# The Nahm–Schmid equations and hypersymplectic geometry

The Nahm–Schmid equations and hypersymplectic geometry Abstract We explore the geometry of the Nahm–Schmid equations, a version of Nahm’s equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang–Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral curves, as well as their relation to hypersymplectic geometry. 1. Introduction Hypersymplectic geometry [11, 15] can informally be thought of as a pseudo-Riemannian analogue of hyperkähler geometry, in which the role of the quaternions H is played instead by the algebra B of the split quaternions. The generators of B (over R ⁠) are denoted i,s,t and satisfy the relations i2=−1,s2=1=t2,is=t=−si. As an R-algebra, B is isomorphic to the associative algebra Mat2(R) of real 2×2 matrices, which is the split real form of the complex algebra Mat2(C) ⁠. A hypersymplectic manifold is then a manifold with an action of B on its tangent bundle which is compatible with a given pseudo-Riemannian metric of split signature; its dimension is necessarily a multiple of four. Definition 1.1 A hypersymplectic manifold is a quintuple (M4k,g,I,S,T) where g is a pseudo-Riemannian metric on M of signature (2k,2k) and I,S,T∈Γ(End(TM)) are parallel skew-adjoint endomorphisms satisfying the relations I2=−1,S2=1=T2,IS=T=−SI. The parallel endomorphism S (and similarly T ⁠) splits TM into its (±1)-eigenbundles, which are of equal rank and integrable in the sense of Frobenius. A hypersymplectic manifold M ⁠, therefore, locally splits (essentially, in a canonical way) as a product M+×M− ⁠, where the two factors have the same dimension 2k ⁠. For this reason, the endomorphism S is also called a local product structure in the literature. Given a hypersymplectic manifold, we obtain a triple of symplectic forms ωI=g(I·,·),ωS=g(S·,·),ωT=g(T·,·) (1.1) in analogy to the hyperkähler situation. The holonomy group of a hypersymplectic manifold is the real symplectic group Sp(2k,R) ⁠, which in particular implies that hypersymplectic manifolds are Ricci-flat. Hypersymplectic and hyperkähler structures have the same complexification, so it is not surprising that many results and constructions from hyperkähler geometry have hypersymplectic analogues. For example, one can show just like in the hyperkähler case that the endomorphisms I,S,T are parallel and integrable if and only if the associated 2-forms ωI,ωS,ωT are all closed. In fact, the hypersymplectic structure, that is, the quadruple (g,I,S,T) ⁠, is completely determined by the corresponding triple of symplectic forms (1.1). For instance, we have S=ωT−1◦ωI ⁠, if we interpret the ωi’s as isomorphisms ωi:TM→T*M ⁠. There is also a hypersymplectic quotient construction analogous to (but more pathological than) the hyperkähler quotient of [16]. Quotients of this sort constitute a basic source of hypersymplectic structures, and they may occur in dimensional reduction from split signature in various contexts in mathematical physics—see for example [17] for an application to the study of sigma-models with extended supersymmetry. The basic result is the following: Proposition 1.2. ([15]). Let (M,g,I,S,T)be a hypersymplectic manifold with an action of a Lie group Gwhich is Hamiltonian with respect to each symplectic form ωi,i∈{I,S,T} ⁠, with hypersymplectic moment map μ=(μI,μS,μT):M→g*⊗R3.Assume that the G-action is free and proper on μ−1(0) ⁠, that 0is a regular value of μ ⁠, and that the metric restricted to the G-orbits in μ−1(0)is non-degenerate. Then the quotient μ−1(0)/Gcarries in a natural way a hypersymplectic structure. In many situations, one can obtain a smooth quotient manifold μ−1(0)/G ⁠. However, this will typically carry a hypersymplectic structure only on the complement of a degeneracy locus. On the other hand, if G acts freely and properly on μ−1(0) ⁠, then the non-degeneracy assumption implies the smoothness of the quotient μ−1(0)/G just like in the proof of the hyperkähler version of the construction. It is our aim in this paper to apply the hypersymplectic quotient construction in an infinite-dimensional setting to obtain a hypersymplectic structure on a suitable open subset of the product manifold G×g3 ⁠. Our approach is closely analogous to Kronheimer’s construction of a hyperkähler metric on T*GC in [19]. Building in part on work by Matsoukas [24], we interpret G×g3 as a moduli space of solutions to the Nahm–Schmid equations (see Section 2 below), a ‘hypersymplectic version’ of Nahm’s equations [25] which can be viewed as a dimensional reduction of the anti-self-dual Yang–Mills equations in split signature. To our best knowledge, these equations first arose in the work of Schmid on deformations of complex manifolds [29], which predates the advent of Nahm’s equations in gauge theory [4, 25]. The article is organized as follows. In Section 2, we introduce the Nahm–Schmid equations as a dimensional reduction of the anti-self-dual Yang–Mills equations in split signature, and derive some immediate properties of their solutions with values in a compact Lie algebra g ⁠. The heart of the article is Section 3, where we examine the hypersymplectic geometry of the (framed) moduli space of solutions over the unit interval. We describe the corresponding degeneracy locus and investigate the complex and product structures of our moduli space. The results that we obtain are akin to those in [27], where the moduli space of solutions to the gauge-theoretic harmonic map equations was studied; note that these equations also arise by dimensional reduction of the anti-self-dual Yang–Mills equations in split signature. Then in Section 4 we consider the Nahm–Schmid equations from the point of view of integrable systems [2, 3]. We discuss the relevant twistor space (as a particular case of a more general construction by Bailey and Eastwood [5]), as well as a Lax-pair formulation of the equations, in particular the associated spectral curve. Finally, in Section 5, we discuss other moduli spaces of solutions to the Nahm–Schmid equations. We show that the moduli spaces of solutions on a half-line (with the limit satisfying a certain stability condition) is a hypersymplectic analogue of the hyperkähler metrics on adjoint orbits in gC ⁠. We also briefly discuss hypersymplectic quotients of the moduli space of solutions over the unit interval by subgroups of G×G ⁠. 2. The Nahm–Schmid equations We use R2,2 to denote R4 equipped with the pseudo-Riemannian metric dx02+dx12−dx22−dx32 ⁠. Let G be a compact Lie group with Lie algebra g ⁠, on which we choose an Ad-invariant inner product ⟨·,·⟩ ⁠. Solutions to the Nahm–Schmid equations will correspond to G-connections ∇=d+∑i=03Tidxi on R2,2 that are anti-self-dual (with respect to the standard orientation given by dx0∧dx1∧dx2∧dx3 ⁠) and also invariant under translation in the variables x1,x2,x3 ⁠. That is, the connection matrices Ti depend only on the (real) coordinate x0 ⁠. We recall that the anti-self-duality equations on R2,2 are [∇0,∇1]=−[∇2,∇3],[∇0,∇2]=−[∇1,∇3],[∇0,∇3]=[∇1,∇2], where ∇i=∂∂xi+Ti,i=0,1,2,3. Since we assume that the Ti only depends on x0 (which we shall also denote by t ⁠), the partial derivatives in the xi-directions with i=1,2,3 play no role in the anti-self-duality equations. We may, therefore, replace the covariant partial derivatives in these equations by ∇0=ddt+T0and∇i=Ti,i=1,2,3. Thus, we arrive at the following definition. Definition 2.1. Let g be the Lie algebra of a compact Lie group. A quadruple of g-valued functions (T0,T1,T2,T3) ⁠, Ti:R→g satisfies the Nahm–Schmid equations if T˙1+[T0,T1]=−[T2,T3],T˙2+[T0,T2]=[T3,T1],T˙3+[T0,T3]=[T1,T2]. (2.1) The solutions to the Nahm–Schmid equations are invariant under gauge transformations, that is, functions u:R→G acting by u.(T0,T1,T2,T3)≔(uT0u−1−u˙u−1,uT1u−1,uT2u−1,uT3u−1). (2.2) Note that we can always find a gauge transformation that solves the ODE uT0u−1−u˙u−1=0. This allows us to transform any solution (T0,T1,T2,T3) of (2.1) into a solution with T0=0 ⁠. For such a solution T=(0,T1,T2,T3) ⁠, the triple (T1,T2,T3) satisfies the reduced Nahm–Schmid equations T˙1=−[T2,T3],T˙2=[T3,T1],T˙3=[T1,T2]. (2.3) An important property of this system of equations is the following. Proposition 2.2 Let (T0,T1,T2,T3)be a solution to the Nahm–Schmid equations. Then the following gauge-invariant quantities are conserved: ∥T1∥2+∥Ti∥2,i=2,3and⟨Ti,Tj⟩,i≠j,i,j≥1. Proof This follows from a direct computation, using the equations and the invariance of the inner product.□ We shall give a natural interpretation of this statement in Section 4. An immediate consequence, which is in contrast to the behaviour of solutions of the usual Nahm equations [25], is a global existence result for solutions that can be brought to the reduced form T above (see also [24]): Corollary 2.3 Any solution of the reduced Nahm–Schmid equations (2.3) exists for all time. Proof Proposition 2.2 shows that we have a conserved quantity C=2∥T1∥2+∥T2∥2+∥T3∥2; note that for any t∈R one has ∥Ti(t)∥2≤C. Thus, the solution is uniformly bounded. The assertion now follows from the fact that if a solution to an ODE only exists for finite time, then it has to leave every compact set. □ Remark 2.4. Put on g×g×g the indefinite metric coming from the identification with g⊗R1,2 and consider the function ϕ:g×g×g→R,ϕ(ξ1,ξ2,ξ3)≔⟨[ξ1,ξ2],ξ3⟩. The right-hand side of the reduced Nahm–Schmid equations (2.3) is then the negative of the gradient of ϕ ⁠, with respect to this metric. We know from Proposition 2.3 that solutions to this gradient flow are bounded, and so exist for all times. It follows moreover that the non-trivial trajectories of this flow are contained in the compact submanifolds MC={(ξ1,ξ2,ξ3)∣2∥ξ1∥2+∥ξ2∥2+∥ξ3∥2=C}⊂g3, for suitable C>0 ⁠. 3. The moduli space over [0,1] In this section, we study solutions of the Nahm–Schmid equations over a compact interval U⊂R ⁠, which we fix to be U=[0,1] for convenience. Again, G denotes any compact Lie group and g stands for its Lie algebra. 3.1. The moduli space as a manifold We denote by A the space of all quadruples of C1-functions on U with values in g ⁠, that is, the (affine) Banach manifold A={(T0,T1,T2,T3):U→g⊗R4∣TidifferentiableofclassC1}≅C1(U,g)⊗R4 equipped with the norm ∥T∥C1≔∑i=03∥Ti∥C1, where we use the usual C1-norm ∥Ti∥C1≔∥Ti∥sup+∥T˙i∥sup on each component. Proposition 3.1. The set of solutions to the Nahm–Schmid equations is a smooth Banach submanifold of A ⁠. Proof This follows just like the analogous statement in the hyperkähler case [19]. Let μ:A→C0(U,g)⊗R3 denote the difference of the RHS and the LHS in (2.1), so that the set of solutions to the Nahm–Schmid equations is given by μ−1(0) ⁠. Owing to the implicit function theorem, we have to check that, for any solution T=(T0,T1,T2,T3)∈μ−1(0) ⁠, the linearization dμT:TTA→C0(U,g)⊗R3 has a bounded right-inverse. That is, to ζ=(ζ1,ζ2,ζ3)∈C0(U,g)⊗R3 we must associate a solution X=(X0,X1,X2,X3)∈TTA of the system of linear ODEs X˙1+[T0,X1]+[X0,T1]+[T2,X3]+[X2,T3]=ζ1X˙2+[T0,X2]+[X0,T2]−[T3,X1]−[X3,T1]=ζ2X˙3+[T0,X3]+[X0,T3]−[T1,X2]−[X1,T2]=ζ3. We note, just like in [19], that for any ζ this system has a unique solution with X0≡0 ⁠, Xi(0)=0 ⁠, i=1,2,3 ⁠, and deduce the existence of the required right-inverse.□ We now want to quotient by a suitable gauge group. We consider the Banach Lie group G≔C2(U,G) of gauge transformations acting on A ⁠. This has a normal subgroup consisting of gauge transformations that are equal to the identity at the endpoints of U ⁠: G00={u∈C2(U,G)∣u(0)=1G=u(1)}. The Lie algebras of G and G00 are given by Lie(G)=C2(U,g) and Lie(G00)={ξ∈C2(U,g)∣ξ(0)=0=ξ(1)}. The action of G (and also of G00 ⁠) on A is smooth and given by (2.2). Furthermore: Lemma 3.2. The gauge group G00acts freely and properly on A ⁠. Proof Suppose we have u∈G00 and T∈A such that u.T=T ⁠. Looking at the T0-component, we see that u solves the initial value problem u˙u−1=uT0u−1−T0,u(0)=1G. Its unique solution is the constant map u(t)≡1G and hence the action is free. To see that it is proper suppose that we have a sequence (Tm)⊂A converging to T∈A and a sequence (um)⊂G00 such that umTm→T˜∈A ⁠. Then umT0mum−1−u˙mum−1→T˜0. (3.1) Since T0m also converges to T0 in the C1-topology, it follows first from the Arzelà–Ascoli theorem that a subsequence of um converges in the C0-topology, and then a repeated use of (3.1) shows that the convergence is actually in the C2-topology. Thus, the action of G00 is proper.□ We are now ready to introduce our main object of study. Definition 3.3 The moduli space of solutions to the Nahm–Schmid equations is M≔{T∈A∣Tsolvesequations(2.1)}/G00. (3.2) Theorem 3.4. Mis a smooth Banach manifold diffeomorphic to G×g×g×g ⁠. Proof Owing to Lemma 3.2, M ⁠, equipped with the quotient topology, is Hausdorff. Let T=(T0,T1,T2,T3) be a solution of the Nahm–Schmid equations. We need to construct a slice to the action of G00 at T ⁠. Let u0 be the unique gauge transformation such that T0=−u˙0u0−1 and u0(0)=1G ⁠. Write ANS for the set of solutions to the Nahm–Schmid equations and define ST={S=(S0,S1,S2,S3)∈ANS∣S0=u0X0u0−1−u˙0u0−1,X0∈g,∥X0∥<ϵ}, where ϵ>0 will be determined later. If u.S∈ST for an S∈ST and a u∈G00 ⁠, then g=u0−1uu0∈G00 and gX0g−1−g˙g−1=Y0∈g ⁠. The unique solution with g(0)=1 is exp(−tY0)exp(tX0) and hence g(t)≡1G if ϵ is small enough. It follows then that u(t)≡1G and the map ST×G00→ANS,(S,u)↦u.S (3.3) is injective for ϵ small enough. The inverse map is given explicitly as follows. Let (R0,R1,R2,R3)∈ANS and let v∈G be the unique gauge transformation such that R0=−v˙v−1 and v(0)=1G ⁠. Let X0∈g satisfy expX0=u0−1(1)v(1) ⁠. Then p(t)=e−tX0u0(t)v(t)−1 is an element of G00 and p−1.R∈ST ⁠. It is easy to check that both the map (3.3) and its inverse are smooth, and hence ANS/G00 is a smooth Banach manifold. The diffeomorphism with G×g×g×g is given by Ψ:M→G×g×g×g,T↦(u0(1),T1(0),T2(0),T3(0)), where u0 is the unique gauge transformation such that T0=−u˙0u0−1 and u0(0)=1G ⁠. The inverse is given by Φ:G×g×g×g→M,(γ,ξ1,ξ2,ξ3)↦uγ.Tξ, where uγ∈G is an arbitrary gauge transformation such that uγ(0)=1G and uγ(1)=γ∈G ⁠, and Tξ=(0,T1,T2,T3) with (T1,T2,T3) being the unique solution of the reduced Nahm–Schmid equations (2.3) with initial conditions Ti(0)=ξi ⁠. We note that uγ is unique up to multiplication by an element of G00 ⁠.□ Remark 3.5 It follows from the proof that the tangent space to the space of solutions ANS at a solution T splits as VT⊕HT ⁠, where VT is the tangent space to the G00-orbit through T ⁠, and HT consists of quadruples (X0,X1,X2,X3) which solve the linearized Nahm equations with X0=u0ξu0−1 for some ξ∈g ⁠. In other words, X0 is an arbitrary solution of X˙0+[T0,X0]=0 ⁠. Thus the tangent space TTM can be identified with the set of solutions (X0,X1,X2,X3) to the system of linear equations X˙1+[T0,X1]+[X0,T1]+[T2,X3]+[X2,T3]=0X˙2+[T0,X2]+[X0,T2]−[T3,X1]−[X3,T1]=0X˙3+[T0,X3]+[X0,T3]−[T1,X2]−[X1,T2]=0X˙0+[T0,X0]=0. (3.4) 3.2. Group actions We now want to examine various symmetries of the construction (3.2). Since G00⊂G is a closed normal subgroup, we obtain an action of G/G00≅G×G on M given by gauge transformations with arbitrary values at t=0 and t=1 ⁠. Explicitly, (u1,u2)∈G×G acts by a gauge transformation u∈G such that u1=u(0),u2=u(1) ⁠. Any two choices of such a u differ by an element of G00 ⁠, which acts trivially on M ⁠. Furthermore, we have an action of the Lorentz group SO(1,2) on the moduli space. An element A=[Aij]i,j=13∈SO(1,2) acts on (T0,T1,T2,T3)∈g⊗(R⊕R1,2) naturally by A.(T0,T1,T2,T3)=(T0,∑j=13A1jTj,∑j=13A2jTj,∑j=13A3jTj). It is easy to check by direct computation that this action preserves Equations (2.1) and commutes with the action of G ⁠. Hence, it descends to an SO(1,2)-action on M that commutes with the (G×G)-action described above. The following result follows straightforwardly from examining the proof of Theorem 3.4; see [10] for an analogous result on the corresponding Nahm moduli space. Proposition 3.6 The bijection Ψ:M→G×g×g×gfrom Theorem3.4is compatible with the actions of SO(1,2)and G×Gin the following way: The action of G×Gon Mand the action (γ,ξ1,ξ2,ξ3)↦(u2γu1−1,u1ξ1u1−1,u1ξ2u1−1,u1ξ3u1−1),(u1,u2)∈G×Gon G×g×g×gare intertwined by Ψ ⁠. The action of SO(1,2)on Mand the action (γ,ξ1,ξ2,ξ3)↦(γ,∑j=13A1jξj,∑j=13A2jξj,∑j=13A3jξj)on G×g×g×gare intertwined by Ψ ⁠. 3.3. Explicit solutions for G=SU(2) Here we will exhibit an example of non-trivial su(2)-valued solutions, which can be found using an Ansatz analogous to the one used to calculate Nahm data for centred SU(2) magnetic 2-monopoles in Euclidean R3 (cf. (8.155) in [23]). It will turn out that this Ansatz essentially yields all solutions in the su(2)-case. Setting T0=0,Tj(t)=fj(t)ej,j=1,2,3, (3.5) where {ej}j=13 is a standard basis for su(2) ⁠, that is, [ei,ej]=ek if (ijk) is a cyclic permutation of (123) ⁠, then the Nahm–Schmid equations yield the following system of ODEs for the functions fj ⁠: f˙1=−f2f3,f˙2=f3f1,f˙3=f1f2. The general solution to this system can be expressed in terms of Jacobi elliptic functions [9] with arbitrary modulus κ∈[0,1] ⁠: f1(t)=aκsnκ(at+b),f2(t)=aκcnκ(at+b),f3(t)=−adnκ(at+b), where a,b∈R are arbitrary constants. For κ=0 ⁠, this yields with dn0≡1 the trivial solution T3=constant,T1=T2=0 ⁠. For κ=1 we obtain sn1(t)=tanh(t),cn1(t)=dn1(t)=sech(t) ⁠. The solution corresponding to fixed κ,a,b is smooth for all t∈R ⁠, as we expected from Corollary 2.3. For κ∈(0,1) it is also periodic with period 4K(κ)/a ⁠, where K is the complete elliptic integral of the first kind [9]. If we fix an invariant inner product on su(2) such that the basis {e1,e2,e3} is orthonormal, then we see that the conserved quantities ∥T1∥2+∥Ti∥2 are f12+f22=a2κ2,f12+f32=a2. (3.6) Note that for κ=1 the conserved quantities coincide, and the solution is of course not periodic in this case. Matsoukas has shown in his DPhil thesis [24] that in fact any solution to the Nahm–Schmid equations with values in su(2) may be put into this form. To see this, first gauge T0 away by a gauge transformation in G≅{1G}×G⊂G×G ⁠, that is, a gauge transformation which equals the identity at t=0 ⁠. Then use the SO(1,2)-action to put the solution (T1,T2,T3) into standard form such that ⟨Ti,Tj⟩=0 whenever i≠j ⁠. Now the vector space su(2) is three-dimensional, so it follows that for generic t∈R the elements T1(t),T2(t),T3(t) form an orthogonal basis. The Nahm–Schmid equations imply that Ti(t) and T˙i(t)=±[Tj(t),Tk(t)] are linearly dependent for each t ⁠. It follows that the direction of Ti does not vary with time for each i ⁠. This implies that there exists an orthonormal basis of su(2) ⁠, say the standard one used above, such that the Tj are of the form Tj(t)=fj(t)ej and hence reduce to the Ansatz used above to obtain explicit solutions. We summarize this discussion as follows. Proposition 3.7 (cf. [24]) Let Mbe the moduli space of su(2)-valued solutions to the Nahm–Schmid equations on [0,1]modulo the gauge group G00 ⁠. Let T∈M ⁠. Then the SO(1,2)×G-orbit of Tcontains an SO(3)∩SO(1,2)≅SO(2)-orbit of solutions of the form (3.5), with respect to a standard orthonormal basis of su(2) ⁠. 3.4. Hypersymplectic interpretation At a point T=(T0,T1,T2,T3)∈A ⁠, the tangent space to A has the description TTA=C1(U,g)⊗R4. On the R4≅B factor, we have the split-quaternionic structure I,S,T induced from multiplication by −i,s,t on B from the right. Explicitly: I(X0,X1,X2,X3)=(X1,−X0,−X3,X2) (3.7) S(X0,X1,X2,X3)=(X2,X3,X0,X1) (3.8) T(X0,X1,X2,X3)=(X3,−X2,−X1,X0). (3.9) We endow A=C1(U,g)⊗R2,2 with the indefinite metric given by the tensor product of the L2-metric and the standard metric on R2,2 ⁠. On tangent vectors X=(X0,X1,X2,X3) and Y=(Y0,Y1,Y2,Y3) ⁠, we have the formula g(X,Y)=∫U∑i=03ηii⟨Xi(t),Yi(t)⟩dt. (3.10) Here we wrote ηii for the diagonal coefficients of the metric on R2,2 ⁠, that is, η00=η11=1,η22=η33=−1 ⁠. This gives A the structure of a flat infinite-dimensional hypersymplectic manifold with symplectic forms ωI(·,·)=g(I·,·),ωS(·,·)=g(S·,·),ωT(·,·)=g(T·,·). The action of the group of gauge transformations G preserves this flat hypersymplectic structure. By calculating ddθ∣θ=0exp(θξ).(T0,T1,T2,T3) for ξ∈Lie(G) ⁠, we see that the fundamental vector fields associated to the action of G (and hence also for G00 ⁠) at a point T=(T0,T1,T2,T3)∈A are given by XTξ=(−ξ˙+[ξ,T0],[ξ,T1],[ξ,T2],[ξ,T3]). Proposition 3.8 The action of the group G00on Apreserves the hypersymplectic structure and the hypersymplectic moment map at T=(T0,T1,T2,T3)∈Ais given by μI(T)=−T˙1−[T0,T1]−[T2,T3],μS(T)=T˙2+[T0,T2]−[T3,T1],μT(T)=T˙3+[T0,T3]−[T1,T2].Hence, we can write the moduli space Mof solutions to the Nahm–Schmid equations formally as the hypersymplectic quotient: M=μ−1(0)/G00. Proof We only exhibit the calculation for μI ⁠, as the other two equations are obtained analogously. Let T∈A ⁠, ξ∈Lie(G00) ⁠. First observe that IXTξ=I(−ξ˙+[ξ,T0],[ξ,T1],[ξ,T2],[ξ,T3])=([ξ,T1],ξ˙−[ξ,T0],−[ξ,T3],[ξ,T2]). Thus, we can calculate for Y∈TTA ⁠, using integration by parts and the boundary condition ξ(0)=0=ξ(1) ⁠: ωI(Xξ,Y)=g(IXTξ,Y)=∫01⟨[ξ,T1],Y0⟩+⟨ξ˙−[ξ,T0],Y1⟩+⟨[ξ,T3],Y2⟩−⟨[ξ,T2],Y3⟩dt=⟨ξ,Y1⟩∣01+∫01⟨ξ,−ξ˙−[T0,Y1]−[Y0,T1]−[T2,Y3]−[Y2,T3]⟩dt=∫01⟨ξ,−ξ˙−[T0,Y1]−[Y0,T1]−[T2,Y3]−[Y2,T3]⟩dt. The assertion follows.□ Lemma 3.9 Let T∈A ⁠. The orthogonal complement to the tangent space to the G00-orbit through Twith respect to the indefinite metric gis given by tangent vectors (X0,X1,X2,X3)∈TTAsatisfying the equation X˙0+∑i=03ηii[Ti,Xi]=0. (3.11) Proof This is a standard calculation using integration by parts and the boundary conditions defining Lie(G00) ⁠. Let ξ∈Lie(G00) ⁠, (X0,X1,X2,X3)∈TTA ⁠. Then g(XTξ,X)=∫01⟨−ξ˙+[ξ,T0],X0⟩+∑i=13ηii⟨[ξ,Ti],Xi⟩dt=∫01⟨ξ,X˙0⟩+∑i=03ηii⟨ξ,[Ti,Xi]⟩dt. The result follows.□ Definition 3.10. Consider the Banach manifold μ−1(0)⊂A of solutions to the Nahm–Schmid equations. The degeneracy locus D is the set of points T∈μ−1(0) such that the metric g restricted to the tangent space of the G00-orbit through T is degenerate. Lemma 3.11. The actions of the gauge group Gand of SO(1,2)both preserve the degeneracy locus D ⁠. Proof This follows immediately from the fact that both G and SO(1,2) act by isometries.□ Proposition 3.12. ([24]). The degeneracy locus Dconsists of exactly those solutions to the Nahm–Schmid equations such that the boundary value problem d2ξdt2+[T˙0,ξ]+2[T0,ξ˙]+∑i=03ηii[Ti,[Ti,ξ]]=0,ξ∈Lie(G00)has a non-trivial solution. Proof The tangent space to the orbit through T is spanned by the values of the fundamental vector fields XTξ ⁠. We thus need to find ξ∈Lie(G00) such that the tangent vector XTξ∈TTA satisfies Equation (3.11). Plugging XTξ=(−ξ˙+[ξ,T0],[ξ,T1],[ξ,T2],[ξ,T3]) into that equation yields the desired result.□ Proposition 3.13 The complement μ−1(0)⧹Dof the degeneracy locus consists exactly of those solutions T∈μ−1(0)for which the linear operator ΔT:Lie(G00)→C0(U,g),ξ↦d2ξdt2+[T˙0,ξ]+2[T0,ξ˙]+∑i=03ηii[Ti,[Ti,ξ]] (3.12)is an isomorphism. In particular, μ−1(0)⧹Dis open. Proof Owing to the Arzelà–Ascoli theorem, the operator ΔT is a compact perturbation of the operator d2dt2:Lie(G00)→C0(U,g) ⁠, which is an isomorphism. Hence ΔT is Fredholm of index zero. Thanks to Proposition 3.12, ΔT is injective if T is not contained in the degeneracy locus. Hence it must be an isomorphism.□ An analogous argument to the one outlined in [19, pp. 4–5] (see also [10]) for the Nahm case yields then the following result. Theorem 3.14 The submanifold M0≔(μ−1(0)⧹D)/G00of Mis a smooth hypersymplectic manifold. The tangent space TTM0can be identified with the set of solutions Xto the system of linear equations (cf. (3.4)) X˙1+[T0,X1]+[X0,T1]+[T2,X3]+[X2,T3]=0X˙2+[T0,X2]+[X0,T2]−[T3,X1]−[X3,T1]=0X˙3+[T0,X3]+[X0,T3]−[T1,X2]−[X1,T2]=0X˙0+[T0,X0]+[T1,X1]−[T2,X2]−[T3,X3]=0,and the hypersymplectic structure is given by (3.7–3.9). The last equation above says that X is orthogonal to the G00-orbit, as we know from Lemma 3.9. We can obtain some more quantitative information about the degeneracy locus. Proposition 3.15 Let T∈μ−1(0)be a solution such that 2sup(∥T2(t)∥2+∥T3(t)∥2)<π2 ⁠. Then Tdoes not belong to the degeneracy locus D ⁠. This holds in particular for any solution with T2≡0≡T3 ⁠. Proof Since the degeneracy locus is invariant under the action of G ⁠, we may restrict attention to solutions such that T0≡0 ⁠. A solution T lies in the degeneracy locus if and only if there is a nontrivial solution ξ∈Lie(G00) to the linear boundary value problem ξ¨+(ad(T1)2−ad(T2)2−ad(T3)2)(ξ)=0,ξ(0)=0=ξ(1). (3.13) Let us write this ODE as ξ¨+Aξ=0 ⁠. Since ad(Ti) is skew-symmetric with respect to the invariant inner product, we have for any ξ∈C0([0,1],g) ⟨Aξ,ξ⟩=−∥[T1,ξ]∥2+∥[T2,ξ]∥2+∥[T3,ξ]∥2≤∥[T2,ξ]∥2+∥[T3,ξ]∥2. Using the pointwise inequality ∥[ξ,η]∥2≤2∥ξ∥2∥η∥2 ⁠, this implies ⟨Aξ,ξ⟩≤2sup(∥T2(t)∥2+∥T3(t)∥2)∥ξ∥2. Setting M≔2sup(∥T2(t)∥2+∥T3(t)∥2) ⁠, we can conclude that f(t)=∥ξ∥2 satisfies the differential inequality f¨+Mf≥0 ⁠. Let g(t)≔f˙(0)Msin(Mt) for t∈U ⁠, which solves g¨+Mg=0 with initial condition g(0)=0 ⁠, g˙(0)=f˙(0) ⁠. Consider the function ϕ(t)=f(t)/g(t) ⁠. Provided that M<π2 ⁠, ϕ is well-defined and differentiable on [0,1] ⁠. The inequality f¨+Mf≥0 and the equality g¨+Mg=0 imply that f¨g−g¨f≥0 on [0,1] and integrating this inequality shows that ϕ is monotonically increasing there. Since ϕ(0)=1 ⁠, ϕ ⁠, and hence ξ ⁠, cannot have another zero on (0,1] ⁠. Thus, if M<π2 ⁠, then the boundary value problem ξ¨+Aξ=0 ⁠, ξ(0)=ξ(1)=0 has only the trivial solution ξ≡0 ⁠.□ On the other hand we have Proposition 3.16 Let T=(T0,T1,T2,T3)be a non-constant solution of the Nahm–Schmid equations on [0,1]such that, for some A∈SO(1,2) ⁠, one of the components T˜i=(AT)i ⁠, i=1,2,3 ⁠, vanishes at t=0and t=1 ⁠. Then Tbelongs to the degeneracy locus D ⁠. Proof Without loss of generality we can assume that T0≡0 ⁠, A=1 and that T1 is not constant and vanishes at 0 and 1 ⁠. It is easy to check that ξ=T1 satisfies (3.13).□ Example 3.17 If G=SU(2) we can say more about Equation (3.12) and the degeneracy locus. For a solution in the standard form T0=0,Tj(t)=fj(t)ej ⁠, j=1,2,3 ⁠, we have (adTj)2(ξ)=−fj2⟨ξ,ej⟩ej and hence Equation (3.13) is diagonal in the standard basis of su(2) ⁠: ξ¨1=−(f22+f32)ξ1ξ¨2=(f12−f32)ξ2ξ¨3=(f12−f22)ξ3, where ξ=∑i=13ξiσi ⁠. Using (3.6), we can rewrite these equations as ξ¨1=(2f12−a2−κ2a2)=a2(2κ2snκ2(at+b)−1−κ2)ξ1ξ¨2=(2f12−a2)ξ2=a2(2κ2snκ2(at+b)−1)ξ2ξ¨3=(2f12−κ2a2)ξ3=a2κ2(2snκ2(at+b)−1)ξ3. These equations have particular solutions ξ1=snκ(at+b) ⁠, ξ2=cnκ(at+b) and ξ3=dnκ(at+b) (cf. Proposition 3.16). Since dnκ does not vanish on the real line, the Sturm Separation Theorem [13, Cor. XI.3.1] implies that the third equation cannot have a non-trivial solution vanishing at two points. On the other hand snκ (resp. cnκ ⁠) vanishes at points 2mK (resp. (2m+1)K ⁠), m∈Z ⁠. Thus, a solution T with b∈KZ ⁠, a∈2KZ belongs to the degeneracy locus D ⁠, while, owing again to the Sturm Separation Theorem, any solution with [a,a+b] properly contained in [mK,(m+1)K] does not belong to D ⁠. Corollary 3.18 For any non-abelian compact Lie algebra g ⁠, the degeneracy locus D⊂Mis non-empty. Proof Let ρ:su(2)→g be a (non-trivial) Lie algebra homomorphism and let (T0,T1,T2,T3) be an su(2)-valued solution to the Nahm–Schmid equations such that T1 is not constant and vanishes at t=0,1 ⁠. Then (ρ(T0),ρ(T1),ρ(T2),ρ(T3))∈D by Proposition 3.16.□ 3.5. Complex structures Denote by gC=g⊗C the complexification of g ⁠. We may identify the complex manifold (A,I) with C1(U,gC))⊗C2 via T=(T0,T1,T2,T3)↦(α,β),whereα≔T0−iT1,β≔T2+iT3. Writing a tangent vector to C1(U,gC)⊗C2 as (a,b) ⁠, the holomorphic symplectic form ωIC=ωS+iωT is given by ωIC((a1,b1),(a2,b2))=∫01⟨b1,a2⟩−⟨a1,b2⟩dt. The vanishing of the hypersymplectic moment map is the same as saying that the complex equation μC≔μS+iμT=0 and the real equation μI=0 are satisfied simultaneously. In the case of the Nahm–Schmid equations, this gives β˙+[α,β]=0, (3.14) α˙+α˙*+[α,α*]−[β,β*]=0. (3.15) Let GC be the complexification of G ⁠, that is, GC is a complex Lie group with Lie algebra gC and maximal compact subgroup G ⁠. The complex equation is invariant under the action of the complexified gauge group GC=C2(U,GC) and its normal subgroup G00C≔{u∈GC∣u(0)=1GC=u(1)} acting by u.α=uαu−1−u˙u−1,u.β=uβu−1. A computation similar to the one in Proposition 3.8 shows that the complex equation is the vanishing condition for the moment map of the action of G00C with respect to ωIC ⁠. We write the moduli space of solutions to the complex equation modulo G00C as N≔{(α,β):U→gC×gC∣β˙+[α,β]=0}/G00C. One can show that this is a smooth Banach manifold. Just like in the proof of Theorem 3.4 we obtain a diffeomorphism Φ:N→GC×gC≅T*GC,(α,β)↦(u0(1),β(0)), where u0∈GC is the unique complex gauge transformation with α=−u˙0u0−1 such that u0(0)=1GC ⁠. The space N is in a natural way a complex-symplectic quotient, and it can be checked that Φ is holomorphic and pulls back the canonical symplectic form on T*GC to the symplectic form ωIC ⁠. We obviously have a natural map M→N ⁠, since any solution to the Nahm–Schmid equations gives a solution to the complex equation. The question to ask at this point is the following: given a solution to the complex equation β˙+[α,β]=0 ⁠, does its G00C-orbit contain a solution to the real equation, and to what extent is this solution unique? Using work of Donaldson [12], Kronheimer answers this question affirmatively for the usual Nahm equations by showing that every G00C-orbit of a solution to the complex equation contains a unique G00-orbit of solutions to the real equation. In the hyperkähler case, it thus turns out that the map M→N is a bijection, inducing, therefore, a hyperkähler structure on T*GC ⁠. We think of g as a subalgebra of u(n) for some n∈N ⁠. Let us write the real equation as μI(α,β)=0 ⁠. We now aim at describing how this equation behaves under complex gauge transformations. To simplify calculations, we define operators on C∞([0,1],gC) by ∂¯α=ddt+[α,·],∂α=ddt−[α*,·],∂¯β=[β,·],∂β=−[β*,·]. The following two lemmas are then obtained by straightforward calculations. Lemma 3.19 μI(α,β)=[∂α,∂¯α]−[∂β,∂¯β],as operators on C∞([0,1],Cn) ⁠. Lemma 3.20 Let u∈GCbe a complex gauge transformation. Then ∂¯u.α=u◦∂¯α◦u−1,∂u.α=(u*)−1◦∂α◦u*∂¯u.β=u◦∂¯β◦u−1,∂u.β=(u*)−1◦∂β◦u*. Using this description of the real moment map, it is now easy to work out the behaviour of the real equation under complex gauge transformations. Lemma 3.21 Let u∈GCbe a complex gauge transformation, and put h≔u*u ⁠. Then u−1(μI(u.α,u.β))u=μI(α,β)−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh)). Now given a solution (α,β) to the complex equation, we want to find a self-adjoint and positive h∈G00C solving the boundary value problem μI(α,β)−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh))=0,h(0)=1=h(1). (3.16) Existence and uniqueness will then imply that the complex gauge transformation u=h1/2 takes (α,β) to a solution of the real equation. Lemma 3.22 Let T=(T0,T1,T2,T3)be a solution to the Nahm–Schmid equations, that is, a solution to the complex equation with μI(α,β)=0 ⁠. Then the linearization of the boundary value problem −∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh))=0is given by the operator −ΔTin (3.12). Proof Suppose that we have a one-parameter family h:(−ϵ,ϵ)×[0,1]→GC of self-adjoint solutions h(s,t)=exp(iξ(s,t)),h(0,t)=1,h(s,0)=1=h(s,1) to the boundary value problem (3.16), where ξ(s,t)(−ϵ,ϵ)×[0,1]→g ⁠. Write ξ′ for the partial derivative of ξ with respect to s at s=0 ⁠. Note that the condition h(s,0)=1=h(s,1) implies that ξ′(0)=0=ξ′(1) ⁠, that is, ξ′∈Lie(G00) ⁠. We compute the linearization of (3.16) for (α,β) with μI(α,β)=0 and denote the linear operator obtained in this way by L:Lie(G00)→C0([0,1],g) ⁠. Explicitly, Lξ′=−idds∣s=0(−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh))). Using the equation 0=μI(α,β)=T˙1+[T0,T1]+[T2,T3] and the Jacobi identity, we see Lξ′=−idds∣s=0(−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh)))=−∂¯α(∂αξ′)+∂¯β(∂βξ′)=−(ddt+[α,·])(ξ˙′−[α*,ξ′])−[β,[β*,ξ]]=−ξ¨′+[α˙*,ξ′]+[α*,ξ˙′]−[α,ξ˙′]+[α,[α*,ξ′]]−[β,[β*,ξ′]]=−ξ¨′−[T˙0,ξ′]−i[T˙1,ξ′]−[T0+iT1,ξ˙′]−[T0−iT1,ξ˙′]−[T0−iT1,[T0+iT1,ξ′]]+[T2+iT3,[T2−iT3,ξ′]]=−ξ¨′−[T˙0,ξ′]+i[[T0,T1]+[T2,T3],ξ′]−[T0+iT1,ξ˙′]−[T0−iT1,ξ˙′]−[T0−iT1,[T0+iT1,ξ′]]+[T2+iT3,[T2−iT3,ξ′]]=−ξ¨′−2[T0,ξ˙′]−[T˙0,ξ′]−[T0,[T0,ξ′]]−[T1,[T1,ξ′]]+[T2,[T2,ξ′]]+[T3,[T3,ξ′]]=−ΔTξ′. □ Theorem 3.23 Let (α,β)be a solution to the Nahm–Schmid equations, and assume that (α,β)is not contained in the degeneracy locus D ⁠. Then there exists a constant ϵ>0such that for any solution (α˜,β˜)of the complex equation with ∥α˜−α∥C1+∥β˜−β∥C1<ϵthere exists a unique u∈G00Cclose to the identity which is self-adjoint such that u.(α˜,β˜)solves the real equation. Proof Consider the map F:Lie(G00)×A→C0(U,g) given by F(ξ,(α˜,β˜))=μI(α˜,β˜)−∂¯α(eiξ(∂αe−iξ))+∂¯β(eiξ(∂βe−iξ)), which satisfies F(0,(α,β))=0 ⁠. The partial derivative of F at the point (0,(α,β)) in the Lie(G00)-direction is the operator ΔT ⁠. Owing to the assumption, this is an isomorphism. Hence, the assertion follows from the implicit function theorem.□ In particular, we note that this result applies to solutions T such that 2sup(∥T2∥2+∥T3∥2)<π2, owing to Proposition 3.15. Corollary 3.24 Consider the complex symplectic manifold (M0,I,ωIC) ⁠. Then there exists an open neighbourhood of the set {T∈M∣T2=T3=0}which is isomorphic to a suitable open neighbourhood of the zero section in T*GCwith its canonical complex symplectic structure. 3.6. Product structures In this subsection, we investigate the paracomplex structure [21, Chapter V] of the moduli space M ⁠. As a first step, we identify the space of field configurations A ⁠, equipped with the product structure S ⁠, as a parakähler manifold [21]. Let B≔{ddt+A∣A∈C1(U,g)} be the space of G-connections on the interval U ⁠. Then we can naturally identify T*B={(ddt+A,B)∣A,B∈C1(U,g)}≅C1(U,g)2. (3.17) Proposition 3.25 The map P:(A,S)→(T*B×T*B,id⊕(−id))given by (T0,T1,T2,T3)↦((T0+T2,T1+T3),(T0−T2,T1−T3)) (3.18)is a diffeomorphism respecting the product structures. Proof Clearly, the map P is a diffeomorphism. It is also straightforward to check by direct calculation that dP◦S=(1⊕(−1))◦dP. □ Now P is not just a diffeomorphism. We observe that it intertwines the symplectic form ωI and the product of canonical symplectic forms on T*B×T*B ⁠. Proposition 3.26 Consider the symplectic form Ω≔ωT*B⊕ωT*Bon T*B×T*B ⁠. Then ωI=P*Ω. Proof The symplectic form ωI is given at an element T∈A by ωI(X,Y)=−g(X0,Y1)+g(X1,Y0)−g(X2,Y3)+g(X3,Y2), where X=(X0,X1,X2,X3),Y=(Y0,Y1,Y2,Y3)∈TTA are two tangent vectors at T ⁠. The symplectic form Ω is given by Ω(V,W)=g(V0,W1)−g(V1,W0)+g(V2,W3)−g(V3,W2), with V=(V0,V1,V2,V3),W=(W0,W1,W2,W3)∈T(T*B×T*B)=T(T*B)⊕T(T*B) ⁠. Now P is a linear map in the description (3.18), and this implies that dP maps a tangent vector X at T∈A to a tangent vector V at P(T)∈T*B×T*B of the form V=(X0+X2,X1+X3,X0−X2,X1−X3). Plugging V,W as above, for some X,Y tangent to T∈A ⁠, into the formula for Ω ⁠, gives the result claimed, that is Ω(V,W)=ωI(X,Y). □ In the coordinates on T*B×T*B provided by the description (3.17), the Nahm–Schmid equations are then equivalent to the system B˙1+[A1,B1]=0B˙2+[A2,B2]=0A˙1−A˙2+[12(A1+A2),A1−A2]−[B1,B2]=0, (3.19) when we set A1≔T0+T2,A2≔T0−T2,B1≔T1+T3,B2≔T1−T3. We may view the first two equations in (3.19) as a single g⊕g-valued ‘paracomplex’ equation, and the third equation as a ‘real’ equation. It is evident that the paracomplex equation is invariant under what one may call paracomplex gauge transformations, that is, elements of G×G acting componentwise as (u1,u2)·(ddt+Ai,Bi)i=12=(ddt+ui−1Aiui+ui−1u˙i,ui−1Biui)i=12. Let us consider the moduli space  given by ≔{(A,B)∈T*B∣B˙+[A,B]=0}/G00. We define, in analogy with Theorem 3.4, a bijection →G×g,(A,B)↦(u(1),B(0)), where again u is the unique gauge transformation that satisfies A=−u˙u−1 and u(0)=1G ⁠. We may view  as a symplectic quotient of T*B by the action of the group G00 ⁠. This map then identifies  and T*G as symplectic manifolds. By applying this construction on each factor of T*B×T*B and composing with the map P from above, we obtain a map from the moduli space of solutions to the Nahm–Schmid equations to ×≅T*G×T*G ⁠. The paracomplex equation in the system (3.19) may be interpreted as saying that, for i=1,2 ⁠, the connections ∇i≔(ddt+Ai)dt+Bidx on the adjoint bundle over Euclidean R2 are flat. We note that any solution to the paracomplex equation with A1=A2 and B1=B2 automatically solves the real equation. The corresponding solutions to the Nahm–Schmid equations satisfy T2=0=T3 ⁠. We can, moreover, rewrite the real equation as A˙1−A˙2+[A2,A1−A2]+[B2,B1−B2]=0, which says that the two flat connections ∇1,∇2 are in Coulomb gauge with respect to each other, that is d∇2∗(∇1−∇2)=0. This gives a natural interpretation of the Nahm–Schmid equations in terms of the geometry of the infinite-dimensional Riemannian manifold T*B/G00 ⁠, with Riemannian metric given by the usual L2-metric. The L2-metric equips the affine space T*B with the structure of a flat infinite-dimensional Riemannian manifold, so geodesic segments γ:[0,1]→T*B starting at (A1,B1) with initial velocity (a1,b1) are just straight lines: γ(τ)=(A1,B1)+τ(a1,b1),τ∈[0,1]. The group G00 acts by isometries and geodesic segments on the quotient T*B/G00 can be lifted (at least locally) to horizontal geodesics on T*B ⁠, that is, geodesics, the velocity vector of which is orthogonal to the G00 orbits. A geodesic γ as above will be horizontal exactly when (a1,b1) satisfies the Coulomb gauge condition. This means that a solution to the Nahm–Schmid equations on [0,1] can be interpreted as a horizontal geodesic segment in T*B ⁠, the endpoints of which lie in the space of solutions to the equation B˙+[A,B]=0. Now a calculation similar to the one at the end of the previous section shows that if ∇1,∇2 defines a solution to the Nahm–Schmid equations, then the linearization of the equation d∇1∗(eξ.(∇2)−∇1)=0 is again ΔTξ=0. Thus, elements of the degeneracy locus correspond to horizontal geodesic segments, the endpoints of which are conjugate. The degeneracy locus D ⁠, therefore, has a natural interpretation in terms of the cut-locus of the infinite-dimensional Riemannian manifold T*B/G00 ⁠. Given a pair of points in  that are sufficiently close, an application of the implicit function theorem shows that there exists a unique gauge transformation close to the identity that puts them into Coulomb gauge. It follows that we can identify an open subset of M0=(μ−1(0)⧹D)/G00 with a neighbourhood of the diagonal inside T*G×T*G ⁠. In summary, we have the following observation: Proposition 3.27 Consider the symplectic manifold (M0,ωI)with local product structure S ⁠. The map Pdefined in Proposition3.25induces a symplectomorphism between a neighbourhood of the subset {T∈M∣T2=T3≡0}and a neighbourhood of the diagonal in (T*G×T*G,ωT*G⊕ωT*G) ⁠. This symplectomorphism is compatible with the local product structure Son M0and the obvious product structure on T*G×T*G ⁠. 4. Twistor space and spectral curves This section describes how general constructions and results on paraconformal manifolds and on integrable systems with spectral parameter apply to the moduli space of solutions of the Nahm–Schmid equations. 4.1. Twistor space Bailey and Eastwood [5] gave a very general construction of twistor space in the holomorphic category and the usual twistor space constructions for real manifolds can be viewed as applying a particular real structure to a Bailey–Eastwood twistor space. In the case of hypersymplectic manifolds, this works as follows. The complexification of the algebra of split quaternions is the same as the complexification of the usual quaternions. Thus on the complexified tangent bundle TCM of a hypersymplectic manifold (M,g,I,S,T) ⁠, we obtain three anti-commuting endomorphisms I,J=iS,K=iT which all square to −1 ⁠. For any point p=(a,b,c) on the sphere S2⊂R3 ⁠, we consider the (−i)-eigenspace Ep of the endomorphism aI+bJ+cK ⁠. This is an integrable distribution of TCM ⁠, since I,S,T are integrable. Since a hypersymplectic manifold is real-analytic, it has a complexification MC ⁠. The holomorphic tangent bundle of MC restricted to M is naturally identified with TCM ⁠, and we can extend the endomorphisms I,J,K uniquely to a neighbourhood of M in MC ⁠. For each p∈S2 ⁠, we obtain an integrable holomorphic distribution Ep in T1,0MC ⁠, and these combine to give an integrable holomorphic distribution E on MC×CP1 ⁠. The twistor space Z of M ⁠, as defined by Bailey and Eastwood, is the space of leaves of all E ⁠. Let us discuss particular features of this construction in the case of hypersymplectic manifolds. First of all, we observe that, if a≠0 ⁠, then the equation (aI+biS+ciT)u=−iu can be rewritten as (a−1I+ca−1S−ba−1T)u=−iu ⁠. The endomorphism a−1I+ca−1S−ba−1T is, as can be readily checked, a complex structure on M ⁠. Thus for a≠0 ⁠, the space of leaves of Ep is just M equipped with the corresponding complex structure a−1I+ca−1S−ba−1T ⁠. If a=0 ⁠, Ep is the complexification of the (−1)-eigenspace (in TM ⁠) of the product structure bS+cT on M ⁠, and its space of leaves does not have to be Hausdorff. If, however, the space of leaves of the (−1)-eigenspace of bS+cT is a manifold Mb,c+ ⁠, then the space of leaves of E(0,b,c) on MC can be identified (possibly after making MC smaller) with the complexification of Mb,c+ ⁠. Thus, in this case, Z is a manifold. This is the case on a neighbourhood of the subset {T∈M∣T2=T3≡0} ⁠, as shown in Proposition 3.27. We shall soon see that the moduli space M of Nahm–Schmid equations has a globally defined twistor space. The space Z is clearly a complex manifold, and the projection onto S2≃CP1 is holomorphic. It has a natural anti-holomorphic involution σ obtained from the real structure on MC and the inversion τ with respect to the circle a=0 (that is, ∣ζ∣=1 ⁠) on CP1 ⁠, σ(m,ζ)=(m¯,1ζ¯). The manifold M can be now recovered as a connected component of the Kodaira moduli space of real sections (that is, equivariant with respect to τ on CP1 and σ on Z ⁠) the normal bundle of which splits as a sum of O(1)’s. The endomorphisms I,S,T are also easily recovered, since the fibre of Z over any (a,b,c) with a≠0 is canonically biholomorphic to (M,a−1I+ca−1S−ba−1T) ⁠, and any real section in our connected component of the Kodaira moduli space meets such a fibre in exactly one point. Observe that, over the circle a=0 ⁠, it is no longer true that real sections separate points of the fibres. All we can say that such sections locally separate points in (Z(0,b,c))σ×(Z(0,−b,−c))σ≃Mb,c+×M−b,−c+, which is equivalent to (M,bS+cT) being locally isomorphic to Mb,c+×M−b,−c+ ⁠. Remark 4.1 Unlike in the case of hyperkähler manifolds, the twistor space is not uniquely determined by M ⁠, since over the circle ∣ζ∣=1 only the τ-invariant part of the fibres of Z→CP1 is uniquely determined. Even more generally, we shall refer to any complex manifold Z with a holomorphic submersion onto CP1 and an anti-holomorphic involution σ covering the inversion τ as a twistor space. It should be pointed out, however, that if we start with such a generalized Z ⁠, and apply the construction given at the beginning of this section to a connected component M of the Kodaira moduli space of real sections of Z ⁠, we shall in general not recover Z ⁠. All we can say is that the resulting twistor space ZM of M is equipped with a local biholomorphism ϕ:ZM→Z ⁠, which is a fibre map (over CP1 ⁠) and which intertwines the real structures. To recover the metric from Z ⁠, we need one more piece of data: a twisted holomorphic symplectic form along the fibres of Z→CP1 ⁠. If we extend the metric g to the complexified tangent bundle TCM and define ωI=g(I·,·) ⁠, ωJ=g(J·,·) ⁠, ωK=g(K·,·) ⁠, then it follows easily that the O(2)-valued 2-form ω+=ωJ+iωK+2ζωI−ζ2(ωJ−iωK) vanishes on vectors in Eζ ⁠. Here ζ is the standard complex affine coordinate on CP1 related to p=(a,b,c) via (a,b,c)=(1−ζζ¯1+ζζ¯,−(ζ+ζ¯)1+ζζ¯,i(ζ¯−ζ)1+ζζ¯). It follows that ω+ descends to fibres of the projection Z→CP1 ⁠. Since J=iS and K=iT ⁠, we can rewrite ω+ in terms of the symplectic structures ωI,ωS,ωT on M ⁠: ω+=i(ωS+iωT−2iζωI−ζ2(ωS−iωT)). We can of course replace ω+ by −iω+ and hence, from now on, we shall consider the following holomorphic symplectic form along the fibres of Z ⁠: ω+=ωS+iωT−2iζωI−ζ2(ωS−iωT). (4.1) Remark 4.2 Away from the locus ∣ζ∣≠1 ⁠, the twistor space can be also constructed in analogy with Hitchin’s original construction of the twistor space of a hyperkähler manifold in [16]. First, we identify CP1⧹{∣ζ∣=1} with the two-sheeted hyperboloid C={(x1,x2,x3)∈R3∣x12−x22−x32=1} via x=(x1,x2,x3)=(1+ζζ¯1−ζζ¯,i(ζ¯−ζ)1−ζζ¯,ζ+ζ¯1−ζζ¯). Then we define an almost complex structure on the product Zo≔M×C ⁠, which at (m,x) is equal to (x1I+x2S+x3T)⊕J0 ⁠, where J0 is the standard complex structure on CP1 ⁠. An argument completely analogous to the one in [16] shows that this almost complex structure is integrable, and Zo becomes a complex manifold fibring over C ⁠. Unlike in the case of hyperkähler geometry, there is, however, no easy way to recover M from Zo ⁠. Assume now that a Lie group G acts on M preserving the hypersymplectic structure, and that this action extends to a global action of a complexification GC of G ⁠. If the G-action is Hamiltonian for ωI,ωS,ωT ⁠, then the action of GC is Hamiltonian for the symplectic form (4.1), and we obtain an associated moment map μ+=μS+iμT−2iζμI−ζ2(μS−iμT). The fibrewise complex-symplectic quotient Z¯ of Z by GC (assuming it is manifold) is then a twistor space in the sense of Remark 4.1. The space of real sections of Z¯ which descend from Z is the hypersymplectic quotient of M by G (note, however, that the hypersymplectic quotient does not need to be a hypersymplectic manifold everywhere: some of these sections may have a wrong normal bundle). We shall now apply the ideas above to the infinite-dimensional context of the Nahm–Schmid equations. The twistor space of the flat infinite-dimensional hypersymplectic Banach manifold A is formally the total space of C1(U,gC)⊗C2⊗O(1) ⁠. The fibrewise moment map for the complexified gauge group G00 with respect to ω+ is the Lax equation discussed in Section 3.5. Thus, each fibre of the quotient twistor space is isomorphic to gC×GC ⁠. A computation completely analogous to that in [19] shows that the twistor space Z of M is obtained by gluing two copies of C×gC×GC by means of the transition function (ζ,η,g)↦(1ζ,ηζ2,gexp(−η/ζ)). The real structure is given by σ(ζ,η,g)=(1/ζ¯,η¯/ζ¯2,(g*)−1exp(−η¯/ζ¯)), and the complex symplectic form along the fibres is the standard symplectic form on T*GC ⁠. Real sections of Z are obtained from real sections of C1(U,gC)⊗C2⊗O(1) via the fibrewise quotient with respect to ω+ ⁠. If we write α=T0−iT1,β=T2+iT3, then the moment maps of Proposition 3.8 can be written as 2iμI(T)=[ddt+α,−ddt+α*]−[β,β*]=α˙+α˙*+[α,α*]−[β,β*](μS+iμT)(T)=[ddt+α,β]=β˙+[α,β](μS−iμT)(T)=[−ddt+α*,β*]=−β˙*+[α*,β*]. Putting the terms together, we obtain for μ+ μ+(T)=μS+iμT−2iζμI−ζ2(μS−iμT)=[ddt+α−ζβ*,β−ζ(−ddt+α*)]. Equivalently, we can modify this to μ+(T)=[ddt+α−ζβ*,β−ζ(α+α*)+ζ2β*]. Thus, the real sections of Z obtained in this way are precisely gauge equivalence classes of solutions to the Nahm–Schmid equations, that is, points of M ⁠. However, only the sections corresponding to M⧹D have normal bundle with the required splitting property. 4.2. The spectral curve The expression for μ+(T) obtained above means that the Nahm–Schmid equations are equivalent to the following single ODE with a complex parameter: ddt(β−ζ(α+α*)+ζ2β*)=[β−ζ(α+α*)+ζ2β*,α−ζβ*]. (4.2) This is an example of a Lax equation with spectral parameter: if we write T(ζ)=β−ζ(α+α*)+ζ2β*,T+(ζ)=α−ζβ*, then (4.2) becomes dT(ζ)dt=[T(ζ),T+(ζ)]. (4.3) This equation has been studied extensively (see for example [1]), and so we can simply state the relevant results. Here we shall restrict ourselves to the case g=u(n) ⁠. First of all, (4.3) implies that the spectrum of T(ζ) is constant in t ⁠, so that the characteristic polynomial of T(ζ) defines a conserved quantity of the Nahm–Schmid system: Σ={(ζ,η);det(η−T(ζ))=0}. This is an algebraic curve (possibly singular) of (arithmetic) genus (n−1)2 in the total space of the line bundle O(2) over CP1 ⁠, invariant under the involution σ(ζ,η)=(1/ζ¯,η¯/ζ¯2). (4.4) In particular, we observe that the coefficient of ζ2 in trT(ζ)2 is the conserved quantity C=2∥T1∥2+∥T2∥2+∥T3∥2 introduced in Corollary 2.3. The cokernel of η−T(ζ) is a semistable sheaf F on Σ with χ(F)=n2−n (and such that H0(Σ,F⊗O(−1))=0 ⁠). The Lax equation corresponds then to the linear flow t→F⊗Lt ⁠, where L is a line bundle on TCP1≃∣O(2)∣ with the transition function exp(μ(ζ,η)) and μ is polynomial in ζ−1 and η determined by the condition that the positive powers of ζ in μ(ζ,T(ζ)) and in −T+(ζ) coincide [1]. In our case, it follows that μ(ζ,η)=η/ζ ⁠. The sheaves arising from solutions to the Nahm–Schmid equations satisfy several further conditions. Consider the case of a smooth spectral curve Σ ⁠. Then the cokernel of η−T(ζ) is one-dimensional for each (ζ,η)∈Σ and, consequently, F is a line bundle. Moreover, F(−1)=F⊗O(−1) is a line bundle of degree g−1 not in the theta divisor. The fact that F is obtained from the matrix polynomial T(ζ) imposes two further conditions: E=F(−1) satisfies the reality condition E⊗τ(E)≃KS≃O(2n−4) ⁠, where τ is the induced anti-holomorphic involution on Jacg−1(Σ) ⁠; the Hitchin metric [14, Section 6] on H0(Σ,F) is positive definite. It is perhaps worth pointing out that the fact that the solutions to the Nahm–Schmid equations exist for all time, implies that the closure of the set of line bundles satisfying the above conditions does not intersect the theta divisor in Jacg−1(Σ) ⁠. Example 4.3 (Spectral curves for g=su(2) ⁠). We calculate the spectral curve associated with a solution (T0,T1,T2,T3)=(0,f1e1,f2e2,f3e3) as in (3.5). With e1=12(i00−i),e2=12(01−10),e3=12(0ii0), (4.5) we get T(ζ)=12(−2f1ζf2(1−ζ2)−f3(1+ζ2)f2(ζ2−1)−f3(1+ζ2)2f1ζ). A direct computation yields det(η−T)=η2−12(2f11+f22+f32)ζ2+14(f22−f32)(1+ζ4). Taking f1(t)=aκsnκ(at+b),f2(t)=aκcnκ(at+b),f3(t)=−adnκ(at+b) ⁠, we can write this as det(η−T)=η2−a22(1+κ2)ζ2+a24(κ2−1)(1+ζ4). In this case, g=1 and we can identify Jac0(Σ) with Σ itself. If we choose the origin (trivial bundle) O to correspond to a τ-invariant point on Σ ⁠, then the induced anti-holomorphic involution on Jac0(Σ)≃Σ is just τ ⁠. According to the above characterization of the sheaves F ⁠, they correspond to points x∈Σ such that x+τ(x)=O ⁠. This is a union of two circles in Σ ⁠, and the Nahm–Schmid flow lives on the circle that does not contain O ⁠. We also observe that the points where f1 or f2 vanish (that is, points for which t lies in −b+(K(κ)/a)Z ⁠) correspond to the even theta characteristics of Σ ⁠. 5. The ‘positive’ part of the moduli space We continue to assume that g=u(n) ⁠. To any triple T1,T2,T3∈u(n) ⁠, we associate, as in the previous section, the matrix polynomial T(ζ)=T2+iT3+2iT1ζ+(−T2+iT3)ζ2 ⁠. It follows that T(ζ)ζ−1 is hermitian for each ζ with ∣ζ∣=1 ⁠, and we denote by X+ the set of matrix polynomials T(ζ) such that T(ζ)ζ−1 is positive definite for each ζ with ∣ζ∣=1 ⁠. Observe that, for n=1 ⁠, this is the subset of R1,2 where x2−zz¯>0 ⁠, x>0 ⁠. It follows from a theorem of Rosenblatt [28] (see also [22] for a simple proof) that any T(ζ)∈X+ can be factorized as T(ζ)=(A+B*ζ)(B+A*ζ), (5.1) where det(B+A*ζ)≠0 for ∣ζ∣≤1 ⁠. The only freedom in the factorization is the following action of U(n) ⁠: (A,B)↦(Ag−1,gB) and, consequently, the factorization is unique if we assume, in addition, that B is hermitian. The following estimate is one reason for our interest in X+ ⁠. Lemma 5.1 If (T1,T2,T3)∈X+ ⁠, then ∥T2+iT3∥≤2∥T1∥ ⁠. Proof We have ∥T2+iT3∥=∥AB∥≤∥A∥∥B∥≤∥2iT1∥1/2∥2iT1∥1/2=2∥T1∥. □ We now consider the Nahm–Schmid flow on X+ ⁠. Definition 5.2 The positive part M+ of the moduli space M consists of solutions (T0(t),T1(t),T2(t),T3(t)) to the Nahm–Schmid equations such that (T1(t),T2(t),T3(t))∈X+ for some t (equivalently for all t ⁠). Suppose that T1(t),T2(t),T3(t) satisfy the Nahm–Schmid equations (with T0≡0 ⁠) and that we have a factorization (5.1) depending smoothly on t (we can always find a smooth factorization, given its uniqueness under the assumptions B=B* ⁠, det(B+A*ζ)≠0 for ∣ζ∣≤1 ⁠). We can then rewrite Equation (3.14) as (A˙−12B*BA+12ABB*)B+A(B˙−12A*AB+12BAA*)=0, and Equation (3.15) as ((A˙−12B*BA+12ABB*)A*+B*(B˙−12A*AB+12BAA*))H=0, where the superscript H denotes the hermitian part of a matrix. Thus, A(t),B(t) satisfy the following equations: A˙=12(B*BA−ABB*)+a, (5.2) B˙=12(A*AB−BAA*)+b, (5.3) where a(t),b(t) satisfy aB+Ab=0,aA*+Aa*+b*B+B*b=0 (5.4) for all t ⁠. Lemma 5.3 Let (a,b)be a solution of (5.4) and suppose that det(B+A*ζ)≠0for ∣ζ∣≤1 ⁠. Then there exists a unique ρ∈u(n)such that a=Aρ ⁠, b=−ρB ⁠. Proof The assumption implies that B is invertible. We can infer from the first equation in (5.4) that there exists a unique ρ∈gl(n,C) such that a=Aρ ⁠, b=−ρB ⁠. The second equation can be now rewritten as (B*)−1A(ρ+ρ*)A*B−1=ρ+ρ*. The conclusion follows from the following lemma:□ Lemma 5.4 Let Xbe a square complex matrix, any two eigenvalues of which satisfy λ1λ¯2≠1 ⁠. Then the only solution of the equation X*hX=hwith hhermitian is h=0 ⁠. Proof We may assume that X is in Jordan form. The equation under consideration means that X is an isometry of the sesquilinear form ⟨,⟩ defined by h ⁠. Let Jk(λ1),Jl(λ2) be two (not necessarily distinct) Jordan blocks of X with e1,…,ek and f1,…,fl the corresponding bases of cyclic X-modules, that is, Xe1=λ1e1 ⁠, Xei=λ1ei+ei−1 for 1<i≤k ⁠, and similarly for the fj ⁠. Since λ¯1λ2≠1 ⁠, ⟨e1,f1⟩=0 ⁠. Now it follows inductively that ⟨ei,fj⟩=0 for all i≤k,j≤l ⁠. Thus, h=0 ⁠.□ We can, therefore, rewrite (5.2) and (5.3) as A˙−Aρ=12(B*BA−ABB*),B˙+ρB=12(A*AB−BAA*). These equations are invariant under the following action of U(n)-valued gauge transformations: A↦Ag−1,B↦gB,ρ↦gρg−1−g˙g−1. In particular, we can use the gauge freedom to make ρ identically 0 and obtain the following equations for A,B ⁠: A˙=12(B*BA−ABB*), (5.5) B˙=12(A*AB−BAA*). (5.6) These equations are the split signature version of the Basu–Harvey–Terashima equations [7, 30]. We conclude: Theorem 5.5 Let (T1(t),T2(t),T3(t)) ⁠, t∈R ⁠, be a solution to the reduced Nahm–Schmid equations (2.3) belonging to M+ ⁠. Then there exist smooth functions A,B:R→Matn,n(C)satisfying (5.5) and (5.6) such that T1(t)=−i2(A(t)A*(t)+B*(t)B(t)),T2(t)+iT3(t)=A(t)B(t). (5.7)Conversely, if A(t),B(t)satisfy (5.5–5.6), then T1(t),T2(t),T3(t)given by (5.7) satisfy (2.3). Remark 5.6 The quantity tr(AA*+B*B) is an invariant of the flow (5.5) and (5.6). It is, therefore, a flow on a sphere. The flow (5.5) and (5.6) is preserved under the involution A↔B ⁠. In particular, there is a subset of solutions with A=B ⁠, that is, satisfying A˙=12(A*A2−A2A*)=12[A*A+AA*,A]. As an application, we obtain additional information about the product structures of M+ ⁠. Proposition 5.7 The map (M+¯,S)→T*G×T*Gdefined in Section3.6is proper. Proof The map under consideration is given by ϕ(T0(t),T1(t),T2(t),T3(t))=(T1(0)+T3(0),u1(1),T1(0)−T3(0),u2(1)), where u1(t) (resp. u2(t) ⁠) satisfies T0+T2=−u˙1u1−1 ⁠, u1(0)=1 (resp. T0−T2=−u˙2u2−1 ⁠, u2(0)=1 ⁠). Let K be a compact set in T*G×T*G ⁠. Then ∥T1(0)∥ is bounded on ϕ−1(K)∩M+¯ by a constant cK ⁠. Lemma 5.1 implies that ∥T2(0)∥2+∥T3(0)∥2≤4cK2 ⁠, and hence the conserved quantity C of Proposition 2.2 is bounded by 6cK2 ⁠. The Nahm–Schmid equations and the Arzelà–Ascoli theorem imply now that ϕ−1(K) is compact in M ⁠.□ 6. Other moduli spaces of solutions to the Nahm–Schmid equations We want to briefly discuss two other moduli spaces: solutions on a half-line and solutions on [0,1] which are symmetric at both ends. 6.1. The moduli space of solutions on a half-line The reduced Nahm–Schmid equations have the form x˙=V(x) ⁠, where x=(T1,T2,T3)∈g3 and V is the vector field given by V(x)=([T3,T2],[T3,T1],[T1,T2]) ⁠. The critical points of V consist of commuting triples (T1,T2,T3) ⁠. Let (τ1,τ2,τ3) be a regular commuting triple, that is, the centralizer of the triple is a Cartan subalgebra h ⁠. Let g=⨁λ∈h*gλ be the root decomposition of g with respect to h ⁠. A simple calculation shows that the differential DV at (τ1,τ2,τ3) preserves the subspace Uλ=gλ⊕gλ⊕gλ and DV∣Uλ has eigenvalues 0,±(λ(τ2)2+λ(τ3)2−λ(τ1)2)1/2 ⁠. Thus, as long as (adτ2)2+(adτ3)2−(adτ1)2 has no negative eigenvalues, the differential of V at the critical point (τ1,τ2,τ3) has no imaginary eigenvalues. We shall call such a triple (irrespectively of being regular or not) stable. Observe that, in particular, the triple (τ1,0,0) is stable. General results about the asymptotic behaviour of the solutions of ODEs (see for example [18] for a discussion of this in the context of Nahm’s equations) imply that any solution to the Nahm–Schmid equations on [0,∞) ⁠, the limit of which is a regular stable triple, approaches this limit exponentially fast, that is, there exists an η>0 such that ∥Ti(t)−τi∥≤const·e−ηt ⁠. Any η smaller than the smallest positive eigenvalue of D(τ1,τ2,τ3)V will do. Let (τ1,τ2,τ3) be a regular stable triple and let η>0 be as above. We introduce (cf. [20]) the Banach space Ω1 consisting of C1-maps f:[0,∞)→g such that the norm ∥f∥Ω1≔supt∥eηtf(t)∥+supt∥eηtf′(t)∥ (6.1) is finite. Let us write τ0=0 and set Aτ1,τ2,τ3≔{(T0,T1,T2,T3):[0,∞)→g⊗R4∣Ti−τi∈Ω1,i=0,1,2,3}. The Banach group G={u:[0,∞)→G∣g(0)=1,g˙g−1∈Ω1,Ad(g)τi∈Ω1,i=1,2,3} acts smoothly on the Banach manifold Aτ1,τ2,τ3 via gauge transformations and we define Mτ1,τ2,τ3={T∈A∣TsolvesEquations(2.1)}/G. As in [20], Mτ1,τ2,τ3 can be also described as the set of solutions to the reduced Nahm–Schmid equations such that limt→∞Ti(t)=Ad(g0)τi ⁠, i=1,2,3 ⁠, for some g0∈G ⁠. Theorem 6.1 Mτ1,τ2,τ3is a smooth Banach manifold diffeomorphic to the tangent bundle T(G/T)of G/T ⁠, where Tis a maximal torus in G ⁠. Proof The fact that Mτ1,τ2,τ3 is a smooth Banach manifold is proved analogously to Theorem 3.4, once we replace the constant X0 in the definition of the slice ST with X0(t)=ve−2ηt ⁠, v∈g ⁠. To identify Mτ1,τ2,τ3 ⁠, we observe that the space of solutions to the reduced Nahm–Schmid equations in a small neighbourhood of the critical point (τ1,τ2,τ3) and converging to this critical point is identified with the sum of negative eigenspaces of D(τ1,τ2,τ3)V (since the triple is stable). By virtue of the arguments given at the beginning of this section, this space is isomorphic to p=⨁λ>0gλ⊕g−λ ⁠. Since every solution to the Nahm–Schmid equations exists for all time, we conclude that the submanifold of Mτ1,τ2,τ3 consisting of solutions with T0≡0 is diffeomorphic to p ⁠. Now the remark made just before the statement of the theorem shows that Mτ1,τ2,τ3 is diffeomorphic to G×Tp≃T(G/T) ⁠.□ We now would like to view Mτ1,τ2,τ3 as the hypersymplectic quotient of Aτ1,τ2,τ3 by G ⁠. As for the moduli space of solutions on an interval, the hypersymplectic structure is not defined on the degeneracy locus, that is, on the set of solutions where the linear operator (3.12), acting on Lie(G) ⁠, has non-trivial kernel. We have: Proposition 6.2 Let (τ1,τ2,τ3)be a regular stable triple. There exists a neighbourhood U of G/T (which is the G-orbit of the constant solution (0,τ1,τ2,τ3) ⁠) in Mτ1,τ2,τ3 ⁠, which is disjoint from the degeneracy locus D ⁠. Proof It is enough to show that the constant solution itself does not belong to D ⁠. As in the proof of Proposition 3.15, we are asking whether the exists a non-zero ξ∈Lie(G) such that ξ¨=((adτ2)2+(adτ3)2−(adτ1)2)(ξ). We can diagonalize this equation and conclude, since the triple is stable, that ∥ξ∥2 is a convex function. Since ξ(0)=0 and ξ has a finite limit at t=∞ ⁠, such an ξ must be identically zero.□ As for the moduli space M ⁠, we can conclude that, away from the degeneracy locus, Mτ1,τ2,τ3 is a smooth hypersymplectic manifold. Its complex and product structures are described by the following result, proved by combining the arguments of Sections 3.5 and 3.6 and of [20]. Theorem 6.3 There exists a neighbourhood Uof G/Tin Mτ1,τ2,τ3not intersecting the degeneracy locus Dsuch that: if τ2+iτ3is a regular semisimple element of gC ⁠, then (U,I)is biholomorphic to an open neighbourhood of Ad(G)(τ2+iτ3)in the complex adjoint orbit of τ2+iτ3 ⁠; if τ2=τ3=0 ⁠, then (U,I)is biholomorphic to an open neighbourhood of GC/Bin T(GC/B) ⁠, where Bis a Borel subgroup of GC ⁠; with respect to the product structure S ⁠, Uis isomorphic to an open neighbourhood of the diagonal G-orbit of (τ1+τ3,τ1−τ3)in O+×O− ⁠, where O±is the adjoint G-orbit of τ1±τ3 ⁠. Remark 6.4 With extra care, one should be able to show, as in [8], that in the case when τ2+iτ3 is a nonzero non-regular element of gC ⁠, U is biholomorphic to a neighbourhood of the zero section in a vector bundle over GC/P ⁠, where P is an appropriate parabolic subgroup. Other complex and product structures of U⊂Mτ1,τ2,τ3 are easily identified by observing that such a complex or product structure W can be rotated via an element A∈SO(1,2) to W0=I or W0=S ⁠, and that (U,W) is isomorphic to (U′,W0) ⁠, where U′ is the corresponding open subset of MA(τ1,τ2,τ3) ⁠. 6.2. Hypersymplectic quotients by compact subgroups We return to M ⁠, the moduli space of solutions on [0,1] ⁠. As observed in Section 3.2, M possesses a hypersymplectic action of G×G ⁠, given by allowing gauge transformations with arbitrary values at t=0 and t=1 ⁠. The hypersymplectic moment map (μI,μS,μT) for this action is computed from the calculation in the proof of Proposition 3.8 and yields μI(T)=(−T1(0),T1(1)),μS(T)=(T2(0),−T2(1)),μT(T)=(T3(0),−T3(1)). We can now consider hypersymplectic quotients of M by any subgroup K of G×G ⁠, to be denoted by N ⁠. If we decompose g⊕g orthogonally with respect to the Ad-invariant metric as k⊕m ⁠, where k=Lie(K) ⁠, then the hypersymplectic quotient is the moduli space of solutions to the Nahm–Schmid equations on [0,1] ⁠, such that (Ti(0),Ti(1))∈m ⁠, i=1,2,3 ⁠, modulo gauge transformations such that (g(0),g(1))∈K ⁠. These moduli spaces can, of course, be singular, and even if they are smooth, their hypersymplectic structure may degenerate at some point. An interesting example is the case G=U(n) ⁠, K=O(n)×O(n) (or G=SU(n) ⁠, K=SO(n)×SO(n) ⁠). The moduli space N consists of solutions such that Ti(0) and Ti(1) are symmetric for i≥1 ⁠, modulo orthogonal gauge transformations. This means that, at t=0,1 ⁠, the quadratic matrix polynomial T(ζ) considered in Section 4.2 is a symmetric matrix for each ζ ⁠. It is not hard to see (cf. [6, Proposition 1.4]) that, under the correspondence between T(ζ) and semistable sheaves F on the spectral curve considered in Section 4.2, such symmetric matrix polynomials correspond to theta characteristics, that is, the sheaf E=F(−1) satisfies E2≃ωΣ ⁠, where ωΣ is the dualizing sheaf of S (here we assume that F is an invertible sheaf on Σ ⁠, that is, that the dimension of any eigenspace of T(ζ) ⁠, ζ∈CP1 ⁠, is equal to 1 ⁠). Thus, solutions to the Nahm–Schmid equations in N correspond to flows between two theta characteristics in the direction Lt ⁠, where L is the line bundle on TCP1 with transition function exp(η/ζ) ⁠. Denote by Ft the sheaf corresponding to T(ζ)(t) ⁠. Thus, F1=L⊗F0 ⁠. Moreover, both F0(−1) and F1(−1) are theta characteristics, that is, they both square to ωΣ ⁠. It follows that L∣Σ2≃O ⁠. This condition is well known to hold for spectral curves of SU(2)-monopoles and it is, therefore, tempting to consider N as a hypersymplectic analogue of the hyperkähler monopole moduli space. To see that such an interpretation requires caution, consider N for G=SU(2) ⁠. As we have seen in Section 3.3, modulo rotations in SO(1,2) ⁠, solutions to the reduced Nahm–Schmid equations can be written as Ti(t)=fi(t)ei ⁠, where the fi are given by Jacobi elliptic functions. With the ei given by (4.5), such a solution is symmetric at t if and only if f2(t)=0 ⁠. Thus, the condition that such a solution belongs to N is equivalent to f2(0)=f2(1)=0 ⁠. Proposition 3.16 implies then, however, that T belongs to the degeneracy locus. Thus, for G=SU(2) ⁠, the hypersymplectic structure is degenerate at every point of N ⁠. Acknowledgements Parts of this work are contained in the third author’s DPhil thesis [26] and he wishes to thank his supervisor, Prof. Andrew Dancer, for his guidance and support. The second author would like to thank Paul Norbury for early discussions related to material contained in this paper, and FCT for financial support under the Grant BPD/14544/03. References 1 M. R. Adams , J. Harnad and J. C. Hurtubise , Isospectral Hamiltonian flows in finite and infinite dimensions. II. Integration of flows , Comm. Math. Phys. 134 ( 1990 ), 555 – 585 . Google Scholar Crossref Search ADS 2 M. Adler and P. van Moerbeke , Completely integrable systems, Euclidean Lie algebras, and curves , Adv. Math. 38 ( 1980 ), 267 – 317 . Google Scholar Crossref Search ADS 3 M. Adler and P. van Moerbeke , Linearization of Hamiltonian systems, Jacobi varieties and representation theory , Adv. Math. 38 ( 1980 ), 318 – 379 . Google Scholar Crossref Search ADS 4 M. F. Atiyah and N. J. Hitchin , The Geometry and Dynamics of Magnetic Monopoles, M. B. Porter Lectures , Princeton University Press , Princeton, NJ , 1988 . 5 T. N. Bailey and M. G. 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Kovalev , Nahm’s equations and complex adjoint orbits , Quart. J. Math. Oxf. Ser. (2) 47 ( 1996 ), 41 – 58 . Google Scholar Crossref Search ADS 19 P. B. Kronheimer , A Hyperkähler structure on the cotangent bundle of a complex Lie group, MSRI preprint, 1988 . 20 P. B. Kronheimer , A hyper-kählerian structure on coadjoint orbits of a semisimple complex group , J. Lond. Math. Soc. 42 ( 1990 ), 193 – 208 . Google Scholar Crossref Search ADS 21 P. Liebermann , Sur le problème d'équivalence de certaines structures infinitésimales , C. R. Acad. Sc. Paris 233 ( 1951 ), 27 – 120 . 22 A. N. Malyshev , Factorization of matrix polynomials , Siberian Math. J. 23 ( 1982 ), 136 – 146 . 23 N. Manton and P. Sutcliffe , Topological Solitons , Cambridge University Press, Cambridge , 2004 . 24 T. Matsoukas , Hypersymplectic Quotients, DPhil Thesis, University of Oxford, 2010 . 25 W. Nahm , All self-dual multimonopoles for arbitrary gauge groups, Preprint TH.3172-CERN (unpublished) 1981 . 26 M. Röser , The ASD Equations in Split Signature and Hypersymplectic Geometry, DPhil thesis, University of Oxford, 2012 . 27 M. Röser , Harmonic maps and hypersymplectic geometry , J. Geom. Phys. 78 ( 2014 ), 111 – 126 . Google Scholar Crossref Search ADS 28 M. Rosenblatt , A multidimensional prediction problem , Arkiv Mat. 3 ( 1958 ), 407 – 424 . Google Scholar Crossref Search ADS 29 W. Schmid , Variation of Hodge structure: the singularities of the period mapping , Invent. Math. 22 ( 1973 ), 211 – 319 . Google Scholar Crossref Search ADS 30 S. Terashima , On M5-branes in N = 6 Membrane Action , JHEP 0808 ( 2008 ), 080 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. 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# The Nahm–Schmid equations and hypersymplectic geometry

The Quarterly Journal of Mathematics, Volume 69 (4) – Dec 1, 2018
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### Abstract

Abstract We explore the geometry of the Nahm–Schmid equations, a version of Nahm’s equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang–Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral curves, as well as their relation to hypersymplectic geometry. 1. Introduction Hypersymplectic geometry [11, 15] can informally be thought of as a pseudo-Riemannian analogue of hyperkähler geometry, in which the role of the quaternions H is played instead by the algebra B of the split quaternions. The generators of B (over R ⁠) are denoted i,s,t and satisfy the relations i2=−1,s2=1=t2,is=t=−si. As an R-algebra, B is isomorphic to the associative algebra Mat2(R) of real 2×2 matrices, which is the split real form of the complex algebra Mat2(C) ⁠. A hypersymplectic manifold is then a manifold with an action of B on its tangent bundle which is compatible with a given pseudo-Riemannian metric of split signature; its dimension is necessarily a multiple of four. Definition 1.1 A hypersymplectic manifold is a quintuple (M4k,g,I,S,T) where g is a pseudo-Riemannian metric on M of signature (2k,2k) and I,S,T∈Γ(End(TM)) are parallel skew-adjoint endomorphisms satisfying the relations I2=−1,S2=1=T2,IS=T=−SI. The parallel endomorphism S (and similarly T ⁠) splits TM into its (±1)-eigenbundles, which are of equal rank and integrable in the sense of Frobenius. A hypersymplectic manifold M ⁠, therefore, locally splits (essentially, in a canonical way) as a product M+×M− ⁠, where the two factors have the same dimension 2k ⁠. For this reason, the endomorphism S is also called a local product structure in the literature. Given a hypersymplectic manifold, we obtain a triple of symplectic forms ωI=g(I·,·),ωS=g(S·,·),ωT=g(T·,·) (1.1) in analogy to the hyperkähler situation. The holonomy group of a hypersymplectic manifold is the real symplectic group Sp(2k,R) ⁠, which in particular implies that hypersymplectic manifolds are Ricci-flat. Hypersymplectic and hyperkähler structures have the same complexification, so it is not surprising that many results and constructions from hyperkähler geometry have hypersymplectic analogues. For example, one can show just like in the hyperkähler case that the endomorphisms I,S,T are parallel and integrable if and only if the associated 2-forms ωI,ωS,ωT are all closed. In fact, the hypersymplectic structure, that is, the quadruple (g,I,S,T) ⁠, is completely determined by the corresponding triple of symplectic forms (1.1). For instance, we have S=ωT−1◦ωI ⁠, if we interpret the ωi’s as isomorphisms ωi:TM→T*M ⁠. There is also a hypersymplectic quotient construction analogous to (but more pathological than) the hyperkähler quotient of [16]. Quotients of this sort constitute a basic source of hypersymplectic structures, and they may occur in dimensional reduction from split signature in various contexts in mathematical physics—see for example [17] for an application to the study of sigma-models with extended supersymmetry. The basic result is the following: Proposition 1.2. ([15]). Let (M,g,I,S,T)be a hypersymplectic manifold with an action of a Lie group Gwhich is Hamiltonian with respect to each symplectic form ωi,i∈{I,S,T} ⁠, with hypersymplectic moment map μ=(μI,μS,μT):M→g*⊗R3.Assume that the G-action is free and proper on μ−1(0) ⁠, that 0is a regular value of μ ⁠, and that the metric restricted to the G-orbits in μ−1(0)is non-degenerate. Then the quotient μ−1(0)/Gcarries in a natural way a hypersymplectic structure. In many situations, one can obtain a smooth quotient manifold μ−1(0)/G ⁠. However, this will typically carry a hypersymplectic structure only on the complement of a degeneracy locus. On the other hand, if G acts freely and properly on μ−1(0) ⁠, then the non-degeneracy assumption implies the smoothness of the quotient μ−1(0)/G just like in the proof of the hyperkähler version of the construction. It is our aim in this paper to apply the hypersymplectic quotient construction in an infinite-dimensional setting to obtain a hypersymplectic structure on a suitable open subset of the product manifold G×g3 ⁠. Our approach is closely analogous to Kronheimer’s construction of a hyperkähler metric on T*GC in [19]. Building in part on work by Matsoukas [24], we interpret G×g3 as a moduli space of solutions to the Nahm–Schmid equations (see Section 2 below), a ‘hypersymplectic version’ of Nahm’s equations [25] which can be viewed as a dimensional reduction of the anti-self-dual Yang–Mills equations in split signature. To our best knowledge, these equations first arose in the work of Schmid on deformations of complex manifolds [29], which predates the advent of Nahm’s equations in gauge theory [4, 25]. The article is organized as follows. In Section 2, we introduce the Nahm–Schmid equations as a dimensional reduction of the anti-self-dual Yang–Mills equations in split signature, and derive some immediate properties of their solutions with values in a compact Lie algebra g ⁠. The heart of the article is Section 3, where we examine the hypersymplectic geometry of the (framed) moduli space of solutions over the unit interval. We describe the corresponding degeneracy locus and investigate the complex and product structures of our moduli space. The results that we obtain are akin to those in [27], where the moduli space of solutions to the gauge-theoretic harmonic map equations was studied; note that these equations also arise by dimensional reduction of the anti-self-dual Yang–Mills equations in split signature. Then in Section 4 we consider the Nahm–Schmid equations from the point of view of integrable systems [2, 3]. We discuss the relevant twistor space (as a particular case of a more general construction by Bailey and Eastwood [5]), as well as a Lax-pair formulation of the equations, in particular the associated spectral curve. Finally, in Section 5, we discuss other moduli spaces of solutions to the Nahm–Schmid equations. We show that the moduli spaces of solutions on a half-line (with the limit satisfying a certain stability condition) is a hypersymplectic analogue of the hyperkähler metrics on adjoint orbits in gC ⁠. We also briefly discuss hypersymplectic quotients of the moduli space of solutions over the unit interval by subgroups of G×G ⁠. 2. The Nahm–Schmid equations We use R2,2 to denote R4 equipped with the pseudo-Riemannian metric dx02+dx12−dx22−dx32 ⁠. Let G be a compact Lie group with Lie algebra g ⁠, on which we choose an Ad-invariant inner product ⟨·,·⟩ ⁠. Solutions to the Nahm–Schmid equations will correspond to G-connections ∇=d+∑i=03Tidxi on R2,2 that are anti-self-dual (with respect to the standard orientation given by dx0∧dx1∧dx2∧dx3 ⁠) and also invariant under translation in the variables x1,x2,x3 ⁠. That is, the connection matrices Ti depend only on the (real) coordinate x0 ⁠. We recall that the anti-self-duality equations on R2,2 are [∇0,∇1]=−[∇2,∇3],[∇0,∇2]=−[∇1,∇3],[∇0,∇3]=[∇1,∇2], where ∇i=∂∂xi+Ti,i=0,1,2,3. Since we assume that the Ti only depends on x0 (which we shall also denote by t ⁠), the partial derivatives in the xi-directions with i=1,2,3 play no role in the anti-self-duality equations. We may, therefore, replace the covariant partial derivatives in these equations by ∇0=ddt+T0and∇i=Ti,i=1,2,3. Thus, we arrive at the following definition. Definition 2.1. Let g be the Lie algebra of a compact Lie group. A quadruple of g-valued functions (T0,T1,T2,T3) ⁠, Ti:R→g satisfies the Nahm–Schmid equations if T˙1+[T0,T1]=−[T2,T3],T˙2+[T0,T2]=[T3,T1],T˙3+[T0,T3]=[T1,T2]. (2.1) The solutions to the Nahm–Schmid equations are invariant under gauge transformations, that is, functions u:R→G acting by u.(T0,T1,T2,T3)≔(uT0u−1−u˙u−1,uT1u−1,uT2u−1,uT3u−1). (2.2) Note that we can always find a gauge transformation that solves the ODE uT0u−1−u˙u−1=0. This allows us to transform any solution (T0,T1,T2,T3) of (2.1) into a solution with T0=0 ⁠. For such a solution T=(0,T1,T2,T3) ⁠, the triple (T1,T2,T3) satisfies the reduced Nahm–Schmid equations T˙1=−[T2,T3],T˙2=[T3,T1],T˙3=[T1,T2]. (2.3) An important property of this system of equations is the following. Proposition 2.2 Let (T0,T1,T2,T3)be a solution to the Nahm–Schmid equations. Then the following gauge-invariant quantities are conserved: ∥T1∥2+∥Ti∥2,i=2,3and⟨Ti,Tj⟩,i≠j,i,j≥1. Proof This follows from a direct computation, using the equations and the invariance of the inner product.□ We shall give a natural interpretation of this statement in Section 4. An immediate consequence, which is in contrast to the behaviour of solutions of the usual Nahm equations [25], is a global existence result for solutions that can be brought to the reduced form T above (see also [24]): Corollary 2.3 Any solution of the reduced Nahm–Schmid equations (2.3) exists for all time. Proof Proposition 2.2 shows that we have a conserved quantity C=2∥T1∥2+∥T2∥2+∥T3∥2; note that for any t∈R one has ∥Ti(t)∥2≤C. Thus, the solution is uniformly bounded. The assertion now follows from the fact that if a solution to an ODE only exists for finite time, then it has to leave every compact set. □ Remark 2.4. Put on g×g×g the indefinite metric coming from the identification with g⊗R1,2 and consider the function ϕ:g×g×g→R,ϕ(ξ1,ξ2,ξ3)≔⟨[ξ1,ξ2],ξ3⟩. The right-hand side of the reduced Nahm–Schmid equations (2.3) is then the negative of the gradient of ϕ ⁠, with respect to this metric. We know from Proposition 2.3 that solutions to this gradient flow are bounded, and so exist for all times. It follows moreover that the non-trivial trajectories of this flow are contained in the compact submanifolds MC={(ξ1,ξ2,ξ3)∣2∥ξ1∥2+∥ξ2∥2+∥ξ3∥2=C}⊂g3, for suitable C>0 ⁠. 3. The moduli space over [0,1] In this section, we study solutions of the Nahm–Schmid equations over a compact interval U⊂R ⁠, which we fix to be U=[0,1] for convenience. Again, G denotes any compact Lie group and g stands for its Lie algebra. 3.1. The moduli space as a manifold We denote by A the space of all quadruples of C1-functions on U with values in g ⁠, that is, the (affine) Banach manifold A={(T0,T1,T2,T3):U→g⊗R4∣TidifferentiableofclassC1}≅C1(U,g)⊗R4 equipped with the norm ∥T∥C1≔∑i=03∥Ti∥C1, where we use the usual C1-norm ∥Ti∥C1≔∥Ti∥sup+∥T˙i∥sup on each component. Proposition 3.1. The set of solutions to the Nahm–Schmid equations is a smooth Banach submanifold of A ⁠. Proof This follows just like the analogous statement in the hyperkähler case [19]. Let μ:A→C0(U,g)⊗R3 denote the difference of the RHS and the LHS in (2.1), so that the set of solutions to the Nahm–Schmid equations is given by μ−1(0) ⁠. Owing to the implicit function theorem, we have to check that, for any solution T=(T0,T1,T2,T3)∈μ−1(0) ⁠, the linearization dμT:TTA→C0(U,g)⊗R3 has a bounded right-inverse. That is, to ζ=(ζ1,ζ2,ζ3)∈C0(U,g)⊗R3 we must associate a solution X=(X0,X1,X2,X3)∈TTA of the system of linear ODEs X˙1+[T0,X1]+[X0,T1]+[T2,X3]+[X2,T3]=ζ1X˙2+[T0,X2]+[X0,T2]−[T3,X1]−[X3,T1]=ζ2X˙3+[T0,X3]+[X0,T3]−[T1,X2]−[X1,T2]=ζ3. We note, just like in [19], that for any ζ this system has a unique solution with X0≡0 ⁠, Xi(0)=0 ⁠, i=1,2,3 ⁠, and deduce the existence of the required right-inverse.□ We now want to quotient by a suitable gauge group. We consider the Banach Lie group G≔C2(U,G) of gauge transformations acting on A ⁠. This has a normal subgroup consisting of gauge transformations that are equal to the identity at the endpoints of U ⁠: G00={u∈C2(U,G)∣u(0)=1G=u(1)}. The Lie algebras of G and G00 are given by Lie(G)=C2(U,g) and Lie(G00)={ξ∈C2(U,g)∣ξ(0)=0=ξ(1)}. The action of G (and also of G00 ⁠) on A is smooth and given by (2.2). Furthermore: Lemma 3.2. The gauge group G00acts freely and properly on A ⁠. Proof Suppose we have u∈G00 and T∈A such that u.T=T ⁠. Looking at the T0-component, we see that u solves the initial value problem u˙u−1=uT0u−1−T0,u(0)=1G. Its unique solution is the constant map u(t)≡1G and hence the action is free. To see that it is proper suppose that we have a sequence (Tm)⊂A converging to T∈A and a sequence (um)⊂G00 such that umTm→T˜∈A ⁠. Then umT0mum−1−u˙mum−1→T˜0. (3.1) Since T0m also converges to T0 in the C1-topology, it follows first from the Arzelà–Ascoli theorem that a subsequence of um converges in the C0-topology, and then a repeated use of (3.1) shows that the convergence is actually in the C2-topology. Thus, the action of G00 is proper.□ We are now ready to introduce our main object of study. Definition 3.3 The moduli space of solutions to the Nahm–Schmid equations is M≔{T∈A∣Tsolvesequations(2.1)}/G00. (3.2) Theorem 3.4. Mis a smooth Banach manifold diffeomorphic to G×g×g×g ⁠. Proof Owing to Lemma 3.2, M ⁠, equipped with the quotient topology, is Hausdorff. Let T=(T0,T1,T2,T3) be a solution of the Nahm–Schmid equations. We need to construct a slice to the action of G00 at T ⁠. Let u0 be the unique gauge transformation such that T0=−u˙0u0−1 and u0(0)=1G ⁠. Write ANS for the set of solutions to the Nahm–Schmid equations and define ST={S=(S0,S1,S2,S3)∈ANS∣S0=u0X0u0−1−u˙0u0−1,X0∈g,∥X0∥<ϵ}, where ϵ>0 will be determined later. If u.S∈ST for an S∈ST and a u∈G00 ⁠, then g=u0−1uu0∈G00 and gX0g−1−g˙g−1=Y0∈g ⁠. The unique solution with g(0)=1 is exp(−tY0)exp(tX0) and hence g(t)≡1G if ϵ is small enough. It follows then that u(t)≡1G and the map ST×G00→ANS,(S,u)↦u.S (3.3) is injective for ϵ small enough. The inverse map is given explicitly as follows. Let (R0,R1,R2,R3)∈ANS and let v∈G be the unique gauge transformation such that R0=−v˙v−1 and v(0)=1G ⁠. Let X0∈g satisfy expX0=u0−1(1)v(1) ⁠. Then p(t)=e−tX0u0(t)v(t)−1 is an element of G00 and p−1.R∈ST ⁠. It is easy to check that both the map (3.3) and its inverse are smooth, and hence ANS/G00 is a smooth Banach manifold. The diffeomorphism with G×g×g×g is given by Ψ:M→G×g×g×g,T↦(u0(1),T1(0),T2(0),T3(0)), where u0 is the unique gauge transformation such that T0=−u˙0u0−1 and u0(0)=1G ⁠. The inverse is given by Φ:G×g×g×g→M,(γ,ξ1,ξ2,ξ3)↦uγ.Tξ, where uγ∈G is an arbitrary gauge transformation such that uγ(0)=1G and uγ(1)=γ∈G ⁠, and Tξ=(0,T1,T2,T3) with (T1,T2,T3) being the unique solution of the reduced Nahm–Schmid equations (2.3) with initial conditions Ti(0)=ξi ⁠. We note that uγ is unique up to multiplication by an element of G00 ⁠.□ Remark 3.5 It follows from the proof that the tangent space to the space of solutions ANS at a solution T splits as VT⊕HT ⁠, where VT is the tangent space to the G00-orbit through T ⁠, and HT consists of quadruples (X0,X1,X2,X3) which solve the linearized Nahm equations with X0=u0ξu0−1 for some ξ∈g ⁠. In other words, X0 is an arbitrary solution of X˙0+[T0,X0]=0 ⁠. Thus the tangent space TTM can be identified with the set of solutions (X0,X1,X2,X3) to the system of linear equations X˙1+[T0,X1]+[X0,T1]+[T2,X3]+[X2,T3]=0X˙2+[T0,X2]+[X0,T2]−[T3,X1]−[X3,T1]=0X˙3+[T0,X3]+[X0,T3]−[T1,X2]−[X1,T2]=0X˙0+[T0,X0]=0. (3.4) 3.2. Group actions We now want to examine various symmetries of the construction (3.2). Since G00⊂G is a closed normal subgroup, we obtain an action of G/G00≅G×G on M given by gauge transformations with arbitrary values at t=0 and t=1 ⁠. Explicitly, (u1,u2)∈G×G acts by a gauge transformation u∈G such that u1=u(0),u2=u(1) ⁠. Any two choices of such a u differ by an element of G00 ⁠, which acts trivially on M ⁠. Furthermore, we have an action of the Lorentz group SO(1,2) on the moduli space. An element A=[Aij]i,j=13∈SO(1,2) acts on (T0,T1,T2,T3)∈g⊗(R⊕R1,2) naturally by A.(T0,T1,T2,T3)=(T0,∑j=13A1jTj,∑j=13A2jTj,∑j=13A3jTj). It is easy to check by direct computation that this action preserves Equations (2.1) and commutes with the action of G ⁠. Hence, it descends to an SO(1,2)-action on M that commutes with the (G×G)-action described above. The following result follows straightforwardly from examining the proof of Theorem 3.4; see [10] for an analogous result on the corresponding Nahm moduli space. Proposition 3.6 The bijection Ψ:M→G×g×g×gfrom Theorem3.4is compatible with the actions of SO(1,2)and G×Gin the following way: The action of G×Gon Mand the action (γ,ξ1,ξ2,ξ3)↦(u2γu1−1,u1ξ1u1−1,u1ξ2u1−1,u1ξ3u1−1),(u1,u2)∈G×Gon G×g×g×gare intertwined by Ψ ⁠. The action of SO(1,2)on Mand the action (γ,ξ1,ξ2,ξ3)↦(γ,∑j=13A1jξj,∑j=13A2jξj,∑j=13A3jξj)on G×g×g×gare intertwined by Ψ ⁠. 3.3. Explicit solutions for G=SU(2) Here we will exhibit an example of non-trivial su(2)-valued solutions, which can be found using an Ansatz analogous to the one used to calculate Nahm data for centred SU(2) magnetic 2-monopoles in Euclidean R3 (cf. (8.155) in [23]). It will turn out that this Ansatz essentially yields all solutions in the su(2)-case. Setting T0=0,Tj(t)=fj(t)ej,j=1,2,3, (3.5) where {ej}j=13 is a standard basis for su(2) ⁠, that is, [ei,ej]=ek if (ijk) is a cyclic permutation of (123) ⁠, then the Nahm–Schmid equations yield the following system of ODEs for the functions fj ⁠: f˙1=−f2f3,f˙2=f3f1,f˙3=f1f2. The general solution to this system can be expressed in terms of Jacobi elliptic functions [9] with arbitrary modulus κ∈[0,1] ⁠: f1(t)=aκsnκ(at+b),f2(t)=aκcnκ(at+b),f3(t)=−adnκ(at+b), where a,b∈R are arbitrary constants. For κ=0 ⁠, this yields with dn0≡1 the trivial solution T3=constant,T1=T2=0 ⁠. For κ=1 we obtain sn1(t)=tanh(t),cn1(t)=dn1(t)=sech(t) ⁠. The solution corresponding to fixed κ,a,b is smooth for all t∈R ⁠, as we expected from Corollary 2.3. For κ∈(0,1) it is also periodic with period 4K(κ)/a ⁠, where K is the complete elliptic integral of the first kind [9]. If we fix an invariant inner product on su(2) such that the basis {e1,e2,e3} is orthonormal, then we see that the conserved quantities ∥T1∥2+∥Ti∥2 are f12+f22=a2κ2,f12+f32=a2. (3.6) Note that for κ=1 the conserved quantities coincide, and the solution is of course not periodic in this case. Matsoukas has shown in his DPhil thesis [24] that in fact any solution to the Nahm–Schmid equations with values in su(2) may be put into this form. To see this, first gauge T0 away by a gauge transformation in G≅{1G}×G⊂G×G ⁠, that is, a gauge transformation which equals the identity at t=0 ⁠. Then use the SO(1,2)-action to put the solution (T1,T2,T3) into standard form such that ⟨Ti,Tj⟩=0 whenever i≠j ⁠. Now the vector space su(2) is three-dimensional, so it follows that for generic t∈R the elements T1(t),T2(t),T3(t) form an orthogonal basis. The Nahm–Schmid equations imply that Ti(t) and T˙i(t)=±[Tj(t),Tk(t)] are linearly dependent for each t ⁠. It follows that the direction of Ti does not vary with time for each i ⁠. This implies that there exists an orthonormal basis of su(2) ⁠, say the standard one used above, such that the Tj are of the form Tj(t)=fj(t)ej and hence reduce to the Ansatz used above to obtain explicit solutions. We summarize this discussion as follows. Proposition 3.7 (cf. [24]) Let Mbe the moduli space of su(2)-valued solutions to the Nahm–Schmid equations on [0,1]modulo the gauge group G00 ⁠. Let T∈M ⁠. Then the SO(1,2)×G-orbit of Tcontains an SO(3)∩SO(1,2)≅SO(2)-orbit of solutions of the form (3.5), with respect to a standard orthonormal basis of su(2) ⁠. 3.4. Hypersymplectic interpretation At a point T=(T0,T1,T2,T3)∈A ⁠, the tangent space to A has the description TTA=C1(U,g)⊗R4. On the R4≅B factor, we have the split-quaternionic structure I,S,T induced from multiplication by −i,s,t on B from the right. Explicitly: I(X0,X1,X2,X3)=(X1,−X0,−X3,X2) (3.7) S(X0,X1,X2,X3)=(X2,X3,X0,X1) (3.8) T(X0,X1,X2,X3)=(X3,−X2,−X1,X0). (3.9) We endow A=C1(U,g)⊗R2,2 with the indefinite metric given by the tensor product of the L2-metric and the standard metric on R2,2 ⁠. On tangent vectors X=(X0,X1,X2,X3) and Y=(Y0,Y1,Y2,Y3) ⁠, we have the formula g(X,Y)=∫U∑i=03ηii⟨Xi(t),Yi(t)⟩dt. (3.10) Here we wrote ηii for the diagonal coefficients of the metric on R2,2 ⁠, that is, η00=η11=1,η22=η33=−1 ⁠. This gives A the structure of a flat infinite-dimensional hypersymplectic manifold with symplectic forms ωI(·,·)=g(I·,·),ωS(·,·)=g(S·,·),ωT(·,·)=g(T·,·). The action of the group of gauge transformations G preserves this flat hypersymplectic structure. By calculating ddθ∣θ=0exp(θξ).(T0,T1,T2,T3) for ξ∈Lie(G) ⁠, we see that the fundamental vector fields associated to the action of G (and hence also for G00 ⁠) at a point T=(T0,T1,T2,T3)∈A are given by XTξ=(−ξ˙+[ξ,T0],[ξ,T1],[ξ,T2],[ξ,T3]). Proposition 3.8 The action of the group G00on Apreserves the hypersymplectic structure and the hypersymplectic moment map at T=(T0,T1,T2,T3)∈Ais given by μI(T)=−T˙1−[T0,T1]−[T2,T3],μS(T)=T˙2+[T0,T2]−[T3,T1],μT(T)=T˙3+[T0,T3]−[T1,T2].Hence, we can write the moduli space Mof solutions to the Nahm–Schmid equations formally as the hypersymplectic quotient: M=μ−1(0)/G00. Proof We only exhibit the calculation for μI ⁠, as the other two equations are obtained analogously. Let T∈A ⁠, ξ∈Lie(G00) ⁠. First observe that IXTξ=I(−ξ˙+[ξ,T0],[ξ,T1],[ξ,T2],[ξ,T3])=([ξ,T1],ξ˙−[ξ,T0],−[ξ,T3],[ξ,T2]). Thus, we can calculate for Y∈TTA ⁠, using integration by parts and the boundary condition ξ(0)=0=ξ(1) ⁠: ωI(Xξ,Y)=g(IXTξ,Y)=∫01⟨[ξ,T1],Y0⟩+⟨ξ˙−[ξ,T0],Y1⟩+⟨[ξ,T3],Y2⟩−⟨[ξ,T2],Y3⟩dt=⟨ξ,Y1⟩∣01+∫01⟨ξ,−ξ˙−[T0,Y1]−[Y0,T1]−[T2,Y3]−[Y2,T3]⟩dt=∫01⟨ξ,−ξ˙−[T0,Y1]−[Y0,T1]−[T2,Y3]−[Y2,T3]⟩dt. The assertion follows.□ Lemma 3.9 Let T∈A ⁠. The orthogonal complement to the tangent space to the G00-orbit through Twith respect to the indefinite metric gis given by tangent vectors (X0,X1,X2,X3)∈TTAsatisfying the equation X˙0+∑i=03ηii[Ti,Xi]=0. (3.11) Proof This is a standard calculation using integration by parts and the boundary conditions defining Lie(G00) ⁠. Let ξ∈Lie(G00) ⁠, (X0,X1,X2,X3)∈TTA ⁠. Then g(XTξ,X)=∫01⟨−ξ˙+[ξ,T0],X0⟩+∑i=13ηii⟨[ξ,Ti],Xi⟩dt=∫01⟨ξ,X˙0⟩+∑i=03ηii⟨ξ,[Ti,Xi]⟩dt. The result follows.□ Definition 3.10. Consider the Banach manifold μ−1(0)⊂A of solutions to the Nahm–Schmid equations. The degeneracy locus D is the set of points T∈μ−1(0) such that the metric g restricted to the tangent space of the G00-orbit through T is degenerate. Lemma 3.11. The actions of the gauge group Gand of SO(1,2)both preserve the degeneracy locus D ⁠. Proof This follows immediately from the fact that both G and SO(1,2) act by isometries.□ Proposition 3.12. ([24]). The degeneracy locus Dconsists of exactly those solutions to the Nahm–Schmid equations such that the boundary value problem d2ξdt2+[T˙0,ξ]+2[T0,ξ˙]+∑i=03ηii[Ti,[Ti,ξ]]=0,ξ∈Lie(G00)has a non-trivial solution. Proof The tangent space to the orbit through T is spanned by the values of the fundamental vector fields XTξ ⁠. We thus need to find ξ∈Lie(G00) such that the tangent vector XTξ∈TTA satisfies Equation (3.11). Plugging XTξ=(−ξ˙+[ξ,T0],[ξ,T1],[ξ,T2],[ξ,T3]) into that equation yields the desired result.□ Proposition 3.13 The complement μ−1(0)⧹Dof the degeneracy locus consists exactly of those solutions T∈μ−1(0)for which the linear operator ΔT:Lie(G00)→C0(U,g),ξ↦d2ξdt2+[T˙0,ξ]+2[T0,ξ˙]+∑i=03ηii[Ti,[Ti,ξ]] (3.12)is an isomorphism. In particular, μ−1(0)⧹Dis open. Proof Owing to the Arzelà–Ascoli theorem, the operator ΔT is a compact perturbation of the operator d2dt2:Lie(G00)→C0(U,g) ⁠, which is an isomorphism. Hence ΔT is Fredholm of index zero. Thanks to Proposition 3.12, ΔT is injective if T is not contained in the degeneracy locus. Hence it must be an isomorphism.□ An analogous argument to the one outlined in [19, pp. 4–5] (see also [10]) for the Nahm case yields then the following result. Theorem 3.14 The submanifold M0≔(μ−1(0)⧹D)/G00of Mis a smooth hypersymplectic manifold. The tangent space TTM0can be identified with the set of solutions Xto the system of linear equations (cf. (3.4)) X˙1+[T0,X1]+[X0,T1]+[T2,X3]+[X2,T3]=0X˙2+[T0,X2]+[X0,T2]−[T3,X1]−[X3,T1]=0X˙3+[T0,X3]+[X0,T3]−[T1,X2]−[X1,T2]=0X˙0+[T0,X0]+[T1,X1]−[T2,X2]−[T3,X3]=0,and the hypersymplectic structure is given by (3.7–3.9). The last equation above says that X is orthogonal to the G00-orbit, as we know from Lemma 3.9. We can obtain some more quantitative information about the degeneracy locus. Proposition 3.15 Let T∈μ−1(0)be a solution such that 2sup(∥T2(t)∥2+∥T3(t)∥2)<π2 ⁠. Then Tdoes not belong to the degeneracy locus D ⁠. This holds in particular for any solution with T2≡0≡T3 ⁠. Proof Since the degeneracy locus is invariant under the action of G ⁠, we may restrict attention to solutions such that T0≡0 ⁠. A solution T lies in the degeneracy locus if and only if there is a nontrivial solution ξ∈Lie(G00) to the linear boundary value problem ξ¨+(ad(T1)2−ad(T2)2−ad(T3)2)(ξ)=0,ξ(0)=0=ξ(1). (3.13) Let us write this ODE as ξ¨+Aξ=0 ⁠. Since ad(Ti) is skew-symmetric with respect to the invariant inner product, we have for any ξ∈C0([0,1],g) ⟨Aξ,ξ⟩=−∥[T1,ξ]∥2+∥[T2,ξ]∥2+∥[T3,ξ]∥2≤∥[T2,ξ]∥2+∥[T3,ξ]∥2. Using the pointwise inequality ∥[ξ,η]∥2≤2∥ξ∥2∥η∥2 ⁠, this implies ⟨Aξ,ξ⟩≤2sup(∥T2(t)∥2+∥T3(t)∥2)∥ξ∥2. Setting M≔2sup(∥T2(t)∥2+∥T3(t)∥2) ⁠, we can conclude that f(t)=∥ξ∥2 satisfies the differential inequality f¨+Mf≥0 ⁠. Let g(t)≔f˙(0)Msin(Mt) for t∈U ⁠, which solves g¨+Mg=0 with initial condition g(0)=0 ⁠, g˙(0)=f˙(0) ⁠. Consider the function ϕ(t)=f(t)/g(t) ⁠. Provided that M<π2 ⁠, ϕ is well-defined and differentiable on [0,1] ⁠. The inequality f¨+Mf≥0 and the equality g¨+Mg=0 imply that f¨g−g¨f≥0 on [0,1] and integrating this inequality shows that ϕ is monotonically increasing there. Since ϕ(0)=1 ⁠, ϕ ⁠, and hence ξ ⁠, cannot have another zero on (0,1] ⁠. Thus, if M<π2 ⁠, then the boundary value problem ξ¨+Aξ=0 ⁠, ξ(0)=ξ(1)=0 has only the trivial solution ξ≡0 ⁠.□ On the other hand we have Proposition 3.16 Let T=(T0,T1,T2,T3)be a non-constant solution of the Nahm–Schmid equations on [0,1]such that, for some A∈SO(1,2) ⁠, one of the components T˜i=(AT)i ⁠, i=1,2,3 ⁠, vanishes at t=0and t=1 ⁠. Then Tbelongs to the degeneracy locus D ⁠. Proof Without loss of generality we can assume that T0≡0 ⁠, A=1 and that T1 is not constant and vanishes at 0 and 1 ⁠. It is easy to check that ξ=T1 satisfies (3.13).□ Example 3.17 If G=SU(2) we can say more about Equation (3.12) and the degeneracy locus. For a solution in the standard form T0=0,Tj(t)=fj(t)ej ⁠, j=1,2,3 ⁠, we have (adTj)2(ξ)=−fj2⟨ξ,ej⟩ej and hence Equation (3.13) is diagonal in the standard basis of su(2) ⁠: ξ¨1=−(f22+f32)ξ1ξ¨2=(f12−f32)ξ2ξ¨3=(f12−f22)ξ3, where ξ=∑i=13ξiσi ⁠. Using (3.6), we can rewrite these equations as ξ¨1=(2f12−a2−κ2a2)=a2(2κ2snκ2(at+b)−1−κ2)ξ1ξ¨2=(2f12−a2)ξ2=a2(2κ2snκ2(at+b)−1)ξ2ξ¨3=(2f12−κ2a2)ξ3=a2κ2(2snκ2(at+b)−1)ξ3. These equations have particular solutions ξ1=snκ(at+b) ⁠, ξ2=cnκ(at+b) and ξ3=dnκ(at+b) (cf. Proposition 3.16). Since dnκ does not vanish on the real line, the Sturm Separation Theorem [13, Cor. XI.3.1] implies that the third equation cannot have a non-trivial solution vanishing at two points. On the other hand snκ (resp. cnκ ⁠) vanishes at points 2mK (resp. (2m+1)K ⁠), m∈Z ⁠. Thus, a solution T with b∈KZ ⁠, a∈2KZ belongs to the degeneracy locus D ⁠, while, owing again to the Sturm Separation Theorem, any solution with [a,a+b] properly contained in [mK,(m+1)K] does not belong to D ⁠. Corollary 3.18 For any non-abelian compact Lie algebra g ⁠, the degeneracy locus D⊂Mis non-empty. Proof Let ρ:su(2)→g be a (non-trivial) Lie algebra homomorphism and let (T0,T1,T2,T3) be an su(2)-valued solution to the Nahm–Schmid equations such that T1 is not constant and vanishes at t=0,1 ⁠. Then (ρ(T0),ρ(T1),ρ(T2),ρ(T3))∈D by Proposition 3.16.□ 3.5. Complex structures Denote by gC=g⊗C the complexification of g ⁠. We may identify the complex manifold (A,I) with C1(U,gC))⊗C2 via T=(T0,T1,T2,T3)↦(α,β),whereα≔T0−iT1,β≔T2+iT3. Writing a tangent vector to C1(U,gC)⊗C2 as (a,b) ⁠, the holomorphic symplectic form ωIC=ωS+iωT is given by ωIC((a1,b1),(a2,b2))=∫01⟨b1,a2⟩−⟨a1,b2⟩dt. The vanishing of the hypersymplectic moment map is the same as saying that the complex equation μC≔μS+iμT=0 and the real equation μI=0 are satisfied simultaneously. In the case of the Nahm–Schmid equations, this gives β˙+[α,β]=0, (3.14) α˙+α˙*+[α,α*]−[β,β*]=0. (3.15) Let GC be the complexification of G ⁠, that is, GC is a complex Lie group with Lie algebra gC and maximal compact subgroup G ⁠. The complex equation is invariant under the action of the complexified gauge group GC=C2(U,GC) and its normal subgroup G00C≔{u∈GC∣u(0)=1GC=u(1)} acting by u.α=uαu−1−u˙u−1,u.β=uβu−1. A computation similar to the one in Proposition 3.8 shows that the complex equation is the vanishing condition for the moment map of the action of G00C with respect to ωIC ⁠. We write the moduli space of solutions to the complex equation modulo G00C as N≔{(α,β):U→gC×gC∣β˙+[α,β]=0}/G00C. One can show that this is a smooth Banach manifold. Just like in the proof of Theorem 3.4 we obtain a diffeomorphism Φ:N→GC×gC≅T*GC,(α,β)↦(u0(1),β(0)), where u0∈GC is the unique complex gauge transformation with α=−u˙0u0−1 such that u0(0)=1GC ⁠. The space N is in a natural way a complex-symplectic quotient, and it can be checked that Φ is holomorphic and pulls back the canonical symplectic form on T*GC to the symplectic form ωIC ⁠. We obviously have a natural map M→N ⁠, since any solution to the Nahm–Schmid equations gives a solution to the complex equation. The question to ask at this point is the following: given a solution to the complex equation β˙+[α,β]=0 ⁠, does its G00C-orbit contain a solution to the real equation, and to what extent is this solution unique? Using work of Donaldson [12], Kronheimer answers this question affirmatively for the usual Nahm equations by showing that every G00C-orbit of a solution to the complex equation contains a unique G00-orbit of solutions to the real equation. In the hyperkähler case, it thus turns out that the map M→N is a bijection, inducing, therefore, a hyperkähler structure on T*GC ⁠. We think of g as a subalgebra of u(n) for some n∈N ⁠. Let us write the real equation as μI(α,β)=0 ⁠. We now aim at describing how this equation behaves under complex gauge transformations. To simplify calculations, we define operators on C∞([0,1],gC) by ∂¯α=ddt+[α,·],∂α=ddt−[α*,·],∂¯β=[β,·],∂β=−[β*,·]. The following two lemmas are then obtained by straightforward calculations. Lemma 3.19 μI(α,β)=[∂α,∂¯α]−[∂β,∂¯β],as operators on C∞([0,1],Cn) ⁠. Lemma 3.20 Let u∈GCbe a complex gauge transformation. Then ∂¯u.α=u◦∂¯α◦u−1,∂u.α=(u*)−1◦∂α◦u*∂¯u.β=u◦∂¯β◦u−1,∂u.β=(u*)−1◦∂β◦u*. Using this description of the real moment map, it is now easy to work out the behaviour of the real equation under complex gauge transformations. Lemma 3.21 Let u∈GCbe a complex gauge transformation, and put h≔u*u ⁠. Then u−1(μI(u.α,u.β))u=μI(α,β)−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh)). Now given a solution (α,β) to the complex equation, we want to find a self-adjoint and positive h∈G00C solving the boundary value problem μI(α,β)−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh))=0,h(0)=1=h(1). (3.16) Existence and uniqueness will then imply that the complex gauge transformation u=h1/2 takes (α,β) to a solution of the real equation. Lemma 3.22 Let T=(T0,T1,T2,T3)be a solution to the Nahm–Schmid equations, that is, a solution to the complex equation with μI(α,β)=0 ⁠. Then the linearization of the boundary value problem −∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh))=0is given by the operator −ΔTin (3.12). Proof Suppose that we have a one-parameter family h:(−ϵ,ϵ)×[0,1]→GC of self-adjoint solutions h(s,t)=exp(iξ(s,t)),h(0,t)=1,h(s,0)=1=h(s,1) to the boundary value problem (3.16), where ξ(s,t)(−ϵ,ϵ)×[0,1]→g ⁠. Write ξ′ for the partial derivative of ξ with respect to s at s=0 ⁠. Note that the condition h(s,0)=1=h(s,1) implies that ξ′(0)=0=ξ′(1) ⁠, that is, ξ′∈Lie(G00) ⁠. We compute the linearization of (3.16) for (α,β) with μI(α,β)=0 and denote the linear operator obtained in this way by L:Lie(G00)→C0([0,1],g) ⁠. Explicitly, Lξ′=−idds∣s=0(−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh))). Using the equation 0=μI(α,β)=T˙1+[T0,T1]+[T2,T3] and the Jacobi identity, we see Lξ′=−idds∣s=0(−∂¯α(h−1(∂αh))+∂¯β(h−1(∂βh)))=−∂¯α(∂αξ′)+∂¯β(∂βξ′)=−(ddt+[α,·])(ξ˙′−[α*,ξ′])−[β,[β*,ξ]]=−ξ¨′+[α˙*,ξ′]+[α*,ξ˙′]−[α,ξ˙′]+[α,[α*,ξ′]]−[β,[β*,ξ′]]=−ξ¨′−[T˙0,ξ′]−i[T˙1,ξ′]−[T0+iT1,ξ˙′]−[T0−iT1,ξ˙′]−[T0−iT1,[T0+iT1,ξ′]]+[T2+iT3,[T2−iT3,ξ′]]=−ξ¨′−[T˙0,ξ′]+i[[T0,T1]+[T2,T3],ξ′]−[T0+iT1,ξ˙′]−[T0−iT1,ξ˙′]−[T0−iT1,[T0+iT1,ξ′]]+[T2+iT3,[T2−iT3,ξ′]]=−ξ¨′−2[T0,ξ˙′]−[T˙0,ξ′]−[T0,[T0,ξ′]]−[T1,[T1,ξ′]]+[T2,[T2,ξ′]]+[T3,[T3,ξ′]]=−ΔTξ′. □ Theorem 3.23 Let (α,β)be a solution to the Nahm–Schmid equations, and assume that (α,β)is not contained in the degeneracy locus D ⁠. Then there exists a constant ϵ>0such that for any solution (α˜,β˜)of the complex equation with ∥α˜−α∥C1+∥β˜−β∥C1<ϵthere exists a unique u∈G00Cclose to the identity which is self-adjoint such that u.(α˜,β˜)solves the real equation. Proof Consider the map F:Lie(G00)×A→C0(U,g) given by F(ξ,(α˜,β˜))=μI(α˜,β˜)−∂¯α(eiξ(∂αe−iξ))+∂¯β(eiξ(∂βe−iξ)), which satisfies F(0,(α,β))=0 ⁠. The partial derivative of F at the point (0,(α,β)) in the Lie(G00)-direction is the operator ΔT ⁠. Owing to the assumption, this is an isomorphism. Hence, the assertion follows from the implicit function theorem.□ In particular, we note that this result applies to solutions T such that 2sup(∥T2∥2+∥T3∥2)<π2, owing to Proposition 3.15. Corollary 3.24 Consider the complex symplectic manifold (M0,I,ωIC) ⁠. Then there exists an open neighbourhood of the set {T∈M∣T2=T3=0}which is isomorphic to a suitable open neighbourhood of the zero section in T*GCwith its canonical complex symplectic structure. 3.6. Product structures In this subsection, we investigate the paracomplex structure [21, Chapter V] of the moduli space M ⁠. As a first step, we identify the space of field configurations A ⁠, equipped with the product structure S ⁠, as a parakähler manifold [21]. Let B≔{ddt+A∣A∈C1(U,g)} be the space of G-connections on the interval U ⁠. Then we can naturally identify T*B={(ddt+A,B)∣A,B∈C1(U,g)}≅C1(U,g)2. (3.17) Proposition 3.25 The map P:(A,S)→(T*B×T*B,id⊕(−id))given by (T0,T1,T2,T3)↦((T0+T2,T1+T3),(T0−T2,T1−T3)) (3.18)is a diffeomorphism respecting the product structures. Proof Clearly, the map P is a diffeomorphism. It is also straightforward to check by direct calculation that dP◦S=(1⊕(−1))◦dP. □ Now P is not just a diffeomorphism. We observe that it intertwines the symplectic form ωI and the product of canonical symplectic forms on T*B×T*B ⁠. Proposition 3.26 Consider the symplectic form Ω≔ωT*B⊕ωT*Bon T*B×T*B ⁠. Then ωI=P*Ω. Proof The symplectic form ωI is given at an element T∈A by ωI(X,Y)=−g(X0,Y1)+g(X1,Y0)−g(X2,Y3)+g(X3,Y2), where X=(X0,X1,X2,X3),Y=(Y0,Y1,Y2,Y3)∈TTA are two tangent vectors at T ⁠. The symplectic form Ω is given by Ω(V,W)=g(V0,W1)−g(V1,W0)+g(V2,W3)−g(V3,W2), with V=(V0,V1,V2,V3),W=(W0,W1,W2,W3)∈T(T*B×T*B)=T(T*B)⊕T(T*B) ⁠. Now P is a linear map in the description (3.18), and this implies that dP maps a tangent vector X at T∈A to a tangent vector V at P(T)∈T*B×T*B of the form V=(X0+X2,X1+X3,X0−X2,X1−X3). Plugging V,W as above, for some X,Y tangent to T∈A ⁠, into the formula for Ω ⁠, gives the result claimed, that is Ω(V,W)=ωI(X,Y). □ In the coordinates on T*B×T*B provided by the description (3.17), the Nahm–Schmid equations are then equivalent to the system B˙1+[A1,B1]=0B˙2+[A2,B2]=0A˙1−A˙2+[12(A1+A2),A1−A2]−[B1,B2]=0, (3.19) when we set A1≔T0+T2,A2≔T0−T2,B1≔T1+T3,B2≔T1−T3. We may view the first two equations in (3.19) as a single g⊕g-valued ‘paracomplex’ equation, and the third equation as a ‘real’ equation. It is evident that the paracomplex equation is invariant under what one may call paracomplex gauge transformations, that is, elements of G×G acting componentwise as (u1,u2)·(ddt+Ai,Bi)i=12=(ddt+ui−1Aiui+ui−1u˙i,ui−1Biui)i=12. Let us consider the moduli space  given by ≔{(A,B)∈T*B∣B˙+[A,B]=0}/G00. We define, in analogy with Theorem 3.4, a bijection →G×g,(A,B)↦(u(1),B(0)), where again u is the unique gauge transformation that satisfies A=−u˙u−1 and u(0)=1G ⁠. We may view  as a symplectic quotient of T*B by the action of the group G00 ⁠. This map then identifies  and T*G as symplectic manifolds. By applying this construction on each factor of T*B×T*B and composing with the map P from above, we obtain a map from the moduli space of solutions to the Nahm–Schmid equations to ×≅T*G×T*G ⁠. The paracomplex equation in the system (3.19) may be interpreted as saying that, for i=1,2 ⁠, the connections ∇i≔(ddt+Ai)dt+Bidx on the adjoint bundle over Euclidean R2 are flat. We note that any solution to the paracomplex equation with A1=A2 and B1=B2 automatically solves the real equation. The corresponding solutions to the Nahm–Schmid equations satisfy T2=0=T3 ⁠. We can, moreover, rewrite the real equation as A˙1−A˙2+[A2,A1−A2]+[B2,B1−B2]=0, which says that the two flat connections ∇1,∇2 are in Coulomb gauge with respect to each other, that is d∇2∗(∇1−∇2)=0. This gives a natural interpretation of the Nahm–Schmid equations in terms of the geometry of the infinite-dimensional Riemannian manifold T*B/G00 ⁠, with Riemannian metric given by the usual L2-metric. The L2-metric equips the affine space T*B with the structure of a flat infinite-dimensional Riemannian manifold, so geodesic segments γ:[0,1]→T*B starting at (A1,B1) with initial velocity (a1,b1) are just straight lines: γ(τ)=(A1,B1)+τ(a1,b1),τ∈[0,1]. The group G00 acts by isometries and geodesic segments on the quotient T*B/G00 can be lifted (at least locally) to horizontal geodesics on T*B ⁠, that is, geodesics, the velocity vector of which is orthogonal to the G00 orbits. A geodesic γ as above will be horizontal exactly when (a1,b1) satisfies the Coulomb gauge condition. This means that a solution to the Nahm–Schmid equations on [0,1] can be interpreted as a horizontal geodesic segment in T*B ⁠, the endpoints of which lie in the space of solutions to the equation B˙+[A,B]=0. Now a calculation similar to the one at the end of the previous section shows that if ∇1,∇2 defines a solution to the Nahm–Schmid equations, then the linearization of the equation d∇1∗(eξ.(∇2)−∇1)=0 is again ΔTξ=0. Thus, elements of the degeneracy locus correspond to horizontal geodesic segments, the endpoints of which are conjugate. The degeneracy locus D ⁠, therefore, has a natural interpretation in terms of the cut-locus of the infinite-dimensional Riemannian manifold T*B/G00 ⁠. Given a pair of points in  that are sufficiently close, an application of the implicit function theorem shows that there exists a unique gauge transformation close to the identity that puts them into Coulomb gauge. It follows that we can identify an open subset of M0=(μ−1(0)⧹D)/G00 with a neighbourhood of the diagonal inside T*G×T*G ⁠. In summary, we have the following observation: Proposition 3.27 Consider the symplectic manifold (M0,ωI)with local product structure S ⁠. The map Pdefined in Proposition3.25induces a symplectomorphism between a neighbourhood of the subset {T∈M∣T2=T3≡0}and a neighbourhood of the diagonal in (T*G×T*G,ωT*G⊕ωT*G) ⁠. This symplectomorphism is compatible with the local product structure Son M0and the obvious product structure on T*G×T*G ⁠. 4. Twistor space and spectral curves This section describes how general constructions and results on paraconformal manifolds and on integrable systems with spectral parameter apply to the moduli space of solutions of the Nahm–Schmid equations. 4.1. Twistor space Bailey and Eastwood [5] gave a very general construction of twistor space in the holomorphic category and the usual twistor space constructions for real manifolds can be viewed as applying a particular real structure to a Bailey–Eastwood twistor space. In the case of hypersymplectic manifolds, this works as follows. The complexification of the algebra of split quaternions is the same as the complexification of the usual quaternions. Thus on the complexified tangent bundle TCM of a hypersymplectic manifold (M,g,I,S,T) ⁠, we obtain three anti-commuting endomorphisms I,J=iS,K=iT which all square to −1 ⁠. For any point p=(a,b,c) on the sphere S2⊂R3 ⁠, we consider the (−i)-eigenspace Ep of the endomorphism aI+bJ+cK ⁠. This is an integrable distribution of TCM ⁠, since I,S,T are integrable. Since a hypersymplectic manifold is real-analytic, it has a complexification MC ⁠. The holomorphic tangent bundle of MC restricted to M is naturally identified with TCM ⁠, and we can extend the endomorphisms I,J,K uniquely to a neighbourhood of M in MC ⁠. For each p∈S2 ⁠, we obtain an integrable holomorphic distribution Ep in T1,0MC ⁠, and these combine to give an integrable holomorphic distribution E on MC×CP1 ⁠. The twistor space Z of M ⁠, as defined by Bailey and Eastwood, is the space of leaves of all E ⁠. Let us discuss particular features of this construction in the case of hypersymplectic manifolds. First of all, we observe that, if a≠0 ⁠, then the equation (aI+biS+ciT)u=−iu can be rewritten as (a−1I+ca−1S−ba−1T)u=−iu ⁠. The endomorphism a−1I+ca−1S−ba−1T is, as can be readily checked, a complex structure on M ⁠. Thus for a≠0 ⁠, the space of leaves of Ep is just M equipped with the corresponding complex structure a−1I+ca−1S−ba−1T ⁠. If a=0 ⁠, Ep is the complexification of the (−1)-eigenspace (in TM ⁠) of the product structure bS+cT on M ⁠, and its space of leaves does not have to be Hausdorff. If, however, the space of leaves of the (−1)-eigenspace of bS+cT is a manifold Mb,c+ ⁠, then the space of leaves of E(0,b,c) on MC can be identified (possibly after making MC smaller) with the complexification of Mb,c+ ⁠. Thus, in this case, Z is a manifold. This is the case on a neighbourhood of the subset {T∈M∣T2=T3≡0} ⁠, as shown in Proposition 3.27. We shall soon see that the moduli space M of Nahm–Schmid equations has a globally defined twistor space. The space Z is clearly a complex manifold, and the projection onto S2≃CP1 is holomorphic. It has a natural anti-holomorphic involution σ obtained from the real structure on MC and the inversion τ with respect to the circle a=0 (that is, ∣ζ∣=1 ⁠) on CP1 ⁠, σ(m,ζ)=(m¯,1ζ¯). The manifold M can be now recovered as a connected component of the Kodaira moduli space of real sections (that is, equivariant with respect to τ on CP1 and σ on Z ⁠) the normal bundle of which splits as a sum of O(1)’s. The endomorphisms I,S,T are also easily recovered, since the fibre of Z over any (a,b,c) with a≠0 is canonically biholomorphic to (M,a−1I+ca−1S−ba−1T) ⁠, and any real section in our connected component of the Kodaira moduli space meets such a fibre in exactly one point. Observe that, over the circle a=0 ⁠, it is no longer true that real sections separate points of the fibres. All we can say that such sections locally separate points in (Z(0,b,c))σ×(Z(0,−b,−c))σ≃Mb,c+×M−b,−c+, which is equivalent to (M,bS+cT) being locally isomorphic to Mb,c+×M−b,−c+ ⁠. Remark 4.1 Unlike in the case of hyperkähler manifolds, the twistor space is not uniquely determined by M ⁠, since over the circle ∣ζ∣=1 only the τ-invariant part of the fibres of Z→CP1 is uniquely determined. Even more generally, we shall refer to any complex manifold Z with a holomorphic submersion onto CP1 and an anti-holomorphic involution σ covering the inversion τ as a twistor space. It should be pointed out, however, that if we start with such a generalized Z ⁠, and apply the construction given at the beginning of this section to a connected component M of the Kodaira moduli space of real sections of Z ⁠, we shall in general not recover Z ⁠. All we can say is that the resulting twistor space ZM of M is equipped with a local biholomorphism ϕ:ZM→Z ⁠, which is a fibre map (over CP1 ⁠) and which intertwines the real structures. To recover the metric from Z ⁠, we need one more piece of data: a twisted holomorphic symplectic form along the fibres of Z→CP1 ⁠. If we extend the metric g to the complexified tangent bundle TCM and define ωI=g(I·,·) ⁠, ωJ=g(J·,·) ⁠, ωK=g(K·,·) ⁠, then it follows easily that the O(2)-valued 2-form ω+=ωJ+iωK+2ζωI−ζ2(ωJ−iωK) vanishes on vectors in Eζ ⁠. Here ζ is the standard complex affine coordinate on CP1 related to p=(a,b,c) via (a,b,c)=(1−ζζ¯1+ζζ¯,−(ζ+ζ¯)1+ζζ¯,i(ζ¯−ζ)1+ζζ¯). It follows that ω+ descends to fibres of the projection Z→CP1 ⁠. Since J=iS and K=iT ⁠, we can rewrite ω+ in terms of the symplectic structures ωI,ωS,ωT on M ⁠: ω+=i(ωS+iωT−2iζωI−ζ2(ωS−iωT)). We can of course replace ω+ by −iω+ and hence, from now on, we shall consider the following holomorphic symplectic form along the fibres of Z ⁠: ω+=ωS+iωT−2iζωI−ζ2(ωS−iωT). (4.1) Remark 4.2 Away from the locus ∣ζ∣≠1 ⁠, the twistor space can be also constructed in analogy with Hitchin’s original construction of the twistor space of a hyperkähler manifold in [16]. First, we identify CP1⧹{∣ζ∣=1} with the two-sheeted hyperboloid C={(x1,x2,x3)∈R3∣x12−x22−x32=1} via x=(x1,x2,x3)=(1+ζζ¯1−ζζ¯,i(ζ¯−ζ)1−ζζ¯,ζ+ζ¯1−ζζ¯). Then we define an almost complex structure on the product Zo≔M×C ⁠, which at (m,x) is equal to (x1I+x2S+x3T)⊕J0 ⁠, where J0 is the standard complex structure on CP1 ⁠. An argument completely analogous to the one in [16] shows that this almost complex structure is integrable, and Zo becomes a complex manifold fibring over C ⁠. Unlike in the case of hyperkähler geometry, there is, however, no easy way to recover M from Zo ⁠. Assume now that a Lie group G acts on M preserving the hypersymplectic structure, and that this action extends to a global action of a complexification GC of G ⁠. If the G-action is Hamiltonian for ωI,ωS,ωT ⁠, then the action of GC is Hamiltonian for the symplectic form (4.1), and we obtain an associated moment map μ+=μS+iμT−2iζμI−ζ2(μS−iμT). The fibrewise complex-symplectic quotient Z¯ of Z by GC (assuming it is manifold) is then a twistor space in the sense of Remark 4.1. The space of real sections of Z¯ which descend from Z is the hypersymplectic quotient of M by G (note, however, that the hypersymplectic quotient does not need to be a hypersymplectic manifold everywhere: some of these sections may have a wrong normal bundle). We shall now apply the ideas above to the infinite-dimensional context of the Nahm–Schmid equations. The twistor space of the flat infinite-dimensional hypersymplectic Banach manifold A is formally the total space of C1(U,gC)⊗C2⊗O(1) ⁠. The fibrewise moment map for the complexified gauge group G00 with respect to ω+ is the Lax equation discussed in Section 3.5. Thus, each fibre of the quotient twistor space is isomorphic to gC×GC ⁠. A computation completely analogous to that in [19] shows that the twistor space Z of M is obtained by gluing two copies of C×gC×GC by means of the transition function (ζ,η,g)↦(1ζ,ηζ2,gexp(−η/ζ)). The real structure is given by σ(ζ,η,g)=(1/ζ¯,η¯/ζ¯2,(g*)−1exp(−η¯/ζ¯)), and the complex symplectic form along the fibres is the standard symplectic form on T*GC ⁠. Real sections of Z are obtained from real sections of C1(U,gC)⊗C2⊗O(1) via the fibrewise quotient with respect to ω+ ⁠. If we write α=T0−iT1,β=T2+iT3, then the moment maps of Proposition 3.8 can be written as 2iμI(T)=[ddt+α,−ddt+α*]−[β,β*]=α˙+α˙*+[α,α*]−[β,β*](μS+iμT)(T)=[ddt+α,β]=β˙+[α,β](μS−iμT)(T)=[−ddt+α*,β*]=−β˙*+[α*,β*]. Putting the terms together, we obtain for μ+ μ+(T)=μS+iμT−2iζμI−ζ2(μS−iμT)=[ddt+α−ζβ*,β−ζ(−ddt+α*)]. Equivalently, we can modify this to μ+(T)=[ddt+α−ζβ*,β−ζ(α+α*)+ζ2β*]. Thus, the real sections of Z obtained in this way are precisely gauge equivalence classes of solutions to the Nahm–Schmid equations, that is, points of M ⁠. However, only the sections corresponding to M⧹D have normal bundle with the required splitting property. 4.2. The spectral curve The expression for μ+(T) obtained above means that the Nahm–Schmid equations are equivalent to the following single ODE with a complex parameter: ddt(β−ζ(α+α*)+ζ2β*)=[β−ζ(α+α*)+ζ2β*,α−ζβ*]. (4.2) This is an example of a Lax equation with spectral parameter: if we write T(ζ)=β−ζ(α+α*)+ζ2β*,T+(ζ)=α−ζβ*, then (4.2) becomes dT(ζ)dt=[T(ζ),T+(ζ)]. (4.3) This equation has been studied extensively (see for example [1]), and so we can simply state the relevant results. Here we shall restrict ourselves to the case g=u(n) ⁠. First of all, (4.3) implies that the spectrum of T(ζ) is constant in t ⁠, so that the characteristic polynomial of T(ζ) defines a conserved quantity of the Nahm–Schmid system: Σ={(ζ,η);det(η−T(ζ))=0}. This is an algebraic curve (possibly singular) of (arithmetic) genus (n−1)2 in the total space of the line bundle O(2) over CP1 ⁠, invariant under the involution σ(ζ,η)=(1/ζ¯,η¯/ζ¯2). (4.4) In particular, we observe that the coefficient of ζ2 in trT(ζ)2 is the conserved quantity C=2∥T1∥2+∥T2∥2+∥T3∥2 introduced in Corollary 2.3. The cokernel of η−T(ζ) is a semistable sheaf F on Σ with χ(F)=n2−n (and such that H0(Σ,F⊗O(−1))=0 ⁠). The Lax equation corresponds then to the linear flow t→F⊗Lt ⁠, where L is a line bundle on TCP1≃∣O(2)∣ with the transition function exp(μ(ζ,η)) and μ is polynomial in ζ−1 and η determined by the condition that the positive powers of ζ in μ(ζ,T(ζ)) and in −T+(ζ) coincide [1]. In our case, it follows that μ(ζ,η)=η/ζ ⁠. The sheaves arising from solutions to the Nahm–Schmid equations satisfy several further conditions. Consider the case of a smooth spectral curve Σ ⁠. Then the cokernel of η−T(ζ) is one-dimensional for each (ζ,η)∈Σ and, consequently, F is a line bundle. Moreover, F(−1)=F⊗O(−1) is a line bundle of degree g−1 not in the theta divisor. The fact that F is obtained from the matrix polynomial T(ζ) imposes two further conditions: E=F(−1) satisfies the reality condition E⊗τ(E)≃KS≃O(2n−4) ⁠, where τ is the induced anti-holomorphic involution on Jacg−1(Σ) ⁠; the Hitchin metric [14, Section 6] on H0(Σ,F) is positive definite. It is perhaps worth pointing out that the fact that the solutions to the Nahm–Schmid equations exist for all time, implies that the closure of the set of line bundles satisfying the above conditions does not intersect the theta divisor in Jacg−1(Σ) ⁠. Example 4.3 (Spectral curves for g=su(2) ⁠). We calculate the spectral curve associated with a solution (T0,T1,T2,T3)=(0,f1e1,f2e2,f3e3) as in (3.5). With e1=12(i00−i),e2=12(01−10),e3=12(0ii0), (4.5) we get T(ζ)=12(−2f1ζf2(1−ζ2)−f3(1+ζ2)f2(ζ2−1)−f3(1+ζ2)2f1ζ). A direct computation yields det(η−T)=η2−12(2f11+f22+f32)ζ2+14(f22−f32)(1+ζ4). Taking f1(t)=aκsnκ(at+b),f2(t)=aκcnκ(at+b),f3(t)=−adnκ(at+b) ⁠, we can write this as det(η−T)=η2−a22(1+κ2)ζ2+a24(κ2−1)(1+ζ4). In this case, g=1 and we can identify Jac0(Σ) with Σ itself. If we choose the origin (trivial bundle) O to correspond to a τ-invariant point on Σ ⁠, then the induced anti-holomorphic involution on Jac0(Σ)≃Σ is just τ ⁠. According to the above characterization of the sheaves F ⁠, they correspond to points x∈Σ such that x+τ(x)=O ⁠. This is a union of two circles in Σ ⁠, and the Nahm–Schmid flow lives on the circle that does not contain O ⁠. We also observe that the points where f1 or f2 vanish (that is, points for which t lies in −b+(K(κ)/a)Z ⁠) correspond to the even theta characteristics of Σ ⁠. 5. The ‘positive’ part of the moduli space We continue to assume that g=u(n) ⁠. To any triple T1,T2,T3∈u(n) ⁠, we associate, as in the previous section, the matrix polynomial T(ζ)=T2+iT3+2iT1ζ+(−T2+iT3)ζ2 ⁠. It follows that T(ζ)ζ−1 is hermitian for each ζ with ∣ζ∣=1 ⁠, and we denote by X+ the set of matrix polynomials T(ζ) such that T(ζ)ζ−1 is positive definite for each ζ with ∣ζ∣=1 ⁠. Observe that, for n=1 ⁠, this is the subset of R1,2 where x2−zz¯>0 ⁠, x>0 ⁠. It follows from a theorem of Rosenblatt [28] (see also [22] for a simple proof) that any T(ζ)∈X+ can be factorized as T(ζ)=(A+B*ζ)(B+A*ζ), (5.1) where det(B+A*ζ)≠0 for ∣ζ∣≤1 ⁠. The only freedom in the factorization is the following action of U(n) ⁠: (A,B)↦(Ag−1,gB) and, consequently, the factorization is unique if we assume, in addition, that B is hermitian. The following estimate is one reason for our interest in X+ ⁠. Lemma 5.1 If (T1,T2,T3)∈X+ ⁠, then ∥T2+iT3∥≤2∥T1∥ ⁠. Proof We have ∥T2+iT3∥=∥AB∥≤∥A∥∥B∥≤∥2iT1∥1/2∥2iT1∥1/2=2∥T1∥. □ We now consider the Nahm–Schmid flow on X+ ⁠. Definition 5.2 The positive part M+ of the moduli space M consists of solutions (T0(t),T1(t),T2(t),T3(t)) to the Nahm–Schmid equations such that (T1(t),T2(t),T3(t))∈X+ for some t (equivalently for all t ⁠). Suppose that T1(t),T2(t),T3(t) satisfy the Nahm–Schmid equations (with T0≡0 ⁠) and that we have a factorization (5.1) depending smoothly on t (we can always find a smooth factorization, given its uniqueness under the assumptions B=B* ⁠, det(B+A*ζ)≠0 for ∣ζ∣≤1 ⁠). We can then rewrite Equation (3.14) as (A˙−12B*BA+12ABB*)B+A(B˙−12A*AB+12BAA*)=0, and Equation (3.15) as ((A˙−12B*BA+12ABB*)A*+B*(B˙−12A*AB+12BAA*))H=0, where the superscript H denotes the hermitian part of a matrix. Thus, A(t),B(t) satisfy the following equations: A˙=12(B*BA−ABB*)+a, (5.2) B˙=12(A*AB−BAA*)+b, (5.3) where a(t),b(t) satisfy aB+Ab=0,aA*+Aa*+b*B+B*b=0 (5.4) for all t ⁠. Lemma 5.3 Let (a,b)be a solution of (5.4) and suppose that det(B+A*ζ)≠0for ∣ζ∣≤1 ⁠. Then there exists a unique ρ∈u(n)such that a=Aρ ⁠, b=−ρB ⁠. Proof The assumption implies that B is invertible. We can infer from the first equation in (5.4) that there exists a unique ρ∈gl(n,C) such that a=Aρ ⁠, b=−ρB ⁠. The second equation can be now rewritten as (B*)−1A(ρ+ρ*)A*B−1=ρ+ρ*. The conclusion follows from the following lemma:□ Lemma 5.4 Let Xbe a square complex matrix, any two eigenvalues of which satisfy λ1λ¯2≠1 ⁠. Then the only solution of the equation X*hX=hwith hhermitian is h=0 ⁠. Proof We may assume that X is in Jordan form. The equation under consideration means that X is an isometry of the sesquilinear form ⟨,⟩ defined by h ⁠. Let Jk(λ1),Jl(λ2) be two (not necessarily distinct) Jordan blocks of X with e1,…,ek and f1,…,fl the corresponding bases of cyclic X-modules, that is, Xe1=λ1e1 ⁠, Xei=λ1ei+ei−1 for 1<i≤k ⁠, and similarly for the fj ⁠. Since λ¯1λ2≠1 ⁠, ⟨e1,f1⟩=0 ⁠. Now it follows inductively that ⟨ei,fj⟩=0 for all i≤k,j≤l ⁠. Thus, h=0 ⁠.□ We can, therefore, rewrite (5.2) and (5.3) as A˙−Aρ=12(B*BA−ABB*),B˙+ρB=12(A*AB−BAA*). These equations are invariant under the following action of U(n)-valued gauge transformations: A↦Ag−1,B↦gB,ρ↦gρg−1−g˙g−1. In particular, we can use the gauge freedom to make ρ identically 0 and obtain the following equations for A,B ⁠: A˙=12(B*BA−ABB*), (5.5) B˙=12(A*AB−BAA*). (5.6) These equations are the split signature version of the Basu–Harvey–Terashima equations [7, 30]. We conclude: Theorem 5.5 Let (T1(t),T2(t),T3(t)) ⁠, t∈R ⁠, be a solution to the reduced Nahm–Schmid equations (2.3) belonging to M+ ⁠. Then there exist smooth functions A,B:R→Matn,n(C)satisfying (5.5) and (5.6) such that T1(t)=−i2(A(t)A*(t)+B*(t)B(t)),T2(t)+iT3(t)=A(t)B(t). (5.7)Conversely, if A(t),B(t)satisfy (5.5–5.6), then T1(t),T2(t),T3(t)given by (5.7) satisfy (2.3). Remark 5.6 The quantity tr(AA*+B*B) is an invariant of the flow (5.5) and (5.6). It is, therefore, a flow on a sphere. The flow (5.5) and (5.6) is preserved under the involution A↔B ⁠. In particular, there is a subset of solutions with A=B ⁠, that is, satisfying A˙=12(A*A2−A2A*)=12[A*A+AA*,A]. As an application, we obtain additional information about the product structures of M+ ⁠. Proposition 5.7 The map (M+¯,S)→T*G×T*Gdefined in Section3.6is proper. Proof The map under consideration is given by ϕ(T0(t),T1(t),T2(t),T3(t))=(T1(0)+T3(0),u1(1),T1(0)−T3(0),u2(1)), where u1(t) (resp. u2(t) ⁠) satisfies T0+T2=−u˙1u1−1 ⁠, u1(0)=1 (resp. T0−T2=−u˙2u2−1 ⁠, u2(0)=1 ⁠). Let K be a compact set in T*G×T*G ⁠. Then ∥T1(0)∥ is bounded on ϕ−1(K)∩M+¯ by a constant cK ⁠. Lemma 5.1 implies that ∥T2(0)∥2+∥T3(0)∥2≤4cK2 ⁠, and hence the conserved quantity C of Proposition 2.2 is bounded by 6cK2 ⁠. The Nahm–Schmid equations and the Arzelà–Ascoli theorem imply now that ϕ−1(K) is compact in M ⁠.□ 6. Other moduli spaces of solutions to the Nahm–Schmid equations We want to briefly discuss two other moduli spaces: solutions on a half-line and solutions on [0,1] which are symmetric at both ends. 6.1. The moduli space of solutions on a half-line The reduced Nahm–Schmid equations have the form x˙=V(x) ⁠, where x=(T1,T2,T3)∈g3 and V is the vector field given by V(x)=([T3,T2],[T3,T1],[T1,T2]) ⁠. The critical points of V consist of commuting triples (T1,T2,T3) ⁠. Let (τ1,τ2,τ3) be a regular commuting triple, that is, the centralizer of the triple is a Cartan subalgebra h ⁠. Let g=⨁λ∈h*gλ be the root decomposition of g with respect to h ⁠. A simple calculation shows that the differential DV at (τ1,τ2,τ3) preserves the subspace Uλ=gλ⊕gλ⊕gλ and DV∣Uλ has eigenvalues 0,±(λ(τ2)2+λ(τ3)2−λ(τ1)2)1/2 ⁠. Thus, as long as (adτ2)2+(adτ3)2−(adτ1)2 has no negative eigenvalues, the differential of V at the critical point (τ1,τ2,τ3) has no imaginary eigenvalues. We shall call such a triple (irrespectively of being regular or not) stable. Observe that, in particular, the triple (τ1,0,0) is stable. General results about the asymptotic behaviour of the solutions of ODEs (see for example [18] for a discussion of this in the context of Nahm’s equations) imply that any solution to the Nahm–Schmid equations on [0,∞) ⁠, the limit of which is a regular stable triple, approaches this limit exponentially fast, that is, there exists an η>0 such that ∥Ti(t)−τi∥≤const·e−ηt ⁠. Any η smaller than the smallest positive eigenvalue of D(τ1,τ2,τ3)V will do. Let (τ1,τ2,τ3) be a regular stable triple and let η>0 be as above. We introduce (cf. [20]) the Banach space Ω1 consisting of C1-maps f:[0,∞)→g such that the norm ∥f∥Ω1≔supt∥eηtf(t)∥+supt∥eηtf′(t)∥ (6.1) is finite. Let us write τ0=0 and set Aτ1,τ2,τ3≔{(T0,T1,T2,T3):[0,∞)→g⊗R4∣Ti−τi∈Ω1,i=0,1,2,3}. The Banach group G={u:[0,∞)→G∣g(0)=1,g˙g−1∈Ω1,Ad(g)τi∈Ω1,i=1,2,3} acts smoothly on the Banach manifold Aτ1,τ2,τ3 via gauge transformations and we define Mτ1,τ2,τ3={T∈A∣TsolvesEquations(2.1)}/G. As in [20], Mτ1,τ2,τ3 can be also described as the set of solutions to the reduced Nahm–Schmid equations such that limt→∞Ti(t)=Ad(g0)τi ⁠, i=1,2,3 ⁠, for some g0∈G ⁠. Theorem 6.1 Mτ1,τ2,τ3is a smooth Banach manifold diffeomorphic to the tangent bundle T(G/T)of G/T ⁠, where Tis a maximal torus in G ⁠. Proof The fact that Mτ1,τ2,τ3 is a smooth Banach manifold is proved analogously to Theorem 3.4, once we replace the constant X0 in the definition of the slice ST with X0(t)=ve−2ηt ⁠, v∈g ⁠. To identify Mτ1,τ2,τ3 ⁠, we observe that the space of solutions to the reduced Nahm–Schmid equations in a small neighbourhood of the critical point (τ1,τ2,τ3) and converging to this critical point is identified with the sum of negative eigenspaces of D(τ1,τ2,τ3)V (since the triple is stable). By virtue of the arguments given at the beginning of this section, this space is isomorphic to p=⨁λ>0gλ⊕g−λ ⁠. Since every solution to the Nahm–Schmid equations exists for all time, we conclude that the submanifold of Mτ1,τ2,τ3 consisting of solutions with T0≡0 is diffeomorphic to p ⁠. Now the remark made just before the statement of the theorem shows that Mτ1,τ2,τ3 is diffeomorphic to G×Tp≃T(G/T) ⁠.□ We now would like to view Mτ1,τ2,τ3 as the hypersymplectic quotient of Aτ1,τ2,τ3 by G ⁠. As for the moduli space of solutions on an interval, the hypersymplectic structure is not defined on the degeneracy locus, that is, on the set of solutions where the linear operator (3.12), acting on Lie(G) ⁠, has non-trivial kernel. We have: Proposition 6.2 Let (τ1,τ2,τ3)be a regular stable triple. There exists a neighbourhood U of G/T (which is the G-orbit of the constant solution (0,τ1,τ2,τ3) ⁠) in Mτ1,τ2,τ3 ⁠, which is disjoint from the degeneracy locus D ⁠. Proof It is enough to show that the constant solution itself does not belong to D ⁠. As in the proof of Proposition 3.15, we are asking whether the exists a non-zero ξ∈Lie(G) such that ξ¨=((adτ2)2+(adτ3)2−(adτ1)2)(ξ). We can diagonalize this equation and conclude, since the triple is stable, that ∥ξ∥2 is a convex function. Since ξ(0)=0 and ξ has a finite limit at t=∞ ⁠, such an ξ must be identically zero.□ As for the moduli space M ⁠, we can conclude that, away from the degeneracy locus, Mτ1,τ2,τ3 is a smooth hypersymplectic manifold. Its complex and product structures are described by the following result, proved by combining the arguments of Sections 3.5 and 3.6 and of [20]. Theorem 6.3 There exists a neighbourhood Uof G/Tin Mτ1,τ2,τ3not intersecting the degeneracy locus Dsuch that: if τ2+iτ3is a regular semisimple element of gC ⁠, then (U,I)is biholomorphic to an open neighbourhood of Ad(G)(τ2+iτ3)in the complex adjoint orbit of τ2+iτ3 ⁠; if τ2=τ3=0 ⁠, then (U,I)is biholomorphic to an open neighbourhood of GC/Bin T(GC/B) ⁠, where Bis a Borel subgroup of GC ⁠; with respect to the product structure S ⁠, Uis isomorphic to an open neighbourhood of the diagonal G-orbit of (τ1+τ3,τ1−τ3)in O+×O− ⁠, where O±is the adjoint G-orbit of τ1±τ3 ⁠. Remark 6.4 With extra care, one should be able to show, as in [8], that in the case when τ2+iτ3 is a nonzero non-regular element of gC ⁠, U is biholomorphic to a neighbourhood of the zero section in a vector bundle over GC/P ⁠, where P is an appropriate parabolic subgroup. Other complex and product structures of U⊂Mτ1,τ2,τ3 are easily identified by observing that such a complex or product structure W can be rotated via an element A∈SO(1,2) to W0=I or W0=S ⁠, and that (U,W) is isomorphic to (U′,W0) ⁠, where U′ is the corresponding open subset of MA(τ1,τ2,τ3) ⁠. 6.2. Hypersymplectic quotients by compact subgroups We return to M ⁠, the moduli space of solutions on [0,1] ⁠. As observed in Section 3.2, M possesses a hypersymplectic action of G×G ⁠, given by allowing gauge transformations with arbitrary values at t=0 and t=1 ⁠. The hypersymplectic moment map (μI,μS,μT) for this action is computed from the calculation in the proof of Proposition 3.8 and yields μI(T)=(−T1(0),T1(1)),μS(T)=(T2(0),−T2(1)),μT(T)=(T3(0),−T3(1)). We can now consider hypersymplectic quotients of M by any subgroup K of G×G ⁠, to be denoted by N ⁠. If we decompose g⊕g orthogonally with respect to the Ad-invariant metric as k⊕m ⁠, where k=Lie(K) ⁠, then the hypersymplectic quotient is the moduli space of solutions to the Nahm–Schmid equations on [0,1] ⁠, such that (Ti(0),Ti(1))∈m ⁠, i=1,2,3 ⁠, modulo gauge transformations such that (g(0),g(1))∈K ⁠. These moduli spaces can, of course, be singular, and even if they are smooth, their hypersymplectic structure may degenerate at some point. An interesting example is the case G=U(n) ⁠, K=O(n)×O(n) (or G=SU(n) ⁠, K=SO(n)×SO(n) ⁠). The moduli space N consists of solutions such that Ti(0) and Ti(1) are symmetric for i≥1 ⁠, modulo orthogonal gauge transformations. This means that, at t=0,1 ⁠, the quadratic matrix polynomial T(ζ) considered in Section 4.2 is a symmetric matrix for each ζ ⁠. It is not hard to see (cf. [6, Proposition 1.4]) that, under the correspondence between T(ζ) and semistable sheaves F on the spectral curve considered in Section 4.2, such symmetric matrix polynomials correspond to theta characteristics, that is, the sheaf E=F(−1) satisfies E2≃ωΣ ⁠, where ωΣ is the dualizing sheaf of S (here we assume that F is an invertible sheaf on Σ ⁠, that is, that the dimension of any eigenspace of T(ζ) ⁠, ζ∈CP1 ⁠, is equal to 1 ⁠). Thus, solutions to the Nahm–Schmid equations in N correspond to flows between two theta characteristics in the direction Lt ⁠, where L is the line bundle on TCP1 with transition function exp(η/ζ) ⁠. Denote by Ft the sheaf corresponding to T(ζ)(t) ⁠. Thus, F1=L⊗F0 ⁠. Moreover, both F0(−1) and F1(−1) are theta characteristics, that is, they both square to ωΣ ⁠. It follows that L∣Σ2≃O ⁠. This condition is well known to hold for spectral curves of SU(2)-monopoles and it is, therefore, tempting to consider N as a hypersymplectic analogue of the hyperkähler monopole moduli space. To see that such an interpretation requires caution, consider N for G=SU(2) ⁠. As we have seen in Section 3.3, modulo rotations in SO(1,2) ⁠, solutions to the reduced Nahm–Schmid equations can be written as Ti(t)=fi(t)ei ⁠, where the fi are given by Jacobi elliptic functions. With the ei given by (4.5), such a solution is symmetric at t if and only if f2(t)=0 ⁠. Thus, the condition that such a solution belongs to N is equivalent to f2(0)=f2(1)=0 ⁠. Proposition 3.16 implies then, however, that T belongs to the degeneracy locus. Thus, for G=SU(2) ⁠, the hypersymplectic structure is degenerate at every point of N ⁠. Acknowledgements Parts of this work are contained in the third author’s DPhil thesis [26] and he wishes to thank his supervisor, Prof. Andrew Dancer, for his guidance and support. 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For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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The Quarterly Journal of MathematicsOxford University Press

Published: Dec 1, 2018

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