Soliton surfaces associated with CP^{N-1} sigma models
SSoliton surfaces associated with C P N − sigma models A M Grundland and S Post Centre de Recherches Math´ematiques. Universit´e de Montr´eal. Montr´ealCP6128 (QC) H3C 3J7, Canada Department of Mathematics and Computer Sciences, Universit´e du Quebec,Trois-Rivi`eres. CP500 (QC)G9A 5H7, CanadaE-mail: [email protected], [email protected]
Abstract.
Soliton surfaces associated with the C P N − sigma model areconstructed using the Generalized Weierstrass and the Fokas-Gel’fand formulasfor immersion of 2D surfaces in Lie algebras. The considered surfaces are definedusing continuous deformations of the zero-curvature representation of the modeland its associated linear spectral problem. The theoretical framework is discussedin detail and several new examples of such surfaces are presented.
1. Introduction
Integrable models and their continuous deformations under various types of dynamicshave produced much interest and activity in several areas of mathematics, physics andbiology. Furthermore soliton surfaces associated with integrable models, and with the C P N − sigma model in particular, have been shown to play an essential role in manyproblems with physical applications (see e.g. [8, 15, 25, 28, 29]). The possibility ofusing a linear spectral problem (LSP) to represent a moving frame on the surfacehas yielded many new results concerning the intrinsic geometric properties of suchsurfaces (see e.g. [1, 2, 3, 4, 21, 22, 24]). In this vein, it has recently proved fruitfulto extend such characterization of soliton surfaces via their immersion function inLie algebras. The construction of such surfaces, related to the completely integrable C P N − sigma model, has been accomplished by representing the equations of motionfor the model as a conservation law which in turn provides a closed differential for thesurface. This is the so called generalized Weierstrass formula for immersion (GWFI)[14, 19, 20, 23]. Another method for the construction of such soliton surfaces makes useof the conformal invariance of the zero-curvature (or ZS-AKNS) representation of themodel with respect to the spectral parameter. This immersion function is known asthe Sym-Tafel formula for immersion [30, 31] and a remarkable result, to be describedlater in this paper, is that the Sym-Tafel formula and the GWFI coincide in the caseof finite action solutions of the C P N − sigma model.More generally, the LSP for the integrable model and its symmetries has beenemployed by Fokas and Gel’fand [11] and later with Finkel and Liu [12], to constructfamilies of soliton surfaces. Recently, the authors have reformulated and extendedthe Fokas-Gel’fand immersion formula using the formalism of generalized vector fieldsand their action on jet space and have given the necessary and sufficient conditionsfor the existence and explicit integration of soliton surfaces in terms of the symmetrycriterion for vector fields [16, 17, 18]. a r X i v : . [ m a t h - ph ] D ec oliton surfaces associated with C P N − sigma models C P N − sigma model.The basics of the model are reviewed in section 2. Section 3 contains an exposition ofthe GWFI and its relation to the Sym-Tafel formula. The Fokas-Gelfand formula forimmersion, in the formalism of generalized vector fields, and its associated surfacesare presented in section 4. Section 5 gives new examples of the application of suchimmersion formulas to the case of the C P sigma model.
2. The C P N − sigma model: Definition and linear spectral problem Over the past decades, there has been significant progress in the study of generalproperties of the C P N − sigma model and the techniques for finding associated 2D-soliton surfaces immersed in multidimensional space. The most fruitful approach tothis subject has been achieved through the descriptions of the model in terms of rank-one Hermitian projectors. A matrix P is said to be a rank-one Hermitian projectorif P = P, tr ( P ) = 1 , P † = P. (1)The target space of the projector P is determined by a complex line in C N , i.e. bythe one-dimensional vector function f ( ξ + , ξ − ) given by P = f ⊗ f † f † · f , (2)where f is a mapping C ⊇ Ω (cid:51) ξ ± = x ± iy (cid:55)→ f = ( f , f , . . . f N − ) ∈ C N (cid:114) { } . In fact, the formula (2) gives an isomorphism between the equivalence classes of C P N − and the set of rank-one, Hermitian projectors. The projector formalismautomatically encodes the scaling invariance of the vectors and ensures that the mapswill be free from removable singularities which could occur in the unnormalized vectorfields f [13]. Furthermore, the equations of motion and other properties of the modeltake a compact form when written in the projector formalism. C P N − sigma model in the projector formalism In terms of a rank-one Hermitian projector P , the Lagrangian density is given by L ( P ) = tr ( ∂ + P ∂ − P ) (3)and the action functional is S ( P ) = (cid:90) C L ( P ) dξ + dξ − . (4)Here the holomorphic and anti-holomorphic derivatives are defined as ∂ + = ∂∂ξ + = 12 (cid:18) ∂∂x − i ∂∂y (cid:19) , ∂ − = ∂∂ξ − = 12 (cid:18) ∂∂x + i ∂∂y (cid:19) . The Euler-Lagrange (E-L) equation for the model is given by[ ∂ + ∂ − P, P ] = ∅ , (5)which is equivalent to a conservative form ∂ + [ ∂ − P, P ] + ∂ − [ ∂ + P, P ] = ∅ . (6) oliton surfaces associated with C P N − sigma models ∅ is the zero matrix. Another important physical quantity of the model is thetopological charge density q ( P ) = tr ( P ∂ + P ∂ − P − P ∂ − P ∂ + P ) , (7)so called because its integral over the plane, the topological charge, is a total divergenceand hence Q ( P ) = 1 π (cid:90) C q ( P ) dξ + dξ − (8)is an integer [10]. As was first proven by [9, 10], see also [33], any solution of the C P N − sigmamodel, defined on the extended complex plane, with finite action can be written as araising operator acting on a holomorphic vector (or a lowering operator acting on ananti-holomorphic vector). In [27], the authors presented this proof in the projectorformalism. In this context, the raising and lowering operators are given by [13]Π ± ( P ) ≡ (cid:40) ∂ ± P P ∂ ∓ Ptr ( ∂ ± P P ∂ ∓ P ) ∂ ± P P ∂ ∓ P (cid:54) = ∅ , ∅ ∂ ± P P ∂ ∓ P = ∅ , (9)and have the property that a rank-one projector P projects onto an equivalence classin C P N − with a holomorphic representative if and only ifΠ − P = ∅ . (10)Analogously, Π + P = ∅ if and only if P maps on to an equivalence class in C P N − withan anti-holomorphic representative. The operators Π ± are called raising and loweringoperators because they are contracting operators which map between solutions of theE-L equation [13]. They satisfy the following properties. Suppose that a rank-oneHermitian projector P is a solution of the E-L equation with finite action, then:(i) Π ± P is either ∅ or is itself a rank-one Hermitian projector solution of the E-Lequation with finite action.(ii) The raising and lowering operators are mutual inverses for solutions of the E-Lequation, i.e. Π ± Π ∓ P = P, whenever Π ∓ P (cid:54) = ∅ . (11)(iii) The raising and lowering operators are mutually orthogonalΠ j + P Π k + P = δ jk Π k + P, j, k ∈ Z , (12)where the quantities Π j ± P are defined inductively byΠ P = 0 , Π − P = Π − ( P ) , Π n +1+ P = Π + (Π n + P ) , Π n − P = Π − (Π n + P ) . (iv) The raising and lowering operators are contracting operators on P . That is, thereexist natural numbers j, k ∈ N with 0 ≤ j + k ≤ N − j +1 − P = ∅ , Π k +1+ P = ∅ . (v) The images of the finite set of rank-one projectors { Π j − P, . . . Π − P, P, Π + P, . . . Π k + P } (13)form an orthogonal basis for the subspace C j + k +1 ⊂ C N . oliton surfaces associated with C P N − sigma models j + k + 1 < N , then the C P N − model can be embedded in a C P j + k model. Thus, for the remainder of the paper, it is assumed that j + k + 1 = N and so Π j +1 − P = ∅ , Π N − j + P = ∅ , which in turn implies that Π j − P is a holomorphic and Π N − − j + P and anti-holomorphicprojector. It is well known that the C P N − sigma model admits a linear spectral problem [32, 33].Defining matrix functions U ≡
21 + λ [ ∂ + P, P ] , U ≡ − λ [ ∂ − P, P ] , (14)the Euler-Lagrange (E-L) equations are equivalent to∆ (cid:48) ≡ [ ∂ + ∂ − P, P ] = ∂ − U − ∂ + U + [ U , U ] = 0 , (15)which are exactly the compatibility conditions for the LSP of the form ∂ + Φ = U Φ , ∂ − Φ = U Φ , (16)where λ is a complex spectral parameter.In is worth noting that, for the C P N − sigma model defined on Euclidean space,the wave function Φ can be explicitly integrated for an arbitrary solution of the E-L equation with finite action. The wave function Φ is given in terms of the set ofrank-one Hermitian projectors (13) by [13, 33]Φ = I + 4 λ (1 − λ ) k (cid:88) j =1 Π j − P − − λ P , Π k +1 − P = ∅ , (17)where I is the identity matrix on C N . If the spectral parameter λ is purely imaginary,Φ is an element of the group SU ( N ) and the inverse of the wave function is given byΦ − = I − λ (1 + λ ) k (cid:88) j =1 Π j − P −
21 + λ P . (18)The exact form of the wave function Φ will be important later in the construction ofthe soliton surfaces.
3. Generalized Weierstrass representation of surfaces associated with C P N − sigma models From the form of the E-L equation (6), it is possible to construct a closed, skew-Hermitian differential dF = − i ([ ∂ + P, P ] dξ + − [ ∂ − P, P ] dξ − ) . (19)Integrating the differential along a curve γ ⊂ C gives a surface immersed in the su ( N )algebra associated with P, a solution of the C P N − sigma model. These results arepresent in the following theorem [19]. oliton surfaces associated with C P N − sigma models Theorem 1 If P is a rank-1 Hermitian projector solution of the Euler-Lagrangeequations then there exists a surface F whose immersion is defined by F = − i (cid:90) γ [ ∂ + P, P ] dξ + − [ ∂ − P, P ] dξ − , F † = − F ∈ su ( N ) . (20)The immersion function defined as in (20) is called the Generalized Weierstrass formulafor immersion (GWFI). Alternatively, the surface F can be defined by its tangentvectors dFdξ + = − i [ ∂ + P, P ] dFdξ − = i [ ∂ − P, P ] , (21)whose compatibility conditions is exactly the E-L matrix equation (6). The Killingform on su ( N ) , ( X, Y ) = − tr ( X · Y ) , X, Y ∈ su ( N ) (22)gives an inner-product on the tangent vectors to a 2D-surface immersed in su ( N ).With this metric, it is possible to show that the surfaces defined as in (20) areconformally parameterized [20]. The following theorem was proven in [27]. Theorem 2
For a finite action solution, P , of the Euler-Lagrange equations (15),the surface F defined by (20) is conformally parameterized and the conformal factorin the first fundamental form is proportional to the Lagrangian density I ( F ) = 12 L ( P ) dξ + dξ − . (23) Thus, the area of the surface is given by the action functional of the model and, inparticular, the surface will have finite area.
Other geometric quantities of the surface can be written in terms of the physicalproperties of the model. In complex coordinates, the Gaussian curvature becomes K ( F ) = − ∂ + ∂ − ( ln ( L ( P ))) L ( P ) . (24)The mean curvature vector, written in matrix form, is given by H ( F ) = − iL ( P ) [ ∂ + P, ∂ − P ] (25)and thus is traceless. The norm of H can be written in terms of the Lagrangian densityand the topological charge density,( H ( F ) , H ( F )) = 8 L ( P ) tr ([ ∂ + P, ∂ − P ] ) = 4 L ( P ) (cid:0) L ( P ) + 3 q ( P ) (cid:1) . (26)It is possible to express the Willmore functional in terms of the action, the Lagrangiandensity, and the topological charge density, W ( F ) = 12 S ( P ) + 32 (cid:90) R q ( P ) L ( P ) dξ + dξ − . (27)The Euler-Poincare character can be written terms of only the Lagrangian density∆( F ) = − π (cid:90) R ∂ + ∂ − ln ( L ( P )) dξ dξ . (28)It was shown in [20] that the surfaces, defined as in theorem 1 can be integratedexplicitly in terms of the set of orthogonal projectors (13). oliton surfaces associated with C P N − sigma models Theorem 3
Under the assumptions of theorem 1, the surface has an associatedinteger k and holomorphic projector P so that the following holds F = F k ≡ − i P k + 2 k − (cid:88) j =0 P j − kN I , P (cid:96) = Π (cid:96) + P . (29)The inverse formula for the projectors P k in terms of the surface F k (29) has beenderived in [20] P k = F k − i (cid:18) kN − (cid:19) F k − kN (cid:18) kN − (cid:19) I . The exact integrated form of the surfaces (29) allows one to obtain the minimalpolynomials for the surfaces, dimensionality of the spanned subspaces of R N − ≡ su ( N ) , and the angle between the immersion functions F k and F j of the 2D surfaces[20]. An interesting fact is that the GWFI is equivalent to another formula for the immersionof 2D-surfaces in Lie algebras, namely the Sym-Tafel formula for immersion [30, 31].This immersion formula of a 2D surface is given by F ST = α ( λ )Φ − ∂∂λ Φ ∈ g , (30)whenever the tangent vectors ∂ + F ST = a ( λ )Φ − ∂U ∂λ Φ , ∂ − F ST = a ( λ )Φ − ∂U ∂λ Φ (31)are linearly independent. Here a ( λ ) ∈ C is an arbitrary function of λ. From the specific form of Φ (17) and Φ − (18), it is possible to directly computethe Sym-Tafel immersion formula as [20] F ST = 2 α ( λ )1 − λ P k + 2 k − (cid:88) j =0 P j − kN I ∈ su ( N ) , (32)which coincides with the GWFI, up to a multiplicative factor.
4. Fokas-Gel’fand immersion formula for surfaces associated with C P N − sigma models In light of the results obtained for the Generalized Weierstrass formula for immersionand its equivalent form, the Sym-Tafel formula for immersion, it seems worthwhile toconsider the larger class of immersion functions defined via the Fokas-Gel’fand formulafor immersion. Such surfaces were recently considered by the authors in a series ofpapers [16, 17, 18]. The necessary and sufficient conditions for the existence of suchsurfaces was formulated in terms of the symmetry criterion for generalized vector fieldsand their action on jet space [26]. oliton surfaces associated with C P N − sigma models The considered surfaces are defined via infinitesimal deformations of the zero-curvaturecondition ∆[ u ] = u − u + (cid:2) u , u (cid:3) = 0 , (33)with independent variables ξ i , i = 1 , u α , α = 1 , g . In the case of the C P N − sigma modeldefined on Euclidean space, the independent variable take the form ξ + = ξ and ξ − = ξ and the Lie algebra is g = su ( N ) . The PDE (33) can be considered as afunction on the jet space M ≡ ( ξ , ξ , u α , u αJ ) , where the derivatives of u α are givenby ∂ n ∂ξ j . . . ∂ξ j n u α ≡ u αJ , J = ( j , . . . j n ) , j i = 1 , , | J | = n. (34)Define the set of smooth functions on the jet space M to be A ≡ C ∞ ( M ) . Theabbreviated notation f ( ξ , ξ , u α , u αJ ) ≡ f [ u ] ∈ A will be used. The zero-curvaturecondition ∆[ u ] = 0 can be realized as the compatibility conditions for a wave functionΨ[ u ] defined by its tangent vectors D α Ψ = u α Ψ , α = 1 , , (35)where the total derivatives D i are defined as D i = ∂∂ξ i + u α,jJ,i ∂∂u α,jJ , α, i = 1 , . (36)Here u α,j are the components of u α in a basis for g , i.e. u α = u α,j e j with j = 1 ...n. Suppose now that it is possible to parameterize the matrices u α → U α ([ θ ] , λ ) ∈ g in terms of some set of dependent variables θ j ( ξ + , ξ − ) , which depend on theindependent variables ξ ± . The potential matrices U α ([ θ ] , λ ) are assumed to depend ona spectral parameter λ in such a way that the zero-curvature condition is equivalentto an integrable PDE in dependent variables θ j which is independent of the spectralparameter ∆ (cid:48) [ θ ] = D U − D U + (cid:2) U , U (cid:3) = 0 . (37)In the case of the C P N − sigma model, the scalar functions θ j represent thecomponents of the projector P .Define a new jet space N to be the jet space associated with θ j and its derivativesand B to be the space of smooth functions on this jet space, also possibly dependingon the spectral parameter λ. The elements of B are denoted f ( λ, ξ + , ξ − , θ j , θ jJ ) = f ([ θ ] , λ ) ∈ B . Let τ be the mapping between the function spaces A and B defined bytaking τ ( u αJ ) = D J ( U α ([ θ ] , λ )) : τu α ∈ g → U α ([ θ ]) ∈ g ↓ ↓ τ A = C ∞ ( M ) → B = C ∞ ( N × C ) . (38)Because of the structure of jet space, the mapping τ commutes with total derivatives.Under the mapping τ , the LSP becomes D α Φ = U α Φ , α = 1 , , (39) oliton surfaces associated with C P N − sigma models θ ] , λ ) = τ (Ψ[ u ]) and potential matrices U α ([ θ ] , λ ) = τ ( u α ) . This spectral problem was used by Fokas and Gel’fand in [11] to constructimmersion functions for surfaces in Lie algebras whose Gauss-Mainardi-Codazzi(GMC) equations are given by D A − D B + [ A, U ] + [ U , B ] = 0 , (40)an infinitesimal deformation of (37). Here A and B are elements of the Lie algebra g with components in the function space B . The immersion functions are defined bytheir tangent vectors D F = Φ − A Φ , D F = Φ − B Φ . (41)It is straightforward to verify that the compatibility conditions of (41) are satisfiedwhenever the LSP (39) and the GMC equations (40) hold.The possible forms of such matrices A and B have been studied by several authors,e.g. [7, 12, 16, 17, 18]. In this case, there are three types of admissible symmetries: • generalized symmetries of the zero-curvature condition itself ∆[ u ] = 0, • generalized symmetries of the corresponding integrable PDE ∆ (cid:48) [ θ ] = 0, • invariance of the integrable PDE with respect to a conformal transformation inthe spectral parameter. This is the Sym-Tafel formula for immersion.In order to construct such symmetries, it is expedient to consider generalizedfields on jet space. A vector field (cid:126)v Q defined on the jet space M in evolutionary formis given by (cid:126)v Q = Q α,j [ u ] ∂∂u α,j . (42)Note that Q α ≡ Q α,j e j is an element of the Lie algebra g . The prolongation of (cid:126)v Q isdefined in the standard way pr(cid:126)v Q = (cid:126)v Q + D J ( Q α,j [ u ]) ∂∂u α,jJ . (43)A vector field (cid:126)w Q defined on the jet space N in evolutionary form is denoted by (cid:126)w Q = Q j [ θ ] ∂∂θ j , Q j [ θ ] ∈ B . (44)The notation Q is used to distinguish the characteristics of a vector field (cid:126)v Q on M from a vector field (cid:126)w Q on N . The prolongation of (cid:126)w Q is defined in the standard wayby analogy with (43).A generalized vector field (cid:126)v Q on jet space M is said to be a generalized symmetryof the nondegenerate PDE ∆[ u ] = 0 if and only if [26] pr(cid:126)v Q (∆[ u ]) = 0 , whenever ∆[ u ] = 0 . (45)From the definition of the prolongation (43), it is immediate to verify that (45) isequivalent to D Q − D Q + [ Q , u ] + [ u , Q ] = 0 (46)Thus, the g -valued functions on jet space NA = τ ( Q ) , B = τ ( Q ) , (47)satisfy (40) whenever (cid:126)v Q is a symmetry of ∆[ u ] = 0. oliton surfaces associated with C P N − sigma models (cid:126)w Q on jet space N is said to be a generalizedsymmetry of the nondegenerate PDE ∆ (cid:48) [ θ ] = 0 if and only if pr (cid:126)w Q (∆ (cid:48) [ θ ]) = 0 , whenever ∆ (cid:48) [ θ ] = 0 . (48)From the definition of the prolongation (43), it is immediate to verify that (45) isequivalent to D ( pr (cid:126)w Q U ) − D ( pr (cid:126)w Q U ) + [ pr (cid:126)w Q U , U ] + [ U , pr (cid:126)w Q U ] = 0 . (49)Thus, the g -valued functions on jet space NA = pr (cid:126)w Q U , B = pr (cid:126)w Q U , (50)satisfy (40) whenever (cid:126)w Q is a symmetry of ∆ (cid:48) [ θ ] = 0.The third possible choice of A and B is associated with the invariance of ∆ (cid:48) [ θ ] = 0with respect to conformal symmetries of the spectral parameter λ. In this case, thechoice of matrices A and B are given by A = a ( λ ) ∂∂λ U , B = a ( λ ) ∂∂λ U , which satisfy (40) whenever ∆ (cid:48) [ θ ] = 0 is independent of the spectral parameter. Withthis choice of A and B , the tangent vectors given by (41) coincide with those for theSym-Tafel formula for immersion (31). Thus, the surfaces take the integrated formgiven in (30). As in the case of the Sym-Tafel formula for immersion, it is often possible to giveexplicit integrated forms for the immersion function for the surfaces. As proven in[18], any symmetry of the zero-curvature condition ∆[ u ] = 0 can be written in termsof a gauge function S [ u ] and the associated immersion function is given by F = Φ − τ ( S [ u ]) Φ ∈ g . (51)For example, the zero-curvature condition ∆[ u ] = 0 is invariant under a scaling of theindependent and dependent variables associated with the vector field (cid:126)v Q with Q = D (cid:0) ξ u (cid:1) + ξ D (cid:0) u (cid:1) , Q = ξ D (cid:0) u (cid:1) + D (cid:0) ξ u (cid:1) . (52)The gauge associated with this symmetry is [18] S [ u ] = ξ u + ξ u and the corresponding surface is F = Φ − (cid:0) ξ U + ξ U (cid:1) Φ ∈ g . (53)It is straightforward to verify that the tangent vectors to this surface are given as in(41) with A = τ ( Q ) = D (cid:0) ξ U (cid:1) + ξ D (cid:0) U (cid:1) , B = τ ( Q ) = ξ D (cid:0) U (cid:1) + D (cid:0) ξ U (cid:1) . As a second example, consider the following generalized symmetry of ∆[ u ] = 0whose generalized vector field (cid:126)v Q has characteristics Q = u + u + [ u , u ] + [ u , u ] , Q = u + u + [ u , u ] + [ u , u ] . (54)The gauge associated to this symmetry is given by S [ u ] = u + u oliton surfaces associated with C P N − sigma models F = Φ − (cid:0) D U + D U (cid:1) Φ ∈ g . (55)As in the previous case, the tangent vectors to this surface are given as in (41) with A = τ ( Q ) = D U + D U + [ D U , U ] + [ D U , U ] ,B = τ ( Q ) = D U + D U + [ D U , U ] + [ D U , U ] . There also exist soliton surfaces associated with generalized symmetries of theintegrable model, such as in the case of conformal symmetries of the C P N − sigmamodel. As was proven in [16], the surface associated with such symmetries, i.e. with A and B of the form (50), can be integrated explicitly as F = Φ − pr (cid:126)w Q Φ ∈ g , (56)whenever (cid:126)w Q is a generalized symmetry of the LSP in the sense that pr (cid:126)w Q ( D α Φ − U α Φ) = 0 whenever D α Φ − U α Φ = 0 . (57)In general, the requirement (57) is difficult to verify since it requires a knowledgeof the integrated form of the wave function Φ . However, in the case of finite actionsolutions of the C P N − sigma model, the wave function is known explicitly (17) andit has been directly shown that conformal symmetries of the independent variables arealso symmetries of the LSP [16]. Thus, the surface F as in (56) takes the particularform F = Φ − (cid:0) g ( ξ + ) U + g ( ξ − ) U (cid:1) Φ ∈ su ( N ) , (58)associated with vector field (cid:126)w = − g ( ξ + ) ∂ + − g ( ξ − ) ∂ − , which is a conformal symmetry of the the C P N − sigma model.
5. Soliton surfaces associated with the C P sigma model To illustrate these theoretical considerations, examples of soliton surfaces associatedwith the C P sigma model are presented. In this model, the only solutions withfinite action are holomorphic and anti-holomorphic projectors. Nevertheless, as willbe demonstrated below, there are various forms of surfaces which can be constructedin this manner.For these simple examples, the holomorphic projector is chosen as the followingmatrix, based on the Veronese sequence f = (1 , ξ + ), P = 1(1 + | ξ | ) (cid:20) ξ − ξ + | ξ | (cid:21) , | ξ | ≡ ξ + ξ − . (59)The wave function in the LSP for this model is simplyΦ = (cid:18) I − − λ P (cid:19) . (60)The surface corresponding to the Sym-Tafel formula for immersion is written as F ST = i (1 + | ξ | ) (cid:34) ξ − ξ + ξ + ξ − (cid:35) , oliton surfaces associated with C P N − sigma models F ST F P Figure 1.
Surfaces F ST and F P with λ = i/ ξ ± = x ± iy with x and y ∈ [ − , where the scaling factor is chosen as i (1 − λ ) / K = H = 4 , and so is in fact a sphere, as demonstrated in figure 1. The surfaces are representedin terms of their components in the following basis for su (2) e = (cid:20) ii (cid:21) , e = (cid:20) −
11 0 (cid:21) , e = (cid:20) i − i (cid:21) . (61)The surface associated with the (scaling) point symmetry of the zero-curvaturecondition (cid:126)v Q (42) has the integrated form F P = Φ − (cid:0) ξ U + ξ U (cid:1) Φ= 1(1 − λ )(1 + | ξ | ) (cid:34) λ ξ + ξ − λ + 1) ξ + ξ − + 1 − λ ) ξ − λ − ξ + ξ − − − λ ) ξ + − λ ξ + ξ − (cid:35) . This surface, F P , also has constant positive Gaussian and mean curvatures K = − λ , H = − iλ, iλ ∈ R though the surface is not a sphere since it has a boundary. A graph of the surface isshown in figure 1.In the case of the surface associated with the generalized symmetry of the zero-curvature condition (55), the immersion function is given by F G = Φ − (cid:0) D U + D U (cid:1) Φ= 4(1 − λ )(1 + | ξ | ) (cid:34) − λ ( ξ +2 + ξ − ) + ξ +2 − ξ − − ( λ + 1) ξ − + ( λ − ξ + (1 − λ ) ξ +3 + (1 + λ ) ξ − + λ ( ξ +2 + ξ − ) − ξ +2 + ξ − (cid:35) . oliton surfaces associated with C P N − sigma models F G F C Figure 2.
Surfaces F G and F C with λ = i/ ξ ± = x ± iy with x and y ∈ [ − , The surface associated with conformal symmetry of the E-L equation (58) with g ( ξ + ) = 1 is given by F C = Φ − (cid:0) U + U (cid:1) Φ= 1(1 − λ )(1 + | ξ | ) (cid:20) ξ − − ξ + + λ ( ξ + − ξ − ) (1 + λ ) ξ − + 1 − λ ( λ − ξ − − − λ ξ + − ξ − − λ ( ξ + − ξ − ) (cid:21) . In both cases the Gaussian and mean curvatures are rational functions though theyare too involved to write out in an illustrative fashion. Their graphs are given in figure2.
6. Future outlook
The approach for constructing soliton surfaces immersed in Lie algebras describedin this paper has proven to be very effective in revealing geometric properties ofinvestigated classes of surfaces. The future objective is to extend it to more generalsigma models and study surfaces related to the complex Grassmannian models andpossibly to models associated with octonion geometry. Since the field theoreticformulation of soliton surfaces by the Euler-Lagrange equation naturally involves thevariation principle, it also leads in a straightforward way to geometric objects likegeodesics, harmonic maps, minimal surfaces and the like. It also relates closely to theformalism of completely integrable systems, e.g. Lax pairs, conservations laws andHamiltonian structures. So, it is apparent that the methods bring a strong unifyingpotential to the prospective research. oliton surfaces associated with C P N − sigma models Acknowledgments
S Post would like to thank the organizers, Dieter Schuch and Michael Ramek, fortheir kind invitation and support in attending the conference. The research reportedin this paper is supported by NSERC of Canada. S Post acknowledges a postdoctoralfellowship provided by the Laboratory of Mathematical Physics of the Centre deRecherches Math´ematiques, Universit´e de Montr´eal.
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