Differential geometry, gauge theories, and gravity.

*(English)*Zbl 0649.53001
Cambridge Monographs on Mathematical Physics. Cambridge etc.: Cambridge University Press. xii+230 p.; £30.00; $ 49.50 (1987).

According to the preface this book is addressed to graduate students of theoretical physics. As regards mathematical prerequisites only a knowledge of linear algebra and real analysis is presupposed, while the physical applications require some acquaintance with the elements of Yang-Mills theories and general relativity.

Chapter 1 is devoted to a somewhat terse description of exterior algebra on an n-dimensional vector space V. Differential forms are introduced in Chapter 2 in terms of tangent spaces of open subsets of \({\mathbb{R}}^ n\); this includes a discussion of the Poincaré lemma, integration of n- forms, and Stokes’ theorem. The notion of a metric on V is introduced together with the induced metric on the dual space \(V^*\), which gives rise to a description of the Hodge (star) isomorphism and isometries. Chapter 4 begins with a formulation of Maxwell’s equations in terms of the exterior calculus, which is followed by a discussion of gauge theories for scalar matter fields. [Here some knowledge of Lie groups, Lie algebras and their representations is presupposed; these sections are entitled “connection \(=\) potential”, and “curvature \(=\) field strength”, although none of these differential-geometric concepts have been defined at this stage.] In Chapter 5, called “Einstein-Cartan” theory, a metric with Minkowski signature is introduced on open subsets of \({\mathbb{R}}^ 4\) in order to describe gravity. Connections, curvature, and torsion are defined within this particular context, special reference being made to the Hilbert-Einstein action principle and Einstein gauge. Lie derivatives and Lie brackets are introduced in Chapter 6, in which also a brief indication is given of graded Lie algebras.

Differentiable manifolds are introduced in Chapter 7; this gives rise to a discussion of vector fields and forms on manifolds, the partition of unity, and integration of forms. Chapter 8 is devoted to a systematic introduction to the theory of Lie groups, Lie algebras, and their representations. The concept of a fibre bundle is presented in Chapter 9, and is illustrated by three useful examples. This serves to clarify much of the subsequent analysis of connections on principal bundles, holonomy groups, and gauge transformations. In Chapter 10 special solutions of the field equations of gauge theories are analyzed with special reference to the Dirac monopole, the t’Hooft-Polyakov monopole and the Yang-Mills instanton. Clifford algebras and groups form the basis of a discussion in Chapter 11 of the Dirac operator, spin structures, and Kähler fermions. The concluding Chapters 12 and 13 are concerned with anomalies that occur when the symmetries of a classical theory are broken by quantum corrections. Such anomalies are treated in an algebraic manner (as opposed to a quantum field theoretical approach), it being noted that there is a purely algebraic algorithm that classifies infinitesimal anomalies.

It is evident that this volume contains a wealth of significant material, much of which is at the forefront of current research; the book is therefore a most welcome addition to the literature. However, this breadth is achieved, at least to some extent, at the expense of mathematical detail. Many important theorems are stated without proof, and the serious reader must therefore be prepared to consult additional sources.

Chapter 1 is devoted to a somewhat terse description of exterior algebra on an n-dimensional vector space V. Differential forms are introduced in Chapter 2 in terms of tangent spaces of open subsets of \({\mathbb{R}}^ n\); this includes a discussion of the Poincaré lemma, integration of n- forms, and Stokes’ theorem. The notion of a metric on V is introduced together with the induced metric on the dual space \(V^*\), which gives rise to a description of the Hodge (star) isomorphism and isometries. Chapter 4 begins with a formulation of Maxwell’s equations in terms of the exterior calculus, which is followed by a discussion of gauge theories for scalar matter fields. [Here some knowledge of Lie groups, Lie algebras and their representations is presupposed; these sections are entitled “connection \(=\) potential”, and “curvature \(=\) field strength”, although none of these differential-geometric concepts have been defined at this stage.] In Chapter 5, called “Einstein-Cartan” theory, a metric with Minkowski signature is introduced on open subsets of \({\mathbb{R}}^ 4\) in order to describe gravity. Connections, curvature, and torsion are defined within this particular context, special reference being made to the Hilbert-Einstein action principle and Einstein gauge. Lie derivatives and Lie brackets are introduced in Chapter 6, in which also a brief indication is given of graded Lie algebras.

Differentiable manifolds are introduced in Chapter 7; this gives rise to a discussion of vector fields and forms on manifolds, the partition of unity, and integration of forms. Chapter 8 is devoted to a systematic introduction to the theory of Lie groups, Lie algebras, and their representations. The concept of a fibre bundle is presented in Chapter 9, and is illustrated by three useful examples. This serves to clarify much of the subsequent analysis of connections on principal bundles, holonomy groups, and gauge transformations. In Chapter 10 special solutions of the field equations of gauge theories are analyzed with special reference to the Dirac monopole, the t’Hooft-Polyakov monopole and the Yang-Mills instanton. Clifford algebras and groups form the basis of a discussion in Chapter 11 of the Dirac operator, spin structures, and Kähler fermions. The concluding Chapters 12 and 13 are concerned with anomalies that occur when the symmetries of a classical theory are broken by quantum corrections. Such anomalies are treated in an algebraic manner (as opposed to a quantum field theoretical approach), it being noted that there is a purely algebraic algorithm that classifies infinitesimal anomalies.

It is evident that this volume contains a wealth of significant material, much of which is at the forefront of current research; the book is therefore a most welcome addition to the literature. However, this breadth is achieved, at least to some extent, at the expense of mathematical detail. Many important theorems are stated without proof, and the serious reader must therefore be prepared to consult additional sources.

Reviewer: H.Rund

##### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

58A05 | Differentiable manifolds, foundations |

58A10 | Differential forms in global analysis |

81T08 | Constructive quantum field theory |

83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |

53C80 | Applications of global differential geometry to the sciences |