6 Surfaces associated with sigma models and the 0 0 2 sine–Gordon equation n a J A. M. Grundland ∗ 2 1 Centre de Recherches Math´ematiques, Universit´e de Montr´eal, ] C. P. 6128, Succ. Centre-ville, Montr´eal, (QC) H3C 3J7, Canada G Universit´e du Qu´ebec, Trois-Rivi`eres CP500 (QC) G9A 5H7, Canada D . h and t a ˇ m L. Snobl† [ Centre de Recherches Math´ematiques, Universit´e de Montr´eal, 1 C. P. 6128, Succ. Centre-ville, Montr´eal, (QC) H3C 3J7, Canada v 2 and 0 3 Faculty of Nuclear Sciences and Physical Engineering, 1 Czech Technical University, 0 6 Bˇrehov´a 7, 115 19 Prague 1, Czech Republic 0 / h t a m Abstract : v i We present a unified method of construction of surfaces associated X withGrassmanniansigmamodels,expressedintermsofanorthogonal r a projector. This description leads to compact formulae for structural equations of two-dimensional surfaces immersed in the su(N)algebra. InthespecialcaseoftheCP1sigmamodelweobtainconstantnegative Gaussian curvature surfaces. As a consequence, this leads us to an explicit relation between the CP1 sigma model and the sine–Gordon equation. ∗email address: [email protected] †email address: Libor.Snobl@fjfi.cvut.cz 1 Keywords: Sigma models, structural equations of surfaces, Lie alge- bras. PACS numbers: 02.40.Hw, 02.20.Sv, 02.30.Ik 1 Introduction Description of the behaviour of surfaces immersed in Rn in connection withintegrablesystemsleadinmostcasestofundamentalformswhere the coefficients satisfy the Gauss–Weingarten and Gauss–Codazzi– Ricci equations. The study of general properties of these equations and the methods of solving them is a rapidly developing area of mod- ern mathematics. A broad review of recent developments in this sub- ject can be found e.g. in [1, 2] (and references therein). Theideaofinducingsurfacesinthree–dimensionalEuclideanspace from the solutions of two–dimensional linear problems is a very old one. It originates from the work of K. Weierstrass [3] and A. Enneper [4] one and half centuries ago. This subject has been extensively researched (see e.g. [5, 6, 7, 8, 9]). Until very recently, the immersion oftwo–dimensional surfacesobtainedthroughtheWeierstrass formula had only been known in low dimensional Euclidean spaces [10, 11]. The relationship between the immersion of surfaces and Dirac–type systems has been developed over the last decades by many authors [12, 13, 14] with a purpose of extension of the applicability of the immersionformulaandconsequentlyofconstructingmorediversetype of surfaces than those obtained previously by the classical approach. For instance surfaces associated with sigma models provide us with a rich class of geometric objects. The rich character of this formulation makes immersion formula an interesting object of study and various special types of surfaces have been investigated ([15]). This paper is a follow–up of the results obtained in [16, 17, 18] and is concerned with two–dimensional surfaces immersed in multi– dimensional Euclidean spaces obtained from solutions of Grassman- nian sigma models defined on Minkowski space. The heart of the matter is that the equations defining the immersion are formulated directly in terms of matrices taking their values in the Lie algebra su(N). The main advantages of this procedureis that the group anal- ysisoftheimmersionmakesitpossibletoconstructregularalgorithms for finding certain classes of surfaces without referring to any addi- tional considerations. The proposed method proceeds directly from 2 the given sigma model. The task of finding the surfaces is facilitated by the group prop- erties of these models. Our main goal is to provide a self–contained comprehensiveapproachtothissubject. Forthispurposeweformulate the structural equations for the immersion expressed in the Cartan’s language of moving farmes, fundamental forms, the Gauss curvature and the mean curvature vector, using an orthogonal projector satisfy- ingtheEuler–Lagrangeequationsofthegivensigmamodel. Themain advantage oftheprojector approach isthattherearenogaugedegrees offreedominthisdescriptionoftheGrassmanniansigmamodels,com- pared to the use of equivalence classes. Such a description leads to muchsimplerformulaeandallows ustowriteinclosed formquantities which were previously too complicated to be presented. Secondly, we return to study the simplest case of this construc- tion, namely the CP1 sigma model. In this particular case, the result- ing surfaces have negative constant Gaussian curvature. Therefore, we may construct and study the corresponding solutions of the sine– Gordon equation. It turns out that the relation doesn’t necessarily involve the construction of surfaces, i.e. there is a direct reduction from the CP1 sigma model to the the sine–Gordon equation, provided certain regularity conditions on the CP1 solution are met. In order to illustrate this relation we consider solutions of the CP1 sigma model obtained in different ways – a symmetry reduction, B¨acklund transformation – and construct both the correspondingsur- faces and the solutions of the sine–Gordon equations. 2 Grassmannian sigma models and their equations of motion We generalize andsimplify our treatment of sigma models on complex Grassmannian manifolds defined on Minkowski space. The starting point of our generalization lies in the realization that most of the properties of associated surfaces were described using a projector P, P2 = P, P† = P, which satisfies the equations of motion in the form [∂ ∂ P,P] = 0. (2.1) L R 3 Therefore,thedescriptionoftheGrassmanniansigmamodelsinterms of the projector on the corresponding subspace defining the element of the Grassmannian manifold G(m,n) = m dimensionalsubspaces of CN (2.2) { − } is more natural in our context than the description using equivalence classes of the elements of SU(N) G(m,n) = , N = m+n. (2.3) S(U(m) U(n)) × In this formalism, the solution of the model is described as a map P :Ω Aut(CN), P† = P, P2 = P → whereΩ R2 willbeassumedtobeaconnectedandsimplyconnected ⊂ domain of definition of the model. The solution is required to be a stationary point of the action = tr ∂ P.∂ P dξ dξ (2.4) L R L R S { } ZΩ where ξ ,ξ are the light-cone coordinates on the Minkowski space L R R2, i.e. the metric on R2 is written as ds2 = dξ dξ . (2.5) L R We shall denote by ∂ and ∂ the derivatives with respect to ξ and L R L ξ , respectively. R Because the hermitian matrix P is subject to the constraint P2 = P we have to introduce the Lagrange multiplier λ = λ† Aut(CN) into ∈ the action (2.4) = tr ∂ P.∂ P +λ.(P2 P) dξ dξ . (2.6) L R L R S { − } ZΩ By the variation of the action (2.6) we get δλ : P2 P = 0, − δP : 2∂ ∂ P +λ.P +P.λ λ = 0. (2.7) L R − 4 InordertoeliminatetheLagrangemultiplierλfromtheEuler–Lagrange equations we multiply (2.7) from the left (right) by P and subtract them, obtaining (2.1) [∂ ∂ P,P] = 0 L R For later reference let us note that an equivalent version of (2.1) reads ∂ [∂ P,P]+∂ [∂ P,P]= 0 (2.8) L R R L and that as a differential consequence of P2 = P one has ∂ P = ∂ PP +P∂ P, P∂ PP = 0, D = L,R. (2.9) D D D D Let us also introduce a notation for the trace of a product of two derivatives of P p = tr (∂ P.∂ P), (2.10) B1...Bk D1...Dl B1...Bk D1...Dl | wherek,l > 0, B ,...,B ,D ,...,D = L,R.Notethatp = 1 k 1 l B1...Bk D1...Dl | p . D1...DlB1...Bk | Finally, let us mention that projectors satisfying (2.1) may also arise in other models or applications which may not be naturally in- terpreted in terms of Grassmannian manifolds. Nevertheless, the con- struction of surfaces associated with them can be performed in the same way. 3 Surfaces obtained from Grassman- nian sigma model Letusnowdiscusstheanalyticaldescriptionofatwo–dimensionalsur- face immersed in the su(N) algebra, associated with the projector F (2.1). Firstly, weshall constructan exact su(N)–valued 1–form whose “potential” 0–form defines the surface . Next, we shall investigate F the geometric characteristics of the surface . F Let us introduce the scalar product 1 (A,B) = trA.B −2 on su(N) and identify the (N2 1)–dimensional Euclidean space with − the su(N) algebra RN2 1 su(N). (3.1) − ≃ 5 We denote M = [∂ P,P], M = [∂ P,P]. (3.2) L L R R It follows from (2.8) that ∂ M +∂ M = 0. (3.3) L R R L We identify tangent vectors to the surface as follows F = M , = M . (3.4) L L R R X X − Equation (3.3) implies the existence of a closed su(N)–valued 1–form on Ω = dξ + dξ , d = 0. L L R R X X X X Because is closed and Ω is by assumption connected and simply X connected, is also exact. In other words, there exists a well–defined X su(N)–valued function X on Ω that = dX. The matrix function X X is unique up to addition of any constant element of su(N) and we identify the components of X with thecoordinates of thesought–after surface in RN2 1. Consequently, we get − F ∂ X = , ∂ X = . (3.5) L L R R X X The map X is called the Weierstrass formula for immersion. In prac- tise, the surface is found by integration F : X(ξ ,ξ ) = (3.6) L R F X Zγ(ξL,ξR) along any curve γ(ξ ,ξ ) in Ω connecting the point (ξ ,ξ ) Ω with L R L R ∈ an arbitrary chosen point (ξ0,ξ0) Ω. L R ∈ We should investigate the behaviour of the constructed surface under the known symmetries of (2.1). The equation (2.1) is invariant under the following change of independent variables (i.e. the confor- mal transformation) ξ f(ξ ), ξ g(ξ ). (3.7) L L R R −→ −→ Since the surface is written in terms of an integral of a one–form, F such a transformation amounts only to a reparametrization of the surface; asageometric objectitremainsthesame. Anothersymmetry is the transformation † P UPU , U U(N). (3.8) −→ ∈ 6 Theonly effect of such a transformation on the surfaceis a rotation in RN2 1, so again the geometry of the surface is unchanged. Therefore, − the surface associated with a solution of (2.1) characterizes the F symmetry equivalence class of solutions of (2.1). By computation of the scalar products of . , B,D = L,R we B D X X find the components of the induced metric on the surface F J , G G = L LR = (3.9) G , J LR R (cid:18) (cid:19) 1 p p = L|L − L|R . 2 p p (cid:18) − L|R R|R (cid:19) As a consequence of (2.1) we find ∂ J = p = 2tr (∂ PP∂ P) =2tr (∂ PP∂ PP) =0 R L LRL LR L LR L | using the cyclic property of the trace and along with (2.9). Similarly ∂ J = 0. L R The first fundamental form of the surface takes the compact form F I = J (dξ )2 2G dξ dξ +J (dξ )2 L L LR L R R R − 1 = (δ )p dξ dξ (3.10) B,D BD B D − 2 | where summation over repeated indices B,D = L,R applies and δ = 1 if B = D and 0 otherwise. It can be shown using the B,D Schwarz inequality that such a first fundamental form I is positive, and then investigated under which conditions it is positive definite [18], i.e. when the surface is, at least locally, well defined. It is useful to note that conformal transformations of independent variables change the metric (2.5) on R2 but leave invariant the Euler– Lagrange equations (2.1. Since the metric (2.5) is no longer needed in the following we feel free to use such a transformation (3.7) to bring the solution of (2.1) to an equivalent solution (outside singular points where J .J = 0 and consequently the tangent vectors ∂ X,∂ X are L R L R linearly dependent) such that 1 1 J = p = 1, J = p = 1. (3.11) L LL R RR 2 | 2 | Such transformation is for a given solution of (2.1) expressed in terms of quadratures. The first fundamental form now takes a particularly 7 simple form I = (dξ )2 p dξ dξ +(dξ )2, (3.12) L LR L R R − | i.e. thesurfaceisdescribedinChebyshevcoordinates. Inthefollowing we shall always assume that we chose our independent coordinates in this way. This assumption allows us to write a lot of expressions in a closed form; otherwise they would be too complicated to be presented here. Using (3.9) we can write the formula for the scalar curvature [19] as p p p p p LRLR LLRR LLR LRR LR K = 2 | − | | | | . (3.13) 4 (p )2 − (4 (p )2)2 (cid:18) − L|R − L|R (cid:19) 4 The moving frames and the Gauss– Weingarten equations Now wemay formally determine amoving frame on thesurface and F write the Gauss–Weingarten equations. Let P be a solution of (2.1) such that det(G) is not zero in a neighbourhood of a regular point (ξ0,ξ0) in Ω, so that we can assume (3.11). Assume also that the L R surface (3.6) associated with these equations is described by the F moving frame on F ~τ = ( L, R,n3,...,nN2 1)T, X X − where the vectors L, R,n3,...,nN2 1 are identified with matrices X X − as in (3.1) and satisfy the normalization conditions 1 ( , ) = 1, ( , ) = p , ( , )= 1, L L L R LR R R X X X X −2 X X ( ,n ) = ( ,n ) = 0, (n ,n ) = δ . (4.1) L k R k j k jk X X The moving frame satisfies the Gauss–Weingarten equations ∂ = AL +AL +QLn , LXL LXL RXR j j ∂ = H n , L R j j X ∂ n = αL +βL +sL n , L j jXL j XR jk k ∂ = H n , R L j j X ∂ = AR +AR +QRn , RXR LXL RXR j j ∂ n = αR +βR +sRn , (4.2) R j j XL j XR jk k 8 where sL +sL = 0, sR +sR = 0, j,k = 3,...,N2 1, jk kj jk kj − p H +2QL p QL+2H αL = 2 L|R j j , βL = 2 L|R j j, j − 4 (p )2 j − 4 (p )2 LR LR − | − | p QR+2H p H +2QR αR = 2 L|R j j, βR = 2 L|R j j , j − 4 (p )2 j − 4 (p )2 LR LR − | − | and AL,AL,AR,AR are the Christoffel symbols of the second kind L R L R given by p p AL = ΓL = − L|R LL|R, L LL 4 (p )2 LR − | 2p AL = ΓL = − LL|R , R RR 4 (p )2 LR − | 2p AR = ΓR = − RR|L , L LL 4 (p )2 LR − | p p AR = ΓR = − L|R RR|L. (4.3) R RR 4 (p )2 LR − | The explicit form of the coefficients H ,QD (where D = L,R; j = j j 3,...,N2 1)dependsonthechosenorthonormalbasis n3,...,nN2 1 of the spa−ce normal to the surface at the point X(ξ0{,ξ0). They a−re} F L R not completely arbitrary, since using (2.1) and (2.9) we find that they are restricted by the condition (∂ ,∂ ) = (∂ ,∂ ) = 0. L L L R R R L R X X X X The derivation of the Gauss–Weingarten equations is almost the same as in [18], only in better notation, therefore we will not present it here and refer the interested reader to [18]. Due to the current notation using P and Chebyshev coordinates the formulae presented here are significantly simpler. An exampleof amoving frameof thesurface can beconstructed F as follows. Let P be a solution of (2.1). Taking into account that tr(A) = tr(ΦAΦ ), Φ SU(N), (4.4) † ∈ we may employ the adjoint representation of the group SU(N) in order to bring ∂ X,∂ X,n to its simplest possible form. We shall L R a request that Φ(ξ ,ξ ) diagonalizes P(ξ ,ξ ), i.e. L R L R † P(ξ ,ξ ) = Φ(ξ ,ξ )diag(0,...,0,1,...,1)Φ (ξ ,ξ ). (4.5) L R L R L R 9 The transformed derivatives of X have the following block matrix structure (for the sake of brevity we suppress the dependence of Φ,P etc. on ξ ,ξ ). L R 0 ∂ΦP ∂ΦX Φ ∂ XΦ = D . D ≡ † D (∂ΦP)† 0 (cid:18) − D (cid:19) where ∂ΦP is defined by the equality. D Let us choose an orthonormal basis in su(N) in the following form (A ) = i(δ δ +δ δ ), 1 j < k N, (4.6) jk ab ja kb jb ka ≤ ≤ (B ) = (δ δ δ δ ), 1 j <k N, jk ab ja kb jb ka − ≤ ≤ p 2 (C ) = i δ δ pδ δ , 1 p N 1. p ab da db p+1,a p+1,b sp(p+1) − ! ≤ ≤ − d=1 X When a solution Φ of (4.5) is known, the construction of the mov- ing frame proceeds as follows. One finds, using the Gramm-Schmidt orthogonalization procedure, the orthonormal vectors A˜ ,B˜ , a m, j > m, a+j > m+2 aj aj ≤ satisfying the conditions (∂ΦX,A˜ ) = 0, (∂ΦX,B˜ )= 0 D aj D bj and span(∂ΦX,A˜ ,B˜ )j>m,a+j>m+2 = span(A ,B ) , (4.7) D aj aj D=L,R, a m aj aj a m<j ≤ ≤ and then identifies the remaining tilded and untilded matrices A˜ = A , B˜ = B , C˜ = C , jk jk jk jk p p wherea,j mora,j > mandp N 1.Consequently,∂ΦX,A˜ , B˜ , C˜ ≤ ≤ − D jk jk p satisfy the normalization conditions like (4.1). By the invariance of trace under unitary transformation (4.4) we find that the mov- ing frame of the surface in the neighbourhood Υ of point X0 = F X(ξ0,ξ0) L R ∂ X(ξ ,ξ ) = Φ(ξ ,ξ )∂ΦX(ξ ,ξ )Φ (ξ ,ξ ), L L R L R L L R † L R ∂ X(ξ ,ξ ) = Φ(ξ ,ξ )∂ΦX(ξ ,ξ )Φ (ξ ,ξ ), R L R L R R L R † L R nA(ξ ,ξ ) = Φ(ξ ,ξ )A˜ (ξ ,ξ )Φ (ξ ,ξ ), jk L R L R jk L R † L R nB(ξ ,ξ ) = Φ(ξ ,ξ )B˜ (ξ ,ξ )Φ (ξ ,ξ ), jk L R L R jk L R † L R nC(ξ ,ξ ) = Φ(ξ ,ξ )C˜ Φ (ξ ,ξ ). (4.8) p L R L R p † L R 10