Fractional Hopfions in the Faddeev-Skyrme model with a symmetry breaking potential

We construct new solutions of the Faddeev-Skyrme model with a symmetry breaking potential admitting $S^1$ vacuum. It includes, as a limiting case, the usual $SO(3)$ symmetry breaking mass term, another limit corresponds to the potential $m^2 \phi_1^2$, which gives a mass to the corresponding component of the scalar field. However we find that the spacial distribution of the energy density of these solutions has more complicated structure, than in the case of the usual Hopfions, typically it represents two separate linked tubes with different thicknesses and positions. In order to classify these configurations we define a counterpart of the usual position curve, which represents a collection of loops $\mathcal{C}_1, \mathcal{C}_{-1}$ corresponding to the preimages of the points $\vec \phi = (\pm 1 \mp \mu, 0,0)$, respectively. Then the Hopf invariant can be defined as $Q= {\rm link} (\mathcal{C}_1,\mathcal{C}_{-1})$. In this model, in the sectors of degrees $Q=5,6,7$ we found solutions of new type, for which one or both of these tubes represent trefoil knots. Further, some of these solutions possess different types of curves $\mathcal{C}_1$ and $\mathcal{C}_{-1}$.Comment: 22 pages, 129 figure

Exact Self-Dual Skyrmions

We introduce a Skyrme type model with the target space being the 3-sphere S^3 and with an action possessing, as usual, quadratic and quartic terms in field derivatives. The novel character of the model is that the strength of the couplings of those two terms are allowed to depend upon the space-time coordinates. The model should therefore be interpreted as an effective theory, such that those couplings correspond in fact to low energy expectation values of fields belonging to a more fundamental theory at high energies. The theory possesses a self-dual sector that saturates the Bogomolny bound leading to an energy depending linearly on the topological charge. The self-duality equations are conformally invariant in three space dimensions leading to a toroidal ansatz and exact self-dual Skyrmion solutions. Those solutions are labelled by two integers and, despite their toroidal character, the energy density is spherically symmetric when those integers are equal and oblate or prolate otherwise.Comment: 14 pages, 3 figures, a reference adde