On Bogomolny equations in generalized gauged baby BPS Skyrme models
aa r X i v : . [ h e p - t h ] D ec On Bogomolny equations in generalized gaugedbaby BPS Skyrme models
L. T. St¸epie´n ∗ Abstract
Using the concept of strong necessary conditions (CSNC), we deriveBogomolny equations and BPS bounds for two modifications of the gaugedbaby BPS Skyrme model: the nonminimal coupling to the gauge field andk-deformed model. In particular, we study, how the Bogomolny equationsand the equation for the potential, reflect these two modifications. Inboth examples, the CSNC method shows to be a very useful tool.
BPS models i.e., field theories which allow for solitonic solutions and simulta-neously admit a reduction of the full second order static equation of motion toa set of first order equations (Bogomolny or BPS equations [1]-[5]), play a no-table role in current physics. The appearance of Bogomolny equations not onlyleads to exact solutions, which significantly enlarge our understanding of consid-ered non-linear models, but also guarantee existence of a topological Bogomolnybound, which results in a topological stability of solitons carrying a non-trivialvalue of the corresponding topological charge. Therefore, it is of high impor-tance to search for models with the BPS property. In fact, between few knownrather restricted methods (completing to a square [1]-[6], the first order formal-ism [7], on-shell method [8]), there exists a completely general method, whichallows for a systematic derivation (if possible) BPS equations. This method,referred as the concept of strong necessary conditions (CSNC), was originallyproposed and analyzed in [9]-[17], has been very recently further developed byAdam and Santamaria [18], who proposed so called first order Euler-Lagrange (FOEL) formalism. ∗ The Pedagogical University of Cracow, ul. Podchorazych 2, 30-084 Krakow, Poland,e-mail: [email protected]
1n the present work we want to apply the CSNC method to a generalized(gauged) baby BPS Skyrme model. The baby BPS Skyrme model [19]-[23] isa limit of the full baby Skyrme theory [24], where the usual 2+1 dimensionalDirichlet term L = ∂ µ ~φ · ∂ µ ~φ disappears. (The full baby Skyrme model [28], isa planar analogon of the Skyrme model, [25], [26], [27], [28]). Here the iso-vectorfield ~φ is an element of two-dimensional target space S . The resulting Lagrangedensity consists of two remaining parts: the 2 + 1 dimensional Skyrme term L and a non-derivative term i.e., a potential V L BP S bSk = α L + α L ≡ − α (cid:16) ∂ µ ~φ × ∂ ν ~φ (cid:17) − α V ( ~n · ~φ ) (1.1)This model possesses a BPS equation which is saturated by infinitely manytopological solitons (baby Skyrmions) carrying arbitrary value of the pertinenttopological charge Q ∈ π ( S ) [22], [23], [29]. In fact, this equation is a twodimensional analogous of the famous BPS equation for the BPS Skyrme model[30] (see also [31]-[33]), which plays an important role in a possible solution ofthe too large binding energy problem in Skyrme model [34]. The next obviousstep in analysis of the baby BPS Skyrme model was the minimal coupling of itto the Maxwell U (1) gauge field, and study of magnetic properties of resultinggauged baby BPS Skyrme model [35], [37] L g BP S bSk = − α (cid:16) D µ ~φ × D ν ~φ (cid:17) − α V ( ~n · ~φ ) − α m F µν (1.2)where the usual derivatives are change into the covariant ones D µ ~φ = ∂ µ ~φ + A µ ~n × ~φ as there is unbroken U (1) subgroup of the SO (3) iso-rotations of ~φ .It has been shows that this model also revels the BPS property where a BPSequation requires a non-trivial superpotential W which is related to the originalpotential by a target space equation λ W ′ + g λ W = 2 µ V (1.3)where the prime denotes differentiation w.r.t. φ . (Here α = λ , α = µ , α m = g .) In a consequence, the gauging of the model provided some restriction onthe potential as far as one wants to keep the BPS property. For example, theBPS equation cannot be satisfied, if a two vacua potential is assumed. This isin a contrast to the non-gauged case, where all (reasonable) potentials lead tosaturated solutions of the BPS equation i.e., baby BPS Skyrmions.Here we want to further analyze, how the existence and the form of theBPS equation (and superpotential) is affected by 1) non-minimal coupling tothe gauge field and/or by 2) k-extension of the model i..e, if the baby Skyrme 4-derivate term is replaced by a function of it. Both modifications are motivated by2ffective nature of the baby Skyrme model. In such a case, non-standard kineticpart (k-model generalization) as well as addition of a ”dielectric” function to thegauge field part are typically accepted generalizations - see for example, con-dense matter [38] or cosmological applications [39]. Obviously, the lagrangianshould be gauge invariant, however in the case of gauged models investigated inthis paper, we want also to investigate (as in [35]), whether any terms analogicalto Proca terms ([40], [41]), will appear in the expressions for the potentials. The static energy functional of the non-minimally extended model is E = 12 E Z d x (cid:20) λ (cid:16) D ~φ × D ~φ (cid:17) + 2 µ V ( ~φ · ~n ) + 1 g U ( ~φ · ~n ) B (cid:21) (2.1)= 12 E Z d x (cid:20) λ Q + 2 µ V ( φ ) + 1 g U ( φ ) B (cid:21) (2.2)where B ≡ F = ∂ A − ∂ A and Q ≡ ~φ · D ~φ × D ~φ = q + ǫ ij A i ∂ j ( ~n · ~φ ) , (2.3)where q ≡ ~φ · ∂ ~φ × ∂ ~φ (2.4)is the topological density multiplied by 4. Now we apply the CSNC method. At the beginning, for simplicity reasons,we compute the Bogomolny equations in an axially symmetric ansatz, for anydielectric coupling function U : ~φ ( r, φ ) = sin f ( r ) cos kφ sin f ( r ) sin kφ cos f ( r ) , A = 0 , A r = ka ( r ) , A φ = ka ( r ) (2.5)Here r, φ are polar coordinates and k ∈ Z . The magnetic field is B = ka ′ ( r ) r .Then the energy functional reads (we have assumed (for some generality, be-cause we want to investigate, whether any terms analogical to Proca terms, will3ppear) that V = V ( a , h )):˜ H = 2 πE Z dy (cid:20) λ k (1 + a ) h y + µ V ( a , h ) + 12 g U ( h ) k a ,y + F h h y + F a a ,y (cid:21) , (2.6)where we introduced a new base space coordinate y = r as well as a newtarget spaces variable h = (1 − cos f ). In this case, the CSNC method isequivalent to adding only the divergence: dF ( a ,h ) dy (so we have here only so-called, divergent invariant), to the density of the energy functional. Hence, wehave: ∂ ˜ H ∂h : µ V ,h + 12 g k a ,y U ,h + F ,hh h ,y + F ,a h a ,y = 0 , (2.7) ∂ ˜ H ∂a : µ V ,a + 4 λ k (1 + a ) h ,y + F ,a a a ,y + F ,ha h ,y = 0 , (2.8) ∂ ˜ H ∂h ,y : 4 λ k (1 + a ) h ,y + F ,h = 0 , (2.9) ∂ ˜ H ∂a ,y : k g U a ,y + F ,a = 0 (2.10)By using the relations h ,y = − F ,h k λ (1 + a ) , (2.11) a ,y = − g F ,a k U ( h ) , (2.12)obtained from (2.9) - (2.10), we eliminate the derivatives of h and a from(2.7) - (2.8). Hence, we have a system for U, V, F . In the case: U = 1, thesolution is V ( a , h ) = 4 g λ (1 + a ) ( F ,a ) + ( F ′ ,h ) + 8 c µ k λ (1 + a ) µ k λ (1 + a ) , (2.13)where F = F ( a , h ) ∈ C . Of course, the dependance of the potential V on a , disappears, when F ( a , h ) = c (1 + a ) W ( h ) , c = const , so we have theequation with the superpotential W : V = 18 µ k λ [4 g λ c W ( h ) + c W ′ ( h )] (2.14)If U = U ( h ), we obtain two sets of the solutions:4 ( a , h ) = F ( h ) , (2.15) U = U ( h ) ∈ C , (2.16) V ( a, h ) = ( F ′ ,h ) µ k λ (1 + a ) (2.17)and F ( a , h ) = f ( h ) a + f ( h ) , U ( h ) = − f ( h ) g λ ( f ′ ( h )) − f ( h ) k λ µ − c g λ , (2.18) V ( a , h ) = ( f ′ ( h ) − f ′ ( h )) (cid:18) f ′ ( h ) − f ′ ( h )2(1+ a ) − f ′ ( h )1+ a (cid:19) µ k λ + f ( h ) , (2.19)where f k = f k ( h ) ∈ C ( k = 1 , ,
3) are arbitrary functions of h . By elimi-nating of f from V , we get: V ( a , h ) = ( f ′ ( h ) − f ′ ( h )) (cid:18) f ′ ( h ) − f ′ ( h )2(1+ a ) − f ′ ( h )1+ a (cid:19) µ k λ + g f µ k U + 18 µ k λ ( f ′ ) − c g µ k . (2.20)Obviously, if f = f , then in the case of (2.20), the superpotential equationhas the form (when W ( h ) ≡ f ): V ( h ) = g W µ k U + 18 µ k λ ( W ′ ) − c g µ k , c = const, (2.21)which is some generalization of (2.14).Hence, the equations (2.11) - (2.12), are the Bogomolny decomposition forthis case. Now we apply the stereographic projection, for the functional (2.1) ~φ = (cid:20) ω + ω ∗ ωω ∗ , − i · ( ω − ω ∗ )1 + ωω ∗ , − ωω ∗ ωω ∗ (cid:21) , i.e. ω = φ + iφ φ , (2.22)5here ω = ω ( x, y ) ∈ C , x, y ∈ R .Hence, after some rescaling, ( cf [35]) E = Z H d x = Z (cid:26) λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] (1 + ωω ∗ ) + λ U B + V (cid:27) d x, (2.23)where B is magnetic field and B ≡ F = ∂ A − ∂ A . We have assumedhere that V = V ( ω, ω ∗ , A k ) , ( k = 1 , U = U ( ω, ω ∗ ).We make the following gauge transformation of H , on the sum of the invari-ants P n =1 I n , [35] H −→ ˜ H = H + X n =1 I n = 4 λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] (1 + ωω ∗ ) + λ U B + V + λ { G ′ [( iε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] + G B } + X l =1 D l G l +1 , (2.24)where I is given by: I = λ { G ′ [( iε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] + G B } , λ = const ., G = G ( ωω ∗ ) ∈ R is some arbitrary function differentiable at leasttwice. G ′ denotes the derivative of the function G with respect to its argument: ωω ∗ , and I = D x G ( ω, ω ∗ , A , A ) , I = D y G ( ω, ω ∗ , A , A ). Next: x = x, x = y, D x ≡ ddx , D y ≡ ddy , and G l +1 = G l +1 ( ω, ω ∗ , A , A ), ( l = 1 , cf. [35]):˜ H ,ω : − λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] (1 + ωω ∗ ) ω ∗ − N ε pr A p ω ∗ ,r + U ,ω B + V ,ω + λ (cid:26) G ′′ ω ∗ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] − G ′ ε pr A p ω ∗ ,r + G ′ ω ∗ B (cid:27) + X l D l G l +1 ,ω = 0 , (2.25)6 H ,ω ∗ : − λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] (1 + ωω ∗ ) ω − N ε pr A p ω ,r + U ,ω ∗ B + V ,ω ∗ + λ (cid:26) G ′′ ω [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] − G ′ ε pr A p ω ,r + G ′ ωB (cid:27) + X l D l G l +1 ,ω ∗ = 0 , (2.26)˜ H ,A s : − N ( ε sr ( ωω ∗ ) ,r ) + V ,A s − λ G ′ ( ε sr ( ωω ∗ ) ,r ) + X l D l G l +1 ,A s = 0 , (2.27)˜ H ,ω ,s : N ( iε sk ω ∗ ,k − ε ps A p ω ∗ ) + λ G ′ ( iε sk ω ∗ ,k − ε ps A p ω ∗ ) + G s +1 ,ω = 0 , (2.28)˜ H ,ω ∗ ,s : N ( iε js ω ,j − ε ps A p ω ) + λ G ′ ( iε js ω ,j − ε ps A p ω ) + G s +1 ,ω ∗ = 0 , (2.29)˜ H ,A s,r : 2 λ U Bε rs + λ G ε rs + G r +1 ,A s = 0 , (2.30)where N = λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ](1+ ωω ∗ ) and G ′ , G ′′ denote the derivativesof the function G with respect to its argument: ωω ∗ .Now, we consider ω, ω ∗ , A i , ( i = 1 , , G k , ( k = 1 , , cf. [35]): G ′ = − λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] λ (1 + ωω ∗ ) , (2.31) B = − λ U ( λ G + G ,A ) , (2.32) G ,A = − G ,A , G = c A , G = − c A , c = const, (2.33)the equations (2.28) - (2.30) become the tautologies and the candidate forBogomolny decomposition is ( cf. [35]):7 λ [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] λ (1 + ωω ∗ ) = − G ′ , λ U B + λ G + c = 0 . (2.34)Now, the next step is checking, when the equations (2.25) - (2.27) are sat-isfied, if (2.34) hold. Thus, we insert (2.33) and (2.34), into (2.25) - (2.27).Hence, we get some system of partial differential equations for V . It has turnedout that V = V ( ω, ω ∗ ), and the solution of this system, for U = U ( ωω ∗ ), (cf.[35]), is: V ( ω, ω ∗ ) = ( c + λ G ( ωω ∗ )) λ U ( ωω ∗ ) + λ λ Z ((1 + ωω ∗ ) G ′ ω ∗ (2 G ′ + G ′′ (1 + ωω ∗ ))) dω − λ λ Z Z { [(2 + 8 ωω ∗ )( G ′ ) + ((1 + 9 ωω ∗ ) G ′′ + G ′′′ ωω ∗ (1 + ωω ∗ ))(1 + ωω ∗ ) G ′ + ωω ∗ (1 + ωω ∗ ) ( G ′′ ) (1 + ωω ∗ ) dω ] + (1 + ωω ∗ ) G ′ ω (2 G ′ + G ′′ (1 + ωω ∗ )) } dω ∗ . (2.35) Another generalization of the gauged baby BPS Skyrme model is given by thefollowing energy integral E = 12 E Z d x (cid:20) λ F ( Q ) + 2 µ V ( φ ) + 1 g B (cid:21) (3.1)where we have again the magnetic field B ≡ F = ∂ A − ∂ A and thestandard Q derivative part is replaced by an arbitrary, positive definite function G . We call it as k-generalized gauged baby BPS Skyrme model. In fact,k-deformed field theories have been extensively investigated especially in thecontext of topological solitons: kinks in 1+1 dimensions [42]-[44], vortices [45]and monopoles [46]. The next step is to consider the non-gauged version of the k-generalized modelthat is the k-generalized baby BPS Skyrme model E = 12 E Z d x (cid:2) λ G ( q ) + 2 µ V ( φ ) (cid:3) (3.2)8here again we restrict ourselves to the monomial function only G ( q ) = ( q ) n .After making stereographic projection, we have E = Z H d x = Z (cid:20)(cid:18) λ ( ε jk i ω ,j ω ∗ ,k ) (1 + ωω ∗ ) (cid:19) n + V ( ω, ω ∗ ) (cid:21) d x (3.3)We make the gauge transformation and we have˜ H = (cid:18) λ ( ε jk i ω ,j ω ∗ ,k ) (1 + ωω ∗ ) (cid:19) n + V ( ω, ω ∗ ) + iG ( ω, ω ∗ ) ε jk ω ,j ω ∗ ,k + X l =1 D l G l +1 ( ω, ω ∗ )(3.4)The dual-equations are:˜ H ,ω : − n (4 λ ) n ( ε jk i ω ,j ω ∗ ,k ) n (1 + ωω ∗ ) n +1 ω ∗ + V ,ω + iG ,ω ε jk ω ,j ω ∗ ,k + X l =1 D l G l +1 ,ω = 0 , (3.5)˜ H ,ω ∗ : − n (4 λ ) n ( ε jk i ω ,j ω ∗ ,k ) n (1 + ωω ∗ ) n +1 ω + V ,ω ∗ + iG ,ω ∗ ε jk ω ,j ω ∗ ,k + X l =1 D l G l +1 ,ω ∗ = 0 , (3.6)˜ H ,ω ,r : 2 n (4 λ ) n ( ε jk i ω ,j ω ∗ ,k ) n − (1 + ωω ∗ ) n ε rm iω ∗ ,m + iG ε rm ω ∗ ,m + G r +1 ,ω = 0 , (3.7)˜ H ,ω ∗ ,r : 2 n (4 λ ) n ( ε jk i ω ,j ω ∗ ,k ) n − (1 + ωω ∗ ) n ε mr iω ,m + iG ε mr ω ,m + G r +1 ,ω ∗ = 0 . (3.8)The equations (3.7) - (3.8) become the tautologies, if we put:( ε jk i ω ,j ω ∗ ,k ) n − = − (1 + ωω ∗ ) n G n (4 λ ) n , (3.9) G r +1 = const, r = 1 , ω ,k , ω ∗ ,k ( k = 1 , V ( ω, ω ∗ ).We find the solution of this system, for G = G ( ωω ∗ ):9 ( ω, ω ∗ ) = Z (cid:18) (1 + ωω ∗ ) − (cid:18) ω ∗ (cid:18) − n +1) n (cid:18) − λ ( ωω ∗ + 1) n G n (4 λ ) n (cid:19) n n − ( ωω ∗ + 1) − n + (cid:18) − λ ( ωω ∗ + 1) n G n (4 λ ) n (cid:19) n − G ′ (1 + ωω ∗ ) (cid:19)(cid:19)(cid:19) dω + Z n − ωω ∗ + 1) (cid:26) ( ωω ∗ + 1) (cid:20) Z G ( ωω ∗ + 1) ( − G (2 ωω ∗ + 1)+ ωω ∗ ( ωω ∗ + 1) G ′ ) n (cid:18) λ (1 + ωω ∗ ) − n (cid:18) − ( ωω ∗ + 1) n G n (4 λ ) n (cid:19) n − (cid:19) n +2 n +1) n (cid:18) λ (1 + ωω ∗ ) − n (cid:18) − ( ωω ∗ + 1) n G ( ωω ∗ )2 n (4 λ ) n (cid:19) n − (cid:19) n G ′ + (cid:18) − ( ωω ∗ + 1) n G n (4 λ ) n (cid:19) n − ((((6 n − ωω ∗ + 2 n − G ′ ( ωω ∗ )+2 ωω ∗ (1 + ωω ∗ )( n −
12 ) G ′′ ) G + ωω ∗ (1 + ωω ∗ ) G ′ )(1 + ωω ∗ )) dω (cid:21) − (cid:18) n − (cid:19)(cid:18) − n +1 n (cid:18) − ( ωω ∗ + 1) G n (cid:19) n n − + G ′ (cid:18) − ( ωω ∗ + 1) n G n (4 λ ) n (cid:19) n − (1 + ωω ∗ ) (cid:19) ω (cid:27) dω ∗ (3.11)Hence, the Bogomolny equation for the k-deformed ungauged BPS babySkyrme model, has the form: ε mn i ω ,m ω ∗ ,n = (cid:18) − (1 + ωω ∗ ) n G ( ωω ∗ )2 n (4 λ ) n (cid:19) n − , (3.12)if the potential has the form (3.25). Some similar result was obtained in [36]. Now we apply the stereographic projection, for the functional (2.1) ~φ = (cid:20) ω + ω ∗ ωω ∗ , − i · ( ω − ω ∗ )1 + ωω ∗ , − ωω ∗ ωω ∗ (cid:21) , i.e. ω = φ + iφ φ , (3.13)where ω = ω ( x, y ) ∈ C , and x, y ∈ R .We make the gauge transformation of H = (cid:20) λ iε jk ω ,j ω ∗ ,k − ε pr A p ( ωω ∗ ) ,r ) (1+ ωω ∗ ) (cid:21) n + V + λ U B , on the sum of the invariants, and we have:10 = (cid:20) λ ( iε jk ω ,j ω ∗ ,k − ε pr A p ( ωω ∗ ) ,r ) (1 + ωω ∗ ) (cid:21) n + V + λ U B + (3.14) λ { G ′ [( iε jk ω ,j ω ∗ ,k − ε pr A p ( ωω ∗ ) ,r )] + G B } + X l =1 D l G l +1 (3.15)where obviously, B ≡ F = ∂ A − ∂ A and j, k, l, p, r = 1 ,
2. We haveassumed here again that V = V ( ω, ω ∗ , A k ), k = 1 , U = U ( ω, ω ∗ )).So, the invariants are again: I = λ { G ′ [( iε ij ω ,i ω ∗ ,j ) − ε pr A p · ( ωω ∗ ) ,r ]+ G B } ,( λ = const ., G = G ( ωω ∗ ) ∈ R is some arbitrary function differentiable atleast twice, G ′ denotes the derivative of the function G with respect to itsargument: ωω ∗ ), I = D x G ( ω, ω ∗ , A , A ) , I = D y G ( ω, ω ∗ , A , A ). Next: x = x, x = y, D x ≡ ddx , D y ≡ ddy , and G l +1 = G l +1 ( ω, ω ∗ , A , A ), ( l = 1 , H ,ω = 14 n n ( λ ) n − N n − (1 + ωω ∗ ) n (cid:20) − λ N ε pr A p ω ∗ ,r (1 + ωω ∗ ) − λ N ω ∗ (1 + ωω ∗ ) (cid:21) (1 + ωω ∗ ) + V ,ω + λ U ,ω B + λ { G ′′ ω ∗ [ iε jk ω ,j ω ∗ ,k − ε pr A p ( ωω ∗ ) ,r ] − (3.16) G ′ ε pr A p ω ∗ ,r + G ′ ω ∗ B } + X l =1 D l G l +1 ,ω = 0 , ˜ H ,ω ∗ = 14 n n ( λ ) n − N n − (1 + ωω ∗ ) n (cid:20) − λ N ε pr A p ω ,r (1 + ωω ∗ ) − λ N ω (1 + ωω ∗ ) (cid:21) (1 + ωω ∗ ) + V ,ω ∗ + λ U ,ω ∗ B + λ { G ′′ ω [( iε jk ω ,j ω ∗ ,k − ε pr A p ( ωω ∗ ) ,r )] − (3.17) G ′ ε pr A p ω ,r + G ′ ωB } + X l =1 D l G l +1 ,ω ∗ = 0 , ˜ H ,A s : − n (4) n λ n N n − (1 + ωω ∗ ) n ( ε sr ( ωω ∗ ) ,r ) + V ,A s − λ G ′ ( ε sr ( ωω ∗ ) ,r ) + X l =1 D l G l +1 ,A s = 0 , (3.18)˜ H ,ω s = 2 n (4) n λ n N n − (1 + ωω ∗ ) n ( iε sk ω ∗ ,k − ε ps A p ω ∗ )+ (3.19) λ G ′ ( iε sk ω ∗ ,k − ε ps A p ω ∗ ) + G s +1 ,ω = 0 , H ,ω ∗ s = 2 n (4) n λ n N n − (1 + ωω ∗ ) n ( iε js ω ,j − ε ps A p ω )+ (3.20) λ G ′ ( iε js ω ,j − ε ps A p ω ) + G s +1 ,ω ∗ = 0 , ˜ H ,A s,r : 2 λ U Bε rs + G ε rs + G r +1 ,A s = 0 (3.21)where N = [ i · ( ε jk ω ,j ω ∗ ,k ) − ε pr A p · ( ωω ∗ ) ,r ] and G ′ , G ′′ denote the derivativesof the function G with respect to its argument: ωω ∗ .Now, in order to make the system self-consistent, we put G = c A , G = − c A , (3.22) iε jk ω ,j ω ∗ ,k − ε pr A p ( ωω ∗ ) ,r = (cid:18) − λ G ′ n (4 λ ) n (1 + ω + ω ∗ ) n (cid:19) n − , (3.23) B = − λ G + c λ U (3.24)and we insert these relations, into (3.16) - (3.18). Hence, we get the equationsfor V and U . We get the formula for V (if G = G ( ωω ∗ )):12 ( ω, ω ∗ ) = 1 λ U (cid:26) Z n − ωω ∗ + 1) (cid:26) ( ωω ∗ + 1) (cid:20) Z G ′ ( ωω ∗ + 1) ( − G ′ (2 ωω ∗ + 1) + ωω ∗ ( ωω ∗ + 1) G ′′ ) n (cid:18) λ ( ωω ∗ + 1) − n (cid:18) − λ ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19) n − (cid:19) n +2 n +1) n (cid:18) λ (1 + ωω ∗ ) − n (cid:18) − λ ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19) n − (cid:19) n G ′ +2 λ (cid:18) − λ ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19) n − (cid:18)(cid:18)(cid:18)(cid:18) n − (cid:19) ωω ∗ + n − (cid:19) G ′′ + ωω ∗ (1 + ωω ∗ ) (cid:18) n − (cid:19) G ′′′ (cid:19) G ′ +12 ωω ∗ (1 + ωω ∗ )( G ′′ ) (cid:19) (1 + ωω ∗ )) dω (cid:21) − (cid:18) − n +1 n (cid:18) λ (1 + ωω ∗ ) − n (cid:18) − ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19)(cid:19) n n − + λ G ′′ (cid:18) − − λ ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19) n − (1 + ωω ∗ ) (cid:19) ω (cid:18) n − (cid:19)(cid:27) dω ∗ λ U +4 λ U Z (cid:20)(cid:18) − n +1) (cid:18)(cid:18) − λ (1 + ωω ∗ ) − n λ ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19) n − (cid:19) n n + G ′′ (cid:18) − λ ( ωω ∗ + 1) n G ′ n (4 λ ) n (cid:19) n − λ (1 + ωω ∗ ) (cid:19) ω ∗ (1 + ωω ∗ ) − (cid:21) dω ++4 c λ U + ( λ G + c ) (cid:27) . (3.25) In the present paper BPS equations for some generalization of the gauged babyBPS Skyrme model, have been found. This have been performed by applyingthe concept of strong necessary conditions (CSNC).In the case of the non-minimally coupled gauged baby BPS Skyrme model,we found the the new BPS equation is modified by a coupling between themagnetic field B and the dielectric function U , in both cases: for an axiallysymmetric ansatz and for the energy functional expressed by stereographic vari-ables. In the case of the ansatz, the term (in BPS equation), which emerges due13o the gauge coupling (proportional to g ) is modified by 1 /U . This modifica-tion can lead to some new restriction on possible potentials V (and the dielectricfunctions U ), for which the BPS equation has nontrivial topological solutions. U does not depend on the field a (where A φ = ka ( r )), but V depends on it inthe general case. Hence, we have an analogon to Proca theory. However, in thecase of (2.20), if the functions f = f , then V = V ( h ). Another modificationsof the Bogomolny decomposition and the formula for the potential, can be ob-served, if one compares these results (for the case with stereographic variables),with the results obtained for gauged restricted baby BPS Skyrme model withminimal coupling, [35].For k-deformation (given by polynomial function G ), both the Bogomolnyequation for the matter (Skyrme) field, as well as the superpotential equation,are modified.In all these cases of gauged baby BPS Skyrme model, investigated in this pa-per, expressed in stereographic variables, the potential (for which Bogomolnydecomposition exists), does not depend on the gauge field A k ( k = 1 , V , for which solutions of the Bogomolny equations exist, itwould be desirable to study it in detail.Another direction is to investigate a relation between the CSNC constructionand supersymmetry (which always is hidden behind Bogomolny equations [15],[47]).In any case, the CSNC framework proven to be a powerful and strightforwardmethod for derivation of the Bogomolny equations. The author thanks to Dr. Hab. A. Wereszczynski for interesting discussions.
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