Dressing Orbits of Harmonic Maps

F.E. Burstall and F. Pedit

Introduction

At the heart of the modern theory of harmonic maps from a Riemann surface to a Riemannian symmetric space is the observation that, in this setting, the harmonic map equations have a zero curvature representation [19, 24, 28] and so correspond to loops of flat connections. This fact was first exploited in the mathematical literature by Uhlenbeck in her study [24] of harmonic maps ℝ^{2} → G into a compact Lie group G. Uhlenbeck discovered that harmonic maps correspond to certain holomorphic maps, the extended solutions, into the based loop group ΩG and used this to define an action of a certain loop group on the space of harmonic maps. However, the main focus of [24] was on harmonic maps of a 2-sphere and for these maps the action reduces to an action of a finite-dimensional quotient group (see also [1, 9]).

In another direction, the zero curvature representation has been central to recent progress in the understanding of the harmonic map equations as soliton equations, i.e. as completely integrable Hamiltonian PDE . By solving certain Lax flows on loop algebras a rather complete description of all harmonic tori in symmetric spaces and Lie groups has been obtained. Of particular importance in this approach are the harmonic maps of finite type: these arise from Lax flows on finite dimensional subspaces of loop algebras and correspond to linear flows on Jacobians of certain algebraic curves [2, 3, 4, 8, 10, 18]. Among these harmonic maps, we further distinguish those of semisimple finite type (see Section 1.3 below) which are characterised by a semisimplicity condition on their derivative. Semisimple finite type harmonic maps account for all non-conformal harmonic tori in rank one symmetric spaces of compact type [4], all non-isotropic harmonic tori in spheres and complex projective spaces [3], and all doubly periodic solutions to the abelian affine Toda field equations for simple Lie groups [2].

The purpose of this paper is to describe some interactions between these two approaches. Our starting point is the fact that underlying all of the above results is the existence of Iwasawa type decompositions of the loop groups and algebras concerned. On the one hand, the Lax equations mentioned above arise from an Iwasawa decomposition of certain twisted loop algebras via the Adler–Kostant–Symes scheme [5]. On the other hand, the loop group action of Uhlenbeck is essentially the dressing action arising from the Iwasawa decompositions of the corresponding loop groups [9]. Moreover, a bridge beween these constructions is provided by Symes’s formula for the solution of the Lax equations: in this construction, first applied by Symes [21] to solve the open Toda lattice, projection of certain complex geodesics on a factor in the Iwasawa decomposition yields (extended framings of) harmonic maps. This provides a map from a certain subspace of the loop algebra to the space of harmonic maps which intertwines the dressing and adjoint actions.

First author partially supported by DOE grant DE-FG02 86ER25015, NSF grant DMS-9011083 and EEC contract SC1-0105-C.

Second author supported by NSF grant DMS-9205293.

This set-up is very familiar in soliton theory where the dressing method was first developed. Here the idea is to use the dressing action to construct new solutions from old and in many cases the dressing orbits through trivial or vacuum solutions account for all the solutions one is interested in. It is this theme that we develop in the present paper.

We treat (primitive) harmonic maps ℝ^{2} → G/K where G/K is a (k-)symmetric space. In this context, our vacuum solutions are the maps f^{A} : ℝ^{2} → G/K defined by

f^{A}(z) = exp(zA + \overline{zA})K,

where A is an element of the Lie algebra of G satisfying [A, \overline{A}] = 0. These maps (the geometry of which has recently been studied by Jensen–Liao [11] in the case G/K = ℂ P^{n}) are equivariant with respect to actions of the abelian group generated by A and \overline{A}.

Our main results concern the orbit O_{A} of f^{A} under the dressing action. This orbit is infinitedimensional in contrast to those studied by Uhlenbeck and we show that every harmonic map of semisimple finite type lies in some O_{A}. As a special case, we deduce that every non-isotropic harmonic torus in a sphere or complex projective space is dressing equivalent to a vacuum solution.

A distinctive feature of the orbits O_{A} is that they admit a hierarchy of commuting flows (conservation laws). We show that this hierarchy can be used to characterise the harmonic maps of finite type: a harmonic map in O_{A} is of finite type if and only if its orbit under the hierarchy is finite-dimensional.

In all these results, just as in those of [3, 4], essential use is made of the semisimplicity assumption on the derivative of the harmonic maps. In an appendix, we examine the situation when this assumption is dropped and discover intriguing relationships between this case and harmonic maps of finite uniton number in the sense of Uhlenbeck.

Special cases and partial versions of some of our results already exist in the literature: nonconformal harmonic tori in S^{2}—the Gauss maps of contant mean curvature tori—were studied by Dorfmeister–Wu [7] and a similar analysis of non-superminimal minimal tori in S^{4} was performed by Wu [27]. However, even in these cases, our methods are different and we believe them to be more transparent.

Most of this research was carried out while both authors visited the SFB 288 at TU-Berlin and the first author visited GANG at the University of Massachusetts, Amherst. We would like to thank the members of both institutions for their support and hospitality.

Finally, the first author’s understanding of the matters treated herein was enhanced by conversations with Ian McIntosh and Martin Guest for which he takes this opportunity to thank them.

Notation Throughout this work, when a Lie group is denoted by an upper case letter, its Lie algebra will be denoted by the corresponding lower case gothic letter. Thus G is a Lie group with Lie algebra 𝔤.

1 Primitive harmonic maps and their extended framings

We are going to study primitive harmonic maps of a Riemann surface into a k-symmetric space. Such maps (or, rather, their framings) have a zero-curvature representation and so give rise to maps into a loop group. This construction is fundamental for everything that follows so we begin by reviewing this circle of ideas to establish notation and to give a context for our results.

1.1 Primitive harmonic maps

Let G be a compact semisimple Lie group. A (regular) k-symmetric G-space [14] is a coset space N = G/K where (G^{τ})_{0} ⊂ K ⊂ G^{τ} for some automorphism τ : G → G of finite order k ≥ 2.

Example A 2-symmetric space is just a Riemannian symmetric space of compact type.

In general, the k-symmetric spaces form a large class of reductive homogeneous spaces which include the generalised flag manifolds (that is, G/K where K is the centraliser of a torus).

The automorphism τ induces a ℤ_{k}-grading of 𝔤^{ℂ}:

𝔤^{ℂ} = ∑_{ℓ∈ℤ_{k}} 𝔤_{ℓ},  (1.1)

where, setting ω = e^{2πi/k}, 𝔤_{ℓ} is the ω^{ℓ}-eigenspace of (the derivative of) τ. We have 𝔤_{0} = 𝔨^{ℂ}, \overline{𝔤_{ℓ}} = 𝔤_{−ℓ} and

[𝔤_{j}, 𝔤_{ℓ}] = 𝔤_{j+ℓ},

where all arithmetic is modulo k. In particular, defining 𝔪 ⊂ 𝔤 by 𝔪^{ℂ} = ∑_{ℓ∈ℤ_{k}\{0}} 𝔤_{ℓ}, we have a reductive decomposition:

𝔤 = 𝔨 ⊕ 𝔪.  (1.2)

Example When k = 2, 𝔤_{1} = 𝔤_{−1} = 𝔪^{ℂ} so that [𝔪, 𝔪] ⊂ 𝔨 and (1.2) is the familiar symmetric decomposition.

The decomposition (1.1) induces a G-invariant decomposition of the tangent bundle of N which is non-trivial when k > 2. Indeed, set o = eK ∈ N and let p = g · o ∈ N. Then the map 𝔤 → T_{p}N given by

ξ ↦ \frac{d}{dt}|_{t=0} exp tξ · p

is a surjection with kernel Ad g 𝔨 and so gives an isomorphism between Ad g 𝔪 ⊂ 𝔤 and T_{p}N.

Notation If 𝔩 ⊂ 𝔤 is an Ad K-invariant subspace, we denote by [𝔩] the sub-bundle of N × 𝔤 with fibres given by

[𝔩]_{g·o} = Ad g 𝔩.

With this notation, we identify [𝔪] with T N and have a decomposition

T N^{ℂ} = ∑_{ℓ∈ℤ_{k}\{0}}[𝔤_{ℓ}].

Definition [3] A map f : M → N of a Riemann surface into a k-symmetric space is primitive if df(T^{1,0}M) ⊂ [𝔤_{−1}].

Note that when k = 2, [𝔤_{−1}] = T N^{ℂ} and the primitivity condition is vacuous.

The study of primitive maps can be motivated by the following considerations: firstly, they arise naturally as twistor lifts (prolongations) of certain harmonic maps into Riemannian symmetric spaces [3, 5]. Secondly, there is a close relationship between primitive maps and solutions of the affine Toda field equations, both abelian and non-abelian. In particular, there is an essentially bijective correspondence between certain primitive maps into the full flag manifold G/T (modulo the left action of G) and solutions to the abelian affine Toda field equations [2, 5] (in this case, τ is the Coxeter–Killing automorphism).

In [5], it is shown that primitive maps are harmonic with respect to suitable invariant metrics on G/K. Moreover, it is shown that the structure equations for such maps have the same form as those for harmonic maps into 2-symmetric spaces. This motivates the following definition:

Definition [5] A map f : M → N of a Riemann surface into a k-symmetric space is primitive harmonic if k = 2 and f is harmonic or k > 2 and f is primitive.

1.2 Extended framings

Henceforth, we assume that the Riemann surface M is contractible (in our applications, we shall take M = ℝ^{2}). As a consequence, all maps M → G/K have global framings g : M → G. We will study primitive harmonic maps via their framings.

Let π : G → G/K be the coset projection and f : M → G/K a primitive harmonic map with framing g : M → G, thus f = π ∘ g. Let α = g^{−1}dg be the pull-back by g of the Maurer–Cartan form of G. We have a decomposition of α according to the eigenspace decomposition (1.1) of 𝔤^{ℂ}:

α = ∑_{ℓ∈ℤ_{k}} α_{ℓ}

which, for a primitive harmonic map, reduces to

α = α′_{−1} + α_{0} + α′′_{1},

where α′_{−1} is a 𝔤_{−1}-valued (1, 0)-form and α′′_{1} = \overline{α′_{−1}} [5]. Moreover, the condition that f be primitive harmonic amounts to demanding that the Maurer–Cartan equation for α

dα + \frac{1}{2}[α ∧ α] = 0  (1.3)

decouples into three equations:

dα′_{−1} + [α_{0} ∧ α′_{−1}] = 0  (1.4′_{𝔪})
dα_{0} + \frac{1}{2}[α_{0} ∧ α_{0}] + [α′_{−1} ∧ α′′_{1}] = 0  (1.4_{𝔨})
dα′′_{1} + [α_{0} ∧ α′′_{1}] = 0.  (1.4′′_{𝔪})

(For k = 2, the equations (1.4′_{𝔪}) and (1.4′′_{𝔪}) are the harmonic map equations while, for k > 2, they are the projections of (1.3) onto 𝔤_{−1} and 𝔤_{1}.)

For λ ∈ ℂ^{∗}, define a 𝔤^{ℂ} -valued 1-form by

α_{λ} = λ^{−1}α′_{−1} + α_{0} + λα′′_{1}.

By construction, α_{λ} enjoys the following properties:

1. α_{λ=1} = α.

2. For all λ ∈ ℂ^{∗}, τα_{λ} = α_{ωλ}.

3. For all λ ∈ ℂ^{∗}, \overline{α_{λ}} = α_{1/λ̄} so that α_{λ} is 𝔤-valued when λ ∈ S^{1}.

The crucial observation is that α satisfies the equations (1.4) if and only if

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

for all λ ∈ ℂ^{∗}. Thus, for each λ, we can integrate the Maurer–Cartan equations to obtain a map F_{λ} : M → G^{ℂ}, unique up to left translation by a constant, satisfying F_{λ}^{−1}dF_{λ} = α_{λ}. Moreover, in view of the properties of α_{λ} listed above, we may choose the constants of integration to ensure

1. F_{1} = g.

2. For all λ ∈ ℂ^{∗}, τF_{λ} = F_{ωλ}.

3. For all λ ∈ ℂ^{∗}, \overline{F_{λ}} = F_{1/λ̄} where conjugation is the Cartan involution of G^{ℂ} fixing G. In particular, F_{λ} : M → G when λ ∈ S^{1}.

4. For each p ∈ M, λ ↦ F_{λ}(p) is holomorphic on ℂ^{∗}.

Otherwise said, we have defined a map F from M into the group Λ_{hol}G_{τ} given by

Λ_{hol}G_{τ} = \{g : ℂ^{∗} → G^{ℂ} : g is holomorphic and \overline{g(λ)} = g(1/λ̄), g(ωλ) = τg(λ)\}.

This prompts the following definition:

Definition A map F : M → Λ_{hol}G_{τ} is an extended framing if

F^{−1}dF = λ^{−1}α′_{−1} + α_{0} + λα′′_{1},

with α′_{−1} a (1, 0)-form on M, or equivalently, if λF^{−1}∂F is holomorphic at λ = 0 (here F^{−1}∂F is the (1, 0)-part of F^{−1}dF).

We have seen that any primitive harmonic map f admits an extended framing F such that F_{1} is a framing of f. Conversely, it is clear that when F is an extended framing then F_{1} frames a primitive harmonic map. (In fact, F_{λ} frames a primitive harmonic map for each λ ∈ S^{1}).

This correspondence between primitive harmonic maps and extended framings can be made bijective modulo gauge transformations by imposing base-point conditions. Fix p_{0} ∈ M and let

H = {f : M → G/K: f is primitive harmonic with f(p_{0}) = o}

be the space of based primitive harmonic maps. Similarly, let

E = {F : M → Λ_{hol}G_{τ}: F is an extended framing with F (p_{0}) ∈ K}

be the space of based extended framings (here we identify K with the constant elements of Λ_{hol}G_{τ}). It is then straightforward to see that we have a bijective correspondence

H ≅ E/K

where the gauge group K = C^{∞}(M, K) acts by point-wise multiplication on the right.

With all this in place, our constructions involving primitive harmonic maps will be made at the level of the corresponding extended framings. Before turning to this, however, we pause to briefly describe a class of primitive harmonic maps which will be important in the sequel.

1.3 Primitive harmonic maps of finite type

In [5], building on earlier work of several authors [2, 3, 4, 8, 18], we described a method for constructing primitive harmonic maps ℝ^{2} → G/K from commuting Hamiltonian flows on loop algebras.

The loop algebra in question is Λ𝔤_{τ} given by

Λ𝔤_{τ} = {ξ : S^{1} → 𝔤: τξ(λ) = ξ(ωλ)}.

Any ξ ∈ Λ𝔤_{τ} has a Fourier decomposition

ξ = ∑_{n∈ℤ} λ^{n}ξ_{n}

and we distinguish the finite-dimensional subspaces Λ_{d} ⊂ Λ𝔤_{τ} given by

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

Now fix d ≡ 1 mod k. A polynomial Killing field is a map ξ : ℝ^{2} → Λ_{d} satisfying the Lax equation

dξ = [ξ, (λ^{−1}ξ_{−d} + r(ξ_{1−d})) dz + (λξ_{d} + \overline{r(ξ_{1−d})}) dz̄].  (1.4)

Here z is the usual holomorphic co-ordinate on ℝ^{2} and r : 𝔤_{0} → 𝔤_{0} is a certain linear map constructed from an Iwasawa decomposition of 𝔤_{0} (see [5] for more details).

Polynomial Killing fields simultaneously integrate a pair of commuting Hamiltonian vector fields on Λ_{d} and so there is a unique such having any prescribed value at z = 0. A polynomial Killing field gives rise to a primitive harmonic map because α_{λ} defined by

α_{λ} = (λ^{−1}ξ_{−d} + r(ξ_{1−d})) dz + (λξ_{d} + \overline{r(ξ_{1−d})}) dz̄  (1.5)

satisfies the Maurer–Cartan equations. We may therefore integrate to get an extended framing F with F_{λ}^{−1}dF_{λ} = α_{λ} and so a primitive harmonic map. The primitive harmonic maps that arise in this way are said to be of finite type.

There is a necessary condition for a primitive harmonic map f to be of finite type: the Lax equation (1.4) implies that ξ_{−d} : ℝ^{2} → 𝔤_{−1} must take values in a single Ad K^{ℂ} -orbit so that, from (1.5), we see that, for some, and hence every, framing of f, α′_{−1}(∂/∂z) must take values in a single orbit also. For doubly periodic primitive harmonic maps (that is, those which cover a map of a torus), this condition is almost sufficient:

Theorem 1.1 [5] A doubly periodic primitive harmonic map ℝ^{2} → G/K is of finite type if, for any framing, α′_{−1}(∂/∂z) takes values in an Ad K^{ℂ} -orbit of semisimple elements of 𝔤_{−1}.

In view of this, we make the following definition:

Definition A primitive harmonic map ℝ^{2} → G/K is of semisimple finite type if it admits a polynomial Killing field ξ with ξ_{−d} semisimple.

The primitive harmonic maps of semisimple type include:

(i) All doubly periodic non-conformal harmonic maps of ℝ^{2} into a rank one symmetric space [4].

(ii) Twistor lifts of any doubly periodic non-isotropic harmonic map of ℝ^{2} into a sphere or complex projective space [3] (see also [2, 8]). Here G/K is a flag manifold in most cases.

(iii) Twistor lifts of certain doubly periodic harmonic maps of ℝ^{2} into low dimensional complex Grassmannians and quaternionic projective spaces [22, 23].

2 Dressing actions of loop groups

We have seen that primitive harmonic maps correspond to extended framings M → Λ_{hol}G_{τ}. Various loop groups have non-trivial actions on Λ_{hol}G_{τ} which induce actions on extended framings. In this section we define these groups and collect some elementary facts concerning their generalised Birkhoff factorisations. Using these facts, we will be able to define these actions on extended framings as well as provide a simple construction of extended framings via an analogue of the Symes formula for solutions of the Toda lattice [21].

2.1 Decompositions of loop groups

A k-symmetric space is defined by the following ingredients, which we fix once and for all:

1. A compact semisimple group G.

2. An automorphism τ : G → G of finite order k ≥ 2 with fixed set K.

3. The primitive k-th root of unity ω = e^{2πi/k}.

Moreover, we fix an Iwasawa decomposition of the reductive group K^{ℂ}:

K^{ℂ} = KB

where B is a solvable subgroup of K^{ℂ}. Thus any element k ∈ K^{ℂ} can be uniquely written as a product

k = k_{K}k_{B}

with k_{K} ∈ K, k_{B} ∈ B.

The loop groups of interest to us are spaces of smooth maps from a pair of circles in ℂ to G^{ℂ} which are equivariant with respect to τ and satisfy a reality condition. To define them, we fix 0 < ϵ < 1 and partition the Riemann sphere ℙ^{1} = ℂ ∪ {∞} as follows: let C_{ϵ} and C_{1/ϵ} denote the circles of radius ϵ and 1/ϵ about 0 ∈ ℂ and define open sets by

I_{ϵ} = {λ ∈ ℙ^{1} : |λ| < ϵ}, I_{1/ϵ} = {λ ∈ ℙ^{1} : |λ| > 1/ϵ}, E^{(ϵ)} = {λ ∈ ℙ^{1} : ϵ < |λ| < 1/ϵ}.

Now put I^{(ϵ)} = I_{ϵ} ∪ I_{1/ϵ} and C^{(ϵ)} = C_{ϵ} ∪ C_{1/ϵ} so that ℙ^{1} = I^{(ϵ)} ∪ C^{(ϵ)} ∪ E^{(ϵ)}.

We define the group of smooth maps Λ^{ϵ}G_{τ} by

Λ^{ϵ}G_{τ} = \{g : C^{(ϵ)} → G^{ℂ} : g(ωλ) = τg(λ), \overline{g(λ)} = g(1/λ̄), for all λ ∈ C^{(ϵ)}\}.

Here, again, the conjugation is the Cartan involution of G^{ℂ} which fixes G.

Remark Observe that the reality condition \overline{g(λ)} = g(1/λ̄) implies that g ∈ Λ^{ϵ}G_{τ} is completely determined by its values on C_{ϵ} so that we have an isomorphism between Λ^{ϵ}G_{τ} and the group of τ-equivariant maps C_{ϵ} → G^{ℂ}. In particular, Λ^{ϵ}G_{τ} becomes in this way a complex Lie group. In what follows, we shall use this isomorphism to identify elements of Λ^{ϵ}G_{τ} with their restrictions to C_{ϵ}.

We now define some subgroups of G:

Λ_{E}^{ϵ}G_{τ} = {g ∈ Λ^{ϵ}G_{τ}: g extends holomorphically to g : E^{(ϵ)} → G^{ℂ} },
Λ_{I}^{ϵ}G_{τ} = {g ∈ Λ^{ϵ}G_{τ}: g extends holomorphically to g : I^{(ϵ)} → G^{ℂ} }.

By unique continuation, any element g of these subgroups satisfies the reality and equivariance conditions

\overline{g(λ)} = g(1/λ̄), g(ωλ) = τg(λ),

for all λ in its domain of definition. In particular, g ∈ Λ_{I}^{ϵ}G_{τ} has g(0) ∈ K^{ℂ} and we distinguish the subgroup

Λ_{I,B}^{ϵ}G_{τ} = {g ∈ Λ_{I}^{ϵ}G_{τ}: g(0) ∈ B}.

The main tool in all our constructions is the following Iwasawa type decomposition for the complex loop groups Λ^{ϵ}G_{τ} which is due to McIntosh [16].

Theorem 2.1 Multiplication Λ_{E}^{ϵ}G_{τ} × Λ_{I,B}^{ϵ}G_{τ} → Λ^{ϵ}G_{τ} is a diffeomorphism onto.

Remark The important fact here is the surjectivity of the multiplication which amounts to the assertion that a certain Riemann–Hilbert problem is always solvable. That this is indeed the case is a consequence of the reality conditions we have imposed on our loops.

Remark The limiting case of Theorem 2.1 as ϵ → 1 is the more familiar assertion that a loop S^{1} → G^{ℂ} can be factorised as a product of a loop S^{1} → G and a loop which has a holomorphic extension to {|λ| < 1}. This result is due to Pressley–Segal [20] and was extended to the twisted setting by Dorfmeister–Pedit–Wu [6].

As a consequence of Theorem 2.1, we have a diffeomorphism

Λ_{E}^{ϵ}G_{τ} ≅ Λ^{ϵ}G_{τ}/Λ_{I,B}^{ϵ}G_{τ}

so that Λ^{ϵ}G_{τ} and, particularly, Λ_{I}^{ϵ}G_{τ} acts on Λ_{E}^{ϵ}G_{τ}. To describe this action explicitly, note that any g ∈ Λ^{ϵ}G_{τ} has a unique factorisation

g = g_{E}g_{I}

with g_{E} ∈ Λ_{E}^{ϵ}G_{τ} and g_{I} ∈ Λ_{I,B}^{ϵ}G_{τ}. The action of Λ_{I}^{ϵ}G_{τ} on Λ_{E}^{ϵ}G_{τ} is now given by

g#_{ϵ}h = (gh)_{E},  (2.1)

for g ∈ Λ_{I}^{ϵ}G_{τ}, h ∈ Λ_{E}^{ϵ}G_{τ}. We call this action the dressing action on Λ_{E}^{ϵ}G_{τ}.

Remark These ideas fit into the general framework of dressing actions on Poisson–Lie groups described by Lu–Weinstein [15]: there is a Poisson–Lie structure on Λ_{E}^{ϵ}G_{τ} for which Λ^{ϵ}G_{τ} is the double group and Λ_{I,B}^{ϵ}G_{τ} is the dual group. In this context, our dressing action is precisely the right dressing action of the dual group in the sense of Lu–Weinstein made into a left action in the usual way. We shall return to these matters elsewhere.

In our applications to harmonic maps, we need to factor out the constant loops K in Λ_{E}^{ϵ}G_{τ}. This is compatible with the dressing action:

Lemma 2.2 The dressing action (2.1) descends to an action on Λ_{E}^{ϵ}G_{τ}/K.

Proof First observe that K is stable under the dressing action: if g ∈ Λ_{I}^{ϵ}G_{τ} with g_{0} = g(0) and k ∈ K, we use the Iwasawa decomposition of K^{ℂ} to write

gk = (g_{0}k)_{K} ((g_{0}k)_{B}(k^{−1}g_{0}^{−1}gk))

so that

g#_{ϵ}k = (g_{0}k)_{K} ∈ K.

Now, for h ∈ Λ_{E}^{ϵ}G_{τ}, we have

g#_{ϵ}(hk) = (ghk)_{E} = ((gh)_{E}(gh)_{I}k)_{E} = (g#_{ϵ}h)((gh)_{I}#_{ϵ}k)

and (gh)_{I}#_{ϵ}k = ((gh)_{I}(0)k)_{K} ∈ K. □

To see how these constructions vary with ϵ, note that, for 0 < ϵ < ϵ′ < 1, restriction of the holomorphic extensions provides injections

Λ_{I}^{ϵ′}G_{τ}^{ℂ} ⊂ Λ_{I}^{ϵ}G_{τ}, Λ_{E}^{ϵ}G_{τ} ⊂ Λ_{E}^{ϵ′}G_{τ}^{ℂ}

and similarly, for 0 < ϵ < 1, we have

Λ_{hol}G_{τ} ⊂ Λ_{E}^{ϵ}G_{τ}.

Indeed, it is easy to see that

Λ_{hol}G_{τ} = ⋂_{0<ϵ<1} Λ_{E}^{ϵ}G_{τ}.  (2.2)

The dressing actions are compatible with these inclusions because the following generalisation of a result of Guest–Ohnita [9]:

Proposition 2.3 For 0 < ϵ < ϵ′ < 1, g ∈ Λ_{I}^{ϵ′}G_{τ}^{ℂ} ⊂ Λ_{I}^{ϵ}G_{τ} and h ∈ Λ_{E}^{ϵ}G_{τ} ⊂ Λ_{E}^{ϵ′}G_{τ}^{ℂ}, we have

g#_{ϵ′}h = g#_{ϵ}h ∈ Λ_{E}^{ϵ}G_{τ}.

Proof On C^{(ϵ′)}, we have

g#_{ϵ′}h = gh(gh)_{I^{(ϵ′)}}^{−1}

where the left hand side has a holomorphic extension to E^{(ϵ′)} while, since h ∈ Λ_{E}^{ϵ}G_{τ}, the right hand side has a holomorphic extension to I^{(ϵ′)} ∩ E^{(ϵ)}. It now follows from a theorem of Painlevé that g#_{ϵ′}h has a holomorphic extension to E^{(ϵ′)}∪C^{(ϵ′)}∪(I^{(ϵ′)}∩E^{(ϵ)}) = E^{(ϵ)}. Thus g#_{ϵ′}h ∈ Λ_{E}^{ϵ}G_{τ} while (gh)_{I^{(ϵ′)}} ∈ Λ_{I,B}^{ϵ}G_{τ} and the proposition follows from the uniqueness of the factorisation of Λ^{ϵ}G_{τ}. □

In particular, the action of Λ_{I}^{ϵ′}G_{τ}^{ℂ} preserves each Λ_{E}^{ϵ}G_{τ} for 0 < ϵ < ϵ′ < 1 and so, taking (2.2) into account, we conclude

Corollary 2.4 The action of each Λ_{E}^{ϵ}G_{τ} preserves Λ_{hol}G_{τ} and, for 0 < ϵ < ϵ′ < 1, g ∈ Λ_{I}^{ϵ′}G_{τ}^{ℂ} ∈ Λ_{I}^{ϵ}G_{τ} and h ∈ Λ_{hol}G_{τ}, we have

g#_{ϵ′}h = g#_{ϵ}h.

Notation Henceforth, we simply write g#h for the action on Λ_{hol}G_{τ}.

Remark It follows from Corollary 2.4 that we can take a direct limit as ϵ → 0 and so obtain an action on Λ_{hol}G_{τ} of the group of germs at zero of τ-equivariant maps ℂ → G^{ℂ}. This action is very similar to the one discussed by Uhlenbeck [24] although she considers only the subgroup of rational maps ℙ^{1} → G^{ℂ} which satisfy the reality conditions and are holomorphic at zero.

Finally, we note that, by virtue of Lemma 2.2, we have

Corollary 2.5 The dressing action of each Λ_{I}^{ϵ}G_{τ} descends to an action on Λ_{hol}G_{τ}/K.

2.2 Symes formula: point-wise version

In [21], Symes gave a formula for solutions of the Toda lattice in terms of the projection of a one-parameter subgroup onto one of the factors in an Iwasawa decomposition. We shall see that a similar formula holds for extended framings. We begin by describing the space of generators for these one-parameter subgroups.

The Lie algebras of the loop groups of the previous section are simply the corresponding algebras of τ-equivariant maps C^{(ϵ)} → 𝔤^{ℂ} and are denoted Λ^{ϵ}𝔤_{τ}, Λ_{E}^{ϵ}𝔤_{τ} and so on. Throughout this section, we use the reality condition to identify elements of Λ^{ϵ}𝔤_{τ} and its subalgebras with maps C_{ϵ} → 𝔤^{ℂ}.

Define the subspace Λ_{−1,∞}^{ϵ} ⊂ Λ^{ϵ}𝔤_{τ} by

Λ_{−1,∞}^{ϵ} = {ξ ∈ Λ^{ϵ}𝔤_{τ}: λξ has a holomorphic extension to I_{ϵ}}.

Thus Λ_{−1,∞}^{ϵ} consists of those elements of Λ^{ϵ}𝔤_{τ} which extend meromorphically to I^{(ϵ)} with at most simple poles at 0 and ∞.

Observe that Λ_{−1,∞}^{ϵ} is stable under the adjoint action of Λ_{I}^{ϵ}G_{τ} on Λ^{ϵ}𝔤_{τ}: if g ∈ Λ_{I}^{ϵ}G_{τ} and ξ ∈ Λ_{−1,∞}^{ϵ} then λξ extends holomorphically to I_{ϵ} whence

λ Ad g(ξ) = Ad g(λξ)

does also.

Now define Φ_{ϵ} : Λ_{−1,∞}^{ϵ} → Λ_{E}^{ϵ}G_{τ} by

Φ_{ϵ}(ξ) = (exp ξ)_{E},

where the exponential map is defined point-wise: (exp ξ)(λ) = exp_{G^{ℂ}} ξ(λ).

Arguing as in Proposition 2.3, one proves

Proposition 2.6 For 0 < ϵ < ϵ′ < 1 and ξ ∈ Λ_{−1,∞}^{ϵ′} ⊂ Λ_{−1,∞}^{ϵ}, we have

Φ_{ϵ′}(ξ) = Φ_{ϵ}(ξ) ∈ Λ_{E}^{ϵ}G_{τ}.

In particular, Φ_{ϵ′} has image in ⋂_{ϵ<ϵ′} Λ_{E}^{ϵ}G_{τ} = Λ_{hol}G_{τ}.

Notation In view of this, we shall henceforth simply write Φ for Φ_{ϵ}.

A useful property of Φ is that it essentially intertwines the adjoint and dressing actions of Λ_{I}^{ϵ}G_{τ}:

Proposition 2.7 Let g ∈ Λ_{I}^{ϵ}G_{τ} and ξ ∈ Λ_{−1,∞}^{ϵ}. Then

Φ(Ad g ξ) = g#(Φ(ξ)k),

where k = ((exp ξ)_{I}(0)g(0)^{−1})_{K} ∈ K.

Note that when g(0) ∈ B, k = 1 so that Φ truly intertwines the actions of Λ_{I,B}^{ϵ}G_{τ}.

Proof Using exp Ad g ξ = g exp(ξ)g^{−1}, we have

Φ(Ad g ξ) = (exp Ad g ξ)_{E} = (g exp(ξ)g^{−1})_{E}
= (gΦ(ξ)(exp ξ)_{I}g^{−1})_{E} = (gΦ(ξ)kg̃)_{E}
= g#(Φ(ξ)k)

where g̃ ∈ Λ_{I,B}^{ϵ}G_{τ} and k = ((exp ξ)_{I}(0)g(0)^{−1})_{K} so that exp(ξ)_{I}g^{−1} = kg̃. □

In particular, denoting by \overline{Φ} : Λ_{−1,∞}^{ϵ} → Λ_{hol}G_{τ}/K the composition of Φ with the coset projection, we conclude from Proposition 2.7 together with Corollary 2.5:

Corollary 2.8 \overline{Φ} intertwines the adjoint and (descended) dressing actions of Λ_{I}^{ϵ}G_{τ}:

\overline{Φ}(Ad g ξ) = g#\overline{Φ}(ξ),

for ξ ∈ Λ_{−1,∞}^{ϵ}, g ∈ Λ_{I}^{ϵ}G_{τ}.

2.3 Dressing action on extended framings

The relevance of the results of Section 2.1 to primitive harmonic maps is that the point-wise dressing action of Λ_{I}^{ϵ}G_{τ} on Λ_{hol}G_{τ} induces an action on extended framings that preserves gauge orbits and the base-point condition. We therefore arrive at an action of Λ_{I}^{ϵ}G_{τ} on H:

Proposition 2.9 Let g ∈ Λ_{I}^{ϵ}G_{τ} and F : M → Λ_{hol}G_{τ} be an extended framing. Define g#F : M → Λ_{hol}G_{τ} by

(g#F)(p) = g#(F (p)),

for p ∈ M. Then

(i) g#F is also an extended solution.

(ii) If F is based (that is, F ∈ E) then so is g#F.

(iii) If k ∈ K then

g#(F k) = (g#F)k̃  (2.3)

with k̃ ∈ K.

Thus Λ_{I}^{ϵ}G_{τ} acts on H = E/K.

Proof To see that g#F is an extended frame, write

gF = ab

where a = g#F and b : M → Λ_{I,B}^{ϵ}G_{τ}. Then

a^{−1}da = Ad b(F^{−1}dF − b^{−1}db)

so that

λa^{−1}∂a = Ad b(λF^{−1}∂F − λb^{−1}∂b).

Now all ingredients on the right hand side are holomorphic in λ on I_{ϵ} so that λa^{−1}∂a is also whence a is an extended framing.

Now suppose that F ∈ E so that F (p_{0}) ∈ K. In the proof of Lemma 2.2 we saw that the action of Λ_{I}^{ϵ}G_{τ} preserves K so that (g#F)(p_{0}) = g#(F (p_{0})) ∈ K as required.

Finally, (2.3) is an immediate consequence of Lemma 2.2. □

Remark Proposition 2.9 is a simple generalisation to our setting of results of Uhlenbeck [24] and Bergveldt–Guest [1] for harmonic maps into Lie groups.

2.4 Symes formula for extended framings

We now use the map Φ of Section 2.2 to construct based extended solutions ℝ^{2} → Λ_{hol}G_{τ} from complex lines in Λ_{−1,∞}^{ϵ}. (In all that follows, we base maps of ℝ^{2} at 0.)

For η ∈ Λ_{−1,∞}^{ϵ}, define F^{η} : ℝ^{2} → Λ_{hol}G_{τ} by

F^{η}(z) = Φ(zη) = (exp zη)_{E}.

Proposition 2.10 F^{η} is a based extended framing.

Proof We have

exp(zη) = F^{η}b,

with b : ℝ^{2} → Λ_{I,B}^{ϵ}G_{τ} so that

λ(F^{η})^{−1}∂F^{η} = Ad b(λη dz − λb^{−1}∂b).  (2.4)

Again all ingredients on the right hand side are holomorphic near λ = 0 (since η ∈ Λ_{−1,∞}^{ϵ}) so that λ(F^{η})^{−1}∂F^{η} is also whence F^{η} is an extended framing. Finally, F^{η}(0) = 1 ∈ K so that F^{η} ∈ E. □

Observe that primitive harmonic maps that arise from Proposition 2.10 in this way have the same restriction on their framings as the maps of finite type: indeed, let η_{−1} = (λη)(0) and put b_{0} = b|_{λ=0} : ℝ^{2} → B. Now set λ = 0 in (2.4) to conclude that

(F^{η})^{−1}dF^{η} = λ^{−1}α′_{−1} + α_{0} + λα′′_{1}

with

α′_{−1} = Ad b_{0}(η_{−1}) dz.  (2.5)

Otherwise said, α′_{−1}(∂/∂z) takes values in a single B-orbit.

In fact, all primitive harmonic maps of finite type arise from this construction: let f : ℝ^{2} → G/K be a based primitive harmonic map of finite type with polynomial Killing field ξ : ℝ^{2} → Λ_{d} and let η = λ^{d−1}ξ(0) ∈ Λ_{−1,∞}^{ϵ}. The arguments of [5, Section 4.2] carry over directly to our setting to prove:

1. F^{η} is an extended framing for f.

2. ξ = Ad(F^{η})^{−1}ξ(0).

To summarise:

Proposition 2.11 f : ℝ^{2} → G/K is of finite type if and only if it admits an extended framing of the form F^{η} where, for some d ≡ 1 mod k, η ∈ λ^{d−1}Λ_{d} ⊂ Λ_{−1,∞}^{ϵ}

Finally, we note the following useful fact: viewing the Symes formula as a map

Λ_{−1,∞}^{ϵ} → H,

we see that this map intertwines the adjoint and dressing actions of Λ_{I}^{ϵ}G_{τ}. Indeed, denoting by [F^{η}] the gauge equivalence class of F^{η}, Corollary 2.8 gives

Lemma 2.12 For g ∈ Λ_{I}^{ϵ}G_{τ} and η ∈ Λ_{−1,∞}^{ϵ},

[F^{Ad g η}] = g#[F^{η}].

3 Dressing orbits of vacuum solutions

The main philosophy of the dressing construction [26, 28] is to construct non-trivial solutions of partial differential equations by applying a dressing action of a suitable loop group to a trivial or vacuum solution. In this section, we define vacuum solutions for our problem and show that any primitive harmonic map of semisimple finite type lies in the dressing orbit of such a solution.

Some care must be exercised when developing the notion of a vacuum solution of the primitive harmonic map equations: the simplest harmonic maps are the constant maps but we learn from Lemma 2.2 that the constant element of H is preserved by the action of Λ_{I}^{ϵ}G_{τ} so that dressing will give us nothing new. (Similar remarks apply to the harmonic maps of finite uniton number in the sense of Uhlenbeck [24], see the appendix.) However, if one is guided by the relationship between primitive harmonic maps and Toda fields, one is led to the following class of solutions which correspond to constant solutions of the appropriate Toda field equations.

Definition A vacuum solution is an extended framing of the form F^{η_{A}} where η_{A} = λ^{−1}A with A ∈ 𝔤_{−1} and [A, \overline{A}] = 0.

Notice that it follows from the vanishing of [A, \overline{A}] that A is semisimple.

In this simple case, we can perform the Iwasawa decomposition explicitly:

exp(zη_{A}) = exp(zλ^{−1}A + z̄λ\overline{A}) exp(−z̄λ\overline{A}),

on C_{ϵ} so that

F^{η_{A}}(z) = exp(zλ^{−1}A + z̄λ\overline{A})

and the corresponding primitive harmonic map f : ℝ^{2} → G/K

f(z) = exp(zλ^{−1}A + z̄λ\overline{A})o

is equivariant for actions of the abelian group generated by A and \overline{A}.

Denote by O_{A} ⊂ H the Λ_{I,B}^{ϵ}G_{τ}-orbit of [F^{η_{A}}] so that

O_{A} ≅ Λ_{I,B}^{ϵ}G_{τ}/Γ_{I,B}^{ϵ}

where Γ_{I,B}^{ϵ} is the stabiliser of [F^{η_{A}}]:

Γ_{I,B}^{ϵ} = {g ∈ Λ_{I,B}^{ϵ}G_{τ}: g#[F^{η_{A}}] = [F^{η_{A}}]}.

Remark Let us pause to justify our decision to study O_{A} rather than the slightly larger Λ_{I}^{ϵ}G_{τ}- orbit of [F^{η_{A}}]. We shall see in Section 4 that O_{A} admits a hierarchy of commuting flows which can be used to characterise the primitive harmonic maps of finite type. These flows do not extend to the Λ_{I}^{ϵ}G_{τ}-orbit. Moreover, since Λ_{I}^{ϵ}G_{τ} = KΛ_{I,B}^{ϵ}G_{τ} and the dressing action on H of K is merely that induced by the action of K on G/K, we see that we lose nothing essential by adopting this approach.

We will study O_{A} by using the equivariance of the map η → [F^{η}] : Λ_{−1,∞}^{ϵ} → H to replace the dressing action of Λ_{I,B}^{ϵ}G_{τ} on H by the easier adjoint action on Λ_{−1,∞}^{ϵ}. To accomplish this, we must describe the fibres of this map. We begin with the following observation:

Lemma 3.1 Let ζ, η ∈ Λ_{−1,∞}^{ϵ}. Then [F^{ζ}] = [F^{η}] if and only if (λζ)(0) = (λη)(0) and

(ad η)^{n}ζ ∈ Λ_{I}^{ϵ}𝔤_{τ},  (3.1)

for all n ≥ 1.

Proof [F^{ζ}] = [F^{η}] if and only if F^{ζ} = F^{η}k, for some k ∈ K, and, using the definitions of F^{ζ} and F^{η}, it is straight-forward to see that this is the case precisely when

e(z) := exp(−zζ) exp(zη) ∈ Λ_{I}^{ϵ}G_{τ},

for z ∈ ℝ^{2}. This, in turn, is the same as demanding that

e^{−1}de = (− Ad exp(−zη) ζ + η) dz

be Λ_{I}^{ϵ}𝔤_{τ}-valued, that is,

e^{− ad zη}ζ − η ∈ Λ_{I}^{ϵ}𝔤_{τ},

for all z ∈ ℝ^{2}. Expanding this last in powers of z and comparing coefficients proves the lemma.

□

Applying this to the case where η = η_{A} gives:

Proposition 3.2 [F^{ζ}] = [F^{η_{A}}] if and only if (λζ)(0) = A and [ζ, A] = 0.

Proof Write ζ = ∑_{n≥−1} λ^{n}ζ_{n} on C_{ϵ}. Comparing coefficients of λ in (3.1) gives

(ad A)^{n}ζ_{n−1} = 0,

for all n ≥ 1. However, since A is semisimple, ker(ad A)^{n} = ker ad A whence [ζ, A] = 0 as required. □

Taking Lemma 2.12 into account, we obtain the following characterisation of the fibres of Λ_{−1,∞}^{ϵ} → H over O_{A}:

Proposition 3.3 For ζ ∈ Λ_{−1,∞}^{ϵ}, g ∈ Λ_{I,B}^{ϵ}G_{τ}, [F^{ζ}] = g#[F^{η_{A}}] ∈ O_{A} if and only if (λζ)(0) = Ad g(0) A and

[ζ, Ad g A] = 0.

Proof By Lemma 2.12, [F^{ζ}] = g#[F^{η_{A}}] if and only if [F^{Ad g^{−1}ζ}] = [F^{η_{A}}]. From Proposition 3.2, we see that this is the case precisely when

(λ Ad g^{−1}ζ)(0) = A

and

[Ad g^{−1}ζ, A] = 0,

whence the result. □

As an immediate corollary to this result and Proposition 2.11, we characterise the maps of finite type in O_{A}:

Corollary 3.4 g#[F^{η_{A}}] is of finite type if and only if, for some d ≡ 1 mod k, there is ξ ∈ Λ_{d} such that ξ_{−d} = Ad g(0) A and

[ξ, Ad g A] = 0.

For example, the vacuum solutions are themselves of finite type: take ξ = λ^{−1}A + λ\overline{A} ∈ Λ_{1}. These solutions have constant polynomial Killing fields and so correspond to fixed points of the Hamiltonian flows on Λ_{1}.

On the other hand, it is a surprisingly simple matter to construct solutions in O_{A} which are not of finite type as the following example shows:

Example Consider the Riemann sphere as a 2-symmetric space so that G = SU(2), K = S^{1} and B consists of diagonal matrices in SL(2, ℂ) of the form

(\begin{matrix}r & 0 \\ 0 & r^{−1} \end{matrix}) ,

where r > 0. In this case, 𝔤_{0} consists of trace zero diagonal matrices and 𝔤_{1} = 𝔤_{−1} of off-diagonal matrices. We take

A = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) ∈ 𝔤_{1}

so that A = \overline{A} and consider the dressing action of B ⊂ Λ_{I,B}^{ϵ}G_{τ} on the vacuum solution [F^{η_{A}}]. For b ∈ B, we know from Corollary 3.4 that b#[F^{η_{A}}] is of finite type if and only if we can find d ≡ 1 mod k and ξ ∈ Λ_{d} such that ξ_{−d} = Ad b A and

[ξ, Ad b A] = 0.

Writing ξ = ∑_{|n|≤d} λ^{n}ξ_{n}, this would force each [ξ_{n}, Ad b A] to vanish and, in particular, since

\overline{ξ_{−d}} = ξ_{d},

we must have

[\overline{Ad b A}, Ad b A] = 0.

However, when

b = (\begin{matrix}r & 0 \\ 0 & r^{−1} \end{matrix})

we have

[\overline{Ad b A}, Ad b A] = (\begin{matrix}r^{4} − r^{−4} & 0 \\ 0 & r^{−4} − r^{4} \end{matrix}) ≠ 0,

unless r = 1. Thus no b#[F^{η_{A}}] is of finite type unless b = 1. This illustrates the complexity of even the simplest case of the dressing action.

We are now in a position to identify the stabiliser group Γ_{I,B}^{ϵ}:

Theorem 3.5 Γ_{I,B}^{ϵ} is the centraliser in Λ_{I,B}^{ϵ}G_{τ} of λ^{−1}A ∈ Λ_{−1,∞}^{ϵ}:

Γ_{I,B}^{ϵ} = {g ∈ Λ_{I,B}^{ϵ}G_{τ}: Ad g A = A}.

Proof By Proposition 3.3, g ∈ Γ_{I,B}^{ϵ} if and only if Ad g(0) A = A and [Ad g A, A] vanishes. Thus Ad g A takes values in Ad G^{ℂ} A ∩ ker ad A. On the other hand, since A is semisimple,

𝔤^{ℂ} = ker ad A ⊕ [A, 𝔤^{ℂ}]

from which it follows that Ad G^{ℂ} A intersects ker ad A transversely at A so that A is an isolated point of Ad G^{ℂ} A ∩ ker ad A. The continuity of λ ↦ Ad g(λ) A on the connected set I_{ϵ} now guarantees that Ad g(λ) A = A for all λ ∈ I_{ϵ}. □

We conclude from this theorem that O_{A} is diffeomorphic to the adjoint orbit of λ^{−1}A in Λ_{−1,∞}^{ϵ}. In Section 4 we shall make use of this fact to define a hierarchy of commuting flows on O_{A}.

Let us now turn to the main theorem of this section and prove that all primitive harmonic maps of semisimple finite type lie in some O_{A}. We begin with a lemma which was proved in the unpublished thesis of I. McIntosh. For completeness of exposition, we shall give a proof here.

Lemma 3.6 Let X ∈ 𝔤_{−1} be semisimple. Then there is A ∈ Ad B X with [A, \overline{A}] = 0.

Proof We begin by finding an element in Ad K^{ℂ} X with the desired property. For this, let ∥ . ∥ denote the norm on 𝔤_{−1} induced by the Killing inner product ( .) on 𝔤. Since X is semisimple, Ad K^{ℂ} X is closed in 𝔤_{−1} [25] so that the restriction of ∥ . ∥^{2} to Ad K^{ℂ} X attains a minimum at some Y ∈ Ad K^{ℂ} X. Now, for χ ∈ \sqrt{−1}𝔨, we have

0 = \frac{d}{dt}|_{t=0} ∥ Ad exp tχ Y ∥^{2} = ([χ, Y], \overline{Y}) − (Y, [χ, \overline{Y}]) = 2(χ, [Y, \overline{Y}]).

However, [Y, \overline{Y}] ∈ \sqrt{−1}𝔨 and the Killing form is definite there so that [Y, \overline{Y}] vanishes as required. Thus, for some k ∈ K^{ℂ}, we have Ad k X = Y. Now put A = Ad k_{K}^{−1}Y so that A = Ad k_{B} X and [A, \overline{A}] = Ad k_{k}^{−1} [Y, \overline{Y}] = 0. □

With this in hand, we prove:

Theorem 3.7 Let η = ∑_{n≥−1} λ^{n}η_{n} ∈ Λ_{−1,∞}^{ϵ′} with η_{−1} semisimple. Then there is ϵ ≤ ϵ′, g ∈ Λ_{I,B}^{ϵ}G_{τ} and a vacuum solution F^{η_{A}} such that

g#[F^{η}] = [F^{η_{A}}].

In particular, since λ^{d−1}Λ_{d} ⊂ Λ_{−1,∞}^{ϵ′} for all 0 < ϵ′ < 1, we have

Corollary 3.8 Any primitive harmonic map of semisimple finite type is in the Λ_{I,B}^{ϵ}G_{τ}-orbit of a vacuum solution for some 0 < ϵ < 1.

Let us turn to the proof of Theorem 3.7: first, in view of Lemma 3.6, we can find A ∈ Ad B η_{−1} such that [A, \overline{A}] = 0 and after dressing by an element of B, we may assume that η_{−1} = A. By Proposition 3.3, it now suffices to find g ∈ Λ_{I,B}^{ϵ}G_{τ}, some 0 < ϵ ≤ ϵ′, such that

Ad g(0) A = A, [A, Ad g η] = 0.

We shall construct g via the Inverse Function Theorem: since A is semisimple,

𝔤^{ℂ} = ker ad A ⊕ [A, 𝔤^{ℂ}]

and we define ϕ : ker ad A ⊕ [A, 𝔤^{ℂ}] → 𝔤^{ℂ} by

ϕ(x, y) = Ad exp(y) x.

Observe that ϕ is equivariant in the following sense:

ωτϕ(x, y) = ϕ(ωτx, τy),  (3.2)

for all (x, y) ∈ ker ad A ⊕ [A, 𝔤^{ℂ}].

Differentiating ϕ at (A, 0) gives

d_{(A,0)}ϕ(v, w) = v + [w, A],

for (v, w) ∈ ker ad A ⊕ [A, 𝔤^{ℂ}], so that d_{(A,0)}ϕ is an isomorphism. By the holomorphic Inverse Function Theorem, there are open neighbourhoods Ω_{1} of (A, 0) and Ω_{2} of A such that ϕ : Ω_{1} → Ω_{2} is a biholomorphism. Moreover, since (A, 0) is fixed by the order k linear automorphism T : (x, y) → (ωτx, τy), we may assume, shrinking Ω_{1} if necessary, that Ω_{1} is T -stable.

Let (ψ_{1}, ψ_{2}) = ϕ^{−1} : Ω_{2} → Ω_{1} so that, for χ ∈ Ω_{2},

χ = Ad(exp ψ_{2}(χ))ψ_{1}(χ),

or, equivalently,

Ad exp(−ψ_{2}(χ)) χ = ψ_{1}(χ) ∈ ker ad A.  (3.3)

From (3.2) and the T -stability of Ω_{1} we deduce that ψ_{2} has the following equivariance property:

ψ_{2}(ωτχ) = τψ_{2}(χ),  (3.4)

for all χ ∈ Ω_{2}.

Since η ∈ Λ_{−1,∞}^{ϵ′}, λη is holomorphic on I_{ϵ′} with (λη)(0) = A so we can find 0 < ϵ ≤ ϵ′ such that C_{ϵ} ∪ I_{ϵ} ⊂ (λη)^{−1}(Ω_{2}). We may therefore define g : C_{ϵ} ∪ I_{ϵ} → G^{ℂ} by

g(λ) = exp(−ψ_{2}(λη(λ))).

By construction g is holomorphic on I_{ϵ} and g(0) = exp(−ψ_{2}(A)) = 1 ∈ B so that

Ad g(0) A = A.

Moreover, from (3.3), for λ ∈ C_{ϵ}, we have

Ad g(λ) η(λ) = λ^{−1} Ad exp(−ψ_{2}(λη(λ))) λη(λ) = λ^{−1}ψ_{1}(λη(λ)) ∈ ker ad A

so that

[A, Ad g η] = 0

on C_{ϵ}. Thus g will define our desired element of Λ_{I,B}^{ϵ}G_{τ} so long as it satisfies the equivariance condition g(ωλ) = τg(λ). For this, recall that η(ωλ) = τη(λ) so that, using (3.4),

g(ωλ) = exp(−ψ_{2}(ωλη(ωλ))) = exp(−ψ_{2}(ωτλη(λ)))
= exp(−τψ_{2}(λη(λ))) = τg(λ)

as required. This completes the proof of Theorem 3.7.

Remark Corollary 3.8 is an extension of results of Dorfmeister–Wu [7] and Wu [27]. In those papers, a similar result was proved by different methods for G = SU(2) and G = SO(5), respectively, with k-symmetric structure given by the Coxeter–Killing automorphism. Geometrically, these cases correspond to non-conformal harmonic maps into S^{2} (i.e. Gauss maps of constant mean curvature surfaces in ℝ^{3}) and (twistor lifts of) minimal non-superminimal maps into S^{4}. In view of [3, 4], the present result accounts for all harmonic non-isotropic 2-tori in any sphere or complex projective space.

To summarise: our results give a rather complete picture of the dressing orbits through primitive harmonic maps [F^{η}] with (λη)(0) semisimple. Any non-constant map of this type lies in some O_{A} and each such orbit is infinite-dimensional. Moreover, if A and A′ both generate vacuum solutions and lie in the same Ad B-orbit, we clearly have

O_{A} = O_{A′}.

Finally, for w ∈ ℂ^{∗}, we can scale the parameter z on ℝ^{2} to identify O_{A} and O_{wA}. We therefore conclude that the set of essentially different dressing orbits of vacuum solutions is parametrised by the projective Ad B-orbits of semisimple elements of 𝔤_{1}. This is a finite-dimensional family.

We conclude this section with an application of Theorem 3.7 which provides a simplified version of Lemma 3.1. Recall that a semisimple A ∈ 𝔤^{ℂ} is regular if its centraliser ker ad A is abelian.

Proposition 3.9 Let ζ, η ∈ Λ_{−1,∞}^{ϵ} with (λζ)(0) = (λη)(0) regular semisimple. Then [F^{ζ}] = [F^{η}] if and only if [ζ, η] = 0.

Proof In view of Lemma 3.1, only the forward implication requires proof. So suppose that [F^{ζ}] = [F^{η}]. By Theorem 3.7, there is a vacuum solution [F^{η_{A}}] and g ∈ Λ_{I}^{δ}G_{τ}^{C}, for 0 < δ ≤ ϵ, such that

[F^{ζ}] = [F^{η}] = g#[F^{η_{A}}].

Thus, by Proposition 3.3, (λζ)(0) = Ad g(0) A, whence A is regular semisimple, and

[ζ, Ad g A] = [η, Ad g A] = 0.

Thus Ad g^{−1} ζ and Ad g^{−1} η take values in ker ad A and so commute. Thus [ζ, η] = 0 as required. □

Example When τ is the Coxeter–Killing automorphism of a simple group G, then any semisimple element in 𝔤_{−1} is regular [12].

4 Higher flows

It is characteristic of “completely integrable” systems of partial differential equations that the solution set admits a hierarchy of commuting flows. In this section, we shall define such a hierarchy on each dressing orbit O_{A} and characterise the primitive harmonic maps of finite type in O_{A} as precisely those whose orbit under these flows is finite-dimensional.

Notation We denote by [g] the coset of g in Λ_{I,B}^{ϵ}G_{τ}/Γ_{I,B}^{ϵ}.

Recall from Section 3 that we have a diffeomorphism Λ_{I,B}^{ϵ}G_{τ}/Γ_{I,B}^{ϵ} ≅ O_{A} given by

[g] ↦ g#[F^{η_{A}}].

In view of the identification Λ_{I,B}^{ϵ}G_{τ} ≅ Λ_{E}^{ϵ}G_{τ}\Λ^{ϵ}G_{τ}, we have a right action of Λ^{ϵ}G_{τ} on Λ_{I,B}^{ϵ}G_{τ} given by

(g, h) ↦ (gh)_{I}.

We now identify an abelian subgroup of Λ^{ϵ}G_{τ} for which this action descends to one on O_{A}. For this, let 𝔷_{A} ⊂ 𝔤^{ℂ} denote the centre of the centraliser of A and define Z ⊂ Λ^{ϵ}𝔤_{τ} by

Z = {ζ ∈ Λ^{ϵ}𝔤_{τ}: for some N ∈ ℕ, ζ(λ) = ∑_{n≥−N} λ^{n}ζ_{n}, for λ ∈ C_{ϵ}, with ζ_{n} ∈ 𝔷_{A} for all n.}

Moreover, let Z ⊂ Λ^{ϵ}G_{τ} be the abelian subgroup obtained by exponentiating Z.

Since A is semisimple, the centraliser of A in Ad G^{ℂ} is connected [13, Lemma 5] from which it follows that Z commutes with Γ_{I,B}^{ϵ}. Thus, for exp ζ ∈ Z and [g] = [g′] ∈ O_{A}, we have g = g′γ, some γ ∈ Γ_{I,B}^{ϵ}, so that

(g exp ζ)_{I} = (g′γ exp ζ)_{I} = (g′ exp(ζ)γ)_{I} = (g′ exp ζ)_{I}γ

whence

[(g exp ζ)_{I}] = [(g′ exp ζ)_{I}]

and we have proved:

Proposition 4.1 Z acts on O_{A} ≅ Λ_{I,B}^{ϵ}G_{τ}/Γ_{I,B}^{ϵ} by

exp ζ · [g] = [(g exp ζ)_{I}].

Remark It is easy to see that Z ∩ Γ_{I,B}^{ϵ} = Z ∩ Λ_{I,B}^{ϵ}G_{τ} acts trivially on O_{A}. With a little more work, one can also show that any z ∈ Z \ Λ_{I}^{ϵ}G_{τ} acts non-trivially on O_{A}. In particular, when 𝔷_{A} ∩ 𝔤_{0} = {0} (which is the case, for instance, for abelian Toda fields), we deduce that there is an effective action of Z/Z ∩ Λ_{I,B}^{ϵ}G_{τ} on O_{A}.

Our main result concerning the action of Z is the following characterisation of the primitive harmonic maps of finite type in O_{A}:

Theorem 4.2 [F] ∈ O_{A} is of finite type if and only if the Z-orbit of [F] is finite-dimensional.

Remark In their study of the sinh-Gordon equation, Dorfmeister–Wu prove this result by different methods for the case G = SU(2) equipped with the Coxeter–Killing automorphism.

We break up our proof of Theorem 4.2 into a sequence of steps. We begin with a simple technical lemma:

Lemma 4.3 Let d ≥ 1 ∈ ℕ and suppose that ζ ∈ Λ^{ϵ}𝔤_{τ} with λ^{d}ζ holomorphic on I_{ϵ}. Then ζ_{E} ∈ Λ_{d} and

(ζ_{E})_{−d} = (λ^{d}ζ)(0).

Proof On C_{ϵ}, we can write ζ = ∑_{n≥−d} λ^{n}ζ_{n} and it is easy to see that

ζ_{E} = ∑_{−d≤n≤−1} λ^{n}ζ_{n} + (ζ_{0})_{𝔨}+ ∑_{1≤n≤d} λ^{n}\overline{ζ_{−n}}

from which the lemma follows immediately. □

We can now prove one part of the theorem:

Proposition 4.4 If [F] = g#[F^{η_{A}}] ∈ O_{A} has finite-dimensional Z-orbit then [F] is of finite type.

Proof According to Corollary 3.4, it suffices to find d ≡ 1 mod k and ξ ∈ Λ_{d} such that ξ_{−d} = Ad g(0) A and

[ξ, Ad g A] = 0.

By hypothesis, the map Z → T_{[F]}O_{A} given by

ζ ↦ \frac{d}{dt}|_{t=0} exp tζ · [g]

has finite-dimensional image. We use left translation by g to identify T_{[F]}O_{A} with T_{[F^{η_{A}}]}O_{A} ≅ Λ_{I,𝔟}^{ϵ}𝔤_{τ}/G_{I,B} (here G_{I,B} is the Lie algebra of Γ_{I,B}^{ϵ}) and then an easy calculation shows that the resulting map Z → Λ_{I,𝔟}^{ϵ}𝔤_{τ}/G_{I,B} is given by

ζ ↦ Ad g^{−1} (Ad g ζ)_{I} mod G_{I,B}.  (4.1)

Let N ⊂ Z denote the kernel of the linear map (4.1). Since N has finite codimension, there is a monic polynomial p such that ζ = p(λ^{−k})λ^{−1}A ∈ Z lies in N so that

Ad g^{−1} (Ad g ζ)_{I} ∈ G_{I,B}

or, equivalently,

[Ad g^{−1} (Ad g ζ)_{I}, A] = 0.

We therefore see that [(Ad g ζ)_{I}, Ad g A] = 0. On the other hand, since ζ ∈ Z, we have [ζ, A] = 0 and we can conclude that

[(Ad g ζ)_{E}, Ad g A] = 0.  (4.2)

Now set d = k deg p + 1 ≡ 1 mod k and observe that λ^{d}ζ is holomorphic (indeed polynomial) on I_{ϵ} with (λ^{d}ζ)(0) = A. It follows that λ^{d} Ad g ζ is holomorphic on I_{ϵ} with (λ^{d} Ad g ζ)(0) = Ad g(0) A so that, from Lemma 4.3, we see that ξ = (Ad g ζ)_{E} ∈ Λ_{d} with ξ_{−d} = Ad g(0) A. This, taken together with (4.2), establishes the proposition. □

Remark In case that G = SO(5) equipped with the Coxeter–Killing automorphism, this proposition was proved by Wu [27] using different methods.

It remains to prove the converse of Proposition 4.4. For this, we use an argument inspired by ideas of McIntosh [17]. Fix d ≡ 1 mod k and let O_{A}^{(d)} consist of those maps in O_{A} which admit a polynomial Killing field ℝ^{2} → Λ_{d}. O_{A}^{(d)} is the image under Λ_{−1,∞}^{ϵ} → H of a subvariety of λ^{d−1}Λ_{d} and so is finite-dimensional. With this in mind, the proof of Theorem 4.2 is completed by the following

Proposition 4.5 O_{A}^{(d)} is Z-stable.

Proof Let [g] ∈ O_{A}^{(d)} so that there is ξ ∈ Λ_{d} such that (λ^{d}ξ)(0) = Ad g(0) A and

[ξ, Ad g A] = 0.

Let ζ ∈ Z and set

ξ̂ = Ad(g exp ζ)_{I} Ad g^{−1} ξ = Ad(exp Ad g ζ)_{I} ξ.

Clearly we have

[ξ̂, Ad(g exp ζ)_{I} A] = [Ad g^{−1} ξ, A] = 0.

It therefore suffices to show that ξ̂ ∈ Λ_{d} and that ξ̂_{−d} = Ad(g exp ζ)_{I}(0) A to conclude that exp ζ · [g] ∈ O_{A}^{(d)}.

For this, first note that since Ad g^{−1} ξ centralises A,

[ζ, Ad g^{−1} ξ] = 0 whence

Ad(g exp ζ) Ad g^{−1} ξ = Ad g Ad g^{−1} ξ = ξ

so that

Ad(exp Ad g ζ)_{E} Ad(exp Ad g ζ)_{I} ξ = ξ

from which we conclude that

ξ̂ = Ad(exp Ad g ζ)_{I} ξ = Ad(exp Ad g ζ)_{E}^{−1} ξ ∈ Λ_{E}^{ϵ}𝔤_{τ}.

On the other hand,

λ^{d}ξ̂ = Ad(g exp ζ)_{I} Ad g^{−1} λ^{d}ξ

which is holomorphic on I_{ϵ} and

(λ^{d}ξ̂)(0) = Ad(g exp ζ)_{I}(0) Ad g^{−1}(0) ξ_{−d} = Ad(g exp ζ)_{I}(0) A.

It now follows from Lemma 4.3 that ξ̂ has the required properties and the proof is complete. □

Appendix: nilpotency of η_{−1} and finite uniton number

In Section 3, we saw that the primitive harmonic maps [F^{η}] with (λη)(0) semisimple comprise a finite-dimensional family of infinite-dimensional dressing orbits. By contrast, we now consider the case where (λη)(0) is nilpotent (which is necessarily the case if the harmonic map [F^{η}] has finite energy). Here our results are less complete and somewhat confused but examples and partial results indicate that the picture is completely different.

Example Let η = λ^{−1}η_{−1} ∈ Λ_{−1,∞}^{ϵ} with η_{−1} nilpotent so that (ad η_{−1})^{ℓ} = 0, say. Comparing coefficients in (3.1), we conclude from Lemma 3.1 that [F^{ζ}] = [F^{η}] if and only if ζ_{−1} = η_{−1} and

(ad η_{−1})^{n}ζ_{n−1} = 0,

for n < ℓ. From this it is easy to see that g#[F^{η}] = [F^{η}] for all g ∈ Λ_{I,B}^{ϵ}G_{τ} whose ℓ-jet at zero coincides with that of 1. Such g comprise a normal subgroup of Λ_{I,B}^{ϵ}G_{τ} of finite codimension so that the action of Λ_{I,B}^{ϵ}G_{τ} on [F^{η}] reduces to that of a finite-dimensional quotient group.

This is reminiscent of the behaviour of harmonic maps with finite uniton number in the sense of Uhlenbeck [24]. A primitive harmonic map has finite uniton number if it has an extended framing F which can be written as a Laurent polynomial of fixed degree in λ:

F (z) = ∑_{|n|≤d} λ^{n}F_{n}(z),

where we fix a faithful representation of G^{ℂ} to make sense of the Laurent expansion. We call the minimal such d the uniton number of [F].

Concerning these maps, Uhlenbeck [24] and Bergveldt–Guest [1] prove

Proposition A.1 The action of Λ_{I}^{ϵ}G_{τ} preserves uniton number and the action on the space of maps of fixed uniton number reduces to that of a finite dimensional quotient group.

Since the orbits O_{A} are infinite-dimensional, we deduce the following result announced in [5]:

Theorem A.2 No [F^{η}] with (λη)(0) non-zero semisimple and, in particular, no non-constant primitive harmonic map of semisimple finite type, has finite uniton number.

By contrast, we have the following model result which we state for G = SU(2) for simplicity of exposition.

Proposition A.3 Let [F^{η}] : ℝ^{2} → S^{2} be harmonic with η_{−1} nilpotent. Then [F^{η}] has finite uniton number.

Proof Set F = F^{η}. In this setting the reality condition reads

F_{1/λ̄}^{∗} = F_{λ}^{−1}

from which it is easy to see that F is a Laurent polynomial precisely when, for some d, both λ^{d}F_{λ} and λ^{d}F_{λ}^{−1} are holomorphic (as matrix valued functions) at λ = 0. In view of the definition of F, this amounts to demanding that λ^{d} exp(±zη) be holomorphic at zero.

To prove this, we first note that after dressing with a constant element of K^{ℂ}, we may assume that

η_{−1} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) or (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) .

For definiteness, we assume

η_{−1} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix})

and write

η(λ) = (\begin{matrix}a(λ) & λ^{−1} + b(λ) \\ c(λ) & −a(λ) \end{matrix})

where a, b, c are holomorphic on I_{ϵ}. The equivariance condition on η means that a is an even function while b and c are odd. In particular, c(0) = 0.

Now set

γ(λ) = (\begin{matrix}1 & 0 \\ 0 & λ^{−1} \end{matrix})

and observe that

Ad γ(λ) η(λ) = (\begin{matrix}a(λ) & 1 + λb(λ) \\ λ^{−1}c(λ) & −a(λ) \end{matrix})

is holomorphic at λ = 0 since λ^{−1}c(λ) is. Thus

γ(λ) exp(±zη(λ))γ(λ)^{−1}

is holomorphic at zero whence λ exp(±zη(λ)) is also. □

Remark The trick of conjugating with γ amounts to passing between the principal and standard realisations of ΛSL(2, ℂ) in the sense of Wilson [26].

It is not difficult to extend the above argument to numerous other cases which include that where G/K is a full flag manifold with Coxeter–Killing automorphism and η_{−1} is principal nilpotent. In view of this, one might feel tempted (as we were!) to conjecture that any primitive harmonic map [F^{η}] with η_{−1} nilpotent has finite uniton number. However, this is not the case as the following analysis of harmonic maps ℝ^{2} → SU(2) shows.

Set K = SU(2) and put G = K × K. We view K as the Riemannian symmetric space G/K with involution τ : G → G given by (k_{1}, k_{2}) ↦ (k_{2}, k_{1}) and coset projection G → K given by (k_{1}, k_{2}) ↦ k_{1}k_{2}^{−1}.

Define Λ_{hol}K by

Λ_{hol}K = \{k : ℂ^{∗} → K^{ℂ} : k is holomorphic and \overline{k(λ)} = k(1/λ̄)\}.

It is easy to see that Λ_{hol}G_{τ} is given by

Λ_{hol}G_{τ} = {(k(−λ), k(λ)): k ∈ Λ_{hol}K}

so that projection onto the second factor gives an isomorphism Λ_{hol}G_{τ} ≅ Λ_{hol}K. Similarly, we can define subgroups Λ_{E}^{ϵ}K and so on, just by dropping the equivariance conditions on the loops C^{(ϵ)} → K^{ℂ} and thus obtain isomorphisms Λ_{E}^{ϵ}G_{τ} ≅ Λ_{E}^{ϵ}K and so on. Under these isomorphisms, the harmonic map ϕ : M → K corresponding to the extended framing F : M → Λ_{hol}K is given by

ϕ = F_{−1}F_{1}^{−1}

(and, indeed, the extended solution corresponding to ϕ in the sense of Uhlenbeck is the map into the based loop group ΩK given by F_{λ}F_{1}^{−1} : M → ΩK).

Having adopted this view-point, we may think of η ∈ Λ_{−1,∞}^{ϵ} as a holomorphic map η : I_{ϵ} \{0} → 𝔨^{ℂ} with at most a simple pole at zero. We now have

Proposition A.4 [F^{η}] has finite uniton number if and only if det η is holomorphic on I_{ϵ}.

Proof Arguing as in Proposition A.3, we see that [F^{η}] has finite uniton number if and only if, for some d ∈ ℕ, λ^{d} exp(zη) is holomorphic on I_{ϵ}, for all z ∈ ℂ.

For A ∈ 𝔰𝔩(2, ℂ), one has

A^{2} = (− det A)I

from which we deduce, setting D = \sqrt{− det A}, that

exp A = cosh(D)I + \frac{sinh(D)}{D}A.

Thus

exp(zη) = cosh(\sqrt{− det zη})I + \frac{sinh(\sqrt{− det zη})}{\sqrt{− det zη}}zη.  (A.1)

If det η is holomorphic on I_{ϵ}, we see that λ exp(zη) is also since, by hypothesis, λη is holomorphic on I_{ϵ}. Thus, in this case, [F^{η}] has finite uniton number.

Conversely, if det η has a pole at zero, it is easy to see from (A.1) that at least one matrix entry of exp(zη) has an essential singularity at zero so that [F^{η}] has infinite uniton number. □

With this in hand, it is easy to find [F^{η}] with η_{−1} nilpotent and infinite uniton number:

Example Define η ∈ Λ_{−1,∞}^{ϵ} by

η(λ) = (\begin{matrix}0 & λ^{−1} \\ 1 & 0 \end{matrix})

so that

η_{−1} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix})

is nilpotent while det η = −λ^{−1} whence [F^{η}] has infinite uniton number by Proposition A.3.

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(F.E. Burstall) School of Mathematical Sciences, University of Bath, Bath BA2 7AY, England

E-mail address: f.e.burstall@maths.bath.ac.uk

(F. Pedit) Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA

E-mail address: franz@gang.umass.edu