Axially-symmetric topological configurations in the skyrme and faddeev chiral models

By definition, in chiral model the field takes values in some homogeneous space G/H. For example, in the Skyrme model (SM) the field is given by the unitary matrix U ∈ SU(2), and in the Faddeev model (FM) - by the unit 3-vector n ∈ S2. Physically interesting configurations in chiral models are endowed with nontrivial topological invariants (charges) Q taking integer values and serving as generators of corresponding homotopic groups. For SM Q = deg(S3 → S3) and is interpreted as the baryon charge B. For FM it coincides with the Hopf invariant QH of the map S3 → S2 and is interpreted as the lepton charge. The energy E in SM and FM is estimated from below by some powers of charges: ES > const|Q|, EF > const|QH|3/4. We consider static axially-symmetric topological configurations in these models realizing the minimal values of energy in some homotopic classes. As is well-known, for Q = 1 in SM the absolute minimum of energy is attained by the so-called hedgehog ansatz (Skyrmion): U = exp[iΘ(r) σ], σ = (σr)/r, r = |r|, where σ stands for Pauli matrices. We prove via the variational method the existence of axially-symmetric configurations (torons) in SM with |Q| > 1 and in FM with |QH| ≥ 1, the corresponding minimizing sequences being constructed, with the property of * weak convergence in W∞ 1.