Structure of Gauge-Invariant Lagrangians

The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below, we tackle one of these problems: the existence of a finite system of generators of gauge-invariant Lagrangians and how to compute them. More precisely, if \(p:C\rightarrow M\) is the bundle of connections on a principal G-bundle \(\pi :P\rightarrow M\), and G is a semisimple connected Lie group, then a finite number \(L_1,\dotsc ,L_{N^\prime }\) of gauge-invariant Lagrangians defined on \(J^1C\) is proved to exist such that for any other gauge-invariant Lagrangian \(L\in C^\infty (J^1C)\), there exists a function \(F\in C^\infty ({\mathbb {R}}^{N^\prime })\), such that \(L=F(L_1,\dotsc ,L_{N^\prime })\). Several examples are dealt with explicitly.