Modern Differential Geometry in Gauge Theories

Abstract

Yang–Mills Theory “Today there is an amazing confluence of the gauge theories in physics (for the Yang–Mills equations) and the geometrical theory of connections on fiber bundles.” S. Mac Lane in Mathematics: Form and Function (Springer-Verlag, New York, 1986). p. 259. “Gauge theories [have] a direct differential-geometric interpretation in terms of fiber bundles with connection.” M. F. Atiyah in Geometry of Yang–Mills Fields (Accademia Nazionale dei Lincei, SNS, Pisa, 1979). p. 42. The objects of our study in this chapter belong to what we may call the Yang– Mills category (see Section 4.2 for concrete definitions), while the corresponding morphisms are suitable connection-preserving sheaf morphisms (ibid. (4.16)). Now, since the necessary background material for the subject matter at issue has not been systematically developed so far, within the framework of abstract differential geometry (see A. Mallios [VS: Vols. I and II]), which is employed by the present treatise, we give below a detailed exposition of all the relevant issues that will be needed in the sequel. In this context, see also, however, A. Mallios [6: p. 164, Appendix II] for a brief account on the same material. Among the various standard presentations of this subject, see, for instance, T. Petrie–J. Randall [1]. So we start with the ensuing fundamental notions for all the subsequent discussion. 1 The Differential Setting As was the case in Volume I of the present treatise, here too, the adjective “differential” has, of course, only a formal meaning, referring to a particular type of (“differential”) operator connected with the subject matter under consideration, given that no “smooth structure” at all (!) is assumed on our base space X . Thus, the ensuing discussion of this section aims, in effect, to collect together all the “differential operators” that have been employed in the preceding (see [VS: Vols. I and II], yet, Volume I of this treatise), to the extent that they provide the corresponding to our case (generalized, alias abstract ) de Rham complex, along with the respective “connection operators” (see below). All of this will be necessary for our subsequent consideradifferential setup.” tions in Section 2, where we shall define the corresponding “dual (alias, adjoint ) A. Mallios, Modern Differential Geometry in Gauge Theories: Yang–Mills Fields, Volume II, DOI: 10.1007/978-0-8176-4634-9_1, 3 © Birkhauser Boston, a part of Springer Science + Business Media, LLC 2010 4 1 Abstract Yang–Mills Theory So, as usual, we start with a C-algebraized space