Harmonic surfaces in the Cayley plane
aa r X i v : . [ m a t h . DG ] S e p HARMONIC SURFACES IN THE CAYLEY PLANE
NUNO CORREIA, RUI PACHECO, AND MARTIN SVENSSON
Abstract.
We consider the twistor theory of nilconformal harmonic mapsfrom a Riemann surface into the Cayley plane O P = F / Spin(9). By ex-hibiting this symmetric space as a submanifold of the Grassmannian of 10-dimensional subspaces of the fundamental representation of F , techniquesand constructions similar to those used in earlier works on twistor construc-tions of nilconformal harmonic maps into classical Grassmannians can also beapplied in this case. The originality of our approach lies on the use of theclassification of nilpotent orbits in Lie algebras as described by D. Djokovi´c. Introduction
In this paper we consider the twistor theory of nilconformal harmonic maps froma Riemann surface into the Cayley plane O P , the 16-dimensional exceptional Rie-mannian symmetric space F / Spin(9). Exhibiting this manifold as a submanifoldof the Grassmannian of 10-dimensional subspaces of the fundamental representa-tion of F allows us to use techniques and constructions similar to those used inearlier works on twistor constructions of nilconformal harmonic maps into classicalGrassmannians [28], as well as the exceptional Grassmannian G / SO(4) [29].We thus describe the canonical twistor fibrations over O P in terms of refine-ments of the Grassmannian structure of the base manifold. As this structure isdefined in terms of the fundamental representation of F , we may use the algebraicstructure of this representation to show that the nilconformality is actually impliedby the conformality: Proposition 4.2.
Any weakly conformal harmonic map from a Riemann surfaceto O P is nilconformal with nilorder ≤ . Following the general twistor theory for harmonic maps into compact symmetricspaces as developed in [7], we see that there are three canonical twistor fibrationsover O P : T , of dimension 30; T of dimension 40; T of dimension 42. As thedimensions of these twistor fibrations are far higher than for example those of theexceptional Grassmannian G / SO(4) and too high to be handled in a straightfor-ward fashion as in [29], we have chosen an approach significantly different from thatof previous works in our study of the twistor lifts of nilconformal harmonic maps,making use of the classification of nilpotent orbits in Lie algebras as described in[11]. And while there are three different canonical twistor fibrations over O P , onlyone of these, T , a submanifold of the isotropic lines in the fundamental representa-tion of F , is needed to describe the twistor theory of nilconformal harmonic maps. Mathematics Subject Classification.
Primary 58E20; Secondary 53C43.
Key words and phrases. harmonic maps, Riemann surfaces, Cayley plane, Twistor theory.The first and the second author were partially supported by Funda¸c˜ao para a Ciˆencia e Tec-nologia through the project UID/MAT/00212/2019.
More precisely, we show the following result, where M is an arbitrary Riemannsurface: Theorem 4.3.
Any weakly conformal harmonic map ϕ : M → O P admits a J -holomorphic lift into T . The almost complex structure J is the classical non-integrable structure thatensures that the projection of any holomorphic map from a Riemann surface to T to O P is harmonic. As T is a 3-symmetric space, J -holomorphicity for a map isequivalent to the map being primitive [6].We next consider the special case of harmonic maps into O P of finite unitonnumber. Following Burstall and Guest [5], we show how to explicitly constructexamples of S -invariant extended solutions as well as their twistor lifts. When thedomain is a torus, the class of harmonic maps of finite type (which are constructedfrom commuting Hamiltonian flows on finite dimensional subspaces of a loop alge-bra) plays a fundamental role. As a matter of fact, it was shown in [4] that anyharmonic map of a torus T into a compact rank one symmetric space is of finitetype if it is not weakly conformal. On the other hand, any non-constant harmonicmap from T into C P n is either of finite type or finite uniton number but not both[3, 20, 21]. The corresponding statement for H P is known to be false [22], andconsequently it is also false for O P , since H P can be totally geodesically [31]embedded in O P .While our approach using the classification of nilpotent orbits is novel, it isnevertheless a natural approach when studying nilconformal harmonic maps intosymmetric spaces or, more generally, Lie groups. Our work suggests a revisit ofearlier results on this topic using this approach, as well as a path to understand thetwistor theory of harmonic maps for the remaining exceptional symmetric spaces.This paper is arranged as follows: in Section 2 we give a brief introduction to theCayley plane O P , showing how this can be viewed as a particular Grassmannianmanifold related to the exceptional Jordan algebra of hermitian 3 × O P andstudy the properties of twistor lifts of harmonic maps into O P . Nilconformality isintroduced in Section 4 where we use the classification of nilptotent orbits in the Liealgebra of F to show that any nilconformal harmonic map from a Riemann surfaceinto O P admits a twistor lift into T . In Section 5 we give some explicit examples ofharmonic maps of finite uniton number into the Cayley plane. Appendix A collectssome facts about the structure of the Lie algebra f of F and its fundamentalrepresentation and Appendix B gives a very brief introduction to the classificationof nilpotent orbits in Lie algebras and show how this applies to f .A few words about conventions: throughout this paper, we denote the complex-ification V ⊗ R C of a real vector space or vector bundle V by V C . Unless otherwisementioned, the symbol G will always denote a compact semi-simple Lie group withtrivial centre contained in U( n ) for some n , while g will denote its Lie algebra.2. The Cayley plane
In this section we recall some classical facts on the construction of the Cayleyplane O P from an exceptional Jordan algebra. We also exhibit the Cayley planeas a subset of a Grassmannian, a construction we will find useful when studyingthe twistor theory of O P . Our main references are [1, 2, 16]. We denote by O the 8-dimensional division algebra of octonions. For x ∈ O ,we denote by x ∗ the octonionic conjugation of x . The following identities hold forall u, v ∈ O and express the fact that O is an alternative, nicely normed ∗ -algebra [1, 2]:(1) ( uu ) v = u ( uv ) , ( uv ) u = u ( vu ) , ( vu ) u = v ( uu ) , ( uu ∗ ) v = u ( u ∗ v ) , ( uv ) u ∗ = u ( vu ∗ ) , ( vu ∗ ) u = v ( u ∗ u ) . Denote by h ( O ) the 27-dimensional simple formally real Jordan algebra of her-mitian 3 × O , equipped with the product a ◦ b = 12 ( ab + ba ) ( a, b ∈ h ( O )) . We also equip h ( O ) with the inner product h a, b i R = 12 tr( a ◦ b ) ( a, b ∈ h ( O )) . As is well known, the exceptional Lie group F is realised as the automorphismgroup of h ( O ) with respect to the product ◦ . Since F also preserves the trace,we have F ⊂ SO( h ( O )). Moreover, since the elements in F fix the real subspacegenerated by the identity matrix in h ( O ), they preserve also the 26-dimensionalsubspace h ( O ) of the traceless matrices in h ( O ). This space is the fundamentalrepresentation of F , its smallest non-trivial representation, see Appendix A. For a, b ∈ h ( O ), we denote by a · b the traceless part of the product a ◦ b .The Cayley plane O P is the 16-dimensional manifold of matrices P ∈ h ( O )satisfying P = P and tr( P ) = 1. The inner product h· , ·i R on h ( O ) induces a Rie-mannian metric on O P . The isometry group of O P is precisely F , acting tran-sitively with isotropy subgroups conjugated to Spin(9). Hence O P ∼ = F / Spin(9),a compact Riemannian symmetric space of rank 1.For P ∈ O P , let L P denote the linear endomorphism of h ( O ) given by L P ( X ) = P ◦ X ( X ∈ h ( O )) . This gives a eigenspace decomposition(2) h ( O ) = A ( P ) ⊕ A ( P ) ⊕ A ( P ) , where A j ( P ) is the eigenspace associated to the eigenvalue j , with j = 0 , ,
1. Forexample, if P ∈ O P is the diagonal matrix diag(1 , , A ( P ) is just thereal span of P while A ( P ) = ( β z z ∗ γ : β, γ ∈ R , z ∈ O ) ,A ( P ) = ( x yx ∗ y ∗ : x, y ∈ O ) . Y. Huang and N. C. Leung [17] gave a uniform description of all compact Rie-mannian symmetric spaces as Grassmannians. According to their description, O P is the space of all copies of h ( O ) in h ( O ). As we shall see, O P can also be inter-preted as a certain Grassmannian of subspaces of h ( O ). Our proof is independentof that in [17]. N. CORREIA, R. PACHECO, AND M. SVENSSON
Theorem 2.1.
There is an isometry between O P and the Grassmannian Gr a of -dimensional subspaces V of h ( O ) satisfying V · V = V .Proof. Take V ∈ Gr a and let ˆ V be the maximal subalgebra of h ( O ) containing the11-dimensional subalgebra R I ⊕ V . According to Racine’s classification of maximalsubalgebras of exceptional Jordan algebras [25], either ˆ V is isomorphic to the 15-dimensional Jordan algebra h ( H ) or ˆ V is an 11-dimensional subalgebra of the formˆ V P = A ( P ) ⊕ A ( P ) , for some P ∈ O P . However, as the maximal subalgebras of h ( H ) have dimension7 and 9 (see [25]), we must have ˆ V = ˆ V P for some P ∈ O P .Assume now that ˆ V P = ˆ V P ′ for P, P ′ ∈ O P . In view of the orthogonaleigenspace decomposition (2), we must have A ( P ) = A ( P ′ ). On the other hand,by transitivity of the F action on O P , we can without loss of generality assumethat P = P = diag(1 , , A ( P ′ ) = A ( P ) for P ′ ∈ O P is only possible if P ′ = P .Thus, given V ∈ Gr a , we have seen that there exists a unique P ∈ O P suchthat ˆ V = ˆ V P . This defines an injection χ : Gr a → O P mapping this V to P .Conversely, given P ∈ O P , consider the 11-dimensional maximal subalgebraˆ V P = A ( P ) ⊕ A ( P ). Take g ∈ F such that P = g ( P ), where P = diag(1 , , V P = g ( ˆ V P ) and R I ⊂ ˆ V P . Since F preserves R I we also have R I ⊂ ˆ V P . Set V = ˆ V P ∩ ( R I ) ⊥ . It is now easy to see that V ∈ Gr a and χ ( V ) = P , thus χ is also surjective. Since χ is F -invariant, χ is an isometry. (cid:3) Interpreting O P as the submanifold of h ( O ) consisting of the projection ma-trices P with tr( P ) = 1, the tangent plane at P to O P is given by T P O P = { Z ∈ h ( O ) : 2 P ◦ Z = Z } = A ( P ) . On the other hand, given a projection matrix P ∈ O P , let us denote by V P thecorresponding subspace in Gr a . With this identification, a tangent vector Z ∈ T P O P corresponds to an elementˆ Z ∈ Hom( V P , V ⊥ P ) ⊕ Hom( V ⊥ P , V P ) . We can make this correspondence more explicit as follows.
Proposition 2.2.
For any Z ∈ T P O P and X ∈ h ( O ), we have(3) ˆ Z ( X ) = − Z ◦ X + 4 Z ◦ ( P ◦ X ) + 4 P ◦ ( Z ◦ X ) . Proof.
Given a projection matrix P ∈ O P it follows easily that the orthogonalprojection onto V P is given by(4) X X − P ◦ X + 4 P ◦ ( P ◦ X ) ( X ∈ h ( O )) . Differentiating (4) at P in the direction Z ∈ T P O P , we obtain (3). (cid:3) Proposition 2.3.
Suppose that the tangent vector Z ∈ T C P O P is isotropic, thatis, tr( Z ◦ Z ) = 0. Then ˆ Z defined by (3) is nilpotent with nilorder r ≤ Proof.
Without loss of generality we may again assume that P = diag(1 , , Z transforms vectors of V P in vectors of V ⊥ P and vice-versa, it is enough to provethat ˆ Z = 0 when restricted to V ⊥ P . Up to multiplication by a constant, we haveˆ Z ( X ) = Z ◦ ( Z ◦ X ) − Z ◦ ( P ◦ ( Z ◦ X )) − Z ◦ ( Z ◦ ( P ◦ X )) − P ◦ ( Z ◦ ( Z ◦ X )) + P ◦ ( Z ◦ ( P ◦ ( Z ◦ X )))+ P ◦ ( Z ◦ ( Z ◦ ( P ◦ X ))) − Z ◦ ( P ◦ ( Z ◦ X )))+ Z ◦ ( P ◦ ( P ◦ ( Z ◦ X ))) + Z ◦ ( P ◦ ( Z ◦ ( P ◦ X ))) . This can be considerably simplified if we take X ∈ V ⊥ P = A ( P ). In view of A ( P ) ◦ A ( P ) ⊆ A ( P ) ⊕ A ( P ) , A ( P ) ◦ ( A ( P ) ⊕ A ( P )) ⊆ A ( P ) , and recalling that Z ∈ T C P O P = A ( P ) C , we see that, up to multiplication by aconstant, ˆ Z ( X ) = Z ◦ ( Z ◦ X ) for X ∈ V ⊥ P . Henceˆ Z ( X ) = Z ◦ ( Z ◦ ( Z ◦ ( Z ◦ X ) . So, it remains to prove that L Z ( X ) = 0, where L Z is the linear endomorphismof h ( O ) given by the left multiplication by Z . Let us write Z = x yx ∗ y ∗ for some x, y ∈ O ⊗ R C . It is easy to see that Z is isotropic if and only if xx ∗ + yy ∗ =0. Since Spin(9) ⊂ F acts transitively on the space of real lines in T P O P , we canassume without loss of generality that X = Taking also (1) into account, a straightforward calculation now shows that L Z ( X ) =0. (cid:3) The twistor spaces of O P Twistors first appeared in the context of theoretical physics, in particular in thework of Penrose, see for example [23]. For harmonic maps, twistor constructionscan be found already in the works of Chern [10] and Calabi [9], and have sincebeen used by many others as the field has developed. The main idea behind twistorconstructions of harmonic maps into a Riemannian manifold (
N, h ) is to find an(almost) complex manifold (
Z, J ) and a fibration π : Z → N with the followingproperty: for any Riemann surface M and any (almost) holomorphic map ψ : M → Z, the composition π ◦ ψ : M → N is harmonic. If that is the case, then π : ( Z, J ) → ( N, h ) is called a twistor fibration and (
Z, J ) a twistor space for (
N, h ). Usingtwistor fibrations we can thus construct harmonic maps from (almost) holomorphicmaps.F.E. Burstall and J.H. Rawnsley introduced a general twistor theory for har-monic maps into compact symmetric spaces [7]. In this section we will briefly recall
N. CORREIA, R. PACHECO, AND M. SVENSSON their construction, and then study this in the specific context of the compact sym-metric space O P . For notation regarding the weight spaces of the fundamentalrepresentation of F we refer to Appendix A.3.1. Harmonic maps into Lie groups and symmetric spaces.
A map ϕ :( M, g ) → ( N, h ) between two Riemannian manifolds is said to be harmonic if itsatisfies the harmonic map equationtr ∇ d ϕ = 0 , where ∇ is the connection on ϕ − T N ⊗ T ∗ M induced by the Levi-Civita connectionson ( M, g ) and (
N, h ) respectively. As is easily seen when dim M = 2, this equationdepends only on the conformal structure on M . Thus, the concept of a harmonicmap from a Riemann surface to a Riemannian manifold makes sense.Let G be a compact semi-simple Lie group equipped with a bi-invariant metricand M a Riemann surface. For simplicity of exposition, we will assume from nowon that G is semi-simple, has trivial centre and is contained in U( n ) for some n .For a map ϕ : M → G , set A ϕ = 12 ϕ − d ϕ. For any local complex coordinate z on M , we write A ϕ = A ϕz d z + A ϕ ¯ z d¯ z. The integrability equation (5) ( A ϕz ) ¯ z − ( A ϕ ¯ z ) z = 2[ A ϕz , A ϕ ¯ z ]follows easily from the Maurer-Cartan equation. Furthermore, it is easy to see that ϕ is harmonic if and only if(6) ( A ϕz ) ¯ z + ( A ϕ ¯ z ) z = 0 . Define the connection D ϕ = d + A ϕ on the trivial bundle M × C n . The conditionfor ϕ being harmonic can now be written as D ϕ ¯ z A ϕz = A ϕz D ϕ ¯ z , where D ϕ ¯ z = ∂ ¯ z + A ϕ ¯ z . Thus the harmonicity of ϕ is equivalent to A ϕz being aholomorphic endomorphism on the trivial bundle with respect to the holomorphicstructure induced by D ϕ .As is well known, a compact inner symmetric space G/K may be totally geodesi-cally immersed in G as a connected component of √ e = { g ∈ G : g = e } . Whena harmonic map ϕ : M → G takes values in a connected component of √ e , we canwrite ϕ = π ϕ − π ⊥ ϕ , where π ϕ denotes the orthogonal projection onto the vectorbundle, also denoted by ϕ , whose fiber at z is the (+1)-eigenspace of ϕ ( z ).3.2. General twistor theory.
Let g be the Lie algebra of G and t ⊂ g be amaximal torus. Denote by ∆ ⊂ i t ∗ the set of roots, where i = √−
1, and for α ∈ ∆ let g α be the corresponding root space. Fix a set of positive simple rootsΦ + = { α , . . . , α n } with dual basis H , . . . , H n ∈ t so that α i ( H j ) = i δ ij .For any non-empty subset I ⊂ { , . . . , n } consider the canonical element ξ I = P i ∈ I H i . From this we construct a parabolic subalgebra of g C as follows: denote by g ξ I j the j -eigenspace of ad( ξ I / i) and set p I = X j ≥ g ξ I j . Let P I be the corresponding parabolic subgroup and T I = G C /P I = G/P I ∩ G thecorresponding flag manifold.The subset I also defines the inner involution τ I = Ad exp( πξ I ) , which corresponds to the symmetric space N I = G/K , where P I ∩ G ⊆ K . Thecorresponding symmetric decomposition is(7) g = k ⊕ m , k C = X i even g ξ I i , m C = X i odd g ξ I i . Since P I ∩ G ⊂ K we have a homogenous projection p I : T I → N I , which we referto as a canonical fibration [7]. The horizontal and vertical spaces at the base point x ∈ T I (that we identify with the identity coset) of the canonical fibration aregiven by(8) V C ( T I ) = k C ∩ ( g ξ I ) ⊥ , H C ( T I ) = m C . Moreover, the flag manifold T I carries a complex structure J with (1 , x given by T , J T I = P i> g ξ I i . By reversing the orientation of J on the fibres, we obtain another almost complex structure on T I , usually denotedby J :(9) T , J T I = X i even < g ξ I i ⊕ X i odd > g ξ I i . Thus the first summand on the right-hand side is V , J and the second summand is H , J . Proposition 3.1. [7] The canonical homogenous map p I : T I → N I is a twistorfibration with respect to the almost complex structure J .For later use we also note that that P I ∩ G is precisely the fixed set for theautomorphism σ I = Ad exp( 2 πk ξ I )of order k , where k = max { α ( ξ I ) / i : α ∈ Φ + } + 1 . Hence the flag manifold T I also carries a structure of a k -symmetric space. Let ω be the primitive k -th root of unity and for each j = 0 , . . . , k − g jξ I = X i = j mod k g ξ I i be the eigenspace of σ I with eigenvalue ω j . N. CORREIA, R. PACHECO, AND M. SVENSSON
The canonical fibrations of O P . We are now in a position to study thecanonical projections for the Cayley plane. We follow the notation set out in Ap-pendix A. It is easy to see that there are exactly three flag manifolds fiberingcanonically over O P [12, § I = { α } , I = { α } , I = { α , α } . We denote the corresponding flag manifolds T , T and T , respectively. It is easyto see that T = F / (SU(3) × U(1) × SU(2)) T = F / (Spin(7) × U(1)) T = F / (SU(3) × U(1) × U(1)) . These spaces may be interpreted geometrically as follows.
Proposition 3.2. ([19], Proposition 6.6) The flag manifold T is the space of alllines in h ( O ) C that square to zero, i.e., T = { ℓ ∈ P ( h ( O ) C ) : ℓ = 0 } . It is easy so see that any ℓ ∈ T is isotropic, so that T is contained in the quadricof isotropic lines in h ( O ) C . Proposition 3.3. ([19], Proposition 6.7) The flag manifold T is the space of two-dimensional isotropic subspaces C ⊂ h ( O ) C satisfying C = 0 (with respect to theusual matrix product). Proposition 3.4.
The flag manifold T is the space of all pairs ( ℓ, C ), where ℓ ∈ T , C ∈ T and ℓ ⊂ C . Proof.
Given such a pair ( ℓ, C ), the stabilizer of this pair in F is the stabilizer of ℓ in the stabilizer of C . The latter is SU(3) × U(1) × SU(2) where the first factor actstrivially on C and the third factor acts by the standard 2-dimensional representationon C . Since this is transitive on the lines in C with stabilizer U(1), we concludethat F is transitive on such pairs with stabilizer SU(3) × U(1) × U(1). (cid:3)
We may also embed O P as the connected component of √ e = { g ∈ F : g = e } containing exp( πξ I ), where I is one of the sets in (11). This is a totally geodesicembedding of O P into F . Similarly, T I can be identified with the connectedcomponent of k √ e = { g ∈ F : g k = e } containing the elementexp (cid:0) πk ξ I (cid:1) . The Lie group F acts on both the images of O P and T I by conjugation, and p I (cid:0) exp (cid:0) πk ξ I (cid:1)(cid:1) = exp( πξ I ) . This can be made more explicit via the fundamental representation, henceforthdenoted by ρ , of F and f (see Appendix A). Recall the isometry Gr a ∼ = O P from Theorem 2.1. The subspace in Gr a corresponding to exp( πξ I ) is exactly the(+1)-eigenspace of ρ (exp( πξ I ). Denoting this space by V we have ρ (exp( πξ I )) = π V − π V ⊥ , where π W denotes orthogonal projection onto a subspace W . Let us first consider the 3-symmetric space T , the flag manifold of isotropic lines ℓ in h ( O ) C satisfying ℓ = 0. The parabolic group P stabilizes the isotropic line ℓ = W which we thus choose as our base point. We now have ρ ( H ) = , on W ⊕ W ⊕ W ⊕ W i , on W ⊕ W ⊕ W ⊕ W ⊕ W ⊕ W ⊕ W ⊕ W , on W . It follows from the multiplication table in Appendix A that ℓ s = X i ≥− W i is the stabilizer of ℓ , i.e., the maximal subspace satisfying ℓ s · ℓ ⊆ ℓ . The (+1)-eigenspace of ρ (exp( πH )) is ℓ ⊕ ¯ ℓ ⊕ ( ℓ s ∩ ℓ s ) and we thus conclude that the projection p : T → O P is given by(12) p ( ℓ ) = ℓ ⊕ ℓ ⊕ ( ℓ s ∩ ℓ s ) ( ℓ ∈ T ) . We now consider the 4-symmetric space T . The parabolic group P stabilizesthe isotropic 2-dimensional subspace C = W ⊕ W which thus corresponds to ourbase point. We have ρ ( H ) = , on W ⊕ W i , on W ⊕ W ⊕ W ⊕ W ⊕ W ⊕ W , on W ⊕ W ⊕ W , on W ⊕ W . Observe that the stabilizer C s and the annihilator of C , i.e., the maximal subspace C a satisfying C a · C = 0, are given by C s = W − ⊕ X i ≥ W i and C a = X i> ,i =4 W i , respectively. We also have C a = P i ≥ W i and we thus conclude that the projection p : T → O P is given by(13) p ( C ) = ( C a ∩ C ⊥ ) ⊕ ( C s ∩ C s ) ⊕ ( C a ∩ C ⊥ ) ( C ∈ T ) . The flag manifold T is a 6-symmetric space and the parabolic group P ∩ P = P stablizes the pair ( ℓ, C ) = ( W , W ⊕ W ) which we thus take as our basepoint. We have ρ ( H + H ) = , on W i , on W ⊕ W ⊕ W ⊕ W , on W ⊕ W ⊕ W , on W ⊕ W ⊕ W , on W , on W . Let C ℓ be the maximal subspaces satisfying C ℓ · C ⊆ ℓ , so that C ℓ = X i ≥ W i , C ℓ = X i ≥ W i and C ℓ = X i ≥ W i . By identifying the (+1)-eigenspace of ρ (exp( π ( H + H )) we thus conclude that theprojection p : T → O P is given by(14) p ( ℓ, C ) = ( C ∩ ℓ ⊥ ) ⊕ ( C ℓ ∩ C ℓ ⊥ ) ⊕ ( C ℓ ⊥ ∩ C ⊥ ℓ ) ⊕ ( C ∩ ℓ ⊥ ) ⊕ ( C ℓ ∩ C ℓ ⊥ ) (( ℓ, C ) ∈ T ) . Twistor lifts.
In this section we investigate the condition of a map into oneof the T I to be holomorphic with respect to the almost complex structure J in-troduced in the previous section. Using Proposition 3.1 this will then imply har-monicity of the resulting map into O P . To avoid cumbersome notation, we willdenote the trivial bundle M × h ( O ) C by just h ( O ) C . Lemma 3.5.
Let ψ : M → T I = F C /P I be a smooth map and set ϕ = p I ◦ ψ : M → O P . Then ψ is J -holomorphic (so that ϕ is harmonic) if and only if:(i) when I = { } , ψ = C is a rank 2 holomorphic subbundle of h ( O ) C withrespect to D ϕ ¯ z , contained in ker A ϕz .(ii) when I = { } , ψ = ℓ is a holomorphic line subbundle of h ( O ) C with respectto D ϕ ¯ z , contained in ker A ϕz .(iii) when I = { , } , ψ = ( ℓ, C ), ℓ and C are holomorphic subbundles of h ( O ) C with respect to D ϕ ¯ z , with ℓ contained in ker A ϕz and A ϕz ( C ) ⊂ ℓ . Proof.
We will prove the lemma for I = { } , the remaining cases are similar.Assume first that ψ = ℓ is J -holomorphic. Let us fix a point z ∈ M and,without loss of generality, assume that ψ ( z ) = W . In view of (8) and (9), it isclear that ρ ( H , J ) W = 0. We also have ρ ( V , J )( ϕ ( z ) ∩ W ⊥ ) ⊥ W , because V , J is a direct sum of negative root spaces.The condition ρ ( H , J ) W = 0 means that ψ = ℓ lies in ker A ϕz at z , while thecondition ρ ( V , J )( ϕ ( z ) ∩ W ⊥ ) ⊥ W means that D ϕ ¯ z ( ℓ ) ⊂ ℓ . Consequently ℓ is aholomorphic line bundle with respect to D ϕ ¯ z contained in ker A ϕz .For the converse, we can use the Lie theoretic description of the fundamentalrepresentation of F (see Appendix A) to check the following: (a) for each nonzero X ∈ H , J , ρ ( X ) W = 0; (b) for each nonzero X ∈ V , J , ρ ( X )( ϕ ( z ) ∩ W ⊥ ) = W .From (a) we see that if ℓ lies in ker A ϕz , then the component of ℓ z along H , J mustvanish everywhere. From (b) we see that if ℓ is a holomorphic line subbundleof h ( O ) C with respect to D ϕ ¯ z , then the component of ℓ z along V , J must vanisheverywhere. (cid:3) Nilconformal Harmonic Maps
Recall that a harmonic map ϕ : M → G ⊂ U( n ) is said to be nilconformal if A ϕz is a nilpotent element of the Lie algebra g C at each point. It follows fromthe holomorphicity of A ϕz that any harmonic map ϕ : S → G is nilconformal. ByEngel’s theorem we know that A ϕz is also nilpotent at each point as a complex linearendomorphism of C n .The complex algebraic variety of nilpotent elements of g C , which we denoteby N ( g C ), decomposes into a disjoint union of conjugacy classes O X of nilpotentelements under the adjoint action of G C (the nilpotent G C -orbits ). A basic fact fromthe theory of algebraic groups is that each orbit O X is open in its Zariski closure O X , and the latter is a complex algebraic subvariety. Since A ϕz is holomorphic and M has complex dimension 1, we have the following result. Theorem 4.1.
Let ϕ : M → G be a nilconformal harmonic map. Then, off adiscrete subset of M , A ϕz takes values in a single nilpotent G C -orbit O X . The least r such that X r = 0 is the nilorder of X ∈ N ( g C ). If ϕ : M → O X offa discrete set and r is the nilorder of X , we say that ϕ has nilorder r .Consider now a symmetric space G/K totally geodesically immersed in G andlet g = k ⊕ m be the corresponding symmetric decomposition. Suppose that ϕ corresponds to a nilconformal harmonic map with values in G/K . In this case, A ϕz is a holomorphic section of the bundle [ m ] C whose fibre at the point gK isAd g ( m C ) . Let ψ : M → G be a (local) lift of ϕ into G so that ϕ = ψK . ThenAd ψ − ( A ϕz ) takes values in a single conjugacy class of nilpotent elements in m C under the adjoint action of K C . Proposition 4.2.
Any weakly conformal harmonic map from a Riemann surfaceto O P is nilconformal with nilorder r ≤ Proof.
A weakly conformal harmonic map from a Riemann surface M is a conformalimmersion off a discrete set of points where its differential vanishes. Hence the resultfollows from Proposition 2.3. (cid:3) Making use of the classification of nilpotent orbits in Lie algebras as described in[11], we see that there are precisely two such nontrivial nilpotent orbits associatedto the Cayley plane (see Appendix B for more details): if we consider the basepoint V ∈ Gr a ∼ = O P given, in terms of weight spaces, by(15) V = W + W + W + W + W + W − + W − + W − + W − , and the corresponding symmetric decomposition f = k ⊕ m given by (7), with I = { , } , then one of the orbits is represented by X ∈ m C and the other orbitrepresented by X + X ∈ m C , where X and X are nonzero elements of the rootspaces ( f ) α and ( f ) α , respectively.4.1. J -lifts. Suppose that we have a nilconformal harmonic map ϕ : M → O P .Then, off a discrete subset of M , A ϕz takes values in a single F C -orbit of nilpotentelements in f C . More precisely, A ϕz is a section of [ m C ] ∩ O X with X = X or X = X + X . We will show that in both cases (see § § ϕ admits a J -holomorphic lift into T , so that the following holds. Theorem 4.3.
Any weakly conformal harmonic map ϕ : M → O P admits a J -holomorphic lift into T . The case X = X . In this case, X acts as follows with respect to the funda-mental representation: W − W − W − W − W − W − W − W − W − W W W W W W W W W W W and X ( W i ) = 0 otherwise. We note from this thatIm X ⊆ W ⊂ V ⊥ , with V given by (15). Lemma 4.4. X has nilorder 3, dim Im X = 1. Proof.
Clearly, X = 0 and X ( W i ) = 0 for all i = −
4. However, since α is aweight, we cannot have X = 0, for this would imply that W − + X ( W − ) is arepresentation of the sl associated to the root α , which is impossible since theeigenvalues must be symmetric around the origin. (cid:3) Let ( W ) a be the annihilator of W = Im X , i.e., the maximal subspace W of h ( O ) C satisfying W · W = 0. It follows easily that( W ) a = W − + W − + W − + W − + W − + W − + W − + W ′ + W + W + W + W + W + W + W + W , where W ′ = ( W ) a ∩ W . From a simple calculation using the symmetry of thetrilinear form on h ( O ) C it follows easily that W ′ = W ′ . Set V = ( W ) a ∩ ( W ) a ∩ ϕ and observe that V ∩ ¯ V ⊥ = W − + W + W + W . From this we also see that ( V ∩ ¯ V ⊥ ) = 0.Since A ϕz is a holomorphic section of [ m C ] ∩ O X , we can use the standard pro-cedure of filling out zeros [8, Proposition 2.2] at points where ( A ϕz ) does not havemaximal rank in order to make A = Im( A ϕz ) a D ϕz -holomorphic subbundle of ϕ ⊥ .As above, denote by A a the annihilator of A and set D = A a ∩ A a ∩ ϕ. It follows easily that
D ∩ ¯ D ⊥ is a holomorphic, isotropic subbundle of ϕ containedin the kernel of A ϕz . Hence, any holomorphic line subbundle of D ∩ ¯ D ⊥ will sufficeas a twistor lift of ϕ . What remains is therefore only to show that there existsa line subbundle of D ∩ ¯ D ⊥ . If M is non-compact and all holomorphic bundlestherefore holomorphically trivial, finding a holomorphic line subbundle of D ∩ ¯ D ⊥ is certainly possible. When M is compact it is well-known that there exists a non-trivial meromorphic section η of D ∩ ¯ D ⊥ . The zeros and poles of η constitute afinite set, and we can use the standard method to “extend” the line bundle span { η } across this set. Hence we have a holomorphic line subbundle of D ∩ ¯ D ⊥ across theentire surface. This completes the proof.4.1.2. The case X = X + X . In this case, X acts as follows with respect to thefundamental representation: W − W − W − , W W W W − W − W − , W W W W − W − W − , W W W W − W − + W − W W + W W Lemma 4.5. Im X | ϕ ⊥ is a holomorphic line subbundle of ϕ . Proof.
It is easy to see that the root spaces associated to the roots α , α and α + α and their commutators generate a copy of sl inside f , the action of whichon W + W + W + W + W − + W − + W − is precisely its adjoint representation. From a simple calculation we thus see that X ( W − + W − ) = W . (cid:3) Since ℓ = Im X | ϕ ⊥ is easily seen to be in the kernel of A ϕz , this defines a J -holomorphic twistor lift of ϕ into T .This concludes the proof of Theorem 4.3. Remark 4.6.
Recall that( f ) C = ( f ) H − ⊕ ( f ) H − ⊕ ( f ) H ⊕ ( f ) H ⊕ ( f ) H . We see from this that the twistor space T is a 3-symmetric space and at the basepoint x = eK , with K = Spin(7) × U(1) we have(16) T , J T = ( f ) H ⊕ ( f ) H − . In this case, we see from (10) and (16) that ( f ) H coincides with T , J T , whichmeans that a smooth map ℓ : M → T is J -holomorphic if and only if ℓ is a primitive map into T [6]. In particular, ℓ is harmonic with respect to a suitablemetric of T . Since h ( O ) is 26-dimensional, T can be naturally embedded in C P .This embedding is not totally geodesic, and therefore, in general, will not preserveharmonicity.5. Harmonic maps of finite uniton number in O P . As first observed by K. Uhlenbeck [30], equations (5) and (6) can be reformulatedin terms of the flatness of the one-parameter family of connections d+ A ϕλ on M × C n ,where A ϕλ = (1 − λ − ) A ϕz d z + (1 − λ ) A ϕ ¯ z d¯ z ( λ ∈ S ) . When M is simply connected we may integrate A ϕλ to obtain an extended solution Φ : M → Ω G = { γ : S → G (smooth) : γ (1) = e } , satisfying Φ − dΦ = A ϕλ and Φ λ = − = ϕ ; such a map Φ is unique up to left multi-plication by a constant loop.When the Fourier series in λ ∈ S associated to an extended solution has finitelymany terms, the extended solution and the corresponding harmonic map are saidto have finite uniton number . It is well known that any harmonic map from the 2-sphere into a compact Lie group has finite uniton number [30]. Among the extendedsolutions of finite uniton number, the simplest case occurs when Φ : M → Ω G takesvalues in a G -conjugacy class of homomorphisms S → G . Such extended solutionsare said to be S -invariant . Next we shall establish the general form for S -invariantextended solutions corresponding to maps into O P .We denote by Gr ( G ) the Grassmannian model [24] for the loop group Ω G . When G = U( n ), then we simply write Gr = Gr (U( n )). This model associates to eachloop γ ∈ Ω G the closed subspace V ∈ Gr ( G ) of H = L ( S , C n ) defined by V = γ H + , where H + is the closed subspace of H consisting of Fourier series whosenegative coefficients vanish. For example, from [24, Proposition 8.5.1] we knowthat Gr (SO( n )) = (cid:8) V ∈ Gr : ¯ V ⊥ = λV (cid:9) .We can use the complex bilinear product in h ( O ) C to define a product on theHilbert space H of square-summable C ∼ = h ( O ) C -valued functions on the circle:if f, g ∈ H , then ( f · g )( λ ) = f ( λ ) · g ( λ ). The Grassmannian model of Ω F is givenby the following proposition, whose proof we omit since it is analogous to that ofProposition 3.2 in [13] for the G case (see also the proof of Theorem 8.6.2 in [24]). Proposition 5.1.
With respect to the fundamental representation of F , we have: Gr ( F ) = { V ∈ Gr (cid:0) SO(26) (cid:1) : V sm · V sm ⊆ V sm } . where V sm denotes the subspace of smooth functions in V , which is dense in V[24].As Segal [26] has observed, a smooth map Φ : M → Ω G is an extended solution ifand only if W = Φ H + satisfies ∂ z W ⊆ λ − W (the pseudo-horizontality condition)and ∂ ¯ z W ⊆ W ( W is a holomorphic vector subbundle of M × H ), with respect toany local complex coordinate system ( U, z ). Hence, given an S -invariant extendedsolution Φ, we have W = A s λ s + A s +1 λ s +1 + . . . + λ r A r + λ r H + , for some integers s ≤ r , where A s ⊆ A s +1 ⊆ . . . ⊆ A r is a superhorizontal sequenceof holomorphic subbundles of M × C n , i.e., the holomorphic subbundles A i satisfy ∂ z A i ⊆ A i +1 .Burstall and Guest [5] have shown that, after a normalization procedure, if G hastrivial centre, any S -invariant extended solution takes values in the G -conjugacyclass of a homomorphism γ I ( λ ) = exp ( − i ln( λ ) ξ I ) , with ξ I = P i ∈ I H i as defined in Section § O P : I = { } , I = { } , and I = { , } . The representation of the corresponding canonical elements ξ I havebeen described in Section § Theorem 5.2.
Let ϕ : M → O P be a harmonic map associated to an S -invariantextended solution. Then ϕ admits an extended solution Φ such that W = Φ H + : M → Gr ( F ) is given by one of the following forms: (i) W = ℓλ − + ¯ ℓ ⊥ s λ − + ℓ s + ¯ ℓ ⊥ λ + λ H + , where ℓ is a holomorphic subbundle of isotropic lines in h ( O ) C and ℓ s isthe stabilizer of ℓ . In this case, ψ = ¯ ℓ : M → T is a J -holomorphic lift of ϕ . (ii) W = C λ − + C a λ − + ¯ C ⊥ s λ − + C s + ¯ C a ⊥ λ + ¯ C ⊥ λ + λ H + , where C is holomorphic subbundle of isotropic two-planes in h ( O ) C sat-isfying C = 0 , C s its stabilizer and C a its annihilator. In this case, ψ = ¯ C : M → T is a J -holomorphic lift of ϕ . (iii) W = ℓλ − + C λ − + C ℓ λ − + C ℓ λ − + C ℓ λ − + ¯ C ⊥ ℓ + ¯ C ℓ ⊥ λ + ¯ C ℓ ⊥ λ + ¯ C ⊥ λ + ¯ ℓ ⊥ λ + λ H + , where ℓ ⊂ C , ℓ is a holomorphic subbundle of isotropic lines in C , C isholomorphic subbundle of isotropic two-planes in C satisfying C = 0 , and C ℓ is the maximal subbundle satisfying C ℓ · C ⊆ ℓ . In this case, ψ = (¯ ℓ, ¯ C ) : M → T is a J -holomorphic lift of ϕ . Example 5.3.
Following the procedure introduced by Burstall and Guest [5] forobtaining harmonic maps of finite uniton number from a Riemann surface M intoan inner symmetric space in terms of meromorphic functions on M , we give nextan explicit example of a J -holomorphic lift ψ = ¯ ℓ into T of a harmonic map ϕ from S = C ∪ {∞} into O P associated to a S -invariant extended solution in theconjugacy class I = { } .With the same notations of [5], since max { α ( H ) / i : α ∈ Φ + } = 2, any suchharmonic map ϕ admits an extended solution of the formΦ = exp( C ) · γ : S → F , where γ ( λ ) = exp( − i ln( λ ) H ) and C = c + c : S → ( f ) H ⊕ ( f ) H is a meromorphic function satisfying (cid:0) c (cid:1) z − (cid:2) c , (cid:0) c (cid:1) z (cid:3) = 0 . Here c i is the component of C in ( f ) H i , with i = 1 ,
2. The corresponding holo-morphic line bundle ℓ is then given by(17) ℓ = ρ (exp ( C )) W − = (cid:18) Id + ρ ( C ) + ρ ( C ) ρ ( C )
3! + ρ ( C ) (cid:19) W − . For each α with α ( H ) = i, fix X α ∈ ( f ) α ⊂ ( f ) H . Take c = z X α + zX α + α + α , then (cid:0) c (cid:1) z = 2 zX α + X α + α + α . Since [( f ) α , ( f ) α + α + α ] = 0 , we can take c = 0. Fix vectors w i ∈ h ( O ) C suchthat W i = span { w i } and w − = X α w − , w − = X α + α + α w − . Since ρ ( C ) = 0, from (17) we obtain ℓ ( z ) = h w − + z w − + zw − i . Appendix A. F : roots and fundamental representation In this appendix we describe the fundamental representation of F in more Lietheoretic terms, making it possible to connect it to the general twistor theory forharmonic maps into Lie groups and (inner) symmetric spaces. Our main referencesare [1], [16] and [18].Denote by f the Lie algebra of F and let Φ + = { α , α , α , α } be a set ofpositive simple roots, forming the following familiar Dynkin diagram: α α α α The longest root of f is 2 α + 3 α + 4 α + 2 α , which is easily seen to be afundamental dominant weight for f corresponding to the node α in the Dynkindiagram. By including the root α = − (2 α + 3 α + 4 α + 2 α ) we get the extendedDynkin Diagram α α α α α
46 N. CORREIA, R. PACHECO, AND M. SVENSSON
By removing α from this we recover the Dynkin diagram of the subalgebra spin (9)in f . This also shows that, as representations of spin (9) we have f = spin (9) + ∆ , where ∆ is the spin-representation of spin (9). It follwos that that the positiveroots of f are Φ + and α + α , α + α , α + α , α + 2 α , α + α + α , α + α + α ,α + α + 2 α , α + α + α + α , α + 2 α + α , α + 2 α + 2 α ,α + 2 α + 2 α , α + α + 2 α + α , α + α + 2 α + 2 α ,α + 2 α + 2 α + α , α + 2 α + 2 α + 2 α , α + 2 α + 3 α + α ,α + 2 α + 3 α + 2 α , α + 2 α + 4 α + 2 α , α + 3 α + 4 α + 2 α , α + 3 α + 4 α + 2 α . The roots with coefficient 1 in front of α are weights of spin (9) on ∆ , the rest arethe roots of spin (9).Since left multiplication with P = diag(1 , ,
0) on h ( O ) commutes with theaction of spin (9), the decomposition of h ( O ) in (2) is also a decomposition of h ( O ) into spin (9) representations. In fact we have A ( P ) = ∆ and A ( P ) = 1 + λ , where 1 denotes the trivial representation spanned by diag(0 , ,
1) and λ the vectorrepresentation of spin (9). As spin (9) acts trivially on A ( P ) we get h ( O ) = 2 · + λ . Taking the trace free part we get the fundamental representation of F decomposingunder spin (9) as h ( O ) = 1 + ∆ + λ , where the trivial representation is spanned by diag( − , , f : w = α + 2 α + 3 α + 2 α , w = α + 2 α + 3 α + α ,w = α + 2 α + 2 α + α , w = α + α + 2 α + α ,w = α + 2 α + α , w = α + α + α + α , w = α + α + α ,w = α + α , w = α , w = α + α + α , w = α + α , w = α , plus their negatives, all of which have multiplicity 1, and w = 0 with multiplicity2. We will henceforth denote by W i the weight space associated to the weights w i .Note that, for any two weight spaces W i and W j , we have W i · W j ⊆ the weight space associated to the weight w i + w j ,where the corresponding weight space is zero in case w i + w j is not a weight. Thefull multiplication table for the weight spaces is as follows: · W W W W W W W W W W W W W W − W − W − W − W − W − W − W − W − W − W W − W − W − W − W − W − W − W − W W W − W − W − W − W − W − W W W W − W − W − W − W − W − W W W W − W − W − W − W − W − W W W W − W − W − W − W W W W W − W − W − W − W W W W W − W − W − W − W W W W W − W − W W W W W W − W − W W W W W W − W − W W W W W W − W − W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W Table 1.
Multiplication table for the weight spaces
Appendix B. Nilpotent orbits associated to a symmetric space
In this appendix we give a brief outline of the classification of nilpotent orbitsassociated to a symmetric space
G/K , where G is a semisimple compact Lie group.We give only the details needed to justify our results; for a more thorough exposi-tion, including proofs, the reader is referred to [11].Let g be a semisimple compact Lie algebra. Given a non-zero nilpotent element X ∈ g C , we say that the triple { X, Y, H } is a standard triple for X if Y, H ∈ g C satisfy [ H, X ] = 2 X, [ H, Y ] = − Y, [ X, Y ] = H. The
Jacobson-Morozov theorem states that standard triples for a non-zero nilpotentelement always exists. As the triple is a copy of an sl subalgebra in g C , the adjointaction of H on g C will have integer eigenvalues and decompose g C into eigenspaces.Denote by g Hi the eigenspace of H with eigenvalue i ∈ Z . From this, we may formthe Jacobson-Morozov parabolic subalgebra associated to X : p = l ⊕ u , where l = g H and u = X i> g Hi . The nilpotent element X ∈ g C is said to be distinguished (in g C ) if the only Levisubalgebra of g C containing X is g C itself. It turns out that X is distinguished ifand only if dim l = dim u / [ u , u ] . Any parabolic subalgebra p = l ⊕ u ⊂ g C with Levi factor l satisfying this equalityis said to be distinguished (in g C ). It is known that a distinguished parabolic sub-algebra is the Jacobson-Morozov parabolic subalgebra of a distinguished element,and this distinguished element lies in [ u , u ] ⊥ ∩ u = g H .Now, let ˜ l be a Levi subalgebra of g C and ˜ p a distinguished parabolic subalgebraof the semisimple algebra [˜ l , ˜ l ]. The subalgebra ˜ p is the Jacobson-Morozov parabolicsubalgebra of a distinguished element X in [˜ l , ˜ l ]. The element X is also nilpotentin g C . It can be shown that this correspondence between pairs (˜ l , ˜ p ), with ˜ l a Levisubalgebra of g C and ˜ p a distinguished parabolic subalgebra of [˜ l , ˜ l ], is invariantunder the adjoint action of G C . Moreover, the induced correspondence between G C -conjugacy classes of such pairs and nilpotent orbits in g C is one-to-one. Thus,the classification of such pairs (˜ l , ˜ p ) gives a classification of the nilpotent orbits in g C .Assume now that g = k ⊕ m is a symmetric decomposition of g . J. Sekiguchi[27] proved that there is a natural bijection between nilpotent K C -orbits in m C andnilpotent G R -orbits in g R , where G R is the noncompact real form of G C associatedto the compact symmetric space G/K and g R its Lie algebra. These real orbitswere classified by D. Djokovi´c [14, 15].We now apply this to the Cayley plane O P = F / Spin(9) with the symmetricdeomposition f = spin (9) ⊕ m given by (7) with I = { , } . The corresponding non-compact real form of F C is F − and, according to the classification of nilpotent F − -orbits (see [11], p. 151) there are two non-trivial nilpotent Spin(9) C -orbitsin m C . Comparing with the classification of nilpotent F C -orbits in f C (see [11], p128) we can identify these two orbits as follows.(1) One of these orbits intersects the nilpotent F C -orbit that corresponds to theLevi subalgebra ˜ l = ˜ A ∼ = sl (2) with a short f simple root, say α . Clearly,the subalgebra ˜ q of [˜ l , ˜ l ] generated by H and the root space ( f ) α is adistinguished subalgebra. Then, any nonzero element X ∈ ( f ) α ⊂ m C isa representative of this nilpotent orbit.(2) The second intersects the nilpotent F C -orbit that corresponds to the Levisubalgebra ˜ l = ˜ A ∼ = sl (3) with the short f simple roots { α , α } . A distin-guished parabolic subalgebra of [˜ l , ˜ l ] is the subalgebra ˜ q = ˜ l ⊕ ˜ u generated by H , H and the root spaces ( f ) α , ( f ) α and ( f ) α + α . A representative X of this nilpotent orbit lies in[˜ u , ˜ u ] ⊥ ∩ ˜ u = ( f ) α ⊕ ( f ) α . Observe that X must have nonzero components both in ( f ) α and ( f ) α ,otherwise X would not be distinguished in ˜ A . Then X is of the form X = X + X , with X and X nonzero elements of ( f ) α and ( f ) α ,respectively. References [1] J. F. Adams, Lectures on Exceptional Lie Groups. Edited by Zafer Mahmud and MamoruMimura. The University of Chicago Press 1996.[2] John C. Baez, The Octonions, Bull. Amer. Math. Soc. (2002), 145-205[3] F. E. Burstall, Harmonic tori in spheres and complex projective spaces, J. Reine Angew.Math. (1995), 149-177.[4] F.E. Burstall, D. Ferus, F. Pedit, U. Pinkall, Harmonic Tori in symmetric spaces and com-muting Hamiltonian systems on loop algebras, Ann. of Math. (1993), 173–212. [5] F. E. Burstall, M. A. Guest, Harmonic two-spheres in compact symmetric spaces, revisited,Math. Ann. (1997), no. 4, 541572.[6] F. E. Burstall and F. Pedit, Harmonic maps via Adler-Konstant-Symes theory, Harmonicmaps and Integrable Systems (A.P. Fordy and J.C.Wood, eds), Aspects of Mathematics E23,Vieweg, 1994, 221–272.[7] F.E Burstall, J.H. Rawnsley, Twistor Theory for Riemannian Symmetric Spaces with Appli-cations to Harmonic Maps of Riemann Surfaces, Lecture Notes in Mathematics, no. 1424,Springer-Verlag, Berlin, Heidelberg, 1990.[8] F. E. Burstall and J. C. Wood, The construction of harmonic maps into complex Grassman-nians, J. Diff. Geom. (1986), 255–298.[9] E. Calabi, Minimal immersions of surfaces in Euclidean spaces, J. Differential Geom. (1967,111–125.[10] S. S. Chern, Minimal surfaces in an Euclidean space of N dimesions, Differential and combina-torial topology (Symposium in honour of Marston Morse), Princeton Univ. Press, Princeton,NJ 1965, 187–198.[11] D.H. Collingwood, W. M. McGovern Nilpotent Orbits in Semisimple Lie Algebras, Van Nos-trand Reinhold, New York, 1993.[12] N. Correia, R. Pacheco, Harmonic maps of finite uniton number and their canonical elements,Ann. Glob. Anal. Geom. (2015), no. 4, 335-358.[13] N. Correia, R. Pacheco, Harmonic maps of finite uniton number into G , Math. Z. (2012),no. 1-2, 13-32.[14] D. Djokovi´c, Classification of nilpotent elements in simple exceptional real Lie algebras ofinner type and description of their centralizers, J. of Alg. (1988), 503–524.[15] D. Djokovi´c, Classification of nilpotent elements in simple real Lie algebras E and E − description of their centralizers, J. of Alg. (1988), 196–207.[16] F. Reese Harvey, Spinors and Calibrations, Academic Press, San Diego, 1990.[17] Y. Huang and N. C. Leung, A uniform description of compact symmetric spaces as Grass-mannians using the magic square, Math. Ann. , issue 1 (2011), 79–106.[18] J.E. Humphreys, Introduction to Lie algebras and representation theory Springer-Verlag,New York - Berlin, 1978.[19] J. M. Landsberg and L. Manivel, On the projective geometry of rational homogeneousvarieties, Commentarii Mathematici Helvetici , issue 1 (2003), 65–100.[20] Y. Ohnita, S. Udagawa, Harmonic maps of finite type into generalized flag manifolds, andtwistor fibrations, Differential geometry and integrable systems (Tokyo, 2000), 245270, Con-temp. Math., , Amer. Math. Soc., Providence, RI, 2002.[21] R. Pacheco, Twistor fibrations giving primitive harmonic maps of finite type, Int. J. Math.Math. Sci. (2005), 3199–3212.[22] R. Pacheco, On harmonic tori in compact rank one symmetric spaces, Differential Geom.Appl. (2009), 352–361.[23] R. Penrose and W. Rindler, Spinors and space-time, Vol. 2, Spinor and twistor methods inspace-time geometry, Cambridge Univ. Press, Cambridge-New York, 1986.[24] A. Pressley and G. Segal, Loop groups, Oxford Mathematical Monographs, Oxford SciencePublications, The Clarendon Press, Oxford University Press, Oxford, 1986.[25] M. L. Racine, Maximal Subalgebras of Exceptional Jordan Algebras, Journal of Algebra (1977), 12–21.[26] G. Segal, Loop groups and harmonic maps, Advances in homotopy theory (Cortona, 1988),153–164, London Math. Soc. Lecture Notes Ser., 139, Cambridge Univ. Press, Cambridge,1989.[27] J. Sekiguchi, Remarks on nilpotent orbits of a symmetric pair, J. Math. Soc. Japan (1987),127–138.[28] M. Svensson, J. C. Wood, New constructions of twistor lifts for harmonic maps, ManuscriptaMath. (2014), 457–502.[29] M. Svensson, J. C. Wood, Harmonic maps into the exceptional symmetric space G / SO(4),J. Lond. Math. Soc. (2) (2015), 1, 291–319.[30] K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J.Differential Geom. (1987), (1989), 1–50.[31] J. A. Wolf, Elliptic spaces in Grassmann manifolds, Illinois J. Math., (1963), 447-462. Centro de Matem´atica e Aplicac¸˜oes (CMA-UBI), Universidade da Beira Interior,6201 – 001 Covilh˜a, Portugal.
E-mail address : [email protected] Centro de Matem´atica e Aplicac¸˜oes (CMA-UBI), Universidade da Beira Interior,6201 – 001 Covilh˜a, Portugal.
E-mail address : [email protected] Department of Mathematics and Computer Science, University of Southern Den-mark, Campusvej 55, 5230 Odense M, Denmark.
E-mail address ::