Closed-vortex-type solitons with Hopf index

Hopf solitons onS3and3

“…In other words, the integral of the Chern-Simons form is bounded above by an expression involving integrals of F 2 and √ F 2 . A value for C for which this inequality holds, for M = R 3 , is [12,8] …”

“…The task is hampered by the fact that the obvious ansätze are incompatible with the equations of motion [7]. Related to this is the fact that topologically nontrivial configurations can admit at most an axial (one-parameter) symmetry [8]. So the situation is unlike that of the Skyrme model, with target space S 3 , where the Q = 1 soliton is a spherically-symmetric "hedgehog".…”

Class of exact solutions of the Faddeev model

“…Since n appears in the description of Yang-Mills field [15][16][17], the vortex structure of n observed here might suggest the same structure of the Yang-Mills field. We should note that the existence of the vortex solution of the Faddeev model was also discussed by Kundu and Rybakov [18]. We finally note that, although no exact analytic expression was presented, the vortex solution of the Abelian Higgs model was found by Nielsen and Olesen [19] thirty years ago.…”

Hopf solitons from instanton holonomy

“…So one expects, with this normalization, that E satisfies the bound E ≥ N 3/4 ; but only the weaker bound E ≥ cN 3/4 with c = 2 −3/2 3 3/8 has been proved [13,14].…”

“…No configuration with N = 0 can be spherically-symmetric [14], but axial symmetry is allowable. The minimum-energy solutions for N = 1 and N = 2 are indeed axially-symmetric [16].…”