Higgs fields and harmonic maps (Q2779215)

For a compact connected simple group \(G\) and its Lie algebra \(\mathfrak g\), a connection \(A\) over the \(G\)-bundle \(G\times {\mathbb R}^3\) together with a Higgs field \(\Phi \in C^{\infty}({\mathbb R}^3 , \mathfrak g)\) is called a \textit{Yang-Mills-Higgs field} if it is a critical point of the functional: NEWLINE\[NEWLINE {\mathcal A}(A, \Phi) = \int_{{\mathbb R}^3} \left( |F_{A}|^2 + |D_{A}\Phi|^2 \right) dx ,NEWLINE\]NEWLINE where \(F_{A}\) is the curvature of \(A\). This functional is invariant under the gauge action \( A \mapsto A^h = \text{Ad}(h^{-1}) A + h^{-1} dh\) and \(\Phi \mapsto \Phi^{h} = \text{Ad}(h^{-1}) \Phi\), with \(h\in C({\mathbb R}^3, G)\). NEWLINENEWLINENEWLINEThe existence, for any connection \(A\), of a radially constant gauge transformation, allows one to define a Higgs field at infinity \(\Phi_{\infty} : {\mathbb S}^{2}(1) \to \left\{ X\in \mathfrak g , |X|= m \right\}\) (\(|X|= m\) is due to the decay conditions described below). Hence \((A,\Phi)\) defines an element of \(\pi_{2}(G/K)\). NEWLINENEWLINENEWLINEThis article investigates the relationship between Yang-Mills-Higgs fields and the harmonicity or holomorphicity of its associated map \(\Phi_{\infty} : {\mathbb S}^{2}(1) \to G/K\). NEWLINENEWLINENEWLINEThis study is carried out under the two assumptions: Decay conditions: \(|\Phi|= m + O(1/r)\) and \( |F_{A}|, |D_{A}\Phi|= O(1/r^{2})\) (\(r=|x|\)). Symmetry breaking ansatz: The image of \(\Phi_{\infty}\) is contained in an orbit of the \(G\)-adjoint action in \(\mathfrak g\). NEWLINENEWLINENEWLINEFirst, in the case of a compact simple Lie group, it is shown that, under some additional decaying conditions for certain bracket terms, if the maps \(\Phi_{t}(x) = \Phi(tx)\) converge towards \(\Phi_{\infty}\) in the \(C^2\)-topology, then \(\Phi_{\infty}\) is harmonic. If the group \(G\) is \(\text{SU}(2)\), then the symmetry breaking condition is unnecessary. NEWLINENEWLINENEWLINEFurthermore, if the connections \(A_{t}= \lambda_{t}^{*} A\) (where \(\lambda_{t}(x) = tx\)) converge in the \(C^1\)-topology towards \(A_{\infty}\) and \(A_{\infty}\), \(\Phi_{\infty}\) and \(\Phi\) satisfy a bracket condition, then \(\Phi_{\infty}\) is holomorphic. NEWLINENEWLINENEWLINETo use this method to find harmonic representatives in a given homotopy class of \(\pi_{2}(G/K)\), two questions remain open. First, the existence of Yang-Mills-Higgs fields with the required symmetry breaking property and, second, the decaying conditions on the bracket terms.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].