cambridge University Press

Summary

Topological solitons occur in many nonlinear classical field theories. They are stable, particle-like objects, with finite mass and a smooth structure. Examples are monopoles and Skyrmions, Ginzburg-Landau vortices and sigma-model lumps, and Yang-Mills instantons. This book is a comprehensive survey of static topological solitons and their dynamical interactions. Particular emphasis is placed on the solitons which satisfy first-order Bogomolny equations. For these, the soliton dynamics can be investigated by finding the geodesics on the moduli space of static multi-soliton solutions. Remarkable scattering processes can be understood this way. The book starts with an introduction to classical field theory, and a survey of several mathematical techniques useful for understanding many types of topological soliton. Subsequent chapters explore key examples of solitons in one, two, three and four dimensions. The final chapter discusses the unstable sphaleron solutions which exist in several field theories.

Author Biography

Nicholas Manton is Professor of Mathematical Physics in the Department of Applied Mathematics and Theoretical Physics at Cambridge.

Table of Contents

Preface	ix
Introduction	1	(14)
Solitons as particles	1	(2)
A brief history of topological solitons	3	(4)
Bogomolny equations and moduli spaces	7	(1)
Soliton dynamics	8	(2)
Solitons and integrable systems	10	(2)
Solitons - experimental status	12	(2)
Outline of this book	14	(1)
Lagrangians and fields	15	(32)
Finite-dimensional systems	15	(6)
Symmetries and conservation laws	21	(2)
Field theory	23	(5)
Noether's theorem in field theory	28	(3)
Vacua and spontaneous symmetry breaking	31	(1)
Gauge theory	32	(11)
The Higgs mechanism	43	(2)
Gradient flow in field theory	45	(2)
Topology in field theory	47	(28)
Homotopy theory	47	(7)
Topological degree	54	(6)
Gauge fields as differential forms	60	(2)
Chern numbers of abelian gauge fields	62	(5)
Chern numbers for non-abelian gauge fields	67	(2)
Chern-Simons forms	69	(6)
Solitons - general theory	75	(34)
Topology and solitons	75	(7)
Scaling arguments	82	(5)
Symmetry and reduction of dimension	87	(12)
Principle of symmetric criticality	99	(3)
Moduli spaces and soliton dynamics	102	(7)
Kinks	109	(22)
Bogomolny bounds and vacuum structure	109	(2)
φ4 kinks	111	(5)
Sine-Gordon kinks	116	(8)
Generalizations	124	(7)
Lumps and rational maps	131	(27)
Lumps in the O(3) sigma model	131	(10)
Lumps on a sphere and symmetric maps	141	(10)
Stabilizing the lump	151	(7)
Vortices	158	(83)
Ginzburg-Landau energy functions	158	(5)
Topology in the global theory	163	(1)
Topology in the gauged theory	164	(3)
Vortex solutions	167	(8)
Forces between gauged vortices	175	(2)
Forces between vortices at large separation	177	(4)
Dynamics of gauged vortices	181	(16)
Second order dynamics	181	(7)
Gradient flow	188	(5)
First order dynamics	193	(4)
Vortices at critical coupling	197	(5)
Moduli space dynamics	202	(3)
The metric on MN	205	(10)
Two-vortex scattering	215	(5)
First order dynamics near critical coupling	220	(3)
Global vortex dynamics	223	(4)
Varying the geometry	227	(11)
Volume of moduli space	231	(3)
Toroidal geometry - the Abrikosov lattice	234	(2)
Vortices on the hyperbolic plane	236	(2)
Statistical mechanics of vortices	238	(3)
Monopoles	241	(108)
Dirac monopoles	241	(8)
Monopoles as solitons	249	(12)
Bogomolny-Prasad-Sommerfield monopoles	261	(5)
Dyons	266	(3)
The Nahm transform	269	(7)
Construction of monopoles from Nahm data	276	(8)
Spectral curves	284	(9)
Rational maps and monopoles	293	(14)
Alternative monopole methods	307	(2)
Monopole dynamics	309	(5)
Moduli spaces and geodesic motion	314	(19)
Well separated monopoles	333	(6)
SU(m) monopoles	339	(7)
Hyperbolic monopoles	346	(3)
Skyrmions	349	(67)
The Skyrme model	349	(4)
Hedgehogs	353	(3)
Asymptotic interactions	356	(5)
Low charge Skyrmions	361	(4)
The rational map ansatz	365	(8)
Higher charge Skyrmions	373	(9)
Lattices, crystals and shells	382	(7)
Skyrmion dynamics	389	(11)
Generalizations of the Skyrme model	400	(6)
Quantization of Skyrmions	406	(2)
The Skyrme-Faddeev model	408	(8)
Instantons	416	(25)
Self-dual Yang-Mills fields	416	(9)
The ADHM construction	425	(3)
Symmetric instantons	428	(3)
Skyrme fields from instantons	431	(6)
Monopoles as self-dual gauge fields	437	(3)
Higher rank gauge groups	440	(1)
Saddle points - sphalerons	441	(26)
Mountain passes	441	(3)
Sphalerons on a circle	444	(3)
The gauged kink	447	(4)
Monopole-antimonopole dipole	451	(3)
The electroweak sphaleron	454	(9)
Unstable solutions in other theories	463	(4)
References	467	(24)
Index	491