On the geometrized Skyrme and Faddeev models
Abstract.
The higherpower derivative terms involved in both Faddeev and Skyrme energy functionals correspond to –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.
Key words and phrases:
Variation, critical point, stability, harmonic map2000 Mathematics Subject Classification:
53B35, 53B50, 58E20, 58E30, 81T201. Introduction
Common tools in field theory, nonlinear models are known in differential geometry mainly through the problem of harmonic maps between Riemannian manifolds. Namely a (smooth) mapping 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 FaddeevHopf models, which are –models with additional fourthpower 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 with the boundary condition , called skyrmions. Their topological degree is identified with the baryon number. The static (conveniently renormalized) Skyrme energy functional is
(1.1) 
This energy has a topological lower bound [14]: .
In the second one, stated in 1975 by Ludvig Faddeev and Antti J. Niemi [15], the configuration fields are unitary vector fields with the boundary condition . The static energy in this case is given by
(1.2) 
where are coupling constants.
Again the field configurations are indexed by an integer, their Hopf invariant: 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 YangMills 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 energytype functional:
(1.3) 
where , are (smooth) Riemannian manifolds, is a coupling constant and is the second elementary symmetric function of the eigenvalues of with respect to .
Even if the variational problem for the energy has already been treated in [8, 11, 48], very little is known about its solutions. From our point of view, the particularities of case are worth to be outlined for their differential geometric interest in its own and hopefully for providing hints for further investigations on the original physical models.
The present generalization of (1.1) and (1.2) was proposed in [26, 30]. Other generalizations of Skyrme and Faddeev energies are discussed in [4, 5, 18, 24, 39, 50].
The paper is organized as follows. The next section reviews the higher power energies in terms of eigenvalues of the CauchyGreen tensor and some classes of mappings characterized by their distortion. In section 3 the EulerLagrange equations for –energy are derived and general solutions inside these classes are pointed out. The effect of (bi)conformal changes of domain metric is also stated. Section 4 presents the second variation formula and analyses the stability of homothetic and holomorphic solutions. Finally we apply the results of previous sections to the stability of homotheties for the full energy (1.3) and to some old and new ansatze for stationary field configurations. We end with an example of absolute minima for the strongly coupled Faddeev model on and a discussion on possible contactomorphic solutions.
2. Higher power energies and the CauchyGreen tensor
2.1. The CauchyGreen tensor and the geometrical distortion induced by a map
Let a smooth mapping between Riemannian manifolds of dimensions and . The so called first fundamental form of is the symmetric, positive semidefinite 2covariant tensor field on , defined as , cf. [11]. Alternatively, using the musical isomorphism, we can see it as the endomorphism , where denote the adjoint of . When , this corresponds to the (right) CauchyGreen (strain) tensor of a deformation in nonlinear elasticity (we shall maintain this name for in the general case).
The CauchyGreen tensor is always diagonalizable; let , , …, and be its (real, nonnegative) eigenvalues, where everywhere. Recall that are also called principal distortion coefficients of .
The elementary symmetric functions in the eigenvalues of represent a measure of the geometrical distortion induced by the map. They are called principal invariants of and will be denoted by:
or, alternatively:
where is the energy density of and is the volume density of , cf. [13].
Remark 2.1.
At any point of , there is an orthonormal basis of corresponding eigenvectors for at that point. Moreover, according to [35, Lemma 2.3], we have a local orthonormal frame of eigenvector fields, around any point of a dense open subset of . In particular, for such local ”eigenfields” we have: , so are orthogonal with norm .
2.2. Higher power energies
According to [13], up to a half factor, we shall call –energy, the following functional
(2.1) 
Therefore, the generalized energy (1.3) reads
(2.2) 
Let us recall another type of (higher power) energytype functional that will be useful for our further discussion. The energy of a (smooth) map is defined as:
The corresponding EulerLagrange operator/equations are, cf. [42]
where is the tension field of (i.e. the EulerLagrange operator associated to the Dirichlet energy). The solutions of these equations are called pharmonic maps.
In particular, for , we have
(2.3) 
or, equivalently, .
Remark 2.2.
It is easy to see that energy is clearer if we point out that, using Newton’s inequalities, . The relation with the
with equality if and only if . If in addition is of bounded dilation, i.e. , we have also the reversed inequality
2.3. Classes of mappings characterized by their distortion
Let us recall some classes of mappings that will have a particular behaviour with respect to the above mentioned energies. For the contact or symplectic geometry background and corresponding notations we refer the reader to [9].

(Equal eigenvalues.) When and , if , we say that our map is horizontally weakly conformal (HWC) or semiconformal of dilation , cf. [7, p. 46] (when the map is submersive, we shall omit the word ”weakly”). If moreover (at regular points), then the map is called horizontally homothetic (HH); in the particular case when , we call it simply homothetic.
When and , if , we say that our map is (weakly) conformal, cf. [7, p. 40]. If this notion is equivalent to the above one.

(Pairwise equal eigenvalues.) When is endowed with an almost Hermitian structure (so is even), a class of mappings that includes the above ones was defined by , cf. [27]. These maps are called pseudo horizontally weakly conformal maps (PHWC). In this case, cf. [28], the eigenvalues of have multiplicity 2, i.e. , , … , is even, the eigenspaces are invariant with respect to the induced metric almost –structure, , on the domain and is holomorphic, i.e. . When is an almost complex structure we recover the classical case of holomorphic maps.
We have also a corresponding notion of pseudo horizontally homothetic (PHH) map [1, 2]. For submersions, PHH condition reads as , . Standard examples of PHH maps are the holomorphic maps between Kähler manifolds and the  holomorphic maps from a Sasakian manifold (with associated structure ) to a Kähler one.
For a suitable notion of PHWC/holomorphic map between odddimensional manifolds see [38] and references therein. We mention here only the fact that, according to [22], a holomorphic map between two contact metric manifolds, and , has eigenvalues , … ( and is odd). This kind of map is called contact homothety or homothetic transformation. 
(Equal products of paired eigenvalues) When both the domain and codomain of are symplectic manifolds, and , and we say that is a symplectomorphism. According to [32], CauchyGreen’s eigenvalues for a symplectomorphism satisfy … ( is even) and the associated complex structure restricts to an isomorphism between the eigenspaces corresponding to and , for all odd . It is an easy task to rephrase this fact for ( being a function on ) and to see that the common value of products of eigenvalues must be equal to . Notice that the conformal case (1) is included in this case.
Moreover one can extend this result for contactomorphisms, that is for mappings between contact metric manifolds, and , satisfying . In this case we obtainwhere is odd and the eigenvector corresponding to must be , the Reeb vector field on the domain. Notice that the conformal case (1) is no more included in this case, but only contact homotheties.
3. EulerLagrange equations for –energy
As in the previous section, will denote a smooth mapping between Riemannian manifolds. For the rest of the paper we suppose to be compact (unless otherwise stated) and we denote by the volume form of its metric. The distributions and on will be called vertical and horizontal spaces and the projections of a vector field along these distributions will be denoted as and , respectively.
3.1. The first variation formula
Let be a (smooth) variation of with variation vector field , i.e.
In this section, we are looking for critical points of –energy, i.e. mappings that satisfy , for any variation. For simplicity let us call these maps –critical. Analogously, critical maps for the full functional (1.3) will be called –critical.
Remark 3.1.
() To every we associate a vector field on , , defined by:
If is a horizontally conformal (surjective) submersion of dilation , then .
() Denote and , where is the pullback connection in (see [7]). Then it is easy to check that:
;
;
;
is harmonic if and only if .
In a (local) orthonormal eigenvector frame for , using the above remark, we can compute the first derivative of as follows:
Denote . Then:
(3.1) 
Remark 3.2.
Let us rewrite two of the terms that appeared above as:
() ;
() , where the right hand term is defined in an arbitrary orthonormal frame as follows:
Definition 3.1.
We call –tension field of the map the following section of the pullback bundle :
We have obtained the following (cf. also [48])
Proposition 3.1 (The first variation formula).
In particular, a map is –critical if it satisfies the following EulerLagrange equations
(3.2) 
Remark 3.3.
The EulerLagrange operator of has been derived in [48] for all :
where is the Newton tensor. In case, and then we can easily obtain the equation (3.2). Nevertheless, in this particular case, we preferred to derive the first variation ab initio, for the sake of completeness (as it might be difficult to access [48]). Recall also that is elliptic on , cf. [48].
Analogously to the harmonic map problem, EulerLagrange equations can be written (at least for submersions) in the conservative form , where
is the – stressenergy tensor, cf. [48, p. 44].
3.2. Consequences of the first variation formula
Let us take a look firstly to the simplest (nontrivial) case, namely , so that will have (at most) two distinct eigenvalues.
Corollary 3.1.
Let be a submersion taking values in a surface . Let denote the (positive) square roots of eigenvalues of its CauchyGreen tensor and let denote the mean curvature vector field of its fibers. Then is critical if and only if the following equation is satisfied:
(3.3) 
In particular a (local) diffeomorphism is critical if and only if , i.e. it preserves areas up to a constant factor.
Proof.
Let a local orthonormal frame of eigenvector fields for , corresponding to the eigenvalues , and 0, respectively (i.e. ’s span ). Since and are orthogonal, will be –critical iff:
(3.4) 
An easy simplification shows us that:
Before letting the geometry coming in, it worth to see a simple example in the flat case (note that the notion of critical map can be extended also to noncompact domains by imposing the first variation to be zero on any compact subdomain).
Example 3.1.
The Hénon map [20] is given by , where are real parameters and .
It is easy to check that in this case we have:
, so is a critical map.
Let us notice that is moreover harmonic/holomorphic if and only if and , that is when it is an isometry.
More examples of areapreserving maps (up to constants) are to be found in [29]. An elementary example in nonflat case, is given by the map between 2spheres is an integer that gives the degree of the map. , where
Example 3.1 reflects a more general fact, as we can easily check the following:
Remark 3.4.
A holomorphic map is critical if and only if it is homothetic.
This remark suggests us that it might be difficult to find topologically interesting mappings that are both harmonic and critical. However, this is the case of the identity map between 3spheres and standard Hopf map, the only exact solutions known for Skyrme and Faddeev models.
The next corollary follows in full general context the interplay between harmonicity/holomorphicity and criticality. Compare it with the results in [38] referring to another generalization of the Faddeev model.
Corollary 3.2.
Any totally geodesic map is –critical.
A harmonic map is –critical if and only if, at any point it satisfies:
(3.6) 
where is a local orthonormal frame of eigenvectors for around that point.
A pseudo horizontally homothetic harmonic map to a Kähler manifold is –critical if and only if .
In particular, any holomorphic map between Kähler manifolds (or holomorphic from Sasaki to Kähler) which has constant Dirichlet energy density, is also –critical.
Proof.
By definition, a totally geodesic map satisfies . In this case it is known that is parallel and its eigenvalues are constant. Consequently every term in (3.2) cancels.
Recall that, for any smooth map we have the identity (cf. [7, Lemma 3.4.5]):
(3.7) 
where is the stressenergy tensor of the map.
As for any PHWC mapping the eigenvalues of are double, according to a PHH harmonic map must satisfy:
But PHH hypothesis assures precisely that , . Then our conclusion easily follows. ∎
Recall that a map that satisfies is called harmonic. For more details and examples see [33]. Notice also that when and , a map that satisfies must be a harmonic morphism with constant dilation; so essentially it is the Hopf map, according to [6]. But Hopf map is a particular case of BoothbyWang fibration as it will be pointed out below.
As for the interplay between conformality and critical maps, we have (compare again with [38]):
Corollary 3.3.
Let be a horizontally conformal submersion with dilation . Let denote the mean curvature vector field of its fibers. Then is –critical if and only if it is harmonic, that is:
(3.8) 
In particular,
a conformal diffeomorphism is –critical if and only if it is homothetic.
a horizontally homothetic submersion is –critical if and only if it has minimal fibres.
a horizontally conformal submersion onto a fourmanifold is –critical if and only if it has minimal fibres.
Proof.
For a HC submersion we have , where is the horizontal distribution. So the terms involving the CauchyGreen tensor in (3.2) are equal to
Recall that for HC submersions of dilation the tension field is given by [7, Prop. 4.5.3]:
Replacing the two above identities in (3.2) and taking into account that we get the equation (3.8). ∎
Recall that a horizontally conformal submersion that satisfies (3.8) is called harmonic morphism. For more details and examples see [10].
Example 3.2.
A BoothbyWang fibration (see [9]) of a compact, regular contact manifold over a Kähler (or just almost Kähler) manifold is a harmonic critical map, as the total space is endowed with the metric , so that the fibration is a Riemannian submersion (so is constant) with minimal fibers. Note that all Hopf fibrations belong to this class of examples.
As the Skyrme model deals with maps taking values in the 3sphere, let us particularize our EulerLagrange equations to the 3dimensional target case. Analogously to the Corollary 3.1, we can establish:
Corollary 3.4.
Let be a submersion to a threemanifold. Let , , be the (nonzero) eigenvalues of its CauchyGreen tensor and the mean curvature field of its fibers. Then is critical if and only if:
(3.9) 
where .
In particular, if is horizontally conformal, i.e. , notice that (3.9) is equivalent to the equation of 4harmonicity (3.8).
As suggested by the elementary example of Hénon map, the most natural choice in finding critical maps is given by area preserving (up to a constant rescaling) maps. Supposing and , after simplification of a factor in every equation, (3.9) becomes (with ):
(3.10) 
If moreover the eigenvector corresponding to is a complete Killing vector field on , then (its integral curves provide a minimal foliation on ) and (the complementary distribution is totally geodesic). The above system simplifies once more and, after comparing with the analogous system of equations for harmonicity of ([28, (7)]):
(3.11) 
we can conclude as follows:
Theorem 3.1.
Let be a smooth mapping between dimensional Riemannian manifolds. Suppose that the eigenvalues of its CauchyGreen tensor satisfy , and that the eigenvector corresponding to is a (globally defined) Killing vector field. Then the following statements are equivalent:
is harmonic
is –critical
.
In particular, if moreover , then is both harmonic and –critical, so critical configuration for the full Skyrme model with arbitrary coupling constant.
As we mentioned in the previous section a particular class of maps that satisfy the hypothesis is provided by contactomorphisms. So, in particular, we have:
Corollary 3.5.
A contactomorphism from a contact 3manifold to a contact metric 3manifold such that , is –critical if and only if the equation is verified.
Moreover, if and are closed manifolds, then .
See Example 5.7 below for an illustration of the above phenomenon on the Heisenberg group.
3.3. The effect of (bi)conformal changes of metric
Let be an almost submersion and denote as usual , . Recall [7, 34] that under a biconformal change of metric, , the tension field of becomes
(3.12) 
where are nowhere vanishing functions on .
Let denote the weighted gradient (with respect to the CauchyGreen tensor of ) of a function defined on , where are eigenvectors of corresponding to the eigenvalues .
Using a standard technique, similar to the harmonic case we can state the following:
Theorem 3.2.
Under a biconformal change of metric, the tension field of becomes
(3.13) 