Trivial Lagrangians on connections and invariance under automorphisms

Abstract

Let π:P→M be a principal G-bundle and p:C→M the bundle of connections on π. In the present paper the authors study variational problems defined by Lagrangians L:J1C→R. The starting point is the classical theorem of Utiyama which characterizes the gauge-invariant Lagrangians. In Theorem 1 the authors characterize the Lagrangian densities which are invariant under the full algebra of infinitesimal automorphisms (in particular, if dimM is odd, the only invariant density is the zero density). After that, they can conclude that all of them are variationally trivial. This property allows them to enunciate the second problem in the paper: Is every gauge-invariant variationally trivial Lagrangian density invariant under the full algebra of automorphisms? The answer is negative and in Theorem 2 they obtain a characterization of them by the de Rham cohomology of M and the characteristic classes of P. Explicit examples, when G=U(1), U(2) and SU(2), are shown.