Axially-Symmetric Topological Configurations in the Skyrme and Faddeev Chiral Models

Abstract

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 \( U \in SU(2) \), and in the Faddeev model (FM) — by the unit 3-vector \( \mathbf{n} \in S^2 \). 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 \( Q = \deg(S^3 \to S^3) \) and is interpreted as the baryon charge \( B \). For FM it coincides with the Hopf invariant \( Q_H \) of the map \( S^3 \to S^2 \) and is interpreted as the lepton charge. The energy \( E \) in SM and FM is estimated from below by some powers of charges: \( E_S > \mathrm{const}|Q| \), \( E_F > \mathrm{const}|Q_H|^{3/4} \).

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): \( U = \exp[i\theta(r) \sigma \cdot \mathbf{r}], \; \sigma = (\sigma_i)/r, \; r = |\mathbf{r}| \), where \( \sigma \) 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 \( |Q_H| \geq 1 \), the corresponding minimizing sequences being constructed, with the property of weak convergence in \( W_\infty^1 \).