New Chern-Simons densities in both odd and even dimensions Eugen Radu†⋆ and Tigran Tchrakian‡⋆⋄ ‡ School of Theoretical Physics – DIAS,10 Burlington Road, Dublin 4, Ireland 1 ⋆ 1 Department of Computer Science, National University of Ireland Maynooth, Maynooth, Ireland 0 ⋄Theory Division, YerevanPhysics Institute(YerPhI),AM-375 036 Yerevan 36, Armenia 2 n a Abstract J AfterreviewingbrieflythedimensionalreductionofChern–Pontryagindensitie,wedefinenewChern– 6 Simonsdensities expressed in termsof Yang-Mills and Higgs fields. These are definedin all dimensions, 2 includinginevendimensionalspacetimes. Theyareconstructedbysubjectingthedimensionallyreduced ] Chern–Pontryagin densites to furtherdescent by two steps. h t - p 1 Introduction e h The central task of these notes is to explain how to subject n−th Chern–Pontryagin(CP) density C(n), [ 1 1 C(n) = ε TrF F ...F (1.1) v ω(π) M1M2M3M4...M2n−1M2n M1M2 M3M4 M2n−1M2n 8 6 defined on on the 2n−dimensional space, to dimensional descent to IRD by considering (1.1) on the direct 0 product space IRD×S2n−D. The resulting residual density on IRD will be denoted as Ω(n). 5 D . The density C(n) is by construction, a total divergence 1 0 C(n) =∇·Ω(n), (1.2) 1 1 anditturnsoutthatundercertainretrictions,thedimensionaldescendantof(1.2)isalsoatotaldivergence. : v This result will be applied to the construction of Chern-Simons solitons in all dimensions. i Some special choices, or restrictions, are made for practical reasons. Firstly, we have restricted to the X codimensionS2n−D,the (2n−D)−sphere,asthis isthemostsymmetriccompactcosetspacethatisdefined r a bothinevenandinodddimensions. Itcanofcoursebe replacedbyanyothersymmetricandcompactcoset space. Secondly, the gauge field of the bulk gauge theory is chosen to be a 2n−1 ×2n−1 array with complex valued entries. Given our choice of spheres for the codimension, this leads to residual gauge fields on which take their values in the Dirac matrix representation of the residual gauge group SO(D). As a result of the above two choices, it is possible to make the symmetry imposition (namely the di- mensional reduction) such that the residual Higgs field is described by a D−component isovector multiplet. With this choice, the asymptotic gauge fields describe a Dirac-Yang [1, 2, 3] monopoles.. Thecentralresultexploitedhereis thatthe CPdensity C(n) onIRD descendedfromthe 2n−dimensional D bulk CP density C(n) is also a total divergence C(n) =∇·Ω(n,D) (1.3) D like C(n) formally is on the bulk. From the reduced density Ω(n,D) ≡ Ω(n,D), where the index i labels the i coordinateoftheresidualspacex withi=1,2,...,D,onecanidentifyaChern–Simons(CS)densityasthe i 1 D−th component of Ω(n,D). This quantity can then be interpreted as a CS term on (D−1)− dimensional Monkowski space, i.e., on the spacetime (t,IRD−2). The solitons of the corresponding CS–Higgs (CSH) theory can be constructed systematically. Note, that this is not the usual CS term defined in terms of a pure Yang–Mills fieldonodd dimensionalspacetime, but ratherthese new CSterms are definedby boththe YM and, the Higgs fields. Most importantly, the definition of these new CS terms is not restricted to odd dimensional spacetimes, but covers also even dimensional spacetimes. Such CSH solutions have not been studied to date. 2 Dimensional reduction of gauge fields The calculus of the dimensional reduction of Yang–Mills fields employed here is based on the formalism of A. S. Schwarz [4], which is specially transparent due to the choice of displaying the results only at a fixed point of the compactsymmetric codimensionalspace KN (the North or South pole for SN). Our formalism is a straightforwardextension of [4, 5, 6]. 2.1 Descent over SN: N odd For the descent from the bulk dimension 2n=D+N down to odd D (over odd N), the components of the residual connection evaluated at the Noth pole of SN are given by A = A (~x)⊗1I (2.1) i i 1 A = Φ(~x)⊗ Γ . (2.2) I I 2 The unit matrix in (2.1), like the N−dimensional gamma matrix in (2.2), are 212(N−1) ×221(N−1) arrays. Choosing the 2n−1×2n−1 bulk gauge groupto be, say, SU(n−1), allowsthe choice of SO(D) as the gauge group of the residual connection A (x). This choice is made such that the asymptotic connections describe i a Dirac–Yang monopole. Forthesamereason,thechoiceforthemultipletstructureoftheHiggsfieldismadetobelessrestrictive. The (anti-Hermitian) field Φ, which is not necessarily traceless 1, can be and is taken to be in the algebra ofSO(D+1),inparticular,inone or otherof the chiralreprentationsofSO(D+1), D+1here being even. Φ=φabΣ , a=i, D+1 , i=1,2,...,D. (2.3) ab (OnlyintheD =3casedoestheHiggsfieldtakeitsvaluesinthealgebraofSO(3),sincetherepresentations SO(3) coincide with those of chiral SO(4).) In anticipation of the corresponding situation of even D to be given next, one can specialise (2.3) to the a D−component isovector expression of the Higgs field Φ=φiΣ , (2.4) i,D+1 withthepurposeofhavingaunifiednotationforbothevenandoddD,wheretheHiggsfieldtakesitsvalues in the components Σ orthogonal to elements Σ of the algebra of SO(D+1). This specialisation is i,D+1 ij not necessary, and is in fact inappropriate should one consider, e.g., axially symmetric fields. It is however adequatefor the presentationhere andis sufficiently generalto describesphericallysymmetric monopoles2. 1Inpractice,whenconstructingsolitonsolutions,Φistakentobetracelesswithoutlossofgenerality. 2Whileallconcreteconsiderationsinthefollowingarerestrictedtosphericallysymmetricfields,itshouldbeemphasisedthat relaxingspherical symmetry resultsinthe Higgs multiplet getting out of the orthogonal complement Σi,D+1 to Σi,j. Indeed, subjecttoaxialsymmetryonehas Φ=f1(ρ,z)Σαβxˆβ+f2(ρ,z)Σβ,D+1xˆβ+f3(ρ,z)ΣD,D+1, (2.5) where xi = (xα,z), |xα|2 = ρ2 and with xˆα = xα/ρ. Clearly, the term in (2.5) multiplying the basis Σαβ does not occur in (2.4). 2 In(2.1)and(2.2),andeverywherehenceforth,wehavedenotedthecomponentsoftheresidualcoordinates as x =~x. The dependence on the codimension coordinate x is suppressed since all fields are evaluated at i I a fixed point (North or South pole) of the codimension space. The resulting components of the curvature are F = F (~x)⊗1I (2.6) ij ij 1 F = D Φ(~x)⊗ Γ (2.7) iI i I 2 F = S(~x) ⊗Γ , (2.8) IJ IJ whereΓ =−1[Γ ,Γ ]aretheDiracrepresentationmatricesofSO(N),thestabilitygroupofthesymmetry IJ 4 I J group of the N−sphere. In (2.7), D Φ is the covariant derivative of the Higgs field Φ i D Φ=∂ Φ+[A ,Φ] (2.9) i i i and S is the quantity S =−(η21I+Φ2), (2.10) where η is the inverse of the radius of the N−sphere. 2.2 Descent over SN: N even The formulae corresponding to (2.1)-(2.8) for the case of even D are somewhat more complex. The reason is the existence of a chiral matrix Γ , in addition to the Dirac matrices Γ , I = 1,2,...,N. Instead of N+1 I (2.1)-(2.2) we now have A = A (~x)⊗1I+B (~x)⊗Γ i i i N+1 1 1 A = φ(~x)⊗ Γ +ψ(~x)⊗ Γ Γ , I I N+1 I 2 2 where A , B , φ, andψ areagainantihermitianmatrices,but with onlyA being traceless. The factthat B i i i i is not traceless here results in an Abelian gauge field in the reduced system. Anticipating what follows, it is much more transparent to re-express these formulas in the form i A = A(+)(~x)⊗ P +A(−)(~x)⊗ P + a (~x)Γ (2.11) i i + i − 2 i N+1 1 1 A = ϕ(~x)⊗ P Γ −ϕ(~x)†⊗ P Γ , (2.12) I + I − I 2 2 where now P± are the 2N2 ×2N2 projection operators 1 P = (1I±Γ ) . (2.13) ± N+1 2 In (2.11), the residual gauge connections A(±) are anti-Hermitian and traceless 2D2 ×2D2 arrays, and the i Abelian connection ai results directly from the trace of the field Bi. The 2D2 ×2D2 ”Higgs” field ϕ in (2.12) is neither Hermitian nor anti-Hermitian. Again, to achieve the desired breaking of the gauge group, to lead eventuallytotherequisiteHiggsisomultiplet,wechoosethegaugegroupinthebulktobeSU(n−1),where 2n=D+N. The components of the curvaturs are readily calculated to give i F = F(+)(~x)⊗ P +F(−)(~x)⊗ P + f (~x)Γ (2.14) ij ij + ij − 2 ij N+1 1 1 F = D ϕ(~x)⊗ P Γ −D ϕ†(~x)⊗ P Γ (2.15) iI i + I i − I 2 2 F = S(+)(~x)⊗ P Γ +S(−)(~x)⊗ P Γ , (2.16) IJ + IJ − IJ 3 the curvatures in (2.14) being defined by F(±) = ∂ A(±)−∂ A(±)+[A(±),A(±)] (2.17) ij i j j i i j f = ∂ a −∂ a . (2.18) ij i j j i The covariantderivative in (2.15) now is defined as D ϕ = ∂ ϕ+A(+)ϕ−ϕA(−)+ia ϕ (2.19) i i i i i D ϕ† = ∂ ϕ†+A(−)ϕ†−ϕ†A(+)−ia ϕ†, (2.20) i i i i i and the quantities S(±) in (2.16) are S(+) =ϕϕ†−η2 , S(−) =ϕ†ϕ−η2. (2.21) In what follows, we will suppress the Abelian field a , since only when less stringent symmetry than i spherical is imposed is it that it would contribute. In any case, using the formal replacement i A(±) ↔A(±)± a 1I i i 2 i yields the algebraic results to be derived below, in the general case. We now refine our calculus of descent over even codimensions further. We see from (2.11) that A(±) i being 2D2 ×2D2 arrays, that they can take their values in the two chiral representations, repectively, of the algebra of SO(D). It is therefore natural to introduce the full SO(D) connection A(+) 0 A = i . (2.22) i " 0 Aµ(−) # Next, we define the D−component isovector Higgs field 0 ϕ Φ= =φiΓ (2.23) −ϕ† 0 i,D+1 (cid:20) (cid:21) in terms of the Dirac matrix representation of the algebra of SO(D+1), with Γ =−1Γ Γ . i,D+1 2 D+1 i Noteherethe formalequivalencebetweenthe Higgsmultiplet (2.23)inevenD,to the correspondingone (2.4) in odd D. This formal equivalence turns out to be very useful in the calulus employed in following Sections. In contrast with the former case of odd D however, the form (2.23) for even D is much more restrictive. This is because in this case the Higgs multiplet is restrictedto take its values in the components Γ orthogonal to the elements Γ of SO(D) by definition, irrespective of what symmetry is imposed. i,D+1 ij It is clear that relaxing the spherical symmetry here, does not result in Φ getting out of the orthogonal complement of Γ , when D is even. ij From (2.22), follows the SO(D) curvature F(+) 0 F =∂ A −∂ A +[A ,A ]= ij (2.24) ij i j j i i j " 0 Fi(j−) # and from (2.22) and (2.23) follows the covariant derivative 0 D ϕ D Φ=∂ Φ+[A ,Φ]= i . (2.25) µ i i −D ϕ† 0 i (cid:20) (cid:21) From (2.23) there simply follows the definition of S for even D S(+) 0 S =−(η21I+Φ2)= . (2.26) 0 S(−) (cid:20) (cid:21) 4 3 New Chern–Simons terms First, we recall the usual dynamical Chern–Simons in odd dimensions defined in terms of the non-Abelian gauge connection. Topologically massive gauge field theories in 2 + 1 dimensional spacetimes were first introduced in [7, 8]. The salient feature of these theories is the presence of a Chern-Simons (CS) dynamical term. To define a CS density one needs to have a gauge connection, and hence also a curvature. Thus, CS densities can be defined both for Abelian (Maxwell) and non-Abelian (Yang–Mills) fields. They can also be definedforthegravitational[9]fieldsinceinthatsystemtooonehasa(Levi-Civitaorotherwise)connection, akin to the Yang-Mills connection in that it carries frame indices analogous to the isotopic indices of ther YM connection. Here we are interested exclusively in the (non-Abelian) YM case, in the presence of an isovector valued Higgs field. The definition of a Chern-Simons (CS) density follows from the definition of the corresponding Chern- Pontryagin (CP) density (1.1). As stated by (1.2), this quantity is a total divergence and the density Ω(n) = Ω(n) (M = 1,2,...,2n) in that case has (2n)−components. The Chern-Simons density is then M defined as one fixed component of Ω(n), say the 2n−th component, Ω(n) =Ω(n) (3.1) CS 2n which now is given in one dimension less, where M =µ,2n and µ=1,2,...(2n−1). This definition of a (dynamical) CS term holds in all odd dimensional spacetimes (t,IRD), with x = µ (x ,x ), i=1,2,...,D, with D being an even integer. That D must be even is clear since D+2=2n, the 0 i 2n dimensions in which the CP density (1.1) is defined, is itself even. The properties of CS densities are reviewed in [10]. Most remarkably, CS densities are defined in odd (space orspacetime)dimensionsandaregaugevariant. The contexthereis thatofa (2n−1)−dimensional Minkowskian space. It is important to realise that dynamical Chern-Simons theories are defined on space- times with Minkowskian signature. The reason is that the usual CS densities appearing in the Lagrangian are by construction gauge variant, but in the definition of the energy densities the CS term itself does not feature, resulting in a Hamiltonian (and hence energy) being gauge invariant as it should be 3. Of course, the CP densities and the resulting CS densities, can be defined in terms of both Abelian and non-Abelian gauge connections and curvatures. The context of the present notes is the construction of soliton solutions 4, unlike in [7, 8]. Thus in any given dimension, our choice of gauge group must be made withdueregardtoregularity,andthemodelschosenmustbeconsistentwiththeDerrickscalingrequirement for the finiteness of energy. Accordingly, in all but 2+1 dimensions, our considerations are restricted to non-Abelian gauge fields. Clearly, such constructions can be extended to all odd dimensional spacetimes systematically. We list Ω , defind by (3.1), for D =2,4,6, familiar densities CS 2 Ω(2) = ε TrA F − A A (3.2) CS λµν λ µν 3 µ ν (cid:20) (cid:21) 2 Ω(3) = ε TrA F F −F A A + A A A A (3.3) CS λµνρσ λ µν ρσ µν ρ σ 5 µ ν ρ σ (cid:20) (cid:21) 4 2 Ω(4) = ε TrA F F F − F F A A − F A F A CS λµνρστκ λ µν ρσ τκ 5 µν ρσ τ κ 5 µν ρ στ κ (cid:20) 4 8 + F A A A A − A A A A A A . (3.4) µν ρ σ τ κ µ ν ρ σ τ κ 5 35 (cid:21) 3ShouldoneemployaCSdensityonaspacewithEuclideansignature, withtheCSdensityappearinginthe staticHamil- tonianitself,thentheenergywouldnotbegaugeinvariant. Hamiltoniansofthistypehavebeenconsideredintheliterature, e.g.,in[11]. Chern-SimonsdensitiesonEuclideanspaces, definedintermsofthecompositeconnection ofasigmamodel,find applicationasthetopological chargedensitiesofHopfsolitons. 4Thetermsolitonsolutionshereisusedratherloosely,implyingonlytheconstructionofregularandfiniteenergysolutions, withoutinsistingontopological stabilityingeneral. 5 Concerning the choice of gauge groups, one notes that the CS term in D+1 dimensions features the productofDpowersofthe(algebravalued)gaugefield/connectioninfrontoftheTrace,whichwouldvanishif thegaugegroupisnotlargerthanSO(D). Inthatcase,theYMconnectionwoulddescribeonlya’magnetic’ component, with the ’electric’ component necessary for the the nonvanishing of the CS density would be absent. As in [12], the most convenient choice is SO(D+2). Since D is always even, the representation of SO(D+2) are the chiral representation in terms of (Dirac) spin matrices. This completes the definition of the usual non-Abelian Chern-Simons densities in D+1 spacetimes. From(3.2)-(3.4),itisclearthatthe CSdensityisgaugevariant. TheEuler”=Lagrangeequationsofthe CS density is nonetheless gaugeinvariant, such that for the exaples (3.2)-(3.4) the correspondingarbitarry variations are δ Ω(2) = ε F (3.5) Aλ CS λµν µν δ Ω(3) = ε F F (3.6) Aλ CS λµνρσ µν ρσ δ Ω(4) = ε F F F . (3.7) Aλ CS λµνρσκη µν ρσ κη This, and other interesting properties of CS densities are given in [10]. A remarkable property of a CS density is its transformation under the action of an element, g, of the (non-Abelian) gauge group. We list these for the two examples (3.2)-(3.3), 2 Ω(2) →Ω˜(2) = Ω(2)− ε Trα α α −2ε ∂ Trα A (3.8) CS CS CS 3 λµν λ µ ν λµν λ µ ν 2 Ω(3) →Ω˜(3) = Ω(3)− ε Trα α α α α CS CS CS 5 λµνρσ λ µ ν ρ σ 1 1 +2ε ∂ Trα A F − A A + F − A A A λµνρσ λ µ ν ρσ ρ σ ρσ ρ σ ν 2 2 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) 1 − A α A −α α A , (3.9) ν ρ σ ν ρ σ 2 (cid:21) whereα =∂ gg−1, asdistinct fromthe algebravaluedquantityβ =g−1∂ g thatappearsasthe inhomo- µ µ µ µ geneous term in the gauge transformation of the non-Abelian curvature (in our convention). Asseenfrom(3.8)-(3.9),thegaugevariationofΩ consistsofatermwhichisexplicitlyatotaldivergence, CS and, another term ω(n) ≃ε Trα α ...α , (3.10) µ1µ2...µ2n−1 µ1 µ2 µ2n−1 whichiseffectivelytotaldivergence,andinaconcretegrouprepresentationparametrisationbecomesexplicitly totaldivergence. Thiscanbeseenbysubjecting(3.10)tovariationswithrespecttothefunctiong,andtaking into account the Lagrange multiplier term resulting from the (unitarity) constraint g†g =gg† =1I. The volume integaralof the CS density then transforms under a gauge transformationas follows. Given the appropriate asymptotic decay of the connection (and hence also the curvature), the surface integrals in (3.8)-(3.9) vanish. The only contribution to the gauge variation of the CS action/energy then comes from the integral of the density (3.10), which (in the case of Euclidean signature) for the appropriate choice of gauge group yields an integer, up to the angular volume as a multiplicative factor. All above stated properties of the Chern-Simons (CS) density hold irrespective of the signature of the space. Here, the signature is takento be Minkowskian,suchthat the CS density in the Lagrangiandoes not contribute to the energy density directly. As a consequence the energy of the soliton is gauge invariant and does not suffer the gauge transformation (3.8)-(3.9). Should a CS density be part of a static Hamiltonian (ona space ofEuclideansignature),then the energyofthe solitonwouldchangeby a multiple of aninteger. 3.1 New Chern–Simons terms in all dimensions The planto introducea completely new type ofChern-Simonsterm. The usualCSdensities Ω(n),(3.1), are CS defined with reference to the total divergence expression (1.2) of the n−th Chern-Pontryagindensity (1.1), 6 as the 2n−th component Ω(n) of the density Ω(n). Likewise, the new CS terms are defined with reference 2n to the total divergence expression (1.3) of the dimensionally reduced n−th CP density, with the dimension D of the residual space replaced formally by D¯ C(n) =∇·Ω(n,D¯). (3.11) D¯ The densities Ω(n,D¯) can be read off from Ω(n,D) given in Section 5, with the formal replacement D → D¯. The new CS term is now identified as the D¯−th component of Ω(n,D¯). The final step in this identification is to assign the value D¯ = D+2, where D is the spacelike dimension of the D+1 dimensional Minkowski space, with the new Chern-Simons term defined as Ω˜(n,D+1) d=ef Ω(n,D+2) (3.12) CS D+2 The departure of the new CS densitiess from the usual CS densities is stark, and these differ in several essentialrespectsfromtheusualonesdescribedintheprevioussubsection. Themostimportantnewfeatures in question are • The field content of the new CS systems includes Higgs fields in addition to the Yang-Mills fields, as a consequence of the dimensional reduction of gauge fields described in Section 4. It should be emphasised that the appearance of the Higgs field here is due to the imposition of symmetries in the descentmechanism,incontrastwithitspresenceinthemodels[13,14,15]supporting2+1dimensional CSvortices,wherethe Higgsfieldwasintroducedbyhandwiththe expedientofsatisfying the Derrick scaling requirement. • The usualdynamicalCS densities definedwith reference to the n−thCP density live in2n−1dimen- sional Minkowski space, i.e., only in odd dimensional spacetime. By contrast, the new CS densities defined with reference to the n−th CP densities live in D+1 dimensional Minkowski space, for all D subject to 2n−2≥D ≥2, (3.13) i.e., in both odd, as well as even dimensions. Indeed, in any given D there is an infinite tower of new CS densities characterisedby the integer n subject to (3.13). This is perhaps the most important feature of the new CS densities. • The smallest simple group consistent with the nonvanishing of the usual CS density in 2n−1 dimen- sional spacetime is SO(2n), with the gauge connection taking its values in the chiral Dirac represen- tation. By contrast, the gauge groups of the new CS densities in D +1 dimensional spacetime are fixedby the prescriptionofthe dimensionaldescentfromwhichthey result. As per the prescriptionof descent described in Section 4, the gauge group now will be SO(D+2), independently of the integer n,while the Higgs field takesits values in the orthogonalcomplement ofSO(D+2) in SO(D+3). As such, it forms an iso-(D+2)−vector multiplet. • Certain properties of the new CS densities are remarkably different for D even and D odd. – Odd D: Unlike in the usual case (3.2)-(3.3), the new CS terms are gauge invariant. The gauge fields are SO(D+2) and the Higgs are in SO(D+3). D being odd, D+3 is even and hence the fields can be parametrised with respect to the chiral (Dirac) representations of SO(D+3). An important consequence of this is the fact that now, both (electric) A and (magnetic) A fields 0 i lie in the same isotopic multiplets, in contrast to the pseudo−dyons described in the previous section. – Even D: The new CS terms now consist of a gauge variant part expressed only in terms of the gauge field, and a gauge invariant part expressed in terms of both gauge and Higgs fields. The leading, gauge variant, term differs from the corresponding usual CS terms (3.2)-(3.3) only due to the presence of a (chiral) Γ matrix infront of the Trace. The gauge and Higgs fields are D+3 7 againinSO(D+2)andinSO(D+3)respectively,butnow,D beingevenD+3isoddandhence the fields are parametrised with respect to the (chirally doubled up) full Dirac representations of SO(D+3). Hence the appearance of the chiral matrix infront of the Trace. As in the usual CS models, the regular finite energy solutions of the new CS models are not topologically stable. These solutions can be constructed numerically. Before proceeding to display some typcal examples in the Subsection following, it is in order to make a small diversion at this point to make a clarification. The new CS densities proposed are functionals of both the Yang–Mills, and, the ”isovector” Higgs field. Thus, the systems to be described below are Chern- Simons–Yang-Mills-Higgsmodelsinaveryspecificsense,namelythattheHiggsfieldisanintrinsicpartofthe new CS density. This is in contrast with Yang-Mills–Higgs-Chern-Simons or Maxwell–Higgs-Chern-Simons models in 2+1 dimensional spacetimes that have appeared ubiquitously in the literature. It is important to emphasise that the latter are entirely different from the systems introduced here, simply because the CS densities they employ are the usual ones, namely (3.2) or more often its Abelian 5 version Ω(2) =ε A F , U(1) λµν λ µν while the CS densities employed here are not simply functionals of the gauge field, but also of the (specific) Higgsfield. Toputthis inperspective,letus commentonthe wellknownAbelianCS-Higgssolitonsin2+1 dimensions constructed in [13, 14] support self-dual vortices, which happen to be unique inasfar as they are also topologically stable. (Their non-Abelian counterparts [15] are not endowed with topological stability.) The presence of the Higgs field in [13, 14, 15] enables the Derrick scaling requirement to be satisfied by virtue of the presence of the Higgs self-interaction potential. In the Abelian case in addition, it results in the topologicalstability of the vortices. If it were not for the topologicalstability, it wouldnot be necessary to have a Higgs field merely to satisfy the Derrick scaling requirement. That can be achieved instead, e.g., by introducing a negative cosmological constant and/or gravity, as was done in the 4+1 dimensional case studied in [12]. Thus, the involvement of the Higgs field in conventional (usual) Chern-Simons theories is not the only option. The reason for emphasising the optional status of the Higgs field in the usual 2+1 dimensionalChern-Simons–Higgsmodelsis,thatinthenewmodelsproposedheretheHiggsfieldisintrinsic to the definition of the (new) Chern-Simons density itself. 3.2 Examples As discussed above, the new dynamical Chern-Simons densities Ω˜(n,D+1)[A ,Φ] CS µ are characterisedby the dimensionality of the space D and the integer n specifying the dimension 2n of the bulk space from which the relevant residual system is arrived at. The case n=2 is empty, since according to (3.13) the largest spacetime in which a new CS density can be constructed is 2n−2, i.e., in 1+1 dimensional Minkowsky space which we ignore. Thecasen=3isnotempty,andaffordstwonontrivialexamples. Thelargestspacetime2n−2,inwhich a new CSdensity canbe constructedinthis caseis 3+1andthe nextin2+1Minkowskispace. These, are, repectively, Ω˜(3,3+1) = ε Tr F F Φ (3.14) CS µνρσ µν ρσ 2 Ω˜(3,2+1) = ε Trγ −2η2A F − A A +(ΦD Φ−D ΦΦ) F . (3.15) CS µνλ 5 λ µν 3 µ ν λ λ µν (cid:20) (cid:18) (cid:19) (cid:21) 5There are, of course, Abelian CS densities in all odd spacetime dimensions but these do not concern us here since in all D+1dimensionswithD=2n≥4,noregularsolitonscanbeconstructed. 8 The case n=4 affords four nontrivial examples, those in 5+1, 4+1, 3+1 and 2+1 Minkowski space. These are, repectively, Ω˜(4,5+1) = ε Tr F F F Φ (3.16) CS µνρστλ µν ρσ τλ 2 Ω˜(4,4+1) = ε TrΓ A F F −F A A + A A A A CS µνρσλ 7 λ µν ρσ µν ρ σ 5 µ ν ρ σ (cid:20) (cid:18) (cid:19) +D Φ(ΦF F +F ΦF +F F Φ) (3.17) λ µν ρσ µν mn µν mn (cid:21) 2 1 Ω˜(4,3+1) = ε Tr Φ η2F F + Φ2F F + F Φ2F CS µνρσ µν ρσ 9 µν ρσ 9 µν ρσ (cid:20) (cid:18) (cid:19) 2 − (ΦD ΦD Φ−D ΦΦD Φ+D ΦD ΦΦ)F (3.18) µ ν µ ν µ ν ρσ 9 (cid:21) 2 Ω˜(4,2+1) = ε TrΓ 6η4A F − A A CS µνλ 5 λ µν 3 µ ν (cid:26) (cid:18) (cid:19) −6η2(ΦD Φ−D ΦΦ) F λ λ µν + Φ2D ΦΦ−ΦD ΦΦ2 −2 Φ3D Φ−D ΦΦ3 F . (3.19) λ λ λ λ µν (cid:27) (cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3) ItisclearthatinanyD+1dimensionalspacetimeaninfinite towerofCSdensitiesΩ˜(n,D+1) canbe defined, CS for all positive integers n. Of these, those in even dimensional spacetimes are gauge invariant, e.g., (3.14), (3.16)and(3.18),whilethoseinodddimensionalspacetimesaregaugevariant,e.g.,(3.15),(3.17)and(3.19), thegaugevariationsinthesecasesbeinggivenformallyby(3.8)and(3.9),withgreplacedbytheappropriate gauge group here. Static soliton solutions to models whose Lagrangians consist of the above introduced types of CS terms together with Yang-Mills–Higgs (YMH) terms are currently under construction. The only constraint in the choice of the detailed models employed is the requirement that the Derrick scaling requirement be satisfied. Such solutions are constructed numerically. In contrast to the monopole solutions, they are not endowed with topological stability because the gauge group must be larger than SO(D), for which the solutions to the constituent YMH model is a stable monopole. Otherwise the CS term would vanish. 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