Harmonic superconformal maps of surfaces in Hn

Journal of Geometry and Physics 42 (2002) 139–165

Harmonic superconformal maps of surfaces in H n Eduardo Hulett Facultad de Matemática, Astronomía y Física, Ciudad Universitaria, 5000 Córdoba, Argentina Received 7 March 2001

Abstract The class of harmonic superconformal maps from Riemann surfaces into real hyperbolic spaces is considered and harmonic sequences are constructed for these maps. They are used to obtain a rigidity result for such maps and to construct primitive lifts into an auxiliary flag space Fm . It is also shown that superconformal harmonic maps into H 2m and H 2m−1 are locally described by 2D-affine Toda fields associated to the pair (so(2m+1, C), σ ), where σ is the involution determined by the non-compact real form so(1, 2m). Applying the Adler–Kostant–Symes integration scheme to appropriate loop algebras we construct finite type primitive maps ψ : H 2 → Fm , and harmonic superconformal maps f : H 2 → H 2m and hence finite type Toda fields. © 2002 Elsevier Science B.V. All rights reserved. MSC: 43A17; 35Q58 Subj. Class.: Differential geometry Keywords: Harmonic maps; Harmonic sequences; Generalized 2D-affine Toda fields; Loop algebras

1. Introduction In recent years, there has been a great deal of interest in the study of harmonic maps from surfaces to Riemannian symmetric spaces of compact type. The fact that the harmonic map equation for surfaces in these spaces is a kind of completely integrable system, was the starting point for the construction of harmonic maps from compact surfaces of genus one (i.e. two-tori) in compact symmetric spaces and Lie groups using ideas coming from soliton theory. There exist now a well established theory exposed in many articles, e.g. [4,5,7], to name only the most relevant to us. When the target is a symmetric space of non-compact type, the integrable character of the harmonic map equation of surfaces still prevails as was proved by Bobenko [1] for surfaces E-mail address: [email protected] (E. Hulett). 0393-0440/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 9 3 - 0 4 4 0 ( 0 1 ) 0 0 0 8 2 - 1

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E. Hulett / Journal of Geometry and Physics 42 (2002) 139–165

in Rn and H n . However, the maximum principle for (sub)harmonic functions implies that there are no non-constant harmonic maps from compact surfaces into these spaces. From the point of view of Mathematical Physics the interest in harmonic maps of surfaces stems from its relation with σ -models. A classical solution of a σ -model is nothing but a harmonic map into some Riemannian manifold. The study of σ -models on non-compact (pseudo)-symmetric spaces arise for example in solid state physics and has many applications, see for instance [12] and the bibliography therein. The goal of the present article is to begin the study of harmonic superconformal maps from non-compact connected Riemann surfaces into the the real n-dimensional hyperbolic space H n of curvature −1. This class of maps can be regarded the natural counterpart of the harmonic superconformal maps of surfaces into Euclidean spheres S n considered in [6]. Roughly speaking, a superconformal harmonic map is one whose harmonic sequence satisfies certain “orthogonality relations”. In [6] a well-known soliton system called 2D-affine Toda field equations, were used by Bolton et al. to describe it superconformal harmonic maps of surfaces into CPn and S n . Let σ denote the conjugation authomorphism of so(2m + 1, C) respect to the non-compact real form so(1, 2m). Along the paper we shall see that a 2D-affine Toda field associated with the pair (so(2m + 1, C); σ ), locally describes the geometry of harmonic superconformal maps of surfaces into H 2m and H 2m−1 . The paper is organized as follows. In Section 2, we recall some standard linear algebra and introduce some notation. We derive the harmonic map equation for conformal maps of surfaces into H n . It is a semilinear elliptic equation which, as consequence of the maximum principle for subharmonic functions, it has only constant solutions if the domain is a compact surface. Hence, only non-compact surfaces are interesting. A brief Lie-algebraic introduction to generalized 2D-affine Toda field equations is also given. In Section 3, we study complex vector subbundles E of the trivial bundle M ×Cn+1 → M over a Riemann surface M satisfying E ∩ E ⊥q = {0} fiberwise, i.e. they are non-degenerate respect to the pseudo hermitian metric q(z, w) = −z0 w¯ 0 + nk=1 zk w¯ k on Cn+1 . We show that these bundles have simple metric connections and hence can be equipped with a compatible holomorphic structure via the Koszul–Malgrange theorem [8]. Within this framework, we consider harmonic maps f : M → H n for which their consecutive Hopf differentials ηj = hc ((∂ (j ) f/∂zj ), (∂ (j ) f/∂zj )) dz2j , j = 1, . . . , m − 1 vanish, except the last one ηm , where m = [(n + 1)/2]. The vanishing of the first Hopf differential η1 = hc (fz , fz )dz2 is equivalent to conformality of f , while the vanishig of η2 is just conformality of the second fundamental form of f and so on. These conditions allow to generate inductively an ordered sequence of non-degenerate line subbundles of M × Cn+1 : L−m , L−(m−1) , . . . , L−1 , L0 , L1 , L2 , . . . , Lm ,

with

L¯ j = L−j ,

satisfying orthogonallity relations Li ⊥q Lj , 0 < |i−j | ≤ 2m. Where the pseudohermitian metric q is positive on Lj for |j | = 1, . . . , m and negative on L0 . Besides there is a Gramm–Schmidt type algorithm which on every local chart (U, z) generates a meromorphic section fj of Lj such that ηj = hc (fj , fj )dz2j . Our approach is conceptually analogous to that of [3,8], where (euclidean) harmonic sequences are constructed for harmonic maps of surfaces into compact Grassmanians and projective spaces. Applying this construction we show in Proposition 3.4 that a harmonic superconformal map f : M → H 2m which lies fully into a totally geodesic H 2m−1 , has a periodic sequence


141

{Lj }, i.e. Lj +2m = Lj , j ∈ Z. This happens also when the target is the standard sphere S 2m (cf [3]). In Theorem 3.6, we show that a harmonic superconformal map f : M → H 2m is determined up to ambient isometries by the induced metric f ∗ h and the mth Hopf differential ηm . In Section 4, the geometry of the harmonic sequence is used to construct primitive lifts of superconformal harmonic maps f : M → H 2m into an auxiliar flag domain Fm . This is a non-compact analog of some twistor constructions for harmonic maps into spheres and projective spaces (cf. [6]). Also special adapted frames or Toda frames for super conformal harmonic maps f : M → H 2m are considered and the 2D-affine Toda field equations associated with the pair (so(2m + 1, C), so(1, 2m)) are shown to be equivalent to the integrability conditions for the existence of a Toda frame. Finally, in Section 5 after introducing some standard Loop-group machinery, we use the Adler–Kostant–Symes integration scheme to construct “finite type” solutions of the 2D-affine Toda field associated to (so(2m + 1, C), σ ). Here, in contrast with the compact target case (S n , CPn etc.) the solutions of the corresponding Toda equations are only local, i.e. they are not defined on the whole complex plane. Thus, modulo conformal maps of the plane, we obtain a recipe for the construction of finite type superconformal harmonic maps and primitive maps of the Poincare disc H 2 . A related question is under what conditions a superconformal harmonic (respective primitive ) map from H 2 is of finite type. This problem, however, seems to be more difficult and will be considered later.

2. Preliminaries The n-dimensional Hyperbolic space H n is the simply connected real space-form of constant sectional curvature −1. It can be realized as the unbounded sheet of the hyperboloid h(x, x) = −1 in Rn+1 containing the point e0 = (1, 0, . . . , 0), where h(x, y) = −x0 y0 + 1 n n+1 k=1 xk yk , x, y ∈ R1 . The ambient Lorenzian inner product h induces on H n a positive definite metric denoted with h and the Lie group SOo (1, n) acts transitively on H n by isometries. In this way H n becomes an homogeneous symmetric space of non-compact type. Taking the SOo (1, n)-orbit of e0 we have the representation H n = SOo (1, n)/{1} × SO(n). For later we introduce the complex bilinear extension of h to Cn+1 by hc (z, w) = use, n −z0 w0 + k=1 zk wk . Defining q(z, w) =: hc (z, w), ¯ we obtain the Hermitian extension of h to Cn+1 as well, hence q(x, y) = h(x, y) ∀x, y ∈ Rn+1 1 . Recall that a matrix F is in O(1, n) iff FJFT J −1 = I , where J = diag(−1, 1, . . . , 1). Thus, X is in so(1, n) = Lie(O(1, n)) iff XT J + JX = 0. Hence, so(1, n) is the set of matrices X of the form 0 b , b ∈ Rn , A ∈ so(n) X= bT A On the other hand, the map 0 ib , X → LXL¯ = −ibT A

L = diag(i, 1, . . . , 1)

(2.1)

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allows to think of so(1, n) sitting inside so(n+1, C). Hence, so(1, n)C = so(n+1, C), and the conjugation authomorphism σ : so(n+1, C) → so(n+1, C) respect to the non-compact real form so(1, n) is given by 0 −m ¯ 0 m → σ (X) = J · X¯ · J = so(n + 1, C) X = (2.2) −mT A m ¯ T A¯ Notice that the map F → LFL¯ embeds O(1, n) into O(n + 1, C). 2.1. The harmonic map equation Let M be a Riemannian manifold and f : M → H n a smooth map. Then its second fundamental form is given by β(U, V ) = UVf − h(Uf , Vf)f − df (∇UM V ),

U, V ∈ X(M)

The tension field of f is τ (f ) = tr β. If M is a Riemann surface (i.e. an oriented 2D manifold equipped with a conformal equivalence class of Riemannian metrics), and f : M → H n a non-constant map, then f is called weakly-conformal if hc (fz , fz ) ≡ 0 for every complex local coordinate z on M. A non-constant weakly-conformal harmonic map f is called a minimal branched immersion in the literature. Unless otherwise stated we will work with conformal maps f : M → H n such that dfx = 0 ∀x ∈ M. In the conformal case, the induced metric g = f ∗ h can be written in a local complex coordinate (U, z) as g = ρ 2 dz · d¯z for some smooth positive function ρ : U → R. Equivalently, q(fz , fz¯ ) = 0

and

q(fz , fz ) = ρ 2

(2.3)

Proposition 2.1. Harmonic map equation: let f : M → H n be a conformal map from a Riemann surface M. Then f : M → H n is harmonic and hence minimal if and only if on every complex chart (z, U ) ∈ M the following equation holds fz¯z = q(fz , fz )f

(2.4)

Proof. Let X be a tangent vector field on H n , then the Levi-Civita connection ∇ H on H n is given by H Y = Xx Y − h(X, Y )x ∇X x

for every Y ∈ X(H n )

From this, we compute the second fundamental form of f , β(U, V ) = UVf − h(Uf, Vf )f − df(∇UM V )

U, V ∈ X(M)

Respect to a complex chart we have (||fx ||4 /4)tr β = β(∂/∂z, ∂/∂z), thus, (2.4) follows.


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The maximum principle for sub-harmonic functions, imply that the only harmonic maps from a compact Riemann surface M to H n are the constant maps. Therefore, we will treat only non-compact Riemann surfaces. 2.2. 2D-affine Toda fields They are systems of elliptic PDEs which exist in association with complex simple Lie algebras. To introduce these equations we recall some standard facts of Lie algebra theory. Let g be a complex simple Lie algebra, h a Cartan sub algebra and ∆ = ∆(gC , h) the corresponding root system. Choose a set of positive roots ∆+ ⊂ ∆, and let π = {α1 , . . . , αr } ⊂ ∆+ be the corresponding set of simple roots, where r = dim h. Fix a maximal abelian subalgebra t ⊂ h so that tC = h. Being g simple, one has a distinguished maximal root µ and we set π ∗ = π ∪ {−µ}. Thus, π ∗ labels the sets of nodes of the extended Dynkin diagram of g. The generalized 2D-affine Toda field system associated to g is defined by the following system of r = dim h non-linear PDEs. 2Ωz¯z = dα e2α(Ω) Hα , Ω(z, z¯ ) ∈ it (2.5) α∈π ∗

where Hα ∈ it is the dual of α respect to the Killing form of g, and dα are real arbitrary constants. Solutions of (2.5) are called Toda fields. If σ denotes the involution of so(2m + 1, C) determined by the non-compact real form so(1, 2m), we shall see that a 2D-affine Toda field associated to the pair (so(2m + 1, C); σ ), locally describe the geometry of harmonic superconformal maps from a Riemann surface M into H 2m and H 2m−1 .

3. Harmonic sequences Let T → CPn the tautological line bundle, i.e. the complex line bundle whose fiber at the point x ∈ CPn is the complex line x itself. By abuse of notation we denote also by T the restriction to CHn = {[v] ∈ CPn : q(v, v) < 0}. The hermitian ambient metric q induces on T a negative definite hermitian metric qT and Hom(T , T ⊥ ) is equipped with the corresponding tensor product hermitian holomorphic structure. For such a structure there is a compatible connection D on Hom(T , T ⊥ ) induced by the flat connection on CHn × Cn+1 → CHn in the following way: (DX σ )s = πT ⊥ (X(σ ◦ s)) − σ (πT (Xs)),

X ∈ TM,

where s ∈ C ∞ (T ) is a local smooth section of T , σ is a smooth section of Hom(T , T ⊥ ), and πT ⊥ is the projection onto the q-orthogonal complement T ⊥ (note that T ∩ T ⊥ = {0}, hence, πT ⊥ and πT are well defined). There is also a connection-preserving biholomorphic isomorphism: γ

T (1,0) CHn →Hom(T , T ⊥ ),

γ (X)s = πT ⊥ (Xs),

s ∈ C ∞ (T ), X ∈ TM

(3.1)

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which satisfies, c qT ⊗ qT ⊥ (γ z, γ w) = − g(z, w), z, w ∈ T (1,0) CHn 2 where c < 0 is the holomorphic sectional curvature of the Kähler metric g on CHn . There is an equivalent description of the geometry of CPn1 = {[v] ∈ CPn : q(v, v) > 0}. For details, we refer the reader to [8,9]. ⊥ A vector sub bundle E ⊂ M × Cn+1 is non-degenerate respect to q, if Ex ∩ Ex q = {0} ∀x ∈ M. When E is a line subbundle, i.e. complex 1D, then E is non-degenerate iff q(v, v) = 0 for every 0 = v ∈ E. Hence, when the base M is connected, the induced metric qE is always definite with the same sign on M. Consequently, we call a line subbundle E ⊂ M × Cn+1 positive or negative if qE is positive definite or negative definite, respectively on M. Recall now the bijective correspondence between smooth maps ϕ : M → CPn and smooth complex line subbundles of M × Cn+1 given by ϕ ↔ E0 = ϕ ∗ (T ) (3.2) Under (3.2), a smooth map ϕ : M → CHn corresponds to a non-degenerate negative line subbundle E ⊂ M × Cn+1 . Analogously a smooth map ϕ : M → CPn1 corresponds under (3.2) to a non-degenerate positive line subbundle E ⊂ M × Cn+1 . Non-degenerate vector subbundles E ⊂ M × Cn+1 are useful since they have metriccompatible connections. In fact, for such a vector sub bundle, there are unambiguously defined orthogonal projectors πEx : Cn+1 → Ex , x ∈ M, hence, a well defined connection ∇E = πE ◦ d, i.e. (∇E )X s = πE (Xs),

s ∈ C ∞ (E), X ∈ TM

Note that the flat connection on M × Cn+1 → M is given by ∇X s = Xs for s : M → Cn+1 and X ∈ TM. Given a complex vector bundle V over a Riemann surface M with a metric-compatible connection ∇, the well-known theorem of Koszul–Malgrange [8] guarantees the existence of a unique holomorphic structure on V compatible with ∇. A local section s is holomorphic respect to this structure if and only if ∇X s = 0 for every X ∈ T (0,1) M. In what follows, we shall consider every non-degenerate vector sub bundle E ⊂ M×Cn+1 equiped with the induced metric qE and the unique holomorphic structure determined by ∇E = πE ◦ d, via the theorem of Koszul–Malgrange. Hence, a local section s ∈ C ∞ (E) is holomorphic if and only if (∂/∂z)s is q-orthogonal to E for any complex coordinate (U, z) on M. Respect to a complex coordinate we denote (∇E )(∂/∂z) and (∇E )(∂/∂z) simply by (∇E ) , and (∇E ) , respectively. Being E is non-degenerate, there is also a well-defined operator AE : TM ⊗ E → E ⊥ given by AE (X ⊗ s) =: πE ⊥ (Xs),

s ∈ C ∞ (E)

(3.3)

Notice that with our previous definitions AE = ∇E ⊥ . According to the splitting of the complexified tangent bundle TMC into (1, 0) and (0, 1) types, the second fundamental form decomposes relative to a complex chart (z, U ) as AE = AE + AE , where ∂ ∂ AE (s) = πE ⊥ s , AE (s) = πE ⊥ s (3.4) ∂z ∂z


145

Remark 3.1. We shall regard AE and AE simply as bundle maps E → E ⊥ . This of course presupposes working on a specific complex chart but, it is routine calculation to check that all the constructions that follow, are independent of the choice of such a chart. Lemma 3.1. Let E be non-degenerate vector subbundle of M × Cn+1 . Then (AE )∗ = −AE ⊥ . Proof. Let s1 and s2 be smooth local sections of E and E ⊥ , respectively. Then, respect to any complex coordinate z, 0 = q(∂z s1 , s2 ) + q(s1 , ∂z¯ s2 ). Hence, q(AE (s1 ), s2 ) = −q(s1 , AE ⊥ (s2 )) and the Lemma follows. Now let ϕ : M → CPn be the unique smooth map determined by a non-degenerate line sub bundle E. Then, if E is negative, ϕ factors through CHn and if E is positive ϕ factors through CPn1 , respectively. Proposition 3.2. In either case ϕ is a harmonic map if and only if AE is a holomorphic section of Hom(E, E ⊥ ). Equivalently, ϕ is harmonic if and only if AE is antiholomorphic. Proof. Let us prove the statement when E is negative since the positive case is analogous. Let ϕ : M → CHn be the corresponding map. By definition ϕ is harmonic if and only if tr(∇ dϕ) = 0, where ∇ is the connection on Hom(TM, TCHn ) determined by the respective n connections ∇ M and ∇ ϕ = ϕ ∗ (∇ CH ). Extending dϕ to a complex linear map from TMC to TCHn , it follows that ϕ is harmonic if and only if on any complex coordinate (z, U ), ϕ

∇∂z¯ dϕ(∂z ) = 0

(3.5)

or, in an equivalent way, ϕ

∇∂z dϕ(∂z¯ ) = 0

(3.6)

Up to identifications (3.4) and (3.1) we get dϕ(∂z¯ ) = AEϕ and dϕ(∂z ) = AEϕ . The pull-back connection ∇ ϕ corresponds under (3.1) to ∇E . Then ϕ is harmonic if and only if AE is a holomorphic section of Hom(E, E ⊥ ). This is equivalent to ∇E AE = 0. In other words, ϕ is harmonic if and only if AE ◦ ∇E = ∇E ⊥ ◦ AE

(3.7)

For a smooth map f : M → H n define ϕ0 : M → CPn by ϕ0 (x) = [f (x)], i.e. ϕ0 (x) is the complex line in Cn+1 determined by f (x). Since h(f, f ) = −1, we have q(λf, λf ) = −|λ|2 < 0,

∀λ ∈ C∗

Hence, ϕ0 factors through the (open) sub manifold CHn ⊂ CPn . Proposition 3.3. Let f : M → H n be a conformal map. Then f is harmonic iff ϕ0 is harmonic.

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Proof. Let L0 = {(x, v) ∈ M × Cn+1 |v ∈ ϕ0 (x)} = ϕ0∗ (T ) ⊂ M × Cn+1 Then h(f, f ) = −1 implies q(fz , f ) = q(fz¯ , f ) = 0 for any complex coordinate, thus, ∇L 0 (f ) = 0. This says that f is a (global) holomorphic section of L0 . On the other hand, f being conformal harmonic and non-constant, it satisfies fz¯z = q(fz , fz )f on every complex chart (z, U ). This clearly implies that ∇L ⊥ ◦AL0 (f ) = 0. Thus, AL0 is holomorphic, hence, 0 ϕ0 is harmonic. Conversely, if ϕ0 = [f ] is harmonic, then (3.7) holds, and since h(f, f ) = −1, one gets ∇L ⊥ ◦ AL0 (f ) = 0. That is, ∇L ⊥ (fz ) = 0. Hence, fz¯z ∈ L0 , and there is a (local) complex 0

0

function λ such that fz¯z = λf . From this, we conclude that fz¯z = q(fz , fz )f , so is f harmonic.

Let f : M → H n be a non-constant conformal harmonic map, then by the Proposition 3.3 f is a global holomorphic section of L0 and AL0 , AL0 are non-vanishing holomorphic and antiholomorphic sections of Hom(L0 , L⊥ 0 ), respectively. The zeros of AL0 and AL0 are isolated and there are unique line sub bundles (extending through the zeros) L1 , L−1 ⊂ L⊥ 0 satisfying Im(AL0 ) ⊆ L1 ,

Im(AL0 ) ⊆ L−1

Hence, L1 and L−1 are positive line sub bundles of M × Cn+1 such that L¯ 1 = L−1 . For, on any complex chart z : U → C, the complex derivatives fz and fz¯ are local smooth sections of L1 and L−1 , respectively and (outside possibly of a closed subset of isolated points of U ), q(fz , fz ) = q(fz¯ , fz¯ ) > 0. In particular, L1 and L−1 are non-degenerate and they can be equipped with holomorphic structures determined by ∇L1 and ∇L−1 via the theorem of Koszul–Malgrange. Moreover, L0 ⊥ L1 , and L0 ⊥ L−1 because of h(f, f ) = −1,

h(fz , f ) = 0,

h(fz¯ , f ) = 0

(3.8)

The orthogonality of L1 and L−1 follows from the vanishing of the first complex Hopf quadratic differential η1 =: hc (fz , fz ) dz2 . Summing up these simple facts, we conclude that {L−1 , L0 , L1 } is a set of mutually q-orthogonal non-degenerate line sub bundles satisfying L¯ 0 = L0 ,

L¯ 1 = L−1

(3.9)

Let ϕ1 and ϕ−1 : M → CPn be smooth maps which are in one-to-one correspondence with L1 and L−1 , respectively. Then by our preceeding considerations ϕ1 and ϕ−1 factor through the (open) sub manifold CPn1 ⊂ CPn . ⊥ Being both inclusions i : L1 2 inductive steps we have η1 ≡ η2 ≡ · · · ≡ ηk−1 ≡ 0, and on a complex local chart (U, z) in M we have recursively generated Cn+1 -valued maps by the algorithm fj +1 = (∂/∂z)fj − (q((∂/∂z)fj , fj )/||fj ||2 )fj , j = 1, . . . , k − 2, which are positive, i.e. ||fj ||2 > 0 since by construction and the vanishing of the ηj , q(fj , f ) = 0 holds, thus, Re fj , Im fj ∈ Tf H n , and since fj ≡ 0, then q(fj , fj ) > 0. These local generated Cn+1 -valued maps fj give rise to a finite sequence of non-degenerate line sub bundles L−k , L−k+1 , . . . , L−1 , L0 , L1 , . . . , Lk−1 , Lk , all which are positive except L0 which is negative, satisfying q-orthogonality relations, Li ⊥ Lj ,

for 0 < |i − j | ≤ 2k − 1.

If ηk ≡ 0, then its ceros are isolated since it is a holomorphic 2k-differential which measures the orthogonality of L−k and Lk . AL

j

By Lemma 3.1, there are holomorphic bundle maps Lj → Lj +1 , and antiholomorphic AL

j

bundle maps Lj → Lj −1 with (ALj )∗ = −ALj +1 with ALj (fj ) = fj +1 , and ALj (fj ) = fj −1 . Due to the correspondence Lj ↔ ϕj there is a sequence of maps ϕ−k , . . . , ϕ−1 , ϕ0 , ϕ1 , . . . , ϕk which are harmonic into CHn for j = 0, and into CPn1 if j = 0. Note that the finite sequence of Cn+1 -valued maps f−k , f−k+1 , . . . , f−1 , f = f0 , f1 , f2 , . . . , fk is locally generated from f on a complex chart (z, U ) by a Gramm–Schmidt type algorithm namely,  ∂fj ∂  2  fj log||f + || = f  j +1 j   ∂z ∂z (3.18)   ∂fj ||fj ||2   fj −1 =−  ∂ z¯ ||fj −1 ||2 where ||fj ||2 = q(fj , fj ) > 0 for |j | = 0 and ||f ||2 = −1. In particular, this shows directly that each fj is a local meromorphic ssection of the corresponding line bundle Lj .


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In this paper, we distinguish a class of maps caracterized by the non-vanishing of the last Hopf differential. This motivates the following. Definition 3.1. Let f : M → H n be a harmonic (non-constant) map and let m = [(n + 1)/2]. We say that f is, superconformal if η1 ≡ η2 ≡ · · · ≡ ηm−1 ≡ 0, but ηm ≡ 0. Remark 3.3. Using induction and formulas (3.18), it can be shown that the above definition does not depend on a particular local complex coordinate. This is left to the reader.A second class of harmonic maps f : M → H n is obtained if one requires that η1 ≡ η2 ≡ · · · ≡ ηm ≡ 0. These are called isotropic in the literature, and were studied in [9]. It can be shown that isotropic harmonic surfaces in H n can be obtained projecting suitable complex curves into some auxiliar flag space. This will be the matter of a subsequent paper (see [10]). According to the preceeding definition a harmonic superconformal map f : M → H2 is precisely a non-conformal harmonic map, whereas a conformal harmonic map f : M → H 4 is superconformal or isotropic, i.e. η2 ≡ 0. As in the case of the spheres S n in [2], there are two types of superconformal harmonic maps into H 2m namely, those which are linearly full into H 2m , and those which are linearly full into some H 2m−1 totally geodesic immersed in H 2m . The second case is simpler and deserves separate attention. Proposition 3.4. Let f : M → H 2m be a harmonic superconformal map which is linearly full into some totally geodesic copy of H 2m−1 immersed in H 2m . Then the sequence of line bundles Lk of f is periodic of period 2m, i.e. Lk+2m = Lk for every integer k. Proof. By hypothesis, there exist a positive vector n ∈ R2m+1 such that f is full into 1 2m 2m−1 totally geodesic immersed into H 2m where V = n⊥ . V ∩ H , which is a copy of H C In particular, Lk ⊂ V for every k and, Li ⊥ Lj for 0 < |i − j | ≤ 2m − 1. Hence, L−m ⊥ Lj for j = −m + 1, . . . , m − 1. This forces L−m = Lm , since by hypothesis C (⊕m−1 k=−m+1 Lk ) ⊕ Lm = V , hence the 2m-periodicity of the sequence Lk follows. This does not occur however with the harmonic superconformal maps f which are linearly full into H 2m . Applying (3.18) and induction we obtain the following formulae which are analogous as those obtained in [2] for S n . Proposition 3.5. Let f : M → H 2m be a superconformal harmonic map. Then f¯j f−j = (−1)j +1 , for 0 ≤ j ≤ m (3.19) ||fj ||2 Here, we consider the following question: which invariants determine a superconformal harmonic map up to ambient isometries? Theorem 3.6. Let f, fˆ : M → H 2m be harmonic superconformal maps having the same induced metrics and the same mth Hopf differentials. Then there is an isometry Φ of H 2m such that Φ ◦ fˆ = f .

150


Proof. We assume that f and fˆ are full and introduce (globally defined) real forms γj = τj |dz|2 and γˆj = τˆj |dz|2 , where τj = ||fj +1 ||2 ||fj ||−2 and τˆj = ||fˆj +1 ||2 ||fˆj ||−2 . Since by hypothesis f ∗ h = fˆ∗ h we have γ0 = γˆ0 , and γ−1 = γˆ−1 . As a consequence of the Gramm–Schmidt type algorithm (3.18) on a complex chart (U, z) we have ∂2 log||fj ||2 = τj − τj −1 , ∂z∂ z¯

|j | = 0, . . . , m

From this, we deduce that γ−1 and γ0 determine γj for |j | = 1, . . . , m. In particular, τj = τˆj for |j | = 0, 1, . . . , m, hence, ||fj || = ||fˆj || for |j | = 0, 1, . . . , m. On the other hand, by hypothesis we know that q(fi , fj ) = q(fî , fˆj ) = 0, 0 < |i − j | ≤ 2m q(fi , f¯i ) = q(fî , f¯î ) = 0, i = 1, . . . , m − 1 q(fm , f¯m ) = q(fˆm , f¯ˆm ) But {fj , f¯j : j = 0, . . . , m} spans C2m+1 . Hence, there is a matrix A = A(z, z¯ ) ∈ SU(1, 2m) such that fˆj = Afj ,

f¯ˆj = Af¯j ,

j = 0, . . . , m

Thus, A = A¯ and using (3.18), we see that (∂/∂z)A = (∂/∂ z¯ )A = 0, i.e. A is a constant matrix in SO(1, 2m). One can suppose (composing with isometries of H 2m if necessary) that there is a point p0 ∈ M such that f (p0 ) = g(p0 ) = e0 , the first vector of the canonical basis. In particular, Ae0 = e0 thus A ∈ SOo (1, 2m).

4. Primitive lifts 4.1. The flag domain Let Fm be the set of ordered sequences (L0 , L1 , . . . , Lm ), Lm+1 of mutually q-orthogonal , complex lines in C2m+1 , where L0 is the complexification of a negative line in R2m+1 1 and Lj ; j = 1, . . . , m are positive complex lines. The pseudo-orthogonal Lie group SOo (1, 2m) acts transitively on Lm in the usual way g.(L0 , L1 , L2 , . . . , Lm ) = (gL0 , gL1 , gL2 , . . . , gLm ) Let L00 = Ce0 and L0k = C(e2k−1 − ie2k ), where ek is the standard k + 1-basis vector of C2m+1 . Then the isotropy of the point o = (L00 , L01 , . . . , L0m ) ∈ Fm is the mD compact torus cos ϕj − sin ϕj T = diag(1, R(ϕ1 ), R(ϕ2 ), . . . , R(ϕm ))|ϕk ∈ R}; R(ϕ) = sin ϕj cos ϕj


151

Thus, we may identify Fm with the quotient SOo (1, 2m)/T , and there is a natural projection P : Fm → H 2m defined as follows. Given L = (L0 , L1 , . . . , L√m ) ∈ Fm , we may pick q-unit basis vectors Vj ∈ Lj , j = 1, . . . , m and put Vj = (1/ 2)(v2j −1 − iv2j ), determined by the following two for j = 1, . . . , m and v0 , the unique vector in R2m+1 1 conditions: h(v0 , v0 ) = −1 {v0 , v1 , v2 , . . . , v2m } is a positively oriented h-orthonormal basis of R2m+1 1 Let P (L) = v0 ∈ H 2m . Thus, P sends an element [F ] ∈ Fm to the 0th column F0 of F ∈ SOo (1, 2m). According to this definition, a point L ∈ Fm is determined by P (L) and L1 , L2 , . . . , Lm−1 . The Killing form of so(1, 2m) induces a SOo (1, 2m)-invariant pseudo-Riemannian metric on Fm , namely, "X, Y # = 21 tr(X.Y ),

X, Y ∈ To (Fm )

With this metric the projection P : Fm → H 2m becomes a (pseudo)-Riemannian submersion since the metric h on H 2m is given by h(X, Y ) = (1/2)tr(X.Y ), for X, Y ∈ Te0 (H 2m ). There is also a 2m-symmetric structure on Fm defined as follows. Let t be the Lie algebra of T : 0 −u t = {diag(0, θ(u1 ), . . . , θ (um ))|uk ∈ R}, θ (u) = u 0 Set σj [diag(0, θ(u1 ), . . . , θ (um ))] = iuj . Then as positive simple roots we take π+ = {α1 , α2 , . . . , αm }, where α1 = σ1 , α2 = σ2 − σ1 , . . . , αm = σm − σm−1 , and µ = σm + σm−1 is the highest root. 1 m Now pick elements Zk ∈ itm so that αj (Zk ) = δjk and put Z = 2m k=1 Zk . Then τ = Ad(exp2πiZ) is an automorphism of so(2m + 1, C) such that τ 2m = Id. It is easily checked that τ preserves the real form so(1, 2m), and if we denote also by τ the corresponding automorphism induced on SOo (1, 2m) (note that τ is determined up to conjugation by an element of SOo (1, 2m)), then τ (T ) ⊂ T and hence τ determines an isometry τ : Fm → Fm of order 2m. We compute Zk = i diag(0, θ(0), θ (0), . . . , θ(0), θ (−1)k , . . . , θ (−1)). From this one can write down the 2m-symmetry τ = Ad(2π iZ) explicitly,

π 2π (m − 1)π ,R , ..., R , R(π ) (4.1) τ = Ad diag 1, R m m m cos ϕ − sin ϕ where, R(ϕ) = sin ϕ cos ϕ

152


Example 4.1. Note that for m = 2 and 3 the symmetry τ is given, respectively by   1 0 0 0 0 0 0   √   1 3   0   − 0 0 0 0    2 2 1 0 0 0 0   √     3 1     0  0 0 0 0  0  0 0 −1 0     2 2     √  ; Ad   0 1 0 0 0 Ad  3 1   0 0 − 0 0  0 −     2 2     −1 0  √ 0 0 0       3 1 0 0 0 − 0 0    0 0 0 0 −1 2 2     0 0 0 0 0 −1 0    0 0 0 0 0 0 −1 Let ν = eiπ/m . We have, thus, a direct sum decomposition so(2m + 1, C) = ⊕j ∈Z2m mj ,

mj = {X ∈ so(2m + 1, C)|τ (X) = ν j X}

Recall now from (2.2) the conjugation atomorphism σ : so(2m + 1, C) → so(2m + 1, C) respect to the non-compact real form so(1, 2m). We need the following. Proposition 4.1. The automorphisms τ and σ commute, i.e. σ ◦ τ = τ ◦ σ . Proof. Easily follows from the definitions of τ and σ and is left to the reader.

As a consequence the ν j -eigenspaces mj satisfy [mj , mk ] ⊂ mj +k , j, k ∈ Z2m

σ (mj ) = m−j .

(4.2)

Note that since m0 = tC , σ (m0 ) = m0 . Moreover, each mk is Ad(T )-invariant and determines an SOo (1, 2m)-invariant subbundle [mk ] of the complexified tangent bundle T (Fm )C of Fm , namely, [mk ](xT) = Ad(x)(L¯ · mk · L),

x ∈ SOo (1, 2m),

2m where L = diag(i, 1, . . . , 1 ). Hence, we have the following identification T (Fm )C = ⊕k∈Z2m [mk ] On the other hand, we know from the general theory [6] that m1 decomposes into a direct sum, m1 =

m k=0

gαk ,

(4.3)


153

where gαk ⊂ so(2m + 1, C) is the αk -root space, and α0 = −µ. From this we obtain the following. Lemma 4.2. The subspace m1 ⊂ so(2m+1, C) consists of matrices X having the following form,   0 0 ··· 0 0 Q1     −QT 0 Q 0 · · · 0   2 2 1     T 0  0 −Q Q · · · 0 2 3 2      ..    T (4.4) X = 0 0 −Q3 02 . . . .       . .. .. ..   .. . . .       0 ··· ··· 0 02 Qm    T 0 ··· ··· 0 −Qm 02 where Q1 = a1 (−1 , i), Qm =

a

b

ia

ib

Qj = aj

,

−1

i

−i

−1

,

j = 1, . . . , (m − 1);

aj , a, b ∈ C

According to [6] we introduce the following. Definition 4.1. An element X ∈ m1 is called cyclic if a, b, a1 , b2 , . . . , am−1 , bm−1 ∈ C − {0} and |a| + |b| > 0. We refer the reader to Lemma 2.1 of [6] for the characterization of cyclic elements in terms of the homogeneous generators of the ring of Ad(SO(2m + 1, C))-invariant polinomials. Definition 4.2. A map ψ : M → Fm is called τ -primitive if dψ(T (1,0) M) ⊂ [m1 ], and contains a cyclic element, (i.e. dψ(T (1,0) M) is not contained in any sub bundle obtained from m1 by omitting one of the direct summands in (4.3)). Given a super conformal harmonic map f : M → H 2m , we may construct a lift fˆ : M → Fm as follows. Put L0 = L0 , Lj = Lj for j = 1, . . . , (m − 1), and let Lm be determined by the condition P (L0 , L1 , . . . , Lm−1 , Lm ) = f . Then Lm lives in Lm ⊕ L¯ m = ⊥q and define [⊕m−1 −m+1 Lj ] fˆ = (L0 , L1 , . . . , Lm−1 , Lm )

(4.5)

154


Theorem 4.3. Let f : M → H 2m be a harmonic map. If f is super conformal then the lift fˆ : M → Fm given by (4.5) is primitive. Conversely, if ψ : M → Fm is primitive, then f = P ◦ ψ : M → H 2m is a super conformal harmonic map such that fˆ = ψ. Proof. Assume that the harmonic map f is superconformal. Let {Lj }m j =−m , be the sequence of non-degenerate line bundles of f , and fj the corresponding local section of Lj , generated on a local complex chart (U, z) by algorithm (3.18). Define R2m+1 -valued functions F1 , . . . , F2m on U by   fj = √12 ||fj ||(F2j −1 − iF2j ), j = 1, . . . , (m − 1) (4.6)  2 2 with |F2m−1 || = ||F2m || = 1 fm = aF2m−1 − ibF2m , where a and b are real valued functions on U such that ηm |U = (a 2 − b2 )dz2m , hence a 2 − b2 ≡ 0 on U and ||fm ||2 = a 2 + b2 . We can assume that ||fj || > 0, j = 1, . . . , m on U since fj vanish only at isolated points. Define F = (f, F1 , F2 , . . . , F2m−1 , F2m ). Then by (4.6) and (3.18), it follows that F is O(1, 2m)-valued on U and we assume that F is SOo (1, 2m)-valued on U . We have F · o = (Cf, C(F1 − iF2 ), . . . , C(F2m−1 − iF2m )) Put L0 = Cf,

Lj = C(F2j −1 − iF2j ),

j = 1, . . . , m

(4.7)

Note that Lm = Lm , but Lm ⊂ Lm ⊕ Lm . Define fˆ = (L0 , L1 , . . . , Lm ), hence, F · o = fˆ, i.e. F is a frame of fˆ on U . Now using (4.6) and (3.18) we see that F satisfies, ¯ 0 + A1 )L, F−1 Fz = L(A where A1 ∈ m−1 , and A0 ∈ m0 . In particular, fˆz = F · L¯ · A1 L ∈ [m1 ](F · o) = Ad(F)(L¯ · m1 · L) where A1 ∈ m1 is cyclic by (4.6) and by our assumption ||fj || > 0 on U for j = 1, . . . , m. This shows that fˆ is primitive. Conversely, let ψ : M → Fm be a primitive map, ψ = (L0 , L1 , . . . , Lm ), and set f = P ◦ψ. Let (U, z) be a complex chart in M, and F = (F0 , F1 , . . . , F2m ) : U → SOo (1, 2m) a local frame of ψ then, L0 = Cf,

Lj = C(F2j −1 − iF2m ),

j = 1, . . . , m

¯ 0 + A1 )L In particular, F is also a local frame for f , and being ψ primitive, F−1 Fz = L(A with A0 ∈ m0 and A1 ∈ m1 . Therefore, A1 has the form (4.4) where a1 , . . . , am−1 , a, b, are complex functions on U vanishing only at isolated points. From the isotropy of L1 follows that f is conformal and, from the structure of A1 we deduce that ∇L ⊥ ◦ AL0 (f ) = 0, hence, 0

f : M → H 2m is harmonic. This also follows from the general theory in [7] since f is covered by a primitive map.


155

√ By rotating on the normal √ plane spanned by {F1 , F2 } we can get fz = (||fz ||/ 2) (F1 − iF2 ) or a1 = i||fz ||/ 2, (note that such a rotation leaves the complex line L1 = C(F1 − iF2 ) unchanged). Using the structure of A1 it is easy to check that f1 =: fz is a holomorphic local section of L1 and that f2 =: (∂/∂z)f1 − (q((∂/∂z)f1 , f1 )/||fz ||2 )f1 defines an holomorphic local section of L2 . Performing a rotation on the normal plane spanned by {F√ 3 , F4 } if necessary, we can get (keeping the complex line L2 unchanged), f2 = (||f2 ||/ 2)(F3 − iF4 ), and a2 = ||f2 ||/2||f1 ||. Notice that η2 ≡ 0 since L2 is isotropic. Continuing this way and rotating appropriately (if needed) the vectors {F2j −1 , √ F2j }, we generate inductively local holomorphic sections fj =(||fj ||/ 2)(F2j −1 − iF2j ) ∈ Lj , and matrix coefficients aj = ||fj +1 ||/2||fj || for j = 1, . . . , (m − 1). Finally setting fm = (∂/∂z)fm−1 − (q((∂/∂z)fm−1 , fm−1 )/||fm−1 ||2 )fm−1 , and using the structure of the matrix A1 once more, we see that the last coefficients in A1 are given by a=

||fm−1 ||−1 α, √ 2

b=i

||fm−1 ||−1 β, √ 2

where α, β are real functions on U such that fm = αF2m−1 −iβF2m with α 2 −β 2 vanishing only at isolated points. Define Lm = C · fm . After a straightforward calculation, one concludes that fm is a holomorphic section of Lm , and ηm = (α 2 − β 2 )dz2m is a non-vanishing holomorphic 2m-differential on U . Moreover, Lm ⊂ Lm ⊕ Lm . All this clearly says that f is a superconformal harmonic map into H 2m satisfying ψ = fˆ. 4.2. Special frames The proof of the Theorem 4.3 suggests that we may use the information contained in the harmonic sequence of a superconformal minimal map f : M → H 2m to construct special adapted frames of f . In fact, let f : M → H 2m be superconformal harmonic, and z : U → C a local complex coordinate on a simply connected open set U ⊂ M, then using (3.18) we generate positive C2m+1 -valued {fj }|j | = 1, . . . , m. Then by (3.18), hc (fm , fm ) is a non-vanishing holomorphic function on U and we may choose the local coordinate z so that hc (fm , fm ) = 1 on U , hence z is unique up to arbitrary translations and multiplication by a 2m root of unity. A proof of this fact is given in [6]. We introduce real functions uj on U by ||fj || = euj (recall that ||f0 ||2 = −1) and R2m+1 -valued functions F1 , . . . , F2m−1 , F2m on U by setting euj fj = √ (F2j −1 − iF2j ), 2

j = 1, 2, . . . , m − 1

fm = cosh um (F2m−1 ) − i sinh um (F2m ) From (3.18) it follows that F = (f, F1 , . . . , F2m ) is SO(1, 2m)-valued on U . According to (4.1) the change of the frame F on U is expressed by Fz = FL¯ · A · L

(4.8)

156


where A is given by  0 Q1 0 0 ··· 0  0 ··· 0  −QT1 θ (−iu1 ) Q2  0 −QT2 θ (−iu2 ) Q3 ··· 0    .. A= 0 −QT3 θ(−iu3 ) · · · . 0  .. .. .. ..   . . . .   ··· ··· 0 θ(−ium−1 ) Qm 0 0

···

...

−QTm

0

eu1 euj +1 −uj 1 Q1 = i √ (1 − i), Qj = − 2 i 2 e−um−1 cosh um −i sinh um Qm = − √ 2 i cosh um sinh um

−i 1

        ,       

(4.9)

θ(−ium )

for j = 1, . . . , (m − 2),

Setting Ω = i diag(0, θ(−u1 ), θ (−u2 ), . . . , θ (−um )) ∈ it, then A can be decomposed as follows: A = Ωz + eΩ Me−Ω ∈ m0 ⊕ m1 , where



0

  −M1T   0    M= 0  .  ..   0  0

(4.10)

M1

0

0

···

0

02

M2

0

···

0

−M2T

02

M3

0

−M3T

02

.. .

···

..

.

...

...

02

...

···

T −Mm

where 1 M1 = √ (i, 1), 2   −1 0  Mm =  −i 0

 Mj =

−1

1 2 −i

i −1



    0    ..  .   ∈ m1 ,      Mm   02

(4.11)

 ,

j = 1, . . . , (m − 1),

(4.12)


157

Among (local) frames of a primitive map ψ : M → Fm we distinguish a special kind of frames. Following [6] we call a local frame F : U → SOo (1, 2m) a Toda frame if there is a complex coordinate z : U → C and a smooth map Ω : U → it such that F−1 Fz = Ωz + Ad(eΩ )L¯ · M · L ∈ m0 ⊕ L¯ · m1 · L (4.13) Note that L¯ · H · L = H ∀H ∈ m0 . Remark 4.1. Using harmonic sequences we have produced a local Toda frame for a τ -primitive map ψ : M → Fm as can be seen from Eq. (4.10). A Toda frame F is unique up to multiplication by an element of the compact torus T ⊂ O(1, 2m). 4.3. Integrability conditions Here, we determine the integrability conditions for the existence of a (local) Toda frame of a primitive map ψ : M → Fm . To this end we decompose the matrix M ∈ m1 into a sum of root matrices Xj ∈ so(2m + 1, C) defined by   0     0 i 1 ... ..   .    −i 0 0 · · ·     1  1    02 Pj ...  X1 = √  −1 0 0 · · ·  , Xj =  , 2 2   T −Pj 02 ...   ..   . 0 .. . j = 2, . . . , m, where Pj = −E2j −3,2j −1 + iE2j −3,2j − iE2j −2,2j −1 − E2j −2,2j , 02 represents the 2 × 2-zero matrix and Eα,β the square matrix with 1 in the (α, β) place and zeros outside. The last matrix X0 is defined by   0   ..  1 . , X0 =    2 02 P0  −P0T 02 P0 = −E2m−3,2m−1 − iE2m−3,2m − iE2m−2,2m−1 + E2m−2,2m Note that CXj = gαj is the root space corresponding to the simple root αj . Also M = X0 + X1 + · · · + Xm holds. Moreover, [Xj , σ (Xj )] = tr(Xj · σ (Xj )) · Hj (4.14) [Xk , σ (Xl )] = 0, for k = l

158


where Hj ∈ it are given by H1 = 21 i diag(0, θ(−1), 02 , . . . , 02 ) Hj = 21 i diag(0, 02 , . . . , θ(1), θ (−1), 02 , . . . , 02 ), j = 2, . . . , m H0 =

1 2 i diag(0, 02 ,

j

(4.15)

. . . , 02 , θ(1), θ (1)) m

We leave for the reader to check that Hj ∈ it is dual to αj respect to tr X · Y . Our choice of matrices Xj is such that 2, for j = 1 dj := tr(Xj · σ (Xj )) = −2, for j = 0, 2, . . . , m

(4.16)

From (4.16) and (4.17) we obtain the following. Lemma 4.4. Let L¯ · A · L = F−1 Fz where A is the matrix (4.10) and B = σ (A). Then the integrability condition [A, B] = Az¯ − Bz for the existence of a Toda frame F is equivalent to 2Ωz¯z =

m

dj e2αj (Ω) Hj

(4.17)

j =0

where Ω = i diag(0, θ(−u1 ), θ (−u2 ), . . . , θ(−um )) ∈ it. αj (Ω) X by definition of X . Analogously we see that Proof. Ad(eΩ )M = j α∈π ∗ e j Ω −Ω σ (Ad(e )M) = Ad(e )σ (M) = α∈π ∗ eαj (Ω) σ (Xj ). From this and (4.16) the Lemma follows. Note that σ (gαj ) = gα−j . In terms of the real functions uj Eq. (4.17) is equivalent to the following non-linear elliptic system  2u1z¯z = e2(u2 −u1 ) + e2u1       2ujz¯z = e2(uj +1 −uj ) − e2(uj −uj −1 ) , j = 2, . . . , m − 2 (4.18)  2u(m−1)z¯z = e−2um−1 cosh 2um − e2(um−1 −um−2 )      2umz¯z = −e−2um−1 sinh 2um Theorem 4.5. Let (U, z) be a complex coordinate of M with U ⊂ M a simply connected open subset and let Ω : U → it be a smooth map. Then Eq. (4.13) has a real solution F ∈ SOo (1, 2m) if and only if the field Ω = i diag(0, θ (−u1 ), θ (−u2 ), . . . , θ(−um )) satisfies the 2D-affine Toda field Eqs. (4.17). In this case ψ = π ◦ F : U → Fm is τ -primitive and f (F ) is a Toda frame of ψ.


159

Proof. The integrability condition for the existence of a Toda frame F is (∂/∂z)(∂/∂ z¯ )F = (∂/∂ z¯ )(∂/∂z)F which by (4.8) is equivalent to [A, B] = Az¯ − Bz . This in turn is equivalent to system (4.18). The final statement follows from (4.10). Remark 4.2. Eq. (4.17) with real arbitrary constants dj is called by some authors the generalized (elliptic) 2D-affine Toda field associated so(2m + 1, C) [11]. Our choice of constants dj in (4.16) depend essentially on the matrices Xj and the conjugation authomorphism σ (2.2) determined by the non-compact real form so(1, 2m). For this reason we call (4.17) the 2D-affine Toda field associated to the pair (so(2m + 1, C); σ ). Notice that the 2D-affine Toda fields considered in [6] are in fact associated to a pair (g, k) where g is a complex simple Lie algebra and k is a compact real form of g. Remark 4.3. Note that a Toda frame F with field Ω for which um ≡ 0 gives rise to a primitive map ψ which projects onto a superconformal harmonic map f which is full into a totally geodesic H 2m−1 ⊂ H 2m .

5. Existence Here, we apply the Adler–Kostant–Symes integration scheme to construct finite type primitive maps ψ : H 2 → Fm and harmonic superconformal maps f : H 2 → H 2m . The main reference here is the expository article by Burstall and Pedit [7]. We introduce the standard ingredients of the general theory in order to keep the paper selfcontained. Given a primitive map ψ : M → Fm , there is a lift F : M˜ → SOo (1, 2m) defined on the universal covering space of M. Pulling back the left Maurer Cartan form θ of SO(2m+1, C) by the frame F, one obtains the so(2m + 1, C)-valued one-form α = F−1 dF on M˜ which splits into (1, 0) and (0, 1) types, α

α

α0

1 1 ¯ α = A1 dz + (A0 dz + A0 d¯z) + A1 d¯z

where A0 = A¯ 0 : M˜ → m0 , A1 : M˜ → m1 , with A1 cyclic and A1 = σ (A1 ). The compatibility (or integrability) condition for the existence of such frame is given by the so called zero curvature condition or Murer–Cartan equation for F: dα + 21 [α ∧ α] = 0 In terms of the one-form α0 , α1 , α1 the equation above decomposes into dα0 + 21 [α0 ∧ α0 ] = −[α1 ∧ α1 ];

dα1 + [α0 ∧ α1 ] = 0

(5.1)

The key point is that these equations may be reinterpreted by introducing a complex spectral parameter λ ∈ S 1 in α, namely, αλ =: λα1 + α0 + λ−1 α1 ,

λ ∈ S1

160


Then α = α1 + α0 + α1 satisfies (5.1) if and only if the loop of one-form αλ satisfies dαλ + 21 [αλ ∧ αλ ] = 0

∀λ ∈ S 1

(5.2)

˜ we obtain an S 1 -loop of frames Fλ : M˜ → Integrating (5.2) on the simply connected M, SOo (1, 2m) such that F−1 λ dFλ = αλ ,

Fλ (p0 ) = Id

∀λ ∈ S 1

where p0 ∈ M˜ is some fixed point. Hence, one gets an S 1 -loop of τ -primitive maps ψλ = π ◦ Fλ : M˜ → Fm with ψ1 = ψ where ψλ (p0 ) = ψ(p0 ) ∀λ ∈ S 1 . In particular, this shows that primitive maps and hence harmonic superconformal maps exist in S 1 -loops that are called associated families in [7]. In order to get a different perspective of the above situation, we introduce here some Loop algebras and groups following [7]. Let g be the non-compact real form so(1, 2m) of the complex simple Lie algebra so(2m + 1, C). Recall from (2.1) and (2.2) that gC = so(2m + 1, C). The twisted complex loop algebra is defined by 1 C iπm ΛgC λ) = τ ξ(λ) ∀λ ∈ S 1 } τ = {ξ : S → g : ξ(e

There is also the real form of ΛgC τ determined by g: 1 Λgτ = {ξ ∈ ΛgC τ : ξ(λ) ∈ g ∀λ ∈ S }

j Any loop ξ ∈ ΛgC τ has a Laurent expansion ξ(λ) = j ∈Z ξj λ with ξj ∈ mj , i.e. such that τ ξj = ν j ξj (ν = eiπ/m ). Given ξ ∈ ΛgC τ its conjugate respect to the real form Λgτ is defined by σ (ξ )(λ) = σ (ξn )λ¯ n , where σ is the conjugation authomorphism in gC determined by g, see (2.2). In particular, ξk λk ∈ Λgτ iff σ (ξk ) = ξ−k k ∈ Z k∈Z

Note that a map ξ : S 1 → so(2m + 1, C) is in Λgτ iff it satisfies the following conditions: σ (ξ(λ)) = ξ(λ) ∀λ ∈ S 1 τ (ξ(λ)) = ξ(νλ) ∀λ ∈ S 1

(5.3)

The first one is called the reality condition determined by the non-compact real form so(1, 2m). C k Define a subalgebra Λ− gC τ = {ξ ∈ Λgτ : ξ = k≤0 ξk λ , ξ0 ∈ t}, then one has an orthogonal decomposition into complementary (closed) subalgebras, − C ΛgC τ = Λgτ ⊕ Λ gτ

with respect to the weakly non-degenerate invariant real symmetric bilinear form (ξ, η) = 21 Im tr ξ · η, ξ, η ∈ ΛgC τ S1


161

− C Let πK : ΛgC τ → Λgτ be the projection determined by ( , ) and the complement Λ gτ . Then πK is given by

πK (ξ ) = ξ (−) + ξt0 + σ (ξ (−) ), where ξ (−) = j d} Restricting f1 , f2 to Λd , then for ξ ∈ Λd , X1 = [ξ, πK (∇ξ f2 )]

(5.5)

X2 = [ξ, πK (∇ξ f1 )],

where ∇ξ f1 = λ1−d ξ, ∇ξ f2 = iλ1−d ξ are the gradients of f1 , f2 respect to ( , ), respectively. Since fi Poisson-commute [7], their commutator vanish, i.e. [X1 , X2 ] = 0. Note that if d ≡ 1 mod(2m) then the gradients ∇fj defined above are ΛgC τ -valued. Both commuting Hamiltonian vector fields Xi considered above are of Lax form, i.e: X(ξ ) = [ξ, E(ξ )]. Then if ξ(t) is any integral curve of such an X, we have d d (ξ, ξ ) = 2 ξ, ξ = 2([ξ, E(ξ )], ξ ) = −2(E(ξ ), [ξ, ξ ]) = 0 dt dt Note that the bilinear form (5.1) is non-definite on Λgτ and, thus, on Λd . Hence, ξ(t) evolves in a generic non-euclidean sphere contained in Λd and, thus, may not be defined for every t ∈ R. Since [X1 , X2 ] = 0 their flows X1s and X2t commute, X1s ◦ X2t = X2t ◦ X1s for s and t where they are defined. Fixed an initial condition ξo ∈ Λd , define ξ : U ⊂ R2 → Λd by ξ(s, t) = X1s ◦ X2t (ξo ), for all (s, t) in a suficiently small open neighbourhood of (0, 0). Then one verifies that ξ(s, t) satisfies dξ = X1 (ξ ) ds + X2 (ξ ) dt,

ξ(0, 0) = ξo

162


Now we define commuting complex vectorfields V and σ (V ) on ΛC d by X1 = V + σ (V ), X2 = i(V − σ (V )), hence, ξd−1 V (ξ ) = ξ, + λξd 2 (5.6) ξ−d+1 σ (V )(ξ ) = ξ, + λ−1 ξ−d 2 Introducing the complex variable z = s + it and applying the above argument to it follows that the ODE’s system:  ∂   ξ = V (ξ ),   ∂z    ∂ (5.7) ξ = σ (V )(ξ ),    ∂z     ξ(0, 0) = ξ ◦ , has a unique solution ξ : U0 → Λd for any given initial condition ξ ◦ ∈ Λd , where U0 ⊂ R2 is an open neighborhood of (0, 0) ∈ R2 which we can assume to be connected and simply connected. With this solution ξ the Λ1 -valued one-form αλ = λξd dz + 21 (ξd−1 dz + ξ−d+1 d¯z) + λ−1 ξ−d d¯z, satisfies the zero curvature condition or Maurer-Cartan equation: dαλ + 21 [αλ ∧ αλ ] = 0 ∀λ ∈ S 1

(5.8)

Thus, integrating on the simply connected U0 we get an S 1 -loop of frames Fλ : U0 → 1 SOo (1, 2m) with F−1 λ dFλ = αλ ∀λ ∈ S and one can arrange the constants of integration 1 so that Fλ (0, 0) = Id ∀λ ∈ S . 5.1. Integration of the Toda equations Using the above construction it is possible to get solutions of the 2D-affine Toda fields (4.17) associated to the pair (so(2m + 1, C); σ ) as follows. Inspecting the ODE system (5.7) we see that the λd−1 and λd terms satisfy ∂ ξd−1 = [ξd , ξ−d ], ∂z

∂ 1 ξd = [ξd , ξ1−d ], ∂z 2

In particular, the m0 part of F−1 dF is given by A0 =

ξd−1 ξ−d+1 d¯z + dz 2 2

Hence, A0 = i

ξ−d+1 ξd−1 d¯z − i dz 2 2

∂ 1 ξd = − [ξd , ξd−1 ]. ∂z 2

(5.9)


163

Now by (5.9) we have d ∗ A0 = 0. Therefore, the equation dΩ = i ∗ A0 , Ω(0, 0) = 0

(5.10)

has a smooth solution Ω : U0 → it and, hence, Ωz = ξd−1 /2, Ωz¯ = −ξ−d+1 /2 hold on U . If we choose the initial condition ξ ◦ in (5.7) so that ξ ◦ d = M ∈ m1 , then Ad(eΩ )M and ξd both satisfy the same system of ODE’s Yz = − 21 [Y, ξd−1 ] Yz¯ = 21 [Y, ξ−d+1 ]

(5.11)

Y (0, 0) = M Therefore, ξd = Ad(eΩ )M on U0 . Thus, each member of the loop S 1 λ → Fλ obtained by integration of (5.8) is a Toda frame for the τ -primitive map ψλ = π ◦ Fλ : U0 → Fm . In particular, Ω : U0 → it is a solution of the 2D-affine Toda field (4.17). Remark 5.1. Note that since [M, σ (M)] = 0, the null map Ω ≡ 0 can not be neither a solution of (4.17) nor (5.10). According to the Riemann mapping theorem, the connected simply connected open neighborhood U0 of (0, 0) is either the complex plane C or is conformally equivalent to the open unit disc D = {z : |z| < 1}. By applying an earlier result of Sattinger it is possible to rule out the U0 = C possibility at least for, namely, harmonic conformal maps of U0 → H n . Theorem 5.1. Let M be a connected non-compact Riemann surface, and f : M → Hn (n > 2) be a minimal immersion, i.e. a harmonic conformal map. Then the universal covering space M˜ of M is conformaly equivalent to the open unit disc D1 = {z : |z| < 1}. In particular, the theorem is true for minimal superconformal immersions of connected surfaces into H n . Proof. Let us suppose that M˜ is conformaly equivalent to C. Then f˜ = f ◦ π : C → Hn is minimal, where π : C → M is the covering map. Let K(g) be the Gaussian curvature of the induced metric g = f˜∗ h. Respect to the global isothermal coordinate z on C, we have g = e2u dz d¯z for some smooth function u defined on the whole complex plane. In terms of u the curvature is given by K(g) = −(1/2)4e−2u (2uz¯z ). Hence, the globally defined smooth function u must satisfy ∆u = −K(g)e2u ,

164


where ∆ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 . Since f˜ is conformal harmonic, the Gaussian curvature satisfies K(g) ≤ −1. Under such conditions Sattinger [13] proved that there is no solution u of the above equation defined on the whole complex plane. Recall the Poincaré model of H 2 , (D1 ; ds 2 = 4|dz|2 /(1−|z|2 )2 ). Using Theorem 5.1 we may (modulo conformal transformations of the plane) summarize the results of this section in the following. Theorem 5.2. Let m ≥ 2 and d ≡ 1 mod(2m). Recall the commuting vector fields V , σ (V ) defined on Λd by (5.6). (i) There exists a unique solution ξ : H 2 → Λd of the ODE’s system (5.7) with initial condition ξ(0, 0) = ξ ◦ ∈ Λd , with ξ ◦ d = M ∈ m1 . (ii) The Λ1 -valued one-form defined in terms of the solution ξ of (i), αλ = λξd dz + 21 (ξd−1 dz + ξ−d+1 d¯z) + λ−1 ξ−d d¯z satisfies the Maurer–Cartan equation dαλ + (1/2)[αλ ∧ αλ ] = 0 ∀λ ∈ S 1 . Hence, Fλ (0, 0) = Id ∀λ ∈ S 1 gives rise to an S 1 -loop of integration of F−1 λ dFλ = αλ , 2 frames Fλ : H → SOo (1, 2m) for the primitive maps ψλ = π ◦ Fλ : H 2 → Fm , and consequently this produces an S 1 -loop of minimal superconformal immersions fλ = P ◦ ψλ : H 2 → H 2m . (iii) The Fλ are Toda frames of the primitive maps ψλ = π ◦ Fλ : H 2 → Fm . Thus, there exist a solution Ω : H 2 → it to the 2D-affine Toda field Eq. (4.17).

Acknowledgements Research partially supported by SECYT, CONICET, Proyecto FOMEC and Universidad Nacional de Córdoba, Argentina

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