Harmonic maps via Adler–Kostant–Symes theory

F.E. Burstall and F. Pedit

Introduction

Over the past few years significant progress has been made in the understanding of various completely integrable nonlinear PDE (soliton equations) and their relationship to classical problems in differential geometry. It has been shown in a series of recent papers [41, 31, 20, 24, 14, 7, 11] that constant mean and Gauss curvature surfaces, Willmore surfaces, minimal surfaces in spheres and projective spaces and generally harmonic maps from a Riemann surface M into various homogeneous spaces may be described as solutions to various soliton equations. Moreover, these solutions are algebraic in the sense that they are obtained by integrating ordinary differential equations of Lax type which linearise on the Jacobian of an appropriate algebraic curve.

Links between harmonic maps and integrable systems have been known to exist for some time: for instance, Uhlenbeck [54] showed that S^{1}-equivariant harmonic maps ℝ × S^{1} → S^{n} amount to solutions of the Neumann system describing motion on S^{n} in a quadratic potential—a classical completely integrable system (for related results on S^{1}-equivariant harmonic maps, see [32, 23]). However, a significant interaction between differential geometry and soliton theory did not emerge until after Wente’s resolution [58] of the Hopf conjecture on the existence of a constant mean curvature torus in ℝ^{3}.

Let us consider this problem in more detail as it contains the seeds of all subsequent developments in this area. Recall that a surface M in ℝ^{3} has constant mean curvature if and only if its Gauss map ϕ : M → S^{2} is harmonic. We are therefore led to pose the following problem: given a compact Riemann surface M of genus g, find all harmonic maps ϕ from M into the 2-sphere. The harmonicity of ϕ implies that the (2, 0)-part of the quadratic differential (dϕ, dϕ) is holomorphic. If M has genus 0 then this differential vanishes identically, i.e. ϕ is conformal and thus ±holomorphic. In this sense the harmonic map equation reduces to the linear Cauchy–Riemann equation. A similar phenomenon occurs when the target space is replaced by a compact symmetric space where the reduction to the Cauchy–Riemann equation is achieved by various twistorial constructions (see [34, 59] in this volume for further details and references).

The situation changes drastically when M has higher genus. There is no systematic theory when M has genus larger then 1 so we shall concentrate on the case where M is a 2-torus. Here, the differential (dϕ, dϕ)^{(2,0)} either vanishes identically (in which case we again have ±holomorphic maps) or is nowhere zero. In the latter case one can choose a global complex coordinate z so that (dϕ, dϕ)^{(2,0)} = dz^{2} and (dϕ, dϕ)^{(1,1)} = cosh(ω) |dz|^{2}. Then the harmonicity of ϕ amounts to the sinh-Gordon equation for ω, a well known soliton equation. Starting from this observation, Pinkall–Sterling [41] show that all doubly-periodic solutions to the sinh-Gordon equation (and thus all constant mean curvature tori) are obtained by integration of a family of completely integrable finite dimensional systems of ODE. These ODE linearise on the Jacobian of an algebraic curve the study of which enabled Bobenko [6] to show that the solutions can be expressed rather explicitly in terms of theta functions. That this behaviour extends to more general target spaces was indicated in [31] for S^{3}, [24] for S^{4} and culminated in [14] where a rather comprehensive theory of harmonic 2-tori in compact Riemannian symmetric spaces is developed.

One may summarise this theory as follows: the starting point is the basic fact [42, 55, 64, 65] that the harmonic map equation can be reformulated as a zero-curvature equation involving an auxiliary parameter or, in other words, as the Maurer–Cartan equation for a certain loop algebra valued 1-form. One can obtain solutions to these equations by integrating a pair of commuting Hamiltonian vector fields on certain finite-dimensional subspaces of loop algebras. In this way, we get harmonic maps of ℝ^{2} which we call harmonic maps of finite type. Finally, under suitable nondegeneracy assumptions, one shows that these procedures account for all doubly periodic harmonic maps, that is, for all harmonic 2-tori.

It is the purpose of this article to provide a unified account of these ideas and, in particular, the results of [7, 11, 14, 24, 41]. A framework for this is given by the theory of Adler–Kostant–Symes [1, 36, 53]. This theory provides a scheme for producing and integrating non-trivial commuting Hamiltonian flows on Lie algebras. The basic setting for the scheme is this: one has a Lie algebra 𝔤 which admits a (vector space) direct sum decomposition into subalgebras

𝔤 = 𝔨 ⊕ 𝔟. (0.1)

In the presence of a suitable inner product, 𝔨 inherits a Poisson structure from the Lie–Poisson structure on 𝔟^{∗}. With this Poisson structure, 𝔨 has a large supply of Poisson commuting functions provided by the restriction to 𝔨 of the invariant functions on 𝔤. To integrate the Hamiltonian flows so obtained, we use the method of Symes: let G be a Lie group with Lie algebra 𝔤 and let K, B be the subgroups corresponding to 𝔨, 𝔟. Suppose that there is a decomposition

G = K × B (0.2) corresponding to (0.1). Then the Hamiltonian flow through an initial condition ξ_{o} ∈ 𝔨 is given by

Ad k^{−1} ξ_{o},

where k is the projection onto K via (0.2) of a suitable geodesic in G.

We apply this theory to a family of twisted loop algebras which admit a decomposition (0.1) of Iwasawa type. A key point is that the subspaces of Laurent polynomial loops of fixed degree are Poisson submanifolds so that we obtain commuting ODE on finite-dimensional Poisson manifolds. From the Hamiltonian flows on these subspaces we get solutions to zero-curvature equations which give rise to (framings of) harmonic maps of ℝ^{2} into various homogeneous spaces.

In fact, we get in this way both harmonic maps into Riemannian symmetric spaces and primitive maps into k-symmetric spaces. These last are rather special harmonic maps into a class of reductive homogeneous spaces that generalise Riemannian symmetric spaces (the involutions of a Riemannian symmetric space are replaced by automorphisms of order k) and include all flag manifolds. A map into such a space is primitive if it satisfies a first order condition not unlike a Cauchy–Riemann equation defined by the geometry of the k-symmetric space. The motivation for studying primitive maps comes from the fact that they include twistor lifts of minimal maps in spheres and complex projective spaces and also, in case the target is a flag manifold, are in bijective correspondence with (periodic) Toda fields.

The method of Symes also works in this infinite-dimensional loop algebra setting. There are Iwasawa decompositions of the loop groups that correspond to our loop algebras and projection of (complex) geodesics provides maps into loop groups which are essentially the extended solutions in the sense of Uhlenbeck [55] for the harmonic maps we have produced.

Finally, we obtain sufficient conditions on a harmonic 2-torus to arise from our constructions. The basic result here is due to Burstall–Ferus–Pedit–Pinkall [14] and the method of Symes is applied to translate that result to our present setting.

1 The Adler–Kostant–Symes scheme

One of the main themes of this volume is the construction of harmonic maps from commuting Hamiltonian flows on loop algebras. A setting for such results is provided by the celebrated analysis of the (open) Toda lattice by Kostant and Symes [36, 53], where a general scheme is described for producing and integrating commuting Hamiltonian flows on Lie algebras (see also [1]). Since this beautiful circle of ideas may be unfamiliar to Riemannian geometers, we begin by rehearsing the main points of the theory in a form suitable for our applications.

1.1 Poisson structures and Lie algebra decompositions

Let M be a manifold. A Poisson structure on M is a Lie algebra structure on C^{∞}(M), with bracket denoted (f, g) ↦ {f, g}, for which ad f is a derivation over multiplication:

{f, gh} = {f, g}h + g{f, h},

for all f, g, h ∈ C^{∞}(M). As a consequence, each f ∈ C^{∞}(M) gives rise to a Hamiltonian vector field X_{f} by

X_{f}g = {f, g}.

It follows from the Jacobi identity for { , } that f ↦ X_{f} is a Lie algebra homomorphism C^{∞}(M) → C^{∞}(T M).

For our basic example of a Poisson structure, let 𝔟 be a (real) Lie algebra with dual 𝔟^{∗}. When f ∈ C^{∞}(𝔟^{∗}) and x ∈ 𝔟^{∗}, we have df_{x} ∈ 𝔟^{∗∗} ≅ 𝔟 and, using this identification, we define a Poisson structure on 𝔟^{∗} by

{f, g}(x) = ⟨x, [df_{x}, dg_{x}]⟩.

The Ad^{∗}-invariant functions on 𝔟^{∗} commute with respect to this Poisson structure but, unfortunately, they also commute with all other functions on 𝔟^{∗} and so have trivial Hamiltonian flows. To get non-trivial commuting flows, we must introduce some extra structure.

Suppose then that we have a Lie algebra 𝔤 which admits a (vector space) direct sum decomposition into subalgebras:

𝔤 = 𝔨 ⊕ 𝔟.

Suppose further that there is a (real) non-degenerate symmetric bilinear form on 𝔤, denoted ( , ), which is invariant:

([ξ, η], ζ) = −(η, [ξ, ζ]).

The bilinear form induces a musical isomorphism 𝔤 ≅ 𝔤^{∗} and so, by restriction, an isomorphism 𝔨^{∘} ≅ 𝔟^{∗}. Here 𝔨^{∘} is the polar of 𝔨 with respect to ( , ):

𝔨^{∘} = {η ∈ 𝔤 : (η, ξ) = 0 for all ξ ∈ 𝔨}.

Thus 𝔨^{∘} acquires a Poisson structure from that of 𝔟^{∗} and this is the Poisson structure that will be important for us.

To describe the Poisson structure on 𝔨^{∘} explicitly, we introduce some notation. Let π_{𝔨}, π_{𝔟}be the projections onto 𝔨, 𝔟 along 𝔟, 𝔨 respectively. Further, for f ∈ C^{∞}(𝔨^{∘}), let f̃ be some extension of f to 𝔤 and let ∇f̃ denote its gradient with respect to ( , ). It is easy to check that the projected gradient

π_{𝔟}∇f̃

depends only on f. A straightforward calculation now shows that the Poisson bracket on 𝔨^{∘} is given by

{f, g}(ξ) = (ξ, [π_{𝔟}∇f̃_{ξ}, π_{𝔟}∇g̃_{ξ}]). (1.1)

To see the Hamiltonian vector field of f, let π_{𝔨^{∘}}denote projection onto 𝔨^{∘} along 𝔟^{∘}. Then, using the invariance of ( , ), we have

{f, g}(ξ) = ([ξ, π_{𝔟}∇f̃_{ξ}], π_{𝔟}∇g̃_{ξ})
= (π_{𝔨^{∘}}[ξ, π_{𝔟}∇f̃_{ξ}], ∇g̃_{ξ})
= dg̃(π_{𝔨^{∘}}[ξ, π_{𝔟}∇f̃_{ξ}]) = dg(π_{𝔨^{∘}}[ξ, π_{𝔟}∇f̃_{ξ}]).

Thus, for ξ ∈ 𝔨^{∘},

X_{f}(ξ) = π_{𝔨^{∘}}[ξ, π_{𝔟}∇f̃_{ξ}]. (1.2)

1.2 Poisson submanifolds

Let M be a Poisson manifold with bracket { , }_{M}. A submanifold V of M is a Poisson submanifold if

1. V has a Poisson structure { , }_{V};

2. the inclusion i : V ↪ M is a Poisson morphism:

{f ∘ i, g ∘ i}_{V} = {f, g}_{M} ∘ i, (1.3)

for f, g ∈ C^{∞}(M).

Since i^{∗} : C^{∞}(M) → C^{∞}(V) is surjective, it is clear that (1.3) completely determines the Poisson structure on V. Thus a submanifold V is Poisson as soon as the prescription (1.3) gives a well-defined bracket { , }_{V} on C^{∞}(V).

There is a simple condition for this to be the case: a submanifold V is Poisson if and only if the restriction of any Hamiltonian vector field to V is tangent to V (see [57, Lemma 1.1]). In the setting of Section 1.1, we use this and the form of the Hamiltonian vector fields (1.2) to obtain the following

Proposition 1.1 Let V ⊂ 𝔨^{∘} be a linear subspace satisfying

π_{𝔨^{∘}}[V, 𝔟] ⊂ V. (1.4)

Then V is a Poisson submanifold of 𝔨^{∘}.

Such considerations will be important when we come to discuss finite-dimensional subspaces of infinite-dimensional loop algebras.

Remark The above discussion is merely the translation into our setting of the fact that ad^{∗} 𝔟-invariant subspaces of 𝔟^{∗} are Poisson submanifolds of 𝔟^{∗} which in turn follows from the fact that the symplectic leaves of 𝔟^{∗} are precisely the co-adjoint orbits.

1.3 Commuting Hamiltonians

We now come to the main point of our constructions: ad-invariant functions on 𝔤 restrict to Poisson commuting functions on 𝔨^{∘}. Recall that f ∈ C^{∞}(𝔤) is ad-invariant if

df_{ξ}([ξ, 𝔤]) = 0,

or, equivalently,

[∇f_{ξ}, ξ] = 0, (1.5)

for all ξ ∈ 𝔤. If f, g ∈ C^{∞}(𝔤) are two ad-invariant functions, then their restrictions to 𝔨^{∘} have Poisson bracket

{f, g}(ξ) = (ξ, [π_{𝔟}∇f_{ξ}, π_{𝔟}∇g_{ξ}])
= ([ξ, π_{𝔟}∇f_{ξ}], π_{𝔟}∇g_{ξ}).

We now use the invariance of f which implies [ξ, π_{𝔨}∇f_{ξ}] = −[ξ, π_{𝔟}∇f_{ξ}] to get

{f, g}(ξ) = −([ξ, π_{𝔨}∇f_{ξ}], π_{𝔟}∇g_{ξ})
= −(ξ, [π_{𝔨}∇f_{ξ}, π_{𝔟}∇g_{ξ}])
= ([ξ, π_{𝔟}∇g_{ξ}], π_{𝔨}∇f_{ξ})
= −([ξ, π_{𝔨}∇g_{ξ}], π_{𝔨}∇f_{ξ}) = (ξ, [π_{𝔨}∇f_{ξ}, π_{𝔨}∇g_{ξ}])

which last vanishes since [π_{𝔨}∇f_{ξ}, π_{𝔨}∇g_{ξ}] ∈ 𝔨 and ξ ∈ 𝔨^{∘}. As for the corresponding Hamiltonian vector fields, from (1.2) and (1.5), we see that the restriction to 𝔨^{∘} of an ad-invariant f has Hamiltonian vector field

X_{f}(ξ) = π_{𝔨^{∘}}[π_{𝔨}∇f_{ξ}, ξ] = [π_{𝔨}∇f_{ξ}, ξ], (1.6)

where the last equality comes from [𝔨, 𝔨^{∘}] ⊂ 𝔨^{∘}.

1.4 Group decompositions and the method of Symes

We have therefore found a family of commuting flows on 𝔨^{∘} and our next task is to integrate them. For this, we globalise the situation following Symes [53]: let G be a Lie group with Lie algebra 𝔤, let K, B be subgroups with Lie algebras 𝔨, 𝔟 and suppose that multiplication K × B → G is a diffeomorphism onto. Thus any g ∈ G may be uniquely written as a product

g = kb,

with k ∈ K, b ∈ B. We further demand that the adjoint action of G on 𝔤 preserve ( , ) (which follows from the invariance of ( , ) if G is connected) and we consider Ad-invariant functions on 𝔤: that is, functions f ∈ C^{∞}(𝔤) satisfying

f(Ad g ξ) = f(ξ), (1.7) for all ξ ∈ 𝔤, g ∈ G (again, when G is connected, this is equivalent to (1.5)). Differentiating equation (1.7) with respect to ξ, we deduce that ∇f is equivariant in the sense that

∇f_{Ad g ξ} = Ad g ∇f_{ξ}, (1.8)

for all ξ ∈ 𝔤, g ∈ G.

So let f ∈ C^{∞}(𝔤) be Ad-invariant and fix an initial condition ξ_{o} ∈ 𝔨^{∘}. Consider the geodesic g : ℝ → G given by g(t) = exp −t∇f_{ξ_{o}} and factorise to get curves k : ℝ → K, b : ℝ → B with

g(t) = k(t)b(t),

for all t ∈ ℝ. Define ξ : ℝ → 𝔨^{∘} by ξ = Ad k^{−1}ξ_{o}. Then ξ is the integral curve of X_{f} with ξ(0) = ξ_{o}. Indeed,

ξ̇ = −[k^{−1}k̇, ξ]

while

−∇f_{ξ_{o}} = ġg^{−1} = k̇k^{−1} + Ad k ḃb^{−1}

so that taking Ad k^{−1} of both sides and projecting onto 𝔨 gives

k^{−1}k̇ = −π_{𝔨}Ad k^{−1}∇f_{ξ_{o}} = −π_{𝔨}∇f_{ξ},

where the last inequality follows from the equivariance (1.8) of ∇f. Thus

ξ̇ = [π_{𝔨}∇f_{ξ}, ξ].

To summarise: we have seen how to equip 𝔨^{∘} with a Poisson structure for which the restriction of Ad-invariant functions on 𝔤 commute. Moreover, the corresponding Hamiltonian vector fields are of Lax form (1.6) and are integrated using the projection onto K of geodesics in G.

Remark In fact, the invariant inner product ( , ), although present in all our applications, is not strictly necessary in the above development. If one replaces 𝔨^{∘} with 𝔨^{⊥} ⊂ 𝔤^{∗}, the annihilator of 𝔨, then, as above, 𝔨^{⊥} ≅ 𝔟^{∗} and so acquires a Poisson structure with respect to which the restrictions to 𝔨^{⊥} of Ad^{∗}-invariant functions on 𝔤^{∗} commute. Moreover, the Symes method of integration also works in this setting mutatis mutandis.

1.5 The Toda lattice

We conclude this discussion with a brief exposition of the original application of the theory to the open Toda lattice. Besides completing the circle of ideas, this will also give us an opportunity to develop concepts we shall need later.

Let 𝔤 be a complex semisimple Lie algebra and 𝔨 a compact real form. Fix a maximal toral subalgebra 𝔱 ⊂ 𝔨 and set 𝔞 = i𝔱 where i = \sqrt{−1}. Then 𝔥 = 𝔱 ⊕ 𝔞 is a Cartan subalgebra of 𝔤. Let Δ(𝔤, 𝔥) ⊂ 𝔥^{∗} be the root system of 𝔤 with respect to 𝔥 and fix a choice of positive roots Δ^{+} ⊂ Δ(𝔤, 𝔥) with corresponding simple roots α_{1}, …, α_{l}.

The Toda lattice is a Hamiltonian system on T^{∗}𝔞 = 𝔞 × 𝔞^{∗}, equipped with its canonical symplectic structure. The Hamiltonian is given by

H(q, p) = |p|^{2}/2 + ∑_{i=1}^{l} e^{2α_{i}(q)}

where the metric is given by the real part of the Killing form κ of 𝔤. One may think of this system as describing l particles moving on a line with exponential interactions governed by the Dynkin diagram of 𝔤. (Note that the roots are real linear functionals on 𝔞.)

To put this Hamiltonian system into our frame-work, we follow Bloch–Flaschka– Ratiu [5] and use the Iwasawa decomposition of 𝔤: for each root α ∈ Δ(𝔤, 𝔥), let 𝔤^{α} be the corresponding root space and define a nilpotent subalgebra 𝔫 ⊂ 𝔤 by

𝔫 = ∑_{α∈Δ^{+}} 𝔤^{α}.

We then have a direct sum decomposition

𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫,

the Iwasawa decomposition of 𝔤 (cf. [29, page 275]). Moreover, there is a global analogue of this decomposition: let G be a Lie group with Lie algebra 𝔤, K the maximal compact subgroup with Lie algebra 𝔨 and A, N the analytic subgroups with Lie algebras 𝔞, 𝔫. Then multiplication K × A × N → G is a diffeomorphism onto.

Now [𝔞, 𝔫] ⊂ 𝔫 so that 𝔟 = 𝔞 ⊕ 𝔫 is a solvable subalgebra of 𝔤 and we may apply our preceding discussion to 𝔤 = 𝔨 ⊕ 𝔟. As our invariant symmetric form, we take the imaginary part of the Killing form: ( , ) = Im κ. The Killing form is positive definite on 𝔞, negative definite on 𝔨 and vanishes identically on 𝔫 so that both 𝔨 and 𝔟 are isotropic for ( , ) giving

𝔨^{∘} = 𝔨, 𝔟^{∘} = 𝔟.

Thus we have a Poisson structure on 𝔨 ≅ 𝔟^{∗}.

For 1 ≤ i ≤ l, choose non-zero vectors X_{i} ∈ 𝔤^{α_{i}} and consider

O = \{H + ∑_{i=1}^{l} a_{i}(X_{i} + \overline{X}_{i}) : H ∈ 𝔱, a_{i} > 0\}.

It is not too difficult to see that O is a symplectic leaf of the Poisson structure on 𝔨. Indeed, the co-adjoint action of B = AN on 𝔟^{∗} induces an action of B on 𝔨, the orbits of which are the symplectic leaves on 𝔨 and one can show that O is the B-orbit of ∑_{i}(X_{i} + \overline{X}_{i}).

In fact, O is symplectomorphic with the phase space T^{∗}𝔞 of the Toda lattice: use ( , ) to identify 𝔞^{∗} with 𝔱 and define ϕ : T^{∗}𝔞 → O by

ϕ(q, p) = p + ∑_{i=1}^{l} c_{i}e^{α_{i}(q)}(X_{i} + \overline{X}_{i}),

where the constants c_{i} are positive constants to be chosen later. By checking the Poisson brackets of suitable co-ordinate functions, it is straight-forward to prove that ϕ is a Poisson morphism and hence a symplectomorphism for any choice of the constants c_{i}.

Now take as an Ad-invariant function f : 𝔤 → ℝ half the negative of the real part of the Killing form:

f(η) = − Re κ(η, η)/2.

On O, we have

f(H + ∑_{i} a_{i}(X_{i} + \overline{X})) = −κ(H, H)/2 − ∑_{i} a_{i}^{2}κ(X_{i}, \overline{X}_{i})

so that

f ∘ ϕ(q, p) = |p|^{2}/2 − ∑_{i} c_{i}^{2}κ(X_{i}, \overline{X}_{i})e^{2α_{i}(q)}
= |p|^{2}/2 + ∑_{i} c_{i}^{2}|X_{i}|^{2}e^{2α_{i}(q)}.

Thus choosing c_{i} = 1/|X_{i}|, we see that f ∘ ϕ is the Toda Hamiltonian.

The preceding theory may now be applied in a number of ways: the invariant polynomials on 𝔤 restrict to give l functionally independent Poisson commuting conserved quantities on O so that the Toda lattice is completely integrable in the sense of Liouville. Moreover, the method of Symes can be used to integrate the Toda flow leading eventually to explicit formulae for the flows in terms of matrix coefficients of the fundamental representations of G. For this and for further details on the ideas we have been discussing, the Reader is referred to the original papers and the books [28, 40].

2 Twisted loop algebras and zero-curvature equations

2.1 What the Maurer–Cartan equations are for

Let G be a Lie group with Lie algebra 𝔤 and let θ be the (left) Maurer–Cartan form of G. Thus θ is the 𝔤-valued 1-form on G given by

θ_{g}(X) = (g^{−1})_{∗}X ∈ T_{e}G = 𝔤,

for X ∈ T_{g}G. A simple calculation using the left-invariance of θ shows that

dθ + \frac{1}{2}[θ ∧ θ] = 0. (2.1)

These equations are the Maurer–Cartan equations.

Now let ϕ : M → G be a map of a manifold M and set α = ϕ^{∗}θ. Then ϕ pulls back (2.1) to give

dα + \frac{1}{2}[α ∧ α] = 0 (2.2)

so that α also satisfies the Maurer–Cartan equations.

Fundamental to what follows is that a partial converse of this is true. We have the following classical theorem (which is proved, for example, in [52]):

Theorem 2.1 Let α be a 𝔤-valued 1-form on a simply-connected manifold M. Then α = ϕ^{∗}θ for some map ϕ : M → G if and only if α satisfies the Maurer–Cartan equations (2.2).

In this case, ϕ is unique up to left translation by a constant element of G.

Otherwise said, (2.2) amounts to the assertion that the connection d + α on the trivial principal G-bundle M × G → M has vanishing curvature (i.e., is flat) while Theorem 2.1 says that there is a gauge transformation ϕ which gauges this flat connection to the trivial connection. It is for this reason that the Maurer–Cartan equations are often called the zero-curvature equations.

Notation When G is a matrix group, ϕ^{∗}θ = ϕ^{−1}dϕ. In previous sections, we have used this latter notation even when G is not a matrix group and we shall continue to do so.

The application of the above constructions to the theory of harmonic maps begins with two observations. Firstly, it is well known [42, 55, 64, 65] that, for maps of a Riemann surface into suitable homogeneous spaces, the harmonic map equations are equivalent to the flatness of a certain loop of connections, that is, to certain solutions of the Maurer–Cartan equations (2.2) where α is a 1-form with values in an appropriate loop algebra.

On the other hand, solutions of (2.2) arise in the setting of the previous sections from the simultaneous integration of several commuting Hamiltonian flows on 𝔨^{∘}. Indeed, let f^{1}, …, f^{m} be Ad-invariant functions on 𝔤 with corresponding Hamiltonian vector fields X_{i} on 𝔨^{∘}. Fixing an initial condition ξ_{o} ∈ 𝔨^{∘}, we may integrate these vector fields (when they are complete) to get ξ : ℝ^{m} → 𝔨^{∘} satisfying

dξ = X_{i}(ξ) dt^{i};
ξ(0) = ξ_{o}.

Moreover, the Symes method shows that ξ = Ad k^{−1}ξ_{o} where k : ℝ^{m} → K satisfies

k^{−1}dk = −π_{𝔨}Ad k^{−1}∇f_{ξ_{o}}^{i} dt^{i} = −π_{𝔨}∇f_{ξ}^{i} dt^{i}.

In particular, α = −π_{𝔨}∇f_{ξ}^{i} dt^{i} is the pull-back by k of the Maurer–Cartan form on K and so satisfies the Maurer–Cartan equations.

The point now is that when 𝔨 is an appropriate loop algebra and the f^{i} are suitably chosen, the 1-form α is of precisely the right form to produce a harmonic map. Moreover, the map k will be an “extended solution” as described in the contribution of Guest–Ohnita to this volume [27].

2.2 Passage to infinite dimension

Our intention is to carry out this programme for a family of twisted loop algebras. We therefore begin by considering the adjustments that must be made when carrying the Adler–Kostant–Symes scheme over to an infinite-dimensional setting. So let G be a reflexive Banach Lie algebra and suppose we have

(i) a decomposition G = K ⊕ B into closed subalgebras;

(ii) a continuous ad-invariant symmetric bilinear form ( , ) which is nondegenerate in the weak sense that the induced map i : G → G^{∗} is injective;

(iii) K, B are both isotropic for ( , ).

It follows that B is reflexive so that we may canonically identify B with B^{∗∗} and so obtain a Poisson structure on B^{∗} as before. By virtue of the isotropy of K, the injection G → G^{∗} restricts to give an injection K → B^{∗} but this need no longer be an isomorphism and so we cannot transfer the Poisson structure of B^{∗} to K.

However, suppose that V ⊂ K is a linear subspace satisfying

1. π_{K}[V, B] ⊂ V;

2. i(V) is closed and has a closed complement in B^{∗} (trivially true if V is finitedimensional).

In this case, V has a Poisson structure for which i : V → B^{∗} is a Poisson morphism. Indeed, by virtue of the second hypothesis, i^{∗} : C^{∞}(B^{∗}) → C^{∞}(V) is a surjection so it suffices to show that the prescription

{f ∘ i, g ∘ i}_{V} = {f, g}_{B^{∗}} ∘ i

is well-defined. This, in turn, follows from the first hypothesis as in Proposition 1.1.

We can now proceed as before with the caveat that not all functions on G have gradients with respect to ( , ). This prompts the following terminology:

Definition F ∈ C^{∞}(G) is admissible if it has a gradient with respect to ( , ).

For admissible functions F, G ∈ C^{∞}(G), we can compute the Poisson bracket of their restrictions to V as before:

{F, G}(ξ) = (ξ, [π_{B}∇F_{ξ}, π_{B}∇G_{ξ}]),

for ξ ∈ V.

12 F.E. Burstall and F. Pedit

Finally, an admissible function F ∈ C^{∞}(G) is invariant if

[ξ, ∇F_{ξ}] = 0, for all ξ ∈ G. For such functions, the analysis of the previous section goes through as before and is summarised in the following theorem. Theorem 2.2 Let F and G be invariant, admissible functions on G. Then their restrictions to V Poisson commute.

Moreover, the Hamiltonian vector field corresponding to F is given by

X_{F} (ξ) = [π_{K}∇F_{ξ}, ξ], for ξ ∈ V.

2.3 Twisted loop algebras To introduce the loop algebras to which these ideas will be applied, we begin by fixing the following ingredients:

1. A compact semisimple Lie algebra 𝔤.

2. An automorphism τ : 𝔤 → 𝔤 of finite order k with fixed set 𝔨.

3. The primitive k-th root of unity ω = e^{2πi/k}. In addition, we fix an Iwasawa decomposition of 𝔨^{ℂ}:

𝔨^{ℂ} = 𝔨 ⊕ 𝔟, where 𝔟 is a solvable subalgebra of 𝔨^{ℂ}. Such a decomposition exists since 𝔨 is compact so that 𝔨^{ℂ} is reductive.

Define a loop algebra by

Λ𝔤_{τ}^{ℂ} = {ξ : S^{1} → 𝔤^{ℂ} : ξ(ωλ) = τ ξ(λ) for all λ ∈ S^{1}} and equip it with the Sobolev H^{r}-topology for some r > \frac{1}{2}. Thus Λ𝔤_{τ}^{ℂ} is a Banach (indeed, Hilbert) Lie algebra under point-wise bracket. There is an analogue of the Iwasawa decomposition for Λ𝔤_{τ}^{ℂ}: let Λ𝔤_{τ} be the real form

Λ𝔤_{τ} = {ξ ∈ Λ𝔤_{τ}^{ℂ} : ξ : S^{1} → 𝔤} and define a complementary subalgebra by

Λ_{+}𝔤_{τ}^{ℂ} = {ξ ∈ Λ𝔤_{τ}^{ℂ} : ξ extends holomorphically to ξ : D → 𝔤^{ℂ} and ξ(0) ∈ 𝔟}, where D is the disc {|λ| < 1}.

A loop ξ ∈ Λ𝔤_{τ}^{ℂ} has a Fourier decomposition:

ξ = ∑_{n∈ℤ} ξ_{−n}λ^{n}

with each ξ_{n} ∈ 𝔤^{ℂ} satisfying τξ_{n} = ω^{−n}ξ_{n}. (Our convention of labelling the coefficient of λ^{n} by −n is to prevent a welter of negative indices in subsequent formulae.) In terms of this decomposition, the conjugation on Λ𝔤_{τ}^{ℂ} across the real form Λ𝔤_{τ} is given by

\overline{ξ} = ∑_{n∈ℤ} \overline{ξ_{−n}}λ^{−n}

where the conjugation in 𝔤^{ℂ} is with respect to the real form 𝔤. Moreover, our subalgebras are given by

Λ𝔤_{τ} = \{ξ ∈ Λ𝔤_{τ}^{ℂ} : \overline{ξ_{n}} = ξ_{−n}\};
Λ_{+}𝔤_{τ}^{ℂ} = {ξ ∈ Λ𝔤_{τ}^{ℂ} : ξ_{n} = 0 for n > 0; ξ_{0} ∈ 𝔟}.

From this it is clear that

Λ𝔤_{τ}^{ℂ} = Λ𝔤_{τ} ⊕ Λ_{+}𝔤_{τ}^{ℂ}

is a vector space decomposition into closed subalgebras.

Now let κ be the Killing form of 𝔤^{ℂ} and introduce a (weakly) non-degenerate invariant symmetric bilinear form ( , ) on Λ𝔤_{τ}^{ℂ} by taking the imaginary part of the L^{2} inner product:

(ξ, η) = Im ∫_{S^{1}} κ(ξ, η).

Observe that both Λ𝔤_{τ} and Λ_{+}𝔤_{τ}^{ℂ} are isotropic for this form so that we are in the situation of the previous discussion.

Let π_{K}, π_{B} be the projections onto Λ𝔤_{τ}, Λ_{+}𝔤_{τ}^{ℂ} along Λ_{+}𝔤_{τ}^{ℂ}, Λ𝔤_{τ} respectively. To calculate these, write ξ ∈ Λ𝔤_{τ}^{ℂ} as

ξ = ξ^{+} + ξ^{0} + ξ^{−},

where

ξ^{+} = ∑_{n>0} ξ_{−n}λ^{n};  ξ^{0} = ξ_{0} ∈ 𝔨^{ℂ};  ξ^{−} = ∑_{n<0} ξ_{−n}λ^{n}.

Further, write ξ^{0} = ξ_{𝔨}^{0} + ξ_{𝔟}^{0} for the Iwasawa decomposition of ξ^{0} into 𝔨 and 𝔟 parts. Then

ξ = (ξ^{−} + ξ_{𝔨}^{0} + \overline{ξ^{−}}) + (ξ^{+} − \overline{ξ^{−}} + ξ_{𝔟}^{0})

and the first summand is in Λ𝔤_{τ} while the second is in Λ_{+}𝔤_{τ}^{ℂ}. In particular, we have

π_{K}ξ = ξ^{−} + ξ_{𝔨}^{0} + \overline{ξ^{−}}. (2.3)

Recall that finite-dimensional subspaces V ⊂ Λ𝔤_{τ} satisfying

π_{K}[V, Λ_{+}𝔤_{τ}^{ℂ}] ⊂ V inherit a Poisson structure from (Λ_{+}𝔤_{τ}^{ℂ})^{∗}. Our main examples of such subspaces are the spaces of polynomial loops given by

Λ_{d} = {ξ ∈ Λ𝔤_{τ} : ξ_{n} = 0 for all |n| > d},

where d ∈ ℕ. Indeed, if ξ ∈ Λ_{d} and η ∈ Λ_{+}𝔤_{τ}^{ℂ} then λ^{d}ξ extends holomorphically to D so that [λ^{d}ξ, η] does also. This means that [ξ, η] = λ^{−d}[λ^{d}ξ, η] has a pole of order at most d at 0 ∈ D so that

π_{K}[ξ, η] = [ξ, η]^{−} + [ξ, η]_{𝔨}^{0} + \overline{[ξ, η]^{−}} ∈ Λ_{d},

as required. We have therefore proved:

Proposition 2.3 For d ∈ ℕ, Λ_{d} is a Poisson manifold on which the restrictions of invariant admissible functions on Λ𝔤_{τ}^{ℂ} Poisson commute.

2.4 Zero-curvature equations

We now come to our main construction: for d ∈ ℕ, we will integrate a pair of commuting Hamiltonian flows on the finite-dimensional Poisson manifold Λ_{d} and, from that data, construct a Λ𝔤_{τ} -valued 1-form which solves the Maurer–Cartan equations. In Section 3, we shall see how such 1-forms are related to harmonic maps.

Fix d ∈ ℕ with d ≡ 1 mod k. Define f^{1}, f^{2} ∈ C^{∞}(Λ𝔤_{τ}^{ℂ}) by

f^{1}(ξ) = −\frac{1}{2} Im ∫_{S^{1}} λ^{d−1}κ(ξ, ξ), f^{2}(ξ) = −\frac{1}{2} Re ∫_{S^{1}} λ^{d−1}κ(ξ, ξ).

It is easy to see that the f^{i} are invariant admissible functions with gradients given by

∇f_{ξ}^{1} = −λ^{d−1}ξ, ∇f_{ξ}^{2} = −iλ^{d−1}ξ,

(here we have used d − 1 ≡ 0 mod k to ensure that the ∇f^{i} so defined take values in Λ𝔤_{τ}^{ℂ}).

We restrict f^{1}, f^{2} to Λ_{d} and let X_{1}, X_{2} denote the corresponding Hamiltonian vector fields. Thus

X_{1}(ξ) = [ξ, π_{K}λ^{d−1}ξ]; X_{2}(ξ) = [ξ, π_{K}iλ^{d−1}ξ],

for ξ ∈ Λ_{d}. That these vector fields are complete is a consequence of two observations: firstly, the L^{2} inner product

⟨ξ, η⟩ = ∫_{S^{1}} κ(ξ, η).

is negative definite on Λ𝔤_{τ} (and thus on Λ_{d}). Secondly, the X_{i} are of Lax form, i.e., of the form

X(ξ) = [ξ, A(ξ)].

For any integral curve ξ of such an X we have

d/dt|ξ|^{2} = 2⟨ξ̇, ξ⟩
= 2⟨[ξ, A(ξ)], ξ⟩ = −2⟨A(ξ), [ξ, ξ]⟩ = 0

so that ξ takes values in a sphere in Λ_{d} whence X is complete.

Let X_{1}^{s}, X_{2}^{t} be the flows corresponding to the X_{i}. Since f^{1} and f^{2} Poisson commute, the commutator [X_{1}, X_{2}] vanishes so that

X_{1}^{s} ∘ X_{2}^{t} = X_{2}^{t} ∘ X_{1}^{s},

for all s, t ∈ ℝ. Now fix an initial condition ξ_{o} ∈ Λ_{d} and define ξ : ℝ^{2} → Λ_{d} by

ξ(s, t) = X_{1}^{s} ∘ X_{2}^{t}(ξ_{o}).

Then ξ satisfies

dξ = X_{1}(ξ) ds + X_{2}(ξ) dt;  ξ(0) = ξ_{o}.  (2.4)

Our interest in such maps comes from the possibility of constructing loops of flat connections from them:

Theorem 2.4 Let ξ : ℝ^{2} → Λ_{d} be the solution of (2.4) and define a Λ𝔤_{τ} -valued 1-form α on ℝ^{2} by

α = −π_{K}(∇f_{ξ}^{1} ds + ∇f_{ξ}^{2} dt).

Then α solves the Maurer–Cartan equations:

dα + \frac{1}{2}[α ∧ α] = 0.

This theorem can be proved by direct calculation or by recourse to the method of Symes. The first approach is an exercise for the Reader while the second will be carried out in Section 4.

We have thus obtained solutions to the Maurer–Cartan equations by integrating Hamiltonian ordinary differential equations on some Λ_{d}. Our application to harmonic maps will depend on the precise algebraic form of the solutions so obtained and it is to this that we now turn. We have

α = −π_{K}(∇f_{ξ}^{1} ds + ∇f_{ξ}^{2} dt) = π_{K}(λ^{d−1}ξ ds + iλ^{d−1}ξ dt).

This simplifies if we introduce the complex co-ordinate z = s + it on ℝ^{2}:

α = π_{K}(λ^{d−1}ξ dz).

We now use (2.3), together with the fact that ξ takes values in Λ_{d}, to get

α = (λ^{d−1}ξ)^{−} dz + (λ^{d−1}ξ dz)_{𝔨}^{0} + \overline{(λ^{d−1}ξ)^{−}} d\overline{z}
= λ^{−1}ξ_{d} dz + (ξ_{d−1} dz)_{𝔨}+ λξ_{−d} d\overline{z}.

To compute the λ-independent term, we must understand the Iwasawa decomposition of 𝔨^{ℂ}: let 𝔱 be the given maximal torus in 𝔨, 𝔫 the nilpotent subalgebra given by the positive root spaces and set 𝔥 = 𝔱^{ℂ}. Then we have

𝔨^{ℂ} = 𝔫 ⊕ 𝔥 ⊕ \overline{𝔫}, 𝔟 = (i𝔱) ⊕ 𝔫.

Corresponding to this decomposition of 𝔨^{ℂ}, write η ∈ 𝔨^{ℂ} as

η = η_{𝔫}+ η_{𝔥}+ η𝔫.

It is then easy to check that

(η dz)_{𝔨}= (η𝔫+ \frac{1}{2}η_{𝔥}) dz + \overline{(η𝔫+ \frac{1}{2}η_{𝔥})} d\overline{z}. (2.5)

This suggests that we define r : 𝔨^{ℂ} → 𝔨^{ℂ} by

r(η) = η𝔫+ \frac{1}{2}η_{𝔥}.

Then, denoting the (1, 0)-part of α by α′, we conclude that

α′ = (λ^{−1}ξ_{d} + r(ξ_{d−1})) dz. (2.6)

We can now summarise our discussion in the following theorem:

Theorem 2.5 For each d ≡ 1 mod k and ξ_{o} ∈ Λ_{d}, there is a unique solution ξ : ℝ^{2} → Λ_{d} to

\frac{∂ξ}{∂z} = [ξ, λ^{−1}ξ_{d} + r(ξ_{d−1})];  ξ(0) = ξ_{o}  (2.7)

and then the Λ𝔤_{τ} -valued 1-form α given by

α = (λ^{−1}ξ_{d} + r(ξ_{d−1})) dz + (λξ_{−d} + \overline{r(ξ_{d−1})}) d\overline{z} (2.8)

satisfies the Maurer–Cartan equations.

Remark When 𝔤 has rank greater then one, we may generalise these methods to produce Maurer–Cartan solutions on ℂ^{n} by simultaneous integration of 2n commuting flows. Briefly, the idea is to consider ad-invariant polynomials P : 𝔤^{ℂ} → ℂ and define functions f^{1}, f^{2} on Λ𝔤_{τ}^{ℂ} by

f^{1}(ξ) = Im ∫_{S^{1}} λ^{m}P (ξ); f^{2}(ξ) = Re ∫_{S^{1}} λ^{m}P (ξ)

which are easily seen to be admissible and invariant. Any collection of such functions, with P ranging over ad-invariant polynomials, will Poisson commute and so give rise to a solution to the Maurer–Cartan equations. By choosing the exponents m appropriately, one obtains solutions α with the same kind of λ-dependence as in (2.6). From such α one can construct pluriharmonic maps of ℂ^{n}.

We shall not pursue this topic further in this chapter but, instead, refer the Reader to the discussion in [14].

2.5 The r-matrix approach

Elsewhere in this volume (and in the papers [24, 14]), a different approach to the construction of commuting flows on loop algebras is employed, to wit: the r-matrix formalism of Reyman–Semenov-Tian-Shansky [48]. In this section, we take a break from our main development to briefly review this method and show how it gives the same Poisson structures on Λ𝔤_{τ} as the Adler–Kostant–Symes scheme. For brevity of exposition, we shall ignore questions of completeness of metrics, admissibility of functions and so on.

We begin with a (possibly infinite-dimensional) Lie algebra G. A linear map R : G → G is called an r-matrix if the bracket defined by

[ξ, η]_{R} = [Rξ, η] + [ξ, Rη]

satisfies the Jacobi identity. This is easily seen to be the case if R solves the (modified) classical Yang–Baxter equations:

R[ξ, η]_{R} − [Rξ, Rη] = a[ξ, η], (2.9)

for some fixed a ∈ ℂ. Given an r-matrix, G acquires a second Lie algebra structure so that the construction which opens Section 1.1 equips G^{∗} with a second Poisson structure.

Suppose also that G is equipped with an ad-invariant inner product ⟨ , ⟩. We use this to transfer the new Poisson structure from G^{∗} to G and thus arrive at a Poisson structure on G with bracket:

{f, g}_{R}(ξ) = ⟨ξ, [grad f_{ξ}, grad g_{ξ}]_{R}⟩, (2.10)

where grad denotes the gradient with respect to ⟨ , ⟩.

Such Poisson structures enjoy many of the properties of those constructed by the Adler–Kostant–Symes scheme. In particular, ad-invariant functions on G Poisson commute and give rise to solutions of the Maurer–Cartan equations.

Let us show that both these constructions coincide in our setting: define a linear map R : Λ𝔤_{τ} → Λ𝔤_{τ} by

Rξ = iξ^{+} + R_{0}ξ^{0} − iξ^{−}.

Here R_{0} : 𝔨 → 𝔨 is given by

R_{0}η = iη_{𝔫}− iη\overline{n}.

One observes that the ±i-eigenspaces of R are subalgebras of Λ𝔤_{τ}^{ℂ} while the 0eigenspace is the abelian subalgebra 𝔥. From this, it is easy to check that R solves the modified classical Yang–Baxter equations (2.9) with a = −1 and so is an rmatrix.

Recall that Λ𝔤_{τ} carries the ad-invariant, negative definite inner product,

⟨ξ, η⟩ = ∫_{S^{1}} κ(ξ, η) so that R induces a Poisson structure on Λ𝔤_{τ}.

We claim that this Poisson structure is the same as the one described in Section 2.3. To see this, it suffices to compare Hamiltonian vector fields. So let f be some function on Λ𝔤_{τ} and recall that its Hamiltonian vector field with respect to the Adler–Kostant–Symes Poisson structure is given by

π_{K}[ξ, π_{B}∇f_{ξ}] = [ξ, π_{B}∇f_{ξ}] − π_{B}[ξ, π_{B}∇f_{ξ}]
= [ξ, π_{B}∇f_{ξ}] − π_{B}[ξ, ∇f_{ξ}],

for ξ ∈ Λ𝔤_{τ}. Here the last equality is due to the fact that π_{B}[ξ, π_{K}∇f_{ξ}] vanishes since Λ𝔤_{τ} is a subalgebra.

On the other hand, we note that R is skew-symmetric for ⟨ , ⟩ and use this, together with (2.10), to see that the Hamiltonian vector field of f with respect to the R-Poisson structure is given by

[ξ, R grad f_{ξ}] − R[ξ, grad f_{ξ}] = [ξ, (R + i) grad f_{ξ}] − (R + i)[ξ, grad f_{ξ}].

That these vector fields are the same is an immediate consequence of two easily verified facts. Firstly, the relation between the two gradients is given by

grad f_{ξ} = \frac{1}{2i}(∇f_{ξ} − \overline{∇f_{ξ}}),

for ξ ∈ Λ𝔤_{τ}. Secondly, one deduces from (2.3) and (2.5) that

π_{B}ξ = \frac{1}{2i}(R + i)(ξ − \overline{ξ}),

for ξ ∈ Λ𝔤_{τ}^{ℂ}. From this it is clear that, for ξ ∈ Λ𝔤_{τ},

[ξ, π_{B}∇f_{ξ}] − π_{B}[ξ, ∇f_{ξ}] = [ξ, (R + i) grad f_{ξ}] − (R + i)[ξ, grad f_{ξ}]

so that the Poisson structures coincide.

Example Let 𝔤 = 𝔰o(5) and let τ : 𝔤 → 𝔤 be conjugation by

( \begin{matrix}1 \\ & & 1 \\ & −1 \\ & & & −1 \\ & & & & −1 \end{matrix} ) .

Then τ is an automorphism of order 4 with fixed set a maximal torus 𝔱 of 𝔰o(5). The Iwasawa decomposition of 𝔱^{ℂ} is 𝔱^{ℂ} = 𝔱 ⊕ i𝔱 with no 𝔫-part so that the r-matrix in this case is given by

Rξ = iξ^{+} − iξ^{−}.

This is (up to a factor of \frac{1}{2}) the r-matrix used in [24] to construct commuting flows from which all minimal non-superminimal 2-tori are obtained.

Remark In fact the r-matrix formalism constitutes a strict generalisation of the Adler–Kostant–Symes scheme. In the notation of Section 1.1, one may define an rmatrix on 𝔤 by setting R = \frac{1}{2}(π_{𝔟}−π_{𝔨}) and then check that 𝔨^{∘} is a Poisson submanifold of 𝔤 with respect to the Poisson structure obtained from R. Moreover, the induced Poisson structure on 𝔨^{∘} coincides with that provided by the Adler–Kostant–Symes scheme (c.f., [40]).

However, when it can be applied, the Adler–Kostant–Symes scheme has advantages. In particular, the symplectic leaves are known to be the B-orbits on 𝔨^{∘} and so Poisson submanifolds are more readily identified.

3 Harmonic and primitive maps

We now turn from symplectic geometry to Riemannian geometry and apply the theory of the preceding sections to the construction of harmonic maps from ℝ^{2} into certain homogeneous spaces. We begin by describing the class of such spaces with which we shall be concerned.

3.1 Symmetric and k-symmetric spaces

Recall the data of Section 2.3: a compact semisimple Lie algebra 𝔤; an automorphism τ : 𝔤 → 𝔤 of order k with fixed set 𝔨; the primitive k-th root of unity ω = e^{2πi/k}. We have an eigenspace decomposition of 𝔤^{ℂ}:

𝔤^{ℂ} = ∑_{i∈ℤ_{k}} 𝔤_{i}

where 𝔤_{i} is the ω^{i}-eigenspace of τ. Clearly, 𝔤_{0} = 𝔨^{ℂ}, \overline{𝔤_{i}} = 𝔤_{−i} and

[𝔤_{i}, 𝔤_{j}] ⊂ 𝔤_{i+j},

for all i, j ∈ ℤ_{k} (here, of course, all arithmetic is modulo k). In particular, define 𝔪 ⊂ 𝔤 by

𝔪^{ℂ} = ∑_{i∈ℤ_{k}\{0}} 𝔤_{i}.

Then

𝔤 = 𝔨 ⊕ 𝔪 (3.1)

and [𝔨, 𝔪] ⊂ 𝔪 so that (3.1) is a reductive decomposition. Moreover, when k = 2, 𝔪^{ℂ} = 𝔤_{1} so that

[𝔪, 𝔪] ⊂ 𝔨

and, in this case, (3.1) is a symmetric decomposition of 𝔤.

Let G be a compact semisimple Lie group with Lie algebra 𝔤 and suppose that τ exponentiates to give an order k automorphism, also called τ, of G. Further let (G^{τ})_{0} ⊂ K ⊂ G^{τ} so that K has Lie algebra 𝔨.

Consider the coset space N = G/K with base-point o = eK. Define τ̂ : N → N by

τ̂(g · o) = τ(g) · o,

for g ∈ G. Similarly, for x = g · o ∈ N, define τ̂_{x} : N → N by

τ̂_{x} = g ∘ τ̂ ∘ g^{−1}.

Then each τ̂_{x} is a diffeomorphism or order k having x as an isolated fixed point. Moreover, we may use the Killing form of 𝔤 to equip N with a metric for which each of the τ̂_{x} is an isometry so that N has the structure of a (regular) k-symmetric space in the sense of [37]. Of course, the 2-symmetric spaces are just the familiar Riemannian symmetric spaces of compact type.

The algebraic situation in 𝔤 transfers to the reductive homogeneous space N as follows: for x = g · o ∈ N, the map 𝔤 → T_{x}N given by

ξ ↦ \frac{d}{dt}|_{t=0} exp tξ · x

restricts to an isomorphism Ad g 𝔪 → T_{x}N. The inverse map β_{x} : T_{x}N → Ad g 𝔪 ⊂ 𝔤 may be viewed as a 𝔤-valued 1-form β on N which, after [15], we call the Maurer– Cartan form of N.

This prompts in part the following notation: if 𝔩 ⊂ 𝔤^{ℂ} is an Ad K-invariant subspace, let [𝔩] denote the subbundle of the trivial bundle \underline{𝔤}^{ℂ} = N × 𝔤^{ℂ} defined by

[𝔩]_{g·o} = Ad g 𝔩.

In particular, β : T N → [𝔪] is a bundle isomorphism while [𝔨]_{x} is the Lie algebra of the stabiliser of x ∈ N. Moreover, the ω^{i}-eigenspace of dτ̂_{x} at x ∈ N corresponds to [𝔤_{i}]_{x} under the isomorphism β_{x}.

3.2 Harmonic maps

A map ϕ : M → N of Riemannian manifolds is harmonic if it extremizes the energy functional

E(ϕ) = ∫_{D} |dϕ|^{2} dvol (3.2)

on all compact sub-domains D ⊂ M (for an introduction to harmonic maps in this volume, see [59]). In case that the target is a reductive homogeneous space N = G/K, the corresponding Euler–Lagrange equations have a particularly simple form. Indeed, if such an N has metric induced from an invariant metric on 𝔤, we may use these metrics to identify T N with T^{∗}N and 𝔤^{∗} with 𝔤. In this way, T N becomes a symplectic manifold on which G acts symplectically and the moment map for this action is precisely the Maurer–Cartan form β : T N → 𝔤 ≅ 𝔤^{∗}. A Noether analysis of the energy functional (3.2) now gives [44] (c.f. also, [59]):

Lemma 3.1 ϕ is harmonic if and only if the pull-back of the Maurer–Cartan form is co-closed:

d^{∗}ϕ^{∗}β = 0.

We can gain a different perspective on the harmonic map equations by lifting everything to the frame bundle. Let π : G → N = G/K be the coset projection and ϕ : M → N be a map. A framing of ϕ is a map Φ : M → G such that π ∘ Φ = ϕ so that the following diagram commutes:

[figure: M; N; G; ϕ; π; Φ]

Such lifts always exist locally.

So let ϕ : M → N be a map with framing Φ and set α = Φ^{−1}dΦ. Corresponding to the reductive decomposition 𝔤 = 𝔨 ⊕ 𝔪 is a decomposition of α,

α = α_{𝔨}+ α_{𝔪}

and one may easily verify that

ϕ^{∗}β = Ad Φ α_{𝔪} .

We can now express the harmonic map equations in terms of α:

d ∗ ϕ^{∗}β = d(Ad Φ ∗ α_{𝔪} )
= Ad Φ{d ∗ α_{𝔪}+ [α ∧ ∗α_{𝔪} ]},

where we have used d(Ad Φ) = Ad Φ ∘ ad α. Thus ϕ if harmonic if and only if

d ∗ α_{𝔪}+ [α ∧ ∗α_{𝔪} ] = 0. (3.3)

Henceforth, we will restrict attention to the case where M is two-dimensional. Then the energy is conformally invariant so that we may take M to be a Riemann surface. We now have a type decomposition α_{𝔪}= α′_{𝔪}+ α′′_{𝔪} where α′_{𝔪}is an 𝔪^{ℂ}-valued (1, 0)-form with complex conjugate α′′_{𝔪}. Then

∗α_{𝔪}= −iα′_{𝔪}+ iα′′_{𝔪}

so that (3.3) becomes

0 = −(dα′_{𝔪}+ [α ∧ α′_{𝔪}]) + (dα′′_{𝔪}+ [α ∧ α′′_{𝔪}])
= −(dα′_{𝔪}+ [α_{𝔨}∧ α′_{𝔪}] + [α′′_{𝔪}∧ α′_{𝔪}]) + (dα′′_{𝔪}+ [α_{𝔨}∧ α′′_{𝔪}] + [α′_{𝔪}∧ α′′_{𝔪}])
= −(dα′_{𝔪}+ [α_{𝔨}∧ α′_{𝔪}]) + (dα′′_{𝔪}+ [α_{𝔨}∧ α′′_{𝔪}]).  (3.4)

On the other hand, we have the Maurer–Cartan equations for α whose 𝔪- and 𝔨-parts read

dα′_{𝔪}+ [α_{𝔨}∧ α′_{𝔪}] + dα′′_{𝔪}+ [α_{𝔨}∧ α′′_{𝔪}] + [α′_{𝔪}∧ α′′_{𝔪}]_{𝔪}= 0 (3.5)

dα_{𝔨}+ \frac{1}{2}[α_{𝔨}∧ α_{𝔨}] + [α′_{𝔪}∧ α′′_{𝔪}]_{𝔨}= 0. (3.6)

In particular, (3.4) and (3.5) are equivalent to

dα′_{𝔪}+ [α_{k} ∧ α′_{𝔪}] = −\frac{1}{2}[α′_{𝔪}∧ α′′_{𝔪}]_{𝔪} .

Suppose now that [α′_{𝔪}∧ α′′_{𝔪}]_{𝔪}vanishes (certainly true when N is a Riemannian symmetric space since then [𝔪, 𝔪] ⊂ 𝔨). The harmonic map equations combine with the Maurer–Cartan equations to give

dα′_{𝔪}+ [α_{k} ∧ α′_{𝔪}] = dα′′_{𝔪}+ [α_{k} ∧ α′′_{𝔪}] = 0 (3.7)

dα_{𝔨}+ \frac{1}{2}[α_{𝔨}∧ α_{𝔨}] + [α′_{𝔪}∧ α′′_{𝔪}] = 0. (3.8)

Remark These last equations have a gauge-theoretic formulation: let A denote the connection d + α_{𝔨}with curvature F_{A} and (temporarily) denote α′_{𝔪}by Ψ. Then our equations read

d^{A}Ψ = 0
F_{A} = −[Ψ ∧ \overline{Ψ}]

which are the Yang–Mills–Higgs equations for the connection A and the Higgs field Ψ. (See Hitchin [30] for a far-reaching study of these equations when N is the noncompact symmetric space G^{ℂ}/G.)

The equations (3.7) and (3.8) are invariant under an S^{1}-action: for λ ∈ S^{1} and α a 𝔤-valued 1-form, set

λ · α = α_{λ} = λ^{−1}α′_{𝔪}+ α_{𝔨}+ λα′′_{𝔪}. (3.9)

This is clearly an action of S^{1} on 𝔤-valued 1-forms which preserves the solution set of (3.7) and (3.8). In fact, more is true: compare coefficients of λ to see that (3.7) and (3.8) hold for α precisely when

dα_{λ} + \frac{1}{2}[α_{λ} ∧ α_{λ}] = 0

for all λ ∈ S^{1}. Otherwise said, we have written the harmonic map equations as a loop of zero-curvature equations (c.f. [42, 55, 64, 65]).

Conversely, suppose that M is simply connected and let α_{λ} be a loop of 1-forms of the form (3.9) such that

1. [α′_{𝔪}∧ α′′_{𝔪}]_{𝔪}= 0;

2. dα_{λ} + \frac{1}{2}[α_{λ} ∧ α_{λ}] = 0, for all λ ∈ S^{1}.

Then, by Theorem 2.1, for each λ ∈ S^{1} we can find a map Φ_{λ} : M → G such that

Φ_{λ}^{−1}dΦ_{λ} = α_{λ}

and, since α_{λ} satisfies (3.7) and (3.8), we conclude that each ϕ_{λ} = π ∘ Φ_{λ} : M → N is harmonic. In particular, harmonic maps of this kind come in S^{1}-families.

This analysis applies to harmonic maps into any reductive homogeneous space N (with metric induced by one on 𝔤) which satisfy the auxiliary condition

[α′_{𝔪}∧ α′′_{𝔪}]_{𝔪}= 0

(which is the same as demanding that [ϕ^{∗}β′∧ϕ^{∗}β′′] be [𝔨]-valued). This last condition is trivially satisfied when N is a 2-symmetric space and then our discussion is valid for all harmonic maps.

So let us consider the case when N is a Riemannian symmetric space and show how the results of Section 2 provide harmonic maps of ℝ^{2} → N. For this, let τ be the involution determining N so that

𝔤_{0} = 𝔨^{ℂ}; 𝔤_{1} = 𝔪^{ℂ}

and recall the loop algebra Λ𝔤_{τ}. Consider the algebraic structure of α_{λ} in (3.9): the coefficient of λ^{−1} is a 𝔤_{1}-valued (1, 0)-form, the constant (in λ) term is 𝔨-valued while the λ-coefficient is a 𝔤_{1}-valued (0, 1)-form. In particular, α_{λ} may be viewed as a Λ𝔤_{τ} -valued 1-form. Moreover, we see from (2.8) that this is precisely the type of Λ𝔤_{τ} -valued 1-form that arises as the solution to the zero-curvature equations in Theorem 2.5 so that combining that theorem with our present discussion gives

Theorem 3.2 For each d ∈ 2ℕ + 1 and ξ_{o} ∈ Λ_{d}, there is a unique solution to

\frac{∂ξ}{∂z} = [ξ, λ^{−1}ξ_{d} + r(ξ_{d−1})]; ξ(0) = ξ_{o}

and then there is a harmonic map ϕ : ℝ^{2} → N with framing Φ : ℝ^{2} → G satisfying Φ^{−1}∂Φ/∂z = ξ_{d} + r(ξ_{d−1}).

We have therefore succeeded in constructing harmonic maps of ℝ^{2} into Riemannian symmetric spaces N from our commuting Hamiltonian flows on the Poisson manifolds Λ_{d}. We call the maps so obtained harmonic maps of finite type.

When is a harmonic map ℝ^{2} → N of finite type? The framing Φ of such a map has α′_{𝔪}= ξ_{d} dz and this leads to necessary conditions. Indeed, comparing coefficients in the differential equation for ξ gives

dξ_{d} = [ξ_{d}, (r(ξ_{d−1}) − ξ_{d−1}) dz + \overline{r(ξ_{d−1})} d\overline{z}]

which takes values in [ξ_{d}, 𝔨^{ℂ}]. Otherwise said, dξ takes values in the tangent space at ξ_{d} to the Ad K^{ℂ}-orbit through ξ_{d}. We therefore deduce from the uniqueness of solutions to ODE:

Lemma 3.3 ξ_{d} : ℝ^{2} → 𝔪^{ℂ} (and hence α′_{𝔪}(∂/∂z)) takes values in a single Ad K^{ℂ}- orbit in 𝔪^{ℂ}.

This has the effect of ensuring that certain polynomial invariants of a finite type harmonic map are constant: if P : 𝔪^{ℂ} → ℂ is an Ad K^{ℂ}-invariant polynomial of degree r then clearly P (α′_{𝔪}(∂/∂z)) is constant. On the other hand, P gives rise to a G-invariant, and hence parallel, section P̂ of S^{r}T^{∗}N^{ℂ} ≅ S^{r}[𝔪^{ℂ}] by

P̂_{g·o} = P ∘ Ad g^{−1}

so that we have

ϕ^{∗}P̂^{(r,0)} = P (α′_{𝔪}(∂/∂z)) dz^{r} = c dz^{r}

for some constant c.

One important class of harmonic maps which satisfy such constraints is that of doubly periodic harmonic maps or, equivalently, lifts to the universal cover of harmonic 2-tori. Indeed, for any harmonic map, it is known that these polynomial invariants are holomorphic differentials: one may write the harmonic map equations as

∇_{∂/∂\overline{z}}^{ϕ}∂ϕ/∂z = 0,

where ∇^{ϕ} is the pull-back of the Levi-Civita connection on N (see [59]) and then, for any parallel field of polynomials P̂ we have

\frac{∂}{∂\overline{z}} P̂(∂ϕ/∂z, …, ∂ϕ/∂z) = rP̂(∇_{∂/∂\overline{z}}^{ϕ}∂ϕ/∂z, ∂ϕ/∂z, …, ∂ϕ/∂z) = 0.

In particular, when ϕ is doubly periodic, each ϕ^{∗}P̂^{(r,0)} is constant by Liouville’s theorem.

Each common level set of the invariant polynomials on 𝔪^{ℂ} comprises but a finite number of Ad K^{ℂ}-orbits so that harmonic 2-tori come close to satisfying the necessary condition to be of finite type provided by Lemma 3.3. In fact, in this case, that necessary condition is almost sufficient: we shall see in Section 5 that the following theorem is a consequence of the results of [14].

Theorem 3.4 Let ϕ : ℝ^{2} → N be a doubly periodic harmonic map into a Riemannian symmetric space. Suppose that some (and hence every) framing of ϕ has α′_{𝔪}(∂/∂z) taking values in a single semisimple Ad K^{ℂ}-orbit in 𝔪^{ℂ}. Then ϕ is of finite type.

Here an orbit is semisimple if it consists of elements ξ ∈ 𝔪^{ℂ} which are semisimple in the sense that ad ξ is diagonalisable.

Let us conclude this section with an example which gives geometric meaning to the “semisimple orbit” hypothesis. For a rank 1 symmetric space, the semisimple orbits are just the non-zero level sets of the Killing form. The corresponding quadratic form on T N^{ℂ} is the complexified metric h so that the constant holomorphic differential for a doubly periodic map ϕ is just the obstruction to conformality ϕ^{∗}h^{(2,0)}. We therefore conclude Theorem 3.5 A doubly periodic non-conformal harmonic map ϕ : ℝ^{2} → N into a rank 1 symmetric space is of finite type.

While this result explicitly excludes the geometrically interesting case of minimal 2-tori, it is already strong enough to be very useful. For instance, the non-conformal harmonic 2-tori in S^{2} are precisely the Gauss maps of constant mean curvature tori in ℝ^{3} and, in this case, the theorem reproduces the basic result of Pinkall–Sterling [41].

For spheres and complex projective spaces, this result will be improved in Section 3.5.

3.3 Primitive maps

We now extend our theory to treat certain harmonic maps into k-symmetric spaces N where k > 2. Let us begin by introducing the class of harmonic maps that our constructions will produce.

So let N = G/K be a k-symmetric space, k > 2, with automorphism τ and associated eigenspace decomposition

𝔤^{ℂ} = ∑_{i∈ℤ_{k}} 𝔤_{i}.

In particular, 𝔤_{−1} = \overline{𝔤_{1}} and, since k > 2, 𝔤_{1} ∩ 𝔤_{−1} = \{0\}.

Definition A map ϕ : M → N of a Riemann surface is primitive if ϕ^{∗}β′ takes values in [𝔤_{−1}]. Equivalently, ϕ is primitive if and only if any framing Φ has α′_{𝔪} taking values in 𝔤_{−1}.

Example Consider the case k = 3: here we have

𝔪^{ℂ} = 𝔤_{1} ⊕ 𝔤_{−1}

so that N acquires an invariant almost complex structure with T^{(1,0)}N ≅ [𝔤_{−1}]. This is the non-integrable almost complex structure discussed by Salamon [49] who viewed 3-symmetric spaces as twistor spaces for Riemannian symmetric spaces (in this regard, see also [10, 15]). In this setting, primitive maps are just (almost) holomorphic maps.

In general, primitive maps are examples of maps which are f-holomorphic with respect to a horizontal f-structure in the sense of Black [4]. Under rather general conditions, such maps enjoy a number of interesting and useful properties which, in our setting, stem ultimately from the relation

[𝔤_{1}, 𝔤_{−1}] ⊂ 𝔤_{0} = 𝔨^{ℂ}. (3.10)

To be more precise, under the additional assumption that all irreducible subrepresentations of the adjoint representation of K on 𝔪^{ℂ} occur with multiplicity one, Black proves:

1. All primitive maps are equiharmonic, that is, harmonic with respect to any invariant metric on N;

2. If p : N = G/K → G/H is any homogeneous projection and ϕ : M → N is equiharmonic then so is p ∘ ϕ.

Even without such extra assumptions, restricted versions of these results are still available. For this, consider a framing of a primitive map with associated 1-form α. We know that α′_{𝔪}takes values in 𝔤_{−1} so that, in view of (3.10), we have

[α′_{𝔪}∧ α′′_{𝔪}]_{𝔪}= 0.

The projections of the Maurer–Cartan equations for α onto 𝔤_{−1}, 𝔤_{1} and 𝔤_{0} therefore read

dα′_{𝔪}+ [α_{𝔨}∧ α′_{𝔪}] = 0 (3.11)

dα′′_{𝔪}+ [α_{𝔨}∧ α′′_{𝔪}] = 0 (3.12)

dα_{𝔨}+ \frac{1}{2}[α_{k} ∧ α_{k}] + [α′_{𝔪}∧ α′′_{𝔪}] = 0. (3.13)

However, these are just the harmonic maps equations (3.7) and (3.8) and we conclude:

Theorem 3.6 A primitive map ϕ : M → N is harmonic with respect to the metric on N induced by that on 𝔤.

Primitive maps are also well-behaved with respect to projections: let G/H be a reductive homogeneous space with K ⊂ H and reductive decomposition

𝔤 = 𝔥 ⊕ 𝔭

which we assume to be orthogonal and stable under τ. As usual, equip G/H with the metric induced by that on 𝔤 and let p : N = G/K → G/H be the homogeneous projection. We have

Theorem 3.7 If ϕ : M → N is primitive then p ∘ ϕ : M → G/H is harmonic.

Proof. Let 𝔩 be the orthogonal complement of 𝔨 in 𝔥 so that we have a τ-stable decomposition

𝔤 = 𝔨 ⊕ 𝔩 ⊕ 𝔭

with

𝔥 = 𝔨 ⊕ 𝔩; 𝔪 = 𝔩 ⊕ 𝔭.

Both 𝔩^{ℂ} and 𝔭^{ℂ} decompose into eigenspaces of τ and we have various relations between these eigenspaces: [𝔩_{1}, 𝔭_{−1}] ⊂ 𝔤_{0} = 𝔨^{ℂ} but [𝔩^{ℂ}, 𝔭^{ℂ}] ⊂ [𝔥^{ℂ}, 𝔭^{ℂ}] ⊂ 𝔭^{ℂ} so that

[𝔩_{1}, 𝔭_{−1}] = 0. (3.14)

We also have [𝔭_{1}, 𝔭_{−1}] ⊂ 𝔨^{ℂ} so that

[𝔭_{1}, 𝔭_{−1}]_{𝔭}= 0. (3.15)

Any framing for ϕ is one for p ∘ ϕ also so it suffices to prove that, for such a framing,

[α′_{𝔭}∧ α′′_{𝔭}]_{𝔭}= 0; (3.16)

dα′_{𝔭}+ [α_{𝔥}∧ α′_{𝔭}] = 0.

However, α′_{𝔪}= α′_{𝔩}+α′_{𝔭} takes values in 𝔤_{−1} so that α′_{𝔭} is 𝔭_{−1}-valued and (3.16) follows from (3.15). Finally,

dα′_{𝔭}+ [α_{𝔥}∧ α′_{𝔭}] = dα′_{𝔭}+ [α_{𝔨}∧ α′_{𝔭}] + [α′′_{𝔩} ∧ α′_{𝔭}]
= dα′_{𝔭}+ [α_{𝔨}∧ α′_{𝔭}]

in view of (3.14) and this last is just the 𝔭-part of dα′_{𝔪}+ [α_{𝔨}∧ α′_{𝔪}] which vanishes by (3.11). □

In particular, primitive maps may give rise, by projection, to harmonic maps into Riemannian symmetric spaces and it is from this possibility that our principal interest in them derives.

Consider now the loop (3.9) of 1-forms

α_{λ} = λ^{−1}α′_{𝔪}+ α_{𝔨}+ λα′′_{𝔪},

where α = Φ^{−1}dΦ for a framing Φ of a primitive map ϕ. Then α′_{𝔪}is 𝔤_{−1}-valued so that we may view α_{λ} as a Λ𝔤_{τ} -valued 1-form.

Since ϕ is harmonic and [α′_{𝔪}∧ α′′_{𝔪}]_{𝔪}vanishes, the discussion in Section 3.2 applies so that, for each λ ∈ S^{1},

dα_{λ} + \frac{1}{2}[α_{λ} ∧ α_{λ}] = 0.

Thus primitive maps also give rise to zero-curvature equations.

Conversely, suppose that M is simply connected and α_{λ} is a Λ𝔤_{τ} -valued 1-form of the form (3.9) such that

dα_{λ} + \frac{1}{2}[α_{λ} ∧ α_{λ}] = 0.

Then, by Theorem 2.1, for each λ ∈ S^{1}, we can find a map Φ_{λ} : M → G such that

Φ_{λ}^{−1}dΦ_{λ} = α_{λ}

and then ϕ_{λ} = π ∘ Φ_{λ} will be harmonic. In fact, more is true: since

(α_{λ})′_{𝔪}= λ^{−1}α′_{𝔪}

takes values in 𝔤_{−1}, we conclude that each ϕ_{λ} is primitive. In particular, primitive maps of simply connected surfaces come in S^{1}-families.

Finally let us show how the results of Section 2 provide primitive maps ℝ^{2} → N. For this, observe that Λ𝔤_{τ} -valued 1-forms of the form (3.9) are precisely the kind of 1-form that arises as the solution to the zero-curvature equations in Theorem 2.5 so that we get Theorem 3.8 For each d ≡ 1 mod k and ξ_{o} ∈ Λ_{d}, there is a unique solution to

\frac{∂ξ}{∂z} = [ξ, λ^{−1}ξ_{d} + r(ξ_{d−1})]; ξ(0) = ξ_{o}

and then there is a primitive map ϕ : ℝ^{2} → N with framing Φ : ℝ^{2} → G satisfying Φ^{−1}dΦ(∂/∂z) = ξ_{d} + r(ξ_{d−1}).

We call the primitive maps so obtained primitive maps of finite type.

A similar analysis to that in Section 3.2 can be carried out for primitive maps to give sufficient conditions for a primitive map to be of finite type. Just as before, such a map must have ξ_{d} = α′_{𝔪}(∂/∂z) taking values in a single Ad K^{ℂ}-orbit in 𝔤_{−1} and, for doubly periodic maps, this condition is almost sufficient. Indeed, as we shall see in Section 5, the results of [11, 14] imply:

Theorem 3.9 Let ϕ : ℝ^{2} → N be a doubly periodic primitive map into a ksymmetric space, k > 2. Suppose that some (and hence every) framing of ϕ has α′_{𝔪}(∂/∂z) taking values in a single semisimple Ad K^{ℂ}-orbit in 𝔤_{−1}. Then ϕ is of finite type.

We conclude this section with an alternative formulation of the “semisimple orbit” condition which will be useful below. First, the holomorphic differentials argument of Section 3.2 can be extended to primitive maps so that one can conclude that a doubly periodic primitive map of ℝ^{2} has α′_{𝔪}(∂/∂z) taking values in a single common level set of the Ad K^{ℂ}-invariant polynomials on 𝔤_{−1}. On the other hand, Vinberg [56] proves that each such level set contains a single semisimple orbit and that this is the unique closed orbit in the level set. We therefore deduce:

Theorem 3.10 Let ϕ : ℝ^{2} → N be a doubly periodic primitive map into a ksymmetric space, k > 2. Suppose that, for some (and hence every) framing of ϕ, α′_{𝔪}(∂/∂z) is semisimple on a dense subset of M. Then ϕ is of finite type.

3.4 Example: Toda fields

As an illustration of our ideas, we describe some results of Bolton–Pedit–Woodward [7] which relate certain primitive maps into flag manifolds with (periodic) Toda fields (for further details, see [8] in this volume). We begin by introducing the basic ingredients of Toda field theory.

Let 𝔤 be a compact, simple Lie algebra with Killing form κ and, as usual, fix a maximal toral subalgebra 𝔱 and set 𝔞 = i𝔱 so that 𝔥 = 𝔱^{ℂ} = 𝔱⊕𝔞 is a Cartan subalgebra of 𝔤^{ℂ}. We also fix a choice of positive roots Δ^{+} ⊂ Δ(𝔤^{ℂ}, 𝔥) with corresponding simple roots α_{1}, …, α_{l}. Since 𝔤 is simple, we have another distinguished root: the highest root μ and we set

Π = {α_{1}, …, α_{l}, −μ}.

Thus Π is the set of roots labelling the nodes of the extended Dynkin diagram of 𝔤^{ℂ}. From the elementary properties of root systems (see, for example, [33]), we have that if α, β ∈ Π and X_{α} ∈ 𝔤^{α}, X_{−β} ∈ 𝔤^{−β} then

[X_{α}, X_{−β}] = \{ \begin{matrix}0 & if α ≠ β \\ κ(X_{α}, X_{−β})H_{α} & if α = β \end{matrix} (3.17)

where H_{α} ∈ 𝔞 is the Killing dual of α.

With this understood, we can describe Toda field theory. This is a nonlinear Lagrangian field theory where the fields are maps u : ℝ^{2} → 𝔞 and the Lagrangian is given by

∫_{ℝ^{2}} \frac{1}{2}κ(du, du) − ∑_{α∈Π} e^{α(u)} dvol.

The Euler–Lagrange equations for this functional are

d^{∗}du − ∑_{α∈Π} e^{α(u)}H_{α} = 0 (3.18)

and solutions of this are Toda fields.

Remark The alert Reader will notice that our Toda field equations differ from those in [7, 8]. The difference is a matter of normalisation with that in [7, 8] chosen to ensure that zero is a solution. We have given a normalisation which emphasises the relationship with the Toda lattice of Section 1.5.

We can find a zero-curvature formulation of (3.18) via the following ansatz: fix, once and for all, root vectors X_{α} ∈ 𝔤^{α}, α ∈ Π with

κ(X_{α}, \overline{X_{α}}) = −\frac{1}{4}

and set X = ∑_{α∈Π} X_{α}. Now for u : ℝ^{2} → 𝔞, define a loop of 𝔤-valued 1-forms

α_{λ} = λ^{−1}α′_{𝔪}+ α_{𝔱}+ λα′′_{𝔪}

by

α′_{𝔪} = Ad exp(u/2)X dz = ∑_{α∈Π} e^{α(u)/2}X_{α} dz

α_{𝔱} = \frac{i}{2} ∗ du = \frac{1}{2} (\frac{∂u}{∂z} dz − \frac{∂u}{∂\overline{z}} d\overline{z})
α′′_{𝔪} = \overline{α′_{𝔪}}= Ad exp(−u/2)\overline{X} d\overline{z}.

In particular, α_{𝔱} is 𝔱-valued so that [α_{𝔱}∧ α_{t}] vanishes. For any field u, it is easy to verify that

dα′_{𝔪}+ [α_{𝔱}∧ α′_{𝔪}] = 0 = dα′′_{𝔪}+ [α_{𝔱}∧ α′′_{𝔪}]

while, in view of (3.17),

dα_{𝔱}+ \frac{1}{2}[α_{𝔱}∧ α_{𝔱}] + [α′_{𝔪}∧ α′′_{𝔪}] = \frac{i}{2}d ∗ du + ∑_{α∈Π} e^{α(u)}κ(X_{α}, \overline{X_{α}})H_{α} dz ∧ d\overline{z}
= \frac{i}{2}(d ∗ du + ∑_{α∈Π} e^{α(u)}H_{α} ∗ 1)
= −\frac{i}{2} ∗ (d^{∗}du − ∑_{α∈Π} e^{α(u)}H_{α}).

Thus we conclude:

Proposition 3.11 u : ℝ^{2} → 𝔞 is a Toda field if and only if

dα_{λ} + [α_{λ} ∧ α_{λ}] = 0,

for all λ ∈ S^{1}.

Remark There are numerous other (gauge-equivalent) zero-curvature formulations of the Toda field equations (see, for example, [38]). We have chosen this one because it gives real-valued α_{λ}.

We now turn to a geometric interpretation of the Toda field equations. Let G be the adjoint group of 𝔤 (so that, in particular, G has trivial centre) and let T, A be the analytic subgroups with Lie algebras 𝔱, 𝔞. Consider the flag manifold G/T —we equip this with the structure of a k-symmetric space as follows. Let ξ_{j} ∈ 𝔱, 1 ≤ j ≤ l be defined by

α_{i}(ξ_{j}) = \sqrt{−1}δ_{ij}

and set ξ = ∑ ξ_{j}. In particular, α(ξ) ∈ \sqrt{−1}ℤ, for all α ∈ Δ(𝔤^{ℂ}, 𝔥), and we define k ∈ ℕ by

μ(ξ) = \sqrt{−1}(k − 1).

We take τ to be conjugation by exp(−2πξ/k): this is an order k automorphism of G. Then T is the identity component of the fixed set of τ so that G/T is a k-symmetric space.

Remark This is an example of a rather general construction: any generalised flag manifold G^{ℂ}/P, where P is a parabolic subgroup, can be given a canonical ksymmetric structure [15, p. 52].

In the case at hand, observe that in the decomposition of 𝔤^{ℂ} into eigenspaces of τ, we have

𝔤_{0} = 𝔥; 𝔤_{−1} = ∑_{α∈Π} 𝔤^{α}

so that we can immediately deduce the following result from Theorem 2.1 and Proposition 3.11:

Proposition 3.12 Let u : ℝ^{2} → 𝔞 be a Toda field with loop of 1-forms α_{λ}. Then each α_{λ} = Φ_{λ}^{−1}dΦ_{λ} for a framing Φ_{λ} : ℝ^{2} → G of a primitive map ϕ_{λ} : ℝ^{2} → G/T.

We call such a framing of a primitive map a Toda frame.

When does a primitive map ℝ^{2} → G/T have a Toda frame? Observe that, by construction, a Toda frame has α′_{𝔪}(∂/∂z) taking values in the Ad A-orbit of X. In fact, the converse is true: it is easy to see that A acts simply transitively on Ad(A)X while exp : 𝔞 → A is a diffeomorphism so that, if a framing has α′_{𝔪}(∂/∂z) taking values in Ad(A)X then we may take logarithms to find u : ℝ^{2} → 𝔞 satisfying

α′_{𝔪}= Ad exp(u/2)X dz.

Moreover, one can show that α_{𝔱} is determined completely by α′_{𝔪}and the Maurer– Cartan equations so that

α_{𝔱}= \frac{i}{2} ∗ du.

Thus

Lemma 3.13 [7] A frame is Toda if and only if α′_{𝔪}(∂/∂z) takes values in Ad(A)X.

To go further, we must introduce some ideas of Kostant [35]: define the set of cyclic elements C ⊂ 𝔤_{−1} by

C = {∑_{α∈Π} c_{α}X_{α} : c_{α} ∈ ℂ \ {0} }.

Concerning these, Kostant proves

1. any element of C is (regular) semisimple;

2. if P_{1}, …, P_{l} are algebraically independent homogeneous generators of the invariant polynomials on 𝔤^{ℂ} with deg P_{1} ≤ ⋯ ≤ deg P_{l} then

P_{i}|_{𝔤_{−1}}≡ 0 for 1 ≤ i ≤ l − 1

and, for ξ ∈ 𝔤_{−1}, ξ ∈ C if and only if P_{l}(ξ) ≠ 0.

3. For c ≠ 0, T acts simply transitively on the set of Ad A-orbits in P_{l}^{−1}{c}.

From this, we deduce that if a framing Φ satisfies P_{l}(α′_{𝔪}(∂/∂z)) ≡ P_{l}(X) then there is a unique gauge transformation taking Φ to a Toda frame. More generally, when P_{l}(α′_{𝔪}(∂/∂z)) is a non-zero constant, we can make a linear change of co-ordinate cz = w so that P_{l}(α′_{𝔪}(∂/∂w)) ≡ P_{l}(X) and so obtain a Toda frame.

To summarise:

Theorem 3.14 [7] A primitive map ϕ : ℝ^{2} → G/T has a Toda frame (after a linear change of co-ordinate) if and only if, for some (and hence, every) framing we have that P_{l}(α′_{𝔪}(∂/∂z)) is a non-zero constant. Moreover, in this case, the Toda frame is unique.

From the uniqueness assertion, we also get

Corollary 3.15 [7] Let ϕ : ℝ^{2} → G/T be a primitive map with Toda frame Φ and corresponding Toda field u. Then ϕ is doubly periodic if and only if Φ is doubly periodic and then u is doubly periodic.

We remark that the usual holomorphic differentials argument shows that a doubly periodic primitive map has a Toda frame if and only if α′_{𝔪}(∂/∂z) is cyclic at some (and hence, every) point of ℝ^{2}.

We can now apply Theorem 3.9 to obtain such doubly periodic primitive maps from commuting flows. Indeed, a primitive map with Toda frame necessarily satisfies the “semisimple orbit” hypothesis since X is semisimple so that we have

Theorem 3.16 [7] A doubly periodic primitive map ϕ : ℝ^{2} → G/T with non-zero P_{l}(α′_{𝔪}(∂/∂z)) is of finite type.

In particular, we obtain doubly periodic Toda fields this way from Corollary 3.15. However, more is true: while it is not the case that doubly-periodic Toda fields necessarily produce doubly periodic primitive maps (there may be holonomy), Bolton– Pedit–Woodward prove that, for doubly periodic Toda fields, the Λ𝔤_{τ} -valued 1-form α_{λ} always arises as in Theorem 2.5 so that all doubly periodic Toda fields are of finite type.

Examples Let us take G = SO(5) with flag manifold F = SO(5)/SO(2) × SO(2). We have the homogeneous fibration p : F → S^{4} and a homogeneous diffeomorphism between F and the Grassmannian bundle G̃_{2}(T S^{4}) of oriented 2-planes in T S^{4}. Under this indentification, a map ϕ : M → F is primitive if and only if it is the Gauss map of a minimal (i.e., conformal and harmonic) map p ∘ ϕ : M → S^{4} (a result originally due to Eells–Salamon [21]). Moreover, ϕ has a Toda frame precisely when the corresponding minimal surface is not superminimal. In this case, the above results reproduce the analysis of Ferus–Pedit–Pinkall–Sterling [24].

Again, let us take F = SU(n)/T^{n−1} as our flag manifold. A map ϕ : M → F may be viewed as a collection of mutually orthogonal maps ϕ_{1}, …, ϕ_{n} : M → ℂP^{n−1} and then ϕ is primitive with Toda frame if and only if the ϕ_{i} comprise the Frenet frame of a superconformal harmonic map M → ℂP^{n−1} in the sense of [7, 8]. It was for the study of such maps that the methods of [7] were developed.

We shall have more to say about results of this kind in the next section.

We conclude this section by briefly contemplating the geometric significance of open Toda fields as this has received a lot of recent attention in the Physics literature [18, 25, 26, 47, 50]. Open Toda fields are extremals of the modified Lagrangian

∫_{ℝ^{2}} \frac{1}{2}κ(du, du) − ∑_{i=1}^{l} e^{α_{i}(u)} dvol

where there is no longer an exponential interaction involving −μ. Again there is a zero-curvature representation but this time the primitive maps so obtained are holomorphic maps into G/T. In fact, they are superhorizontal holomorphic maps in the sense of [15]. As a consequence, open Toda fields are much simpler objects for which Weierstrass representation formulae are available. See [38] in this volume for a brief discussion of such fields.

3.5 Example: primitive maps and twistor lifts

We have seen how all non-conformal harmonic 2-tori in a rank-1 symmetric space are of finite type. However, this excludes the geometrically interesting case of minimal 2-tori. On the other hand, special cases of the results of the previous section have been used to account for certain minimal 2-tori in S^{4} [24], S^{n} and ℂP^{n} [7] by showing that these tori have lifts which are primitive maps of finite type.

In this section, we present a generalisation of these results which accounts for all harmonic 2-tori in a sphere S^{n}. We begin by recalling some results from the well-developed twistor theory of harmonic maps of a Riemann surface into a sphere.

So let ϕ : M → S^{n} be a harmonic map of a Riemann surface. Let T = ϕ^{−1}T S^{n} with connection ∇ given by the pull-back of the Levi-Civita connection of S^{n}. Let z be a local holomorphic co-ordinate on M, set ∇′ = ∇_{∂/∂z}, ∇′′ = ∇_{∂/∂\overline{z}} and inductively define ∇^{i}ϕ by

∇^{1}ϕ = ∂ϕ/∂z; ∇^{i+1}ϕ = ∇′∇^{i}ϕ.

For x ∈ M, define W_{x}^{j} ⊂ T_{x}^{ℂ} by

W_{x}^{j} = span_{ℂ}{∇_{x}^{i}ϕ : 1 ≤ i ≤ j}.

Clearly, each W_{x}^{j} is defined independently of the choice of holomorphic co-ordinate z.

If ϕ is non-constant and (weakly) conformal then each W_{x}^{1} is isotropic for the complexified metric on T^{ℂ} and is 1-dimensional off a discrete set of points in M. This motivates, in part, the following definition:

Definition The isotropy dimension r of a conformal harmonic map ϕ : M → S^{n} is given by

r = max{j : max_{x} dim_{ℂ} W_{x}^{j} = j and W_{x}^{j} is isotropic for all x}.

We make the convention that a non-conformal map has isotropy dimension zero.

The following facts are well known (c.f. [60]): if ϕ has isotropy dimension r > 0 then

1. For 1 ≤ j ≤ r + 1, there is a bundle W^{j} of T^{ℂ} whose fibre at x coincides with W_{x}^{j} except at a discrete set of points;

2. Each W^{j} is stable under ∇′′;

3. For 1 ≤ j ≤ r, rank W^{j} = j and rank W^{r+1} ≤ r + 1.

We therefore distinguish two possibilities: either W^{r} = W^{r+1} or not^{1}.

1These two possibilities regulate the relationship between the isotropy dimension of ϕ and the isotropy order of ϕ [2] described by Wood [59] in this volume: if ϕ has isotropy dimension r then ϕ has isotropy order 2r if and only if W^{r} = W^{r+1} and isotropy order 2r + 1 otherwise.

In the first case, we see that W^{r} is isotropic and stable under ∇′ so that the (complexified) inner products

(∇^{i}ϕ, ∇^{j}ϕ) ≡ 0

for all i, j ∈ ℕ. Harmonic maps of this kind are variously called pseudo-holomorphic [16], real isotropic [22] or superminimal [9] and were completely classified by Calabi [16] (see also [3]) who proved that all such were projections of horizontal holomorphic curves in the twistor space of S^{n}. Indeed, since W^{r} is stable under both ∇′ and ∇′′, it is parallel so that

(W^{r} ⊕ \overline{W^{r}}) ∩ T

is a parallel sub-bundle of T with (real) rank 2r. It follows that ϕ factors through an equatorial 2r-sphere in S^{n} so that, without loss of generality, we may take 2r = n. Now recall that the twistor space Z on S^{2r} may be viewed as the bundle of isotropic r-planes in the complexification of T S^{2r}. This is a complex manifold: in fact, SO(2r + 1) acts transitively on each of the two connected components of Z and each component is so realised as the generalised flag manifold SO(2r +1)/U(r). It is clear that W^{r} defines a map ψ : M → Z covering ϕ and the condition that W^{r} be parallel is equivalent to the demand that ψ be holomorphic and horizontal with respect to the twistor fibration Z → S^{2r}. For more on twistor spaces of symmetric spaces and their applications to harmonic maps, the Reader is refered to [15], the surveys [12, 13, 46, 61] and the article by Kobak [34] in this volume.

Remark It is clear that the isotropy dimension r of a map ϕ : M → S^{n} must satisfy 2r ≤ n. In case that 2r = n, it follows from the easily verified fact

(∇^{r+1}ϕ, ∇^{i}ϕ) ≡ 0,

for 1 ≤ i ≤ r, that W^{r} = W^{r+1} so that ϕ is superminimal.

Let us now consider the non-superminimal harmonic maps where W^{r} ≠ W^{r+1} or, equivalently,

(∇^{r+1}ϕ, ∇^{r+1}ϕ) ≢ 0.

It is our contention that ϕ is covered by a primitive map into a suitable k-symmetric space and, moreover, that when M is a 2-torus this primitive map is of finite type.

First we describe the relevant k-symmetric space: for 2r < n, let F^{r}(S^{n}) be the bundle over S^{n} with fibre

F_{x}^{r}(S^{n}) = {w_{1} ⊂ ⋯ ⊂ w_{r} ⊂ (T_{x}S^{n})^{ℂ} : each w_{j} is an isotropic j-plane}

Now SO(n + 1) acts transitively on F^{r}(S^{n}) with stabilisers conjugate to

\overbrace{SO(2) × ⋯ × SO(2)}^{r times} ×SO(n − 2r)

so that F^{r}(S^{n}) is a homogeneous space.

Fix a base-point f = (w_{1} ⊂ ⋯ ⊂ w_{r}) ∈ F_{x}^{r}(S^{n}) and orthogonalise to obtain isotropic lines L_{1}, …, L_{r} and real subspaces L_{r+1} and L_{0} = span_{ℂ}{x} so that

(ℝ^{n+1})^{ℂ} = L_{0} ⊕ ∑_{i=1}^{r}(L_{i} ⊕ \overline{L_{i}}) ⊕ L_{r+1}.

Take k = 2r + 2, let ω be the usual primitive k-th root of unity and define Q ∈ O(n + 1) by

Q = ω^{−i} on L_{i}.

Then conjugation by Q is an order k automorphism of SO(n + 1) and the identity component of its fixed set is the stabiliser of f. Thus F^{r}(S^{n}) is a k-symmetric space.

Let ψ : M → F^{r}(S^{n}) be a map with projection ϕ : M → S^{n}. We may view ψ as a flag of isotropic sub-bundles

ψ^{(1)} ⊂ ⋯ ⊂ ψ^{(r)} ⊂ T^{ℂ}

(here again T = ϕ^{−1}T S^{n}) and one can prove

Proposition 3.17 [11] ψ : M → F^{r}(S^{n}) is primitive if and only if

(i) ∇^{1}ϕ is a (local) section of ψ^{(1)};

(ii) each ψ^{(i)} is stable under ∇′′;

(iii) if σ is a local section of ψ^{(i)} then ∇′σ is a local section of ψ^{(i+1)}.

We know from Theorem 3.7 that if ψ is primitive then ϕ is harmonic but, in the present case, we have a converse. Let ϕ : M → S^{n} be a non-superminimal harmonic map of isotropy order r. It is clear from Proposition 3.17 that the bundle of flags W^{1} ⊂ ⋯ ⊂ W^{r} defines a primitive map ψ : M → F^{r}(S^{n}). When is ψ of finite type? We have

Lemma 3.18 [11] A (local) framing of ψ has α′_{𝔪}(∂/∂z) semisimple at x ∈ M if and only if

(∇^{r+1}ϕ, ∇^{r+1}ϕ) ≠ 0

at x ∈ M.

Since W^{r} ≠ W^{r+1}, we deduce that this condition holds on a dense open subset of M and so conclude from Theorem 3.10:

Theorem 3.19 [11] A non-superminimal harmonic map of a 2-torus into a sphere is covered by a primitive map of finite type.

Thus harmonic 2-tori in spheres are completely accounted for: either they are superminimal and so arise as projections of holomorphic curves or they are obtained by integrating commuting flows.

Example We have seen that a non-superminimal map into S^{n} must have isotropy dimension r with 2r < n. Suppose n = 2m and consider maps of isotropy dimension m − 1. Now F^{m−1}(S^{2m}) is the full flag manifold of SO(2m + 1) and its k-symmetric structure is precisely that described in Section 3.4. Moreover, when M is a 2-torus, the primitive maps we have constructed are precisely those with Toda frames.

Again, if n = 2m − 1, the maximal isotropy dimension of a non-superminimal map is m − 1 and, viewing S^{2m−1} as an equator of S^{2m}, we again conclude that such tori are covered by primitive maps with Toda frames.

To summarise: non-superminimal harmonic 2-tori of maximal isotropy order are all obtained from doubly-periodic Toda fields of finite type. This result was first proved by Bolton–Pedit–Woodward [7] (see also [18]) who called such maps superconformal.

Remark A similar analysis can be carried out for harmonic maps into a complex projective space. Again one has a notion of an isotropic or superminimal map (these are the legs of the Frenet frame of a holomorphic curve) and all non-superminimal harmonic maps are covered by primitive maps into flag manifolds with their canonical k-symmetric structure. Moreover, when M is a 2-torus, these primitive maps are of finite type. The Reader is refered to [11] for more details on this and the other results of this section.

4 Loop groups and extended framings

The Adler–Kostant–Symes scheme produces commuting Hamiltonian flows on Lie algebras and we have seen how to apply this scheme to the twisted loop algebras Λ𝔤_{τ}^{ℂ} and so obtain loops of flat connections and hence harmonic and primitive maps. However, in Section 1.4, a method of Symes was described for integrating these flows via projection of geodesics. We now show how this method applies in our setting. In so doing, we will introduce an analogue of Uhlenbeck’s theory [55] of “extended solutions” (see, also, [27] in this volume) and, incidently, obtain a conceptual proof of Theorem 2.4.

Terminology To avoid treating the case of k-symmetric spaces with k = 2 separately, henceforth we shall talk of primitive harmonic maps, conscious of the fact that the primitive condition is vacuous when k = 2 and that the harmonic condition is implied by the primitive condition when k > 2 (Theorem 3.6).

4.1 Iwasawa decomposition of ΛG_{τ}^{ℂ}

We begin by introducing the infinite-dimensional Lie groups that correspond to the Lie algebras Λ𝔤_{τ}^{ℂ}, Λ𝔤_{τ} and Λ_{+}𝔤_{τ}^{ℂ} of Section 2.3. So let G be a compact semisimple Lie group with order k automorphism τ whose fixed set is K. Further, let

K^{ℂ} = KB

be the Iwasawa decomposition of K^{ℂ} corresponding to that of 𝔨^{ℂ}.

Let ΛG_{τ}^{ℂ} be the manifold of loops

ΛG_{τ}^{ℂ} = {γ : S^{1} → G^{ℂ} : γ(ωλ) = τγ(λ) for all λ ∈ S^{1}}.

Thus ΛG_{τ}^{ℂ} is an infinite-dimensional Lie group^{2} under point-wise multiplication. Further, define subgroups by

ΛG_{τ} = {γ ∈ ΛG_{τ}^{ℂ} : γ : S^{1} → G};

Λ_{+}G_{τ}^{ℂ} = {γ ∈ ΛG_{τ}^{ℂ} : γ extends holomorphically to γ : D → G^{ℂ}, γ(0) ∈ B}.

Clearly, ΛG_{τ}^{ℂ} has Lie algebra Λ𝔤_{τ}^{ℂ} and so on.

We now have the following Iwasawa decomposition:

Theorem 4.1 [19] Multiplication ΛG_{τ} × Λ_{+}G_{τ}^{ℂ} → ΛG_{τ}^{ℂ} is a diffeomorphism onto.

This result was deduced by Dorfmeister–Pedit–Wu [19] from the special case τ = id which was proved by Pressley–Segal [43]. The main idea of Pressley–Segal was that the homogeneous space ΛG_{τ}^{ℂ}/Λ_{+}G_{τ}^{ℂ} can be realised as an infinite-dimensional Grassmannian on which one can show that ΛG_{τ} acts transitively (c.f. [27] in this volume).

4.2 Extended framings and a Symes formula

Let α_{λ} = λ^{−1}α′_{𝔪}+α_{0}+λα′′_{𝔪} be a Λ𝔤_{τ} -valued 1-form on a simply-connected surface M which satisfies the Maurer–Cartan equations. We have seen that, for each λ ∈ S^{1}, there is a smooth map Φ_{λ} : M → G framing a primitive harmonic map ϕ_{λ} : M → G/K and satisfying

Φ_{λ}^{−1}dΦ_{λ} = α_{λ}.

Moreover, Φ_{λ} is unique up to left translation by a constant. We may choose these constants so that Φ_{λ}(x) depends smoothly on λ for some (and hence every) x ∈ M and then we may view the Φ_{λ} as a single smooth map Φ̂ : M → ΛG_{τ} such that the following diagram commutes:

[figure: M; G; ΛG_{τ}; Φ_{λ}; ev_{λ}; Φ̂]

2We shall not worry about the topology of ΛGℂ_{τ}: in fact, it suffices to work with the (Fréchet)

C^{∞}-topology [43].

where ev_{λ} : ΛG_{τ} → G is given by evaluating the loop at λ ∈ S^{1}.

We call such maps Φ̂ extended framings. Thus Φ̂ is an extended framing if and only if

(Φ̂^{−1}dΦ̂)_{λ} = α_{λ} = λ^{−1}α′_{𝔪}+ α_{0} + λα′′_{𝔪},

where α′_{𝔪}, α_{0} and α′′_{𝔪} do not depend on λ. Any extended framing gives rise to a loop of primitive harmonic maps. Moreover, any primitive harmonic map of a simply connected M gives rise to a loop of such maps which are covered in this way by an extended framing. (In the case, k = 2 these results are due to Rawnsley [45].)

We can find examples of extended framings by recourse to Theorem 4.1 and the method of Symes. For this, let Λ^{(1)}𝔤_{τ}^{ℂ} ⊂ Λ𝔤_{τ}^{ℂ} consist of those elements η ∈ Λ𝔤_{τ}^{ℂ} for which λη extends holomorphically to D. Thus η ∈ Λ^{(1)}𝔤_{τ}^{ℂ} if and only if

η = ∑_{n≥−1} λ^{n}η_{−n}.

Observe that if g ∈ Λ_{+}G_{τ}^{ℂ} and η ∈ Λ^{(1)}𝔤_{τ}^{ℂ} then Ad g λη = λ Ad g η also extends holomorphically to D so that Λ^{(1)}𝔤_{τ}^{ℂ} is invariant under the adjoint action of Λ_{+}G_{τ}^{ℂ}.

Now choose η_{o} ∈ Λ^{(1)}𝔤_{τ}^{ℂ} and define the (complex) geodesic g : ℝ^{2} → Λ𝔤_{τ}^{ℂ} by

g(z) = exp(zη_{o}).

Using Theorem 4.1, we find maps a : ℝ^{2} → ΛG_{τ}, b : ℝ^{2} → Λ_{+}G_{τ}^{ℂ} such that

g = ab

and we now have

Theorem 4.2 a is an extended framing.

Proof. We argue as in Section 1.4 to see that

η_{o} dz = dg g^{−1} = da a^{−1} + Ad a(db b^{−1})

so that

a^{−1}da = π_{K}(Ad a^{−1}η_{o} dz).

Set η = Ad a^{−1}η_{o} and observe that since

Ad g(z) η_{o} = Ad exp(zη_{o}) η_{o} = η_{o},

we also have

η = Ad b η_{o}

so that η takes values in Λ^{(1)}𝔤_{τ}^{ℂ}. This means that η = ∑_{n≥−1} λ^{n}η_{−n} so that, by (2.3) we have

a^{−1}da = π_{K}(η dz) = λ^{−1}η_{1} dz + (η_{0} dz)_{𝔨}+ λ\overline{η_{1}} d\overline{z} (4.1) and a is an extended framing. □

The extended framings corresponding to finite type primitive harmonic maps all arise in this way: let d ≡ 1 mod k and fix an initial condition ξ_{o} ∈ Λ_{d}. Then η_{o} = λ^{d−1}ξ_{o} ∈ Λ^{(1)}𝔤_{τ}^{ℂ} and we apply Theorem 4.2 to get an extended framing a. Now set ξ = Ad a^{−1}ξ_{o} = λ^{1−d}η. Again we have ξ = Ad b ξ_{o} so that, since λ^{d}ξ_{o} extends holomorphically to the disc, λ^{d}ξ does also and it follows that ξ : ℝ^{2} → Λ_{d}. Now

dξ = [ξ, a^{−1}da]

while, from (4.1) and (2.5), we have

a^{−1}da = π_{K}(λ^{d−1}ξ dz) = (λ^{−1}ξ_{d} + r(ξ_{d−1})) dz + (λξ_{−d} + \overline{r(ξ_{d−1})}) d\overline{z}.

We therefore conclude that

1. ξ is the solution to the Hamiltonian equations (2.7);

2. the 1-form (2.8) constructed from ξ is precisely a^{−1}da (and so solves the Maurer–Cartan equations so that the proof of Theorem 2.4 is complete);

3. a is precisely the extended framing for the finite type primitive harmonic map corresponding to ξ.

This gives an alternative characterisation of the finite type condition in terms of the extended framing which will be useful below. To be precise, we have proved the following theorem:

Theorem 4.3 A primitive harmonic map ϕ : ℝ^{2} → G/K into a k-symmetric space is of finite type if and only if, for some d ≡ 1 mod k, there exists ξ_{o} ∈ Λ_{d} such that, if a : ℝ^{2} → ΛG_{τ}, b : ℝ^{2} → Λ_{+}G_{τ}^{ℂ} are defined by

exp(zλ^{d−1}ξ_{o}) = a(z)b(z),

for z ∈ ℝ^{2}, then a is an extended framing for ϕ so that

ϕ = π ∘ ev_{1} ∘ a,

where π : G → G/K is the coset projection.

Remark These simple results are the starting point of a loop-group theoretic analysis of the harmonic map equations which, to some extent, parallels that of the KdV equation by Segal–Wilson [51]. We shall return to this elsewhere. For a similar approach to Toda fields, the Reader is refered to the contribution of McIntosh in this volume [38] as well as [20, 39, 63].

5 Another approach

To date, the most comprehensive results concerning the construction of harmonic maps from commuting Hamiltonian flows are to be found in [14]. The approach adopted in that paper is similar to the one we have been describing but differs from it in a number of important respects. Among these are:

1. The Hamiltonian flows arise from the r-matrix formalism described in Section 2.5 and not from the Adler–Kostant–Symes scheme.

2. The main objects of study are harmonic maps into a Lie group G rather than into an arbitrary (k-)symmetric space. Since Lie groups are parallelisable, this means that one can treat such maps directly without recourse to a framing. Maps into k-symmetric spaces are then viewed as particular maps into G via a Cartan embedding (see Section 5.3 below).

In the remaining sections of this article, we describe the theory of [14] and its extension to the k-symmetric case in [11]. We will show how it relates to the theory we have been developing above and, in particular, we will see how Theorems 3.4 and 3.9 can be read off a corresponding result in [14].

5.1 Hamiltonian flows in the based loop algebra

We begin by rehearsing Uhlenbeck’s zero-curvature reformulation of the harmonic map equations for maps into a Lie group G [55] and the method of Burstall–Ferus– Pedit–Pinkall [14] for producing solutions to those zero-curvature equations. The Reader will find a full account of these results in Wood’s contribution to this volume [59] and so we shall be brief here, referring the Reader to [59] or [14] for more details.

So let ϕ : M → G be a map of a Riemann surface into a compact Lie group. As usual, set α = ϕ^{−1}dϕ. We may view G as the homogeneous space G/{e} so that ϕ is its own framing and it then follows from Section 3.2 that ϕ is harmonic if and only if α is co-closed:

d^{∗}α = 0. (5.1)

Indeed, in this case, α = α_{𝔪}in (3.3) while [α ∧ ∗α] always vanishes.

Combining (5.1) with the Maurer–Cartan equations for α, one concludes with Uhlenbeck that the loop of 1-forms given by

A_{λ} = \frac{1 − λ^{−1}}{2} α′ + \frac{1 − λ}{2}α′′ (5.2)

satisfies the Maurer–Cartan equations for each λ ∈ S^{1}. Conversely, given a loop of flat connections (5.2) on a simply connected surface M, we integrate to find a harmonic map ϕ : M → G with ϕ^{−1}dϕ = A_{−1}.

Observe that the loop A_{λ} in (5.2) satisfies A_{1} ≡ 0 and so may be viewed as a 1-form with values in the based loop algebra Ω𝔤 given by

Ω𝔤 = {ξ : S^{1} → 𝔤 : ξ(1) = 0}.

View Ω𝔤 as a subalgebra of the free loop algebra Λ𝔤 (which is Λ𝔤_{τ} for τ = id) and set Ω_{d} = Λ_{d} ∩ Ω𝔤. We shall produce flat Ω𝔤-valued 1-forms (5.2) on ℝ^{2} by integrating commuting flows on Ω_{d}. For this, fix d ∈ ℕ and define vector fields X_{1}, X_{2} on Ω_{d} by

\frac{1}{2}(X_{1} − iX_{2})(ξ) = [ξ, 2i(λ^{−1} − 1)ξ_{d}].

As explained in Wood’s contribution [59], there is a Poisson structure on Ω𝔤 which arises from an r-matrix as in Section 2.5. Moreover, with respect to this Poisson structure, the vector fields X_{i} are Hamiltonian vector fields of Ad-invariant functions and so commute. Being of Lax type, they are also complete. Thus, if we fix an initial condition ξ_{o} ∈ Ω_{d}, we may simultaneously integrate the X_{i} to get a unique map ξ : ℝ^{2} → Ω_{d} satisfying

\frac{∂ξ}{∂z} = [ξ, 2i(λ^{−1} − 1)ξ_{d}]; ξ(0) = ξ_{o}.

Then, defining an Ω𝔤-valued 1-form by

A_{λ} = 2i(λ^{−1} − 1)ξ_{d} dz − 2i(λ − 1)ξ_{−d} d\overline{z}

produces a solution to the Maurer–Cartan equations and hence a harmonic map ϕ : ℝ^{2} → G with ϕ^{−1}∂ϕ/∂z = −4iξ_{d}.

Again we shall call the maps so obtained harmonic maps of finite type.

Notation Consistency with our previous development compels us to use slightly different notation from that in Wood’s contribution [59] and, indeed, [14]. In those papers, the spectral parameter λ is the reciprocal of ours while the coefficients ξ_{d} differ by a sign.

Remark One should note that the finite-dimensional subspaces Ω_{d} are not Poisson submanifolds for the Poisson structure on Ω𝔤: they are only invariant for the Hamiltonian flows of Ad-invariant functions. Thus, while the ordinary differential equations that we are interested in evolve on a finite-dimensional space, these spaces are not Poisson manifolds in their own right. This is in contrast with the spaces Λ_{d} ⊂ Λ𝔤_{τ} discussed above.

When is a harmonic map ϕ : ℝ^{2} → G of finite type? Given a harmonic map ϕ with associated Ω𝔤-valued 1-form A_{λ}, we may reformulate the finite type condition as the demand that, for some d ∈ ℕ, there is a map ξ : ℝ^{2} → Ω_{d} satisfying

dξ = [ξ, A_{λ}];  (5.3)
α′ = −4iξ_{d} dz.  (5.4)

As before, a necessary condition for the existence of such a ξ is that ϕ^{−1}∂ϕ/∂z takes values in a single Ad G^{ℂ}-orbit in 𝔤^{ℂ}. The principal result of [14] is that a partial converse is true for doubly periodic harmonic maps:

Theorem 5.1 [14] A doubly periodic harmonic map ϕ : ℝ^{2} → G is of finite type if ϕ^{−1}∂ϕ/∂z takes values in a single semisimple Ad G^{ℂ}-orbit.

We shall see later that Theorems 3.4 and 3.9 are corollaries of this theorem. Meanwhile, let us see how it is proved. The first step is to find a formal solution to (5.3) and (5.4), that is, a formal Laurent series

Y = ∑_{j≥−1} λ^{j}Y_{−j}

with each Y_{j} : ℝ^{2} → 𝔤^{ℂ} such that α′ = −4iY_{1} dz and

dY = [Y, A_{λ}] (5.5)

coefficient-wise. Using the semisimplicity hypothesis (and specifically that fact that η ∈ 𝔤^{ℂ} is semisimple if and only if ad η is invertible on its image) one can find an explicit recursion formula for the Y_{j} and so construct Y. It is an immediate consequence of the recursion formula that if A_{λ} is doubly periodic then so is each Y_{j}.

The second step is where we make essential use of the double periodicity hypothesis. It follows from (5.5) that each Y_{j} is a Jacobi field for ϕ and thus is a solution of a linear elliptic equation. When α is doubly periodic, we view each Y_{j} as being defined on a torus and the compactness of the torus together with standard elliptic theory now implies that the Y_{j} span a finite-dimensional space. Using this, it is easy to construct a solution ξ : ℝ^{2} → Ω_{d}, for some d, from some of the Y_{j}.

5.2 Extended solutions and the Symes method

A variant of the Symes method of Section 4.2 is available for integrating the flows on Ω_{d}. This is based on the following loop group decomposition: set

ΛG^{ℂ} = {γ : S^{1} → G^{ℂ}}

and define subgroups

ΩG = {γ ∈ ΛG^{ℂ} : γ : S^{1} → G and γ(1) = e};
Λ_{+}G^{ℂ} = {γ ∈ ΛG^{ℂ} : γ extends holomorphically to γ : D → G^{ℂ}}.

We have

Theorem 5.2 [43] Multiplication ΩG × Λ_{+}G^{ℂ} → ΛG^{ℂ} is a diffeomorphism onto.

Remark This is very similar to the Iwasawa decomposition in Section 4.1 the main difference being that the uniqueness of the decomposition is ensured by basing the real loops at 1 rather than basing the holomorphic loops at 0.

Consider now a loop of 1-forms on a simply connected surface M:

A_{λ} = \frac{1 − λ^{−1}}{2} α′ + \frac{1 − λ}{2}α′

which solve the Maurer–Cartan equations. For each λ ∈ S^{1}, we integrate to get maps Ψ_{λ} : M → G such that

Ψ_{λ}^{−1}dΨ_{λ} = A_{λ}.

In particular, since A_{1} ≡ 0, Ψ_{1} is constant and we choose the constants of integration so that Ψ_{1} ≡ e and the Ψ_{λ} form a smooth map Ψ̂ : M → ΩG. Such Ψ̂ are the extended solutions of Uhlenbeck [55] (c.f., [27, 59] in this volume). It is clear from Section 5.1 that if Ψ̂ is an extended solution, the map ϕ defined by the commuting diagram

[figure: M; G; ΩG; ϕ; ev_{−1}; Ψ̂]

is harmonic and all harmonic maps M → G of a simply connected M are covered by an extended solution in this way.

The method of Symes used in Section 4.2 can be adapted to this setting. Firstly, if η_{o} ∈ Λ𝔤^{ℂ} is such that λη_{o} ∈ Λ_{+}𝔤^{ℂ} (i.e., λη_{o} extends holomorphically to D) then we define a : ℝ^{2} → ΩG, b : ℝ^{2} → Λ_{+}G^{ℂ} by

exp(zη_{o}) = a(z)b(z),

for z ∈ ℝ^{2}, and we have

Proposition 5.3 a is an extended solution.

Secondly, if ξ_{o} ∈ Ω_{d}, put η_{o} = 2iλ^{d−1}ξ_{o} and define a as above. Then the map ξ : ℝ^{2} → Ω𝔤 given by

ξ = Ad a^{−1}ξ_{o}

has the following properties:

1. ξ takes values in Ω_{d};

2. a^{−1}da = 2i(λ^{−1} − 1)ξ_{d} dz − 2i(λ − 1)ξ_{−d} d\overline{z};

3. dξ = [ξ, a^{−1}da].

Thus ξ is the solution of our Hamiltonian flows on Ω_{d} with initial condition ξ_{o} and a is the extended solution for the corresponding harmonic map of finite type.

The proofs of these assertions are similar to those in Section 4.2 and consequently left to the Reader.

5.3 Maps into k-symmetric spaces and Cartan embeddings

To make contact with the results of Section 3 and, in particular, to obtain primitive harmonic maps into k-symmetric spaces from the constructions of Section 5.1, we introduce a certain embedding of a k-symmetric space G/K into its group of isometries G. This will enable us to view primitive harmonic maps into G/K as maps into G to which the methods of Section 5.1 can be applied.

So let G/K be a k-symmetric space with involution τ and Maurer–Cartan form β : T (G/K) → 𝔤. Define a map ι : G/K → G by

ι(g · o) = τ(g)g^{−1}.

It is clear that ι is well-defined. In fact, it is an immersion, as can be seen from the following

Lemma 5.4 [11] Let θ be the Maurer–Cartan form of G. Then, for x ∈ G/K,

ι^{∗}θ_{x} = τ_{x}β_{x} − β_{x},

where τ_{g·o} = Ad g ∘ τ ∘ Ad g^{−1}.

Moreover, if K is the fixed set of τ, then ι is an embedding. We call ι the Cartan embedding of G/K into G.

When k = 2, the Cartan embedding is well-known to be totally geodesic [17] so that if ϕ : M → G/K is harmonic then ι ∘ ϕ : M → G is also. Thus the methods of Section 5.1 apply to ι ∘ ϕ. This is the approach to harmonic maps into symmetric spaces adopted in [14]. When k > 2, ι is no longer totally geodesic but all is not lost: a primitive map ϕ : M → G/K still gives rise to a loop A_{λ} of the form (5.2) from which ϕ may be recovered.

For this, let ϕ : M → G/K be a primitive harmonic map into a k-symmetric space and set δ = ϕ^{∗}β. The equations (3.7) for a framing of ϕ can be written in terms of δ as

dδ′ − [δ′ ∧ δ′′] = dδ′′ − [δ′ ∧ δ′′] = 0.

However, these are precisely the coefficients of (λ^{−1} − 1) and (λ − 1) in the loop of 1-forms

A_{λ} = (λ^{−1} − 1)δ′ + (λ − 1)δ′′. (5.6)

Thus a primitive map ϕ gives rise to a loop of flat 1-forms A_{λ} of the form (5.2). To recover ψ = ι ∘ ϕ from A_{λ}, observe that from Lemma 5.4 we get

ψ^{∗}θ = (ω^{−1} − 1)δ′ + (ω − 1)δ′′

whence

ψ^{−1}dψ = A_{ω}.

Example Let k = 2. Then ω = −1 and α = ψ^{−1}dψ = −2δ so that the loop A_{λ} is just

\frac{1 − λ^{−1}}{2} α′ + \frac{1 − λ}{2}α′′

which is, of course, the loop associated to the harmonic map ψ = ι∘ϕ by Uhlenbeck.

We have seen in Section 5.1 how to produce loops (5.2) of flat 1-forms from commuting flows on Ω_{d}. When does this procedure give rise to primitive harmonic maps? This is a question with a pleasantly simple answer: it is just a matter of choosing the right initial condition for the flows as the following theorem shows [11] (see [14] for the case k = 2).

Theorem 5.5 Fix d ≡ 1 mod k and choose ξ_{o} ∈ Ω_{d} ∩ Λ𝔤_{τ}. Let ξ : ℝ^{2} → Ω_{d} satisfy

\frac{∂ξ}{∂z} = [ξ, 2i(λ^{−1} − 1)ξ_{d}]; ξ(0) = ξ_{o}

and let ψ : ℝ^{2} → G be the map satisfying

ψ^{−1} \frac{∂ψ}{∂z} = 2i(ω^{−1} − 1)ξ_{d} ψ(0) = 0.

Then ψ = ι ∘ ϕ for a primitive harmonic map ϕ : ℝ^{2} → G/K.

It would be natural to call such maps primitive harmonic maps of finite type. However, this terminology has already been reserved for primitive harmonic maps which possess a framing which arises from commuting flows on Λ_{d} ⊂ Λ𝔤_{τ}. A priori, this is a different condition so, for now, we call the maps provided by Theorem 5.5 primitive harmonic maps of G-finite type. Later we shall see that G-finite type maps are indeed of finite type in the sense of Section 3.

Meanwhile, in view of the relation

ψ^{−1}∂ψ/∂z = (ω^{−1} − 1)δ′(∂/∂z),

a semisimple orbit condition on δ′(∂/∂z) implies one for ψ^{−1}∂ψ/∂z and a simple adaptation of the proof of Theorem 5.1 then gives:

Theorem 5.6 [11] Let ϕ : ℝ^{2} → G/K be a doubly periodic primitive harmonic map such that δ′(∂/∂z) takes values in a single semisimple Ad G^{ℂ}-orbit. Then ϕ is of G-finite type.

If Φ is a framing for such a map, we have

Ad Φα_{𝔪}= δ

so again, an orbit condition on δ′ is implied by one on α′_{𝔪}and we therefore deduce the following analogue of Theorems 3.4 and 3.9:

Theorem 5.7 Let ϕ : ℝ^{2} → G/K be a doubly periodic primitive harmonic map into a k-symmetric space. Suppose that some (and hence every) framing of ϕ has α′_{𝔪}(∂/∂z) taking values in a single semisimple Ad K^{ℂ}-orbit in 𝔤_{−1}. Then ϕ is of G-finite type.

5.4 Finite vs. G-finite type

In this section, we shall show that a primitive harmonic map ϕ : ℝ^{2} → G/K of G-finite type is in fact of finite type. Together with Theorem 5.7, this will provide a proof of Theorems 3.4 and 3.9.

Before embarking on this however, let us pause to consider why this rather indirect method is needed to prove Theorems 3.4 and 3.9. The problem is that the approach of Section 3 deals with framings of ϕ and, moreover, framings of a rather special kind: for ϕ to be of finite type, the map ξ : ℝ^{2} → Λ_{d} produces a framing with several restrictions:

1. α′_{𝔪}(∂/∂z) = ξ_{d} so that, as we have seen, α′_{𝔪}(∂/∂z) takes values in a single Ad K^{ℂ}-orbit.

2. α_{𝔨}(∂/∂z) = r(ξ_{d−1}) so that α_{𝔨}(∂/∂z) takes values in \overline{𝔫} ⊕ 𝔥.

3. In view of the equations for ξ we have

\frac{∂ξ_{d}}{∂z} = [ξ_{d}, (r − 1)ξ_{d−1}]

which amounts to a differential relation between α′_{𝔪}and α′_{𝔨}.

These conditions are quite stringent: for example, for the primitive maps into flag manifolds discussed in Section 3.4 they are equivalent to the demand that the framing be a Toda frame.

Now suppose that ϕ does have such a framing with α doubly periodic. The argument of Theorem 5.1 can easily be adapted to produce the necessary ξ : ℝ^{2} → Λ_{d}. Indeed, this is the approach used in [7, 24] to treat primitive maps of flag manifolds since, in this case, by Corollary 3.15, a doubly periodic map with Toda frame has doubly periodic Toda frame. However, in the general case, we know of no direct method to equip a doubly periodic primitive harmonic map with a doubly periodic frame of the right kind. Hence the need to proceed indirectly.

This said, our proof is a simple adaptation of the Symes method we have been developing. So let ϕ : ℝ^{2} → G/K be a primitive harmonic map of G-finite type with ψ = ι ∘ ϕ : ℝ^{2} → G. We know from Sections 5.2 and 5.3 that the G-finite type condition means that, for some d ≡ 1 mod k, there is ξ_{o} ∈ Ω_{d} ∩ Λ𝔤_{τ} so that the map g : ℝ^{2} → ΛG^{ℂ} z ↦ exp(2iλ^{d−1}ξ_{o}) has a unique factorisation

g = ab (5.7)

a : ℝ^{2} → ΩG, b : ℝ^{2} → Λ_{+}G^{ℂ} with

a_{ω} = ψ.

On the other hand, we may view g as a map ℝ^{2} → ΛG_{τ}^{ℂ} and use the Iwasawa decomposition of Theorem 4.1 to write

g = ãb̃

with ã : ℝ^{2} → ΛG_{τ}, b̃ : ℝ^{2} → Λ_{+}G_{τ}^{ℂ}. We know from Theorem 4.3 that ã is an extended framing of a primitive harmonic map of finite type given by

π ∘ ev_{1} ∘ ã

and it is our contention that this map is ϕ. For this, it suffices to show that

ι ∘ π ∘ ev_{1} ∘ ã = ψ.

From the definition of the Cartan embedding, we get

ι ∘ π ∘ ev_{1} ∘ ã = τ(ã_{1})ã_{1}^{−1} = ã_{ω}ã_{1}^{−1}

where we have also used the symmetry ã_{ωλ} = τ ∘ ã_{λ}. However, by the uniqueness of the decomposition (5.7), we have

a = ãã_{1}^{−1}; b = ã_{1}b̃

so that

ã_{ω}ã_{1}^{−1} = a_{ω} = ψ.

We have therefore proved:

Theorem 5.8 A primitive harmonic map ℝ^{2} → G/K of G-finite type is of finite type.

5.5 Coda: finite type vs. finite uniton number

There is another interesting class of harmonic maps from a Riemann surface to a Lie group G: these are the harmonic maps with finite uniton number discovered by Uhlenbeck [55]. The theory of these maps is fundamental to the twistorial approach to harmonic 2-spheres in symmetric spaces (see [15, 55, 62] and [27, 59] in this volume).

We conclude this article with a simple example which demonstrates that the class of finite type harmonic maps is essentially disjoint from that of maps with finite uniton number.

First let us recall Uhlenbeck’s theory: let ϕ : M → G be a harmonic map of a Riemann surface and suppose that ϕ admits an extended solution Φ : M → ΩG (which is certainly true if M is simply connected). The extended solution is unique up to left multiplication by an constant element of ΩG and we say that ϕ has finite uniton number if this constant can be chosen so that Φ takes values in the subspace of (Laurent) polynomial loops in ΩG of some fixed degree:

Φ_{λ}(z) = ∑_{|m|≤d} λ^{m}T_{m}(z), (5.8)

for z ∈ M. Of course, to make sense of (5.8) we must view G as a matrix group.

Such polynomial loops admit a decomposition into “linear” factors which gives rise to a decomposition of the harmonic map into flag factors as described by Wood [59] in this volume.

The main result of Uhlenbeck is that any extended solution on a compact M can be normalised to take values in such a space of polynomial loops. As a consequence, any harmonic map S^{2} → G has finite uniton number.

The situation for finite type harmonic maps is completely different: one can prove

Theorem 5.9 A finite type harmonic map ϕ : ℝ^{2} → G with ϕ^{−1}∂ϕ/∂z semisimple does not have finite uniton number.

Corollary 5.10 A doubly periodic harmonic map ϕ : ℝ^{2} → G with ϕ^{−1}∂ϕ/∂z taking values in a semisimple orbit does not have finite uniton number.

Note that a harmonic map ϕ : S^{2} → G has ϕ^{−1}∂ϕ/∂z nilpotent by the usual holomorphic differentials argument.

The proof of Theorem 5.9 would take us too far afield so we shall content ourselves with presenting a simple example which turns out to contain the heart of the matter.

Let A ∈ 𝔤^{ℂ} and suppose that [A, \overline{A}] = 0 (from which it follows that A is semisimple). Set ξ_{A} = (λ^{−1} − 1)A + (λ − 1)\overline{A} ∈ Ω_{1} and consider the geodesic g : ℝ^{2} → ΛG^{ℂ} given by

g(z) = exp(zξ_{A}).

It is easy to see that the extended solution obtained by factorising g is given by

a(z) = exp((λ^{−1} − 1)zA + (λ − 1)\overline{zA})

so that the corresponding harmonic map ϕ : ℝ^{2} → G of finite type is given by

ϕ(z) = exp(−2zA − 2\overline{zA}).

Thus ϕ is a product of geodesics with commuting generators. The corresponding map ξ : ℝ^{2} → Ω_{1} is constant so that we have a stationary point for the Hamiltonian flows.

Suppose now that ϕ has finite uniton number. Then there is a constant γ ∈ ΩG for which γa = Φ has polynomial dependence on λ. But fixing z_{1} ≠ z_{2} ∈ ℝ^{2}, this implies that

Φ(z_{1})^{−1}Φ(z_{2}) = a(z_{1})^{−1}a(z_{2})

is a Laurent polynomial in λ. But

a(z_{1})^{−1}a(z_{2}) = exp((λ^{−1} − 1)(z_{2} − z_{1})A + (λ − 1)\overline{(z_{2} − z_{1})A})

which has an essential singularity at 0.

Harmonic maps via Adler–Kostant–Symes theory 49

Bibliography

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

[2] A. Bahy-El-Dien and J.C. Wood, The explicit construction of all harmonic twospheres in G_{2}(R^{n}), J. reine angew. Math. 398 (1989), 36–66.

[3] J. Barbosa, On minimal immersions of S^{2} into S^{2m}, Trans. Amer. Math. Soc. 210 (1975), 75–106.

[4] M. Black, Harmonic maps into homogeneous spaces, Pitman Res. Notes in Math., vol. 255, Longman, Harlow, 1991.

[5] A.M. Bloch, H. Flaschka, and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra., Duke Math. J. 61 (1990), 41–65.

[6] A.I. Bobenko, All constant mean curvature tori in R^{3}, S^{3}, H^{3} in terms of thetafunctions, Math. Ann. 290 (1991), 209–245.

[7] J. Bolton, F. Pedit, and L.M. Woodward, Minimal surfaces and the affine Toda field model, preprint, 1993.

[8] J. Bolton and L.M. Woodward, The affine Toda equations and minimal surfaces, this volume.

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

[10] , Lie groups and twistor spaces, Duke Math. J. 52 (1985), 223–261.

[11] F.E. Burstall, Harmonic tori in spheres and complex projective spaces, in preparation.

[12] , Twistor methods for harmonic maps, Differential Geometry (V.L. Hansen, ed.), Lect. Notes in Math., vol. 1263, Springer, Berlin, Heidelberg, New York, 1987, pp. 55–96.

[13] , Recent developments in twistor methods for harmonic maps, Harmonic mappings, twistors and σ-models (P. Gauduchon, ed.), World Scientific, Singapore, 1988, pp. 158–176.

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

[15] F.E. Burstall and J.H. Rawnsley, Twistor theory for Riemannian symmetric spaces with applications to harmonic maps of Riemann surfaces, Lect. Notes in Math., vol. 1424, Springer, Berlin, Heidelberg, New York, 1990.

[16] E. Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds, Presses de Université de Montréal, Montréal, 1967, pp. 59–81.

[17] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian Geometry, North Holland, Amsterdam, 1975.

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

[19] J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass representation of harmonic maps into symmetric spaces, in preparation.

50 F.E. Burstall and F. Pedit

[20] J. Dorfmeister and H. Wu, Constant mean curvature surfaces and loop groups, J. reine angew. Math., to appear.

[21] J. Eells and S.M. Salamon, Twistorial construction of harmonic maps of surfaces into four manifolds, Ann. Scuola Norm. Sup. Pisa 12 (1985), 589–640.

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

[23] D. Ferus and F. Pedit, S^{1}-equivariant minimal tori in S^{4} and S^{1}-equivariant Willmore tori in S^{3}, Math. Z. 204 (1990), 269–282.

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

[25] K. Fujii, A relation between instantons of Grassmannian σ-models and Toda equations I, II, III, Yokohama preprints, 1992.

[26] J.-L. Gervais and Y. Matsuo, W -geometries, Phys. Lett. B 274 (1992), 309–316.

[27] M.A. Guest and Y. Ohnita, Loop group actions on harmonic maps and their applications, this volume.

[28] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Camb. Univ. Press, Cambridge, 1984.

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

[30] N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987), 59–126.

[31] , Harmonic maps from a 2-torus to the 3-sphere, J. Diff. Geom. 31 (1990), 627–710.

[32] W.Y. Hsiang and H.B. Lawson, Minimal submanifolds of low cohomogeneity, J. Diff. Geom. 5 (1971), 1–38.

[33] J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer, New York, Heidelberg, London, 1972.

[34] P.Z. Kobak, Twistors, nilpotent orbits and harmonic maps, this volume.

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

[36] , The solution to a generalized Toda lattice and representation theory, Adv. Math. 34 (1979), 195–338.

[37] O. Kowalski, Generalized symmetric spaces, Lect. Notes in Math., vol. 805, Springer, Berlin, Heidelberg, New York, 1980.

[38] I. McIntosh, Infinite dimensional Lie groups and the two-dimensional Toda lattice, this volume.

[39] , Global solutions of the elliptic 2D Toda lattice, preprint, 1992.

[40] A.M. Perelomov, Integrable systems of classical mechanics and Lie algebras, Birkhäuser, Basel, 1990.

[41] U. Pinkall and I. Sterling, On the classification of constant mean curvature tori, Ann. of Math. 130 (1989), 407–451.

[42] K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976), 207–221.

Harmonic maps via Adler–Kostant–Symes theory 51

[43] A.N. Pressley and G. Segal, Loop Groups, Oxford Math. Monographs, Clarendon Press, Oxford, 1986.

[44] J.H. Rawnsley, Noether’s theorem for harmonic maps, Diff. Geom. Methods in Math. Phys. (S. Sternberg, ed.), Reidel, Dortrecht, Boston, London, 1984, pp. 197–202.

[45] , Harmonic 2-spheres, Quantum Theories and Geometry (M. Cahen and M. Flato, eds.), Kluwer Acad. Publ., Dordrecht, 1988, pp. 175–189.

[46] , Twistor theory for Riemannian manifolds, New Developments in Lie Theory and Their Applications (J. Tirao and N. Wallach, eds.), Progress in Mathematics, vol. 105, Birkhäuser, Boston, Basel, Stuttgart, 1992, pp. 115–128.

[47] G.L. Rcheulishvili and M.V. Saveliev, Multidimensional nonlinear systems related to the Grassman manifolds BI, DI, Funktsional. Anal. i Prilozhen 21 (1987), 83–84.

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

[49] S.M. Salamon, Harmonic and holomorphic maps, Geometry Seminar L. Bianchi., Lect. Notes Math., vol. 1164, Springer, Berlin, Heidelberg, New York, 1986, pp. 161–224.

[50] M.V. Saveliev, Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms., Teoret. Mat. Fiz. 60 (1984), 9–23.

[51] G. Segal and G. Wilson, Loop groups and equations of KdV type, Pub. Math. de I.H.E.S. 61 (1985), 5–65.

[52] S. Sternberg, Lectures on differential geometry, Chelsea, New York, 1983.

[53] W. Symes, Systems of Toda type, inverse spectral problems and representation theory, Invent. Math. 159 (1980), 13–51.

[54] K. Uhlenbeck, Equivariant harmonic maps into spheres, Harmonic Maps (R.J. Knill, M. Kalka, and H.C.J. Sealey, eds.), Lect. Notes in Math., vol. 949, Springer, Berlin, Heidelberg, New York, 1982, pp. 146–158.

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

[56] E.B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 488–526, 709.

[57] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523–557.

[58] H.C. Wente, Counterexample to a conjecture of H. Hopf, Pac. J. Math. 121 (1986), 193–243.

[59] J.C. Wood, Harmonic maps into symmetric spaces and integrable systems, this volume.

[60] , Holomorphic differentials and classification theorems for harmonic maps and minimal immersions, Global Riemannian Geometry (T.J. Willmore and N.J. Hitchin, eds.), E. Horwood Series in Math. and its Appl., E. Horwood, Chichester, 1984, pp. 168–175.

[61] , Twistor constructions for harmonic maps, Differential Geometry and Differential Equations (C.H. Gu, ed.), Lect. Notes in Math., vol. 1255, Springer, Berlin, Heidelberg, New York, 1988, pp. 130–159.

52 F.E. Burstall and F. Pedit

[62] , Explicit construction and parametrisation of harmonic two-spheres in the unitary group, Proc. Lond. Math. Soc. 58 (1989), 608–624.

[63] H. Wu, Banach manifolds of minimal surfaces in the 4-sphere, Differential Geometry (R.E. Greene and S.T. Yau, eds.), Proc. Symp. Pure Math., vol. 54, Amer. Math. Soc., Providence RI, 1993, pp. 530–539.

[64] V.E. Zakharov and A.V. Mikhaǐlov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Sov. Phys. JETP 47 (1978), 1017–1027.

[65] 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 (1978), 13–22.