Three-dimensional Yang–Mills–Higgs equations in gauge-invariant variables

A formulation of three-dimensional $SU(2)$- and $SO(1, 2)$ Yang–Mills theories in the gauge-invariant variables $G_{ij} =\hbox {tr}^*F_i^*F_j$ is proposed. It is shown that the tensor $G_{ij}$ satisfies three-dimensional Einstein equations with simple right-hand side and $G_{ij}$ playing the role of a metric. This result is generalized to the case of a system of Yang–Mills–Higgs fields.