Minimal surfaces and the affine Toda field model

John Bolton  Franz Pedit^{∗}  Lyndon Woodward

1 Introduction

Minimal immersions or, more generally, harmonic maps of a Riemann surface S into S^{n}, P^{n} and other Riemannian symmetric spaces have been intensively studied over the past twenty five years.

The subject has been given considerable impetus by the interest of mathematical physicists in non-linear σ-models, which are harmonic maps of S^{2} into P^{n}, and in related problems which may be handled using twistor theory and methods of complex geometry.

For example, [EW83], harmonic maps of S^{2} into P^{n} may be characterised as being elements of Frenet frames of holomorphic curves in P^{n} or, equivalently, in twistorial terms as projections of suitable holomorphic horizontal curves in the twistor space SU (n + 1)/T^{n} over P^{n} (see [Bur90]). Indeed, considering SU (n + 1)/T^{n} as the full flag manifold these holomorphic horizontal curves in SU (n + 1)/T^{n} correspond precisely to Frenet frames of holomorphic curves in P^{n}. From an analytic viewpoint one has reduced the non-linear harmonic map equations to the Cauchy-Riemann equations.

A similar resolution of the problem may be found for harmonic maps of S^{2} into S^{n} [Cal67], a complex Grassmannian G_{k}( ^{n}) [Wol88], or a compact Lie group G [Uhl89]. In the latter case, one has to consider holomorphic curves in a finite dimensional stratum of the based loop group of G.

For a (compact) Riemann surface S of higher genus the situation is more complicated. The above constructions give some but not all weakly conformal harmonic maps into S^{n} and P^{n}. Those which can be so constructed are the easiest to deal with. They have strong isotropy properties and are called superminimal or totally isotropic [Bry82], [EW83]. Such maps into S^{2}, S^{3} and S^{4} are particularly wellunderstood. Those into S^{2} are ± holomorphic, those into S^{3} factor through a totally geodesic S^{2}, and those into S^{4} can be described using a Weierstrass representation [Bry82].

Considerable stimulus was given to the study of non-superminimal harmonic maps by the work of Wente [Wen86] on the existence of constant mean curvature tori in ^{3}. These are precisely the immersed tori in ^{3} whose Gauss map ϕ: T^{2} → S^{2} is a non-superminimal harmonic map. Then in [PS89], Pinkall and Sterling showed that all non-superminimal harmonic maps ϕ: T^{2} → S^{2} are of finite type, that is to say are obtained from solutions to a finite dimensional completely integrable system of ordinary differential equations. In analytical terms this involves showing that every doubly periodic solution of the sinh-Gordon equation

w_{z\overline{z}} + sinh 2w = 0 ,  (1.1)

is obtained by solving a finite dimensional system of ordinary differential equations. This is reminiscent of ideas in the theory of soliton equations [FT87], of which (1.1) is an example. Using an algebrogeometric integration algorithm developed for such equations, Bobenko [Bob91] gave parametrizations of all constant mean curvature tori in 3-dimensional space forms using θ-functions.

That such finite type behavior is to be expected for harmonic maps of 2-tori into higher dimensional spheres was indicated by Hitchin’s study of harmonic 2-tori in S^{3} [Hit90], and by the results in [FPPS92] where all minimal non-superminimal 2-tori in S^{4} were constructed from completely integrable Hamiltonian ordinary differential equations in finite dimensional subspaces on a certain loop algebra. In the latter case the minimal surface equations turn out to be the affine Toda field equations for SO(5),

∗partially supported by a grant from the Emory University Research Committee, University of Durham Research Council, Alexander von Humboldt Foundation and the SFB 288

2w_{z\overline{z}} − e^{2w} cosh 2η + e^{2w} = 0
2η_{z\overline{z}} + e^{−2w} sinh η = 0

(1.2)

which is another well-known example of a soliton system [For90]. Note that (1.1) is obtained from (1.2) by putting η = 0. Equations (1.1) and (1.2) have also been studied in [DW], [Wu93] using loop group factorization techniques.

The general theory of harmonic 2-tori in symmetric spaces was developed along similar lines in [BFPP], where, among other things, it was shown that every non-conformal harmonic map of a 2-torus into a rank 1 symmetric space is of finite type. However, the condition of non-conformality excludes the geometrically interesting case of minimal 2-tori.

By contrast, in the present paper we consider harmonic maps into S^{n} and P^{n} which satisfy very strong isotropy conditions just short of superminimality. (Recall that conformality may be expressed as an isotropy condition on first derivatives.) Our approach extends that of [FPPS92] in a way which seeks to make the underlying theory more transparent.

We now describe the basic ideas for the case of harmonic maps into P^{n}, the S^{n} case being similar but rather more technical. Suppose that ϕ: S → P^{n} is a harmonic map of a Riemann surface S into P^{n}, and without loss of generality we suppose that ϕ is linearly full, that is to say that ϕ(S) is not contained in a proper linear subspace of P^{n}. Then, as we recall in section 4, ϕ gives rise to a sequence {ϕ_{ℓ}} of harmonic maps ϕ_{ℓ}: S → P^{n} with ϕ_{0} = ϕ, called the harmonic sequence of ϕ. Adjacent elements in this sequence are orthogonal (with respect to the standard Hermitian inner product on ^{n+1}) by construction and if some k consecutive elements are orthogonal then so are any k consecutive elements [BW86], [BW92], in which case ϕ is said to be k-isotropic. In particular, ϕ is conformal if and only if ϕ is 3-isotropic and, in all cases, ϕ is at most (n + 1)-isotropic. There are two possible ways in which this latter case can arise. If ϕ is superminimal then the harmonic sequence has length n + 1 and is just the Frenet frame of a holomorphic curve (the directrix curve of ϕ) so that in particular ϕ is (n + 1)-isotropic. If ϕ is (n + 1)-isotropic but not superminimal then the harmonic sequence of ϕ is, in an obvious sense, orthogonally periodic of period n + 1 (c.f. remark 4.5) and we say that ϕ is superconformal.

Suppose now that ϕ: S → P^{n} is a superconformal harmonic map. Then there is a map ϕ̃: S → SU (n + 1)/T^{n} of S into the manifold of full flags in ^{n+1} given by

ϕ̃ = (ϕ_{0}, …, ϕ_{n}) .

It follows that if π: SU (n + 1)/T^{n} → P^{n} is the map which assigns to a full flag its initial element, then ϕ = π ∘ ϕ̃. This is the analogue of the twistor lift for superminimal maps mentioned above, but in this case ϕ̃ is neither holomorphic nor horizontal. It is, however, τ-primitive (see definition 2.2) where τ = Ad diag(1, ζ, … ζ^{n}), ζ = e^{2π/n+1}, so that, in particular dϕ̃(T^{(1,0)}S) is contained in the ζ-eigenspace of the endomorphism of the complexified tangent bundle T (SU (n+1)/T^{n}) induced by τ. The significance of τ is that it is the symmetry of smallest order m which exhibits SU (n + 1)/T^{n} as an m-symmetric space [Jim88] with m = n + 1. Conversely, if ψ: S → SU (n + 1)/T^{n} is a τ-primitive map then ϕ = π ∘ ψ is a superconformal harmonic map.

The relationship between superconformal harmonic maps and the affine Toda field model comes about from the fact that (theorem 2.5) any τ-primitive map ψ: S → SU (n + 1)/T^{n} has an essentially unique local framing F: S → SU (n + 1) whose Maurer-Cartan equations

dα + \frac{1}{2}[α ∧ α] = 0, α = F^{−1}dF

can be rewritten as the affine Toda field equations for SU (n + 1)(theorem 2.6). Thus τ-primitive maps, or equivalently superconformal harmonic maps, are the geometrical objects described by the affine Toda field model for SU (n + 1). This, and its analogue for certain other compact simple Lie groups, is the crucial fact on which the whole paper is based.

In a similar fashion, superconformal harmonic maps of S into S^{2n−1} and S^{2n} (see section 5 for the definitions and geometrical interpretation of superconformality in these cases) correspond to τ-primitive maps into SO(2n + 1)/T^{n} and thus give the geometric analogues of the affine Toda field models for these groups. The S^{2n}-case and its relationship to the affine Toda field model of SO(2n + 1) has also been considered in [DS].

We remark that τ-primitive maps provide natural examples of equiharmonic maps, that is maps ψ: S → G/T, where T is a maximal torus of a compact simple Lie group G, which are harmonic with respect to every G-invariant metric on G/T. Such maps have been introduced by Black [Bla91].

Finally, all harmonic maps into P^{1} = S^{2}, all conformal harmonic maps into P^{2}, S^{3} and S^{4}, and all almost complex curves in the nearly Kähler S^{6} are either superminimal or superconformal, and as we show in section 6, the non-superminimal almost complex curves in S^{6} correspond to solutions of the affine Toda field model for the exceptional group G_{2}.

The layout of the paper is as follows. In section 2 we study τ-primitive maps ψ: S → G/T, where G is a compact simple Lie group with maximal torus T. We show (theorems 2.5 and 2.6) that such maps are related to the affine Toda field model for G and show that when S is a torus T^{2}, the corresponding solutions of the affine Toda field equations for G are doubly periodic (corollary 2.7). While reading this section the reader may find it helpful to refer to section 4 where the special case of G = SU (n + 1) is discussed.

In section 3 we construct a large class of τ-primitive maps ψ: ^{2} → G/T by integrating certain Hamiltonian ODE’s on finite dimensional subspaces of the τ-twisted loop algebra of the Lie algebra G of G. This is a standard construction which can be done by using either the r-matrix formalism [RSTS88] or the A-K-S integration scheme [AvM80]. The main result, which follows directly from the developments in [FPPS92] and [BFPP] is that if a τ-primitive map ψ: ^{2} → G/T factors through a 2-torus then ψ is obtained by the above construction, and hence is of finite type. In principle, this gives an explicit method of finding all doubly periodic solutions to the affine Toda field model for G, although it is not easy to write down solutions explicitly. A slightly different approach to solving the affine Toda equations using loop group splittings in the Grassmannian model appears in [McI].

Section 4 deals with superconformal harmonic maps ψ: S → P^{n} and the correspondence between such maps and τ-primitive maps S → SU (n + 1)/T^{n} is established (see theorem 4.6). As a corollary, using the results of section 3, it follows that any superconformal harmonic 2-torus ψ: T^{2} → P^{n} is of finite type. In addition to recovering the results of [PS89] this, together with the results in [EW83], accounts for all harmonic 2-tori in P^{2}.

In section 5 we adapt the methods of section 4 to the case of superconformal harmonic maps into S^{n}. In particular, we recover the results of [FPPS92] on minimal tori in S^{4}.

Finally, in section 6 we consider almost complex curves in the nearly Kähler S^{6}. We recall that S^{6} = G_{2}/SU (3) is a symmetric space for the exceptional Lie group G_{2} and that G_{2} is the group of automorphisms of the nearly Kähler structure. We show that non-superminimal almost complex curves in the nearly Kähler S^{6} correspond to solutions of the affine Toda field model for G_{2}. This is derived from the results of section 5 together with the characterization (proposition 6.1) of non-superminimal almost complex curves as those superconformal harmonic maps ψ: S → S^{6} whose τ-primitive lifts ψ̃: S → SO(7)/T^{3} take values in G_{2}/T^{2}. Thus, in light of the results in section 3, every non-superminimal almost complex 2-torus in S^{6} is of finite type. Together with the results in [BVW] this accounts for all almost complex 2-tori in S^{6}.

Parts of this work were carried out during research visits by the second author to the University of Durham and the SFB 288 at Technische Universität Berlin. We would like to thank the Departments of Mathematics of both universities for their interest and support.

2 Affine Toda fields and harmonic maps

In this section we give a geometric formulation of the (periodic) two dimensional affine Toda field model in terms of special harmonic maps from a connected Riemann surface S into the quotient space G/T of a compact simple Lie group G by its maximal torus T, where G/T is equipped with a G-invariant metric. It is well known that horizontal holomorphic curves into G/T (which are harmonic with respect to any G-invariant metric on G/T) give rise to the open (or non-periodic) affine Toda field equations.

We consider here a rather different class of maps ψ: S → G/T, called τ-primitive maps (see definition 2.2), which are harmonic with respect to any G-invariant metric. These maps are neither holomorphic nor horizontal but are adapted to the canonical m-symmetric space structure on G/T given by a certain inner automorphism τ of G of order m.

We begin by recalling some basic notions about Lie algebras and the structure of G/T, and then relate these ideas to the affine Toda field equations.

Let G = T ⊕ ⨁_{α∈Δ} G^{α} be the root space decomposition of the complexification G of the Lie algebra G of G with maximal torus T. We fix (once and for all) a Cartan-Weyl basis ξ_{α} ∈ G^{α}, α ∈ Δ, of the root spaces. In particular, we have the following relations:

\begin{matrix}\overline{ξ}_{α} = ξ_{−α}, \\ [ξ_{α}, ξ_{−α}] = α^{#} \\ (ξ_{α}, ξ_{β}) = δ_{α,−β} \end{matrix}  (2.1)

where #: T^{∗} → T is the isomorphism induced by the Killing form ( , ) on G which corresponds to “raising indices”. We also fix a set of simple roots α_{1}, …, α_{r}, r=rank G. If θ = ∑_{k=1}^{r} m_{k}α_{k}, m_{k} ∈ ^{+}, denotes the highest root we define α_{0} by α_{0}: = −θ. Then we define

M_{1} = ⨁_{k=0}^{r} G^{α_{k}}  (2.2)

as the sum of the simple root spaces and the root space corresponding to α_{0}. Following [Kos59] we call an element ξ ∈ M_{1} cyclic if

ξ = ∑_{k=0}^{r} a_{k}ξ_{α_{k}}  (2.3)

with all a_{k} ∈ \{0}. The following facts, which are proved in [Kos59], hold for cyclic elements:

Lemma 2.1.

(i) cyclic elements are regular semisimple.

(ii) if P_{1}, …, P_{r}: G → are homogeneous generators of the ring of AdG -invariant polynomials with deg P_{1} < … < deg P_{r} then

P_{k} |_{M_{1}}= 0 for k = 1, …, r − 1,

and ξ ∈ M_{1} is cyclic if and only if P_{r}(ξ) ≠ 0.

(iii) if ξ, ξ̃ ∈ M_{1} are cyclic then there exists t ∈ T with Ad(t)(ξ) = ξ̃ if and only if P_{r}(ξ) = P_{r}(ξ̃).

The simple roots and the height of the highest root θ determine, up to conjugation, an inner automorphism τ: G → G of order m = ∑_{k=0}^{r} m_{k} [Hel78], where by definition m_{0}: = 1. The differential of τ at the identity of G, which we also denote by τ, is an automorphism of G which acts on M_{1} via multiplication by ζ = e^{\frac{2πi}{m}}.

Explicitly we have τ = Ad exp(2πiZ) where Z = \frac{1}{m} ∑_{k=1}^{r} η_{k} where each η_{k} ∈ T and α_{j}(η_{k}) = δ_{jk}. Note that the fixed point set G^{τ} of τ is the maximal torus T of G. In fact, τ is the automorphism of smallest order m which exhibits G/T as an m-symmetric space [Jim88]. This gives us the _{m}-grading

G = ⨁_{k∈ _{m}} M_{k}  (2.4)

where M_{k} = \overline{M_{−k}} is the ζ^{k}-eigenspace of τ on G . The adjoint action of G on G thus leads to a decomposition

T (G/T) = ⨁_{k∈ _{m}\{0}}[M_{k}] ⊂ G/T × G  (2.5)

of the complexified tangent bundle of G/T into a direct sum of subbundles of the trivial G -bundle over G/T where [M_{k}]_{gT} = Adg(M_{k}) ⊂ G . An element of [M_{1}] will be said to be cyclic if it is in the orbit of a cyclic element of M_{1} under the adjoint action of G.

Definition 2.2. A map ψ: S → G/T is called τ-primitive if dψ(T^{(1,0)}S) ⊂ [M_{1}] and contains a cyclic element.

Remark 2.3 (i) In the case where G/T is the 2-sphere S^{2} the above definition puts no condition on ψ. In order not to exclude this case from our discussion, τ-primitive in this case will mean harmonic with dψ(T^{(1,0)}S) containing a semi-simple element.

(ii) It is shown in [Bla91] that τ-primitive maps are equiharmonic, i.e. harmonic with respect to any G-invariant metric on G/T. Moreover, they project to equiharmonic maps under every homogeneous projection π: G/T → G/H hence, in particular, to harmonic maps in the underlying symmetric spaces G/K. In sections 4 and 5 we discuss those harmonic maps into P^{n} and S^{n} which are obtained in this way.

A standard holomorphic differential argument (contained in the proof of the next lemma) shows that τ-primitive maps have cyclic \frac{∂}{∂z} -derivatives except perhaps at a discrete set of points.

Lemma 2.4. Let ψ: S → G/T be τ-primitive. Then, viewing P_{r} ∈ S^{d_{r}}(T (G/T) ) as a symmetric multi-linear form of degree d_{r} = deg P_{r}, P_{r}(dψ)^{(d_{r},0)} is a holomorphic -valued differential of degree d_{r}. In particular, \frac{∂ψ}{∂z} (p) ∈ [M_{1}] is cyclic except possibly at a discrete set of points.

Proof. Let ∇ be the Levi-Civita connection for a G-invariant metric on G/T. Since P_{r}: T (G/T) → is the restriction of an AdG -invariant polynomial on G , P_{r} is ∇-parallel. Thus

\frac{∂}{∂\overline{z}} P_{r}(\frac{∂ψ}{∂z} , …, \frac{∂ψ}{∂z}) = d_{r} · P_{r}(∇_{\frac{∂}{∂\overline{z}}} \frac{∂ψ}{∂z} , \frac{∂ψ}{∂z} , …, \frac{∂ψ}{∂z}).  (2.6)

Since ψ is a harmonic map, ∇_{\frac{∂}{∂\overline{z}}} \frac{∂ψ}{∂z} = 0, so that P_{r}(\frac{∂ψ}{∂z}) is holomorphic. Hence P_{r}(\frac{∂ψ}{∂z}) vanishes only on a discrete set of points. □

We now come to the main results of this section. We show that, locally, τ-primitive maps correspond to solutions of the affine Toda field equations.

By a (local) frame (or framing) of a map ψ: S → G/T we mean a map F: U → G from an open subset U ⊂ S, with π ∘ F = ψ where π: G → G/T is the homogeneous projection. Note that ψ is τ-primitive if and only if

F^{−1} \frac{∂F}{∂z} = A_{0} + A_{1} ∈ M_{0} ⊕ M_{1}  (2.7)

where A_{0} ∈ M_{0}, A_{1} ∈ M_{1} and A_{1} is cyclic at some point. In this case, since F is real

F^{−1} \frac{∂F}{∂\overline{z}} = \overline{A}_{0} + \overline{A}_{1},  (2.8)

so that, since \frac{∂^{2}F}{∂z∂\overline{z}} = \frac{∂^{2}F}{∂\overline{z}∂z}, we obtain the Maurer-Cartan equations namely,

\frac{∂A_{0}}{∂\overline{z}} − \frac{∂\overline{A}_{0}}{∂z} = [A_{1}, \overline{A}_{1}],  (2.9)

\frac{∂A_{1}}{∂\overline{z}} = [A_{1}, \overline{A}_{0}].  (2.10)

These are the integrability conditions for the existence of a real G-valued solution of (2.7).

A local frame will be called a Toda frame if there is a complex coordinate z: U → and a smooth map Ω: U → iT such that

F^{−1} \frac{∂F}{∂z} = \frac{∂Ω}{∂z} + Ad exp(Ω)(B) ∈ M_{0} ⊕ M_{1}.  (2.11)

where B = ∑_{k=0}^{r} \sqrt{m_{k}}ξ_{α_{k}} ∈ M_{1}.

Theorem 2.5. Let ψ: S → G/T be τ-primitive and let p_{0} ∈ S be such that \frac{∂ψ}{∂z} (p_{0}) is cyclic. Then there is a local Toda frame F of ψ around p_{0}. Moreover, the complex coordinate z for the Toda frame is unique up to an arbitrary translation and rotation by a d_{r}-th root of unity, while the Toda frame F is unique up to multiplication by an element of the center Z(G) of G.

Theorem 2.6. Let z: U → be a complex coordinate on a simply connected open subset U of S and let Ω: U → iT be smooth. Then (2.11) has a real solution F if and only if Ω satisfies the affine Toda field equations for G [For90], namely

2\frac{∂^{2}Ω}{∂z∂\overline{z}} + ∑_{k=0}^{r} m_{k}e^{2α_{k}(Ω)}α_{k}^{#} = 0.  (2.12)

In this case, ψ = π ∘ F: U → G/T is τ-primitive and F is a Toda framing of ψ.

Proof of theorem 2.5. Let F: U → G be a framing of ψ: U → G/T over a simply connected domain with complex coordinate z: U → so that \frac{∂ψ}{∂z} (p) is cyclic for all p∈U. Then, writing F^{−1} \frac{∂F}{∂z} = A_{0} + A_{1} as in (2.7), it follows from lemma 2.4 that P_{r}(A_{1}) is holomorphic and nowhere vanishing. Hence there exists a holomorphic change of coordinate such that

P_{r}(A_{1}) = P_{r}(B)

on U. Such a coordinate change is necessarily unique up to an arbitrary translation and rotation by a d_{r}-th root of unity. Using lemma 2.1 (iii), we have a map

η: U → M_{0} = T

with

Ad exp η(B) = A_{1}.

Now write η = Λ + Ω with Λ = \overline{Λ} and Ω = −\overline{Ω}. Regauging F by exp Λ, we obtain a new framing (again called F) for which

Ad exp(Ω)(B) = A_{1}.  (2.13)

It follows that \frac{∂A_{1}}{∂\overline{z}} = [\frac{∂Ω}{∂\overline{z}}, A_{1}], so using (2.10) we obtain

[A_{1}, \overline{A}_{0} + \frac{∂Ω}{∂\overline{z}}] = 0.

Hence, at every point p ∈ U, \overline{A}_{0} + \frac{∂Ω}{∂\overline{z}} is in the centralizer of A_{1}, which is a maximal torus orthogonal to M_{0} [Kos59]. Thus, \overline{A}_{0} = −\frac{∂Ω}{∂\overline{z}}, or equivalently

A_{0} = \frac{∂Ω}{∂z}.  (2.14)

The first part of theorem 2.5 now follows from (2.7), (2.13) and (2.14), so it remains to prove the uniqueness property of the Toda frame. Let F̃: U → G be another framing with

F̃^{−1} \frac{∂F̃}{∂z} = \frac{∂Ω̃}{∂z} + Ad exp(Ω̃)(B)  (2.15)

for some smooth map Ω̃: U → iT . Since any two framings differ by a gauge with values in T we have a map Λ: U → T such that

F̃ = F exp Λ.  (2.16)

It now follows from (2.11), (2.15) and (2.16) that

\frac{∂Ω̃}{∂z} + Ad exp(Ω̃)(B) = \frac{∂Ω}{∂z} + \frac{∂Λ}{∂z} + Ad exp(Ω − Λ)(B).  (2.17)

Equating M_{1} components in (2.17), we have

Ad exp(Ω̃ − Ω + Λ)(B) = B,

or equivalently, from the definition of B,

∑_{k=0}^{r} \sqrt{m_{k}}e^{α_{k}(_{Ω}−Ω+Λ)}ξ_{α_{k}} = ∑_{k=1}^{r} \sqrt{m_{k}}ξ_{α_{k}}.

However, ξ_{α_{0}}, …, ξ_{α_{r}} are linearly independent, so that

α_{k}(Ω̃ − Ω + Λ) ∈ 2πi , k = 0, …, r.

It follows that

Ω̃ = Ω  (2.18)

and that exp Λ is in the center of G. The final statement of theorem 2.5 now follows from (2.16). □

Proof of theorem 2.6. It follows from (2.1), (2.9) and (2.10) that the affine Toda equations (2.12) are the integrability conditions for the existence of a real G-valued solution of (2.11). The final statement of the theorem is clear from (2.7). □

In what follows we will be concerned with τ-primitive maps from a 2-torus.

Corollary 2.7. Let ψ: T^{2} → G/T be a τ-primitive map of a 2-torus. Then there exists a coordinate z: ^{2} → on the universal cover ^{2} → T^{2} which is unique up to an arbitrary translation and rotation by a d_{r}-th root of unity and a Toda frame F: T̃^{2} → G on a 2-torus T̃^{2} covering T^{2} such that (2.11) holds. Moreover, Ω factors through T^{2} and so is a solution of the affine Toda field equations (2.12) defined on T^{2}. Conversely, every τ-primitive map ψ: T^{2} → G/T is obtained from a solution of (2.12) on T^{2}.

Proof. We use the notation in the proof of theorem 2.5 with z being the standard complex coordinate on 2 = . Since P_{r}(dψ)^{(d_{r},0)} is non vanishing and holomorphic on T^{2} and since P_{r}(dψ)^{(d_{r},0)} = P_{r}(A_{1})dz^{d_{r}} we must have that P_{r}(A_{1}) is non-zero constant. After rescaling z by a suitable constant we may assume that P_{r}(A_{1}) = P_{r}(B). The construction in the proof of theorem 2.5 may now be carried out on the universal cover ^{2} → T^{2}. Since a Toda frame is unique up to the center Z(G), which is finite, we have a Toda frame F: T̃^{2} → G defined on a torus T̃^{2} covering T^{2}. That Ω factors through T^{2} follows from (2.18). □

3 Integration of the affine Toda field equations

In the previous section we have seen that τ-primitive maps ψ: T^{2} → G/T of a 2-torus into G/T give rise to solutions of the affine Toda field equations on T^{2} and that any such τ-primitive map comes from some doubly periodic solution of the affine Toda equations. We now describe how all doubly periodic solutions of the affine Toda equations may be obtained by integrating a pair of commuting Hamiltonian flows on a finite dimensional subspace of a certain loop algebra. The treatment we give here extends and complements the ideas in [FPPS92] and [BFPP] from which it derives.

Let ψ: S → G/T be τ-primitive and let F: S̃ → G be a framing of ψ on the universal cover S̃ → S. Then

A = F^{−1}dF = A′′_{1} + A_{0} + A′_{1}  (3.1)

where A_{0}: T S̃ → M_{0}, A_{0} = \overline{A}_{0}, A′_{1}: T^{(1,0)}S̃ → M_{1}, A′′_{1} = \overline{A′_{1}}, and P_{r}(A′_{1}) ≢ 0. The integrability condition for the existence of a frame F: S̃ → G satisfying (3.1) is the Maurer-Cartan equation

dA + \frac{1}{2}[A ∧ A] = 0  (3.2)

which, when split into its M_{0} and M_{1} parts, becomes

\begin{matrix}dA_{0} + \frac{1}{2}[A_{0} ∧ A_{0}] = −[A′_{1} ∧ A′′_{1}] \\ dA′_{1} + [A_{0} ∧ A′_{1}] = 0. \end{matrix}  (3.3)

The crucial observation in dealing with these equations is that they may be reformulated (see [Uhl89], [ZS78]) using a “spectral parameter” as follows:

A = A′′_{1} + A_{0} + A′_{1} satisfies (3.3) if and only if
A^{λ} = λ^{−1}A′′_{1} + A_{0} + λA′_{1} satisfies (3.3) for all λ ∈ S^{1}.

(3.4)

Thus we obtain an S^{1}-family of frames F^{λ}: S̃ → G and, correspondingly, an S^{1}-family of τ-primitive maps ψ^{λ} = π ∘ F^{λ}: S̃ → G/T with ψ = ψ^{1} (if one bases ψ^{λ}(p_{0}) = ψ(p_{0}) for all λ ∈ S^{1} at some point p_{0} ∈ S̃).

These ideas may be conveniently described in terms of certain loop spaces. Let ζ = e^{2πi/m} where m is the order of the automorphism τ: G → G which exhibits G/T as an m-symmetric space as discussed in section 2 and define the (Banach) Lie group ΛG_{τ} of τ-twisted loops in G by

ΛG_{τ} = {g : S^{1} → G | g is H^{1} − smooth, g(ζλ) = τg(λ)}.  (3.5)

The Lie algebra ΛG_{τ} of ΛG_{τ} is then given by

ΛG_{τ} = {ξ: S^{1} → G | ξ is H^{1} − smooth, ξ(ζλ) = τξ(λ)}  (3.6)

which admits the Ad ΛG_{τ} -invariant inner product

(ξ, η) = ∫_{S^{1}}(ξ(λ), η(λ))dλ.  (3.7)

Expanding ξ ∈ ΛG_{τ} in a Laurant series we get

ξ = ∑_{k∈} λ^{k}ξ_{k}, ξ_{k} ∈ M_{k}, \overline{ξ}_{k} = ξ_{−k}.  (3.8)

We also have a filtration of ΛG_{τ} by finite dimensional subspaces of Laurant polynomials of degree d ∈ ,

Λ_{d} = {ξ ∈ ΛG_{τ} | ξ = ∑_{|k|≤d} λ^{k}ξ_{k}}.  (3.9)

Finally we note that there is a bijective correspondence between S^{1}-families of maps F^{λ} : S̃ → G, λ ∈ S^{1}, and maps F̂ : S̃ → ΛG_{τ} given by F^{λ}( ) = F̂( )(λ). The above discussion now may be summarised as follows:

Lemma 3.1. A map ψ : S̃ → G/T is τ-primitive if and only if there exists a map F̂: S̃ → ΛG_{τ} with

F̂^{−1}dF̂ = λ^{−1}A′′_{1} + A_{0} + λA′_{1}: T S̃ → Λ_{1} ⊂ ΛG_{τ}

and P_{r}(A′_{1}) ≢ 0, so that ψ = π ∘ F^{1}.

We now show how to construct a large class of τ-primitive maps ψ : ^{2} → G/T using a version of the A-K-S integration scheme or the r-matrix formalism [AvM80], [RSTS88, FPPS92]. Since the results in [BFPP, FPPS92] will imply that any doubly periodic τ-primitive map is in the above class this shows, in principle, how all τ-primitive maps defined on 2-tori may be constructed.

For each d ≡ 1(m) define a pair of vector fields V, \overline{V} on ΛG_{τ} by

\begin{matrix}V (ξ) = [ξ, i(\frac{ξ_{d−1}}{2} + λξ_{d})] \\ \overline{V} (ξ) = [ξ, (−i)(\frac{ξ_{1−d}}{2} + λ^{−1}ξ_{−d})]. \end{matrix}  (3.10)

It is clear that V and \overline{V} are tangent to Λ_{d} . It follows from the basic integration lemma of [BFPP, FPPS92] that

[V, \overline{V}] = 0  (3.11)

and that the system of ODE’s

\begin{matrix}\frac{∂ξ}{∂z} = V (ξ), \\ \frac{∂ξ}{∂\overline{z}} = \overline{V} (ξ), \\ ξ(0) =ξ^{∘}∈ Λ_{d}, \end{matrix}  (3.12)

has a unique, global, real solution ξ: ^{2} → Λ_{d} for any given real initial condition ξ^{∘}∈ Λ_{d}.

Given such a solution ξ: ^{2} → Λ_{d} of (3.12) the Λ_{1}-valued 1-form Â on ^{2}

Â = i(\frac{ξ_{d−1}}{2} + λξ_{d})dz − i(\frac{ξ_{1−d}}{2} + λ^{−1}ξ_{−d})d\overline{z}  (3.13)

satisfies the Maurer-Cartan equation and thus gives rise to a map F̂: ^{2} → ΛG_{τ} with F̂^{−1}dF̂ = Â.

We note that, since P_{r} is AdG -invariant, it follows that along solutions of (3.12),

\begin{aligned}dP_{r}(ξ) & = d_{r} · P_{r}(dξ, ξ, …, ξ) \\ & = d_{r} · P_{r}([ξ, Â], ξ, …, ξ) = 0, \end{aligned}

and hence P_{r}(ξ) is a constant of the motion. In particular, since ξ ∈ Λ_{d}, it follows by consideration of highest powers of λ that P_{r}(ξ_{d}) is also a constant of the motion so that

P_{r}(ξ_{d}) = P_{r}(ξ^{∘}_{d}).

The following lemma is now immediate from lemma 3.1.

Lemma 3.2. If ξ^{∘}_{d}∈ M_{1} is a cyclic element then, for each λ ∈ S^{1}, ψ^{λ} = π∘F^{λ}: ^{2} → G/T is τ-primitive with framing F^{λ}: ^{2} → G.

The proof of the next lemma shows how solutions of the affine Toda field equations arise from this construction.

Lemma 3.3. Let ξ^{∘}_{d}∈ M_{1} be the cyclic element B of section 2 and let F^{λ}: ^{2} → G, λ ∈ S^{1}, be the corresponding framings constructed above. Then each F^{λ} is a Toda frame for the τ-primitive map ψ^{λ} = π ∘ F^{λ}: ^{2} → G/T and thus there exists Ω: ^{2} → iT ⊂ M_{0} which satisfies equation (2.11) with F = F^{1}. In particular, Ω is a solution of the affine Toda field equations (2.12) for G.

Proof. By considering terms in λ^{d−1} and λ^{d} in (3.12) we have

\begin{matrix}\frac{∂ξ_{d−1}}{∂\overline{z}} = i[ξ_{d}, ξ_{−d}], \\ \frac{∂ξ_{d}}{∂\overline{z}} = −\frac{i}{2}[ξ_{d}, ξ_{1−d}], \\ \frac{∂ξ_{d}}{∂z} = −\frac{i}{2}[ξ_{d}, ξ_{d−1}]. \end{matrix}  (3.14)

Then the M_{0} component A_{0} of (F^{λ})^{−1}dF^{λ} is given by

A_{0} = i\frac{ξ_{d−1}}{2} dz − i\frac{ξ_{1−d}}{2} d\overline{z},

so that

∗A_{0} = −\frac{ξ_{d−1}}{2} dz − \frac{ξ_{1−d}}{2} d\overline{z},

and hence, using (3.14), d ∗ A_{0} = 0. Let Ω: ^{2} → iT be a solution of

dΩ = −i ∗ A_{0}, Ω(0) = 0.

Then

\frac{∂Ω}{∂z} = i\frac{ξ_{d−1}}{2} ,

and hence Ad exp Ω(B) and ξ_{d} both satisfy the same system of ODE’s

\begin{aligned}\frac{∂η}{∂z} & = −\frac{i}{2}[η, ξ_{d−1}] \\ \frac{∂η}{∂\overline{z}} & = −\frac{i}{2}[η, ξ_{1−d}], \end{aligned}  (3.15)

with initial condition η(0) = B. Hence

ξ_{d} = Ad exp Ω(B).

Thus (2.11) holds for F = F^{λ} with λ = 1. □

Remark 3.4 (i) The vector fields V, \overline{V} are canonical Hamiltonian vector fields on ΛG_{τ} with respect to the Poisson structure defined by the r-matrix

R: ΛG_{τ} → ΛG_{τ} , R(λ^{k}ξ_{k}) = \frac{i}{2} sign (k)λ^{k}ξ_{k}

and the Ad ΛG_{τ} -invariant Hamiltonians f: ΛG_{τ} → , f(ξ) = \frac{λ^{1−d}}{2} (ξ, ξ). In A-K-S-language this corresponds to the splitting of ΛG_{τ} into two subalgebras

ΛG_{τ} = ΛG_{τ} ⨁ Λ^{−}

where Λ^{−} = {ξ ∈ ΛG_{τ} | ξ = ∑_{k≤0} λ^{k}ξ_{k}, ξ_{0} ∈ iT }. (See [BP] or [DPW] for more details.)

(ii) Equations (3.12) are in Lax form and hence isospectral. It follows from standard techniques [FPPS92], [Bur] that (3.12) linearise on the Jacobian of their spectral curve. This is where periodicity conditions on the solutions can be discussed. A detailed study of this aspect will be done elsewhere.

Conforming to standard notation [BFPP] a τ-primitive map ψ: ^{2} → G/T obtained via the construction above will be called a τ-primitive map of finite type. The above discussion may be summarised as follows:

Theorem 3.5. A τ-primitive map ψ: ^{2} → G/T is of finite type if and only if there exists an S^{1}-family F̂ = F^{λ}: ^{2} → ΛG_{τ} of Toda frames with ψ = π ∘ F^{1} and a Laurant polynomial solution ξ: ^{2} → Λ_{d}, some d ≡ 1(m), with

\begin{aligned}dξ & = [ξ, F̂^{−1}dF̂], \\ F̂^{−1} \frac{∂F̂}{∂z} & = i(\frac{ξ_{d−1}}{2} + λξ_{d}). \end{aligned}  (3.16)

We call such ξ: ^{2} → Λ_{d} an adapted polynomial Killing field (for the Toda frame F^{λ}).

We now come to the main result of this section:

Theorem 3.6. Let ψ: T^{2} → G/T be a τ-primitive map of a 2-torus. Then ψ is of finite type.

Combining this with corollary 2.7 we obtain the following

Corollary 3.7. Every doubly periodic solution of the affine Toda field equation for G is obtained by integrating the finite dimensional Hamiltonian ODE’s (3.12).

Proof of theorem 3.6. All the crucial ingredients for proving theorem 3.6 are contained in [BFPP], section 2. Hence we will confine ourselves to showing how to adopt our situation to the setup therein. From corollary 2.7 we know that there exists an (essentially) unique Toda frame F^{λ}: ^{2} → G with

(F^{λ})^{−1} \frac{∂F^{λ}}{∂z} = \frac{∂Ω}{∂z} + λAd exp Ω(B)

where Ω: T^{2} → iT . To find an adapted polynomial Killing field for F^{λ} we have to solve

dξ = [ξ, (∂Ω + λAd exp Ω(B))dz + (−\overline{∂}Ω + λ^{−1}Ad exp(−Ω)(\overline{B}))d\overline{z}]  (3.17)

for a Laurant polynomial ξ: ^{2} → Λ_{d}, some d ≡ 1(m), with leading terms

i(\frac{1}{2}ξ_{d−1} + λξ_{d}) = \frac{∂Ω}{∂z} + λAd exp Ω(B).

We start by regauging F^{λ} to a complex frame

F̃ = F^{λ} exp Ω  (3.18)

which has

F̃dF̃^{−1} = (2∂Ω + λB)dz + λ^{−1}Ad exp(−2Ω)(\overline{B})d\overline{z}.  (3.19)

Hence (3.17) becomes

dξ̃ = [ξ̃, F̃^{−1}dF̃]  (3.20)

for the gauged polynomial Killing field ξ̃ = Ad exp(−Ω)(ξ): ^{2} → Λ_{d} which we require to satisfy

i(\frac{ξ̃_{d−1}}{2} + λξ̃_{d}) = 2\frac{∂Ω}{∂z} + λB.

We choose a faithful representation G ↪ U(N) of G into a unitary group and view G ⊂ gℓ(N, ) as a Lie subalgebra of gℓ(N, ). We claim it suffices to find a formal solution Y = ∑_{k≤j} λ^{k}Y_{k}, j ≡ 1(m), Y_{k}: T^{2} → gℓ(N, ), satisfying (3.20) with top term Y_{j} = B. That such a solution Y exists follows verbatim from the proof of theorem 2.4 in [BFPP]: B is regular semisimple (in gℓ(N, )) and (3.20) for Y is precisely equation (3.18) in the proof of theorem 2.4. in [BFPP]. Also note that 2\frac{∂Ω}{∂z}: T^{2} → T is defined on the 2-torus T^{2}, hence the Y_{k} constructed in the proof of theorem 2.4. in [BFPP] as differential polynomials in Ω, will be defined on T^{2}, i.e. Y_{k}: T^{2} → gℓ(N, ). Since we have granted that we have a formal solution to (3.20) defined on T^{2} we use the standard ellipticity argument in [FPPS92] or theorem 2.3 of [BFPP], to obtain a complex, polynomial solution to (3.20),

\begin{aligned}X̃ & : T^{2} → Λgℓ(N, ) \\ X̃ & = ∑_{k=0}^{d} λ^{k}X̃_{k}, d ≡ 1(m), X̃_{d} = B. \end{aligned}

Thus X = Ad exp Ω(X̃) = ∑_{k=0}^{d} λ^{k}X_{k}, d ≡ 1(m), X_{d} = Ad exp(Ω)(B), is a complex polynomial solution to the real equation (3.17). Putting

η̃ = X + \overline{X}

we obtain a real solution of equation (3.17) with η̃_{d} = Ad exp(Ω)(B). Since G is simple we have an adinvariant projection gℓ(N, ) → G which projects η̃ into a Λ_{d}-valued Killing field η: T^{2} → Λ_{d}. Finally we average η using τ to obtain

ξ = (\frac{1}{m} ∑_{k=1}^{m} τ^{k})(η): T^{2} → Λ_{d}

thus solving (3.17) with ξ_{d} = Ad exp(Ω)(B). But the top term in (3.17) gives

[ξ_{d}, \frac{∂Ω}{∂z}] + [ξ_{d−1}, ξ_{d}] = \frac{∂ξ_{d}}{∂z} = [\frac{∂Ω}{∂z}, ξ_{d}]

and hence

[ξ_{d}, 2\frac{∂Ω}{∂z} − ξ_{d−1}] = 0.

But ξ_{d} being cyclic in M_{1} implies

\frac{1}{2}ξ_{d−1} = \frac{∂Ω}{∂z}

which completes the proof. □

4 Superconformal harmonic surfaces in CP^{n}

In this section we give the τ-primitive condition a more geometric flavour by characterizing those harmonic maps into P^{n} which arise from τ-primitive maps into SU (n + 1)/T^{n} under homogeneous projection. The harmonic maps so obtained we call superconformal (definition 4.4), since they turn out to be isotropic of the highest possible order without being totally isotropic, the totally isotropic case having been dealt with by [EW83] and [Cal67], for instance.

We begin by writing down the general theory of section 2 in the special case of G=SU (n + 1), and hence obtain a characterization in lemma 4.1 of those maps ψ: S → SU (n + 1)/T^{n} which are τ-primitive.

The group SU (n + 1) has rank n and a maximal torus T^{n} consists of the diagonal elements of SU (n + 1), so that T is the space of purely imaginary diagonal matrices with zero trace. The complexification of su(n + 1) is sℓ(n + 1, ), and the roots are

{σ_{i} − σ_{j} : i ≠ j; i, j = 0, …, n}

where

σ_{i}(diag(y_{0}, …, y_{n})) = y_{i}, i = 0, …, n.

We take

{σ_{i} − σ_{j} : i > j; i, j = 0, …, n}

to be the positive roots, and α_{j} = σ_{j} − σ_{j−1}, j = 1, …, n, as the positive simple roots. Then θ = σ_{n} − σ_{0} = ∑_{j=1}^{n} α_{j} is the highest root, so we take α_{0} = −θ. For each pair (i, j), i, j ∈ {0, …, n}, i ≠ j, let E_{ij} be the element of sℓ(n + 1, ) whose only non-zero entry is a 1 in the (i, j)-th place. Then E_{ij} spans the root space corresponding to σ_{i} − σ_{j} and

{E_{ij} : i ≠ j; i, j = 0, …, n}

is a Cartan-Weyl basis. Hence, in the notation of section 2, M_{1} is the vector subspace of sℓ(n + 1, ) spanned by E_{1,0}, …, E_{n,n−1}, E_{0,n}. Thus the elements of M_{1} are of the form ∑_{i∈ _{n+1}} a_{i,i−1}E_{i,i−1} where a_{1,0}, …, a_{n,n−1}, a_{0,n} ∈ and B = ∑_{i∈ _{n−1}} E_{i,i−1}.

We are now in a position to identify τ-primitive maps ψ: S → SU (n + 1)/T^{n}. The elements of the flag manifold SU (n + 1)/T^{n} may be written as ordered (n + 1)-tuples (L_{0}, …, L_{n}) of mutually orthogonal 1-dimensional subspaces of ^{n+1}. Then, given a map ψ: S → SU (n + 1)/T^{n}, we write ψ(p) = (L_{0}(p), …, L_{n}(p)), p ∈ S. Thus ψ corresponds to a decomposition of the trivial bundle S × ^{n+1} into an ordered orthogonal direct sum of line subbundles, which we also denote by L_{0}, …, L_{n}.

Lemma 4.1. Let ψ: S → SU (n + 1)/T^{n} and, as above, write ψ = (L_{0}, …, L_{n}). Then ψ is τ-primitive if and only if for each nowhere vanishing local section g_{k} of L_{k},

\begin{matrix}\frac{∂g_{k}}{∂z} ∈ L_{k} ⊕ L_{k+1}, k = 0, …, n − 1, \\ \frac{∂g_{n}}{∂z} ∈ L_{n} ⊕ L_{0}, \end{matrix}

and \frac{∂g_{k}}{∂z} ∈ L_{k} at only a discrete set of points for all k = 0, …, n.

We now recall some elementary facts about harmonic sequences associated to harmonic maps into P^{n}, (see for example [BW92, BW86, Wol88]) and also give the definition of superconformality for harmonic maps into P^{n}.

Let L → P^{n} denote the tautological line bundle whose fibre over x ∈ P^{n} is the line x ⊂ ^{n+1}. There is a bijective correspondence between maps ϕ: S → P^{n} and smooth complex line subbundles of S × ^{n+1} given by ϕ ↔ ϕ^{∗}L.

Let ϕ: S → P^{n} be smooth and let L_{0} = ϕ^{∗}L. Then the standard identification of T P^{n} with Hom(L, L^{⊥}) enables us to regard the derivative dϕ as a map

dϕ: T S ⊗ L_{0} → L_{0}^{⊥},

where ⊥ denotes the orthogonal complement with respect to the standard Hermitian inner product ⟨ , ⟩ on ^{n+1}. Specifically, if X ∈ T S and s is a local section of L_{0} then dϕ(X ⊗ s) is the component of ds(X) orthogonal to L_{0}.

Every complex vector subbundle V of S × ^{n+1} inherits a holomorphic structure for which s is a local holomorphic section if and only if \frac{∂s}{∂\overline{z}} is orthogonal to V. Then ϕ is harmonic if and only if the (1, 0) part ∂ of dϕ is a holomorphic bundle map or equivalently, if and only if the (0, 1) part \overline{∂} of dϕ is an antiholomorphic bundle map.

Assume now that ϕ is harmonic. If ∂(resp. \overline{∂}) is not identically zero then the zeros are isolated and there are uniquely determined line bundles L_{1} ⊇ im(∂)(resp. L_{−1} ⊇ im(\overline{∂})) extending im(∂)(resp. im(\overline{∂})) across the zeros. The maps ϕ_{1}, ϕ_{−1}: S → P^{n} corresponding to L_{1}, L_{−1} are again harmonic hence, proceeding inductively, we obtain a sequence of line bundles, the harmonic sequence,

… L_{−2}, L_{−1}, L_{0}, L_{1}, L_{2} …

and the corresponding sequence

… ϕ_{−2}, ϕ_{−1}, ϕ_{0} = ϕ, ϕ_{1}, ϕ_{2} …

of harmonic maps from S into P^{n}. We also have a sequence of holomorphic bundle maps

∂_{k}: T^{(1,0)}S ⊗ L_{k} → L_{k+1}

and antiholomorphic bundle maps

\overline{∂}_{k}: T^{(0,1)}S ⊗ L_{k} → L_{k−1}.

Let z be a local complex coordinate on S and let f_{0} be a nowhere zero holomorphic local section of L_{0}. Then for each k there is a meromorphic local section f_{k} of L_{k} such that

f_{k+1} = ∂_{k}(\frac{∂}{∂z} ⊗ f_{k}).  (4.1)

Then f_{k+1} is the component of \frac{∂f_{k}}{∂z} orthogonal to f_{k} and we have the following equations, which hold away from the singularities of f_{k}:

\frac{∂f_{k}}{∂z} = f_{k+1} + \frac{∂}{∂z} log |f_{k}|^{2} f_{k},  (4.2)

\frac{∂f_{k}}{∂\overline{z}} = −\frac{|f_{k}|^{2}}{|f_{k−1}|^{2}} f_{k−1}.  (4.3)

Note that, by construction, any two adjacent line bundles in the harmonic sequence are orthogonal.

The following fact, which can be found in [BW86] or [BW92], follows immediatly from (4.2), (4.3).

Proposition 4.2. If some k consecutive bundles in a harmonic sequence are mutually orthogonal then every k consecutive bundles of this sequence are mutually orthogonal.

Definition 4.3. A harmonic map ϕ: S → P^{n} is k-isotropic if k consecutive bundles in the harmonic sequence of ϕ are mutually orthogonal.

We remark that a harmonic map is conformal if and only if it is 3-isotropic [BW92]. Those harmonic maps which are given by the elements of the Frenet frame of a linearly full holomorphic curve in P^{n} are said to be superminimal or totally isotropic [EW83]. The harmonic sequence of such a map is given by the Frenet frame, and in particular there are exactly n + 1 elements in the harmonic sequence, all of which are mutually orthogonal. Every harmonic map of the 2-sphere into P^{n} is totally isotropic.

If a harmonic map ϕ is not totally isotropic then each map ∂_{k}, \overline{∂}_{k} is non-zero and the harmonic sequence \{L_{k}\} is defined for all k ∈ . In particular, \frac{∂f_{k}}{∂z} ∈ L_{k} at only a discrete set of points for each k ∈ .

The results in [BFPP] only apply to non-conformal harmonic 2-tori in P^{n}. This excludes the geometrically interesting case of minimal tori. In this section we describe all harmonic 2-tori in P^{n} which are isotropic of the highest possible order without being totally isotropic.

Definition 4.4. A harmonic map ϕ: S → P^{n} is superconformal if it is (n + 1)-isotropic but not totally isotropic.

Remark 4.5 Proposition 4.2 implies that a harmonic map ϕ: S → P^{n} if and only if ϕ has an orthogonally periodic harmonic sequence (i.e. L_{0}, …, L_{n} are mutually perpendicular with L_{n+1} = L_{0}). Thus, in particular, every harmonic map ϕ: S → P^{1} is either ± holomorphic or superconformal, and every conformal harmonic map ϕ: S → P^{2} is either totally isotropic or superconformal.

Let π: SU (n + 1)/T^{n} → P^{n} be the map which assigns to the flag its first element

π(L_{0}, …, L_{n}) = L_{0}.

The manifolds SU (n+1)/T^{n} and P^{n} have canonical SU (n+1)-invariant Kähler metrics and the map π is an SU (n+1)-equivariant complex analytic Riemannian submersion. We note that every superconformal harmonic map ϕ: S → P^{n} determines a lift ϕ̃: S → SU (n + 1)/T^{n} given by

ϕ̃ = (L_{0}, …, L_{n}).  (4.4)

We are now in a position to state and prove the main theorem of this section.

Theorem 4.6. Let ϕ: S → P^{n} be a superconformal harmonic map. Then the lift ϕ̃: S → SU (n+1)/T^{n} given by (4.4) is τ-primitive. Conversely, if ψ: S → SU (n+1)/T^{n} is τ-primitive then ϕ = π∘ψ: S → P^{n} is a superconformal harmonic map with lift ϕ̃ equal to ψ.

Proof. Let ϕ: S → P^{n} be a superconformal harmonic map with lift ϕ̃ = (L_{0}, …, L_{n}). It follows from (4.2) that if g_{k} is a nowhere vanishing smooth local section of L_{k} then \frac{∂g_{k}}{∂z} ∈ L_{k} ⊕L_{k+1}, k = 0, …, n, and, since ϕ is not totally isotropic, \frac{∂g_{k}}{∂z} ∈ L_{k} at only a discrete set of points . However, since the harmonic sequence is orthogonally periodic, L_{n+1} = L_{0}, so that ϕ̃ is τ-primitive by lemma 4.1.

Conversely, assume that ψ: S → SU (n + 1)/T^{n} is τ-primitive and let ψ = (L_{0}, …, L_{n}). The harmonicity of ϕ = π ∘ ψ follows from the general theory of Black [Bla91] but in this particular case one may give an elementary argument as follows. Let g_{0}, …, g_{n} be local sections of L_{0}, …, L_{n}. Then, since ψ is τ-primitive, \frac{∂g_{r}}{∂z} ∈ L_{r} ⊕ L_{r+1} so that, for r ≠ s, s + 1,

⟨\frac{∂g_{r}}{∂\overline{z}} , g_{s}⟩ = −⟨g_{r}, \frac{∂g_{s}}{∂z} ⟩ = 0.  (4.5)

Now let f_{0} be a local holomorphic section of L_{0} and let f_{1} = ∂_{0}(\frac{∂}{∂z} ⊗ f_{0}), so that f_{1} is the component of \frac{∂f_{0}}{∂z} orthogonal to L_{0}. It follows from (4.5) that \frac{∂f_{0}}{∂\overline{z}} ∈ L_{n}, and hence

\begin{aligned}⟨\frac{∂f_{1}}{∂\overline{z}} , g_{1}⟩ & = ⟨\frac{∂}{∂\overline{z}} \frac{∂f_{0}}{∂z} , g_{1}⟩ \\ & = ⟨\frac{∂}{∂z} \frac{∂f_{0}}{∂\overline{z}} , g_{1}⟩ \\ & = 0. \end{aligned}  (4.6)

Hence, using (4.5) and (4.6), we see that f_{1} is a holomorphic section of L^{⊥}. Thus ∂_{0} is a holomorphic bundle map, so that ϕ is harmonic. Furthermore, lemma 4.1 and the method of construction of the harmonic sequence show that ϕ has L_{0}, …, L_{n} as the first n + 1 elements of its harmonic sequence, so that ϕ is superconformal with lift ϕ̃ = ψ. □

Theorems 4.6 and 3.6 have the following important consequence:

Corollary 4.7. Every superconformal harmonic 2-torus ϕ: T^{2} → P^{n} is of finite type.

In particular, together with the results on superminimal maps [EW83] this accounts for all harmonic 2-tori in P^{1} = S^{2} and P^{2}.

Finally it is interesting to note that the geometry of the harmonic sequence of a superconformal harmonic map ϕ: S → P^{n} may be used to produce directly the Toda framing whose existence was proved in theorem 2.5, thus giving an alternative proof of theorem 4.6 which does not use lemma 4.1. We begin by letting z be a local complex coordinate on a simply connected open subset U of S, and {f_{k}} a sequence of holomorphic sections of the bundles {L_{k}} satisfying the Frenet equations (4.2) and (4.3). Then, since L_{n+1} = L_{0} from remark 4.5, we have that f_{n+1} = αf_{0} for some holomorphic non-vanishing function α: U → . Thus, after a holomorphic change of coordinate we may assume that

f_{n+1} = f_{0}.  (4.7)

Moreover, by (4.3)

\frac{∂}{∂\overline{z}} det(f_{0}, …, f_{n}) = 0

so that, replacing f_{k} by det(f_{0}, …, f_{n})^{− \frac{1}{n+1}} f_{k} we may assume that

det(f_{0}, …, f_{n}) = 1.  (4.8)

Since f_{0}, …, f_{n} are mutually perpendicular it follows that we may define F = (F_{0}, …, F_{n}): U → SU (n+1) by setting F_{k} = e^{−w_{k}}f_{k}, where e^{w_{k}} = |f_{k}|, for k = 0, …, n. It now follows from (4.2), (4.8) and (4.7) that

F^{−1} \frac{∂F}{∂z} = ∂Ω + Ad exp(Ω)(B)

where Ω = diag(w_{0}, …, w_{n}): U → iT , and B = ∑_{i∈ _{n+1}} E_{i,i−1}. Hence F is the required local Toda framing and z is a corresponding complex coordinate.

5 Superconformal harmonic surfaces in S^{n}

Suppose f: S → S^{n} is a smooth map from a connected Riemann surface S, and let π: S^{n} → P^{n} denote the standard Riemannian double covering and i: P^{n} ↪ P^{n} the standard isometric totally geodesic inclusion. Then ϕ = i ∘ π ∘ f: S → P^{n} is harmonic if and only if f is harmonic. (Also ϕ is linearly full if and only if f is linearly full.) As in section 4, we may construct the harmonic sequence of ϕ but now refer to this as the harmonic sequence of f. In this case, f_{0} = f is a global holomorphic section of L_{0} and L_{−k} = \overline{L}_{k} (see [BW92]).

A harmonic map f: S → S^{n} will be called k-isotropic if ϕ = i ∘ π ∘ f: S → P^{n} is k-isotropic in the sense of section 4.

Lemma 5.1. ([BW92]) If, for some positive integer k < \frac{1}{2}n + 1, f: S → S^{n} is (2k − 1)-isotropic then f is 2k-isotropic. If n is even and if f is (n + 1)-isotropic then f is totally isotropic.

Suppose then that f: S → S^{2m} is a harmonic map. If f is (2m + 1)-isotropic then by lemma 5.1 f is totally isotropic (superminimal). We define f to be superconformal if it is 2m-isotropic but not totally isotropic.

Lemma 5.2. If f : S → S^{2m} is a superconformal harmonic map then L_{1}, …, L_{m−1} are mutually orthogonal lines whose span is an isotropic subspace of ^{2m+1} orthogonal to L_{0}. Moreover, L_{m} is not an isotropic line.

Proof. The first part follows from the fact that L_{−(m−1)}, …, L_{m−1} are mutually orthogonal and L_{−k} = \overline{L_{k}} for all k. For the second part, we note that if L_{m} were isotropic then L_{−m}, …, L_{m} would be mutually orthogonal, so that f would be (2m + 1)-isotropic which gives a contradiction to lemma 5.1. □

There are two types of superconformal harmonic maps into S^{2m}, namely those which are

(a) linearly full in S^{2m};

(b) linearly full in a totally geodesic S^{2m−1} in S^{2m}.

In case (b) the harmonic sequence is orthogonally periodic (c.f. remark 4.5) since L_{k+2m} = L_{k} for all k, and any 2m consecutive line bundles L_{k}, …, L_{k+2m−1} are mutually orthogonal. Moreover, L_{m} = \overline{L}_{m} and the corresponding map f̃: S → S^{2m−1} is the polar of S in the sense of [Law70].

Notice that any conformal harmonic (i.e. minimal) map f: S → S^{3} or S^{4} is superconformal or superminimal and so is every almost complex curve f: S → S^{6} (c.f. section 6).

We now consider the theory of section 2 in the case where G = SO(2m + 1). First let _{m} denote the flag manifold whose elements are ordered sequences

X = (X_{0}, …, X_{m})

of mutually orthogonal complex lines in ^{2m+1} where X_{0} is the complexification of a real line in ^{2m+1} and X_{1}, … X_{m} span a maximal isotropic subspace of ^{2m+1}. Note that each element X = (X_{0}, …, X_{m}) ∈ _{m} determines a unique unit vector u_{0} = ρ(X) ∈ ^{2m+1} which spans X_{0}. For if \frac{1}{\sqrt{2}}(u_{2k−1} − iu_{2k}) is a unit basis vector for X_{k} (k = 1, …, m) then u_{0} is the unique unit vector in ^{2m+1} for which u_{0}, …, u_{2m} is a positively oriented orthonormal basis of ^{2m+1}. This defines a projection ρ: _{m} → S^{2m}. We also note that, similarly, X ∈ _{m} is uniquely determined by ρ(X) and X_{1}, …, X_{m−1}.

The group SO(2m+1) acts transitively on _{m} and the stabilizer of each point is a maximal torus. Indeed, for k = 0, …, 2m, let e_{k} be the (k + 1)-st standard basis vector of ^{2m+1} (having a 1 in the (k + 1)-st place and zeros elsewhere) and let C_{0} = e_{0}, C_{k} = (e_{2k−1} − ie_{2k}) for k = 1, …, m. Then the stabilizer of C = (C_{0}, …, C_{m}) ∈ _{m} is the maximal torus

T^{m} = {diag(1, R(θ_{1}), …, R(θ_{m})) | θ_{1}, …, θ_{m} ∈ }

where

R(θ) = ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) .

Thus we may identify _{m} with SO(2m + 1)/T^{m} and ρ with the projection of SO(2m + 1)/T^{m} onto S^{2m} induced by sending an element of SO(2m + 1) to its first column.

Let f: S → S^{2m} be a superconformal harmonic map. Then, by lemma 5.2, there is a lift f̃: S → SO(2m + 1)/T^{m} defined by

f̃ = (X_{0}, …, X_{m})  (5.1)

where X_{k} = L_{k}, k = 0, …, m − 1 and X_{m} is determined by the condition that ρ ∘ f̃ = f. Note that L_{m} is contained in the orthogonal complement X_{m} ⊕ \overline{X_{m}} of L_{−(m−1)} ⊕ ⋯ ⊕ L_{m−1}. This is needed in the proof of theorem 5.4.

As we will show, f̃ is τ-primitive and the process of lifting gives a 1 − 1 correspondence between superconformal harmonic maps f: S → S^{2m} and τ-primitive maps ψ: S → SO(2m + 1)/T^{m}.

We first characterize τ-primitive maps using the root space decomposition for so(2m + 1) as follows: let e_{0}, …, e_{2m} be the standard basis of ^{2m+1}. As a Cartan subalgebra T for so(2m + 1), we take the subspace of so(2m + 1) consisting of those (2m + 1) × (2m + 1) matrices for which each of the subspaces e_{0}, e_{2k−1} ⊕ e_{2k} is an invariant subspace. Thus, these are matrices of the form

diag(0, Θ_{1}, …, Θ_{m}), where Θ_{k} = ( \begin{matrix}0 & −θ_{k} \\ θ_{k} & 0 \end{matrix} ) , k = 1, …, m.

For each k = 1, …, m, let σ_{k}: T → i be defined by σ_{k}(diag(0, Θ_{1}, …, Θ_{m})) = iθ_{k}. Then as positive simple roots we take σ_{1}, σ_{2} − σ_{1}, …, σ_{m} − σ_{m−1}. The highest root is then given by

σ_{m} + σ_{m−1} = 2(σ_{1} + (σ_{2} − σ_{1}) + (σ_{3} − σ_{2}) + ⋯ + (σ_{m−1} − σ_{m−2})) + σ_{m} − σ_{m−1}.

In order to describe the subspace M_{1} of so(2m + 1) we first define u_{0}, …, u_{m} to be the unit vectors in 2m+1 = ^{2m+1} ⊗ given by

u_{0} = e_{0}, u_{k} = \frac{1}{\sqrt{2}}(e_{2k−1} − ie_{2k}), k = 1, …, m.

Then u_{0}, u_{1}, …, u_{m}, \overline{u_{1}}, …, \overline{u_{m}} is a basis of ^{2m+1}. Let V_{1}, …, V_{m+1} ∈ so(2m + 1) be defined as follows:

(a) for k = 1, …, m,

\begin{matrix}V_{k}u_{k−1} = u_{k}, V_{k}u_{ℓ} = 0, ℓ ≠ k − 1 \\ V_{k}\overline{u_{k}} = −\overline{u_{k−1}}, V_{k}\overline{u_{ℓ}} = 0, ℓ ≠ k. \end{matrix}

(b)

V_{m+1}u_{k} = 0, k = 0, …, m − 2
V_{m+1}u_{m−1} = \overline{u_{m}}, V_{m+1}u_{m} = −\overline{u_{m−1}}
V_{m+1}\overline{u_{k}} = 0, k = 1, …, m.

Then V_{1} spans the eigenspace A_{1} corresponding to σ_{1}, V_{k} spans the eigenspace A_{k} corresponding to σ_{k} − σ_{k−1} for k = 2, …, m, and V_{m+1} spans the eigenspace A_{m+1} corresponding to −(σ_{m} + σ_{m−1}). Then

M_{1} = ⨁_{k=1}^{m+1} A_{k}.

We now give a characterization of τ-primitive maps ψ: S → SO(2m + 1)/T^{m}.

Proposition 5.3 The map ψ = (X_{0}, …, X_{m}): S → SO(2m + 1)/T^{m} is τ-primitive if and only if, for each nowhere vanishing local section g_{k} of X_{k} (i) \frac{∂g_{k}}{∂z} ∈ X_{k} ⊕ X_{k+1} for k = 0, …, m − 2, and has a non-zero X_{k+1} component except possibly at a discrete set of points.

(ii) \frac{∂g_{m−1}}{∂z} ∈ X_{m−1} ⊕ X_{m} ⊕ \overline{X}_{m}, and has non-zero X_{m}, \overline{X_{m}} components except possibly at a discrete set of points.

(iii) \frac{∂\overline{g}_{k}}{∂z} ∈ \overline{X}_{k−1} ⊕ \overline{X}_{k} for k = 1, …, m − 1.

Proof. Let g_{0}, …, g_{m} be nowhere vanishing local sections of X_{0}, …, X_{m} respectively, and for k = 0, …, m let G_{k} = g_{k}/|g_{k}|. If (i), (ii), (iii) hold then, for suitable scalars λ_{k}, μ_{k}, α_{k}, β_{k}, ν we may write

\begin{matrix}\frac{∂G_{k}}{∂z} = λ_{k}G_{k} + μ_{k}G_{k+1}, k = 0, …, m − 2, \\ \frac{∂G_{m−1}}{∂z} = λ_{m−1}G_{m−1} + μ_{m−1}G_{m} + ν\overline{G_{m}}, \\ \frac{∂\overline{G_{k}}}{∂z} = λ_{k}\overline{G_{k−1}} + β_{k}\overline{G_{k}}, k = 0, …, m − 1. \end{matrix}  (5.2)

It then follows that for k = 1, …, m − 1, α_{k} = −μ_{k−1}, and for suitable β_{m},

\frac{∂\overline{G_{m}}}{∂z} = −μ_{m−1}\overline{G_{m−1}} + β_{m}\overline{G_{m}}.

Hence, if G̃ is the SO(2m + 1)-valued map given by

G̃ = (G_{0}, \frac{1}{\sqrt{2}}(G_{1} + \overline{G_{1}}), \frac{i}{\sqrt{2}}(G_{1} − \overline{G_{1}}), …, \frac{1}{\sqrt{2}}(G_{m} + \overline{G_{m}}), \frac{i}{\sqrt{2}}(G_{m} + \overline{G_{m}}))

then

G̃^{−1} \frac{∂G̃}{∂z} = A_{0} + A_{1} ∈ M_{0} ⊕ M_{1}

where

A_{1} = ∑_{k=0}^{m−1} μ_{k}V_{k+1} + νV_{m+1}.

The additional assumtions on \frac{∂g_{k}}{∂z} show that each of μ_{0}, …, μ_{m−1}, ν vanish only at isolated points. Hence ψ is τ-primitive. The converse is similar but easier. □

We now state the main result of this section. The proof is similar to that of theorem 4.6, except that we also need the second statement of lemma 5.2.

Theorem 5.4. Let f: S → S^{2m} be a superconformal harmonic map. Then the lift f̃: S → SO(2m+1)/T^{m} given by (5.1) is τ-primitive. Conversely if ψ: S → SO(2m + 1)/T^{m} is τ-primitive, then f = ρ ∘ ψ: S → S^{2m} is a superconformal harmonic map with lift ψ.

We recall from above that superconformal harmonic maps f: S → S^{2m} are either linearly full in S^{2m} or are linearly full in a totally geodesic (2m − 1)-sphere S^{2m} ∩ (N)^{⊥} where N is a unit vector in ^{2m+1}. These latter maps are characterized by the fact that X_{m} ⊕ \overline{X}_{m} contains the constant real line spanned by N (for then X_{0} is orthogonal to N). The real vector in X_{m} ⊕ \overline{X}_{m} orthogonal to N determines the polar of f in the sense of [Law70].

As in the P^{n} case, the above theorem has the following important consequence:

Corollary 5.5. Every superconformal harmonic 2-torus ϕ: T^{2} → S^{2m} is of finite type.

Since every (non-super)minimal 2-torus in S^{4} is superconformal we recover the results in [FPPS92].

Also, as in the P^{n} case, we may produce directly the corresponding Toda framing, hence proving theorem 5.4 without using proposition 5.3. Firstly, given a local complex coordinate z in some simply connected open subset of S we have the sequence of ^{2m+1}-valued functions (4.1) where f_{0} = f. Then from (4.3) and the fact that f is not superminimal it follows that (f_{m}, f_{m}) is a non-zero holomorphic function and hence we may suppose that z is chosen such that (f_{m}, f_{m}) = 1. (The coordinate z is thus uniquely determined up to multiplication by a 2m-th root of unity).

Now define a real positively oriented orthonormal frame E_{0}, …, E_{2m} on U and a real valued function η on U as follows:

E_{0} = f_{0}, \frac{f_{k}}{|f_{k}|} = \frac{1}{\sqrt{2}}(E_{2k−1} − iE_{2k}), k = 1, …, m − 1,

f_{m} = cosh ηE_{2m−1} − i sinh ηE_{2m}.

We note that E_{0}, …, E_{2m} and η are uniquely determined by f_{0} (up to ambiguities involving 2m-th roots of unity in the choice of the coordinate z mentioned above).

A straightforward calculation then shows that for E = (E_{0}, …, E_{2m}) ∈ SO(2m + 1),

E^{−1} \frac{∂E}{∂z} = \frac{∂Ω}{∂z} + Ad exp(Ω)(B)

where iΩ = diag(0, Θ_{1}, …, Θ_{m}) with Θ_{k} = ( \begin{matrix}0 & −w_{k} \\ w_{k} & 0 \end{matrix} ) , e^{w_{k}} = |f_{k}|, k = 1, …, m − 1, Θ_{m} =

( \begin{matrix}0 & −η \\ η & 0 \end{matrix} ) and B = ∑_{k=1}^{m+1} V_{k}.

We note that the case where f: S → S^{2m} is linearly full into a totally geodesic S^{2m−1} is characterized by η ≡ 0.

6 Almost complex curves in S^{6}

We recall that the vector cross-product on ^{7} (see [BVW] for details relevant to the present context and further references) derived from Cayley multiplication on ^{8} may be used to define an almost complex structure on the unit sphere S^{6} in ^{7}.

For any unit vector x ∈ ^{7}, the orthogonal complement (x)^{⊥} admits an orthogonal complex structure J_{x} defined by

J_{x}v = x × v, v ∈ (x)^{⊥}.

Since (x)^{⊥} may be identified with the tangent space T_{x}S^{6}, this defines an orthogonal almost complex structure on S^{6}. This, together with the standard metric on S^{6}, defines a nearly Kähler structure whose automorphism group is the exceptional Lie group G_{2}.

The group G_{2} is the subgroup of SO(7) consisting of those 7 × 7 matrices whose columns u_{1}, …, u_{7} form an orthonormal basis for ^{7} for which

u_{3} = u_{1} × u_{2}, u_{5} = u_{1} × u_{4}, u_{6} = u_{2} × u_{4}, u_{7} = u_{3} × u_{4}.

Such a basis is called a G_{2}-basis. For example, changing the notation of the previous section, the standard basis e_{1}, …, e_{7} (rather than e_{0}, …, e_{6}) of ^{7} is a G_{2}-basis.

The elements of SO(7) which fix e_{1} and commute with J_{e_{1}} on (e_{1})^{⊥} form a subgroup SU (3) ⊂ G_{2} ⊂ SO(7). Care is needed here since

J_{e_{1}}(e_{2} − ie_{3}) = i(e_{2} − ie_{3}), J_{e_{1}}(e_{4} − ie_{5}) = i(e_{4} − ie_{5})

but

J_{e_{1}}(e_{6} − ie_{7}) = −i(e_{6} − ie_{7}).

so that this subgroup SU (3) is a conjugate of the standard one in SO(7).

The maximal torus T^{2} of SU (3) and of G_{2} consists of elements of the form

diag(1, R(θ_{1}), R(θ_{2}), R(θ_{3})), θ_{1} + θ_{2} = θ_{3}.

Moreover, G_{2}/T^{2} ⊂ SO(7)/T^{3}, and, in the notation of section 5, consists of those elements X = (X_{0}, X_{1}, X_{2}, X_{3}) ∈ SO(7)/T^{3} for which X_{1}, X_{2}, (resp. X_{3}) are +i( resp. −i)-eigenspaces for J_{ρ(x)}. In fact the condition on X_{3} follows from those on X_{1} and X_{2}. One may also show that, equivalently, if u_{0} = ρ(X) then there is a basis v_{1}, v_{2}, \overline{v}_{3} of X_{1}, X_{2}, X_{3} respectively such that |v_{k}| = \frac{1}{\sqrt{2}} and

\begin{matrix}u_{0} × v_{k} = iv_{k} \\ v_{k} × v_{ℓ} = ϵ_{kℓm}\overline{v}_{m} \\ v_{k} × \overline{v}_{ℓ} = −\frac{i}{2}δ_{kℓ}u_{0} \end{matrix}  \} k, ℓ, m ∈ \{1, 2, 3\},

where × denotes the extension of the vector cross-product to ^{7}. (c.f. [BVW].)

An almost complex curve in S^{6} is a non-constant smooth map f: S → S^{6} whose differential is complex linear. In terms of a local complex coordinate z = x + iy the condition for f to be almost complex may be written

f × f_{z} = if_{z}

or equivalently, in terms of the notation of section 4, (4.1),

f × f_{1} = if_{1}.  (6.1)

It is easy to check (c.f. [BVW]) that an almost complex curve which is not totally isotropic is a superconformal harmonic map.

Proposition 6.1. Let f: S → S^{6} be superconformal with lift f̃: S → SO(7)/T^{3}. Then f is almost complex if and only if f̃(S) ⊆ G_{2}/T^{2}.

Proof. If f̃(S) ⊆ G_{2}/T^{2} then writing f̃ = (X_{0}, X_{1}, X_{2}, X_{3}) we see that, since f = ρ ∘ f̃ and f_{1} ∈ X_{1}, it follows that (6.1) holds.

Conversely suppose that f: S → S^{6} is an almost complex curve. Then (6.1) holds, and differentiating (6.1) gives

f × f_{2} = if_{2}.

Since f_{k} ∈ X_{k}, (k = 1, 2), the result follows. □

Now we consider the roots of G_{2} in order to consider τ-primitive maps. Recall from section 5 that we may take σ_{1}, σ_{2} −σ_{1}, σ_{3} −σ_{2} as a set of positive simple roots of so(7) and that the highest root is σ_{3} +σ_{2}. From above, we see that the Cartan subalgebra T^{2} of G_{2} is the subspace of the Cartan subalgebra T^{3} of so(7) defined by σ_{1} + σ_{2} = σ_{3}. As a set of positive simple roots of G_{2} we may take σ_{1}, σ_{2} − σ_{1} in which case the highest root is σ_{3} + σ_{2} = σ_{1} + 2σ_{2} = 3σ_{1} + 2(σ_{2} − σ_{1}).

The root spaces of so(7) corresponding to σ_{2} − σ_{1} and −(σ_{3} + σ_{2}) are both contained in G_{2}, while the G_{2}-root space corresponding to σ_{1}(= σ_{3} − σ_{2}) is the intersection with G_{2} of the direct sum of the so(7) root spaces corresponding to σ_{1} and σ_{3} − σ_{2}.

It is now clear that

M_{1}(G_{2}) = M_{1}(SO(7)) ∩ G_{2}.  (6.2)

Alternatively, one may deduce this from the fact that G_{2}/T^{2} is a 6-symmetric submanifold of the 6symmetric space SO(7)/T^{3}, where in both cases the symmetry of order 6 is given by τ = Ad exp(2πiZ) where Z = diag(1, R(\frac{2π}{6} ), R(\frac{4π}{6} ), R(\frac{6π}{6} )), as indicated by the general theory of section 2. For in both cases M_{1} is the e^{2πi/6}-eigenspace of the automorphism of the complexified Lie algebra given by τ.

The following proposition is now clear.

Proposition 6.2. Let ψ: S → G_{2}/T^{2} and let i: G_{2}/T^{2} → SO(7)/T^{3} be the natural inclusion. Then ψ is τ-primitive if an only if i ∘ ψ is τ-primitive.

Propositions 6.1 and 6.2 now immediately give the following theorem.

Theorem 6.3. Let f: S → S^{6} be an almost complex curve which is not totally isotropic. Then the lift f̃: S → G_{2}/T^{2} given by (4.1) is τ-primitive. Conversely, if ψ: S → G_{2}/T^{2} is τ-primitive then f = ρ ∘ ψ: S → S^{6} is almost complex with lift ψ.

As in the previous sections this has the following important consequence.

Corollary 6.4. Every non-superminimal almost complex 2-torus in S^{6} is of finite type.

Together with the results on superminimal almost complex curves in [BVW] this accounts for all almost complex 2-tori in S^{6}.

Finally we consider Toda framings for almost complex curves. As in section 5, we first choose a local complex coordinate z in an open subset U of S such that (f_{3}, f_{3}) = 1. Now define a real positively oriented frame E_{1}, …, E_{7} on U and a real valued function η on U as follows:

E_{1} = f_{0}, \frac{f_{k}}{|f_{k}|} = \frac{1}{\sqrt{2}} (E_{2k} − iE_{2k+1}), k = 1, 2,

f_{3} = − cosh η E_{7} − i sinh η E_{6}.

This is then a G_{2}-framing as shown in [BVW]. Note that the seemingly natural way of writing f_{3} in terms of E_{6}, E_{7} and η (as in section 5) does not give a G_{2}-framing. Moreover (see [BVW]), if |f_{k}| = e^{w_{k}}, k = 1, 2, then w_{1} + w_{2} = η, and writing E = (E_{1}, …, E_{7}) we have

E^{−1} \frac{∂E}{∂z} = \frac{∂Ω}{∂z} + Ad(exp Ω)(B),

where

Ω = i ( \begin{matrix}0 \\ & & w_{1} \\ & −w_{1} \\ & & & & w_{2} \\ & & & −w_{2} \\ & & & & & & η \\ & & & & & −η \end{matrix} ) ∈ iT^{2}

and

B = ( \begin{matrix}& \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ −\frac{i}{\sqrt{2}} & & & \frac{1}{2} & −\frac{i}{2} \\ −\frac{1}{\sqrt{2}} & & & \frac{i}{2} & \frac{1}{2} \\ & −\frac{1}{2} & −\frac{i}{2} & & & \frac{1}{2} & \frac{i}{2} \\ & \frac{i}{2} & −\frac{1}{2} & & & \frac{i}{2} & −\frac{1}{2} \\ & & & −\frac{1}{2} & −\frac{i}{2} \\ & & & −\frac{i}{2} & \frac{1}{2} \end{matrix} ) ∈ M_{1}(G_{2}) .

References

[1] M. Adler and P. van Moerbeke. Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math., 38:267–317, 1980.

[2] M. Black. Harmonic maps into homogeneous spaces. Longman, Harlow, 1991.

[3] A. I. Bobenko. All constant mean curvature tori in ^{3}, S^{3}, H^{3} in terms of theta-functions. Math. Ann., 290:209–245, 1991.

[4] J. Bolton, L. Vrancken, and L. M. Woodward. On almost complex curves in the nearly Kähler 6-sphere. Preprint.

[5] J. Bolton and L. M. Woodward. Congruence theorems for harmonic maps from a Riemann surface into P^{n} and S^{n}. J. London Math. Soc., 45(2):363–376, 1992.

[6] R. L. Bryant. Conformal and minimal immersions of compact surfaces into the 4-sphere. J. Diff. Geom., 17:455–473, 1982.

[7] F. E. Burstall. Harmonic maps and soliton theory. Bath preprint, 1991.

[8] F. E. Burstall. Riemannian twistor spaces and holonomy groups. In Twistors in Mathematics and Physics. C.U.P., 1990. pp 53–70.

[9] F. E. Burstall, D. Ferus, F. Pedit, and U. Pinkall. Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras. To appear in Annals of Math.

[10] F. E. Burstall and F. Pedit. Dressing orbits of harmonic maps. In preparation.

[11] F. E. Burstall and J. C. Wood. The construction of harmonic maps into complex Grassmanians. J. Diff. Geom., 23:255–297, 1986.

[12] E. Calabi. Minimal immersions of surfaces into Euclidean spheres. J. Diff. Geom., 1:111–125, 1967.

[13] A. Doliwa and A. Sym. Minimal surfaces in S^{2m} and Toda systems. Preprint, 1992.

[14] J. Dorfmeister, F. Pedit, and H. Wu. Harmonic surfaces into symmetric spaces via loop group factorization. In preparation.

[15] J. Dorfmeister and H. Wu. Constant mean curvature surfaces and loop groups. To appear in J. reine angew. Math.

[16] J. Eells and J. C. Wood. Harmonic maps from surfaces into projective spaces. Adv. in Math., 49:217–263, 1983.

[17] L. D. Faddeev and L. A. Takhtajan. Hamiltonian methods in the theory of solitons. Springer-Verlag, New York-Heidelberg-Berlin, 1987.

[18] D. Ferus, F. Pedit, U. Pinkall, and I. Sterling. Minimal tori in S^{4}. J. reine angew. Math., 429:1–47, 1992.

[19] A. P. Fordy. Integrable equations associated with simple Lie algebras and symmetric spaces. Manchester Univ. Press, Manchester, New York, 1990. pp 315–337.

[20] S. Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, San Francisco, London, 1978.

[21] N. J. Hitchin. Harmonic maps from a 2-torus to the 3-sphere. J. Diff. Geom., 31:627–710, 1990.

[22] A. Jimenez. Addendum to “Existence of Hermitian n-symmetric spaces and non-commutative naturally reductive spaces”. Math. Z., 197:455–456, 1988.

[23] B. Kostant. The principal three dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math., 81:973–1032, 1959.

[24] H. B. Lawson. Complete minimal surfaces in S^{3}. Annals of Math., 92:335–374, 1970.

[25] I. McIntosh. Global solutions of the elliptic 2d Toda lattice. Preprint.

[26] U. Pinkall and I. Sterling. On the classification of constant mean curvature tori. Annals of Math., 130:407–451, 1989.

[27] A. G. Reyman and M. A. Semenov-Tian-Shansky. Compatible Poisson structures for Lax equations: an r-matrix approach. Phys. Let. A, 130:456–460, 1988.

[28] K. Uhlenbeck. Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom., 30:1–50, 1989.

[29] H. Wente. Counterexample to a conjecture of H. Hopf. Pacific Journal of Math., 121:193–243, 1986.

[30] J. G. Wolfson. Harmonic sequences and harmonic maps of surfaces into complex Grassman manifolds. J. Diff. Geom., 27:161–178, 1988.

[31] H. Wu. Banach manifolds of minimal surfaces in the 4-sphere. In R. Greene & S.-T. Yau, editor, AMS Proc. of Symposia in Pure Math., pages 530–539, 1993.

[32] V. E. Zakharov and A. B. Shabat. Integration of non-linear equations of mathematical physics by the method of inverse scattering, II. Funkts. Anal i Prilozhen, 13:13–22, 1978.