Elliptic operators, connections and gauge transformations

Abstract

A study is made of the action of various Banach Lie groups of principal bundle automorphims (gauge transformations) on corresponding spaces of connections on some principal bundle, using standard theorems of global analysis together with elliptic regularity theorems. A proof of elliptic regularity theorems in Sobolev and Holder norms for linear elliptic partial differential operators with smooth coefficients acting on sections of smooth vector bundles is presented. This proofassumes acquaintance with the theory of tempered distributions and their Fourier transforms and with the theory of compact and Fredholm operators, and also uses results from the papers of Calderon and Zygmund and from the early papers of Hormander on pseuoo-differential operators, but is otherwise intended to be self-contained. Elliptic regularity theorems arc proved for elliptic orcrators with non-smooth coefficients, using only the regularity theorems for elliptic operators with smooth coefficients, together with the Sobolev embedding theorems, the Rellich-Kondrakov theorem and the Sobolev multiplication theorems.For later convenience these elliptic regularity results are presented as a generalization of the analytical aspects of Hodge theory. Various theorems concerning the action of automorphisms on connections are proved, culminating in the slice theorems obtained in chapter VIII. Regularity theorems for Yang-Mills connections and for Yang-Mills Higgs systems arc obtained, In chapter IX analytical properties of the covariant derivative operators associated with a connection arc related to the holonomy group of the connection via a theorem which shows the existence of an upper bound on the length of loop required Lo generate the holonomy group of a connection with compact holonomygroup.