Dual Pairs and Regularization of Kummer Shapes in Resonances

We present an account of dual pairs and the Kummer shapes for $n:m$ resonances that provides an alternative to Holm and Vizman's work. The advantages of our point of view are that the associated Poisson structure on $\mathfrak{su}(2)^{*}$ is the standard $(+)$-Lie--Poisson bracket independent of the values of $(n,m)$ as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of $(n,m)$. A similar result holds for $n:-m$ resonance with a paraboloid and $\mathfrak{su}(1,1)^{*}$. The result also has a straightforward generalization to multidimensional resonances as well.


Kummer Shapes and Dual Pairs in Resonances.
Hamiltonian systems with resonant symmetry have been studied quite extensively from many different perspectives. Resonant symmetry crops up in many different forms of S 1 symmetries. Although it is one of the simplest symmetries geometrically, it is not only rich in examples and applications but also possesses interesting mathematical structures; see, e.g., Holm [7,, Dullin et al. [3], Haller [6,Chapter 4], and references therein.
From the geometric point of view, Churchill et al. [2], Kummer [11,12,13] made a seminal contribution by introducing what is now often referred to as the Kummer shapes. Recently Holm and Vizman [8] discovered a Poisson-geometric structure behind the Kummer shapes by finding a dual pair of Poisson maps (see, e.g., Weinstein [16] and Ortega and Ratiu [15,Chapter 11]) in n : m resonances.
1.2. Main Results and Outline. We build on the work of Holm [7,Chapter 4] and Holm and Vizman [8] to provide an alternative view of the dual pair constructed in [8] as well as of the Kummer shapes in n : m, n : −m, and multidimensional resonances.
Our approach is to relate n : m resonance with any (n, m) ∈ N 2 with the 1 : 1 resonance case; this relationship along with the dual pair from [8] (see also Golubitsky et al. [5]) for 1 : 1 resonance naturally gives rise to the dual pair for n : m resonance; see Theorem 2.1. Our dual pair for n : m resonances is slightly different from that of [8]. Specifically, the Poisson structure on su(2) * in our dual pair is the standard (+)-Lie-Poisson structure regardless of the values of (n, m) ∈ N 2 . This is in contrast to the Poisson structure in [8] that depends on the values of (n, m) ∈ N 2 . An advantage of this result is that the reduced dynamics in su(2) * becomes a standard Lie-Poisson dynamics.
A byproduct of this construction is that the Kummer shapes-which usually arise as various shapes such as beet, lemon, onion, turnip, etc. depending on the values of n and m [7, Section 4.4.2]-are all "regularized" to become a sphere. Section 2.6 shows that a similar approach works between n : −m resonance and 1 : −1 resonance. In this case, again all the Kummer shapes are regularized to become a paraboloid.
We also show, in Section 3, that the argument for n : m resonances easily generalizes to multidimensional resonances.

Kummer Shapes and Dual Pairs in n : m Resonances
We first briefly review Hamiltonian dynamics with n : m resonant symmetry following Holm [7,Chapter 4] and Holm and Vizman [8]. We then find a Poisson map that provides a bridge between n : m resonances and the 1 : 1 resonance using a change of variables introduced in [7, Section A. 5.4]. This Poisson map naturally gives rise to a dual pair of Poisson maps for n : m resonances with the standard (+)-Lie-Poisson bracket on su(2) * by relating it to the dual pair for 1 : 1 resonance from Golubitsky et al. [5] and Holm and Vizman [8]. This gives an alternative account of the dual pairs in n : m resonances that is slightly different from those in Holm and Vizman [8]. In fact, the Kummer shapes [2,[11][12][13] turn out to be spheres regardless of the values of n and m. We work out an example to illustrate this result, as well as extend the result to n : −m resonances.
The associated Poisson bracket is Let n, m ∈ N be a pair of natural numbers and consider the following S 1 -action on C 2 × : (e iθ , (a 1 , a 2 )) → (e inθ a 1 , e imθ a 2 ) =: Ψ n:m θ (a).
The corresponding infinitesimal generator is defined for any ω ∈ T 1 S 1 ∼ = R as follows: where "c.c." stands for the complex conjugate of the preceding terms. This is essentially equivalent to the dynamics of two harmonic oscillators with frequencies n and m: One also sees that this is the Hamiltonian vector field corresponding to the function (n|a 1 | 2 + m|a 2 | 2 )/2.
This change or coordinates is briefly mentioned in Holm [7,Section A.5.4], and is a slight modification of the change of variables introduced in [7, Section 4.4], where √ m and √ n are m and n respectively instead. Note that the map is not one-to-one and hence is not invertible in general.
Let b = (b 1 , b 2 ) be the coordinates for the second copy of C 2 × , and equip C 2 × = {(b 1 , b 2 )} with the same symplectic structure Ω C 2 × defined in (1) above, and hence with the same Poisson bracket as the above, i.e., Then it is straightforward calculations (see the proof of Proposition 3.1 below) to see that f n:m is a Poisson map, i.e., One also sees that f n:m is a local symplectomorphism with respect to Ω C 2 × as well, i.e., for any a ∈ C 2 × , there exists an open neighborhood U of a in C 2 × such that f n:m | U : U → f n:m (U ) is symplectic. In fact, f n:m is a local diffeomorphism because those distinct points a 1 ,ã 1 ∈ C × such that a m 1 /( are on the same circle (i.e., |a 1 | = |ã 1 |) but are separated by angles 2kπ/m with k = 1, . . . , m − 1; the same goes with the second portion of f n:m . The (local) symplecticity follows from similar coordinate calculations as above; again see the proof of Proposition 3.1 below for more details.
Let us also define R n:m : C 2 × → R by R n:m (a) := 1 2 Clearly it satisfies R n:m = R 1:1 •f n:m , and nmR n:m is the Hamiltonian function whose corresponding vector field gives (3), i.e., R n:m is essentially the momentum map corresponding to the action (2). Now consider the following natural action of the special unitary group SU(2) on C 2 × : It is then clear that R 1:1 is invariant under the action, i.e., R 1: The momentum map J 1:1 : C 2 × → su(2) * corresponding to the above action is then given by See Lemma 3.2 below for a generalization of this result and a proof. Note that we also identified su(2) ∼ = su(2) * with R 3 as follows: where we identified su(2) * with su(2) via the inner product on su(2) and hence su(2) * is identified with R 3 using the identification su(2) ∼ = R 3 above. Hence here. Note that the above Poisson bracket satisfy for any even permutation (i, j, k) of (1, 2, 3). Since the SU(2)-action Φ defined in (6) is a left action and the momentum map J 1:1 : C 2 × → su(2) * is equivariant, J 1:1 is a Poisson map (see, e.g., Marsden and Ratiu [14,Theorem 12.4.1]) with respect to the Poisson bracket (5) and (8), i.e., for any F, G ∈ C ∞ (su(2) * ), In fact, Holm and Vizman [8] (see also [5]) showed that R 1:1 and J 1:1 form a dual pair of Poisson maps: that is, (ker T a R 1:1 ) Ω = ker T a J 1:1 for any a ∈ C 2 × .
2.4. n : m Resonance Invariants. Let us combine the map f n:m from (4) and the momentum map J 1:1 from (7) to define In coordinates, we have These are essentially the "invariants" (of (3) but not necessarily invariants of a general Hamiltonian system in n : m resonance) from [7, Proposition 4.4.1 on p. 266] although the expressions are slightly different.
Note that J n:m is also slightly different from the corresponding map Π in Holm and Vizman [8] as well. This difference leads to an alternative construction of a dual map as well as different Kummer shapes as we shall see in the next subsection.

Dual Pairs and Kummer Shapes.
We are now ready to describe our account of dual pairs and Kummer shapes in n : m resonances. Specifically, our result identifies a relationship between the dual pair (9) of the 1 : 1 resonance and n : m resonances as well as the momentum map origin of the dual pairs of Poisson maps for n : m resonances.
Theorem 2.1. The Poisson maps R n:m : C 2 × → R and J n:m : C 2 × → su(2) * + are a dual pair for any pair of natural numbers (n, m) ∈ N 2 , i.e., for any a ∈ C 2 × , ker T a R n:m and ker T a J n:m are symplectic orthogonal complements to each other. Moreover, the dual pair of Poisson maps for n : m resonances is related to the dual pair of momentum maps R 1:1 and J 1:1 as is shown in the diagram below.
Proof. We know from Holm and Vizman [8, Theorem 3.1] that the bottom part constitutes a dual pair: For any b ∈ C 2 × , ker T b R 1:1 and ker T b J 1:1 are symplectic orthogonal complements to each other with respect to Ω C 2 × , i.e., (ker T b R 1:1 ) Ω = ker T b J 1:1 . However, since R n:m = R 1:1 • f n:m , we see that, for any a ∈ C 2 × , T a R n:m = T fn:m(a) R 1:1 • T a f n:m .
Now recall that f n:m is a local diffeomorphism; so we have ker T a R = (T a f n:m ) −1 (ker T fn:m(a) R 1:1 ). Similarly, ker T a J n:m = (T a f n:m ) −1 (ker T fn:m(a) J 1:1 ) because J n:m = J 1:1 • f n:m . Since f n:m is a local symplectomorphism with respect to Ω C 2 × , we conclude that (ker T a R n:m ) Ω = ker T a J n:m for any a ∈ C 2 × .
Basic results on dual pairs (see Weinstein [16] and Ortega and Ratiu [15,Chapter 11]) imply that the image J n:m (R −1 n:m (r)) of the level set R −1 n:m (r) of R n:m at any r > 0 under the map J n:m is a symplectic leaf in the image of J n:m in su(2) * . This is what Holm [7,Section 4.4] refers to as an orbit manifold or Kummer shape.
What does the Kummer shape look like in this setting? It is well known that SU(2) is a double cover of SO(3) and the coadjoint action of SU(2) in su(2) * ∼ = R 3 is written as rotations in R 3 by corresponding elements in SO(3), and hence the coadjoint orbit in su(2) * ∼ = R 3 are spheres; these are the symplectic leaves in su(2) * or the Kummer shape here. In fact, setting µ = J n:m (a), we see that µ 2 1 + µ 2 2 + µ 2 3 = R n:m (a) 2 . Therefore, for any pair (n, m) ∈ N 2 , the Kummer shape J n:m (R −1 n:m (r)) is a sphere without the north and south poles (which correspond to those cases with a 2 = 0 and a 1 = 0 respectively that were removed from the outset). To summarize: Corollary 2.2 (Regularization of Kummer shape). The Kummer shape formed in su(2) * using the dual pair from Theorem 3.3 is the sphere with radius R n:m (a) centered at the origin with the north and south poles removed for any (n, m) ∈ N 2 . Remark 2.3. This result is seemingly contradictory to those from [7, Section 4.4.2] and [8] that the Kummer shapes take all kinds of different pinched spheres such as beet, lemon, onion, turnip, etc. depending on the values of n and m. The reason for this apparent contradiction is that our definition of the Poisson map J n:m is slightly different from theirs, and the map regularizes or un-pinches these various Kummer shapes in their setting to spheres.
As stated above, an advantage of our setting is that the Poisson structure in su(2) * is simple and standard-the (+)-Lie-Poisson structure on su(2) * -as well as independent of n and m, whereas the Poisson structure from [7,8]  Example 2.4 (1:2 resonance). We consider the dynamics in C 2 × with respect to the symplectic structure (1) and the Hamiltonian H(a) = Re(a 2 1ā 2 ). The Hamiltonian system i X H Ω C 2 × = dH yieldṡ a 1 = 2iā 1 a 2 ,ȧ 2 = i a 2 1 .
Clearly the Hamiltonian H has the 1 : 2 resonant symmetry, i.e., H • Ψ 1:2 θ = H for any e iθ ∈ S 1 (see (2) for the definition of the action Ψ), and thus R 1:2 (a) = 1 2 is conserved along the dynamics. On the other hand, the map J 1:2 : C 2 × → su(2) * takes the form Let us define the Hamiltonian h : su(2) * → R by h • J 1:2 = H. This yields where µ = µ 2 1 + µ 2 2 + µ 2 3 . Then the Kummer shape is defined by µ = r for the constant r := R 1:2 (a 0 ) defined by the initial condition a 0 ∈ C 2 × for the above dynamics. Now, Theorem 2.1 implies that setting µ = (µ 1 , µ 2 , µ 3 ) = J 1:2 (a) ∈ su(2) * reduces the dynamics to a Lie-Poisson dynamics in su(2) * -more specifically on the coadjoint orbit or the Kummer shape µ = c-with respect to the Lie-Poisson bracket (8) and the above Hamiltonian h. In fact, the Lie-Poisson equation (10) yieldṡ on the Kummer shape µ = r. The orbit of the above Lie-Poisson dynamics is given by the intersection of the sphere µ = r and the level set of the Hamiltonian h; see Fig. 1. On the other hand, the standard Kummer shape in the 1:2 resonance would be a "turnip" [7,Section 4.4.2], i.e., one of the poles of the sphere is pinched, and the Poisson bracket in the reduced space su(2) * is not the standard Lie-Poisson bracket; see Holm and Vizman [8].   (a 1 , a 2 )) → (e inθ a 1 , e −imθ a 2 ).
on C 2 × equipped with (1). However, equivalently, one may redefineā 2 as a 2 and instead consider the action Ψ n:m given in (2) on C 2 × equipped with the symplectic form It is a straightforward computation as in n : m resonances to check that f n:m is a local symplectomorphism with respect to Ω 1:−1 as well as that f n:m is Poisson with respect to the corresponding Poisson bracket: Defining We also define R n:−m : C 2 × → R as R n:−m (a) := 1 2 and consider the natural action of SU(1, 1) on (C 2 × , Ω 1:−1 ). Then the corresponding momentum map J 1:−1 : C 2 × → su(1, 1) * is given by It is clearly equivariant and thus J 1:−1 is a Poisson map with respect to Ω 1:−1 su(1, 1) * The Kummer shape in this case is a paraboloid for any (n, m) ∈ N 2 . In fact, setting µ = J n:−m (a), we have µ 2 3 − µ 2 1 − µ 2 2 = R n:−m (a) 2 .
3. Generalization to Multi-dimensional Resonance 3.1. Setup. Let a = (a 1 , . . . , a d ) be coordinates for C d × , and generalize the symplectic form (1) to C d × as follows: where Im(ā j da j ).
The associated Poisson bracket is We can also generalize the map f n:m introduced in (4) earlier as follows: Proposition 3.1. Given a multi-index of natural numbers n := (n 1 , . . . , n d ) ∈ N d , let us define {ν j } j∈{1,...,d} ⊂ N by and consider the map Then f n is a Poisson map as well as a local symplectomorphism.
Proof. let b = (b 1 , . . . , b d ) be the coordinates for the second copy of C d × .
Then the map f n is written as b = f n (a), and one sees that, for any j ∈ {1, . . . , d}, where the summation on j is not assumed. This implies that, for any F, as well asb j db j − b j db j =ā j da j − a j dā j ⇐⇒ Im(b j db j ) = Im(ā j da j ). The former equality implies and hence f n is Poisson, whereas the latter implies that f n -which is a local diffeomorphism although it is not globally one-to-one-locally leaves Θ C d × invariant and hence Ω C d × as well.

Momentum Maps. Let us consider the
(e iθ , a) → (e in 1 θ a 1 , . . . , e in d θ a d ) =: Ψ n θ (a). (14) It is clear that Ψ n (·) leaves the canonical one-form for any e iθ ∈ S 1 , and hence is symplectic with respect to Ω C d × . The corresponding momentum map is where N := d j=1 n j and we defined R n : Clearly we have R n = R 1 • f n . Let us also consider a natural SU(d)-action on C d × , i.e., and find an expression for the corresponding momentum map for the special case n = 1 := (1, . . . , 1) ∈ N d : Lemma 3.2. The momentum map J 1 : C d × → su(d) * corresponding to the above SU(d)-action (16) is given by It is a Poisson map with respect to Ω C d × and the (+)-Lie-Poisson bracket on su(d) * .
Proof. Let us first find the momentum mapJ : C d × → u(d) * corresponding to the U(d)-action defined the same manner as (16). Let ξ ∈ u(d) be arbitrary. Then the corresponding infinitesimal generator is given by ξ C d × (b) = ξb. Since this action clearly leaves Θ C d × invariant, the momentum mapJ is defined by where we define an inner product on u(d) as follows: We may then identify u(d) * with u(d) and su(d) * with su(d) via the above inner product. Now, where we used the fact that ξ * = −ξ and hence b * ξb is a pure imaginary number. So we havẽ J(b) = i bb * . Now note that the action Φ in (16) is the induced subgroup action of of the above U(d)-action. Let ι : su(d) → u(d) be the inclusion and ι * : u(d) * → su(d) * be its dual. Then the momentum map J 1 is given by J 1 = ι * •J; see, e.g., Marsden and Ratiu [14,Exercise 11.4.2].
By definition, the dual map ι * : u(d) * → su(d) * satisfies ι * (µ), ξ = µ, ι(ξ) = µ| su(d) , ξ , and hence ι * (µ) = µ| su(d) . It is easy to see that the orthogonal complement of su(d) in u(d) in terms of the above inner product is given by Therefore, using the identification u(d) * ∼ = u(d) and su(d) * ∼ = su(d), the dual map ι * is given by the orthogonal projection onto su(d): Therefore, we obtain 3.3. Dual Pairs. Now we are ready to generalize Theorem 2.1 to the above multi-dimensional setting. Let su(d) * + denote su(d) * equipped with the (+)-Lie-Poisson bracket on su(d) * , and define J n : C d × → su(d) * + as J n := J 1 • f n . Then we have the following generalization: Theorem 3.3. The Poisson maps R n : C d × → R and J n : C d × → J 1 (C d × ) ⊂ su(d) * + are a dual pair for any multi-index n ∈ N d of d natural numbers, i.e., for any a ∈ C d × , ker T a R n and ker T a J n are symplectic orthogonal complements to each other. Moreover, the dual pair of Poisson maps for the n resonances is related to the dual pair of momentum maps R 1 and J 1 for the 1-resonance as is shown in the diagram below.
Proof. First consider the special case with n = 1. We note in passing that this case is also treated in Cariñena et al. [ Therefore, both R 1 and J 1 are Poisson maps; particularly the latter is Poisson with respect to the canonical Poisson bracket (13) on C d × and the (+)-Lie-Poisson bracket on su(d) * . One also sees that SU(d) acts on the level sets of R 1 transitively via the above action Φ as follows: The level set R −1 1 (r) of R 1 with any r > 0 is a (2d − 1)-dimensional sphere in C d × (those points corresponding to the removed origins of the copies of C × are removed) centered at the (removed) origin, and thus SU(d) acts on each level set transitively. It is also clear that every point in C d × is a regular point of R 1 and J 1 ; notice that the codomain of J 1 is restricted to the image J 1 (C d × ) in su(d) * + . Therefore, by Theorem 2.1 of [8], R 1 and J 1 constitute a dual pair. The extension to an arbitrary n ∈ N d is a simple generalization of the proof of Theorem 2.1 using Proposition 3.1 as the above diagram shows: Note that we have R n = R 1 • f n and J n = J 1 • f n here.
The Hamiltonian H has 1 : 1 : 2 resonant symmetry, i.e., H • Ψ n θ = H with n = (1, 1, 2) for any e iθ ∈ S 1 (see (14) for the definition of the action Ψ), and thus We also identify su(3) * with su(3) as well just as described in the proof of Lemma 3.2. The map f n : C 3 × → C 3 × is defined as