Real Forms of Holomorphic Hamiltonian Systems
aa r X i v : . [ m a t h . S G ] O c t Real Forms of Holomorphic Hamiltonian Systems
Philip Arathoon & Marine FontaineSeptember 2020
Abstract
By complexifying a Hamiltonian system one obtains dynamics on a holomorphic sym-plectic manifold. To invert this construction we present a theory of real forms which notonly recovers the original system but also yields different real Hamiltonian systems whichshare the same complexification. This provides a notion of real forms for holomorphicHamiltonian systems analogous to that of real forms for complex Lie algebras. The the-ory behaves well with respect to reduction and integrability. For instance, the integrablesystem for the spherical pendulum on T ∗ S can be complexified and admits a compact,integrable real form on S × S . This produces a ‘unitary trick’ for integrable systemswhich curiously requires an essential use of hyperk¨ahler geometry. Keywords : Hamiltonian dynamics, integrable systems, hyperk¨ahler geometry
Background and Outline
Real analytic Hamiltonian systems are closely related to complex holomorphic Hamiltoniansystems. Indeed, if we treat the variables in a real analytic system as being complex weobtain a holomorphic system. For this reason it is not uncommon to treat these two conceptsequivalently. However, what this perspective overlooks is the possibility that many differentand distinct real analytic systems might each complexify to the same system. This motivates atheory of real forms for holomorphic Hamiltonian systems which will allow us to treat differentreal Hamiltonian systems as real forms of the same complex system.This idea is not new and has been considered before in [18, 30, 16, 17]. However, theirapproach is limited to systems on C n with real subspaces R n as real forms. We generalise thisto dynamics on holomorphic symplectic manifolds and introduce a wider definition for whatit means to be a real form. This has the advantage of extending the scope of the theory toinclude a greater variety of dynamical systems and also brings it into closer contact with ideasin differential geometry. Below we give an outline of the work contained within.
1. Introduction
A real analytic manifold can be complexified to give a complex manifold [42, 39, 29]. Theoriginal manifold appears as a totally real submanifold of half dimension, and thus, any suchsubmanifold with this property shall be considered a real form of the complex manifold. Ifthe original manifold possesses an analytic symplectic form then the complexification will bea holomorphic symplectic manifold. Exactly as with ordinary Hamiltonian dynamics one can1onsider a Hamiltonian vector field generated by a holomorphic function and investigate thedynamics generated by its flow.We distinguish those real forms upon which the holomorphic symplectic form is either purelyreal or pure imaginary; what we shall call a real- or imaginary-symplectic form. If such a realform is invariant under the flow of a holomorphic Hamiltonian then we justify why the restricteddynamics on this real form can be considered to be a real form of the holomorphic Hamiltoniansystem.
2. Reduction
If a holomorphic Hamiltonian systems admits some symmetry it is natural to ask how thissymmetry might manifest on a real form. A more precise formulation of this question is to askhow the complex and real symplectic reduced spaces might be related. We choose to address thisin terms of Poisson reduction, which in turn requires us to generalise holomorphic symplecticgeometry to the setting of holomorphic Poisson geometry (see [31] and the references therein).For when the symmetry of the holomorphic symplectic manifold is in a certain sense com-patible with respect to a real form, we show the extent to which a reduced space of the real formcan itself be considered a real form of a complex reduced space. Our notion of compatibilityin the presence of a real structure is identical to the work [36] and has much in common with[19].To demonstrate these results we introduce a guiding example which finds use throughoutthe paper: the example of the complex 2-sphere CS , by which we mean the affine variety x + y + z = 1in C . This is the complexification of the ordinary real sphere, and as such, is an example of aholomorphic symplectic manifold. We exhibit this complex sphere as a holomorphic symplecticreduced space for an action of C on C . Certain subspaces R ⊂ C are compatible real-symplectic forms with respect to this group action, and hence, descend to give examples ofreal-symplectic forms on CS .
3. Branes
A hyperk¨ahler manifold defines a holomorphic symplectic form for each choice of complexstructure. Naturally this provides us with a stock of examples of holomorphic symplecticmanifolds. In addition to this, complex-Lagrangian submanifolds taken with respect to onecomplex structure give real- and imaginary-symplectic forms with respect to two other complexstructures, respectively. This gives us a nice trick to generate examples of real- and imaginary-symplectic forms. We adopt the terminology taken from ideas in string theory and refer to suchsubmanifolds as branes.This trick is implemented for the example of the Eguchi-Hanson space whose constructionwe obtain through hyperk¨ahler reduction on the biquaternions [24, 20]. For separate choicesof complex structure the underlying holomorphic symplectic manifold can be either CS or T ∗ CP . In this way we demonstrate that complex-Lagrangian submanifolds of T ∗ CP , suchas the zero-section or a fibre, correspond to real-symplectic forms of CS . We also obtain anidentification between CS and T ∗ CP which turns out to be an essential ingredient for thespherical pendulum example in the following section.2 . Integrability The generalisation of an integrable system to the holomorphic category is straightforward, con-sidered for instance in [12] and extensively discussed for the real analytic category in [41, 1].Our main question is to ask whether holomorphic integrability is equivalent to integrabilityon a real form. We answer in the affirmative and show that if a real form admits an analyticintegrable system then this may be extended to a holomorphic integrable system in a neigh-bourhood of the real form. Conversely, if a holomorphic Hamiltonian system is integrable inthe holomorphic sense, then any real form of the dynamics is integrable in the real sense.This provides us with a powerful method for generating new integrable systems from old:begin with a real analytic integrable system, complexify it and obtain real integrable systems ondifferent real forms. This can be demonstrated for the classical integrable system of the sphericalpendulum. By using the hyperk¨ahler geometry from the previous section we identify T ∗ S with CS , whose complexification then gives a holomorphic integrable system on CS × CS . Thisadmits a real integrable system on the real form S × S which can, in a sense, be considereda compact real form of the spherical pendulum. A holomorphic symplectic manifold is a complex manifold M equipped with a closed and non-degenerate holomorphic 2-form Ω. This gives an isomorphism between each tangent space andits dual by identifying df ∈ T ∗ p M with the tangent vector X f ∈ T p M satisfyingΩ( X f , Y ) = h df, Y i for all Y ∈ T p M . In this way we may associate to a holomorphic function f a Hamiltonianvector field X f . If we separate the holomorphic symplectic form into its real and imaginaryparts we obtain real symplectic forms ω R and ω I on M , where Ω = ω R + iω I . If we write thecomplex structure on each tangent space as I then these two forms are related by ω R ( I ( X ) , Y ) = − ω I ( X, Y ) . (1.1) Proposition 1.1.
Let f be a holomorphic function defined on a holomorphic symplectic man-ifold ( M, Ω) and write f = u + iv as its decomposition into real and imaginary parts. TheHamiltonian vector field X f is equivalently the Hamiltonian vector field of u on ( M, ω R ) , andof v on ( M, ω I ) .Proof. This follows immediately by expanding Ω( X f , Y ) = h df, Y i into real and imaginaryparts as ω R ( X f , Y ) + iω I ( X f , Y ) = h du, Y i + i h dv, Y i . This proposition shows that the Hamiltonian vector field generated by a holomorphic func-tion defines a bihamiltonian structure on M with respect to the two symplectic forms ω R and ω I [1, 37]. 3o generalise the concept of real structures on complex vector spaces we say that a realstructure on a complex manifold M is an involution R whose derivative is everywhere conjugate-linear. If non-empty, the fixed-point set of R can be considered a real form of M . More generally,a real form of a complex manifold shall mean a totally real submanifold of half dimension. Definition 1.1.
A real form N of ( M, Ω) is called a real-symplectic form if the restriction ofΩ to N is purely real, and an imaginary-symplectic form if this restriction is purely imaginary.Furthermore, a real structure R is said to be • a real-symplectic structure if R ∗ Ω = Ω (conjugate-symplectic) • and an imaginary-symplectic structure if R ∗ Ω = − Ω (anti-conjugate-symplectic)Assuming they are non-empty, the fixed-point sets of real- and imaginary-symplectic struc-tures are real- and imaginary-symplectic forms, respectively.
Proposition 1.2.
A totally real submanifold N of ( M, Ω) is a real-symplectic form if andonly if it is a Lagrangian submanifold with respect to ( M, ω I ) . This implies N is a symplecticsubmanifold of ( M, ω R ) . Likewise, N is an imaginary-symplectic form if and only if it isLagrangian with respect to ( M, ω R ) , and this implies it is a symplectic submanifold of ( M, ω I ) .Proof. The first implication is immediate from the definitions. If T p N is a Lagrangian subspacewith respect to ω I then from (1.1) it follows that I ( T p N ) is the orthogonal complement to T p N with respect to ω R . As I ( T p N ) ∩ T p N = { } we see that T p N is a symplectic subspace of T p M with respect to ω R for all p ∈ N . The case for imaginary-symplectic forms is similar. Example 1.1 (Cotangent lift of a real structure) . Let C be a complex manifold together witha real structure r . We can lift this to a real structure R on the cotangent bundle by setting h R ( η ) , X i = h η, r ∗ X i for η ∈ T ∗ p C and for all X ∈ T r ( p ) C . This satisfies R ∗ λ = λ where λ is the canonical one-form.Therefore, we may lift real structures on C to real-symplectic structures on ( T ∗ C, Ω can ). Thefixed-point set ( T ∗ C ) R is canonically symplectomorphic to T ∗ C r . Suppose we have a dynamical system on ( M, Ω) generated by a holomorphic Hamiltonian f .Given a real form N ⊂ M we would like to be able to describe a real Hamiltonian system on N which can, in a sense, be said to be a real form of the holomorphic system ( M, Ω , f ). This raisestwo questions: what real forms N should we consider, and what should be the correspondingreal Hamiltonian?In answer to the first question we shall insist that N is a real- or imaginary-symplectic form.For the sake of brevity we shall throughout this paper mostly make reference to real-symplecticforms, however the imaginary case is entirely similar. From Proposition 1.2 the restriction of ω R to N is symplectic. If we write this restriction as b ω R then we have the real symplecticmanifold ( N, b ω R ).We now turn to the second question. In order for the dynamical system on M to descendmeaningfully to a dynamical system on N we must suppose that N is invariant under the flowgenerated by f . If we decompose the function into its real and imaginary parts as f = u + iv then4y Proposition 1.1 the flow of f is equivalently the Hamiltonian flow of u on ( M, ω R ). For p ∈ N the Hamiltonian vector field X u belongs to T p N if and only if du yields zero when evaluated onthe orthogonal complement to T p N with respect to ω R . From the proof of Proposition 1.2 thiscomplement is I ( T p N ), and therefore we require h du, I ( T p N ) i = 0. Since f is holomorphic theCauchy-Riemann equations show us that this is equivalent to h dv, T p N i = 0. Theorem 1.3.
Let ( M, Ω) be a holomorphic symplectic manifold. A real-symplectic form N ⊂ M is invariant under the Hamiltonian flow generated by a holomorphic function f = u + iv ifand only if v is locally constant on N . In this case, the flow on N is identical to the Hamiltonianflow generated by the restriction of u to ( N, b ω R ) . For the situation described in this theorem we can speak of the dynamical system ( N, b ω R , u )as being a real form of the holomorphic system ( M, Ω , f ). The principal application we havein mind is to identify different real Hamiltonian systems by recognising that they are bothreal forms of the same holomorphic system. For example, one can show that many dynamicalsystems in spherical and hyperbolic geometry are both real forms of the same complexifiedsystem, such as geodesic flows or central force problems. A holomorphic Poisson manifold is a complex manifold P equipped with a holomorphic section π of V T P with the property that the bracket { f, g } ( x ) = π x ( df, dg )between locally defined holomorphic functions satisfies the Jacobi identity and is thus a complex-valued Poisson bracket. Such a Poisson structure defines a fibrewise complex-linear map ♯ : T ∗ P → T P which sends df ∈ T ∗ x P to the vector ♯df ∈ T x P which satisfies h dg, ♯df i = π x ( df, dg ) (2.1)for all dg ∈ T ∗ x P . Exactly as with the real situation, the image of ♯ defines a complex involutive(generalised) distribution called the characteristic distribution. One may define a holomorphicsymplectic form Ω on each leaf O of the distribution byΩ( ♯df, ♯dg ) = π ( df, dg ) . (2.2)This form is non-degenerate and well defined thanks to ker ♯ = (Im ♯ ) ◦ and is a closed holomor-phic form as a consequence of the Jacobi identity. The leaves of the characteristic distributionare therefore immersed holomorphic symplectic manifolds.By decomposing a complex one-form into its real and imaginary parts df = du + idv weestablish two real isomorphisms between the spaces of complex-linear forms on T x P and real-linear forms on T x P . We can then define two real Poisson structures π R and π I on P bysetting π ( df , df ) = π R ( du , du ) + iπ I ( dv , dv ) . (2.3)Applying (2.1) gives ♯df = ♯ R du = ♯ I dv. (2.4)5onsequently, the characteristic distributions coincide, and with the aid of (2.2) we see thatthe real and imaginary parts of the holomorphic symplectic form Ω on a leaf O are preciselythe real symplectic forms ω R and ω I induced by the Poisson structures π R and π I , respectively.In light of Proposition 1.2, the generalisation of Definition 1.1 requires a review of theappropriate analogues of submanifolds in symplectic geometry to Poisson geometry [43]. • A submanifold N of ( P, π I ) is coisotropic if ♯ I ( T N ◦ ) ⊂ T N . • A submanifold N of ( P, π R ) is called pointwise Poisson-Dirac if ♯ R ( T x N ◦ ) ∩ T x N = { } for every x ∈ N . For any dU ∈ T ∗ x N this implies the existence of a unique du ∈ T ∗ x P which projects to dU and for which ♯ R ( du ) ∈ T x N . In this way we may define a bivectorΠ R on N by Π R ( dU , dU ) = π R ( du , du ) . (2.5)If Π R varies smoothly across N then N is a Poisson-Dirac submanifold of ( P, π R ). Definition 2.1.
A real form N ⊂ P is a real-Poisson form of ( P, π ) if N is a coisotropicsubmanifold of ( P, π I ). A real structure R on a holomorphic Poisson manifold ( P, π ) will becalled a real-Poisson structure if π R ( x ) ( R ∗ df , R ∗ dg ) = π x ( df, dg ) (2.6)holds at all x and for all df, dg ∈ T ∗ x P . The notation R ∗ df denotes the conjugate-adjoint, whichfor all X ∈ T R ( x ) P satisfies h R ∗ df , X i = h df, R ∗ X i . (2.7)If non-empty, the fixed-point set N of a real-Poisson structure R is a real-Poisson form. Tosee this consider the complex-linear form df = du + idv for dv ∈ T x N ◦ . From the definitions wehave R ∗ df = df , and so it follows from (2.6) that π ( df , df ) is purely real for dv , dv ∈ T x N ◦ .Therefore 0 = π I ( dv , dv ) = h dv , ♯ I dv i for all dv ∈ T x N ◦ . The vector ♯ I dv must thereforebelong to T x N , and hence, N is a coisotropic submanifold of ( P, π I ). Proposition 2.1. If N is a real-Poisson form of a holomorphic Poisson manifold ( P, π ) then N is a Poisson-Dirac submanifold of ( P, π R ) . Furthermore, if the intersection between N anda holomorphic symplectic leaf O is a submanifold of O then N ∩ O is a real-symplectic form of ( O , Ω) .Proof. If N is a real-Poisson form then the subspace T x N ∩ T x O is a coisotropic subspace of( T x O , ω I ). On the other hand, since N is a real form and O a complex submanifold we musthave ( T x N ∩ T x O ) ⊕ I ( T x N ∩ T x O ) ⊂ T x O . Therefore, the dimension of T x N ∩ T x O must be less than or equal to half the dimension of T x O . Yet since this is a coisotropic subspace it must be exactly half the dimension, and hence,a Lagrangian subspace. It follows from Proposition 1.2 that if N ∩ O is a submanifold of O then it is Lagrangian with respect to ( O , ω I ) and therefore a real-symplectic form of ( O , Ω).Since T x N ∩ T x O is Lagrangian with respect to ( T x O , ω I ) it follows from (1.1) that it issymplectic with respect to ( T x O , ω R ) and therefore, N is pointwise Poisson-Dirac with respectto ( P, ω R ). To show that Π R in (2.5) varies smoothly it suffices to show that the map T ∗ N → T ∗ P | N which sends dU ∈ T ∗ x N to du ∈ T ∗ x P is smooth [11]. As T x P = T x N ⊕ I ( T x N ) we maysmoothly extend dU to a form on T x P by setting it to equal zero on I ( T x N ). We claim that6his extension is precisely du . By complex linearity of the form df = du + idv we see that du vanishing on I ( T x N ) implies dv vanishes on T x N , and hence dv ∈ T x N ◦ . Let dw ∈ T x N ◦ bearbitrary. From (2.4) we obtain h dw, ♯ R du i = h dw, ♯ I dv i which must equal zero as ♯ I dv ∈ T x N since N is coisotropic, and so ♯ R du ∈ T x N as desired. Example 2.1 (Complex Lie algebras as holomorphic Poisson manifolds and their real forms) . The prototypical example of a holomorphic Poisson manifold is the dual of a complex Liealgebra g ∗ equipped with the Kostant-Kirilov-Souriau (KKS) Poisson bracket π η ( df, dg ) = h η, [ df, dg ] i where the one-forms df and dg on g ∗ are canonically taken to be elements in the Lie algebra g upon which the Lie bracket [ , ] is defined. Let ρ ∗ be a real form on g which is also a Liealgebra automorphism with non-empty fixed-point set g ρ . In other words, g ρ is a real formof g in the Lie algebraic sense. This involution lifts to a real structure ρ ∗ on g ∗ given by theconjugate-adjoint as in (2.7). This defines a real Poisson structure on g ∗ whose fixed-point setmay be identified with the dual of g ρ . Let (
M, ω ) be a holomorphic symplectic manifold and G a complex Lie group which acts on M by holomorphic symplectomorphisms. The quotient topology on the orbit space M/G isnot nice in general, so we shall suppose that the orbit map P : M → M/G is a holomorphicsubmersion between complex manifolds. By virtue of G acting symplectically, the Poissonbracket between G -invariant functions is again G -invariant. This allows us to define a uniquePoisson structure e π on M/G for which the projection map is a Poisson map. The space(
M/G, e π ) is the (holomorphic) Poisson reduced space [35].Consider the fixed-point set M R of a real-symplectic structure R on M . We would liketo understand how this behaves with respect to Poisson reduction. In the presence of a realstructure it is reasonable to expect some degree of compatibility between the involution R andthe action of G . If we suppose that R maps G -orbits into G -orbits then it descends to a realstructure on M/G which we shall denote by e R . As the Poisson bracket between G -invariantfunctions is again G -invariant, it follows from (2.6) and the definition of e π that e R is a real-Poisson structure on ( M/G, e π ). This compatibility condition can be ensured if the followingequivariant definition holds. Definition 2.2.
A holomorphic group action of a complex Lie group G on a complex manifold M equipped with a real structure R will be called R -compatible with respect to a real groupstructure ρ on G if R ( g · m ) = ρ ( g ) · R ( m ) (2.8)for all g ∈ G and m ∈ M .For such a compatible group action the real Lie group G ρ acts symplectically on the realform ( M R , b ω R ). If in addition, we suppose the orbit map p : M R → M R /G ρ is also a submer-sion between smooth manifolds then we can equally consider the real Poisson reduced space( M R /G ρ , b Π R ). The following proposition establishes the relation between the two possiblechoices of real reduced space, ( M R /G ρ , b Π R ) and (( M/G ) e R , e Π R ).7 heorem 2.2. There is a Poisson map Ψ from ( M R /G ρ , b Π R ) into (( M/G ) e R , e Π R ) . This mapis an immersion with discrete fibres and image P ( M R ) . Moreover, if G -acts freely on M R thenthis map is an injection, and hence, P ( M R ) is an immersed Poisson submanifold of ( M/G ) e R .Proof. The map Ψ sends the G ρ -orbit through x ∈ M R to the G -orbit through x . The commu-tativity of the square below tells us that the image of Ψ is P ( M R ). M R MM R /G ρ M/G ιp P Ψ (2.9)Since P ◦ ι = Ψ ◦ p is smooth it follows from an application of the Submersion Theoremthat Ψ is a smooth map.For x in M R consider the tangent vector ξ · x for ξ ∈ g . If ξ · x belongs to T x M R then R ∗ ( ξ · x ) = ξ · x . By taking the infinitesimal version of (2.8) we have ξ · x = ρ ∗ ( ξ ) · x . Itfollows that ξ · x is also the tangent vector generated by the element ( ξ + ρ ∗ ( ξ )) /
2. However,this element is clearly fixed by ρ ∗ , and so belongs to the Lie algebra g ρ of G ρ . It follows thatthe tangent vector ξ · x belongs to T x ( G ρ · x ), which establishes T x M R ∩ T x ( G · x ) = T x ( G ρ · x ) . (2.10)Consequently, the intersection of a G -orbit with M R is the discrete union of G ρ -orbits and sothe fibres of Ψ are discrete. For x ∈ M R suppose g · x also belongs to M R for some g ∈ G . By(2.8) we have g · x = ρ ( g ) · x , and therefore, if G acts freely on M R , then ρ ( g ) = g . Therefore( G · x ) ∩ M R = G ρ · x , which implies that Ψ is an injection.Let x ( t ) be a curve in M R with tangent vector X at x = x (0) and suppose Ψ ∗ p ∗ X is zeroin T P ( x ) ( M/G ). This implies X ∈ T x ( G · x ) ∩ T x M R which from (2.10) shows that p ∗ X = 0,and hence, Ψ is an immersion.Let U and U be real functions on ( M/G ) e R . Since this is a Poisson-Dirac submanifoldof ( M/G, e π R ) there exist extensions u and u on M/G whose Hamiltonian vector fields on(
M/G ) e R are tangent to ( M/G ) e R and for which { U , U } e Π R ( P ( x )) = { u , u } e π R ( P ( x )) = { u ◦ P, u ◦ P } π R ( x ) (2.11)for any x ∈ M R . In the last equality we have used the fact that P is a Poisson map. Theextensions u and u are e R -invariant. As P ◦ R = e R ◦ P the functions u ◦ P and u ◦ P arealso R -invariant, and therefore their Hamiltonian vector fields are tangent to M R . Since M R is a symplectic submanifold with respect to ω R the right-hand side above is equal to { u ◦ ( P ◦ ι ) , u ◦ ( P ◦ ι ) } b π R ( x )where b π R is the Poisson bivector for ( M R , b ω R ). As P ◦ ι = Ψ ◦ p we can use the fact that p isPoisson to rewrite this as { u ◦ Ψ , u ◦ Ψ } b Π R ( p ( x )) . By comparing this to the left-hand side in (2.11) and writing P ( x ) = Ψ( p ( x )) it follows fromsurjectivity of p that Ψ is Poisson. 8 emark 2.1. If the action of G is free and admits an equivariant holomorphic momentummap then the holomorphic symplectic leaves of ( M/G, e π ) are connected components of theorbit-reduced spaces for the Hamiltonian action of G on M . If in addition, the G -action is R -compatible with respect to a real form ρ , then the action of G ρ on M R is also free andHamiltonian and the symplectic leaves of ( M R /G ρ , b Π R ) are the connected components of theorbit-reduced spaces. The previous theorem tells us that Ψ( M R /G ρ ) is an immersed Poissonsubmanifold of (( M/G ) e R , e Π R ). This implies that Ψ( M R /G ρ ) is a union of symplectic leaves in( M/G ) e R , and since Ψ is Poisson, it restricts to a symplectomorphism between leaves in M R /G ρ and leaves in ( M/G ) e R . It follows that Ψ restricted to a G ρ -orbit-reduced space on M R gives asymplectomorphism between this space and a real-symplectic form of a G -orbit-reduced spaceon M . In this section we will present a detailed example which exhibits CS as a holomorphic sym-plectic reduced space for an action of GL ( C ) on T ∗ C . An advantage of this example is thatthe holomorphic Poisson reduced space is shown to be a Poisson submanifold of gl (2; C ) ∗ . Thisallows us to approach real forms of CS from two different perspectives: as fixed-point sets in g (2; C ) ∗ arising from real forms of GL (2; C ) as in Example 2.1, or as compatible real forms in T ∗ C which descend through the reduction as in Theorem 2.2.The bilinear form h p, q i = p T q identifies T ∗ C with C ⊕ C equipped with the standardholomorphic symplectic form Ω (( q , p ) , ( q , p )) = p T q − p T q . The cotangent lift of the standard representation of GL (2; C ) is g · ( q, p ) = ( gq, g − T p )and is Hamiltonian with momentum map P : C ⊕ C −→ gl (2; C ); ( q, p ) qp T . Here we have identified gl (2; C ) with its dual using the trace form.Consider the action of the subgroup which acts on C as scalar multiplication by z ∈ C \{ } .This has momentum map µ = ι ∗ ◦ P where ι ∗ is the projection gl (2; C ) → C , which is simplythe trace. If we remove the origin this action is free and P serves as an orbit map. Since themomentum map P is Poisson, the reduced Poisson structure on the quotient coincides with theKKS bracket on gl (2; C ).The symplectic leaves of gl (2; C ) are its coadjoint orbits and generically these are level sets ofthe two Casimirs, the trace and determinant. For ζ = 0 the orbit-reduced space µ − ( ζ ) /GL ( C )consists of those ξ = P ( q, p ) with det ξ = 0 and Trace ξ = µ ( q, p ) = ζ . As an affine variety thisreduced space µ − ( ζ ) /GL ( C ) = (cid:26) ξ = 12 (cid:18) t + ix y + iz − y + iz t − ix (cid:19) | det ξ = 0 , Trace ξ = ζ (cid:27) (2.12)is a complex 2-sphere given by t = ζ and x + y + z = − ζ . For ζ = i we denote this complexsphere by CS and for ζ = − i CS .The following proposition is a direct application of the material from the previous subsec-tions to this example. 9 roposition 2.3. On T ∗ C ∼ = C ⊕ C we have real-symplectic structures R , S , and T givenbelow. The action of GL ( C ) is compatible with these real structures with respect to the realforms ρ , σ , and τ , respectively. The real-symplectic structures descend to the quotient P ( C ⊕ C \ { } ) ⊂ gl (2; C ) to give real-Poisson structures e R , e S , and e T , respectively. R ( q, p ) = ( q, p ) ρ ( z ) = z e R ( ξ ) = ξS ( q, p ) = ( ip, iq ) σ ( z ) = 1 /z e S ( ξ ) = − ξ † T ( q, p ) = ( I p, I q ) τ ( z ) = 1 /z e T ( ξ ) = I ξ † I where I = diag ( i, − i ) . These involutions arise from real forms of the group GL (2; C ) , and their fixed-point sets are gl (2; R ) , u (2) , and u (1 , , respectively. The reduced space i CS is e R -invariant, and CS isboth e S - and e T -invariant. The corresponding reduced real-symplectic forms are the one-sheetedhyperboloid S × R ⊂ i CS , the sphere S ⊂ CS , and the two-sheeted hyperboloid H ⊔ H ⊂ CS . A hyperk¨ahler manifold is a Riemannian manifold (
M, g ) equipped with three complex struc-tures, I , J , and K , which satisfy the usual quaternionic relations, and for which ( M, g ) isK¨ahler with respect to each of them. Recall that a complex manifold (
M, I ) equipped witha Riemannian metric g is K¨ahler if the associated K¨ahler form ω ( X, Y ) = g ( I ( X ) , Y ) is asymplectic form on M .A hyperk¨ahler manifold ( M, g, I, J, K ) defines a holomorphic symplectic form on M withrespect to each complex structure. If we denote the K¨ahler forms by ω ( X, Y ) = g ( I ( X ) , Y ) , ω ( X, Y ) = g ( J ( X ) , Y ) , ω ( X, Y ) = g ( K ( X ) , Y )then Ω = ω + iω , Ω = ω + iω , and Ω = ω + iω each define holomorphic symplecticforms on M with respect to the complex structures, I , J , and K , respectively. Proposition 3.1.
A submanifold N of a hyperk¨ahler manifold M is a complex-Lagrangiansubmanifold of ( M, I, Ω ) if and only if it is an imaginary-symplectic form of ( M, J, Ω ) and areal-symplectic form of ( M, K, Ω ) .Proof. It suffices to consider the tangent space T p M to a point p ∈ N . If N is complex-Lagrangian with respect to ( M, I, Ω ) then it is Lagrangian with respect to ω and ω . Thisimplies that g ( J ( X ) , Y ) and g ( K ( X ) , Y ) are zero for all X, Y ∈ T p N , which means J ( T p N ) = K ( T p N ) is the orthogonal complement to T p N . Consequently, T p N is a real form with respectto both J and K , and so the first implication follows from Proposition 1.2.Conversely, if N is an imaginary-symplectic form of ( M, J, Ω ) and a real-symplectic formof ( M, K, Ω ) then it is Lagrangian with respect to both ω and ω . This implies J ( T p N )and K ( T p N ) are both equal to the orthogonal complement to T p N , and so N is a complexsubmanifold of ( M, I ) since I ( T p N ) = J K ( T p N ) = T p N . Definition 3.1. A brane of a hyperk¨ahler manifold shall mean a submanifold which is complex-Lagrangian with respect to some holomorphic symplectic form associated to the hyperk¨ahlerstructure. 10ranes appear in string theory as special submanifolds of hyperk¨ahler manifolds [26] andsome attention has been given to how these appear as fixed-point sets of certain involutions [15,8]. This is very much related to our notion of real- and imaginary-symplectic structures. Fromthe point of view of finding real- and imaginary-symplectic forms the advantage of looking athyperk¨ahler manifolds is that it gives us an alternative class of submanifolds to look for, namelycomplex-Lagrangian submanifolds. The study of such complex Lagrangian submanifolds is welldeveloped [22, 32].In the next section we shall construct an example which equips the complex 2-sphere witha hyperk¨ahler structure and show that complex-Lagrangian submanifolds of T ∗ CP correspondto both real-symplectic forms of CS and imaginary-symplectic forms of i CS . The algebra C ⊗ R H of biquaternions consists of elements of the form1 u + Iv + J w + Kz (3.1)where { , I, J, K } denotes the standard basis of H and u, v, w, z ∈ C . This may be endowedwith a hyperk¨ahler structure by setting the metric to be | u | + | v | + | w | + | z | and the complexstructures to be left multiplication by I , J , and K . The centre of this algebra is C , and thus,multiplication by the circle U (1) ⊂ C preserves the three complex structures and the metric. Lemma 3.2.
For a distinguished complex structure on the biquaternions, say for instance leftmultiplication by I , the holomorphic symplectic manifold ( C ⊗ R H , I, Ω = ω + iω ) may beidentified with C ⊕ C equipped with the standard complex and holomorphic-symplectic structureby identifying the biquaternion u + Iv + J w + Kz with the pair q = 1 √ (cid:18) u + ivw + iz (cid:19) , p = 1 √ (cid:18) w + iz − u − iv (cid:19) . In addition, this identifies the metric with the standard metric | q | + | p | and sends the circleaction to the action ( e iθ q, e − iθ p ) . For a different choice of complex structure one simply performsthe appropriate cyclic permutation of ( v, w, z ) . The circle action is isometric and Hamiltonian with respect to each of the K¨ahler forms ω , ω , and ω , with respective momentum maps µ , µ , and µ into u (1) ∗ ∼ = i R . We can thereforeconsider a hyperk¨ahler reduced space given by the quotient M = ( µ − ( i ) ∩ µ − (0) ∩ µ − (0)) /U (1) . Theorem 3.3.
Let ( M, g, I, J, K ) be the hyperk¨ahler reduced space of the circle action on thebiquaternions for the regular value ( i, , ∈ u (1) ∗ ⊗ R . We have the following holomorphicsymplectormorphisms:1. ( M, I, Ω ) ∼ = ( T ∗ CP , i, Ω can ) ,2. ( M, J, Ω ) ∼ = ( i CS , i, Ω KKS ) ,3. ( M, K, Ω ) ∼ = ( CS , i, Ω KKS ) .Here the complex 2-sphere is taken as a coadjoint orbit of gl (2; C ) ∗ where Ω KKS is the Kostant-Kirilov-Souriau form. roof. Using Lemma 3.2 we identify the biquaternions with I distinguished with C ⊕ C . Thecircle action admits a holomorphic extension to the action of GL ( C ) given in Section 2.3 withmomentum map µ = µ + iµ . The corresponding holomorphic symplectic structure on thereduced space is that inherited by the quotient µ − (0) s /GL ( C ) (3.2)where µ − (0) s is the set of all points in µ − (0) whose GL ( C )-orbit intersects µ − ( i ). In thisbasis µ ( q, p ) = i ( | q | − | p | ), and so the set of such stable points µ − (0) s are those ( q, p ) with q = 0 and p T q = 0. This quotient may be identified with T ∗ CP by associating the orbitthrough ( q, p ) with the covector p ∈ T ∗ [ q ] CP = Hom([ q ] , C / [ q ]) ∗ ∼ = ( C / [ q ]) ∗ = [ q ] ◦ . Observethat the canonical one-form on T ∗ CP pulls back to the canonical one-form on C ⊕ C (seealso Example 3.8 in [20]).For a different choice of distinguished complex structure we apply the same argument. For J we must consider µ − ( − s where µ = µ + iµ , and for K we have µ − ( i ) s where µ = µ + iµ .In both cases, all points in these fibres are stable, and so the equivalent quotient constructionin (3.2) is the holomorphic symplectic reduced space in (2.12). Armed with Theorem 3.3 and Proposition 3.1 we are able to exhibit relationships betweenbranes of T ∗ CS , i CS , and CS . Thanks to our work in Proposition 2.3 we already have afew examples of real-symplectic forms for i CS and CS . In addition to these we also have thefollowing: • By lifting the standard real structure on CP to the cotangent bundle as in Example 1.1we obtain the real-symplectic form T ∗ RP of T ∗ CP . • The zero section and fibres of T ∗ CP are complex-Lagrangian submanifolds. • Any one-dimensional complex submanifold of the complex-sphere x + y + z = − ζ isa complex-Lagrangian submanifold. If, for instance, we set x = 0, then we obtain thecomplex circle CS as a complex-Lagrangian submanifold of CS , and i CS of i CS .The real structures, e R , e S , and e T in Proposition 2.3 descend from real structures R , S , and T on C ⊕ C . In the proof of Theorem 3.3 we identified this C ⊕ C with the biquaternions usingLemma 3.2. Therefore we may interpret these involutions as involutions on the biquaternions.In addition to these we also have the real structure on T ∗ CP with fixed-point set T ∗ RP . Inthe proof of Theorem 3.3 the space T ∗ CP is identified with the quotient (3.2). This givesanother real structure U on C ⊕ C which can also be identified with an involution on thebiquaternions.We can now use the proof of Theorem 3.3 to describe how these involutions each descendto the reduced space. The results are given in Table 1. We use the notation ( I , J , K ) =( iσ , iσ , iσ ) where the σ denote the usual Pauli matrices.In this table each column represents one of the four involutions R , S , T , and U on the bi-quaternions. Each row details the corresponding involution on C ⊕ C for each complex struc-ture, I , J , and K . To distinguish between each identification of C ⊕ C with the biquaternionswe write, for instance, C I ⊕ C I to denote the correspondence given in Lemma 3.2 for the givencomplex structure I . Where applicable, we also include the corresponding involution on the12educed space inside gl (2; C ) ∗ as in (2.12) and the fixed-point set. The imaginary-symplecticforms have been omitted. Proposition 3.4.
On the hyperk¨ahler reduced space M we have the following relationshipsbetween select branes.1. The zero section CP and a pair of antipodal fibres C ⊔ C are complex-Lagrangian subman-ifolds of ( T ∗ CP , i, Ω can ) which correspond to the real-symplectic forms S and H ⊔ H of ( CS , i, Ω KKS ) .2. The complex-Lagrangian submanifold i CS of ( i CS , i, Ω KKS ) corresponds to the real-symplectic form T ∗ RP of ( T ∗ CP , i, Ω can ) .3. The complex-Lagrangian submanifold CS of ( CS , i, Ω KKS ) corresponds to the real-symplecticform S × R of ( i CS , i, Ω KKS ) . Remark 3.1.
Complex-Lagrangian submanifolds of hyperk¨ahler manifolds are very closelyrelated to special Lagrangian submanifolds of Calabi-Yau manifolds [21]. An almost Calabi-Yau manifold is a K¨ahler manifold ( M (2 n ) , I, ω ) together with a non-vanishing holomorphic n -form Θ [25]. A submanifold N is said to be special Lagrangian if it is Lagrangian withrespect to ω and if Im Θ | N = 0. Given a hyperk¨ahler manifold M consider the K¨ahler manifold( M, K, ω ) together with the holomorphic 2 n -formΘ = Ω ∧ · · · ∧ Ω | {z } n times If N is a complex-Lagrangian submanifold of ( M, I, Ω ) then it is a special Lagrangian subman-ifold of ( M, K, ω , Θ). Therefore, complex-Lagrangian submanifolds of T ∗ CP correspond tospecial Lagrangian submanifolds of CS . Such submanifolds of the complex sphere have beenconsidered before, for instance in [40, 2], and the wider study of special Lagrangian submanifoldsis itself well studied [23, 4]. The hyperk¨ahler structure we have constructed on M is known as the Eguchi-Hanson metricon T S [14]. A consequence of Theorem 3.3 is that it defines hyperk¨ahler structures on T ∗ CP , i CS , and CS . By identifying these spaces with M we implicitly obtain an identification ofspaces Φ : CS −→ T ∗ CP . (3.3)The map Φ is quite curious as it is not a biholomorphism with respect to the standard complexstructures. Instead, it pushes forward the standard structure on CS to an alternative structureon T ∗ CP arising from the hyperk¨ahler structure. Furthermore, by considering the K¨ahler form ω we see that Φ is a symplectomorphism in the sense that Φ ∗ Re(Ω can ) = Im(Ω
KKS ).More generally, it is known that the regular coadjoint orbits G c /T c of a semisimple complexLie group G c are hyperk¨ahler and that they are diffeomorphic to T ∗ ( G/T ) where G is thecompact form of G c and T a maximal torus [28, 6]. The explicit map Φ between G c /T c and T ∗ ( G/T ) admits a general construction in special circumstances [7], however it is in generalsurprisingly difficult to obtain explicitly [12]. For our purposes the map Φ will be used later in13
S T U C I ⊕ C I ( − i K q, i K p ) ( q, − p ) ( − i K q, i K p ) ( q, p ) T ∗ CP · CP C ⊔ C , T ∗ RP C J ⊕ C J ( q, p ) ( − p, − q ) ( − i J p, i J q ) ( − i K p, − i K q ) ξ ξ † − J ξ † J − K ξ T K i CS S × R · · i CS C K ⊕ C K ( i K p, i K q ) ( ip, iq ) ( I p, I q ) ( − i I q, i I p ) − K ξ T K − ξ † I ξ † I I ξ ICS CS S H ⊔ H · Table 1: Brane involutions on the biquaternions.an important example for the spherical pendulum. For this reason we must here deduce someproperties of Φ.Firstly, we claim that Φ is SU (2)-equivariant. To see this consider right multiplication byordinary unit quaternions on the biquaternions. Applying Lemma 3.2 reveals that this acts on C ⊕ C as g · ( q, p ) = ( gq, gp ) for g ∈ SU (2). This descends through the hyperk¨ahler reductionto give the cotangent lift of SU (2) to T ∗ CP and the adjoint action of SU (2) ⊂ GL (2; C ) on CS . For each of these spaces there is a single SU (2)-invariant generator. For T ∗ CP this isthe modulus | η | of a covector, and for CS it is Trace( ξ † ξ ). Lemma 3.5.
With respect to the identification Φ the SU (2) -invariant generators on T ∗ CP and CS satisfy | η | = | x | + | y | + | z | . (3.4) Proof.
As in the proof of Theorem 3.3 we identify T ∗ CP with the U (1)-orbits through ( q, p )in C I ⊕ C I where p T q = 0 and | q | − | p | = 1. One can show that | η | is equal to 4 | q | | p | whichmay be rewritten as Γ − | q | + | p | . From Lemma 3.2 we see thatΓ = | u | + | v | + | w | + | z | on the biquaternions. This is also equal to | q | + | p | but for ( q, p ) ∈ C K ⊕ C K where now p T q = i and | q | = | p | . For ξ = qp T the generator Trace( ξ † ξ ) is equal to | q | | p | , which in thiscase may be rewritten as Γ /
4. Using the notation in (2.12) allows us to express Γ − T ∗ CP corresponds to a connected component H of the fixed-point set of e T in CS . Therefore, for some SU (2)-equivariant identification of CP with S , the bundle map π ◦ Φ : CS → S is determined, up to some action of SU (2),by the property that it be SU (2)-equivariant and whose fibre over a given point is H . If we14emporarily denote elements ( x, y, z ) of CS by x then one can see that such a map is given byx p | Im(x) | Re(x) . (3.5) Definition 4.1.
A collection of holomorphic functions f , . . . , f n on a holomorphic symplecticmanifold M (2 n ) defines a holomorphic integrable system if the functions Poisson commute withrespect to the complex Poisson bracket and if the derivatives are linearly independent in anopen dense subset.The regular fibres of a holomorphic integrable system are complex-Lagrangian submani-folds. These fibres are Lagrangian with respect to both ω R and ω I , and therefore the real andimaginary parts of the f k define real integrable systems with respect to both ω R and ω I . Thisis an example of a bi-integrable system and has been observed numerous times before [5, 13].In the ordinary setting of real integrable systems a great deal of effort can be spent checkinga collection of integrals are linearly independent almost everywhere. In this respect workingwithin the rigid category of analytic functions has an advantage: if any two such functions agreein an open set then they must be identical thanks to the Identity Theorem. More generally,any subset which enjoys this property is sometimes referred to as a key set [3, 27]. If we havean analytic real form N of M , then every open subset of N is also a key set. This can beseen by working in an analytic chart C n in which N appears as R n and considering the seriesexpansion [33]. These ideas hold more generally for holomorphic forms, and it is by applyingthese arguments to df ∧ · · · ∧ df k that we prove the following Lemma 4.1.
Let f , . . . , f k be holomorphic functions defined on a connected, complex manifold M . If the df j are linearly independent somewhere then they are independent everywhere in anopen dense subset of M . If, in addition, N is an analytic real form of M then the df j are alsolinearly independent in an open dense subset of N . Proposition 4.2.
Suppose u , . . . , u n is an integrable system of analytic functions on an ana-lytic real-symplectic form ( N, b ω R ) . The holomorphic extensions of these functions defined in aneighbourhood of N is a holomorphic integrable system.Proof. Let f , . . . , f n be the holomorphic extensions defined in some neighbourhood of N . Sincethe f j are purely real on N their Hamiltonian vector fields are tangent to N , and so since N isLagrangian with respect to ω I we have { f j , f k } ( p ) = Ω( X f j , X f k ) = ω R ( X u j , X u k ) = { u j , u k } R ( p ) = 0for p ∈ N . As N is a key set it follows that { f j , f k } must be zero everywhere in the neighbour-hood of N . Finally, if the du j are independent at p ∈ N then so are the df j .We now consider a partial converse to this proposition. Suppose f , . . . , f n is a holomorphicintegrable system on M . By restricting the real and imaginary parts of these functions to aconnected and analytic real-symplectic form N we obtain a collection of 2 n real functions. Let15 denote the algebra of functions on N generated by taking the Poisson bracket with respectto b ω R between these functions. Fix some p ∈ N for which the dimension ofSpan { dw p | w ∈ A } is maximal. We can then select k functions w , . . . , w k belonging to A whose derivatives forma basis of this space at p . These functions are analytic, and so, since their derivatives arelinearly independent at some point they must be independent on an open dense set U of N .From the maximality assumption the fibres of w , . . . , w k must coincide with the fibres of A in U . The fibres of A are subsets of the level sets of f , . . . , f n restricted to N . However, as theregular fibres of the holomorphic integrable system are Lagrangian with respect to ω R , and as N is symplectic with respect to ω R , any submanifold contained to these intersections must beisotropic with respect to ( N, b ω R ). The algebra A is therefore a complete algebra on ( N, b ω R ) inthe sense defined in [10].This construction might appear underwhelming. Indeed, as the regular fibres of f , . . . , f n and N are both real 2 n -dimensional submanifolds in M , the generic transversal intersectionbetween them is a point. A foliation of N into points qualifies as a non-commutative integrablesystem, albeit not an interesting one. On the other hand, it is important to appreciate thesignificance of this to dynamics. Suppose the flow of a holomorphic Hamiltonian on M admitsa holomorphic integrable system f , . . . , f n . If the flow leaves N invariant then A is an algebraof first integrals. A result of [10] shows that this implies integrability in the standard sense. Theorem 4.3.
Suppose the flow of a holomorphic Hamiltonian on ( M, Ω) admits a holomorphicintegrable system. If the flow leaves an analytic real-symplectic form N invariant then thecorresponding real Hamiltonian system on ( N, b ω R ) is integrable. Remark 4.1.
If we have two analytic real-symplectic forms N and N of M then we canuse these ideas to obtain something reminiscent of the unitary trick from Lie theory but fordynamical systems. Suppose N admits a real analytic integrable system, which by Proposi-tion 4.2 complexifies to a holomorphic integrable system on a neighbourhood of N which weshall suppose contains N . If the flow of the complexified Hamiltonian preserves N then byTheorem 4.3 this provides a separate real integrable system on N . This technique can be usedto obtain new integrable systems from old. For instance, an application of this idea can be usedto show that the real forms of the complex Neumann system considered in [34] are integrable. As we remarked earlier, the level sets of an arbitrary holomorphic integrable system f , . . . , f n and a real-symplectic form will typically intersect transversally in a point. In practice it isreasonable to expect some compatibility between f = ( f , . . . , f n ) → C n and a real-symplecticstructure R . Let us temporarily refer to the integrable system f as being R -compatible if R acts on the fibres of f . Equivalently, f ◦ R factors through f .Suppose the fixed-point set M R is non-empty. The fixed-point set of a real structure isanalytic [38], and so by Lemma 4.1 we may find a regular point x of f belonging to M R . From R -compatibility, the derivatives d ( f k ◦ R ) = R ∗ df k at x belong to the complex subspace spanned by the df k in T ∗ x M . Therefore, the involution df k R ∗ df k defines a linear real structure on this subspace. We may suppose without any loss16f generality that the integrals f , . . . , f n are such that their derivatives at x are fixed by thisinvolution. Now consider the alternative integrals g , . . . , g n given by g k = f k + f k ◦ R. (4.1)The integrable system g = ( g , . . . , g n ) → C n is regular at x , and so by Lemma 4.1, is regularin a dense subset of M . The integrable systems f and g are therefore functionally equivalent.By noting that g ◦ R = g we are led to the following amended definition. Definition 4.2.
Let g be a complex Lie algebra. A map µ : M → g ∗ on a holomorphicsymplectic manifold ( M, Ω) equipped with a real-symplectic structure R is called R -compatible if there exists a linear real-Poisson structure ρ ∗ on g ∗ which satisfies µ ◦ R = ρ ∗ ◦ µ. (4.2)The integrals in (4.1) are each purely real when restricted to M R . They therefore definea real analytic integrable system on M R . This generalises the result of [18] to holomorphicsymplectic manifolds.In keeping with the previous definition we shall present this result in terms of a com-plex abelian Lie algebra g . It is an exercise to show that the real-Poisson structure ρ ∗ is theconjugate-adjoint of a real structure ρ ∗ in the Lie algebraic sense on g . The image µ ( M R ) iscontained to Fix ρ ∗ which consists of all complex-linear forms on g which are purely real onFix ρ ∗ = g ρ . As in Example 2.1 this may canonically be identified with ( g ρ ) ∗ . Theorem 4.4.
Let µ : M → g ∗ be a holomorphic integrable system on a holomorphic symplecticmanifold ( M, Ω) equipped with a real-symplectic structure R with non-empty fixed-point set M R .If the map µ is R -compatible then the restriction b µ = µ | M R : M R → Fix ρ ∗ ∼ = ( g ρ ) ∗ (4.3) defines a real analytic integrable system on ( M R , b ω R ) . More generally, if the Hamiltonian vector fields generated by a map µ : M → g ∗ are completethen we can consider µ to be the momentum map for the action of a connected, complex group G . The real structure ρ ∗ on g can also be exponentiated to give a real group structure ρ on G .In this situation, the R -compatibility of µ implies the action of G is R -compatible with respectto ρ in the sense of Definition 2.2. To see this note that (4.2) implies R ∗ ( X ξ ) = X ρ ∗ ξ (4.4)where X ξ denotes the Hamiltonian vector field generated by ξ ∈ g . This is the infinitesimalversion of the equivariant condition in (2.8). We can therefore integrate this expression to findthat the action of G on M is R -compatible with respect to ρ . From Table 1 the involution e U ( ξ ) = I ξ I defines an imaginary-symplectic structure on CS .We can twist this involution on the product CS × CS to get another imaginary-symplecticstructure Υ : ( ξ , ξ ) (cid:16) e U ( ξ ) , e U ( ξ ) (cid:17) (4.5)17hose fixed-point set is a conjugate-diagonal copy of CS given by n ( ξ, e U ( ξ )) | ∀ ξ ∈ CS o ⊂ CS × CS . (4.6)The imaginary part of the holomorphic symplectic form restricts to this to give Im(Ω KKS ) on CS . We can then use the symplectomorphism Φ : ( CS , Im(Ω
KKS )) → ( T ∗ CP , Re(Ω can )) in(3.3) to identify this imaginary-symplectic form of CS × CS with ( T ∗ S , ω can ).On the other hand, the real-symplectic structureΣ : ( ξ , ξ ) (cid:16) e S ( ξ ) , e S ( ξ ) (cid:17) (4.7)has fixed-point set S × S . In principle we can perform a ‘unitary trick’ by complexifying anintegrable system on T ∗ S to obtain a holomorphic integrable system on CS × CS , and then,provided this is Σ-compatible, use Theorem 4.4 to derive a real integrable system on S × S .This can be thought of as a ‘compact real form’ of the integrable system on T ∗ S . Theorem 4.5.
The holomorphic integrable system ( H, J ) on CS × CS given by H = 12 ( x x + y y − z z ) + z − z p ( x + x ) + ( y + y ) + ( z − z ) J = z + z i is both Σ - and Υ -compatible. The corresponding real integrable system on the fixed-point set of Υ is sent via Φ to the spherical pendulum on T ∗ S where H is the energy and J the angularmomentum about the vertical.Proof. The kinetic energy term on CS is obtained from Lemma 3.5. For the potential energywe require the bundle map π ◦ Φ : CS → S given in (3.5). If we suppose the vertical isabout (0 , ,
1) then the third component in the right-hand side of (3.5) gives the height of thependulum. Finally, the angular momentum J on T ∗ S generates rotations about the vertical.Since Φ : CS → T ∗ S is SU (2)-equivariant this corresponds to the U (1) ⊂ SU (2) action on CS which fixes (0 , , z .We thus have H = 12 ( | x | + | y | + | z | ) + z + z p − ( x − x ) − ( y − y ) − ( z − z ) J = z − z i for the pull-back of the integrable system on T ∗ S to CS . To extend H and J to holomorphicfunctions on CS × CS we use the fact that e U ( x, y, z ) = ( x, y, − z ) and our identification of CS with the fixed-point set of Υ given in (4.6).For reference we provide in Figure 1 the energy-momentum bifurcation diagram for thecorresponding real integrable systems on T ∗ S and S × S . The black points correspond torank zero critical points of ( H, J ). The nature of the critical points of the momentum map issignificant to the study of classifications of integrable systems [9, 44, 45]. Each non-degeneratecritical point of rank zero corresponds to a Cartan subalgebra of the symplectic Lie algebra.In this respect the study of holomorphic integrable systems affords an advantage over its real18 HT ∗ S JHS × S Figure 1: Bifurcation diagram for the spherical pendulum and its ‘compact real form’.counterpart: unlike the real symplectic Lie algebra, every Cartan subalgebra of the complexsymplectic Lie algebra is equivalent up to conjugacy. This implies that for a holomorphicintegrable system, all non-degenerate rank zero critical points are locally the same.It is intriguing to contemplate how this might contribute to the task of classification, similarin spirit to that in [ ? ]. In particular, it would be interesting to complexify an integrable systemand use Theorem 4.4 to obtain a family of separate integrable systems and see how theyare related. Many questions now arise. How are the bifurcation diagrams related, and howwill the nature of the critical points vary on each real form? Furthermore, does every realanalytic integrable system admit a compact real integrable form as is the case for the sphericalpendulum? Acknowledgements
Philip Arathoon is funded by an EPSRC Doctoral Prize Award hosted by the University ofManchester and Marine Fontaine is supported by the FWO-EoS Project G0H4518N.We would like to extend special thanks to James Montaldi for carefully reading a draft ofthis paper and for suggesting many helpful comments throughout its preparation.
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P. Arathoon,
University of Manchester [email protected]
M. Fontaine,
University of Antwerp [email protected]@uantwerpen.be