NUPHB114675114675S0550-3213(19)30161-010.1016/j.nuclphysb.2019.114675The AuthorsHigh Energy Physics – TheorySkyrme and Faddeev models in the low-energy limit of 4d Yang–Mills–Higgs theoriesOlafLechtenfeldab⁎olaf.lechtenfeld@itp.uni-hannover.deAlexander D.Popovaalexander.popov@itp.uni-hannover.deaInstitut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, GermanyInstitut für Theoretische PhysikLeibniz Universität HannoverAppelstraße 2Hannover30167GermanyInstitut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, GermanybRiemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, GermanyRiemann Center for Geometry and PhysicsLeibniz Universität HannoverAppelstraße 2Hannover30167GermanyRiemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany⁎Corresponding author.Editor: Stephan StiebergerAbstractFirstly, we consider U(Nc) Yang–Mills gauge theory on R3,1 with Nf>Nc flavours of scalar fields in the fundamental representation of U(Nc). The moduli space of vacua is the Grassmannian manifold Gr(Nc,Nf). It is shown that for strong gauge coupling this 4d Yang–Mills–Higgs theory reduces to the Faddeev sigma model on R3,1 with Gr(Nc,Nf) as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we consider a Yang–Mills–Higgs model with Nf=2Nc and a Higgs potential breaking the flavour group U(Nf)=U(2Nc) to U+(Nc)×U−(Nc), realizing the simplest A2⊕A2-type quiver gauge theory. The vacuum moduli space of this model is the group manifold Uh(Nc) which is the quotient of U+(Nc)×U−(Nc) by its diagonal subgroup. When the gauge coupling constant is large, this 4d Yang–Mills–Higgs model reduces to the Skyrme sigma model on R3,1 with Uh(Nc) as target. Thus, both the Skyrme and the Faddeev model arise as effective field theories in the infrared of Yang–Mills–Higgs models.1Introduction and summaryIn 1975, Faddeev introduced a (3+1)-dimensional SU(2)/U(1) coset sigma model that includes a term quartic in derivatives to stabilize classical solutions [1]. This model is similar to the Skyrme model [2], which features maps from R3,1 into SU(2). Despite their similarity, these models are quite different from one another. Topological solitons of the Skyrme model have a point-like core and are supposed to describe baryons and nuclei (see e.g. [3] for a review and [4–6] for some recent works). On the other hand, solitons in the Faddeev model take the form of stable knotted strings characterized by the Hopf charge (homotopy class of maps S3→S2). It is conjectured that Faddeev-model solitons describe glueballs (see e.g. [7–9] for reviews).The standard Skyrme model [2] supposedly describes pions. Other mesons can be incorporated into an extended 4d Skyrme model, which is obtained from 5d Yang–Mills theory on an AdS-type manifold M5 with boundary ∂M5=R3,1 as derived from D-brane configurations in string theory and the holographic approach [10] (see e.g. [11–13] for reviews). This extended Skyrme model also arises in the adiabatic limit of the 5d Yang–Mills system on R3,1×I, where I is a short interval [14].11The adiabatic approach was used in field theory for the first time by Manton [15]. For a review of this approach see [16,17]; brief discussions can be found e.g. in [18–21]. Similarly, also an extended 4d Faddeev model can emerge in a low-energy limit of 5d maximally supersymmetric Yang–Mills theory with its five adjoint scalars [22]. In contrast to the extended Skyrme model, for the extended Faddeev model one needs to keep one of the five adjoint scalars and must modify the fifth dimension from I to the half-line R+. The boundary conditions required for the reduction to R3,1 are encoded in Nahm equations along the fifth dimension [23,24], which reduce to a “baby” Nahm equation on R+ for one adjoint scalar [22].Quantum chromodynamics (QCD) as well as Yang–Mills theory are strongly coupled in the infrared limit, and hence the perturbative expansion for them breaks down. In the absence of a quantitative understanding of non-perturbative QCD, convenient alternatives at low energy are provided by effective models among which nonlinear sigma models play an important role, especially the Skyrme and Faddeev models. Both models are the standard two-derivative sigma models on R3,1 with a compact Lie group G and a coset space G/H as target spaces, respectively, completed with a four-derivative term which stabilizes classical solutions against scaling. In the Faddeev model, H is a closed subgroup of G such that G/H is a coadjoint orbit.As we discussed above, both Skyrme and Faddeev models can be obtained as low-energy limit of 5d Yang–Mills–Higgs (YMH) theories on the classical level. On the other hand, in the strong-coupling or infrared limit, many YMH models on Rd−1,1 with d≥2 reduce to standard two-derivative sigma models governing maps from Rd−1,1 to a moduli space of Higgs vacua. In other words, YMH theories flow in the infrared to sigma models on the same space Rd−1,1 (see e.g. [25] and references therein). For YMH models which are bosonic parts of supersymmetric QCD in d=4, these classical moduli spaces are non-trivial Kähler or hyper-Kähler manifolds [25–27]. Here we will show that the four-derivative Skyrme term also naturally appears in these four-dimensional YMH models in the framework of the adiabatic approach.22Some steps in the derivation of Skyrme terms from YMH models were taken in [28,29], but for a different class of YMH models and without using the adiabatic method. To summarize, we demonstrate that both the Skyrme model and the Faddeev model occupy an infrared corner of 4d YMH models related with N=2 supersymmetric QCD.2Yang–Mills–Higgs modelNotation. On Minkowski space R3,1∋xμ with the metric (ημν)=diag(−1,1,1,1), we consider U(Nc) gauge theory with Nf flavours of scalar fields in the fundamental representation of U(Nc), combined in an Nc×Nf matrix Φ. A gauge potential A=Aμdxμ and the Yang–Mills field F=dA+A∧A take values in the Lie algebra u(Nc). Its components read Fμν=∂μAν−∂νAμ+[Aμ,Aν], where ∂μ:=∂/∂xμ and μ,ν=0,1,2,3. For the generators Iıˆ of the gauge group U(Nc) we use the standard normalization tr(IıˆIȷˆ)=−12δıˆȷˆ.Transformations of fields. The covariant derivative of the complex Higgs field Φ in the bi-fundamental representation of U(Nc)×U(Nf) with Nf>Nc reads(2.1)DμΦ=∂μΦ+AμΦ since the U(Nf) flavour group acts on Φ only by global transformations(2.2)Φ↦Φgf. We denote by G the infinite-dimensional group C∞(R3,1,U(Nc)) of gauge transformations which are parametrized by gc(x)∈G for x∈R3,1. Then A and Φ are transformed as(2.3)A↦Agc=gcAgc−1+gcdgc−1andΦ↦Φgc=gcΦ. For the infinitesimal action of G we have(2.4)A↦δϵcA=dϵc−[A,ϵc]andδϵcΦ=ϵcΦwithgc=exp(ϵc). Similarly, for the U(Nf) flavour symmetry we have(2.5)δϵfA=0andδϵfΦ=Φϵf, where ϵc∈LieG=C∞(R3,1,u(Nc)) and ϵf∈u(Nf).Lagrangian. We consider the Yang–Mills–Higgs (YMH) action functional(2.6)S=−∫R3,1d4x{tr(12e2Fμν†Fμν+DμΦ(DμΦ)†)+e24V(Φ)}, where † denotes Hermitian conjugation, e is the gauge coupling constant, and(2.7)V(Φ)=tr(M21Nc−ΦΦ†)2 is the Higgs potential with a mass parameter M. The Lagrangian from (2.6) is related with the bosonic part of the Lagrangian for N=2 supersymmetric QCD, and such Lagrangians are often considered in the literature (see e.g. [25,30] and references therein).The energy density H of YMH configurations described by (2.6) is(2.8)H=tr(1e2F0a†F0a+D0Φ(D0Φ)†+12e2Fab†Fab+DaΦ(DaΦ)†)+e24V(Φ), where a,b=1,2,3. Here both V(Φ) and H are positive-semidefinite and gauge-invariant functions. They are also U(Nf)-invariant.Vacua. A YMH vacuum configuration (Aˆ,Fˆ,Φˆ) is defined by the vanishing of the energy density (2.8). This is achieved by(2.9)Fˆμν=0,DˆμΦˆ=0andV(Φˆ)=0, where the last equation defines the Higgs vacuum manifold. Denote by M˜ the space of solutions of (2.9) with Aˆ=0=Fˆ and Φˆ∈Mat(Nc,Nf;C) (complex Nc×Nf matrices) such that(2.10)ΦˆΦˆ†=M21Nc, i.e. M˜ is the space of solutions to (2.10). The group U(Nc) acts freely on M˜ by left multiplication, Φˆ↦gcΦˆ. It is not difficult to show [31] that M˜ is fibred over the Grassmannian(2.11)Gr(Nc,Nf)=[U(Nc)×U(Nf−Nc)]\U(Nf)=:M with the projection(2.12)π:M˜⟶U(Nc)M and the group U(Nc) as fibres.It is important to distinguish between the Higgs field Φ depending on x∈R3,1 and the vacua Φˆ∈Mat(Nc,Nf;C), which solve (2.10). The moduli space of vacua M is the Grassmannian (2.11), any element of which can be obtained from a reference vacuum Φˆ0. We choose Φˆ0=(1Nc0Nc×(Nf−Nc)) so that the isotropy group of Φˆ0 for the right U(Nf) action is(2.13)U(Nc)×U(Nf−Nc)={gf∈U(Nf):Φˆ0gf=gcΦˆ0for somegc∈U(Nc)}. It is obvious [31] that such gf have the form diag(gc,gf−c) with gf−c∈U(Nf−Nc). In other words, the right action of the isotropy group U(Nc)×U(Nf−Nc) on Φˆ0 is equivalent to the left action of the gauge group U(Nc), and we simply have(2.14)M˜=U(Nf−Nc)\U(Nf).3Moduli space of vacuaGeometry of Gr(Nc,Nf). The space M˜ in (2.12) parametrizes all vacua for the model (2.6), and the Grassmannian M in (2.11) and (2.12) parametrizes gauge inequivalent vacua, i.e. the vacuum moduli space. Both M˜ and M are homogeneous spaces with a right action of U(Nf). Note that right cosets can be changed to left cosets by interchanging Φˆ with Φˆ†.Let m be the tangent space to the Grassmannian M at the fixed point Φˆ0. Then we have the splitting(3.1)u(Nf)=m⊕u(Nc)⊕u(Nf−Nc), and m˜=m⊕u(Nc) can be identified with the tangent space of M˜ at any given point. For u(Nf) we choose a basis(3.2){Ii}={Iı¯,Iıˆ,Ii′}with{ı¯=1,…,dimm=2Nc(Nf−Nc),ıˆ=dimm+1,…,dimm+Nc2,i′=dimm+Nc2+1,…,dimm+Nc2+(Nf−Nc)2, so that Iı¯, Iıˆ and Ii′ form orthogonal bases for m, u(Nc) and u(Nf−Nc), respectively. One can associate to Ii vector fields Vi on U(Nf) and a basis {ei}={eı¯,eıˆ,ei′} of one-forms which is dual to {Vi}, i.e. Vi⌟ej=δij. These one-forms obey the Maurer-Cartan equations(3.3)deı¯=−fȷˆk¯ı¯eȷˆ∧ek¯−fı¯j′k¯ej′∧ek¯,deıˆ=−12fȷ¯k¯ıˆeȷ¯∧ek¯−12fȷˆkˆıˆeȷˆ∧ekˆ,dei′=−12fȷ¯k¯i′eȷ¯∧ek¯−12fj′k′i′ej′∧ek′, where we used the fact that Gr(Nc,Nf) is a symmetric space.The Grassmannian M=Gr(Nc,Nf) supports an orthonormal frame of one-forms {eı¯} locally giving the U(Nf)-invariant metric as(3.4)dsM2=δı¯ȷ¯eı¯eȷ¯=δı¯ȷ¯eαı¯eβȷ¯dXαdXβ=:gαβdXαdXβforα,β=1,…,2Nc(Nf−Nc), where {Xα} is a set of real local coordinates of a point X∈Gr(Nc,Nf), and ∂α=∂/∂Xα will denote derivatives with respect to them.Canonical connection. On the principal U(Nc)-bundle (2.12) there exists a unique U(Nf)-equivariant connection, the so-called canonical connection (see e.g. [32–35]),(3.5)AGr=AαGrdXα=eıˆIıˆ=eαıˆIıˆdXα taking values in u(Nc). It satisfies both Yang–Mills and generalized instanton equations on Gr(Nc,Nf) [33–35]. The curvature of the canonical connection (3.5) in the bundle (2.12) follows as(3.6)FGr=12FαβGrdXα∧dXβ=−12fȷ¯k¯ıˆIıˆeȷ¯∧ek¯=−12fȷ¯k¯ıˆIıˆeαȷ¯eβk¯dXα∧dXβ. Variation of Φˆ. By letting Xα run over M we obtain a local section Φˆ(Xα) of the bundle (2.12). The infinitesimal changes of this section are given by the covariant derivatives (cf. [18,19])(3.7)δαΦˆ=∂αΦˆ+AαGrΦˆ, where AαGr are the components of the connection (3.5) in the principal U(Nc) bundle (2.12).4Faddeev model in the infrared limit of 4d YMHDependence on xμ. Now we return to Yang–Mills–Higgs theory on R3,1. In Section 3 we described the moduli space M=Gr(Nc,Nf) of vacua for the YMH model (2.6)–(2.8). For small exitations around M, in the strong gauge-coupling limit e2≫1, the Higgs field Φ(x) can be considered as a map(4.1)Φ:R3,1→Gr(Nc,Nf) since for e2≫1 it should be at a minimum of the Higgs potential (2.7). The moduli-space approximation then postulates that all fields depend on the spacetime coordinates x={xμ} only via coordinates Xα=Xα(x) on M (see e.g. [15–21] and references therein). By substituting Φ(Xα(x)) and A(Xα(x)) into the initial action (2.6), we obtain an effective field theory describing small fluctuations around the vacuum moduli space M.Two-derivative part of effective action. Multiplying (3.7) by ∂μXα, we obtain(4.2)∂μΦ=(∂μXα)∂αΦ=(∂μXα)δαΦ−ϵμΦwithϵμ=(∂μXα)AαGr, where ϵμ∈u(Nc) is the pull-back of AGr from Gr(Nc,Nf) to R3,1. It immediately follows that(4.3)DμΦ=∂μΦ+AμΦ=(∂μXα)δαΦ+(Aμ−ϵμ)Φ. We see that DμΦ are tangent33This is a key requirement of the adiabatic approach. It is necessary for the description of small fluctuations around the initial moduli space when the dynamical fields are collective coordinates (see e.g. [18,19,36]). to C∞(R3,1,Gr(Nc,Nf)) if(4.4)Aμ=ϵμ. Substituting (4.3) with (4.4) into the (2.6), we obtain(4.5)Skin=−∫R3,1d4xημνtr{DμΦ(DνΦ)†}=−M22∫R3,1d4xημνgαβ∂μXα∂νXβ, where(4.6)gαβ=2M2tr{δαΦ(δβΦ)†}=δı¯ȷ¯eαı¯eβȷ¯ are the components of the metric (3.4) on Gr(Nc,Nf) pulled back to R3,1, so gαβ(Xγ(x)) now depend on x. We introduced the mass scale M from (2.7) into (4.6) to render gαβ dimensionless. Thus, this part of the action (2.6) reduces to the standard non-linear sigma model on R3,1 with the Grassmannian Gr(Nc,Nf) as its target.Four-derivative part of effective action. As discussed earlier, the potential term in (2.6) vanishes since Φ(x) takes values in the manifold M=Gr(Nc,Nf) of gauge-inequivalent vacua. For calculating the first term in (2.6), we use (4.4), and for the curvature of A=Aμdxμ we obtain(4.7)F=dA+A∧A=12Fμνdxμ∧dxν=−12fȷ¯k¯ıˆIıˆeαȷ¯eβk¯∂μXα∂νXβdxμ∧dxν, allowing one to extract the components Fμν. Substituting (4.7) into (2.6) we arrive at(4.8)SFad=−12e2∫R3,1d4xtr(Fμν†Fμν)=−14e2∫R3,1d4xδıˆȷˆfl¯k¯ıˆfm¯n¯ȷˆeαl¯eβk¯eγm¯eδn¯∂μXα∂νXβ∂μXγ∂νXδ, where ∂μ:=ημσ∂σ. Thus, in the infrared limit the Yang–Mills–Higgs action (2.6) is reduced to the Faddeev action,(4.9)Seff=−∫R3,1d4x{M22gαβ∂μXα∂μXβ+14e2δıˆȷˆfl¯k¯ıˆfm¯n¯ȷˆeαl¯eβk¯eγm¯eδn¯∂μXα∂νXβ∂μXγ∂νXδ} for scalar fields Xα with values in the Grassmannian Gr(Nc,Nf).5A⊕2A2-quiver gauge theoryFields. It is possible to obtain not only the Faddeev model but also the standard Skyrme model from Yang–Mills–Higgs theory in four dimensions. To achieve this, we should consider a 4d YMH model with a group manifold, say U(N), as the moduli space M of vacua. The simplest way to do this is to specialize the model (2.6) to Nf=2Nc=:2N but with a potential different from (2.7). We parametrize(5.1)Φ=:(ϕ−,ϕ+)withϕ±∈Mat(N,N;C). Thus, we have a u(N)-valued gauge field F, an N×2N complex Higgs field Φ=(ϕ−,ϕ+), the group of gauge transformations G=C∞(R3,1,U(N)) and transformations (2.2)–(2.5) for Nf=2Nc=2N.Action. We consider the Yang–Mills–Higgs (YMH) action functional(5.2)S=−∫R3,1d4x{tr(12e2Fμν†Fμν+DμΦ(DμΦ)†)+e24V(Φ)}, and the two-term potential(5.3)V(Φ)=tr(m21N−ϕ−ϕ−†)2+tr(m21N−ϕ+ϕ+†)2 with a mass parameter m. This action can be obtained from A+2⊕A2− quiver gauge theory (see e.g. [37,38] and references therein) corresponding to a direct sum of quivers(5.4)A2±:CN⟶ϕ±CN, where four copies of CN at four vertices carry the fundamental U(N) representation, and the arrows ϕ± denote maps between them.The form (5.3) of the Higgs potential breaks the flavour group U(2N) to the subgroup G=U−(N)×U+(N). Let {Ii} be a basis of the Lie algebra g=LieG=u−(N)⊕u+(N) realized as 2N×2N block-diagonal matrices with the normalization tr(IiIj)=−12δij for i=1,…,2N2. The covariant derivative in (5.2) reads(5.5)DμΦ=(Dμϕ−,Dμϕ+)withDμϕ±=∂μϕ±+Aμϕ±, with a u(N)-valued gauge potential A=Aμdxμ.Vacua. The energy density of YMH configurations described by the action (5.2) has the form (2.8) with V(Φ) given by (5.3). The vacuum configurations are defined by (2.9), which implies(5.6)ϕˆ−ϕˆ−†=m21Nandϕˆ+ϕˆ+†=m21N. Equations (5.6) are solved by some(5.7)(ϕˆ−,ϕˆ+)∈U−(N)×U+(N)=M˜ subject to global gauge transformations(5.8)(ϕˆ−,ϕˆ+)↦(hϕˆ−,hϕˆ+)forh∈U(N). The group U(N) acts freely on the vacuum manifold M˜ by left multiplication, and one can define the projection(5.9)π:M˜⟶U(N)Mvia(ϕˆ−,ϕˆ+)⟼(1N,ϕˆ)withϕˆ=ϕˆ−−1ϕˆ+. Hence, the moduli space of vacua(5.10)M=U(N)\[U−(N)×U+(N)] is diffeomorphic to the group manifold U(N), any element of which can be obtained from a reference vacuum Φˆ0. We choose 1mΦˆ0=(1N,1N) so that the isotropy group for the right G action is(5.11)U(N)={g∈G:Φˆ0g=hΦˆ0for someh∈U(N)}. It is obvious that g=diag(h,h), i.e. the isotropy group is(5.12)diag(G)≅Udiag(N)=U(N)=:H, and the global gauge transformations form the stability subgroup in a realization of the group manifold U(N) as the coset space H\G in (5.10). This is also seen from the fact that (hϕˆ−)−1(hϕˆ+)=ϕˆ−−1ϕˆ+, i.e. ϕˆ is inert under the action of H. From (5.9) it follows the decomposition(5.13)g=u−(N)⊕u+(N)=m⊕h=m⊕u(N)diagwithh={(η,η)|η∈u(N)}.Geometry of H\G. The geometry of a group manifold considered as a homogeneous space has some characteristic features (see e.g. [31,39,40]) which we briefly describe here. In the split (5.13), m is not necessarily orthogonal to h with respect to the Cartan–Killing form. In fact, there are three natural reductive decompositions of g with the following versions of m:(5.14)m0={(−θ,θ)},m−={(−θ,0)},m+={(0,θ)},withθ∈u(N). The first case yields H\G as a symmetric space with m0 orthogonal to h. With the choice m+ or m− the coset (5.10) becomes a nonsymmetric homogeneous manifold. Obviously, m≅u(N) in all three cases. The choices of m0, m− and m+ correspond to the gauges ϕˆ−=ϕˆ+†, ϕˆ+=m1N and ϕˆ−=m1N, respectively, which determine different coset representatives, i.e. sections of the bundle (5.9) with M˜=G and M=H\G.We split the basis of g according to the decomposition (5.13),(5.15){Ii}={Iı¯,Iıˆ}with{ı¯=1,…,N2form,ıˆ=N2+1,…,2N2forh. We have an orthonormal frame of one-forms {eı¯} on H\G, the metric (3.4) with α,β=1,…,N2 and the canonical connection Acan=eıˆIıˆ=eαıˆIıˆdXα for all three cases m0, m− and m+. However, the Maurer–Cartan equations depend on the case:(5.16)m0:deı¯=−fȷˆk¯ı¯eȷˆ∧ek¯anddeıˆ=−12fȷ¯k¯ıˆeȷ¯∧ek¯−12fȷˆkˆıˆeȷˆ∧ekˆ,m−:deı¯=−fȷˆk¯ı¯eȷˆ∧ek¯+12fȷ¯k¯ı¯eȷ¯∧ek¯anddeıˆ=−12fȷˆkˆıˆeȷˆ∧ekˆ,m+:deı¯=−fȷˆk¯ı¯eȷˆ∧ek¯−12fȷ¯k¯ı¯eȷ¯∧ek¯anddeıˆ=−12fȷˆkˆıˆeȷˆ∧ekˆ. Furthermore, on the group manifold (5.10) one can introduce a family of connections(5.17)AU(N)ϰ=ϰeıˆIıˆ=ϰeıˆαIıˆdXα=:AαϰdXαwithϰ∈R with curvature(5.18)FU(N)ϰ=12ϰ(ϰ−1)fȷˆkˆıˆIıˆeȷˆ∧ekˆ−12ϰfȷ¯k¯ıˆIıˆeȷ¯∧ek¯. For the cases m± the last term in (5.18) vanishes. The connection (5.17) is the unique G-equivariant family of connections on the bundle (5.9) [31,39].Variation of Φˆ. In the following we adopt the gauge 1mϕˆ−=1N fixing m=m+, so fȷ¯k¯ıˆ=0 in (5.18). Then, abbreviating ϕˆ+≡ϕˆ,(5.19)AU(N)ϰ=ϰϕˆ(∂αϕˆ−1)dXα⇒Aϰα=ϰϕˆ∂αϕˆ−1, and letting Xα run over M≅U(N) we obtain a local section Φˆ(Xα)=m(1N,ϕˆ(Xα)) of the bundle (5.9). Infinitesimal changes of this section are given by the covariant derivatives (cf. [18,19])(5.20)δαΦˆ=∂αΦˆ+AαϰΦˆ=m(Aαϰ,∂αϕˆ+Aαϰϕˆ).6Skyrme model in the infrared limit of 4d YMH theoryThe derivation of the Skyrme model as an effective theory for the 4d YMH model (5.2) is similar to the derivation of the Faddeev model from the YMH action (2.6). The main difference is that now the vacuum moduli space M=H\G=U(N) is a group manifold, whose geometry was described in Section 5. According to the philosophy of the adiabatic method, we assume that the gauge potential A=Aμdxμ and the Higgs field Φ depend on the R3,1 coordinates x only via real coordinates Xα=Xα(x) on U(N), and we substitute A(Xα(x)) and Φ(Xα(x)) into the action (5.2) by using results of Section 5.Kinetic term. Multiplying (5.20) by ∂μXα, we obtain(6.1)DμΦ=∂μXαδαΦ+(Aμ−ϵμ)Φ, where ϵμ=(∂μXα)Aαϰ∈u(N) is the pull-back of AU(N)ϰ from U(N) to R3,1. To render (DμΦ)Φ† tangent to C∞(R3,1,U(N)), we choose(6.2)Aμ=ϵμ=ϰ(∂μXα)ϕ∂αϕ−1=ϰϕ∂μϕ−1, where ϕ is a U(N)-valued function. Notice that (5.19) and (5.20) imply(6.3)δαΦ=−m(ϰ(∂αϕ)ϕ†,(ϰ−1)∂αϕ)⇒DμΦ=−m(ϰ(∂μϕ)ϕ†,(ϰ−1)∂μϕ). Substituting (6.1)–(6.3) into (5.2), we obtain(6.4)Skin=−∫R3,1d4xημνtr{DμΦ(DνΦ)†}=14fπ2∫R3,1d4xημνtr(RμRν) with(6.5)Rμ:=ϕ∂μϕ−1and14fπ2=(ϰ2+(ϰ−1)2)m2, where fπ may be interpreted as the pion decay constant. Thus, this part of the action (5.2) reduces to the standard non-linear sigma model on R3,1 with a U(N) target space.Skyrme term. For calculating the F2-terms in (5.2) we employ (6.2) and find(6.6)F=dA+A∧A=ϰ(ϰ−1)ϕdϕ−1∧ϕdϕ−1=12ϰ(ϰ−1)[Rμ,Rν]dxμ∧dxν since Aμ=ϰϕ∂μϕ−1=ϰRμ after the pull-back to R3,1. Substituting (6.6) into (5.2), we obtain(6.7)SSky=−12e2∫R3,1d4xtr(Fμν†Fμν)=132ζ2∫R3,1d4xημληνσtr([Rμ,Rν][Rλ,Rσ]), where(6.8)132ζ2=ϰ2(ϰ−1)28e2, and ζ is the dimensionless Skyrme parameter. Hence, in the infrared limit the Yang–Mills–Higgs action (5.2) is reduced to the action of the Skyrme model,(6.9)Seff=∫R3,1d4x{fπ24ημνtr(RμRν)+132ζ2ημληνσtr([Rμ,Rν][Rλ,Rσ])}. Thus, both Skyrme and Faddeev models appear as effective field theories in the infrared of Yang–Mills–Higgs models.AcknowledgementsThis work was partially supported by the Deutsche Forschungsgemeinschaft grant LE 838/13. 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