Abstract
Self-duality is a very important concept in the study and applications of topological solitons in many areas of Physics. The rich mathematical structures underlying it lead, in many cases, to the development of exact and non-perturbative methods. We present a generalization of the Yang-Mills-Higgs system by the introduction of scalar fields assembled in a symmetric and invertible matrix h of the same dimension as the gauge group. The coupling of such new fields to the gauge and Higgs fields is made by replacing the Killing form, in the contraction of the group indices, by the matrix h in the kinetic term for the gauge fields, and by its inverse in the Higgs field kinetic term. The theory is conformally invariant in the three dimensional space R^3. An important aspect of the model is that for practically all configurations of the gauge and Higgs fields the new scalar fields adjust themselves to solve the modified self-duality equations. We construct solutions using a spherically symmetric ans\"atz and show that the 't Hooft-Polyakov monopole becomes a self-dual solution of such modified Yang-Mills-Higgs system. We use an ans\"atz based on the conformal symmetry to construct vacuum solutions presenting non-trivial toroidal magnetic fields.
Highlights
Topological solitons play a fundamental role in the study of nonlinear phenomena in many areas of science
There is a class of topological solitons that deserves a special attention as they reveal deeper mathematical structures in the theory, which may lead to the development of some exact and nonperturbative methods
They make the static sector of the theory conformally invariant in the three-dimensional space R3, and that plays an important role in many aspects of the theory, especially in the construction of solutions
Summary
Topological solitons play a fundamental role in the study of nonlinear phenomena in many areas of science. That may happen even in the cases where there is no lower bound on the energy or Euclidean action By exploring such ideas it was possible to develop the concept of generalized self-dualities where one can construct, from one single topological charge, a large class of field theories possessing self-dual sectors [4]. The introduction of the scalar fields hab brings in some novel features They make the static sector of the theory conformally invariant in the three-dimensional space R3, and that plays an important role in many aspects of the theory, especially in the construction of solutions. That is guaranteed in most of the cases, but as we will show, it is possible to use the conformal symmetry to build an Ansatz to construct vacuum solutions with vanishing energy and topological charge, and presenting nontrivial magnetic fields in toroidal configurations. VI, we present our conclusions, and in the Appendix we show that the modified YangMills-Higgs system is conformally invariant in the threedimensional space R3
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