  For a given closed two-form, we introduce the cone Yang-Mills functional
which is a Yang-Mills-type functional for a pair $(A,B)$, a connection one-form
$A$ and a scalar $B$ taking value in the adjoint representation of a Lie group.
The functional arises naturally from dimensionally reducing the Yang-Mills
functional over the fiber of a circle bundle with the two-form being the Euler
class. We write down the Euler-Lagrange equations of the functional and present
some of the properties of its critical solutions, especially in comparison with
Yang-Mills solutions. We show that a special class of three-dimensional
solutions satisfy a duality condition which generalizes the Bogomolny monopole
equations. Moreover, we analyze the zero solutions of the cone Yang-Mills
functional and give an algebraic classification characterizing principal
bundles that carry such cone-flat solutions when the two-form is
non-degenerate.
