Gauge-Invariant Characterization of Yang–Mills–Higgs Equations

Abstract

Let C → M be the bundle of connections of a principal G-bundle P → M over a pseudo-Riemannian manifold (M,g) of signature (n+, n−) and let E → M be the associated bundle with P under a linear representation of G on a finite-dimensional vector space. For an arbitrary Lie group G, the O(n+,n-) × G-invariant quadratic Lagrangians on J1(C ×M E) are characterized. In particular, for a simple Lie group the Yang–Mills and Yang–Mills–Higgs Lagrangians are characterized, up to an scalar factor, to be the only O(n+, n−) × G-invariant quadratic Lagrangians. These results are also analyzed on several examples of interest in gauge theory.