Grundlehren der mathematischen Wissenschaften 307 A Series of Comprehensive Studies in Mathematics Editors M. Artin S. S. Chern 1. Coates 1. M. Frohlich H. Hironaka F. Hirzebruch L. Hormander C. C. Moore 1. K. Moser M. Nagata W. Schmidt D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt S.Watanabe Managing Editors M. Berger B. Eckmann S. R. S. Varadhan Albert S. Schwarz Quantum Field Theory and Topology With 30 Figures Springer-Verlag Berlin Heidelberg GmbH Albert S. Schwarz Department of Mathematics 565 Kerr Hall University of California Davis, CA 95616, USA Translators: Eugene Yankowsky Silvio Levy Geometry Center 1300 South Second St. Minneapolis, MN 55454, USA Title of the original Russian edition: Kvantovaya teoriya polya i topologiya. Nauka, Moscow 1989. An expanded version of the last third of the Russian edition is being published separately in English under the title: Topology for Physicists. Mathematics Subject Classification (1991): 81Txx ISBN 978-3-642-08130-9 ISBN 978-3-662-02943-5 (eBook) DOI 10.1007/978-3-662-02943-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993 Softcover reprint of the hardcover 18t edition 1993 Typesetting: Camera-ready copy produced from the translation editor's output file using a Springer TEX macro package 41/3140-543210 Printed on acid-free paper Preface In recent years topology has firmly established itself as an important part of the physicist's mathematical toolkit. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics-suffice it to say that topological ideas play an important role in one of the suggested explanations of high-temperature superconductivity. Topology is also used in the analysis of another remarkable discovery of recent years, the quantum Hall effect. The main focus of this book is on the results of quantum field theory that are obtained by topological methods, but some topological aspects of the theory of condensed matter are also discussed. The topological concepts and theorems used in physics are very diverse, and for this reason I have included a substantial amount of purely mathematical information. This book is aimed at different classes of readers-from students who know only basic calculus, linear algebra and quantum mechanics, to specialists in quantum field theory and mathematicians who want to learn about the physical applications of topology. Part I can be considered as an introduction to quantum field theory: it dis cusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III pro vides the mathematical prerequisites for the understanding of Parts I and II. Readers with a weak background in group theory and topology might profit from skimming Part III before reading the first two parts. For a more extensive introduction to topology, the reader is directed to Topology for Physicists [59]. Parts I and II contain some references to that book: the reference T1.3, for ex ample, means Chapter 1, Section 3 of Topology for Physicists. In spite of these references, this book tries to be self-contained; the quantum field theorist who has looked through Part III and the Definitions and Notations at the-begin ning of the book will likely be able to read Part II without trouble. Likewise, mathematicians can use Part I to familiarize themselves with the foundations of quantum field theory, and then proceed with Part II for a study of the physical applications of topology. Text in smaller print can be skipped without detriment to the main flow of ideas. Here is a more detailed outline of Part II. Chapter 8, devoted to topo logically stable defects in condensed matter, is essentially independent of the VI Preface rest of the book. Chapters 9-15 and 18 are devoted to topological integrals of motion and topologically nontrivial particles, such as magnetic monopoles. In Chapters 16 and 17 we analyze symmetric gauge fields, and apply the results to the study of magnetic monopoles. Topologically nontrivial strings are the subject of Chapters 19 and 20. Nonlinear fields and multivalued action inte grals occupy Chapters 21 and 22, which are closely related. Functional integrals and their applications to quantum theory are discussed in Chapters 23 and 24, the results being used in Chapter 25 to study the quantization of gauge theo ries. Chapters 26-28 contain mathematical information on elliptic operators and their determinants, to be used in the succeeding chapters. Chapter 29 studies quantum anomalies, and Chapters 30-32 instantons in gauge theories. Finally, Chapter 33 considers functional integrals in theories with fermions, and the re sults are applied in Chapter 34 to the study of the instanton contribution in quantum chromodynamics. For a minimum understanding of the applications of topology to quantum field theory, I suggest reading Chapters 9-13, 23-25 and 30. Readers interested in topologically nontrivial particles and strings can then proceed to read Chap ters 14, 19 and 21, and as supplementary material Chapters 18 (which depends on 14), 20 (based on 14 and 19) and 22 (dependent on 21). Reapers interested in instantons should turn to Chapters 31-34, after having read Chapters 26-29 for mathematical background, if necessary. The decision to keep this book to a manageable size and to a relatively elementary level meant that many fascinating physical results obtained through topological methods could not be included. In particular, I have not discussed the rapidly developing theory of strings, in spite of the crucial role that topology plays in it. The questions discussed in this book were discussed with many of my col leagues. I am particularly grateful to L. D. Faddeev, S. P. Novikov and A. M. Polyakov, who made important contributions to the field covered in the book. I take this opportunity to thank my former students, V. A. Fateev, I. V. Frolov, D. B. Fuchs and Yu. S. Tyupkin, and especially M. A. Baranov and A. A. Rosly, who gave me invaluable assistance. I would also like to thank my wife, L. M. Kissina, for her understanding and help. A. S. Schwarz Contents Introduction ....... 1 Definitions and Notations 6 Part I The Basic Lagrangians of Quantum Field Theory 11 1 The Simplest Lagrangians 13 2 Quadratic Lagrangians 17 3 Internal Symmetries ... 19 4 Gauge Fields ....... 24 5 Particles Corresponding to Nonquadratic Lagrangians 28 6 Lagrangians of Strong, Weak and Electromagnetic Interactions 30 7 Grand Unifications . . . . . . . . . . . . . . . . . . . . . . . . . 37 Part II Topological Methods in Quantum Field Theory 41 8 Topologically Stable Defects .......... 43 9 Topological Integrals of Motion . . . . . . . . . 56 10 A Two-Dimensional Model. Abrikosov Vortices 62 11 't Hooft-Polyakov Monopoles . . . . . . . . . . 68 12 Topological Integrals of Motion in Gauge Theory 74 13 Particles in Gauge Theories . . . . . . . . . . . . 80 14 The Magnetic Charge ............... 83 15 Electromagnetic Field Strength and Magnetic Charge in Gauge Theories . . . . . . . . . 89 16 Extrema of Symmetric Functionals . . . . . . . . 94 17 Symmetric Gauge Fields . . . . . . . . . . . . . . 97 18 Estimates of the Energy of a Magnetic Monopole 104 19 Topologically Non-Trivial Strings . 109 20 Particles in the Presence of Strings 115 21 Nonlinear Fields ....... 122 22 Multivalued Action Integrals 128 23 Functional Integrals ..... 132 24 Applications of Functional Integrals to Quantum Theory 138 25 Quantization of Gauge Theories ............. 146 VIII Contents 26 Elliptic Operators .................... 158 27 The Index and Other Properties of Elliptic Operators 163 28 Determinants of Elliptic Operators 169 29 Quantum Anomalies . . . . . . . . . . 173 30 Instantons ............... 183 31 The Number of Instanton Parameters 194 32 Computation of the Instanton Contribution 199 33 Functional Integrals for a Theory Containing Fermion Fields 207 34 Instantons in Quantum Chromodynamics .......... 216 Part III Mathematical Background 221 35 Topological Spaces 223 36 Groups ................. 225 37 Gluings ................. 229 38 Equivalence Relations and Quotient Spaces 233 39 Group Representations ........... 235 40 Group Actions ................ 241 41 The Adjoint Representation of a Lie Group 245 42 Elements of Homotopy Theory ... 247 43 Applications of Topology to Physics .... 257 Bibliographical Remarks 261 References 263 Index ........... 269 Introduction Topology is the study of continuous maps. From the point of view of topology, two spaces that can be transformed into each other without tearing or gluing are equivalent. More precisely, a topological equivalence, or homeomorphism, is a continuous bijection whose inverse, too, is continuous. For example, every convex, bounded, closed subset of n-dimensional space that is not contained in an (n - 1) -dimensional subspace is homeomorphic to an n-dimensional ball. The boundary of such a set is homeomorphic to the boundary of an n-ball, that is, an (n - I)-dimensional sphere. For continuity to have a meaning, it is sufficient that there be a concept of distance between any two points in the space. Such a rule for assigning a distance to each pair of points is called a metric, and a space equipped with it is a metric space. But a space doesn't have to have a metric in order for continuity to make sense; it is enough that there be a well-defined, albeit qualitative, notion of points being close to one another. The existence of this notion, which is generally formalized in terms of neighborhoods or limits, makes the space into a topological space. Topological spaces are found everywhere in physics. For example, the con figuration space and the phase space of a system in classical mechanics are equipped with a natural topology, as is the set of equilibrium states of a system at a given temperature, in statistical physics. In quantum field theory there arise infinite-dimensional topological spaces. All this opens up possibilities for using topology in physics. Of course, the primary focus of interest to a physicist-quantitative descriptions of physical phenomena-cannot be reduced to topology. But qualitative features can be understood in terms of topology. If a physical system, and consequently the as sociated topological space, depends on a parameter, it may happen that the space's topology changes abruptly for certain values of the parameter; this change in topology is reflected in qualitative changes in the system's behav ior. For instance, critical temperatures (those where a phase transition occurs) are characterized by a change in the topology of the set of equilibrium states. Physicists are interested not only in topological spaces, but even more so in the topological properties of continuous maps between such spaces. These maps are generally fields of some form-for example, a nonzero vector field defined on a subset of n-dimensional space can be thought of as a map from this set into the set of nonzero vectors. 2 Introduction Most important are the homotopy invariants of continuous maps. A number, or some other datum, associated with a map is called a homotopy invariant of the map if it does not change under infinitesimal variations of the map. More precisely, a homotopy invariant is something that does not change under a continuous deformation, or homotopy, of the map. A continuous deformation can be thought of as the accumulation of infinitesimal variations. For example, given a nonzero, continuous vector field on the complement of a disk D in the plane, we can compute an integer called the rotation number or index of the field, which tells how many times the vector turns as one goes around a simple loop encircling D. More formally, let the vector field q,(x, y) have components (lPi(x, y), !li2(x, y)), and assume without loss of generality that D contains the origin. We can form the complex-valued function !Ii(r, cp) = !iiI (r cos cp, r sin cp) + i!li2(r cos cp, r sin cp) and write it in the form !Ii(r,cp) = A(r, cp)eia(r,<p) , where a(r,cp) is continuous: this is because the field is nowhere zero (so eia(r,<p) is well-defined and continuous) and the domain of the field avoids the origin (so a branch of the log can be chosen continuously for a). The index n is defined by 271'n = a(r, 271') - a(r, 0). It is easy to see that the index is a homotopy invariant: by definition, it changes continuously with the field, but being an integer it can only vary discretely. Therefore it cannot change at all under continuous variations of the field. In particular, the radial field q,(x, y) = (x, y) has index n = 1, because !Ii(r, cp) = rei<p; this agrees with the intuitive idea that as you go around the origin the field turns around once. The tangential field !P(x, y) = (-y, x) also has index n = 1, because !Ii(r, cp) = rei<p+1r/2. The field !P(x, y) = (x2 - y2, 2xy) has index 2, because !Ii(r, cp) = r2e2i<p. Maps that can be continuously deformed into one another are called homo topic, and a continuous family of deformations going from one to the other is a homotopy between the two. Homotopic maps are also said to belong to the same homotopy class. For fields we often talk about a topological type instead of a homotopy class. By definition, any homotopy invariant has the same value for all maps in a homotopy class. Conversely, it may happen that if a certain homotopy invariant has the same value for two maps, the maps belong to the same homotopy class: we then say that the invariant characterizes such maps up to homotopy. For example, the index is sufficient to characterize the homotopy class of nonzero vector fields defined away from the origin: two vector fields having the same index can be continuously deformed into one another, without ever vanishing. (This is somewhat harder to prove than the fact that the index is a homotopy invariant.) If a nonzero vector field defined outside a disk can 'be extended continuously to a nonzero field on the whole plane, its index is zero and the field is said to be topologica.lly trivial. Generally, n can be interpreted as the algebraic number of singular points that appear when the field is extended to the interior of the disk. (Singular points are those at which the field vanishes as well as points where the field is undefined.) Introduction 3 There are many physical interpretations for the mathematical results just discussed. For instance, the plane with a given vector field may represent the phase space of a system with one degree of freedom, in which case the field determines the dynamics of the system. The topology of the problem then gives information about the equilibrium positions (points where the vector field van ishes). Or the vector field may represent a magnetization field, and the singular points can be interpreted as defects in a ferromagnet. A complex-valued function might represents the wave function of a superconductor (the order parameter), and its singular points vortices in the semiconductor. In field theory, both classical and quantum, topological invariants can be considered as integrals of motion: if we can assign to each field with a finite energy a number that does not change under continuous variations of the field, this number is an integral of motion, because it does not vary with time as the field changes continuously. In particular, topological integrals of motion can arise in theories that admit a continuum of classical vacuums (a classical vacuum is the classical analogue of the ground state). One can capture the asymptotic behavior of the field at infinity by defining a map from a "sphere at infinity" into the space of classical vacuums, and any homotopy invariant of this map is a topological integral of motion for the system, This works whether the classical vacuums form a linear space or a manifold such as a sphere. An example of a topological integral of motion is the magnetic charge. The magnetic charge of a field in a domain V is defined as (411")-1 times the flux of the magnetic field strength H over the boundary of V. If the field is defined everywhere inside V, the relation div H = 0 implies that the magnetic charge is zero. That is the situation in electromagnetism. But in grand unification theories (theories that account for electromagnetic, weak and strong interactions), the electromagnetic field strength is not defined everywhere, and it is possible to have fields with nonzero magnetic charge. In fact, it turns out that such fields always exist, and one concludes that in grand unification theories there exist particles that carry magnetic charge (magnetic monopoles). The simplest and most important applications of topology to physics have to do with homotopy theory. But another branch of topology, homology theory, also plays an important role. Homology theory can be applied either directly (for example, in analyzing multiple integrals arising from Feynman diagrams) or as a technical means for building homotopy invariants. Homology theory is closely linked with the multidimensional generalizations of Green's formula, of Gauss's divergence theorem and of Stokes' theorem. Such generalizations are generally formulated most conveniently in the language of exterior forms, that is, sums of antisymmetrized products of differentials. The basic concepts of homology theory are cycles, which can be seen as closed objects (that is, curves, surfaces, etc., having no boundary), and bound aries. Boundaries are cycles that are homologous to zero, or homologically triv ial. For example, if r is a closed curve in three-space, the complement R3 \ r contains one-dimensional cycles that are not homologous to zero---that is, that r. cannot bound a surface that avoids This intuitive statement can be proved