On the geometrized Skyrme and Faddeev models

Abstract

Abstract. The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to σ 2 -energy, introduced by Eells and Sampson in [13]. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined. IntroductionCommon tools in field theory, non-linear σ-models are known in differential geometry mainly through the problem of harmonic maps between Riemannian manifolds. Namely a (smooth) mapping ϕ : (M, g) → (N, h) is harmonic if it is critical point for the Dirichlet energy functional [13],a generalization of the kinetic energy of classical mechanics. Less discussed from differential geometric point of view are Skyrme and Faddeev-Hopf models, which are σ-models with additional fourth-power derivative terms (for an overview including recent progress concerning both models, see [25]).The first one was proposed in the sixties by Tony Skyrme [37], to model baryons as topological solitons (see [31]) of pion fields. Meanwhile it has been shown [47] to be a low energy effective theory of quantum chromodynamics that becomes exact as the number of quark colours becomes large. Thus baryons are represented by energy minimising, topologically nontrivial maps ϕ :with the boundary condition ϕ({|x| → ∞}) = I 2 , called skyrmions. Their topological degree is identified with the baryon number. The static (conveniently renormalized) Skyrme energy functional isThis energy has a topological lower bound [14]: E Skyrme (ϕ) ≥ 6π 2 |degϕ|. In the second one, stated in 1975 by Ludvig Faddeev and Antti J. Niemi [15], the configuration fields are unitary vector fields ϕ : R 3 → S 2 ⊂ R 3 with the boundary condition ϕ({|x| → ∞}) = (0, 0, 1). The static energy in this case is given bywhere c 2 , c 4 are coupling constants. Again the field configurations are indexed by an integer, their Hopf invariant : Q(ϕ) ∈ π 3 (S 2 ) ∼ = Z and the energy has a topological lower bound: cf. [45]. Although this model can be viewed as a constrained variant of the Skyrme model, it exhibits important specific properties, e.g. it allows knotted solitons. Moreover, in [16] it has been proposed that it arises as a dual description of strongly coupled SU (2) Yang-Mills theory, with the solitonic strings (possibly) representing glueballs. See also [17] for an alternative approach to these issues.Both models rise the same kind of topologically constrained minimization problem: find out static energy minimizers in each topological class (i.e. of prescribed baryon number or Hopf invariant). We can give an unitary treatment for both if we take into account that they are particular cases of the following energy-type functional:

On a Stability Property of Skyrme-Related Energy Functionals

“…For the identity map of S 3 (of unit radius) this reads κ ≥ 1 2 and one knows that it is also a sufficient condition [11,12]; see also [22].…”

$${\sigma_2}$$ σ 2 -Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

“…The second motivation is Skyrme model which is a new model describing strong interactions of quantum fields and σ 2 -energy appears as a part of this physical model. Skyrme's idea was to add to the standard Dirichlet energy an additional stabilizing term which is the σ 2 -energy and it would prevent stationary fields from being singular as it happens for harmonic maps (see, e.g., [4,18,19]).…”

A note on higher-charge configurations for the Faddeev-Hopf model

“…We shall refer to E σ2 as strongly coupled energy. The critical points for the full energy, or σ 1,2 -critical maps, are characterized by the equations: [4,19,24], with C ϕ = dϕ t • dϕ ∈ End(T M ) denoting the Cauchy-Green tensor of ϕ. Obvious solutions for (1.2) are those maps that are both harmonic and σ 2 -critical.…”

“…This is the case for the standard Hopf map (S 3 , can) → (S 2 , can), the only exact solution between (round) spheres known until now. But this situation seems very rare and a heuristic reason for this, given in [19], is that while the prototype for harmonic maps (from a Riemann surface to C) is provided by a holomorphic/conformal map, the prototype of σ 2 -critical maps is an area-preserving map. In this 2-dimensional context, a map encompasses both conditions if and only if it is homothetic.…”

“…In this 2-dimensional context, a map encompasses both conditions if and only if it is homothetic. For a further analysis of transversally (to the Reeb foliation) area-preserving maps between 3-dimensional contact manifolds and various stability results, see [19].…”