Asymptotically flat solutions of the Bogomol'nyi equations with solvable gauge group (Q1382672)

In the Euclidean space \(\mathbb{R}^3\) the author considers a pair \((\widehat\varphi, \widehat A_a)\) consisting of a scalar function \(\widehat\varphi\) (called the Higgs field) and a gauge field \(\widehat A_a\) corresponding to the gauge group \(G\). The symbol \(\widehat{\;}\)\ stands for a matrix representation of the Lie algebra of the Lie group \(G\). The author studies a special type of solutions of the Bogomol'nyi equations \(D_a \widehat\varphi = \widehat B_a\), where \(D_a \widehat\varphi\) is the gauge derivative of the Higgs field, \(\widehat F_{ab}\) is the curvature tensor of the gauge field \(\widehat A_a\), and \(B_a={1\over 2} \varepsilon_{abc}\widehat F_{ab}\). He considers the Bogomol'nyi equations with the Lie algebra of the harmonic oscillator \(ho({\mathbb{N}},\mathbb{R})\), which is quasinilpotent and admits an invariant nondegenerate bilinear form. The author proves the following Theorem. Every asymptotically flat monopole solution of the Bogomol'nyi equations with gauge group \(HO({\mathbb{N}}, \mathbb{R})\) is trivial, i.e., \(\widehat\varphi\equiv const\) and \(\widehat A\equiv 0\) for a certain gauge.