The geometry of gauge fields
In this rather extensive article the authors endeavour to present a complete and systematic description of the mathematical background of gauge field theories. Some basic knowledge of the theory of differentiable manifolds is assumed, and accordingly the discussion of the exterior differential calculus and tensor bundles is quite terse. This is allowed by a brief description in Chapters 2 and 3 of principal and associated bundles, and the theory of connections. Chapter 4 is devoted to some aspects of algebraic topology, differential complexes, and cohomology theory. An outline of the basic concepts of gauge field theories is presented in Chapter 5, with special reference to the space of gauge connections and a variational approach to coupled field equations. In Chapter 6 special Yang-Mills fields and instantons are introduced, and a description is given of the moduli space of the complete (8k-3)-parameter family of instanton solutions over \(S^ 4\) with gauge group SU(2) and instanton number k. The final Chapter 7 deals with coupled Yang-Mills-Higgs fields, the notion of dimensional reduction, and monopole solutions. This brief summary does not reflect fully the unusually wide scope of the article under review. Clearly the omission in the latter of proofs and elaboration of many basic results is inevitable. However, a very detailed list of references serves to provide careful documentation.
scientific article; zbMATH DE number 4113439
scientific article