MONOPOLES AND THE GIBBONS-MANTON METRIC

ROGER BIELAWSKI

Abstract. We show that, in the region where monopoles are well separated, the L^{2}-metric on the moduli space of n-monopoles is exponentially close to the T^{n}-invariant hyperkähler metric proposed by Gibbons and Manton. The proof is based on a description of the Gibbons-Manton metric as a metric on a certain moduli space of solutions to Nahm’s equations, and on twistor methods. In particular, we show how the twistor description of monopole metrics determines the asymptotic metric.

The moduli space M_{n} of (framed) static SU(2)-monopoles of charge n, i.e. solutions to Bogomolny equations d_{A}Φ = ∗F, carries a natural hyperkähler metric [1]. The geodesic motion in this metric is a good approximation to the dynamics of low energy monopoles [26, 33]. For the charge n = 2 the metric has been determined explicitly by Atiyah and Hitchin [1], and it follows from their explicit formula that when the two monopoles are well separated, the metric becomes (exponentially fast) the Euclidean Taub-NUT metric with a negative mass parameter. It was also shown by N. Manton [27] that this asymptotic metric can be determined by treating well-separated monopoles as dyons. The equations of motion for a pair of dyons in ℝ^{3} are found to be equivalent to the equations for geodesic motion on Taub-NUT space.

For an arbitrary charge n, it was shown in [3] that, when the individual monopoles are well-separated, the L^{2}-metric is close (as inverse of the separation distance) to the flat Euclidean metric. Gibbons and Manton [14] have then calculated the Lagrangian for the motion of n dyons in ℝ^{3} and shown that it is equivalent to the Lagrangian for geodesic motion in a hyperkähler metric on a torus bundle over the configuration space C̃_{n}(ℝ^{3}). This metric is T^{n}-invariant and has a simple algebraic form. Gibbons and Manton have conjectured, by analogy with the n = 2 case, that the exact n-monopole metric differs from their metric by an exponentially small amount as the separation gets large. We shall prove this conjecture here.

Our strategy is as follows. We construct certain moduli space M̃_{n} of solutions to Nahm’s equations which carries a T^{n}-invariant hyperkähler metric. Using twistor methods we identify this metric as the Gibbons-Manton metric. Finally, we show that the metrics on M̃_{n} and M_{n} are exponentially close. This proof adapts equally well to the asymptotic behaviour of SU(N)-monopole metrics with maximal symmetry breaking, as will be shown elsewhere.

The asymptotic picture can be explained in the twistor setting. We recall that a monopole is determined (up to framing) by a curve S - the spectral curve - in T ℂP^{1}, which satisfies certain conditions [16]. One of these is triviality of the line bundle L^{−2} over S, and a nonzero section of this bundle is the other ingredient needed to determine the metric [19, 1]. Asymptotically we have now the following situation. When the individual monopoles become well separated the spectral curve of the n-monopole degenerates (exponentially fast) into the union of spectral curves S_{i} of individual monopoles, while the section of L^{−2} becomes (also exponentially fast) n meromorphic sections of L^{−2} over the individual S_{i}. The zeros and poles of these sections occur only at the intersection points of the curves S_{i}. This information (and the topology of the asymptotic region of M_{n}) is, as we show in the last section, sufficient to conclude that the asymptotic metric is the Gibbons-Manton metric.

1991 Mathematics Subject Classification. 53C25, 81T13.

The article is organized as follows. In sections 1 and 2 we recall the definitions of the Gibbons-Manton and monopole metrics. In section 3 we introduce the moduli space M̃_{n} of solutions to Nahm’s equations and give heuristic arguments why the metric on M̃_{n} should be exponentially close to the monopole metric. In section 4, as a preliminary step to study M̃_{n} we introduce yet another moduli space of solutions to Nahm’s equations, somewhat simpler than M̃_{n}. In that section we also discuss the relation with Kronheimer’s metrics on G^{ℂ}/T^{ℂ}, where G is a compact semisimple Lie group and T ≤ G is a maximal torus. In section 5 we identify M̃_{n} as a differential, complex, and finally complex-symplectic manifold. In section 6 we calculate the twistor space of M̃_{n} and identify its hyperkähler metric as the Gibbons-Manton metric. In section 7 we finally show that the monopole metric and the metric on M̃_{n} are exponentially close. Short section 8 shows how one can read off the Gibbons-Manton metric, as the asymptotic form of the monopole metric, from the twistor description of the latter.

1. The Gibbons-Manton metric

The Gibbons-Manton metric [14] is an example of 4n-dimensional (pseudo)- hyperkähler metric admiting a tri-Hamiltonian (hence isometric) action of the ndimensional torus T^{n}. Such metrics have particularly nice properties and were studied by several authors [25, 18, 32]. The Gibbons-Manton metric was described as a hyperkähler quotient of a flat quaternionic vector space by Gibbons and Rychenkova in [15]. We recall here this description, which we slightly modify to better suit our purposes. We start with flat hyperkähler metrics g_{1} and g_{2} on M_{1} = (S^{1} × ℝ^{3})^{n} and M_{2} = ℍ^{n(n−1)/2}. We consider a pseudo-hyperkähler metric on the product manifold M = M_{1} × M_{2} given by g = g_{1} − g_{2}. The complex structures on ℍ are given by the right multiplication by quaternions i, j, k. The metric g_{1} is invariant under the obvious action (by translations) of T^{n} = (S^{1})^{n} and the metric g_{2} is invariant under the left diagonal action of T^{n(n−1)/2}. We consider a homomorphism ϕ : T^{n(n−1)/2} → T^{n} given by

(t_{ij})_{i<j} ↦ ( ∏_{j=i+1}^{n} t_{ij} ∏_{j=1}^{i−1} t_{ji}^{−1})_{i=1,... ,n} .

This defines an action of T^{n(n−1)/2} on M = M_{1} × M_{2} by t · (m_{1}, m_{2}) = (ϕ(t) · m_{1}, t · m_{2}). Gibbons and Rychenkova have shown that the hyperkähler quotient of (M, g) by this action of T^{n(n−1)/2} is the Gibbons-Manton metric.

We remark that, if we choose coordinates (t_{i}, x_{i}) on M_{1}, t_{i} ∈ S^{1} and x_{i} ∈ ℝ^{3}, and quaternionic coordinates q_{ij}, i < j, on ℍ^{n(n−1)/2}, then the moment map equation are:

\frac{1}{2}q_{ij}iq̄_{ij} = x_{i} − x_{j}.  (1.1)

As long as x_{i} ≠ x_{j} for i ≠ j, the torus T^{n(n−1)/2} acts freely on the zero-set of the moment map. The quotient of this set by T^{n(n−1)/2} is a smooth hyperkähler manifold which we denote by M_{GM}. The action of T^{n} on M_{1} induces a free tri-Hamiltonian action on M_{GM} for which the moment map is just (x_{1}, …, x_{n}). This makes M_{GM} into a T^{n}-bundle over the configuration space C̃_{n}(ℝ^{3}) of n distinct points in ℝ^{3}. We shall now determine this bundle. We recall that a basis of H_{2}(C̃_{n}(ℝ^{3}), ℤ) is given by the n(n − 1)/2 2-spheres

S_{ij}^{2} = {(x_{1}, …, x_{n}) ∈ ℝ^{3} ⊗ ℝ^{n}; |x_{i} − x_{j}| = const, x_{k} = const if k ≠ i, j}(1.2)

where i < j. We have

Proposition 1.1. The hyperkähler moment map for the action of T^{n} makes M_{GM} into a T^{n}-bundle over C̃_{n}(ℝ^{3}) determined by the element (s_{1}, …, s_{n}) of H^{2}(C̃_{n}(ℝ^{3}), ℤ^{n}) given by

s_{k}(S_{ij}^{2}) = \begin{cases}−1 & if k = i \\ 1 & if k = j \\ 0 & otherwise. \end{cases}

Proof. From the formula (1.1) it follows that restricting the bundle to a fixed S_{ij}^{2} is equivalent to considering the case n = 2. In other words s_{k}(S_{ij}^{2}) = 0 if k ≠ i, j and we have to consider only one quaternionic coordinate q_{ij}. The zero-set of the moment map is \frac{1}{2}q_{ij}iq̄_{ij} = x_{i} − x_{j} and the circle S^{1} by which we quotient acts by t · (q_{ij}, (t_{i}, x_{i}), (t_{j}, x_{j})) = (tq_{ij}, (tt_{i}, x_{i}), (t^{−1}t_{j}, x_{j})). The quotient can be obtained by setting t_{i} = 1 and the induced action of the i-th generator s_{i} of T^{n} is then given by left multiplication by s_{i}^{−1} on q_{ij}. Since the map q_{ij} → \frac{1}{2}q_{ij}iq̄_{ij} with the left action of S^{1} on {q_{ij} ∈ ℍ; |q_{ij}| = 1} is the Hopf bundle, it follows that s_{i}(S_{ij}^{2}) = −1. A similar argument shows that s_{j}(S_{ij}^{2}) = 1.

In particular, (\overline{t}, \overline{x}) = (t_{i}, x_{i}) form local coordinates on M_{GM}. The metric tensor can be then written in the form [32]:

g = Φd\overline{x} · d\overline{x} + Φ^{−1}(d\overline{t} + A)^{2},

where the matrix Φ and the 1-form A depend only on the x_{i} and satisfy certain linear PDE’s. In particular, Φ determines the metric. For the Gibbons-Manton metric

4Φ_{ij} = \begin{cases}1 − ∑_{k≠i} \frac{1}{∥x_{i}−x_{k}∥} & if i = j \\ \frac{1}{∥x_{i}−x_{j}∥} & if i ≠ j. \end{cases}

2. Nahm’s equations and monopole metrics

We shall recall in this section the description of the L^{2}-metric on the moduli space of charge n SU(2)-monopoles in terms of Nahm’s equations. A proof that the Nahm transform [30, 16] between the two moduli spaces is an isometry was given by Nakajima in [31].

One starts with the space A of quadruples (T_{0}, T_{1}, T_{2}, T_{3}) of smooth 𝔲(n)-valued functions on (−1, 1) such that T_{1}, T_{2}, T_{3} have simple poles at ±1 with residues \frac{1}{2}ρ(σ_{i}), i = 1, 2, 3, where ρ : 𝔰𝔲(2) → 𝔲(n) is the standard irreducible n-dimensional representation of 𝔰𝔲(2) and σ_{i} are the Pauli matrices. Equipped with the L^{2}-norm (given by a biinvariant inner product on 𝔲(n)), A becomes a flat quaternionic affine space. There is an isometric and triholomorphic action of the gauge group G of U(n)-valued functions g : [−1, 1] → U(n) which are 1 at ±1:

T_{0} ↦ Ad(g)T_{0} − ġg^{−1}
T_{i} ↦ Ad(g)T_{i} ,  i = 1, 2, 3.  (2.1)

The zero-set of the hyperkähler moment map for this action is then described by Nahm’s equations [30]:

Ṫ_{i} + [T_{0}, T_{i}] + \frac{1}{2} ∑_{j,k=1,2,3} ϵ_{ijk}[T_{j}, T_{k}] = 0 , i = 1, 2, 3.  (2.2)

The quotient of the space of solutions by G is the a smooth hyperkähler manifold M_{n} of dimension 4n. By the above mentioned result of Nakajima, M_{n} is the moduli space of (framed) charge n SU(2)-monopoles. With respect to any complex structure M_{n} is biholomorphic to the space of based rational maps of degree n on ℂP^{1} [13]. If we replace U(n) by = SU(n) (resp. by P SU(n)) in the above description, we obtain the moduli space of strongly centered (resp. centered) SU(2)-monopoles of charge n.

Remark 2.1. A similar construction can be done for any compact Lie group G. We require ρ : 𝔰𝔲(2) → 𝔤 to be a Lie algebra homomorphism whose image lies in the regular part of 𝔤. We obtain a smooth hyperkähler manifold of dimension 4 rank G which can be identified with a totally geodesic submanifold of certain moduli space of SU(N)-monopoles (with a minimal symmetry breaking). Alternatively, as a complex manifold, it is a desingularization of (𝔥^{ℂ} × T^{ℂ}) /W where T^{ℂ} is a maximal torus in G^{ℂ}, 𝔥^{ℂ} its Lie algebra, and W the corresponding Weyl group [6].

The tangent space to M_{n} can be described as the space of solutions to the linearized Nahm’s equations and satisfying the condition of being orthogonal (in the L^{2}-metric) to vectors arising from infinitesimal gauge transformations. In other words the tangent space to M_{n} at a solution (T_{0}, T_{1}, T_{2}, T_{3}) can be identified with the set of solutions (t_{0}, t_{1}, t_{2}, t_{3}) to the following system of linear equations:

\begin{matrix}ṫ_{0} + [T_{0}, t_{0}] + [T_{1}, t_{1}] + [T_{2}, t_{2}] + [T_{3}, t_{3}] = 0, \\ ṫ_{1} + [T_{0}, t_{1}] − [T_{1}, t_{0}] + [T_{2}, t_{3}] − [T_{3}, t_{2}] = 0, \\ ṫ_{2} + [T_{0}, t_{2}] − [T_{1}, t_{3}] − [T_{2}, t_{0}] + [T_{3}, t_{1}] = 0, \\ ṫ_{3} + [T_{0}, t_{3}] + [T_{1}, t_{2}] − [T_{2}, t_{1}] − [T_{3}, t_{0}] = 0. \end{matrix}  (2.3)

The metric is defined by

∥(t_{0}, t_{1}, t_{2}, t_{3})∥^{2} = \frac{1}{2} ∫_{−1}^{1} ∑_{0}^{3} ∥t_{i}∥^{2}.  (2.4)

The three anti-commuting complex structures can be seen by writing a tangent vector as t_{0} + it_{1} + jt_{2} + kt_{3}.

3. The asymptotic moduli space

We shall now construct a one-parameter family of moduli spaces M̃_{n}(c), c ∈ ℝ, of solutions to Nahm’s equations carrying (pseudo-)hyperkähler metrics. We shall see later on that these metrics are the Gibbons-Manton metric with different mass parameters.

We consider the subspace Ω_{1} of exponentially fast decaying functions in C^{1}[0, ∞], i.e.:

Ω_{1} = {f : [0, ∞] → 𝔲(n); ∃_{η>0} sup_{t} (e^{ηt}∥f(t)∥ + e^{ηt}∥df/dt∥) < +∞} . (3.1)

As in the previous section, ρ : 𝔰𝔲(2) → 𝔲(n) is the standard irreducible n-dimensional representation of 𝔰𝔲(2) (in particular, ρ(σ_{1}) is a diagonal matrix). We denote by 𝔥 the (Cartan) subalgebra of 𝔲(n) consisting of diagonal matrices. Let Ã_{n} be the space of C^{1}-functions (T_{0}, T_{1}, T_{2}, T_{3}) defined on (0, +∞] and satisfying (cf. [23]):

(i) T_{1}, T_{2}, T_{3} have simple poles at 0 with res T_{i} = \frac{1}{2}ρ(σ_{i}); (ii) T_{i}(+∞) ∈ 𝔥 for i = 0, …, 3; (iii) (T_{1}(+∞), T_{2}(+∞), T_{3}(+∞)) is a regular triple, i.e. its centralizer is 𝔥; (iv) (T_{i}(t) − T_{i}(+∞)) ∈ Ω_{1} for i = 0, 1, 2, 3.

Next we shall define the relevant gauge group. The Lie algebra of our gauge group G(c) is the space of C^{2}-paths ρ : [0, +∞) → 𝔲(n) such that

(i) ρ(0) = 0 and ρ̇ has a limit in 𝔥 at +∞; (ii) (ρ̇ − ρ̇(+∞)) ∈ Ω_{1}, and [τ, ρ] ∈ Ω_{1} for any regular element τ ∈ 𝔥; (iii) cρ̇(+∞) + lim_{t→+∞}(ρ(t) − tρ̇(+∞)) = 0. It is the Lie algebra of the Lie group

G(c) = {g : [0, +∞) →U(n); g(0) = 1, s(g) := lim ġg^{−1} ∈ 𝔥, (τ − Ad(g)τ) ∈ Ω_{1},

(ġg^{−1} − s(g)) ∈ Ω_{1}, exp(cs(g)) lim (g(t) exp(−ts(g))) = 1} .

Remark . The last condition in the definition of G(c) means that g(t) is asymptotic to exp(ht − ch) for some diagonal h.

We introduce a family of metrics on Ã_{n}. Let (t_{0}, t_{1}, t_{2}, t_{3}) be a tangent vector to the space Ã_{n} at a point (T_{0}, T_{1}, T_{2}, T_{3}). The functions t_{i} are now regular at 0, i = 0, …, 3. We put

∥(t_{0}, t_{1}, t_{2}, t_{3})∥_{c}^{2} = c ∑_{0}^{3} ∥t_{i}(+∞)∥^{2} + ∫_{0}^{+∞} ∑_{0}^{3} (∥t_{i}(s)∥^{2} − ∥t_{i}(+∞)∥^{2}) ds.(3.2)

We observe that the group G(c) acting by (2.1) preserves the metric ∥·∥_{c} and the three complex structure of the flat hyperkähler manifold Ã_{n}. We define M̃_{n}(c) as the (formal) hyperkähler quotient of Ã_{n} by G(c) (with respect to the metric ∥ · ∥_{c}). The zero set of the moment map is given by the equations (2.2) (here the condition (iii) in the definition of Lie(G(c)) is essential) and so M̃_{n}(c) is defined as the moduli space of solutions to Nahm’s equations:

M̃_{n}(c) = {solutions to (2.2) in Ã_{n}} /G(c).

Remark . If c > 0, then the metric (3.2) on M̃_{n}(c) will be seen to be positive definite if (T_{1}(+∞), T_{2}(+∞), T_{3}(+∞)) is sufficiently far from the walls of Weyl chambers. On the other hand, if c < 0, then the metric will be shown to be everywhere negative definite. Therefore, for c < 0 we should really replace ∥ · ∥_{c} with its negative; it is, however more convenient to consider the metrics ∥ · ∥_{c}.

We observe that sending a solution T_{i} to the solution rT_{i}(rt) for any r > 0 induces a homothety of factor r between M̃_{n}(c) and M̃_{n}(rc).

Before we begin the detailed study of M̃_{n}(c), let us explain why we expect this metric to be exponentially close to the monopole metric. It is known [4] that the solutions to Nahm’s equations on (0, 2) corresponding to a well-separated monopole are exponentially close to being constant away from the boundary points (i.e. on any [ϵ, 2−ϵ]). The same is true for solutions on the half line (0, +∞): as long as the triple (T_{1}(+∞), T_{2}(+∞), T_{3}(+∞)) is regular, the solutions are exponentially close to being constant away from 0 [23] (it is helpful to notice that the space of regular triples is the same as the space C̃_{n}(ℝ^{3}) of distinct points in ℝ^{3}). Our strategy is to take two solutions, on half-lines (0, ∞) and (−∞, 2) with the same values at ±∞, cut them off at t = 1 and use this non-smooth solution on (0, 2) (with correct boundary behaviour) to obtain an exact solution to the monopole Nahm data. The exact solution will differ from the approximate one by an exponentially small amount. Furthermore the part of the half-line solutions which we have cut off is exponentially close to being constant and, for c = 1, contributes an exponentially small amount to the metric ∥ · ∥_{c} (all estimates are uniform and can be differentiated). This can be seen from the fact that we can rewrite (3.2) as

∥(t_{0}, t_{1}, t_{2}, t_{3})∥_{c}^{2} = ∫_{0}^{c} ∑_{0}^{3} ∥t_{i}(s)∥^{2} + ∫_{c}^{+∞} ∑_{0}^{3} (∥t_{i}(s)∥^{2} − ∥t_{i}(+∞)∥^{2}) ds.(3.3)

The first term, together with the corresponding term for the solution on (−∞, 2), is exponentially close to the monopole metric (for c = 1).

4. Moduli space of regular semisimple adjoint orbits

In order to obtain information about M̃_{n}(c) we need to consider first another moduli space of solutions to Nahm’s equations, defined analogously, except that we require the solutions to be smooth at t = 0. This space, which can be defined for an arbitrary compact Lie group G, is of some interest as all hyperkähler structures on G^{ℂ}/T^{ℂ} (here T^{ℂ} is a maximal torus) due to Kronheimer [23] can be obtained from it as hyperkähler quotients (see Theorem 4.3 below). A reader who is primarily interested in monopoles should think of G as U(n).

Let us first recall how Kronheimer constructs hyperkähler metrics on G^{ℂ}/T^{ℂ}. Let 𝔥 be the Lie algebra of T^{ℂ} and let (τ_{1}, τ_{2}, τ_{3}) ∈ 𝔥^{3} be a regular triple, i.e. one whose centralizer is 𝔥. For a fixed η > 0, consider the Banach space

Ω_{1}^{η} = {f : [0, ∞] → 𝔤; sup_{t} (e^{ηt}∥f(t)∥ + e^{ηt}∥df/dt∥) < +∞}

with the norm ∥f∥ = sup_{t} (e^{ηt}∥f(t)∥ + e^{ηt}∥df/dt∥). Define A^{η}(τ_{1}, τ_{2}, τ_{3}) as the space of C^{1}-functions (T_{0}, T_{1}, T_{2}, T_{3}) : (0, +∞] → 𝔤 which satisfy:

{T_{0}(t), (T_{i}(t) − τ_{i}) ; i = 1, 2, 3} ⊂ Ω_{1}^{η}.

Define also G^{η} by replacing Ω_{1} with Ω_{1}^{η} in the definition of G given in the previous section. Kronheimer shows then that for small enough η

M(τ_{1}, τ_{2}, τ_{3}) = {solutions to (2.2) in A^{η}(τ_{1}, τ_{2}, τ_{3})} /G^{η}

equipped with the L^{2} metric is a smooth hyperkähler manifold, diffeomorphic to G^{ℂ}/T^{ℂ}. Futhermore, if (τ_{2}, τ_{3}) is regular, then M(τ_{1}, τ_{2}, τ_{3}) is biholomorphic, with respect to the complex structure I, to the complex adjoint orbit of τ_{2} + iτ_{3}.

We observe that the union of all M(τ_{1}, τ_{2}, τ_{3}) has a natural topology and it is, in fact, a smooth manifold. We shall show now that there is a T -bundle over this union which carries a (pseudo)-hyperkähler metric. We define the space A_{G} by omitting the condition (i) in the definition of Ã_{n} in the previous section. Instead we require that the T_{i} are smooth at t = 0 for i = 0, 1, 2, 3. We define M_{G}(c), c ∈ ℝ, as the (formal) hyperkähler quotient of A_{G} by G(c) with respect to the metric (3.2). We have:

Proposition 4.1. M_{G}(c) equipped with the metric (3.2) is a smooth hyperkähler manifold. The tangent space at a solution (T_{0}, T_{1}, T_{2}, T_{3}) is described by the equations (2.3).

We remark that the metric 3.2 may be degenerate at some points. However the hypercomplex structure is defined everywhere.

Proof. Define M_{G}^{η}(c) by replacing Ω with Ω^{η} in the definition of M_{G}(c). By the exponential decay property of solutions to Nahm’s equations ([23], Lemma 3.4), a neighbourhood of a particular element in M_{G}(c) is canonically identified with its neighbourhood in M_{G}^{η}(c) for small enough η. Therefore we can use the transversality arguments of [23], Lemma 3.8 and Proposition 3.9 (with a slight modification due to condition (iii) in the definition of Lie(G(c))) to deduce the smoothness. The fact that the metric is hyperkähler is, formally, the consequence of the fact that M_{G}(c) is a hyperkähler quotient. One can, in fact, check directly that the three Kähler forms are closed. We shall also, later on, identify the complex structures and the complex symplectic forms proving their closedness.

We observe now that the action on A_{G} of gauge transformations which are asymptotic to exp(−th + λh), h ∈ 𝔥, λ ∈ ℝ, induce a free isometric action of T = exp(𝔥) on M_{G}(c). In fact this action is tri-Hamiltonian and a simple calculation shows

Proposition 4.2. The hyperkähler moment map μ = (μ_{1}, μ_{2}, μ_{3}) for the action of T on M_{G}(c) is given by μ_{i}(T_{0}, T_{1}, T_{2}, T_{3}) = T_{i}(+∞) for i = 1, 2, 3.

As an immediate corollary we have:

Theorem 4.3. Let (τ_{1}, τ_{2}, τ_{3}) be a regular triple in 𝔥^{3}. The hyperkähler quotient μ^{−1}(τ_{1}, τ_{2}, τ_{3})/T of M_{G}(c) by the torus T is isometric to Kronheimer’s M(τ_{1}, τ_{2}, τ_{3}).

We have also a tri-Hamiltonian action of G on M_{G}(c) given by the gauge transformations with arbitrary values at t = 0. The hyperkähler moment map for this action is (T_{1}(0), T_{2}(0), T_{3}(0)).

We have two other group actions on M_{G}(c). There is a free isometric and triholomorphic action of the Weyl group W = N(T)/T given by the gauge transformations which become constant (and in W) exponentially fast.

Finally there is a free isometric SU(2)-action which rotates the complex structures. As a consequence it has a globally defined Kähler potential for each Kähler form (cf. [18]). The potential for ω_{2} (or ω_{3}) is given by the moment map for the action of a circle in SU(2) which preserves I. This is easily seen to be

K_{J} = c ∑_{i=2}^{3} ∥T_{i}(+∞)∥^{2} + ∫_{0}^{+∞} ∑_{i=2}^{3}(∥T_{i}(s)∥^{2} − ∥T_{i}(+∞)∥^{2})ds.

Remark 4.4. There is a similar (pseudo)-hyperkähler manifold with a torus action such that the hyperkähler quotients by this torus are isometric to Kronheimer’s ALE-metrics on the minimal resolution of a given Kleinian singularity ℂ^{2}/Γ [24]. This manifold is defined as M_{G} except that the T_{i} have poles at t = 0 with the residues defined by a subregular homomorphism 𝔰𝔲(2) → 𝔤 (cf. [6, 5]).

Remark 4.5. One can observe that M_{G}(0) is a cone metric (with the ℝ_{>0}-action given by T_{i}(t) ↦ rT_{i}(rt)) and in fact, it is an ℍ^{∗}-bundle over a pseudo-quaternion-Kähler manifold (cf. [34]).

5. M̃_{n}(c) as a manifold

We now return to the space M̃_{n}(c) defined in section 3. Our first task is to show that this space is smooth. We shall show that M̃_{n}(c) is a smooth hyperkähler quotient of the product of the space M_{U(n)}(c−1) considered in the previous section and of another moduli space of solutions to Nahm’s equations. This latter space, denoted by N_{n}, is given by 𝔲(n)-valued solutions to Nahm’s equations defined on (0, 1] smooth at t = 1 and with the same poles as M̃_{n}(c) at t = 0. The gauge group consists of gauge transformations which are identity at t = 0, 1. Equipped with the metric (2.4) this is a smooth hyperkähler manifold [6, 11]. It admits a tri-Hamiltonian action of U(n) given by gauge transformations with arbitrary values at t = 1. In addition, we consider the space M_{U(n)}(c − 1) defined in the previous section. We identify it this time with the space of solutions on [1, +∞] via the map T_{i}(t) ↦ T_{i}(t + 1) (so that the gauge transformations behave now, near +∞, as elements of G(c)).

It is easy to observe that the space M̃_{n}(c) is the hyperkähler quotient of N_{n} × M_{U(n)}(c−1) by the diagonal action of U(n) (cf. [6]; the moment map equations simply match the functions T_{1}, T_{2}, T_{3} at t = 1; after that, quotienting by G means that the remaining gauge transformations are smooth at t = 1). Using this description of M̃_{n}(c) we can finally show

Proposition 5.1. M̃_{n}(c) equipped with the metric (3.2) is a smooth hyperkähler manifold. The tangent space at a solution (T_{0}, T_{1}, T_{2}, T_{3}) is described by the equations (2.3).

Proof. Since the metric (3.2) may be degenerate, we still have to show that the moment map equations on N_{n}×M_{U(n)}(c−1) are everywhere transversal. Consider a particular point in M_{U(n)}(c−1) which we represent by a solution m = (T_{0}, T_{1}, T_{2}, T_{3}) with T_{0}(+∞) = 0 and T_{i}(+∞) = τ_{i}, i = 1, 2, 3. Let μ be the hyperkähler moment map for the action of G on N_{n} × M_{U(n)}. We observe that the image of dμ_{|_{m}} contains the image of dμ′_{|_{m}}, μ′ being the hyperkähler moment map for the action of G on N_{n} ×M(τ_{1}, τ_{2}, τ_{3}) (Kronheimer’s definition of M(τ_{1}, τ_{2}, τ_{3}) was recalled in the previous section). The metric on N_{n}×M(τ_{1}, τ_{2}, τ_{3}) is non-degenerate and, as G acts freely, dμ′_{|_{m}} is surjective. Thus dμ is surjective at each point in N_{n} × M_{U(n)}(c − 1) and M̃_{n}(c) is smooth.

We observe that, as in the case of M_{U(n)}(c), M̃_{n}(c) has isometric actions of the torus T^{n} (defined as the diagonal subgroup of U(n)), of the symmetric group S_{n}, and of SU(2). In particular, the hyperkähler moment map for the action of T^{n} is still given by the values of T_{1}, T_{2}, T_{3} at infinity (cf. Proposition 4.2).

We can describe the topology of M̃_{n}(c):

Proposition 5.2. M̃_{n}(c) is a principal T^{n}-bundle over the configuration space C̃_{n}(ℝ^{3}) of n distinct points in ℝ^{3}.

We postpone identifying this bundle until the next section (Proposition 6.3).

Proof. The space C̃_{n}(ℝ^{3}) is the space of regular triples in the subalgebra of diagonal matrices and the moment map μ for the action of T^{n} gives us a projection M̃_{n}(c) → C̃_{n}(ℝ^{3}). Let us consider a fixed regular triple (τ_{1}, τ_{2}, τ_{3}) and all elements of M̃_{n}(c) with T_{i}(+∞) = τ_{i}, i = 1, 2, 3, i.e. μ^{−1}(τ_{1}, τ_{2}, τ_{3}). For each such solution we can make T_{0} identically 0 by some gauge transformation g with g(0) = 1. This is defined uniquely up to the action of G × T^{n} and so the set of T^{n}-orbits projecting via μ to (τ_{1}, τ_{2}, τ_{3}) can be identified with the set of solutions to Nahm’s equations with T_{0} ≡ 0, T_{1}, T_{2}, T_{3} having the appropriate residues at t = 0 and being conjugate to τ_{1}, τ_{2}, τ_{3} at infinity. By the considerations at the beginning of this section this space is the hyperkähler quotient of N_{n} × M(τ_{1}, τ_{2}, τ_{3}) by U(n). The arguments of [6] show that the corresponding complex-symplectic quotient can be identified with the intersection of a regular semisimple adjoint orbit of GL(n, ℂ) with the slice to the regular nilpotent orbit. This intersection is a single point. Finally, in order to identify in this case the hyperkähler quotient with the complex-symplectic one we can adapt the argument in the proof of Proposition 2.20 in [20].

Our next task is to describe the complex structure of M̃_{n}(c) (because of the action of SU(2) all complex structures are equivalent). As usual (cf. [13]), if we choose a complex structure, say I, we can introduce complex coordinates on the moduli space of solutions to Nahm’s equations by writing α = T_{0} + iT_{1} and β = T_{2} + iT_{3}. The Nahm equations can be then written as one complex and one real equation:

\frac{dβ}{dt} = [β, α]  (5.1)
\frac{d}{dt}(α + α^{∗}) = [α^{∗}, α] + [β^{∗}, β].  (5.2)

By the remark made at the beginning of this section, M̃_{n}(c) is the hyperkähler quotient of the product manifold N_{n}×M_{U(n)}(c−1). We shall show that as a complex symplectic manifold M̃_{n}(c) is the complex-symplectic quotient of N_{n}×M_{U(n)}(c−1). Let us recall the complex structure of N_{n} [13, 19, 6, 12]. Let e_{1}, …, e_{n} denote the standard basis of ℂ^{n}. There is a unique solution w_{1} of the equation

\frac{dw}{dt} = −αw  (5.3)

with

lim_{t→0} (t^{−(n−1)/2}w_{1}(t) − e_{1}) = 0.  (5.4)

Setting w_{i}(t) = β^{i−1}(t)w_{1}(t), we obtain a solution to (5.3) with

lim_{t→0} (t^{i−(n+1)/2}w_{i}(t) − e_{i}) = 0.

The complex gauge transformation g(t) with g^{−1} = (w_{1}, …, w_{n}) makes α identically zero and sends β(t) to the constant matrix

B(β_{1}, …, β_{n}) = (\begin{matrix}0 & … & 0 & (−1)^{n+1}S_{n} \\ 1 & ... & & (−1)^{n}S_{n−1} \\ & ... & ... & ⋮ \\ 0 & … & 1 & S_{1} \end{matrix}) .  (5.5)

Here β_{i} denote the (constant) eigenvalues of β(t) and S_{i} is the i-th elementary symmetric polynomial in {β_{1}, …, β_{n}}. The mapping (α, β) → (g(1), B) gives a biholomorphism between (N_{n}, I) and Gl(n, ℂ) × ℂ^{n} [6].

We describe the complex structure of M̃_{n}(c) as follows:

Proposition 5.3. There exists a T^{n}-equivariant biholomorphism between M̃_{n}(c) and an open subset of

(∐_{𝔫}{[g, b] ∈ Gl(n, ℂ) ×_{N} (𝔡 + 𝔫); gbg^{−1} is of the form (5.5)}) /∼,

where 𝔡 denotes diagonal matrices, the union is over unipotent algebras 𝔫 (with respect to 𝔡) and N = exp 𝔫. Furthermore, the relation ∼ is given as follows: [g, d + n] ∼ [g′, d′ + n′] if and only if n ∈ 𝔫, n′ ∈ 𝔫′, and either 𝔫′ ⊂ 𝔫 and there exists an m ∈ N such that gm^{−1} = g′, Ad(m)(d + 𝔫) = d′ + 𝔫′ or vice versa (i.e. 𝔫 ⊂ 𝔫′ etc.).

Remark . It will follow from the description of the twistor space that this biholomorphism is actually onto. Proving this right now would require showing that the T^{n}-action on M̃_{n}(c) extends to the global action of (ℂ^{∗})^{n}. This, in turn, requires showing existence of solutions to a mixed Dirichlet-Robin problem on the half-line - something that seems quite tricky.

Proof. Fix a unipotent algebra 𝔫 and consider the set of all solutions (α, β) = (T_{0}+iT_{1}, T_{2}+iT_{3}) on [1, +∞) such that the intersection of the sum of positive eigenvalues of ad(iT_{1}(+∞)) with C(β(+∞)) is contained in 𝔫. Let M(𝔫; c−1) be the corresponding subset of M_{U(n)}(c). We observe that, since (T_{1}(+∞), T_{2}(+∞), T_{3}(+∞)) is a regular triple, the projection of T_{1}(+∞) onto 𝔡^{ℂ} ∩ C(β(+∞)) is a regular element, and so 𝔫 contains the unipotent radical of a Borel subalgebra of C(β(+∞)) for any element of M(𝔫; c − 1). Using gauge freedom, we always make T_{0}(+∞) = 0 and, by Proposition 4.1 of Biquard [8], such a representative is of the form g(α(+∞), β(+∞) + Ad(exp{−α(+∞)t})n), where n ∈ 𝔫 and g is a bounded Gl(n, ℂ)-valued gauge transformation. The transformation g is defined modulo exp{−α(+∞)t}g_{0} exp{α(+∞)t} with g_{0} ∈ P = exp(𝔡 + 𝔫). Since T_{0}(+∞) = 0 and T_{0} is decaying exponentially fast, g has a limit (in T^{ℂ}) at +∞. If we replace g(t) by g′(t) = g(t)g(+∞)^{−1} exp{−α(+∞)t + cα(+∞)}, then (α, β) = g′(0, β(+∞) + n′) for an n′ ∈ 𝔫. The transformation g′, which satisfies (at infinity) the boundary condition of an element of G(c − 1)^{ℂ}, is now defined modulo constant gauge transformations in N. Moreover g′(1) is independent of G(c − 1) and we obtain a map ϕ : M(𝔫) → Gl(n, ℂ) ×_{N} (𝔡 + 𝔫) by sending (α, β) to (g′(1), β(+∞) + n′). Considering the infinitesimal version of this construction shows that ϕ is holomorphic.

Since ϕ is U(n)-equivariant, it is (locally) Gl(n, ℂ)-equivariant. We can adapt the argument of Proposition 2.20 in [19] to show that M̃_{n}(c) is the complex-symplectic quotient of N_{n} ×M_{U(n)}(c−1) by (local action of) Gl(n, ℂ). Let us restrict attention to N_{n} × M(𝔫). The complex symplectic moment map at the point (g, B) of N_{n} is −g^{−1}Bg (here g ∈ Gl(n, ℂ) and B is of the form (5.5)) and the complex symplectic moment map at the point corresponding to [g′, β_{d} + n] is g′(β_{d} + n)g^{′−1} (here β_{d} is diagonal and n ∈ 𝔫). The moment map equation for the diagonal action of Gl(n, ℂ) is g^{−1}Bg = g′(β_{d} + n)g^{′−1}. If we now quotient by Gl(n, ℂ), i.e. send g to identity, we shall end up with the set of [g′, b] ∈ Gl(n, ℂ) ×_{N} (𝔡 + 𝔫) such that g′bg^{′−1} = B (B is determined by the diagonal part of b). This identifies the charts described in this proposition. By going through the procedure we can conclude that the charts for different 𝔫 are matched as claimed.

So far we have shown that there is a holomorphic map ϕ from M̃_{n}(c) to the manifold M described in the statement. We still have to show that ϕ is 1-1. By construction our map is T^{n}-equivariant, and so (ℂ^{∗})^{n}-equivariant (where the action is defined). Since the (ℂ^{∗})^{n}-action on M is free, it is free on M̃_{n}(c). Furthermore the (ℂ^{∗})^{n}-action on M leaves invariant sets of the form M ∩ (Gl(n, ℂ) ×_{N} (d + 𝔫)), d ∈ 𝔡. Each such set is single orbit of (ℂ^{∗})^{n} and so ϕ is 1-1.

The above description of M̃_{n}(c) is rather complicated. We remark that the open dense subset where β(+∞) is regular corresponds to 𝔫 = 0, i.e. to

{(β_{d}, g); β_{d} = diag(β_{1}, …, β_{n}), β_{i} ≠ β_{j} if i ≠ j, gβ_{d}g^{−1} = B(β_{1}, …, β_{n})}.

We shall denote the corresponding subset of M̃_{n}(c) by M̃_{n}^{reg}(c). We observe that an element g of Gl(n, ℂ) which sends diag(β_{1}, …, β_{n}) to B(β_{1}, …, β_{n}) is of the form

g = V (β_{1}, …, β_{n})^{−1} diag(u_{1}, …, u_{n})  (5.6)

where u_{i} ≠ 0 and V (β_{1}, …, β_{n}) is the Vandermonde matrix, i.e. V_{ij} = (β_{i})^{j−1}. We can calculate the complex symplectic form ω = ω_{2} + iω_{3} on M̃_{n}^{reg}(c):

Proposition 5.4. The complex symplectic form ω on M̃_{n}^{reg}(c) is given, in coordinates β_{i}, u_{i}, i = 1, …, n, by

∑_{i=1}^{n} \frac{du_{i}}{u_{i}} ∧ dβ_{i} − ∑_{i<j} \frac{dβ_{i} ∧ dβ_{j}}{β_{i} − β_{j}} .  (5.7)

Proof. First, we calculate ω on the subset of M_{U(n)}(c − 1) where β(+∞) is regular. This subset is biholomorphic to Gl(n, ℂ) × {regular elements of 𝔥^{ℂ}} and according to the proof of Proposition 5.3, an element (α, β) of this set corresponding to (g, β_{d}) ∈ Gl(n, ℂ) × 𝔥 can be written as (α, β) = (−ġ(t)g^{−1}, g(t)β_{d}g(t)^{−1}), where g(t) is a complex gauge transformation with g(0) = g. Therefore a tangent vector (a(t), b(t)) at (α, β) can be written as

(a, b) = (−gρ̇g^{−1}, g(b_{d} + [ρ, β_{d}])g^{−1}),  (5.8)

where ρ is dual to g^{−1}dg and b_{d} is dual to dβ_{d}. The complex symplectic form on M_{U(n)}(c − 1) is given by

ω = (c − 1) tr(dα(+∞) ∧ dβ(+∞)) + ∫_{0}^{+∞} tr(dα ∧ dβ − dα(+∞) ∧ dβ(+∞)).

For two tangent vectors (a, b) and (â, b̂), corresponding, via (5.8), to (ρ, b_{d}) and (ρ̂, b̂_{d}) we obtain

ω = − tr(b_{d}ρ̂ − ρb̂_{d} − [ρ, ρ̂]β_{d}),

where ρ = ρ(0), ρ̂ = ρ̂(0). To calculate the symplectic form on M̃_{n}^{reg}(c) it remains to substitute (5.6) for g. Let us write u for diag(u_{1}, …, u_{n}). Then ρ becomes dual to u^{−1}du − u^{−1}dV V^{−1}u. Let us write ν for the tangent vector dual to u^{−1}du and Υ for the tangent vector dual to dV V^{−1}. Since ν is diagonal and the i-th row of Υ is of the form b_{i}s (here we write b_{d} = diag(b_{1}, …, b_{n})), for a covector s, we can write ω as

ω = − tr(b_{d}ν̂ − νb̂_{d} − [Υ, Υ̂]β_{d}).

It remains to calculate tr[Υ, Υ̂]β_{d}. Let us write W_{ij} for the (i, j)-th entry of V^{−1}, i.e.

W_{ij} = (−1)^{n−i}S_{n−i}(β_{1}, …, β̂_{j}, …, β_{n})/ ∏_{k≠j}(β_{j} − β_{k}),  (5.9)

S_{k} being the k-th elementary symmetric polynomial (S_{0} = 1). We calculate the (i, i)-th entry of [Υ, Υ̂] as

∑_{j}(b_{i}b̂_{j} − b̂_{i}b_{j}) (∑_{k}(k − 1)β_{i}^{k−2}W_{kj}) (∑_{k}(k − 1)β_{j}^{k−2}W_{ki}) .

This means that

tr[Υ, Υ̂]β_{d} = ∑_{i<j}(b_{i}b̂_{j}−b̂_{i}b_{j})(β_{i}−β_{j}) (∑_{k}(k − 1)β_{i}^{k−2}W_{kj}) (∑_{k}(k − 1)β_{j}^{k−2}W_{ki}) .

Formula (5.7) will be proven if we can show (for i ≠ j) the following identity:

(∑_{k}(k − 1)β_{i}^{k−2}W_{kj}) (∑_{k}(k − 1)β_{j}^{k−2}W_{ki}) = \frac{−1}{(β_{i} − β_{j})^{2}} .  (5.10)

According to (5.9) we have

∑_{k}(k − 1)β_{i}^{k−2}W_{kj} = \frac{∑_{k}(k − 1)β_{i}^{k−2}(−1)^{n−k}S_{n−k}(β_{1}, …, β̂_{j}, …, β_{n})}{∏_{s≠j}(β_{j} − β_{s})}.(5.11)

We compute the numerator of this expression. We set p = n − 1 and (a_{1}, …, a_{p}) = (β_{1}, …, β̂_{j}, …, β_{n}). Then the numerator can be written as

∑_{s=0}^{p}(p − s)(−1)^{s}a_{i}^{p−1−s}S_{s}(a_{1}, …, a_{p}) = \frac{d}{dt} (∑_{s=0}^{p}(−1)^{s}S_{s}t^{p−s})_{t=a_{i}} .

Since ∑ S_{s}t^{s} = ∏(1 + a_{s}t), we can rewrite the expression under the derivative as

∑_{s=0}^{p}(−1)^{s}S_{s}t^{p−s} = ∏_{s=0}^{p}(t − a_{s}).

Taking the derivative and substituting a_{i} for t, finally gives

∑_{s=0}^{p}(p − s)(−1)^{s}a_{i}^{p−1−s} = ∏_{s≠i}(a_{i} − a_{s}).

Going back to (5.11), we have

∑_{k}(k − 1)β_{i}^{k−2}W_{kj} = \frac{∏_{s≠i,j}(β_{i} − β_{s})}{∏_{s≠j}(β_{j} − β_{s})},

from which (5.10) follows.

Remark 5.5. Setting

p_{i} = u_{i}/ ∏_{j>i}(β_{i} − β_{j}),

the formula (5.7) can be rewritten as

ω = ∑_{i=1}^{n} \frac{dp_{i}}{p_{i}} ∧ dβ_{i}.

6. The twistor space and the metric on M̃_{n}(c)

We shall now identify the twistor space Z(c) of M̃_{n}(c). As a first step, we observe, after Hitchin et al. [18], that the hyperkähler moment map μ for the T^{n}- action defines a moment map, also denoted by μ, for the complex-symplectic form along the fibers Z(c) → ℂP^{1}. This μ is a map from Z(c) to O(2) ⊗ ℂ^{n}. We shall first identify the open subset Z^{reg}(c) of Z(c) defined as the set

Z^{reg}(c) = μ^{−1} (O(2) ⊗ ℂ^{n} − O(2) ⊗ Δ) ,  (6.1)

where Δ is the generalized diagonal in ℂ^{n}. In terms of the coordinates (β_{1}, …, β_{n}) and (u_{1}, …, u_{n}) given by (5.6), Z^{reg}(c) has the following description:

Proposition 6.1. Z^{reg}(c) is obtained by taking two copies of ℂ×(ℂ^{n} −Δ)×(ℂ^{∗})^{n} with coordinates (ζ, β_{i}, u_{i}) and (ζ̃, β̃_{i}, ũ_{i}), i = 1, …, n, and identifying over ζ ≠ 0 by

ζ̃ = ζ^{−1}
β̃_{i} = ζ^{−2}β_{i}
ũ_{i} = ζ^{−(n−1)} exp{−cβ_{i}/ζ}u_{i}.

The real structure is given by

ζ ↦ −1/ζ̄
β_{i} ↦ −β̄_{i}/ζ̄^{2}
u_{i} ↦ ū_{i}^{−1} (1/ζ̄)^{n−1} ∏_{j≠i}(β̄_{i} − β̄_{j})e^{cβ̄_{i}/ζ̄}.

Finally, the complex symplectic form along the fibers is given by (5.7).

Proof. For any hyperkähler moduli space of solutions to Nahm’s equations one can trivialize the twistor space by choosing an affine coordinate ζ on ℂP^{1} and then putting η = β + (α + α^{∗})ζ − β^{∗}ζ^{2}, u = α − β^{∗}ζ for ζ ≠ ∞, and η̃ = β/ζ^{2} + (α + α^{∗})/ζ − β^{∗}, ũ = −α^{∗} − β/ζ for ζ ≠ 0. Then, over ζ ≠ 0, ∞, we have η̃ = η/ζ^{2}, ũ = u − η/ζ. Moreover, the real structure is ζ ↦ −1/ζ̄, η ↦ −η^{∗}/ζ̄^{2}, u ↦ −u^{∗} + η^{∗}/ζ̄ (cf. [12, 9].

We now have to go through the procedure in the proof of Proposition 5.3 to describe Z^{reg} in coordinates (ζ, β_{i}, u_{i}) and (ζ̃, β̃_{i}, ũ_{i}). First we describe the twistor space of N_{n} in coordinates (g, B) and (g̃, B̃) defined right after (5.5) (cf. [12]). Going through the procedure assigning (g, B) to (α, β) we see that B̃ = B(β_{1}/ζ^{−2}, …, β_{n}/ζ^{−2}). On the other hand g is given by g = g(1) where g(t) is a complex gauge transformation such that \frac{d}{dt}g^{−1} = −ug^{−1}. This means that g(t) makes u identically zero. We observe that exp{−Bt/ζ}g(t) makes ũ identically zero and η̃ into B/ζ^{2}. The initial value for the solution g^{−1} depends on ζ and so we can write g̃(t) = U exp{−Bt/ζ}g(t) for some constant matrix U. If we are to get the form (5.5), we must have U = U′d(ζ), where

d(ζ) = diag (ζ^{−(n−1)}, ζ^{−(n−3)}, …, ζ^{n−1}) .  (6.2)

In addition U′ commutes with B(β_{1}/ζ^{−2}, …, β_{n}/ζ^{−2}). Moreover, the initial value for the equation \frac{d}{dt}g^{−1} = −αg^{−1} depends only on the residues of u, η, ũ, η̃ and therefore U′ does not depend on B. Since the initial values belong to SU(n), we also have U′ ∈ SU(n). It follows that U′ belongs to the center of SU(n). This is only an ambiguity in the choice of trivialization and it does not affect the twistor space. Similar considerations show that the real structure sends B(β_{1}, …, β_{n}) to B(−β̄_{1}/ζ̄^{−2}, …, −β̄_{n}/ζ̄^{−2}) and g to r(ζ) exp{B^{∗}/ζ̄} (g^{∗})^{−1} where

r_{ij}(ζ) = \begin{cases}0 & if & i + j ≠ n + 1 \\ (−1)^{j−1}ζ̄^{n+1−2j} & if & i + j = n + 1 \end{cases} .

This time the remaining ambiguity is given by a real element in the center of SU(n), i.e. −1 if n is even.

We now go through a similar procedure for the subset of M_{U(n)}(c − 1) where β(+∞) is regular. We have assigned in the proof of Proposition 5.3 to each element of this set a pair (g, β(+∞). We already know how β(+∞) changes (as it is given by the complex moment map for a torus action). The proof of Proposition 5.3 shows that the other coordinates, g on {ζ ≠ ∞} and g̃ on {ζ ≠ 0}, are related by g̃ = g exp{−(c − 1)β(+∞)/ζ}. The real structure sends g to (g^{∗})^{−1} exp{(c − 1)β(+∞)^{∗}/ζ̄}.

Finally we have to go to the complex-symplectic quotient as in the proof of Proposition 5.3. We end up with (g, β_{d}) and (g̃, β̃_{d}) where β_{d} = diag(β_{1}, …, β_{n}) and gβ_{d}g^{−1} = B(β_{1}, …, β_{n}) (and similarily for (g̃, β̃_{d})). We see that β_{i} and β̃_{i} are related as stated and g̃ = d(ζ) exp{−B/ζ}g exp{−(c − 1)β_{d}/ζ}. Since exp{−B/ζ}g = g exp{−β_{d}/ζ}, g̃ = d(ζ)g exp{−cβ_{d}/ζ}. If we now go to the coordinates u_{i}, ũ_{i} defined by (5.6), we see that they change as required, since the (i, j)-th entry of V^{−1} is given by (5.9) and the β_{i} change as prescribed (i.e. as sections of O(2)). A similar argument shows that the real structure is, up to a sign, the one described in the statement (it is enough to compare the last row in r(ζ)(V^{−1} diag{u_{i}})^{∗−1} diag{e^{cβ̄_{i}/ζ̄}} and in V^{−1}(−β̄_{1}/ζ̄^{−2}, …, −β̄_{n}/ζ̄^{−2}) diag{u′_{i}}). We shall see shortly (Proposition 6.2) that the negative of the real structure described in the statement does not admit any sections (a section would be equivalent to a complex number with imaginary modulus). The formula for the complex symplectic structure is a direct consequence of Proposition 5.4.

We now wish to find the full twistor space and the metric on M̃_{n}(c) and this means finding a family of real sections. We know their projections to O(2) ⊗ ℂ^{n}: they are given by (β + (α + α^{∗})ζ − β^{∗}) (+∞) (cf. [18]) and are parameterized by n distinct points in ℝ^{3} with coordinates (x_{i}, Re z_{i}, Im z_{i}), i = 1, …, n, where x_{i} = \sqrt{−1}T_{1}(+∞), z_{i} = β(+∞). In other words we have n curves S_{i} = \{(ζ, η); η = z_{i} +2x_{i}ζ −z̄_{i}ζ^{2}} in T ℂP^{1} (here η is the fiber coordinate). According to Proposition 6.1 the u_{i} coordinate of a real section of Z(c) changes as a non-zero section of the bundle L^{c}(k − 1) (with the transition function ζ^{k−1}e^{cη/ζ} from ∞ to 0) over S_{i}. This is true only away from the intersection points of the curves S_{i} and we have to understand what happens to the section at these points. Two curves S_{i} = {(ζ, η); η = z_{i} + 2x_{i}ζ − z̄_{i}ζ^{2}} and S_{j} = {(ζ, η); η = z_{j} + 2x_{j}ζ − z̄_{j}ζ^{2}} intersect in a pair of distinct points a_{ij} and a_{ji}, where

a_{ij} = \frac{(x_{i} − x_{j}) + r_{ij}}{z̄_{i} − z̄_{j}} , r_{ij} = \sqrt{(x_{i} − x_{j})^{2} + |z_{i} − z_{j}|^{2}}.  (6.3)

We have:

Proposition 6.2. The real sections of the twistor space Z(c) of M̃_{n}(c) are given, over ζ ≠ ∞, by (β_{1}(ζ), …, β_{n}(ζ), u_{1}(ζ), …, u_{n}(ζ)), where

β_{i}(ζ) = z_{i} + 2x_{i}ζ − z̄_{i}ζ^{2},

u_{i}(ζ) = A_{i} ∏_{j≠i}(ζ − a_{ji})e^{c(x_{i}−z̄_{i}ζ)},

where (x_{i}, z_{i}), i = 1, …, n, are distinct points in ℝ×ℂ and A_{i} are complex numbers satisfying

A_{i}Ā_{i} = ∏_{j≠i}(x_{i} − x_{j} + r_{ij}).

Remark . Given Proposition 5.2, this finally shows that the biholomorphism of Proposition 5.3 is onto.

Proof. Consider a real section s of Z(c) (corresponding to a solution (T_{0}, T_{1}, T_{2}, T_{3})) which projects to a given real section (β_{1}(ζ), …, β_{n}(ζ)) of O(2)⊗ℂ^{n}. For a generic section the intersection points of the β_{s} are all distinct. We consider the point a_{ji} at which β_{i} intersects β_{j} and let us assume that no other β_{s} intersect there. We recall that \sqrt{−1}T_{1}(ζ) = \frac{1}{2}(α + α^{∗}) − β^{∗}ζ and, hence, \sqrt{−1}T_{1}(ζ)(+∞)_{ss} = x_{s} − z̄_{s}ζ. This means that \sqrt{−1}T_{1}(a_{ji})(+∞)_{jj} < \sqrt{−1}T_{1}(a_{ji})(+∞)_{ii}, and so, with respect to the complex structure corresponding to a_{ji} ∈ ℂP^{1}, the solution (T_{0}, T_{1}, T_{2}, T_{3}) belongs to the chart described in Proposition 5.3 with 𝔫 generated by the matrix with the only non-zero entry having coordinates (i, j). Let us write s as (β_{i}(ζ), u_{i}(ζ)), i = 1, …, n, in a neighbourhood of a_{ji}, ζ ≠ a_{ji} (notice that the procedure of Proposition 6.1 does assign well-defined complex numbers u_{1}(ζ), …, u_{n}(ζ) to each ζ ≠ a_{ji}). According to the proof of Proposition 5.3 there is an element m(ζ) ∈ N = exp 𝔫 such that the following expression

V (β_{1}(ζ), …, β_{n}(ζ))^{−1} diag(u_{1}(ζ), …, u_{n}(ζ))m(ζ)

has an invertible limit at ζ = a_{ji}. Let W_{kl}(ζ) denotes the (k, l)-th entry of V (β_{1}(ζ), …, β_{n}(ζ))^{−1} and let p(ζ) denote the only non-zero non-diagonal entry of diag(u_{1}(ζ), …, u_{n}(ζ))m(ζ) (p(ζ) is the (i, j)-th entry). We then have that W_{kj}u_{j} + W_{ki}p and W_{ki}u_{i} have a finite limit at ζ = a_{ji}, for all k = 1, …, n. From the formula (5.9) a finite limit for W_{ni}u_{i} implies that u_{i}(a_{ji}) = 0, while the nonvanishing of the last row of V^{−1} diag(u_{s})m means that a_{ji} is a single zero of u_{i}. If more than two sections β_{s}(ζ) meet at a_{ji} the considerations are similar but involve larger 𝔫. We can conclude the a_{ji} contribute precisely n − 1 zeros of u_{i} (counting multiplicities) and, given Proposition 6.1, this proves the formula for u_{i}(ζ) as soon as we show that u_{i} has no other zeros, or, equivalently, no poles. To prove this latter statement it is enough to show that u_{j} does not have a pole at a_{ji}. We go back to the situation when 𝔫 is one-dimensional, and where we concluded that W_{kj}u_{j} + W_{ki}p has a finite limit at ζ = a_{ji} for all k = 1, …, n. We can write W_{nj}u_{j} + W_{ni}p as (fu_{j} + gp)/(β_{i} − β_{j}) where f and g have finite limits at ζ = a_{ji}. We then have

W_{n−1,j}u_{j} + W_{n−1,i}p = − (fu_{j}(∑_{s≠j} β_{s}) + gp(∑_{s≠i} β_{s})) \frac{1}{β_{i} − β_{j}}

which can be rewritten as

−fu_{j} − (∑_{s≠i} β_{s})\frac{fu_{j} + gp}{β_{i} − β_{j}}.

Since the second term has a finite limit, so does fu_{j} and hence u_{j}. Again, if more than two sections β_{s}(ζ) meet at a_{ji} the considerations are similar but involve larger 𝔫. Thus we have shown the second formula of the statement. The last formula follows from the reality condition and the fact that a_{ji} = −1/ā_{ij} (this calculation also eliminates the ±1 ambiguity in the choice of the real structure in the proof of 6.1).

We can finally identify M̃_{n}(c) as a T^{n}-bundle over the configuration space C̃_{n}(ℝ^{3}) of n distinct points x_{i} in ℝ^{3}.

Proposition 6.3. M̃_{n}(c) is equivalent to the T^{n}-bundle described in Proposition 1.1.

Proof. From the last formula in Proposition 6.2 it follows that A_{i} ≠ 0 if, for all j ≠ i, z_{i} ≠ z_{j} or x_{i} > x_{j}. On the other hand, if we put

A_{I} = A_{i} ∏_{j∈I} a_{ji},

for any subset I of {j; j ≠ i}, then we have

A_{I} Ā_{I} = ∏_{\begin{matrix}j≠i \\ j∉I \end{matrix}}(x_{i} − x_{j} + r_{ij}) ∏_{j∈I}(x_{j} − x_{i} + r_{ij}).

Let us choose sets I_{1}, …, I_{n} such that I_{i} ⊂ {j; j ≠ i} and j ∈ I_{i} ⇔ i ∉ I_{j}. Define U(I_{1}, …, I_{n}) as the complement of the subset {(x_{i}, z_{i})_{i=1,...,n}; I_{i}^{c} = {j; z_{i} = z_{j} and x_{i} < x_{j}}} (I_{i}^{c} denotes the complement of I_{i} in {j; j ≠ i}). The sets U(I_{1}, …, I_{n}) cover C̃_{n}(ℝ^{3}) and over each of them the bundle M̃_{n}(c) is trivialized by coordinates (x_{i}, z_{i}, A_{I_{i}}/|A_{I_{i}}|). To determine the bundle, choose i < j. The bundle restricted to S_{ij}^{2} is given by the transition function from U(I_{1}, …, I_{n}) where j ∉ I(i) to U(I′_{1}, …, I′_{n}) where I′_{i} = I_{i} ∪ {j}, I′_{j} = I_{j} − {i}, I′_{k} = I_{k} for k ≠ i, j. Let ϕ_{k} be the transition function for the k-th generator of T^{n}, i.e. the transition function from A_{I_{k}}/|A_{I_{k}}| to A_{I′_{k}}/|A_{I′_{k}}|. We see that ϕ_{k} = 1 if k ≠ i, j, and ϕ_{i} = a_{ji}/|a_{ji}|, ϕ_{j} = |a_{ji}|/a_{ji}. Therefore ϕ_{i} = (z_{j} − z_{i})/|z_{j} − z_{i}| and ϕ_{j} = ϕ_{i}^{−1}. It remains to identify the circle bundle over the sphere x^{2} + |z|^{2} = const given by the transition function z/|z| from the region U_{0} = {z ≠ 0 or x > 0} to the region U_{1} = {z ≠ 0 or x < 0}. Let us write the unit 3-sphere as {(u, v) ∈ ℂ^{2}; |u|^{2} + |v|^{2} = 1}. The Hopf bundle is given the S^{1} action t · (u, v) = (tu, t^{−1}v) and the projection S^{3} → S^{2} by the map x = |u|^{2} − |v|^{2}, z = 2uv. Over U_{0} this bundle is trivialized by (x, z, u/|u|) and over U_{1} by (x, z, |v|/v). The transition function is |z|/z. Thus [ϕ_{i}] = −1 ∈ H^{1}(S_{ij}^{2}, S^{1}).

We can now calculate the metric on M̃_{n}(c). By the remark at the end of section 3, it is enough to know the metric for c = −1, 0, 1, as the others are obtained by homothety. We shall calculate the metric for c = 1. The metric for c = −1 is the everywhere negative definite version of the Gibbons-Manton metric (this can be seen from the c = 1 calculation) and the one for c = 0 is the negative-definite cone metric over a 3-Sasakian manifold.

Theorem 6.4. M̃_{n}(1) is isomorphic, as a hyperkähler manifold, to the Gibbons-Manton manifold M_{GM} defined in section 1.

Proof. We know from the previous proposition that the two spaces are diffeomorphic. We shall show that the twistor description of M̃_{n}(1) and of the Gibbons-Manton metric coincide. We recall from section 1 that the latter is a hyperkähler quotient of M = M_{1} × M_{2} by a torus, where M_{1} = (S^{1} × ℝ^{3})^{n} and M_{2} = ℍ^{n(n−1)/2}. With respect to any complex structure M_{1} = (ℂ^{∗})^{n} × ℂ^{n} and M_{2} = ℂ^{n(n−1)/2} × ℂ^{n(n−1)/2}. Let us write the corresponding complex coordinates as (p_{i}, β_{i}), i = 1, …, n, on M_{1} and as (v_{ij}, w_{ij}), i < j, on M_{2}. The complex-symplectic forms corresponding to metrics g_{1} and g_{2} are given by

∑_{i=1}^{n} \frac{dp_{i}}{p_{i}} ∧ dβ_{i}  (6.4)
∑_{i<j} dv_{ij} ∧ dw_{ij}.  (6.5)

The real sections of the twistor space Z_{1} of M_{1} are written, over ζ ≠ ∞, as

β_{i}(ζ) = z_{i} + 2x_{i}ζ − z̄_{i}ζ^{2},  p_{i}(ζ) = B_{i}e^{x_{i}−z̄_{i}ζ},  (6.6)

where B_{i}B̄_{i} = 1. The real sections of the twistor space Z_{2} of M_{2} are (cf. [2], chapter 13.F):

v_{ij}(ζ) = C_{ij}(ζ − a_{ij}),  w_{ij}(ζ) = D_{ij}(ζ − a_{ji}),  (6.7)

where a_{ij}, a_{ji} are roots of v_{ij}w_{ij} = z_{ij} + 2x_{ij}ζ − z̄_{ij}ζ^{2} for some (x_{ij}, z_{ij}) ∈ ℝ × ℂ, i.e.

a_{ij} = \frac{x_{ij} + \sqrt{x_{ij}^{2} + |z_{ij}|^{2}}}{z̄_{ij}} , a_{ji} = \frac{x_{ij} − \sqrt{x_{ij}^{2} + |z_{ij}|^{2}}}{z̄_{ij}}

and

C_{ij}C̄_{ij} = −x_{ij} + \sqrt{x_{ij}^{2} + |z_{ij}|^{2}}, D_{ij}D̄_{ij} = x_{ij} + \sqrt{x_{ij}^{2} + |z_{ij}|^{2}}.

Here the paricular choice of sections is forced either by the fact the metric is positive definite or by requiring that the S^{1}-action t · (v_{ij}, w_{ij}) = (tv_{ij}, t^{−1}w_{ij}) determines the Hopf bundle over the 2-sphere x_{ij}^{2} + |z_{ij}|^{2} = 1 (this calculation was done in the proof of Proposition 6.3). To obtain the twistor description of the Gibbons-Manton metric we have to perform the complex-symplectic quotient construction along the fibers of Z_{1} ⊕ Z_{2} with respect to the difference of the forms (6.4) and (6.5). As in section 1, the moment map equations are v_{ij}w_{ij} = β_{i} − β_{j} and so the a_{ij}, a_{ji} are given by (6.3). Since we already know that the manifolds are diffeomorphic, it is sufficient to determine the metric on an open dense subset, e.g. on the set where all v_{ij} are non-zero. Quotienting this set by (ℂ^{∗})^{n(n−1)/2} is equivalent to sending

all v_{ij} to 1. This is achieved by acting by the element (v_{ij})^{−1} of (ℂ^{∗})^{n(n−1)/2}. By the description of the torus action given in section 1, this sends p_{i}(ζ) to

B_{i} \frac{∏_{j<i} C_{ji}(ζ − a_{ji})}{∏_{j>i} C_{ij}(ζ − a_{ij})}e^{x_{i}−z̄_{i}ζ} = E_{i} \frac{∏_{j<i}(ζ − a_{ji})}{∏_{j>i}(ζ − a_{ij})}e^{x_{i}−z̄_{i}ζ},  (6.8)

where

E_{i}Ē_{i} = \frac{∏_{j<i}(x_{i} − x_{j} + r_{ij})}{∏_{j>i}(x_{j} − x_{i} + r_{ij})}.  (6.9)

These and the β_{i} give the real sections for the Gibbons-Manton metric and the symplectic form is (6.4). We now compare this with the description of Z(1) given in Proposition 6.2. According to Remark 5.5 we should set p_{i} = u_{i}/ ∏_{j>i}(β_{i} − β_{j}) in order to have the same symplectic form. We obtain

p_{i}(ζ) = \frac{A_{i}}{∏_{j>i}(z̄_{j} − z̄_{i})} \frac{∏_{j<i}(ζ − a_{ji})}{∏_{j>i}(ζ − a_{ij})}e^{x_{i}−z̄_{i}ζ}.

All we have to do is to compare is the norm of A_{i}/ ∏_{j>i}(z̄_{j} − z̄_{i}) with the norm of E_{i}. We have, from Proposition 6.2 and equation (6.9),

\frac{A_{i}Ā_{i}}{∏_{j>i} |z_{j} − z_{i}|^{2}} = \frac{∏_{j≠i}(x_{i} − x_{j} + r_{ij})}{∏_{j>i} |z_{j} − z_{i}|^{2}} = ∏_{j<i}(x_{i} − x_{j} + r_{ij}) ∏_{j>i} \frac{(x_{i} − x_{j} + r_{ij})}{|z_{j} − z_{i}|^{2}}

= \frac{∏_{j<i}(x_{i} − x_{j} + r_{ij})}{∏_{j>i}(x_{j} − x_{i} + r_{ij})} = E_{i}Ē_{i},

which proves the theorem.

We shall finish the section with a remark that Propositions 6.2 and 6.3 can be generalized to define hyperkähler metrics on a class of T^{n}-bundles over C̃_{n}(ℝ^{3}). We have:

Theorem 6.5. Let P be a T^{n}-bundle over C̃_{n}(ℝ^{3}) determined by an element (s_{1}, …, s_{n}) of H^{2}(C̃_{n}(ℝ^{3}), ℤ^{n}) satisfying s_{k}(S_{ij}^{2}) = 0 if k ≠ i, j and s_{i}(S_{ij}^{2}) = −s_{j}(S_{ij}^{2}). Then P carries a family of (pseudo)-hyperkähler metrics such that the real sections of the twistor space are given, over ζ ≠ ∞, by (β_{1}(ζ), …, β_{n}(ζ), u_{1}(ζ), …, u_{n}(ζ)), where

β_{i}(ζ) = z_{i} + 2x_{i}ζ − z̄_{i}ζ^{2},

u_{i}(ζ) = A_{i} ∏_{j≠i}(ζ − a_{ji})^{s_{ij}}e^{c(x_{i}−z̄_{i}ζ)},

where c is a real constant, (x_{i}, z_{i}), i = 1, …, n, are distinct points in ℝ × ℂ, s_{ij} = |s_{i}(S_{ij}^{2})|, and A_{i} are complex numbers satisfying

A_{i}Ā_{i} = ∏_{j≠i}(x_{i} − x_{j} + r_{ij})^{s_{ij}}.  □

This description determines a hypercomplex structure on P. A (pseudo)-hyperkähler metric can be then calculated using any complex-symplectic form along the fibers, given as a section of Λ^{2}T_{F}^{∗} ⊗O(2), e.g. the form (5.7). These metrics will correspond to the motion of n dyons in ℝ^{3} interacting in different ways (cf. [14]).

Remark . The calculation of the metric given above shows that the Taub-NUT metric (cf. [2]) has two very different descriptions in terms of Nahm’s equaations:

1) it is the metric on the totally geodesic submanifold M̃_{2}^{0}(−1) of M̃_{2}(−1) defined by considering 𝔰𝔲(2)-valued solutions to Nahm’s equations and SU(2)-valued gauge transformations; 2) it is the metric on the moduli space of SU(3)-monopoles of charge (1, 1) [10, 29].

7. Asymptotic comparison of the metrics

We shall now show that the Gibbons-Manton metric and the monopole metric are asymptotically exponentially close. The asymptotic region, where the individual monopoles are separated, of the monopole space M_{n} is diffeomorphic to P/S_{n}, where P is a torus bundle over the configuration space C̃_{n}(ℝ^{3}) and S_{n} the symmetric group. The bundle P is not, however, the bundle of Proposition 6.3. Rather, as we shall see shortly, it is the quotient of that bundle by a (ℤ_{2})^{n}-subgroup of T^{n}. In other words it is the bundle determined by an s ∈ H^{2}(C̃_{n}(ℝ^{3}), ℤ^{n}) with all s_{k} being twice of those in Proposition 6.3.

We shall compare the metric on M_{n} with the metric on the hyperkähler quotient of M̃_{n}(1) × M̃_{n}(1) by the diagonal T^{n}-action. We do this in order to have solutions to Nahm’s equations with poles at both ends of the interval [−1, 1]. For any c, c′, let us write M̃_{n}(c, c′) for the hyperkähler quotient of M̃_{n}(c) × M̃_{n}(c′) by the diagonal action of T^{n}. The action of T^{n} given by t · (m, m′) = (tm, m′) induces a tri-Hamiltonian action of T^{n} on M̃_{n}(c, c′) which makes M̃_{n}(c, c′) into a T^{n}-bundle over C̃_{n}(ℝ^{3}). We have

Lemma 7.1. M̃_{n}(c, c′) is isomorphic, as a hyperkähler manifold, to M̃_{n}(c+c′)/(ℤ_{2})^{n}, where (ℤ_{2})^{n} = {t ∈ T^{n}; t^{2} = 1}.

Proof. Let μ, μ′ be the moment maps for the action of T^{n} on M̃_{n}(c), M̃_{n}(c′) respectively. The moment map for the diagonal T^{n}-action on the product is μ + μ′. If we go back to the proof of Proposition 6.3 and use the same notation, we can see that the zero-set of this moment map is a (T^{n} × T^{n})-bundle over C̃_{n}(ℝ^{3}) which restricted to each S_{ij}^{2} is given by transition functions (ϕ_{1}, …, ϕ_{n}, ϕ_{1}^{−1}, …, ϕ_{n}^{−1}) (the point being that U(I′_{1}, …, I′_{n}) = −U(I_{1}, …, I_{n})). Hence, if we quotient by T^{n}, by sending the second T^{n} to 1 over each U(I_{1}, …, I_{n}), we end up with a T^{n}-bundle for which the transition functions are ϕ_{k}^{2}, k = 1, …, n. This proves the differentialgeometric part of the statement. To obtain the isometry we repeat this argument for the twistor space of M̃_{n}(c) × M̃_{n}(c′), performing the complex-symplectic quotient along the fibers as in the proof of Theorem 6.4.

¿From now on, we shall consider M̃_{n}(1, 1) with half (compare the formula (2.4)) of the metric given by the above lemma. In other words, locally the metric is still the Gibbons-Manton metric.

We can identify M̃_{n}(1, 1) with the moduli space of pairs ((T_{0}, T_{1}, T_{2}, T_{3}), (T′_{0}, T′_{1}, T′_{2}, T′_{3})) of solutions to Nahm’s equations, defined respectively on [−1, ∞] and on [−∞, 1], such that T_{i}(+∞) = T′_{i}(−∞) for i = 0, 1, 2, 3, and the residues of T_{i} at −1 and of T′_{i} at +1, i = 1, 2, 3, define the standard n-dimensional irreducible representation of 𝔰𝔲(2). The group of gauge transformations G(1, 1) is now defined as pairs (g, g′) such that g(t + 1), g′(−t + 1) ∈ G(c) for some c and s = lim_{t→+∞} ġg^{−1} = lim_{t→−∞} ġ′g^{′−1}. The tangent space consists of pairs ((t_{0}, t_{1}, t_{2}, t_{3}), (t′_{0}, t′_{1}, t′_{2}, t′_{3})) defined on [−1, ∞] and on [−∞, 1], respectively, with t_{i}(+∞) = t′_{i}(−∞) and satisfying equations (2.3). The metric on M̃_{n}(1, 1) can be written as

\frac{1}{2} ∑_{0}^{3} ∥t_{i}(+∞)∥^{2} + \frac{1}{2} ∫_{−1}^{+∞} ∑_{0}^{3} (∥t_{i}(s)∥^{2} − ∥t_{i}(+∞)∥^{2}) ds +

\frac{1}{2} ∑_{0}^{3} ∥t′_{i}(−∞)∥^{2} + \frac{1}{2} ∫_{−∞}^{1} ∑_{0}^{3} (∥t′_{i}(s)∥^{2} − ∥t′_{i}(−∞)∥^{2}) ds.

We can rewrite this as

\frac{1}{2} ∫_{0}^{+∞} ∑_{0}^{3} (∥t_{i}(s)∥^{2} − ∥t_{i}(+∞)∥^{2}) ds + \frac{1}{2} ∫_{−∞}^{0} ∑_{0}^{3} (∥t′_{i}(s)∥^{2} − ∥t_{i}(+∞)∥^{2}) ds

+ \frac{1}{2} ∫_{−1}^{0} ∑_{0}^{3} ∥t_{i}(s)∥^{2}ds + \frac{1}{2} ∫_{0}^{1} ∑_{0}^{3} ∥t′_{i}(s)∥^{2}ds. (7.1)

Let us fix complex structure, say I and write as in section 5, α for T_{0} + iT_{1}, β for T_{2} + iT_{3}. We write an element of M̃_{n}(1, 1) as a pair ((α_{−}, β_{−}), (α_{+}, β_{+})). We shall write β_{i} for the (i, i)-th entry of β_{−}(+∞) = β_{+}(−∞) and denote by M̃_{n}^{reg}(1, 1) the subset of M̃_{n}(1, 1) where all β_{i} are distinct. Similarily, we write M_{n}^{reg} for the subset of (α, β) in (M_{n}, I) where the eigenvalues of β are distinct. We shall prove:

Theorem 7.2. There exists a biholomorphism ϕ from M̃_{n}^{reg}(1, 1)/S_{n} to M_{n}^{reg} such that

|ϕ^{∗}g − g′| = O(e^{−cR})  (7.2)

where g, g′ denote the monopole and Gibbons-Manton metric respectively, c = c(n) is a constant, and R is the separation distance of particles in C_{n}(ℝ^{3}), i.e.

R = min{|x_{i} − x_{j}|; i ≠ j}.  (7.3)

The same estimate holds for the Riemannian curvature tensor.

Since such a biholomorphism will be defined for any complex structure and the union of M̃_{n}^{reg}(1, 1) for different complex structures is all of M̃_{n}(1, 1), we conclude that the monopole and the Gibbons-Manton metrics are exponentially close in the asymptotic region of the monopole moduli space.

The remainder of the section is devoted to proving this theorem. We need the following lemma:

Lemma 7.3. Let C > 0. The space M̃_{n}^{reg}(1) is biholomorphic to the quotient of the space of solutions (α, β) to the equation (5.1) which have the correct boundary behaviour at t = 0 and are constant (hence diagonal) for t ≥ C by the group of complex gauge transformations g : [0, +∞) → Gl(n, ℂ) with g(0) = 1 and g(t) = exp(ht − h) for some diagonal h for t ≥ C.

Proof. Let (α, β) be an element of M̃_{n}^{reg}(1) and let α_{d} = α(+∞), β_{d} = β(+∞). According to the proof of Proposition 5.3, there is unique complex gauge transformation g defined on [C/2, +∞) with g(+∞) = 1 such that (α, β) = g(α_{d}, β_{d}). Let ĝ : [C/2, ∞) → Gl(n, ℂ) be a smooth path with the values and the first derivatives of ĝ and g coinciding at t = C/2 and with ĝ(t) = 1 and for t ≥ C. We obtain a solution (α̂, β̂) to the complex Nahm equation (5.1) by setting

(α̂, β̂)(t) = \begin{cases}(α, β)(t) & if t < C \\ ĝ(t)(α_{d}, β_{d}) & if t ≥ C. \end{cases}  (7.4)

This is a solution of the type described in the statement of this lemma. The proof of 5.3 shows further that it is only g(C/2) exp{(1 − C/2)α_{d}} (and a solution to (5.1) on [0, C/2]) that determines the element of M̃_{n}^{reg}(1). Therefore we obtain a well defined holomorphic map from M̃_{n}^{reg}(1) to the moduli space described in the statement. Let us define the inverse map. Let (α̂, β̂) be an element of the moduli space described in the statement. As in [23] we can find a bounded complex gauge transformation g_{0} such that g_{0}(α̂, β̂) is an element of M̃_{n}^{reg}(1). We can assume that g_{0} has a limit h at +∞ (this follows from the convexity property of g_{0} [13], since we can assume that g_{0}(t) is hermitian for all t). According to Proposition 6.2 the action of T^{n} on M̃_{n}(1) extends to a global action (ℂ^{∗})^{n} with respect to the complex structure I (or any other). Let (α, β) be the element of M̃_{n}^{reg}(1) obtained from g_{0}(α̂, β̂) by the action of h^{−1} ∈ (ℂ^{∗})^{n}. Then (α, β) = g(α̂, β̂) and g ∈ G^{ℂ}(1). This gives the inverse mapping.

We can now construct a biholomorphism between M̃_{n}^{reg}(1, 1)/S_{n} and M_{n}^{reg}. From the above lemma, M̃_{n}^{reg}(1, 1) is biholomorphic to the quotient of the space of pairs ((α_{−}, β_{−}), (α_{+}, β_{+})) defined on [−1, +∞) and on (−∞, 1] respectively such that (α_{−}, β_{−})(t + 1) and (α_{+}, β_{+})(1 − t) are as in the above lemma, (α_{−}, β_{−})(+∞) = (α_{+}, β_{+})(−∞) by the group of pairs (g_{−}, g_{+}) with g_{−}(−1) = g_{+}(1) = 1 and such that there are diagonal h, p with g_{−}(t) = exp(th−p) for t > −r, g_{+}(t) = exp(th−p) for t < r (r ∈ (0, 1) is fixed but arbitrary). We define a solution (α̂, β̂) to the complex Nahm equation (5.1) on (−1, 1) by

(α̂, β̂)(t) = \begin{cases}(α_{−}, β_{−})(t) & if t < 0 \\ (α_{+}, β_{+})(t) & if t ≥ 0. \end{cases}  (7.5)

The G^{ℂ}-orbit of this solution (see section 2 for the definition of G) contains a unique element of M_{n} [13, 20]. Furthermore, the action of a (g_{−}, g_{+}) translates into the action of g ∈ G^{ℂ}, where g(t) = g_{−}(t) for t < 0 and g(t) = g_{+}(t) for t ≥ 0. Therefore we have a well defined holomorphic map ϕ_{r} from M̃_{n}^{reg}(1, 1) to M_{n}. If we now have an element (α, β) of M_{n}^{reg}, we can diagonalize β on [−r, r] and make α diagonal and constant on [−r, r]. Let (α̃, β̃) be the resulting solution to the complex Nahm equation. We obtain an element of M̃_{n}^{reg}(1, 1) by setting

(α_{−}, β_{−})(t) = \begin{cases}(α̃, β̃)(t) & for t < 0 \\ (α̃, β̃)(0) & for t ≥ 0 \end{cases}

and similarily for (α_{+}, β_{+}). This defines the inverse to ϕ_{r} up to the ordering of eigenvalues of β. In other words ϕ_{r} induces a biholomorphism between M̃_{n}^{reg}(1, 1)/S_{n} and M_{n}^{reg}. Furthermore, for a fixed element ((α_{−}, β_{−}), (α_{+}, β_{+})) of M̃_{n}^{reg}(1, 1) and two parameters r, r′, the resulting (α̂, β̂) of (7.5) are G^{ℂ}-equivalent and therefore ϕ_{r}, ϕ_{r′} induce the same biholomorphism ϕ.

Let us now prove the estimate (7.3). Fortunately, much of the analysis has been already done in [3]. First of all, we recall ([23], Lemma 3.4) that solutions to Nahm’s equations which have a regular triple as a limit at infinity, approach this limit exponentially fast, of order O (e^{−cR}) (that is T_{1}, T_{2}, T_{3} do and we can always make T_{0} to have such decay by using the gauge freedom). The proofs of Propositions 3.11 – 3.14 in [3] show that the same holds for tangent vectors (t_{0}, t_{1}, t_{2}, t_{3}). Let us now see what happens to a tangent vector v under the map ϕ. The gauge transformations (g, g′) which make the element ((α_{−}, β_{−}), (α_{+}, β_{+})) of M̃_{n}^{reg}(1, 1) constant and equal to the common value at infinity on [−1 + C/2, +∞) and (−∞, 1 − C/2] are exponentially close to the identity. In the next stage of the construction of ϕ - the formula (7.4) - we have smoothed out the solutions which can be again done by gauge transformations exponentially close to 1. Therefore the resulting tangent vector v̂ is exponentially close to the original one in the metric (7.1). We have then restricted the solutions (formula (7.5)) to obtain a solution (α̂, β̂) to the complex Nahm equation on [−1, 1]. Let p denote this operation of restriction. The first line of the formula (7.1) is exponentially small and therefore the norm of v̂ in (7.1) and the norm of dp(v̂) in (2.4) are exponentially close. The solution (α̂, β̂) will not satisfy the real Nahm equation, however, we will have

F (α̂, β̂) := \frac{d}{dt} (α̂ + α̂^{∗}) + [α̂, α̂^{∗}] + [β̂, β̂^{∗}] = O(e^{−cR}).

Lemma 2.10 in [13] implies now that we can solve the real equation by a complex gauge transformation bounded as O(e^{−cR}). We can now show that the vector dϕ(v) tangent to M_{n} (which is obtained from dp(v̂)) is exponentially close to dp(v̂) by following the analysis of section 3 in [3] step by step, replacing the O(1/R) estimates by O(e^{−cR}). This proves the estimate (7.3). For the curvature estimates we do the same using the analysis of section 4 in [3]. This proves Theorem 7.2.

8. Twistor description of monopoles and the Gibbons-Manton metric

We shall show in this section how the twistor description of monopole metrics determines the asymptotic metric. We recall [13] that the moduli space of nmonopoles is biholomorphic to the space of based rational maps p(z)/q(z) on ℂP^{1} of degree n (based means that deg p < deg q). On the set, where the roots β_{1}, …, β_{n} of q(z) are distinct, these roots and the values p_{i} = p(β_{i}) of p form local coordinates and the complex-symplectic form can be written as [1]:

∑_{i=1}^{n} dβ_{i} ∧ \frac{dp_{i}}{p_{i}} .  (8.1)

The metric is determined by the real sections p(z, ζ)/q(z, ζ). Their description is provided in [19]. The denominator q(z, ζ) is given by a curve S - the spectral curve of the monopole - in T ℂP^{1} [16]. This curve satisfies several conditions one of which is the triviality of the line bundle L^{−2} restricted to S and Hurtubise [19] shows that the numerator p(z, ζ) is given by a nonzero section of this bundle (the values p_{i}(ζ) are given by the values of this section at the intersection points β_{i}(ζ) of S with T_{ζ}ℂP^{1}.

What happens when the individual monopoles separate? First of all, the spectral curve approaches the union of spectral curves of individual monopoles exponentially fast [4]. These curves S_{i} are of the form η_{i} = z_{i} + 2x_{i}ζ − z̄_{i}ζ^{2}, i = 1, …, n, where (x_{i}, Re z_{i}, Im z_{i}) are locations of 1-monopoles (particles). What happens to the section of L^{−2}? We make a heuristic assumption (which we know to be true from section 6) that the section acquires zeros and poles at the intersection points of the S_{i} (more precisely the only singularities of p_{i}(ζ) occur at the intersection points of S_{i} with other S_{j}). As we shall see this is sufficient to determine the asymptotic metric.

First of all the real structure on the bundle L^{−2} is u ↦ ū^{−1}e^{−2η̄/ζ̄} and therefore if p_{i} has a zero at one of the points of S_{i} ∩S_{j}, then it has a pole of the same order at the other, and vice versa. Furthermore, since the metric and hence the real sections are invariant under the action of the symmetric group, we must have

p_{i}(ζ) = A_{i} ∏_{j≠i} (\frac{ζ − a_{ij}}{ζ − a_{ji}} )^{k} e^{−2(x_{i}−z̄_{i}ζ)}, i = 1, …, n,

where a_{ij}, a_{ji} are the two points in S_{i} ∩ S_{j} given by (6.3) and k is an integer. The reality condition implies that

A_{i}Ā_{i} = ∏_{j≠i} a_{ji}^{k}ā_{ji}^{k}.

One can now calculate the asymptotic metric, using (8.1). The sign of k will determine the signature, while |k| is simply a constant multiple. The actual value of k is determined by the topology of the asymptotic region of M_{n}, and comparing with Proposition 6.3 and the remarks at the beginning of section 7 we conclude that k = 1 (in the coordinates of Proposition 6.2, p_{i} = ∏_{j;j≠i}(β_{i} − β_{j})/u_{i}^{2}).

We remark that the above analysis can be easily done for other compact Lie groups G. The twistor description of metrics on moduli spaces of G-monopoles with maximal symmetry breaking is known from the work of Murray [28] and Hurtubise and Murray [21, 22] and from this the asymptotic metric can be calculated. We shall do the exact analysis in the case of G = SU(N) in a subsequent paper.

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Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, 53225 Bonn, Germany

   E-mail address: rogerb@mpim-bonn.mpg.de