Towards a unified treatment of Yang-Mills and Higgs fields

Starting from a noncommutative algebra $\mathcal{A}$ of the form $\mathcal{C}\otimes \mathcal{M}$, where $\mathcal{C}$ is the algebra of smooth functions on space-time and $\mathcal{M}$ is the algebra of $n\times n$ Hermitian matrices, we construct an exterior algebra of differential forms over $\mathcal{A}$. We use the one-forms of this algebra to describe Yang-Mills and Higgs fields on a similar footing and construct a Lagrangian from its two-forms. We show how, in the resulting geometrical description, a Higgs potential that leads to spontaneous symmetry breaking arises naturally. We discuss the application of this formalism to the bosonic sectors of the standard electroweak theory and a grand-unified model based on SU(5)⊗U(1).