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Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin

Abstract

We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable ‘free’ system carried by a spin extension of the Heisenberg double of the \({\mathrm{U}}(n)\) Poisson–Lie group. The Poisson structure on the unreduced real phase space \({\mathrm{GL}}(n,{\mathbb {C}})\times {\mathbb {C}}^{nd}\) is the direct product of that of the Heisenberg double and \(d\ge 2\) copies of a \({\mathrm{U}}(n)\) covariant Poisson structure on \({\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}\) found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars–Schneider system and we prove its degenerate integrability.

Introduction

The unbroken interest in integrable many-body systems of Calogero–Moser–Sutherland [8], [38], [54] and Ruijsenaars–Schneider (abbreviated RS) [50] types is due to their ubiquity in physical applications and rich web of connections to important areas of mathematics [3], [12], [40], [49], [55]. The same can be said about spin extensions of these models, which currently attract attention [5], [6], [10], [14], [15], [16], [28], [33], [42], [45], [46], [47], [51], [57].

Two kinds of spin many-body models are studied in the literature. Those that feature only ‘collective spin variables’ belonging to some group theoretic phase space such as a coadjoint orbit, and those that have ‘spin-vectors’ embodying internal degrees of freedom of the interacting particles. The former type of models arise rather naturally in harmonic analysis and its classical mechanical counterpart [12], [13], [21], [22], [34], [47]. The latter type of models, built on ‘individual spins,’ were introduced at the non-relativistic level by Gibbons and Hermsen [24], and their ‘relativistic’ generalization was later put forward by Krichever and Zabrodin [32].

In fact, in 1995 Krichever and Zabrodin introduced a family of spin RS models at the level of equations of motion and posed the question of their Hamiltonian structure and integrability. These models have rational, trigonometric/hyperbolic and elliptic versions and are usually studied in the holomorphic category. The elliptic model encodes the dynamics of the poles of elliptic solutions of the 2D non-Abelian Toda lattice [32], and a special hyperbolic degeneration is related to affine Toda solitons [7]. The existence of a Hamiltonian structure was established by Krichever [31] in the general case based on a universal construction that is hard to make explicit (see also [53]). The rational case was treated via Hamiltonian reduction by Arutyunov and Frolov [4] in 1997, utilizing a ‘spin extension’ of the holomorphic cotangent bundle of \({\mathrm{GL}}(n,{\mathbb {C}})\). More than twenty years later, there appeared two different treatments of the holomorphic trigonometric/hyperbolic models: by Chalykh and Fairon [10] based on double brackets and quasi-Hamiltonian structures, and by Arutyunov and Olivucci [5] based on Hamiltonian reduction of a spin extension of the Heisenberg double [52] of the standard factorizable Poisson–Lie group structure on \({\mathrm{GL}}(n,{\mathbb {C}})\). In the present paper, we shall deal with the trigonometric real form of the models of [32] utilizing the Heisenberg double of the Poisson–Lie group \({\mathrm{U}}(n)\), which is a natural generalization of \(T^* {\mathrm{U}}(n)\). The spin extension of the Heisenberg double that we consider is based on a \({\mathrm{U}}(n)\) covariant Poisson structure on \({\mathbb {C}}^n\) introduced by Zakrzewski [56].

Although the holomorphic systems are of great interest from several viewpoints, it is not easy to extract from them the features of the dynamics of the real forms, which should also be investigated. For motivation, it perhaps suffices to recall that all pioneering papers of the subject [8], [38], [50], [54] are devoted to point particles moving along the real line or circle.

The \({\mathbb {C}}\)-valued dynamical variables of the Krichever–Zabrodin model are ‘particle positions’ \(x_i\) (\(i=1,\dots , n)\) together with d-component row vectors \(c_i\) and column vectors \(a_i\). The composite spin variables \(F_{ij}\) are built from these individual spins according to the rule

$$\begin{aligned} F_{ij}:= c_i \cdot a_j := \sum _{\alpha =1}^d c_i^\alpha a_j^\alpha , \end{aligned}$$
(1.1)

and the equations of motion can be written in first-order form as follows:

$$\begin{aligned} {\dot{x}}_i = F_{ii}, \quad {\dot{a}}_i^\alpha = \lambda _i a_i^\alpha + \sum _{k\ne i} V(x_{ik}) a_k^\alpha F_{ki} , \quad {\dot{c}}_j^\alpha = - \lambda _j c_j^\alpha - \sum _{k\ne j} V(x_{kj}) c_k^\alpha F_{jk}, \nonumber \\ \end{aligned}$$
(1.2)

where \(x_{ik}:= x_i - x_k\). In the elliptic case, the ‘potential’ is given by \(V(x)= \zeta (x) - \zeta (x+ \gamma )\) with the Weierstrass zeta-function and an arbitrary complex ‘coupling constant’ \(\gamma \ne 0\). The model admits hyperbolic/trigonometric degenerations for which one has \(V^{\text {hyp}}(x) = \coth (x) - \coth (x+\gamma )\) and \(V^{\text {rat}}(x) = x^{-1} - (x+\gamma )^{-1}\). The parameters \(\lambda _i\) in (1.2) are arbitrary. This is a hallmark of gauge invariance, and thus, it is natural to declare that the ‘physical observables’ are invariant with respect to arbitrary rescalings

$$\begin{aligned} a_i \mapsto \Lambda _i^{-1} a_i, \quad c_i \mapsto \Lambda _i c_i, \end{aligned}$$
(1.3)

where the \(\Lambda _i\) may depend on the dynamical variables as well. One way to deal with this ambiguity is to impose a gauge fixing condition. Note also the interesting feature of the model that the spins \(a_i, c_i\) are not purely internal degrees of freedom, since they directly encode the velocities through the equations of motion \({\dot{x}}_i = F_{ii}\).

In the trigonometric real form of our interest, we put \(x_j:= \frac{1}{2}q_j\), where the \(q_j\) are real and are regarded as angles. In other words, we deal with particles located on the unit circle at the points \(Q_j:= \exp ({\mathrm{i}}q_j)\). The spins \(c_i\) and \(a_i\) are complex conjugates of each other, and we parametrize them as

$$\begin{aligned} c_i^\alpha = v(\alpha )_i, \quad a_i^\alpha =\overline{ v}(\alpha )_i, \end{aligned}$$
(1.4)

where \(v(-)_i\) is regarded as a d-component row vector. For each \(\alpha \), \(v(\alpha )\) is also viewed as an n-component column vector, and thus, \(F = \sum _\alpha v(\alpha ) v(\alpha )^\dagger \) is an n by n Hermitian matrix. The potential V is now chosen to be

$$\begin{aligned} V(x) := \cot (x) - \cot (x - {\mathrm{i}}\gamma ) \end{aligned}$$
(1.5)

with a real, positive coupling constant \(\gamma \). The gauge transformations are given by arbitrary \(\Lambda _i \in {\mathrm{U}}(1)\) and accordingly we have \(\lambda _i \in {\mathrm{i}}{\mathbb {R}}\). It can be checked that these reality constraints are consistent with the equations of motion (1.2). We remark in passing that they imply the second-order equation

$$\begin{aligned} \frac{1}{2}\ddot{q}_i = \sum _{j\ne i} F_{ij}F_{ji} \left[ V\left( \frac{q_{ij}}{2}\right) - V\left( \frac{q_{ji}}{2}\right) \right] = \sum _{j\ne i} \vert F_{ij} \vert ^2 \frac{2\cot (\frac{q_{ij}}{2})}{1+\sinh ^{-2}(\gamma )\,\sin ^2(\frac{q_{ij}}{2})}. \nonumber \\ \end{aligned}$$
(1.6)

The equations of motion as given above are local in the sense that one does not know on what phase space their flow is complete, which is required for an integrable system. Neglecting this issue, let us assume that \(\sum _\alpha v(\alpha )_i \ne 0\) for all i, which permits us to impose the conditions

$$\begin{aligned} {{\mathcal {U}}}_i:= \sum _\alpha v(\alpha )_i >0. \end{aligned}$$
(1.7)

Note that (1.7) is a gauge fixing in disguise, since it amounts to setting the phase of \({{\mathcal {U}}}_i\) to 0. Then, consistency with the requirement \(\mathfrak {I}({{\mathcal {U}}}_i) =0\) can be used to uniquely determine the \(\lambda _i\), and one finds the gauge fixed equations of motion

$$\begin{aligned} \frac{1}{2}{\dot{q}}_i = F_{ii}, \qquad {\dot{v}}(\alpha )_j = {\mathrm{i}}\eta _j v(\alpha )_j - \sum _{\ell \ne j} F_{jl} v(\alpha )_l V\left( \frac{q_{lj}}{2}\right) , \end{aligned}$$
(1.8)

with

$$\begin{aligned} {\mathrm{i}}\eta _j = \frac{1}{2}\sum _{\ell \ne j} \frac{{{\mathcal {U}}}_l}{{{\mathcal {U}}}_j} \left[ F_{jl} V\left( \frac{q_{lj}}{2}\right) + F_{lj} V\left( \frac{q_{jl}}{2}\right) \right] . \end{aligned}$$
(1.9)

In this paper, we develop a Hamiltonian reduction approach to the real, trigonometric spin RS model specified above. In particular, this yields a phase space on which all flows of interest are complete. An open dense subset of the phase space will be associated with the gauge fixing condition (1.7), and on this submanifold we shall determine the explicit form of the Poisson brackets that generate the equations of motion (1.8) by means of the Hamiltonian

$$\begin{aligned} {{\mathcal {H}}}= \sum _{i} F_{ii}. \end{aligned}$$
(1.10)

We shall also prove the degenerate integrability of the model by displaying \((2nd -n)\) independent, real-analytic integrals of motion that form a polynomial Poisson algebra whose n-dimensional Poisson center contains \({{\mathcal {H}}}\). These results will be derived by using the \(q_i\) and gauge fixed versions of the ‘dressed spins’ \(v(\alpha )\) as coordinates on the reduced phase space obtained from Hamiltonian reduction. However, we will put forward another remarkable set of variables as well, which consists of canonical pairs \(q_i, p_i\) and ‘reduced primary spins’ \(w^\alpha \) that decouple from q and p under the reduced Poisson bracket. On the overlap of their dense domains, the relation between the two sets of variables can be given explicitly, but the formula is very involved. The drawback of the variables \(q, p, w^\alpha \) is that in terms of them \({{\mathcal {H}}}\) and the equations of motion become complicated.

Notice that the Newton equations (1.6) imply the conservation of the sum of the velocities \({\dot{q}}_i\), which gives the Hamiltonian via (1.8) and (1.10), and \({\dot{q}}_i\) is nonnegative by (1.8). The same features appear in the spinless chiral RS model [50] defined by the Hamiltonian

$$\begin{aligned} {{\mathcal {H}}}_{\text {RS}}^+= \sum _i e^{2\theta _i} \prod _{j\ne i} \left[ 1+ \frac{\sinh ^2\gamma }{1 + \sin ^2\frac{q_i- q_j}{2}}\right] ^{\frac{1}{2}}, \end{aligned}$$
(1.11)

with Darboux coordinatesFootnote 1\(q_i, \theta _j\). The second-order equations of motion for \(q_i\) generated by this Hamiltonian reproduce the \(d=1\) special case of (1.6). To see this, note that \(F_{ij} F_{ji} = F_{ii} F_{jj}\) if \(d=1\), and substitute \(\dot{q_i}=2 F_{ii}\) from (1.8) into (1.6). In fact, the spinless RS model results from the \(d=1\) special case of our Hamiltonian reduction: in this case \(w^1\) becomes gauge equivalent to a constant vector and one derives the model utilizing also a canonical transformation between qp mentioned above and \(q,\theta \) [19]. Thus, the spin RS systems of [32] are generalizations of the chiral RS model. We follow the general practice in dropping ‘chiral’ from their name.

Our result on the degenerate integrability of the system is not surprising, since the same property holds in the complex holomorphic case [5], [10] and it also holds generically for large families of related spin many-body models obtained by Hamiltonian reduction [44], [45], [46]. Despite these earlier results, the degenerate integrability of our specific real system cannot be obtained directly. Therefore, it requires a separate treatment, and we shall exhibit the desired integrals of motion in explicit form.

Here is an outline of this work and its main results. In Sect. 2, we present the master phase space \({{\mathcal {M}}}\) which is an extension of the Heisenberg double of \({\mathrm{U}}(n)\) by a space of primary spins. The latter space is formed of \(d\ge 2\) copies of \({\mathbb {C}}^n\) endowed with a \({\mathrm{U}}(n)\) covariant Poisson bracket and a (Poisson–Lie) moment map, see Proposition 2.1. We also introduce the ‘free’ degenerate integrable system on \({{\mathcal {M}}}\) that will be reduced. In Sect. 3, we define the Hamiltonian reduction and progress toward the description of the corresponding reduced phase space \({{\mathcal {M}}}_{\text {red}}\), which is a real-analytic symplectic manifold of dimension 2nd. In particular, we exhibit two models of dense open subsets of \({{\mathcal {M}}}_{\text {red}}\); the first one is used in the subsequent sections to derive the real form of the trigonometric spin RS system described above, while the second one allows us to prove that \({{\mathcal {M}}}_{\text {red}}\) is connected and it leads to a concise formula for the reduced symplectic form (see Theorem 3.14 and Corollary 3.15). The second model will also be used for recovering the Gibbons–Hermsen system through a scaling limit (see Remark 3.16). In Sect. 4, we characterize the projection of a family of free Hamiltonian vector fields of \({{\mathcal {M}}}\) onto \({{\mathcal {M}}}_{\text {red}}\), and show in Corollary 4.3 that one of these projections reproduces the equations of motion (1.8). Then, in Sect. 5, we obtain the reduced Poisson bracket presented in Theorem 5.8. This offers an alternative way to derive the equations of motion (1.8), and we also provide a formula for the Poisson bracket of the Lax matrix that generates the commuting reduced Hamiltonians, see Proposition 5.10. In Sect. 6, we demonstrate the degenerate integrability of the real trigonometric spin RS system, with the final result formulated as Theorem 6.7. Section 7 concludes this work and gathers open questions. There are four appendices devoted to auxiliary results and proofs.

Note on conventions. The sign function \(\text {sgn}\) is such that \(\text {sgn}(i-k)\) is \(+1\) if \(i>k\), \(-1\) if \(i<k\), and 0 for \(i=k\). Similarly to Kronecker’s delta function, we define for any condition \(\text {c}\) the symbol \(\delta _{\text {c}}\) which equals \(+1\) if \(\text {c}\) is satisfied, and 0 otherwise. For example, \(\delta _{(j<l\le k)}\) equals \(+1\) if \(j<l\) and \(l\le k\), while it is 0 if one of those two conditions is not satisfied.

Heisenberg Double, Primary Spins and ‘Free’ Integrable System

Eventually, we shall obtain the real, trigonometric spin RS system by reduction of an ‘obviously integrable’ system on the phase space \({{\mathcal {M}}}:= M \times {\mathbb {C}}^{n\times d}\), where M is the Heisenberg double of the Poisson–Lie group \({\mathrm{U}}(n)\) and \({\mathbb {C}}^{n\times d}\) is the space of the so-called primary spin variables. In this section, we present a quick overview of these structures, to be used in the subsequent sections. More details can be found in the references [16], [19], [29], [30], [35], [52] and in “Appendix A.”

The Heisenberg Double and Its Models

Let us start with the real vector space direct sum

$$\begin{aligned} \mathfrak {gl}(n,{\mathbb {C}})= {\mathfrak {u}}(n) + {\mathfrak {b}}(n), \end{aligned}$$
(2.1)

where \({\mathfrak {b}}(n)\) denotes the Lie algebra of upper triangular complex matrices having real entries along the diagonal, and the unitary Lie algebra \({\mathfrak {u}}(n)\) consists of the skew-Hermitian matrices. These are isotropic subalgebras with respect to the non-degenerate, invariant bilinear form of \(\mathfrak {gl}(n,{\mathbb {C}})\) given by

$$\begin{aligned} \langle X, Y \rangle := \mathfrak {I}\text {tr}(XY), \quad \forall X,Y \in \mathfrak {gl}(n,{\mathbb {C}}), \end{aligned}$$
(2.2)

which means that we have a Manin triple at hand. Then define

$$\begin{aligned} R:= \frac{1}{2} \left( P_{{\mathfrak {u}}(n)} - P_{{\mathfrak {b}}(n)}\right) , \end{aligned}$$
(2.3)

using the projection operators with ranges \({\mathfrak {u}}(n)\) and \({\mathfrak {b}}(n)\), associated with the decomposition (2.1). For any \(X\in \mathfrak {gl}(n,{\mathbb {C}})\), we may write

$$\begin{aligned} X= X_{{\mathfrak {u}}(n)} + X_{{\mathfrak {b}}(n)} \quad \hbox {with}\quad X_{{\mathfrak {u}}(n)} = P_{{\mathfrak {u}}(n)}(X),\, X_{{\mathfrak {b}}(n)} = P_{{\mathfrak {b}}(n)}(X). \end{aligned}$$
(2.4)

As a manifold, M is the real Lie group \({\mathrm{GL}}(n,{\mathbb {C}})\), and for any smooth real function \(f\in C^\infty ({\mathrm{GL}}(n,{\mathbb {C}}))\) we introduce the \(\mathfrak {gl}(n,{\mathbb {C}})\)-valued derivatives \(\nabla f\) and \(\nabla 'f\) by

$$\begin{aligned} \langle \nabla f(K), X\rangle := \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=0}f(e^{tX} K), \quad \langle \nabla ' f(K), X\rangle := \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=0}f(Ke^{tX}),\quad \forall X\in \mathfrak {gl}(n,{\mathbb {C}}), \nonumber \\ \end{aligned}$$
(2.5)

where K denotes the variable running over \({\mathrm{GL}}(n,{\mathbb {C}})\). The commutative algebra of smooth real functions, \(C^\infty (M)\), carries two natural Poisson brackets provided by

$$\begin{aligned} \{ f, h\}_{\pm } := \langle \nabla f, R \nabla h \rangle \pm \langle \nabla ' f, R \nabla ' h \rangle . \end{aligned}$$
(2.6)

The minus bracket makes \({\mathrm{GL}}(n,{\mathbb {C}})\) into a real Poisson–Lie group, while the plus one corresponds to a symplectic structure on M. The former is called the Drinfeld double Poisson bracket and the latter the Heisenberg double Poisson bracket [52].

The real Poisson brackets can be extended to complex functions by requiring complex bilinearity. Then, the real Poisson brackets can be recovered if we know all Poisson brackets between the matrix elements of K and its complex conjugate \(\overline{K}\). In the case of the Drinfeld double, we have

$$\begin{aligned} \{ K_{ij}, K_{kl}\}_- ={\mathrm{i}}K_{kj} K_{il} \left[ \delta _{ik} + 2 \delta _{(i>k)} - \delta _{lj} - 2 \delta _{(l>j)}\right] , \end{aligned}$$
(2.7)

and

$$\begin{aligned} \{ K_{ij}, \overline{K}_{kl} \}_- = {\mathrm{i}}K_{ij} \overline{K}_{kl} [ \delta _{ik} - \delta _{jl}] + 2 {\mathrm{i}}\left[ \delta _{ik} \sum _{\beta > i} K_{\beta j} \overline{K}_{\beta l} - \delta _{jl} \sum _{\alpha <j} K_{i\alpha } \overline{K}_{k\alpha }\right] . \nonumber \\ \end{aligned}$$
(2.8)

Consider the subgroup \(\text {B}(n)< {\mathrm{GL}}(n,{\mathbb {C}})\) of upper triangular matrices having positive entries along the diagonal, and the unitary subgroup \({\mathrm{U}}(n) < {\mathrm{GL}}(n,{\mathbb {C}})\). These subgroups correspond to the subalgebras in (2.1). It is well known that both \({\mathrm{U}}(n)\) and \(\text {B}(n)\) are Poisson submanifolds of the Drinfeld double \(({\mathrm{GL}}(n,{\mathbb {C}}), \{\ ,\ \}_-)\). We denote their inherited Poisson structures by \(\{\ ,\ \}_U\) and \(\{\ ,\ \}_B\), which makes them Poisson–Lie groups.

The Poisson brackets on \(C^\infty ({\mathrm{U}}(n))\) and on \(C^\infty (\text {B}(n))\) admit the following description. For any real function \(\phi \in C^\infty ({\mathrm{U}}(n))\) introduce the \({\mathfrak {b}}(n)\)-valued derivatives \(D\phi \) and \(D'\phi \) by

$$\begin{aligned} \langle D \phi (g), X\rangle := \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=0}\phi (e^{tX} g), \quad \langle D' \phi (g), X\rangle := \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=0}\phi (ge^{tX}), \,\, \forall X\in {\mathfrak {u}}(n), \nonumber \\ \end{aligned}$$
(2.9)

and for any \(\chi \in C^\infty (\text {B}(n))\) similarly introduce the \({\mathfrak {u}}(n)\)-valued derivatives \(D\chi \) and \(D'\chi \). Then, we have

$$\begin{aligned} \{\phi _1, \phi _2\}_U(g) = -\left\langle D'\phi _1(g), g^{-1} (D\phi _2(g)) g \right\rangle , \quad \forall g\in {\mathrm{U}}(n), \end{aligned}$$
(2.10)

where the conjugation takes place inside \({\mathrm{GL}}(n,{\mathbb {C}})\). Similarly

$$\begin{aligned} \{ \chi _1, \chi _2\}_B(b)= \left\langle D' \chi _1(b), b^{-1} (D\chi _2(b)) b \right\rangle , \quad \forall b\in \text {B}(n). \end{aligned}$$
(2.11)

The opposite signs in the last two formulae are due to our conventions.

By the Gram–Schmidt process, every element \(K\in {\mathrm{GL}}(n,{\mathbb {C}})\) admits the unique decompositions

$$\begin{aligned} K = b_L g_R^{-1} = g_L b_R^{-1}\quad \hbox {with}\quad b_L, b_R \in \text {B}(n),\, g_L, g_R\in {\mathrm{U}}(n), \end{aligned}$$
(2.12)

and K can be recovered also from the pairs \((g_L, b_L)\) and \((g_R, b_R)\), by utilizing the identity

$$\begin{aligned} b_L^{-1} g_L = g_R^{-1} b_R. \end{aligned}$$
(2.13)

These decompositions give rise to the maps \(\Lambda _L, \Lambda _R\) into \(\text {B}(n)\) and \(\Xi _L, \Xi _R\) into \({\mathrm{U}}(n)\),

$$\begin{aligned} \Lambda _L(K):= b_L, \quad \Lambda _R(K):= b_R, \quad \Xi _L(K):= g_L,\quad \Xi _R(K):= g_R. \end{aligned}$$
(2.14)

Then, we obtain the maps from \({\mathrm{GL}}(n,{\mathbb {C}})\) onto \({\mathrm{U}}(n)\times \text {B}(n)\),

$$\begin{aligned} (\Xi _L, \Lambda _R),\quad (\Xi _R, \Lambda _L),\quad (\Xi _L, \Lambda _L),\quad (\Xi _R, \Lambda _R), \end{aligned}$$
(2.15)

which are all (real-analytic) diffeomorphisms. In particular, we shall use the diffeomorphism

$$\begin{aligned} m_1:= (\Xi _R, \Lambda _R): {\mathrm{GL}}(n,{\mathbb {C}})\rightarrow {\mathrm{U}}(n) \times \text {B}(n) \end{aligned}$$
(2.16)

to transfer the Heisenberg double Poisson bracket to \(C^\infty ({\mathrm{U}}(n) \times \text {B}(n))\). The formula of the resulting Poisson bracket [16], called \(\{\ ,\ \}_+^1\), can be written as follows:

$$\begin{aligned} \{{{\mathcal {F}}}, {{\mathcal {H}}}\}_+^1(g,b)= & {} \left\langle D_2' {{\mathcal {F}}}, b^{-1} (D_2{{\mathcal {H}}}) b \right\rangle -\left\langle D'_1{{\mathcal {F}}}, g^{-1} (D_1{{\mathcal {H}}}) g\right\rangle \nonumber \\&+ \left\langle D_1{{\mathcal {F}}}, D_2{{\mathcal {H}}}\right\rangle -\left\langle D_1 {{\mathcal {H}}}, D_2{{\mathcal {F}}}\right\rangle , \end{aligned}$$
(2.17)

for any \({{\mathcal {F}}}, {{\mathcal {H}}}\in C^\infty ({\mathrm{U}}(n)\times \text {B}(n))\). The derivatives on the right-hand side are taken at \((g,b)\in {\mathrm{U}}(n)\times \text {B}(n)\); the subscripts 1 and 2 refer to derivatives with respect to the first and second arguments. As an application, one can determine the Poisson brackets between the matrix elements of \((g,b):= (g_R, b_R)\) on the Heisenberg double, which gives

$$\begin{aligned} \{ g_{lm}, b_{jk} \}_+ = {\mathrm{i}}\delta _{jl} g_{lm} b_{jk} + 2{\mathrm{i}}\delta _{(j<l\le k)}\, g_{jm} b_{lk}, \end{aligned}$$
(2.18)

and

$$\begin{aligned} \{ g_{lm}, \overline{b}_{jk} \}_+ = {\mathrm{i}}\delta _{jl} g_{lm} \overline{b}_{jk} + 2{\mathrm{i}}\delta _{jl} \sum _{j<\beta \le k}\, g_{\beta m} \overline{b}_{\beta k}. \end{aligned}$$
(2.19)

The same formulae are valid w.r.t. \(\{\ ,\ \}_+^1\), and this was used for the computation.

Observe from the formula (2.17) that both \(\Xi _R\) and \(\Lambda _R\) are Poisson maps w.r.t. the Heisenberg double Poisson bracket and the Poisson brackets \(\{\ ,\ \}_U\) and \(\{\ ,\ \}_B\), respectively. The same is true regarding the maps \(\Xi _L\) and \(\Lambda _L\). A further property that we use later is that

$$\begin{aligned} \{ \Lambda _L^*(\chi _1), \Lambda _R^*(\chi _2)\}_+ = 0, \quad \forall \chi _1, \chi _2 \in C^\infty (\text {B}(n)). \end{aligned}$$
(2.20)

Note, incidentally, that \(\Xi _L\) and \(\Xi _R\) enjoy the analogous identity.

The Heisenberg double admits another convenient model as well. This relies on the diffeomorphism between \(\text {B}(n)\) and the manifold \({{\mathfrak {P}}}(n) = \exp ({\mathrm{i}}{\mathfrak {u}}(n))\) of positive definite Hermitian matrices, defined by \(b \mapsto L:=bb^\dagger \). Then, we have the diffeomorphism

$$\begin{aligned} m_2: {\mathrm{GL}}(n,{\mathbb {C}})\rightarrow {\mathrm{U}}(n) \times {{\mathfrak {P}}}(n), \quad m_2:= (\Xi _R, \Lambda _R \Lambda _R^\dagger ). \end{aligned}$$
(2.21)

For \(\psi \in C^\infty ({{\mathfrak {P}}}(n))\), define the \({\mathfrak {u}}(n)\)-valued derivative \(d \psi \) by

$$\begin{aligned} \langle d \psi (L), X\rangle := \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=0}\psi (L + t X), \,\,\forall X\in {\mathrm{i}}{\mathfrak {u}}(n). \end{aligned}$$
(2.22)

This definition makes sense since \((L + t X)\in {{\mathfrak {P}}}(n)\) for small t. By using \(m_2\), one can transfer the Poisson bracket \(\{\ ,\ \}_+\) to \(C^\infty ({\mathrm{U}}(n) \times {{\mathfrak {P}}}(n))\). The resulting Poisson bracket is called \(\{\ ,\ \}_+^2\) and is given [16] by the following explicit formula:

$$\begin{aligned} \{F, H\}_+^2 (g,L)= & {} 4\left\langle L d_2F, \left( L d_2 H \right) _{{\mathfrak {u}}(n)}\right\rangle -\left\langle D'_1 F , g^{-1} (D_1 H) g \right\rangle \nonumber \\&+ 2\left\langle D_1F, L d_2H \right\rangle -2\left\langle D_1 H, L d_2F\right\rangle , \end{aligned}$$
(2.23)

for any \(F,H\in C^\infty ({\mathrm{U}}(n) \times {{\mathfrak {P}}}(n))\), where the derivatives with respect to the first and second arguments are taken at \((g,L)\in {\mathrm{U}}(n)\times {{\mathfrak {P}}}(n)\). The subscript \({\mathfrak {u}}(n)\) refers to the decomposition defined in (2.4).

We end this review of the Heisenberg double by recalling the symplectic form, denoted \(\Omega _M\), that corresponds to the non-degenerate Poisson structure \(\{\ ,\ \}_+\). It can be displayed [1] as

$$\begin{aligned} \Omega _M=\frac{1}{2}\mathfrak {I}\text {tr}(\hbox {d} \Lambda _L \Lambda _L^{-1}\wedge \hbox {d} \Xi _L \Xi _L^{-1})+ \frac{1}{2}\mathfrak {I}\text {tr}( \hbox {d}\Lambda _R \Lambda _R^{-1} \wedge \hbox {d}\Xi _R \Xi _R^{-1}). \end{aligned}$$
(2.24)

Here, \(d\Lambda _L\) collects the exterior derivatives of the components of the matrix valued function \(\Lambda _L\). To be clear about our conventions, we remark that the wedge does not contain \(\frac{1}{2}\), and the Hamiltonian vector field \(X_h\) of h satisfies \(d h = \Omega _M(\ , X_h)\) and \(\{f,h\}_+ = df(X_h) = \Omega _M(X_h, X_f)\).

The Primary Spin Variables

We begin by recalling that the real, trigonometric spin Sutherland model of Gibbons–Hermsen [24] type can be derived via Hamiltonian reduction of \(T^*{\mathrm{U}}(n) \times {\mathbb {C}}^{n\times d}\), where \({\mathbb {C}}^{n\times d} \simeq {\mathbb {R}}^{2n \times d}\) carries its canonical Poisson structure. In particular, if the elements of \({\mathbb {C}}^{n\times d}\) are represented as a collection of \({\mathbb {C}}^n\) column vectors

$$\begin{aligned} w^1, w^2, \dots , w^d, \end{aligned}$$
(2.25)

then the d different copies pairwise Poisson commute. The symmetry group underlying the reduction is \({\mathrm{U}}(n)\), which acts on \({\mathbb {C}}^n\) in the obvious manner,

$$\begin{aligned} {{\mathcal {A}}}^{(n)}: {\mathrm{U}}(n) \times {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n \quad \hbox {given by}\quad {{\mathcal {A}}}^{(n)}(g,w):= g w. \end{aligned}$$
(2.26)

For our generalization, it is natural to require this to be a Poisson action, i.e., \({{\mathcal {A}}}^{(n)}\) should be a Poisson map with respect to the Poisson structure (2.10) on \({\mathrm{U}}(n)\) and a suitable Poisson structure on \({\mathbb {C}}^n\). A further requirement is that the \({\mathrm{U}}(n)\)-action should be generated by a moment map.

Specialized to \({\mathrm{U}}(n)\) with the Poisson structure (2.10), the notion of moment map that we use can be summarized as follows.Footnote 2 Suppose that we have a Poisson manifold \(({{\mathcal {P}}}, \{\ ,\ \}_{{\mathcal {P}}})\) and a Poisson map \(\Lambda : {{\mathcal {P}}}\rightarrow \text {B}(n)\), where \(\text {B}(n)\) is endowed with the Poisson structure (2.11). Then, for any \(X\in {\mathfrak {u}}(n)\), the following formula defines a vector field \(X_{{{\mathcal {P}}}}\) on \({{\mathcal {P}}}\):

$$\begin{aligned} {{\mathcal {L}}}_{X_{{{\mathcal {P}}}}}({{\mathcal {F}}}) \equiv X_{{{\mathcal {P}}}}[{{\mathcal {F}}}] :=\left\langle X, \{{{\mathcal {F}}}, \Lambda \}_{{\mathcal {P}}}\Lambda ^{-1} \right\rangle , \qquad \forall {{\mathcal {F}}}\in C^\infty ({{\mathcal {P}}}), \end{aligned}$$
(2.27)

where the Poisson bracket is taken with every entry of the matrix \(\Lambda \) and \({{\mathcal {L}}}_{X_{{\mathcal {P}}}}\) denotes Lie derivative along \(X_{{{\mathcal {P}}}}\). The map \(X \mapsto X_{{{\mathcal {P}}}}\) is automatically a Lie algebra anti-homomorphism, representing an infinitesimal left action of \({\mathrm{U}}(n)\). If it integrates to a global action of \({\mathrm{U}}(n)\), then the resulting action is Poisson, i.e., the action map \({{\mathcal {A}}}: {\mathrm{U}}(n) \times {{\mathcal {P}}}\rightarrow {{\mathcal {P}}}\) is Poisson. In the situation just outlined, \(\Lambda \) is called the (Poisson–Lie) moment map of the corresponding Poisson action.

In the next proposition, we collect the key properties of a Poisson structure on \({\mathbb {C}}^n\), which is a special case of the \({\mathrm{U}}(n)\) covariant Poisson structures found by Zakrzewski [56].

Proposition 2.1

The following formula defines a Poisson structure on \({\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}\):

$$\begin{aligned}&\{w_i,w_l\}={\mathrm{i}}\, \mathrm{sgn}(i-l) w_i w_l, \qquad \forall 1\le i,l \le n, \end{aligned}$$
(2.28)
$$\begin{aligned}&\{w_i,\overline{w}_l\}={\mathrm{i}}\, \delta _{il} (2+|w|^2) +{\mathrm{i}}w_i \overline{w}_l+ {\mathrm{i}}\, \delta _{il} \sum _{r=1}^n \mathrm{sgn}(r-i) |w_r|^2\,. \end{aligned}$$
(2.29)

These formulae imply

$$\begin{aligned} \{ \overline{w}_i, \overline{w}_l\} = \overline{\{w_i, w_l\}}, \end{aligned}$$
(2.30)

which means that the Poisson bracket of real functions is real. With respect to this Poisson bracket, the action (2.26) of \({\mathrm{U}}(n)\) with (2.10) is Poisson and is generated by the moment map \({{\mathbf {b}}}: {\mathbb {C}}^n \rightarrow \mathrm{B(n)}\) given by

$$\begin{aligned} {{\mathbf {b}}}_{jj}(w) = \sqrt{{{\mathcal {G}}}_j/{{\mathcal {G}}}_{j+1}},\qquad {{\mathbf {b}}}_{ij}(w) = \frac{w_i \overline{w}_j}{ \sqrt{{{\mathcal {G}}}_j {{\mathcal {G}}}_{j+1}}}, \qquad \forall 1\le i<j\le n\nonumber \\ \end{aligned}$$
(2.31)

with

$$\begin{aligned} {{\mathcal {G}}}_j = 1 + \sum _{k=j}^n \vert w_k \vert ^2,\quad {{\mathcal {G}}}_{n+1}:= 1, \end{aligned}$$
(2.32)

The map \({{\mathbf {b}}}\) satisfies the identity

$$\begin{aligned} {{\mathbf {b}}}(w) {{\mathbf {b}}}(w)^\dagger = {{\varvec{1}}}_n + w w^\dagger . \end{aligned}$$
(2.33)

The Poisson structure is non-degenerate, and the corresponding symplectic form is given by

$$\begin{aligned} \Omega _{{\mathbb {C}}^n} = \frac{{\mathrm{i}}}{2}\sum _{k=1}^n\frac{1}{{{\mathcal {G}}}_k} \hbox {d}w_k\wedge \hbox {d}{\overline{w}}_k + \frac{{\mathrm{i}}}{4}\sum _{k=1}^{n-1}\frac{1}{{{\mathcal {G}}}_k{{\mathcal {G}}}_{k+1}} \hbox {d}{{\mathcal {G}}}_{k+1} \wedge \left( {\overline{w}}_k \hbox {d}w_k - w_k \hbox {d}{\overline{w}}_k\right) . \nonumber \\ \end{aligned}$$
(2.34)

A variant of the factorization formula (2.33) (without connection to Zakrzewski’s Poisson bracket) was found earlier by Klimčík, as presented in an unpublished initial version of [19]. For convenience, we give a self-contained proof of the proposition in “Appendix .”

Definition 2.2

The pairwise Poisson commuting \(w^1,\dots , w^d\) with each copy subject to the Poisson brackets (2.28), (2.29) are called primary spin variables. The Poisson space obtained in this manner is denoted \(\left( {\mathbb {C}}^{n\times d}, \{\ ,\ \}_{{\mathcal {W}}}\right) \), and we shall also use the notation

$$\begin{aligned} W:= (w^1, \dots , w^d). \end{aligned}$$
(2.35)

The corresponding symplectic form, \(\Omega _{{\mathcal {W}}}\), is the sum of d-copies of \(\Omega _{{\mathbb {C}}^n}\) (2.34), one for each variable \(w^\alpha \), \(\alpha =1,\dots , d\).

The Unreduced ‘Free’ Integrable System

Let \({\mathfrak {H}}\) be an Abelian Poisson subalgebra of the Poisson algebra of (smooth, real-analytic, etc.) functions on a symplectic manifold \({{\mathcal {M}}}\) of dimension 2N, such that all elements of \({\mathfrak {H}}\) generate complete Hamiltonian flows. Assume that the functional dimension of \({\mathfrak {H}}\) is rFootnote 3 and that there exists also a Poisson subalgebra \({{\mathfrak {C}}}\) of the functions on \({{\mathcal {M}}}\) whose functional dimensions is \((2N -r)\) and its center contains \({\mathfrak {H}}\). Then \({\mathfrak {H}}\) is a called an integrable system with Hamiltonians \({\mathfrak {H}}\) and algebra of constants of motion \({{\mathfrak {C}}}\). Liouville integrability is the \(r=N\), \({{\mathfrak {C}}}= {\mathfrak {H}}\), special case. One calls the system degenerate integrable (or non-commutative integrable, or superintegrable) if \(r<N\). In the degenerate case, similarly to Liouville integrability, the flows of the Hamiltonians belonging to \({\mathfrak {H}}\) are linear in suitable coordinate systems on the joint level surfaces of \({{\mathfrak {C}}}\). For further details of this notion and its variants, and for the generalization of the Liouville–Arnold theorem, we refer to the papers [27], [37], [39], [45] and to Section 11.8 of the book [48].

Consider the Heisenberg double \((M, \{\ ,\ \}_+)\) and the space of primary spins \(({\mathbb {C}}^{n\times d}, \{\ ,\ \}_{{{\mathcal {W}}}})\) introduced in Sect. 2.2. Define

$$\begin{aligned} {{\mathcal {M}}}:= M \times {\mathbb {C}}^{n\times d} \end{aligned}$$
(2.36)

and equip it with the product Poisson structure, \(\{\ ,\ \}_{{\mathcal {M}}}\), which comes from the symplectic form

$$\begin{aligned} \Omega _{{{\mathcal {M}}}} = \Omega _M + \Omega _{{{\mathcal {W}}}}. \end{aligned}$$
(2.37)

Let \(C^\infty (\text {B}(n))^{{\mathrm{U}}(n)}\) denote those functions on \(\text {B}(n)\) that are invariant with respect to the dressing action of \({\mathrm{U}}(n)\) on \(\text {B}(n)\), operating as

$$\begin{aligned} \text {Dress}_g(b):= \Lambda _L( g b), \qquad \forall (g,b) \in {\mathrm{U}}(n) \times \text {B}(n). \end{aligned}$$
(2.38)

It is well known that these invariant functions form the center of the Poisson bracket \(\{\ ,\ \}_B\) (2.11). Extend all maps displayed in (2.14) to \({{\mathcal {M}}}\) in the trivial manner, for example, by setting

$$\begin{aligned} \Lambda _R(K,W):= \Lambda _R(K) \quad \hbox {with}\quad (K,W) \in M \times {\mathbb {C}}^{n\times d}. \end{aligned}$$
(2.39)

Then

$$\begin{aligned} {\mathfrak {H}}:= \Lambda _R^*\bigl ( C^\infty \bigl (\text {B}(n)\bigr )^{{\mathrm{U}}(n)}\bigr ) \end{aligned}$$
(2.40)

is an Abelian Poisson subalgebra of \(C^\infty ({{\mathcal {M}}})\). We call the elements of \({\mathfrak {H}}\) ‘free Hamiltonians’ since their flows are easily written down explicitly. Indeed, for \(H= \Lambda _R^*(h)\) the flow sends the initial value (K(0), W(0)) to

$$\begin{aligned} (K(t), W(t)) = \left( K(0) \exp \left( - t Dh(b_R(0))\right) , W(0)\right) . \end{aligned}$$
(2.41)

It follows that \(b_R\) and \(b_L\) are constants along the flow and we have the ‘free motion’ on \({\mathrm{U}}(n)\) given by

$$\begin{aligned} g_R(t) = \exp \left( t Dh(b_R(0))\right) g_R(0). \end{aligned}$$
(2.42)

The functional dimension of \({\mathfrak {H}}\) is n, and for independent generators one may take

$$\begin{aligned} H_k:= \Lambda _R^*(h_k) \quad \hbox {with}\quad h_k(b) = \frac{1}{2k}\text {tr}(bb^\dagger )^k, \quad k=1,\dots , n. \end{aligned}$$
(2.43)

The invariance of these functions follows from the useful identity

$$\begin{aligned} (\text {Dress}_g(b)) (\text {Dress}_g(b))^\dagger = g (bb^\dagger ) g^{-1}. \end{aligned}$$
(2.44)

The system is degenerate integrable, with \({{\mathfrak {C}}}\) taken to be the algebra of all constants of motion, which are provided by arbitrary smooth functions depending on \(b_L, b_R\) and W. From the decomposition (2.12), we get

$$\begin{aligned} b_R b_R^\dagger = g_R \left( b_L^{-1} (b_L^{-1})^\dagger \right) g_R^{-1}, \end{aligned}$$
(2.45)

and this entails n relations between the functions of \(b_L\) and \(b_R\). Thus the functional dimension of \({{\mathfrak {C}}}\) is \(2N - n\), with \(N= n^2 + nd\), as required. It is worth noting that the joint level surfaces of \({{\mathfrak {C}}}\) are compact, since they can be viewed as closed subsets of \({\mathrm{U}}(n)\).

There are several ways to enlarge \({\mathfrak {H}}\) into an Abelian Poisson algebra of functional dimension N, i.e., to obtain Liouville integrability of the Hamiltonians in \({\mathfrak {H}}\). However, there is no canonical way to do so. Degenerate integrability is a stronger property than Liouville integrability, since it restricts the flows of the Hamiltonians to smaller level surfaces. For these reasons, we shall not pay attention to Liouville integrability in this paper.

Defining the Reduction and Solving the Moment Map Constraint

We first describe a Poisson action of \({\mathrm{U}}(n)\) on \({{\mathcal {M}}}\) and use it for defining the reduction of the free integrable system. Then we shall deal with two parametrizations of the ‘constraint surface,’ which is obtained by imposing the moment map constraint of Eq. (3.17) below.

Definition of the Reduction

Let us start by introducing the following Poisson map \(\Lambda :{{\mathcal {M}}}\rightarrow \text {B}(n)\),

$$\begin{aligned} \Lambda (K,W) = \Lambda _L(K) \Lambda _R(K) {{\mathbf {b}}}(w^1) {{\mathbf {b}}}(w^2) \cdots {{\mathbf {b}}}(w^d), \end{aligned}$$
(3.1)

using the notations (2.14), (2.31). The Poisson property of \(\Lambda \) holds since all the factors are separately Poisson maps, and their matrix elements mutually Poisson commute. The infinitesimal action generated by \(\Lambda \), via the formula (2.27), integrates to a global Poisson action of \({\mathrm{U}}(n)\) on \({{\mathcal {M}}}\). This action turns out to have a nice form in terms of the new variables on \({{\mathcal {M}}}\) given below.

Definition 3.1

For \(\alpha =1,\dots , d\), introduce

$$\begin{aligned} b_\alpha := {{\mathbf {b}}}(w^\alpha ), \qquad B_\alpha := b_R b_1 b_2 \cdots b_\alpha , \quad B_0:=b_R, \end{aligned}$$
(3.2)

and define the dressed spins \(v(\alpha )\) and the half-dressed spins \(v^\alpha \) by the equalities

$$\begin{aligned} v(\alpha ):= B_{\alpha -1} w^\alpha =: b_R v^\alpha . \end{aligned}$$
(3.3)

Lemma 3.2

The new variables on \({{\mathcal {M}}}\) given by

$$\begin{aligned} g_R, b_R, v(1),\dots , v(d) \end{aligned}$$
(3.4)

are related by a diffeomorphism of \({\mathrm{U}}(n) \times \mathrm{B(n)} \times {\mathbb {C}}^{n\times d}\) to the variables

$$\begin{aligned} g_R, b_R, w^1,\dots , w^d. \end{aligned}$$
(3.5)

Proof

We have the relations,

$$\begin{aligned} w^1 = b_R^{-1} v(1), \,\, w^2 = {{\mathbf {b}}}(w^1)^{-1} b_R^{-1} v(2),\dots , w^d = {{\mathbf {b}}}(w^{d-1})^{-1} \cdots {{\mathbf {b}}}(w^1)^{-1} b_R^{-1} v(d), \nonumber \\ \end{aligned}$$
(3.6)

which can be used to reconstruct the variables (3.5) from those in (3.4). \(\square \)

The statement analogous to Lemma 3.2 for the variables \((g_R, b_R, v^1,\dots , v^d)\) also holds. Later in the paper, we shall use the following identities enjoyed by the half-dressed spins and the dressed spins, which are direct consequences of (2.33). These identities and the subsequent proposition actually motivated the introduction of these variables.

Lemma 3.3

With the above notations, one has the identities

$$\begin{aligned} (b_1 b_2 \cdots b_d) (b_1 b_2 \cdots b_d)^\dagger = {{\varvec{1}}}_n + \sum _{\alpha =1}^d v^\alpha (v^\alpha )^\dagger , \quad B_d B_d^\dagger = b_R b_R^\dagger + \sum _{\alpha =1}^d v(\alpha ) v(\alpha )^\dagger . \nonumber \\ \end{aligned}$$
(3.7)

Proposition 3.4

The moment map \(\Lambda : {{\mathcal {M}}}\rightarrow \mathrm{B(n)}\) given by (3.1) generates the action \({{\mathcal {A}}}: {\mathrm{U}}(n) \times {{\mathcal {M}}}\rightarrow {{\mathcal {M}}}\) that operates as follows:

$$\begin{aligned} {{\mathcal {A}}}_\eta : \left( g_R, b_R, v(1),\dots , v(d)\right) \mapsto \left( {\tilde{\eta }} g_R {\tilde{\eta }}^{-1}, \mathrm{Dress}_{{\tilde{\eta }}}(b_R), {\tilde{\eta }} v(1), \dots , {\tilde{\eta }} v(d)\right) , \quad \forall \eta \in {\mathrm{U}}(n), \nonumber \\ \end{aligned}$$
(3.8)

where \({\tilde{\eta }} = \Xi _R(\eta b_L)^{-1}\) with \(b_L = \Lambda _L \circ m_1^{-1}(g_R, b_R)\) using (2.16). In other words, \(b_L = \Lambda _L(K)\) with \(K\in M\) parametrized by the pair \((g_R, b_R)\).

Proof

In order to avoid clumsy formulae and the introduction of further notations, in what follows we identify the variables \(g_R, b_R, B_\alpha , w^\alpha \) and so on with the associated evaluation functions on M, \({{\mathcal {M}}}\) and \({\mathbb {C}}^n\). We shall also use the infinitesimal dressing action corresponding to (2.38), which has the form

$$\begin{aligned} \text {dress}_X(b) = b (b^{-1} X b)_{{\mathfrak {b}}(n)}, \qquad \forall X\in {\mathfrak {u}}(n),\,\, b\in \text {B}(n). \end{aligned}$$
(3.9)

For any \(X\in {\mathfrak {u}}(n)\), denote \(X_{{\mathcal {M}}}\), \(X_M\), \(X_{{\mathbb {C}}^n}\) the vector fields associated with the moment maps \(\Lambda : {{\mathcal {M}}}\rightarrow \text {B}(n)\), \(\Lambda _L \Lambda _R: M \rightarrow \text {B}(n)\) and \({{\mathbf {b}}}: {\mathbb {C}}^n \rightarrow \text {B}(n)\), respectively. The formula of \(X_M\) is known [19], [29] and \(X_{{\mathbb {C}}^n}\) can be read off from Sect. 2.2. In fact, we have

$$\begin{aligned} {{\mathcal {L}}}_{X_M} g_R = [(b_L^{-1} X b_L)_{{\mathfrak {u}}(n)}, g_R], \quad {{\mathcal {L}}}_{X_M} b_R = \text {dress}_{(b_L^{-1} X b_L)_{{\mathfrak {u}}(n)}} (b_R) \end{aligned}$$
(3.10)

and

$$\begin{aligned} {{\mathcal {L}}}_{X_{{\mathbb {C}}^n}} w= X w, \quad {{\mathcal {L}}}_{X_{{\mathbb {C}}^n}} {{\mathbf {b}}}= \text {dress}_{X}({{\mathbf {b}}}). \end{aligned}$$
(3.11)

By using these, application of the definition (2.27) to the real and imaginary parts of the evaluation functions gives

$$\begin{aligned} {{\mathcal {L}}}_{X_{{\mathcal {M}}}} w^\alpha = \left( (b_L B_{\alpha -1})^{-1} X b_L B_{\alpha -1}\right) _{{\mathfrak {u}}(n)}\, w^\alpha ,\quad {{\mathcal {L}}}_{X_{{\mathcal {M}}}} B_\alpha = \text {dress}_{ (b_L^{-1} X b_L)_{{\mathfrak {u}}(n)}}(B_\alpha ). \nonumber \\ \end{aligned}$$
(3.12)

From the last two equalities, we obtain

$$\begin{aligned} {{\mathcal {L}}}_{X_{{\mathcal {M}}}} v(\alpha ) = (b_L^{-1} X b_L)_{{\mathfrak {u}}(n)}\, v(\alpha ). \end{aligned}$$
(3.13)

In conclusion, we see that the vector field \(X_{{\mathcal {M}}}\) is encoded by the formula (3.13) together with

$$\begin{aligned} {{\mathcal {L}}}_{X_{{\mathcal {M}}}} g_R = [(b_L^{-1} X b_L)_{{\mathfrak {u}}(n)}, g_R], \qquad {{\mathcal {L}}}_{X_{{\mathcal {M}}}} b_R = \text {dress}_{(b_L^{-1} X b_L)_{{\mathfrak {u}}(n)}} (b_R), \nonumber \\ \end{aligned}$$
(3.14)

which follow from (3.10) and the structure of \(\Lambda \) (3.1). The completion of the proof now requires checking that the formula (3.8) indeed gives a left-action of \({\mathrm{U}}(n)\) on \({{\mathcal {M}}}\), whose infinitesimal version reproduces the vector field \({{\mathcal {M}}}\) found above. These last steps require some lines but are fully straightforward, and thus, we omit further details. \(\square \)

Remark 3.5

The action (3.8) on \({{\mathcal {M}}}\) is called (extended) quasi-adjoint action, since if we forget the \(v(\alpha )\), then it becomes the quasi-adjoint action on M that goes back to [29]. At any fixed \((b_R, g_R)\), the map \(\eta \mapsto {\tilde{\eta }}\) that appears in (3.8) is a diffeomorphism on \({\mathrm{U}}(n)\), and thus, the quasi-adjoint action and the so-called obvious action have the same orbits. The obvious action, denoted \(A: {\mathrm{U}}(n) \times {{\mathcal {M}}}\rightarrow {{\mathcal {M}}}\), operates as follows:

$$\begin{aligned} A_g(g_R,b_R,v) := (g g_R g^{-1}, \text {Dress}_g(b_R), g v), \qquad \forall g\in {\mathrm{U}}(n),\, (g_R,b_R,v)\in {{\mathcal {M}}}, \nonumber \\ \end{aligned}$$
(3.15)

where

$$\begin{aligned} v:= (v(1), \dots , v(d)) \quad \hbox {and}\quad gv:= (g v(1),\dots , g v(d)). \end{aligned}$$
(3.16)

We are interested in the reduction of \({{\mathcal {M}}}\) defined by imposing the moment map constraint

$$\begin{aligned} \Lambda = e^\gamma {{\varvec{1}}}_n \quad \hbox {with a fixed constant}\quad \gamma >0. \end{aligned}$$
(3.17)

The corresponding reduced phase space is

$$\begin{aligned} {{\mathcal {M}}}_{\text {red}} = \Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)/{\mathrm{U}}(n). \end{aligned}$$
(3.18)

According to Remark 3.5, it does not matter whether we use the quasi-adjoint or the obvious action for taking the quotient.

Denote \(C^\infty ({{\mathcal {M}}})^{{\mathrm{U}}(n)}\) the \({\mathrm{U}}(n)\) invariant functions on \({{\mathcal {M}}}\). We may identify \(C^\infty ({{\mathcal {M}}}_\text {red})\) as the restriction of \(C^\infty ({{\mathcal {M}}})^{{\mathrm{U}}(n)}\) to the ‘constraint surface’ \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\). Then, \(C^\infty ({{\mathcal {M}}}_\text {red})\) is naturally a Poisson algebra, with bracket denoted \(\{\ ,\ \}_\text {red}\). This is obtained by using that the Poisson bracket of any two invariant functions is again invariant, and its restriction to \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) depends only on the restrictions of the two functions themselves. One sees this relying on the first class [25] character of the constraints that appear in (3.17).

Since the elements of \({\mathfrak {H}}\) (2.40) are \({\mathrm{U}}(n)\) invariant, they give rise to an Abelian Poisson subalgebra, \({\mathfrak {H}}_\text {red}\), of \(C^\infty ({{\mathcal {M}}}_\text {red})\). The flows of the elements of \({\mathfrak {H}}_\text {red}\) on \({{\mathcal {M}}}_\text {red}\) result by projection of the free flows (2.41), see Sect. 4.

Remark 3.6

By using that (3.17) can be written equivalently as \(\Lambda \Lambda ^\dagger = e^{2 \gamma } {{\varvec{1}}}_n\), it is not difficult to see that the triple \((g_R, b_R, v)\) belongs to \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) if and only if it satisfies

$$\begin{aligned} e^{2\gamma } g_R^{-1} (b_R b_R^\dagger ) g_R - (b_R b_R^\dagger ) = \sum _{\alpha =1}^d v(\alpha ) v(\alpha )^\dagger . \end{aligned}$$
(3.19)

We notice that the set of the triples \((g_R, b_Rb_R^\dagger , v)\) subject to (3.19) is a subset of the set \({{\mathcal {M}}}_{n,d,q}^\times \) defined in [10], which contains the elements \((X,Z, {{\mathcal {A}}}_1,\ldots , {{\mathcal {A}}}_d, {{\mathcal {B}}}_1,\ldots , {{\mathcal {B}}}_d)\) satisfying

$$\begin{aligned} q^{-1} X Z X^{-1} - Z = \sum _{\alpha =1}^d {{\mathcal {A}}}_\alpha {{\mathcal {B}}}_\alpha \end{aligned}$$
(3.20)

and the invertibility conditions

$$\begin{aligned} \Bigl (Z + \sum _{\alpha =1}^k {{\mathcal {A}}}_\alpha {{\mathcal {B}}}_\alpha \Bigr )\in {\mathrm{GL}}(n,{\mathbb {C}}), \quad \forall k=1,\dots , d, \end{aligned}$$
(3.21)

where q is a nonzero complex constant, \(X, Z \in {\mathrm{GL}}(n,{\mathbb {C}})\), \({{\mathcal {A}}}_\alpha \in {\mathbb {C}}^{n\times 1}\) and \({{\mathcal {B}}}_\alpha \in {\mathbb {C}}^{1\times n}\), for \(\alpha =1,\dots , d\). (These are equations (4.3) and (4.4) in [10].) It is known that if q is not a root of unity, then the action of \({\mathrm{GL}}(n,{\mathbb {C}})\) on \({{\mathcal {M}}}_{n,d,q}^\times \), defined by

$$\begin{aligned} g.(X,Z, {{\mathcal {A}}}_\alpha , {{\mathcal {B}}}_\alpha ) := (g X g^{-1}, g Z g^{-1}, g {{\mathcal {A}}}_\alpha , {{\mathcal {B}}}_\alpha g^{-1}), \quad \forall g\in {\mathrm{GL}}(n,{\mathbb {C}}), \nonumber \\ \end{aligned}$$
(3.22)

is free. As explained in [9], [10], this goes back to results in representation theory [11]. Direct comparison of (3.19) and (3.20), and of the corresponding group actions, shows that the \({\mathrm{U}}(n)\) action (3.15) on our ‘constraint surface’ \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) is free. In our case the invertibility conditions (3.21) hold in consequence of the following identities that generalize (3.7):

$$\begin{aligned} b_R b_R^\dagger + \sum _{\alpha =1}^k v(\alpha ) v(\alpha )^\dagger = \left( b_R {{\mathbf {b}}}(w^1)\cdots {{\mathbf {b}}}(w^k) \right) \left( b_R {{\mathbf {b}}}(w^1)\cdots {{\mathbf {b}}}(w^k)\right) ^\dagger . \nonumber \\ \end{aligned}$$
(3.23)

Because \({\mathrm{U}}(n)\) acts freely on it, \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) is an embedded submanifold of \({{\mathcal {M}}}\), and \({{\mathcal {M}}}_\text {red}\) (3.18) is a smooth symplectic manifold, whose Poisson algebra coincides with \(\left( C^\infty ({{\mathcal {M}}}_\text {red}), \{\ ,\ \}_\text {red}\right) \) presented in the preceding paragraph. Furthermore, since \(({{\mathcal {M}}}, \Omega _{{{\mathcal {M}}}})\) is actually a real-analytic symplectic manifold and the formulae of the \({\mathrm{U}}(n)\) action and the moment map are all given by real-analytic functions, \({{\mathcal {M}}}_\text {red}\) is also a real-analytic symplectic manifold. For the underlying general theory, the reader may consult [41] and also “Appendix ” in [18].

Remark 3.7

We will eventually prove the degenerate integrability of the reduced system by taking advantage of the following functions on \({{\mathcal {M}}}\):

$$\begin{aligned} I_{\alpha \beta }^k := \text {tr}\left( v(\alpha )v(\beta )^\dagger L^k \right) = v(\beta )^\dagger L^k v(\alpha ) \,, \quad 1\le \alpha ,\beta \le d,\,\, k\ge 0\,, \end{aligned}$$
(3.24)

where

$$\begin{aligned} L := b_R b_R^\dagger . \end{aligned}$$
(3.25)

The identity (2.44) shows that L transforms by conjugation, and therefore, these integrals of motion are invariant under the \({\mathrm{U}}(n)\) action (3.15) on \({{\mathcal {M}}}\). Their real and imaginary parts descend to real-analytic functions on the reduced phase space.

Solution of the Constraint in Terms of Q and Dressed Spins

Our fundamental task is to describe the set of \({\mathrm{U}}(n)\) orbits in the ‘constraint surface’ \(\Lambda ^{-1}(e^{\gamma } {{\varvec{1}}}_n)\). For this purpose, it will be convenient to label the points of \({{\mathcal {M}}}\) by \(g_R\), L and \(v= (v(1), \dots , v(d))\) using that L is given by (3.25). In the various arguments, we shall also employ alternative variables.

Since \(g_R\) can be diagonalized by conjugation, we see from the form of the \({\mathrm{U}}(n)\) action (3.8) (or (3.15)) that every \({\mathrm{U}}(n)\) orbit lying in the constraint surface intersects the set

$$\begin{aligned} {{\mathcal {M}}}_0 := \Lambda ^{-1}( e^\gamma {{\varvec{1}}}_n) \cap \Xi _R^{-1}({\mathbb {T}}^n), \end{aligned}$$
(3.26)

where \({\mathbb {T}}^n\) is the subgroup of diagonal matrices in \({\mathrm{U}}(n)\). Below,

$$\begin{aligned} Q:= \text {diag}(Q_1,\dots , Q_n) \end{aligned}$$
(3.27)

stands for an element of \({\mathbb {T}}^n\), and \({\text {Ad}}_{Q^{-1}}\) denotes conjugation by \(Q^{-1}\). For any \(\gamma \in {\mathbb {R}}^*\), \(\left( e^{2\gamma } {\text {Ad}}_{Q^{-1}} - {\text {id}}\right) \) is an invertible linear operator on \(\mathfrak {gl}(n,{\mathbb {C}})\), which preserves the subspace of Hermitian matrices. After this preparation, we present a useful characterization of \({{\mathcal {M}}}_0\).

Proposition 3.8

If \((Q,L,v) \in {{\mathcal {M}}}_0\) (3.26), then L can be expressed in terms of Q and v as follows:

$$\begin{aligned} L = \left( e^{2\gamma } \mathrm{Ad}_{Q^{-1}} - {\text {id}}\right) ^{-1} \left( \sum _{\alpha =1}^d v(\alpha ) v(\alpha )^\dagger \right) . \end{aligned}$$
(3.28)

For the matrix elements of L, this gives

$$\begin{aligned} L_{ij} = \frac{F_{ij}}{ e^{2\gamma } Q_j Q_i^{-1} - 1} \quad \hbox {with}\quad F:=\sum _{\alpha =1}^d v(\alpha ) v(\alpha )^\dagger . \end{aligned}$$
(3.29)

Conversely, if the Hermitian matrix L given by the formula (3.28) is positive definite, then \((Q,L,v)\in {{\mathcal {M}}}_0\).

Proof

If \(g_R= Q \in {\mathbb {T}}^n\), then we have

$$\begin{aligned} b_L Q^{-1} = Q^{-1} Q b_L Q^{-1} = Q^{-1} b_R^{-1}, \end{aligned}$$
(3.30)

showing that \(b_L = Q^{-1} b_R^{-1} Q\). Therefore, on \({{\mathcal {M}}}_0\) the moment map \(\Lambda \) (3.1) reads

$$\begin{aligned} \Lambda (Q, L, v) = Q^{-1} b_R^{-1} Q b_R {{\mathbf {b}}}(w^1) {{\mathbf {b}}}(w^2) \cdots {{\mathbf {b}}}(w^d), \end{aligned}$$
(3.31)

where \(b_R\) and the \(w^\alpha \) are viewed as functions of L and v, given by the invertible relations of Eq. (3.25) and Definition 3.1. We substitute this into the following equivalent form of the moment map constraint (3.17),

$$\begin{aligned} \Lambda (Q, L, v) \Lambda (Q, L, v)^\dagger = e^{2\gamma } {{\varvec{1}}}_n, \end{aligned}$$
(3.32)

and thus obtain the requirement

$$\begin{aligned} (b_R b_1 b_2 \cdots b_d) ( b_Rb_1 b_2 \cdots b_d)^\dagger = e^{2\gamma } Q^{-1} L Q \quad \hbox {with}\quad b_\alpha = {{\mathbf {b}}}(w^\alpha ). \end{aligned}$$
(3.33)

By using Lemma 3.3 and the definitions of L and F, this in turn is equivalent to

$$\begin{aligned} e^{2 \gamma } Q^{-1} L Q - L = F. \end{aligned}$$
(3.34)

It follows that if \((Q,L,v)\in {{\mathcal {M}}}_0\), then L is given by the formula (3.28).

To deal with the converse statement, notice that L as given by the formula (3.28) is Hermitian and automatically satisfies (3.34), but its positive definiteness is a non-trivial condition on the pair (Qv). Suppose that L (3.28) is positive definite. Then, there exists a unique \(b_R\in \text {B}(n)\) for which \(L = b_R b_R^\dagger \). Defining \(v^\alpha := b_R^{-1} v(\alpha )\) (cf. (3.3)), we can convert (3.34) into

$$\begin{aligned} e^{2\gamma } b_R^{-1} Q^{-1} b_R Q (b_R^{-1} Q^{-1} b_R Q)^\dagger = {{\varvec{1}}}_n + \sum _{\alpha =1}^d v^\alpha (v^\alpha )^\dagger . \end{aligned}$$
(3.35)

Then, there exists unique \(w^1,\dots , w^d\) from \({\mathbb {C}}^n\) for which (3.3) holds, and by (3.2) and (3.7) these variables satisfy

$$\begin{aligned} {{\varvec{1}}}_n + \sum _{\alpha =1}^d v^\alpha (v^\alpha )^\dagger = ({{\mathbf {b}}}(w^1)\cdots {{\mathbf {b}}}(w^d)) ( {{\mathbf {b}}}(w^1) \cdots {{\mathbf {b}}}(w^d))^\dagger . \end{aligned}$$
(3.36)

Inserting into (3.35), and using (3.31), we see that (3.35) implies the constraint Eq. (3.32), whereby the proof is complete. \(\square \)

We have the following consequence of Proposition 3.8.

Corollary 3.9

Let L(Qv) be given by (3.28) and define the set

$$\begin{aligned} {{\mathcal {P}}}_0 := \{ (Q,v) \in {\mathbb {T}}^n \times {\mathbb {C}}^{n\times d}\mid L(Q,v) \,\hbox {is positive definite}\,\}. \end{aligned}$$
(3.37)

The formula (3.28) establishes a bijection between \({{\mathcal {M}}}_0\) (3.26) and \({{\mathcal {P}}}_0\), which is an open subset of \({\mathbb {T}}^n \times {\mathbb {C}}^{n\times d}\).

Let us call Q regular if it belongs to

$$\begin{aligned} {\mathbb {T}}^n_\text {reg}:= \{ Q = \text {diag}(Q_1,\dots , Q_n)\mid Q_i\in {\mathrm{U}}(1),\, Q_i \ne Q_j \,\,\, \forall i\ne j\}. \end{aligned}$$
(3.38)

Define

$$\begin{aligned} {{\mathcal {M}}}_0^\text {reg}:= \{ (Q, L(Q,v), v)\in {{\mathcal {M}}}_0 \mid Q \in {\mathbb {T}}^n_\text {reg}\}, \end{aligned}$$
(3.39)

where L(Qv) is specified by (3.28). Notice that any \(g \in {\mathrm{U}}(n)\) for which \(A_g\) (3.15) maps an element of \({{\mathcal {M}}}_0^\text {reg}\) to \({{\mathcal {M}}}_0^\text {reg}\) must belong to the normalizer \({{\mathcal {N}}}(n)\) of \({\mathbb {T}}^n\) inside \({\mathrm{U}}(n)\). Therefore, the quotient of the regular part of the constraint surface by \({\mathrm{U}}(n)\), denoted \({{\mathcal {M}}}^\text {reg}_\text {red}\), can be identified as

$$\begin{aligned} {{\mathcal {M}}}^\text {reg}_\text {red}= {{\mathcal {M}}}_0^\text {reg}/{{\mathcal {N}}}(n), \end{aligned}$$
(3.40)

where the quotient refers to the restriction of the obvious \({\mathrm{U}}(n)\) action (3.15) to the subgroup \({{\mathcal {N}}}(n) < {\mathrm{U}}(n)\).

Let us call \(Q\in {\mathbb {T}}^n\) admissible if \(Q\in \Xi _R^{-1}\left( \Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\right) \). In the next section, we present an alternative procedure for solving the moment map constraint (3.17), which will show that all elements of \({\mathbb {T}}^n_\text {reg}\) are admissible.

Remark 3.10

Equation (3.34) implies \((e^{2\gamma } - 1) \text {tr}(L) = \text {tr}(F)\), and this can hold for a nonzero real \(\gamma \) only if \(\gamma >0\) and \(\text {tr}(F)>0\), since L must be positive definite and \(\text {tr}(F) \ge 0\) by the definition (3.29). This is why we assumed that \(\gamma >0\). It would be desirable to describe the elements of the set \({{\mathcal {P}}}_0\) (3.37) explicitly. For \(d=1\) the solution of this problem can be read off from [19]. On account of the next two observations, we expect that the structure of \({{\mathcal {P}}}_0\) is very different for \(d<n\) and for \(d\ge n\). First, let us notice that \(Q={{\varvec{1}}}_n\) is not admissible if \(d<n\), since in this case the rank of L given by the formula (3.28) is at most d, while the rank of any positive definite L is n. Second, note that if \(d\ge n\), then we can arrange to have \(F={{\varvec{1}}}_n\) by suitable choice of v. Let \(v_0\) be such a choice. Then, \(L({{\varvec{1}}}_n, v_0)\) is a positive multiple of \({{\varvec{1}}}_n\), and therefore, there is an open neighborhood of \(({{\varvec{1}}}_n, v_0)\) in \({\mathbb {T}}^n \times {\mathbb {C}}^{n\times d}\) that belongs to \({{\mathcal {P}}}_0\).

Solution of the Constraint in Terms of Qp and Primary Spins

Now we return to using the variables \(g_R, b_R\) and W (2.35) for labeling the points of \({{\mathcal {M}}}\). We can uniquely decompose every element \(b\in \text {B}(n)\) as the product of a diagonal matrix, \(b_0\), and an upper triangular matrix, \(b_+\), with unit diagonal. Applying this to \(b= b_R\), we write

$$\begin{aligned} b_R= b_0 b_+, \end{aligned}$$
(3.41)

and introduce also

$$\begin{aligned} S(W):= {{\mathbf {b}}}(w^1) {{\mathbf {b}}}(w^2) \cdots {{\mathbf {b}}}(w^d)=: S_0(W) S_+(W). \end{aligned}$$
(3.42)

The moment map constraint on \({{\mathcal {M}}}_0\) (3.26) reads

$$\begin{aligned} \Lambda (Q,b_R, W) = Q^{-1} b_R^{-1} Q b_R S(W) = Q^{-1} b_+^{-1} Q b_+ S_0(W) S_+(W) = e^\gamma {{\varvec{1}}}_n. \nonumber \\ \end{aligned}$$
(3.43)

Since \(b_0\) drops out from the formula of \(\Lambda \), it is left arbitrary, and we parametrize it as

$$\begin{aligned} b_0 = e^p \quad \hbox {with}\quad p= \text {diag}(p_1,\ldots , p_n), \quad p_i \in {\mathbb {R}}. \end{aligned}$$
(3.44)

A crucial observation is that (3.43) can be separated according to the diagonal and strictly upper-triangular parts, since it is equivalent to the two requirements

$$\begin{aligned} S_0(W) = e^\gamma {{\varvec{1}}}_n \end{aligned}$$
(3.45)

and

$$\begin{aligned} Q^{-1} b_+^{-1} Q b_+ S_+(W) = {{\varvec{1}}}_n. \end{aligned}$$
(3.46)

The constraint (3.45) is responsible for a reduction of the primary spin variables. Next we make a little detour and present a general analysis of such reductions.

Let us introduce the map \(\phi : {\mathbb {C}}^{n\times d} \rightarrow {\mathfrak {b}}(n)_0\) by writing

$$\begin{aligned} S_0(W):= \exp (\phi (W)), \end{aligned}$$
(3.47)

and notice from Remark A.5 that \(\phi \) is the moment map for the ordinary Hamiltonian action of \({\mathbb {T}}^n\) on the symplectic manifold \(({\mathbb {C}}^{n\times d}, \Omega _{{\mathcal {W}}})\) of the primary spins. Here, the dual of the Lie algebra of the torus \({\mathbb {T}}^n< {\mathrm{U}}(n)\) is identified with the space \({\mathfrak {b}}(n)_0\) of real diagonal matrices. The torus action in question is given by

$$\begin{aligned} \tau \cdot (w^1,\dots , w^d) = (\tau w^1,\dots , \tau w^d), \qquad \forall \tau \in {\mathbb {T}}^n. \end{aligned}$$
(3.48)

Taking any moment map value from the range of \(\phi \),

$$\begin{aligned} \Gamma := \text {diag}(\gamma _1,\dots , \gamma _n), \end{aligned}$$
(3.49)

we define the reduced space of primary spins:

$$\begin{aligned} {\mathbb {C}}^{n\times d}_\text {red}(\Gamma ):= \phi ^{-1}(\Gamma )/{\mathbb {T}}^n. \end{aligned}$$
(3.50)

Proposition 3.11

The moment map \(\phi : {\mathbb {C}}^{n\times d} \rightarrow {\mathfrak {b}}(n)_0\) defined by (3.47) with (3.42) is proper, i.e., the inverse image of any compact set is compact. Fixing any moment map value \(\Gamma \) for which \(\gamma _j>0\) for all j, the reduced spin–space (3.50) is a smooth, compact and connected symplectic manifold of dimension \(2n(d-1)\).

Proof

We first prove that the map \(\phi \) is proper. Since the compact sets of Euclidean spaces are the bounded and closed sets, and since \(\phi \) is continuous, it is enough to show that the inverse image of any bounded subset of \({\mathfrak {b}}(n)_0\simeq {\mathbb {R}}^n\) is a bounded subset of \({\mathbb {C}}^{n\times d}\). Due to the definition of \(\phi \) and Eq. (2.31), the formula of \(\phi =\text {diag}(\phi _1,\dots , \phi _n)\) is determined by the equality

$$\begin{aligned} \exp (2 \phi _j(W)) = \prod _{\alpha =1}^d \frac{{{\mathcal {G}}}_j(w^\alpha )}{{{\mathcal {G}}}_{j+1}(w^\alpha )} = \prod _{\alpha =1}^d \left[ 1 + \frac{|w_j^\alpha |^2}{{{\mathcal {G}}}_{j+1}(w^\alpha )}\right] . \end{aligned}$$
(3.51)

The second equality shows that \(\phi _j(W)\ge 0\). On the other hand, using the first equality and that \({{\mathcal {G}}}_{n+1}=1\), we see that

$$\begin{aligned} W\in \phi ^{-1}(\text {diag}(\gamma _1,\dots , \gamma _n)) \end{aligned}$$
(3.52)

if and only if

$$\begin{aligned} \prod _{\alpha =1}^d {{\mathcal {G}}}_j(w^\alpha ) = \exp ( 2\sum _{k=j}^n \gamma _k), \qquad \forall j=1,\dots , n. \end{aligned}$$
(3.53)

Now, if \(\text {diag}(\gamma _1,\dots , \gamma _n)\) is from a bounded set, then \( \sum _{k=1}^n \gamma _k \le C\) with some constant C. By using this and the \(j=1\) special case of (3.53), we obtain

$$\begin{aligned} 1 + \sum _{\alpha =1}^d |w^\alpha |^2 \le \prod _{\alpha =1}^d (1 + |w^\alpha |^2)= \prod _{\alpha =1}^d {{\mathcal {G}}}_1(w^\alpha ) \le e^{2C}, \end{aligned}$$
(3.54)

which implies that the inverse image of any bounded set is bounded.

If \(\gamma _j >0\) for all j, then we see from the formula (3.51) that for any \(W\in \Phi ^{-1}(\Gamma )\) and for each \(1\le j \le n\) there must exist an index \(1\le \alpha (j) \le d\) such that

$$\begin{aligned} w_j^{\alpha (j)}\ne 0. \end{aligned}$$
(3.55)

This implies immediately that the action (3.48) of \({\mathbb {T}}^n\) is free on \(\phi ^{-1}(\Gamma )\), and therefore \({\mathbb {C}}^{n\times d}_\text {red}(\Gamma )\) (3.50) is a smooth symplectic manifold of dimension \(2n(d-1)\). Since the moment map \(\phi \) is proper, all its fibers \(\phi ^{-1}(\Gamma )\) are compact, and by Theorem 4.1 in [26] they are also connected. Hence, \({\mathbb {C}}^{n\times d}_\text {red}(\Gamma )\) is also compact and connected. \(\square \)

Remark 3.12

If \(\gamma _j >0\) for all j, then \({\mathbb {C}}^{n\times d}_\text {red}(\Gamma )\) is actually a real-analytic symplectic manifold. To cover it with charts, for any map \(\mu : \{1,\dots , n\} \rightarrow \{ 1,\dots , d\}\) we introduce the set

$$\begin{aligned} X(\mu ):= \{ W \in \phi ^{-1}(\Gamma )\mid w_j^{\mu (j)} \ne 0,\,\forall j\}. \end{aligned}$$
(3.56)

Then, the reduced spin-space is the union of the open subsets \(Y(\mu ):= X(\mu )/{\mathbb {T}}^n\), and a model of \(Y(\mu )\) is provided by

$$\begin{aligned} Z(\mu ):= \{ W \in \phi ^{-1}(\Gamma )\mid w_j^{\mu (j)} >0,\, \forall j\}. \end{aligned}$$
(3.57)

One can specify coordinates on \(Z(\mu )\) by solving the constraints (3.53) for the \(w_j^{\mu (j)}\) in terms of the remaining free variables, the \(w_j^\alpha \) with \(\alpha \ne \mu (j)\), which take their values in a certain open subset of \({\mathbb {C}}^{n(d-1)}\). It is an interesting exercise to fill out the details and to also write down the reduced symplectic form by using these charts.

If \(d=1\), then the reduced spin-space consists of a single point. This is also true in the trivial case for which \(\gamma _j = 0\) for all j. If some of the \(\gamma _j\) are zero and the others are positive, then the moment map constraint \(\phi (W)=\Gamma \) leads to a stratified symplectic space. Finally, note that for the case corresponding to Eq. (3.45) \(\gamma _j = \gamma >0\) for all j.

Now returning to our main problem, it is useful to recast (3.46) in the form

$$\begin{aligned} b_+ S_+(W) = Q^{-1} b_+ Q. \end{aligned}$$
(3.58)

By using the principal gradation of \(n\times n\) matrices, this equation can be solved recursively for \(b_+\) if \(S_+(W)\) and Q are given, with Q regular. In fact, the following lemma is obtained by a word-by-word application of the arguments of Section 5 in [15]; hence, we omit the proof.

Lemma 3.13

Suppose that \(S_+=S_+(W)\) and Q are given, with \(Q\in {\mathbb {T}}^n_\mathrm{reg}\). Then, Eq. (3.58) admits a unique solution for \(b_+\), denoted \(b_+(Q,W)\). Using the notation

$$\begin{aligned} {{\mathcal {I}}}^{a, a+ j} = \frac{1}{Q_{a+j} Q_a^{-1} - 1}, \quad a=1,\dots , n-1, \end{aligned}$$
(3.59)

and placing the matrix indices in the upstairs position, we have

$$\begin{aligned} b_+^{a,a+1} = {{\mathcal {I}}}^{a, a+1} S_+^{a,a+1}, \end{aligned}$$
(3.60)

and for \(k=2,\dots , n-a\) we have

$$\begin{aligned} b_+^{a, a+k} = {{\mathcal {I}}}^{a, a+k} S_+^{a, a+k} + \sum _{\begin{array}{c} m=2,\dots , k\\ (i_1,\dots , i_m)\in {\mathbb {N}}^m\\ i_1 + \dots + i_m = k \end{array} } \prod _{\alpha =1}^m {{\mathcal {I}}}^{a, a + i_1 +\dots + i_\alpha } S_+^{a + i_1 + \dots + i_{\alpha -1}, a+ i_1 + \dots + i_\alpha }. \nonumber \\ \end{aligned}$$
(3.61)

Now we restrict ourselves to the regular part of \({{\mathcal {M}}}_0\), stressing that it is defined without reference to any particular parametrization:

$$\begin{aligned} {{\mathcal {M}}}_0^\text {reg}\equiv \Lambda ^{-1}( e^\gamma {{\varvec{1}}}_n) \cap \Xi _R^{-1}({\mathbb {T}}^n_\text {reg}). \end{aligned}$$
(3.62)

Any gauge transformation that maps an element of \({{\mathcal {M}}}_0^\text {reg}\) to \({{\mathcal {M}}}_0^\text {reg}\) is given by the obvious action (3.15) of the normalizer \({{\mathcal {N}}}(n)\) of \({\mathbb {T}}^n\) inside \({\mathrm{U}}(n)\). The normalizer has the normal subgroup \({\mathbb {T}}^n\), and the corresponding factor group is the permutation group

$$\begin{aligned} S_n = {{\mathcal {N}}}(n)/{\mathbb {T}}^n. \end{aligned}$$
(3.63)

Consequently, we have

$$\begin{aligned} {{\mathcal {M}}}_\text {red}^\text {reg}= {{\mathcal {M}}}_0^\text {reg}/{{\mathcal {N}}}(n) = ({{\mathcal {M}}}_0^\text {reg}/{\mathbb {T}}^n)/S_n. \end{aligned}$$
(3.64)

It is plain that \({{\mathcal {M}}}_\text {red}^\text {reg}\) is a dense, open subset of the reduced phase space, and the above consecutive quotients show that \({{\mathcal {M}}}_0^\text {reg}/{\mathbb {T}}^n\) is an \(S_n\) covering spaceFootnote 4 of this dense open subset.

Theorem 3.14

By solving the moment map constraint for \(b_R\) in the form \(b_R= e^p b_+(Q,W)\) as explained above, the manifold \({{\mathcal {M}}}_0^\mathrm{reg}\) (3.62) can be identified with the model space

$$\begin{aligned} {\tilde{{{\mathcal {P}}}}}_0^\mathrm{reg}:={\mathbb {T}}^n_\mathrm{reg} \times {\mathfrak {b}}(n)_0 \times \phi ^{-1}(\Gamma )= \{ (Q, p, W)\mid Q\in {\mathbb {T}}^n_\mathrm{reg},\,\, p\in {\mathfrak {b}}(n)_0,\,\, W\in \phi ^{-1}(\Gamma ) \},\nonumber \\ \end{aligned}$$
(3.65)

where \(\Gamma = \gamma {{\varvec{1}}}_n\). Utilizing this model, the covering space \({{\mathcal {M}}}_0^\mathrm{reg}/{\mathbb {T}}^n\) of the regular part of the reduced phase space becomes identified with the symplectic manifold

$$\begin{aligned} T^*{\mathbb {T}}^n_\mathrm{reg} \times {\mathbb {C}}^{n\times d}_\mathrm{red}(\Gamma ) \end{aligned}$$
(3.66)

equipped with its natural product symplectic structure.

Proof

The restriction of the action (3.15) to \({\mathbb {T}}^n\) translates into the action

$$\begin{aligned} A_\tau (Q,p, W) = (Q,p,\tau \cdot W), \qquad \tau \in {\mathbb {T}}^n, \end{aligned}$$
(3.67)

on the model space \({\tilde{{{\mathcal {P}}}}}_0^\text {reg}\), from which we obtain the identification \({{\mathcal {M}}}_0^\text {reg}/{\mathbb {T}}^n \simeq T^* {\mathbb {T}}^n_\text {reg}\times {\mathbb {C}}^{n\times d}_\text {red}(\Gamma )\) at the level of manifolds. Let \(\xi _1: {\tilde{{{\mathcal {P}}}}}_0^\text {reg}\rightarrow {{\mathcal {M}}}\) and \(\xi _2: \phi ^{-1}(\Gamma ) \rightarrow {\mathbb {C}}^{n\times d}\) denote the natural inclusions, and write \(Q_j = e^{{\mathrm{i}}q_j}\). Then, a simple calculation gives

$$\begin{aligned} \xi _1^* (\Omega _{{\mathcal {M}}}) = \sum _{j=1}^n dp_j \wedge dq_j + \xi _2^* (\Omega _{{\mathcal {W}}}), \end{aligned}$$
(3.68)

which proves the claimed identification at the level of symplectic manifolds. \(\square \)

Corollary 3.15

The dense open submanifold \({{\mathcal {M}}}_\mathrm{red}^\mathrm{reg}\subseteq {{\mathcal {M}}}_\mathrm{red}\) is connected, and consequently, \({{\mathcal {M}}}_\mathrm{red}\) is also connected.

Proof

Since \(\phi ^{-1}(\Gamma )\) is connected by Proposition 3.11, the connected components of \({\tilde{{{\mathcal {P}}}}}_0^\text {reg}\) correspond to the connected components of \({\mathbb {T}}^n_\text {reg}\). It is well known (see, e.g., “Appendix ”) that any two connected components of \({\mathbb {T}}^n_\text {reg}\) are related by permutations. Thus, \({{\mathcal {M}}}_\text {red}^\text {reg}\) is the continuous image of a single connected component of \({\tilde{{{\mathcal {P}}}}}_0^\text {reg}\), implying its connectedness. The proof is finished by recalling that if a dense open subset of a topological space is connected, then the space itself is connected. \(\square \)

Remark 3.16

In this long remark, we explain how the (trigonometric real form of the) spin Sutherland model of Gibbons and Hermsen [24] can be obtained from our construction via a scaling limit. For this, we introduce a positive parameter \(\epsilon \) and replace the variables p by \(\epsilon \, p\) and W by \(\epsilon ^{\frac{1}{2}} W\), while keeping Q unchanged. The formulae (2.31) imply

$$\begin{aligned} {{\mathbf {b}}}( \epsilon ^{\frac{1}{2}} w)_{kk}= 1 + \frac{1}{2} \epsilon |w_k|^2 + \text {o}(\epsilon ),\,\, \forall k \quad \hbox {and}\quad {{\mathbf {b}}}(\epsilon ^{\frac{1}{2}} w)_{ij} = \epsilon w_i \overline{w}_j + \text {o}(\epsilon ), \,\, \forall i<j.\nonumber \\ \end{aligned}$$
(3.69)

By using this, we see that the matrix \(b=b_+\) in (3.41), given in explicit form by Lemma 3.13, has the expansion

$$\begin{aligned} b(Q,p, \epsilon ^{\frac{1}{2}} W)_{ij} = \epsilon (Q_j Q_i^{-1} -1)^{-1} \sum _{\alpha =1}^d w_i^\alpha \overline{w}_j^\alpha + \text {o}(\epsilon ), \quad \forall i<j. \end{aligned}$$
(3.70)

Then, for \(L= b_R b_R^\dagger \) with \(b_R= \exp ({\epsilon p}) b(Q,\epsilon p, \epsilon ^{\frac{1}{2}} W)\), we find

$$\begin{aligned} \text {tr}(L^{\pm 1}) = n \pm 2 \epsilon \, \text {tr}(p) + 2 \epsilon ^2 \text {tr}( p^2) + \epsilon ^2 \sum _{i<j} \frac{ \vert ( w_i^\bullet , w_j^\bullet )\vert ^2 }{ \vert Q_j Q_i^{-1} - 1\vert ^2} + \text {o}(\epsilon ^2), \end{aligned}$$
(3.71)

where \(w_i^{\bullet }\in {\mathbb {C}}^d\) with components \(w_i^\alpha \), and \((w_i^\bullet , w_j^\bullet ):= \sum _{\alpha =1}^d w_i^\alpha \overline{w}_j^\alpha \). Therefore, we obtain

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \frac{1}{8 \epsilon ^2}(\text {tr}(L) + \text {tr}(L^{-1}) -2n) = \frac{1}{2} \text {tr}( p^2) + \frac{1}{32} \sum _{i\ne j} \frac{\vert ( w_i^\bullet , w_j^\bullet )\vert ^2 }{ \sin ^2\frac{q_i-q_j}{2}}, \end{aligned}$$
(3.72)

which is just the standard Hamiltonian of the (real, trigonometric) Gibbons–Hermsen model.

Replacing \(\gamma \) by \(=\epsilon \gamma \) and taking the limit, the residual constraint (3.45) gives \( (w_j^\bullet , w_j^\bullet ) = 2 \gamma \). Then, rescaling not only the variables but also the symplectic form (3.68), one gets

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \epsilon ^{-1} \left( \xi ^*\Omega _{{{\mathcal {M}}}}\right) = \sum _{j=1}^n dp_j \wedge dq_j + \frac{{\mathrm{i}}}{2} \sum _{j=1}^n \sum _{\alpha =1}^d d w_j^\alpha \wedge d\overline{w}_j^\alpha , \end{aligned}$$
(3.73)

which reproduces the symplectic form of the Gibbons–Hermsen model.

It is known [19] that the standard spinless RS Hamiltonian [50] can be derived as the reduction of \(\text {tr}(L) + \text {tr}(L^{-1})\) in the \(d=1\) case. For \(d\ge 2\), we shall show (see Corollary 4.3 and Corollary 5.9) that the Hamiltonian of the (real, trigonometric) spin RS model of Krichever of Zabrodin [32] is the reduction of \(\text {tr}(L)\). As was already discussed in Introduction, the term chiral spin RS model could have been a more fitting name for the model of [32], but we follow the literature in dropping ‘chiral’ in this context.

In this subsection, we have derived an almost complete description of the reduced system. We have established that \(T^* {\mathbb {T}}^n_\text {reg}\times {\mathbb {C}}^{n\times d}_\text {red}(\Gamma )\) is an \(S_n\) covering space of a dense, open subset of the reduced phase space, and we can write down the Hamiltonians \(\text {tr}(L^k)\) by using the explicit formula \(b_R = e^p b_+(Q,W)\). Why is the paper not finished at this stage? Well, one reason is that although Qp and W are very nice variables for presenting the reduced symplectic form, they are not the ones that feature in the Krichever–Zabrodin equations of motion, which we wish to reproduce in our setting. In fact, the usage of the dressed spins \(v(\alpha )\) (3.3) will turn out indispensable for this purpose. (Notationwise, we took this into account already in Eq. (1.8).) Another, closely related, reason is that the action of permutations is practically intractable in terms of the primary spins.Footnote 5 More precisely, the action on the components of Q is the obvious one, but on p and W it is known only in an implicit manner, via the realization of these variables as functions of \(L=b_R b_R^\dagger \) and the dressed spins, on which the full \({\mathrm{U}}(n)\) action, and thus also the permutation action, is governed by the simple formula (3.15). In short, both the formula \(b_+(Q,W)\) and the change of variables from QpW to Q and dressed spins \(v(\alpha )\) are complicated, and for some purposes the latter will prove to be more convenient variables.

The Reduced Equations of Motion

Below, we first present a characterization of the projection of the Hamiltonian vector fields of arbitrary elements of \({\mathfrak {H}}\) (2.40) to the dense open submanifold \({{\mathcal {M}}}^\text {reg}_\text {red}\) (3.40) of the reduced phase space \({{\mathcal {M}}}_\text {red}\) (3.18). Then, we reproduce the trigonometric real form of the equations of motion (1.2) of [32] as the simplest special case of the reduced dynamics.

As in Sect. 3.2, we parametrize the points of \({{\mathcal {M}}}\) by \(g_R\), v and \(L = b_R b_R^\dagger \). For any \(h\in C^\infty (\text {B}(n))^{{\mathrm{U}}(n)}\) we put

$$\begin{aligned} {{\mathcal {V}}}(L):= D h(b_R). \end{aligned}$$
(4.1)

Denoting the Hamiltonian vector field of \(H= \Lambda _R^*(h)\) by \(X_H\) and viewing \(g_R\), \(v(\alpha )\) and L as evaluation functions, in correspondence to (2.41), we have

$$\begin{aligned} X_H[g_R] = {{\mathcal {V}}}(L) g_R,\quad X_H[v(\alpha )]=0, \quad X_H[L]=0. \end{aligned}$$
(4.2)

It is clear that \(X_H\) admits a well-defined projection on \({{\mathcal {M}}}_\text {red}\), which encodes the reduced dynamics. Of course, one may add any infinitesimal gauge transformation to the vector field \(X_H\) without modifying its projection on \({{\mathcal {M}}}_\text {red}\), i.e., instead of \(X_H\) one may equally well consider any \(Y_H\) of the form

$$\begin{aligned}&Y_H[g_R] = {{\mathcal {V}}}(L) g_R + [Z(g_R, L, v), g_R],\nonumber \\&Y_H[v(\alpha )] = Z(g_R, L, v) v(\alpha ), \nonumber \\&Y_H[L] = [Z(g_R, L, v), L], \end{aligned}$$
(4.3)

with arbitrary \(Z(g_R, L, v)\in {\mathfrak {u}}(n)\). It is also clear that one may use the restriction of \(Y_H\) to \({{\mathcal {M}}}_0\) (3.26) for determining the projection, and Z can be chosen in such a manner to guarantee the tangency of the restricted vector field to \({{\mathcal {M}}}_0\).

Let us consider the vector space decomposition

$$\begin{aligned} {\mathfrak {u}}(n)= {\mathfrak {u}}(n)_0 + {\mathfrak {u}}(n)_\perp , \end{aligned}$$
(4.4)

where \({\mathfrak {u}}(n)_0\) and \({\mathfrak {u}}(n)_\perp \) consist of diagonal and off-diagonal matrices, respectively. Accordingly, for any \(T \in {\mathfrak {u}}(n)\) we have

$$\begin{aligned} T = T_0 + T_\perp , \qquad T= {\mathfrak {u}}(n)_0,\,\, T_\perp \in {\mathfrak {u}}(n)_\perp . \end{aligned}$$
(4.5)

Using \({\text {Ad}}_Q(T)= Q T Q^{-1}\), the restriction of the operator \(({\text {Ad}}_Q - {\text {id}})\) to \({\mathfrak {u}}(n)_\perp \) is invertible, and we define

$$\begin{aligned} K(Q,L):= \left( ({\text {Ad}}_Q - {\text {id}})\vert _{{\mathfrak {u}}(n)_\perp }\right) ^{-1} {{\mathcal {V}}}(L)_\perp . \end{aligned}$$
(4.6)

More explicitly, setting \(Q = \exp ({\mathrm{i}}q)\), we have \(K_{kk}=0\) and

$$\begin{aligned} K(Q,L)_{kl} = - \frac{1}{2} {{\mathcal {V}}}(L)_{kl} - \frac{{\mathrm{i}}}{2} {{\mathcal {V}}}(L)_{kl} \,\cot \left( \frac{q_k - q_l}{2}\right) , \quad \forall k\ne l. \end{aligned}$$
(4.7)

Proposition 4.1

For any \(H\in {\mathfrak {H}}\) (2.40), applying the previous notations, the following formulae yield a vector field \(Y_H^0\) on \({{\mathcal {M}}}_0^\mathrm{reg}\) (3.39),

$$\begin{aligned}&Y_H^0[Q] = {{\mathcal {V}}}(L)_0 Q ,\nonumber \\&Y_H^0[v(\alpha )] =\bigl ( K(Q,L) + {{\mathcal {Z}}}(Q, L, v)\bigr ) v(\alpha ), \nonumber \\&Y_H^0[L] = [K(Q,L) + {{\mathcal {Z}}}(Q, L, v), L], \end{aligned}$$
(4.8)

for any \({{\mathcal {Z}}}(Q, L, v)\in {\mathfrak {u}}(n)_0\) (4.4), with \(L=L(Q,v)\) given by (3.28). The vector field \(Y_H^0\) admits a well-defined projection on \({{\mathcal {M}}}_\mathrm{red}^\mathrm{reg} \subset {{\mathcal {M}}}_\mathrm{red}\) (3.40), which coincides with the corresponding restriction of the projection of the Hamiltonian vector field \(X_H\) (4.2) to \({{\mathcal {M}}}_\mathrm{red}\).

Proof

As a consequence of \(({\text {Ad}}_Q -{\text {id}}) K(Q,L) = {{\mathcal {V}}}(L)_\perp \), we have the identity

$$\begin{aligned} {{\mathcal {V}}}(L) Q + [ K(Q,L) ,Q] = {{\mathcal {V}}}(L)_0 Q. \end{aligned}$$
(4.9)

This shows that \(Y_H^0\) is obtained by restricting \(Y_H\) (4.3) to \({{\mathcal {M}}}_0^\text {reg}\), where

$$\begin{aligned} Z(Q,L,v) = K(Q,L) + {{\mathcal {Z}}}(Q,L,v). \end{aligned}$$
(4.10)

This choice of Z guarantees that the restricted vector field is tangent to \({{\mathcal {M}}}_0^\text {reg}\). The fact that \({{\mathcal {Z}}}\) is left undetermined reflects the residual \({{\mathcal {N}}}(n)\) gauge transformations acting on \({{\mathcal {M}}}_0^\text {reg}\). \(\square \)

Remark 4.2

Only the first two relations in (4.8) are essential, since the third one follows from them via the formula (3.28) of \(L=L(Q,v)\). Now take an initial value \((Q^0, L^0, v^0)\in {{\mathcal {M}}}_0^\text {reg}\) and \(\epsilon >0\) (\(\epsilon =\infty \) is allowed) such that

$$\begin{aligned} g_R(t) = \exp (t {{\mathcal {V}}}(L^0)) Q^0 \in {\mathrm{U}}(n)_\text {reg}\quad \hbox {for}\quad -\epsilon<t < \epsilon , \end{aligned}$$
(4.11)

where the elements of \({\mathrm{U}}(n)_\text {reg}\) have n distinct eigenvalues. Notice from (2.42) that \(g_R(t)\) describes the unreduced solution curve and that a small enough \(\epsilon \) will certainly do. Then, for \(- \epsilon< t < \epsilon \) there exists a unique smooth curve \(\eta (t) \in {\mathrm{U}}(n)\) for which

$$\begin{aligned} Q(t):= \eta (t) g_R(t) \eta (t)^{-1} \in {\mathbb {T}}^n_\text {reg}\quad \hbox {and}\quad \eta (0)={{\varvec{1}}}_n,\quad \left( {\dot{\eta }}(t) \eta (t)^{-1}\right) _0 =0. \nonumber \\ \end{aligned}$$
(4.12)

It is easy to see that \((Q(t), L(t), v(\alpha )(t))\) given by the above Q(t) and

$$\begin{aligned} L(t) = \eta (t) L^0 \eta (t)^{-1}, \quad v(\alpha )(t) = \eta (t) v(\alpha )^0 \end{aligned}$$
(4.13)

yields the integral curve of the vector field (4.8) with \({{\mathcal {Z}}}=0\). We here used the property \({{\mathcal {V}}}(g L g^{-1}) = g {{\mathcal {V}}}(L) g^{-1}\) (\(\forall g\in {\mathrm{U}}(n), L\in {{\mathfrak {P}}}(n)\)), which follows from the definition (4.1). The auxiliary conditions imposed in (4.12) fix the ambiguity of the ‘diagonalizer’ \(\eta (t)\) of \(g_R(t)\). The reduction approach leads to this solution algorithm naturally, but we should stress that an analogous algorithm was found long ago by Ragnisco and Suris [43] using a direct method.

Corollary 4.3

Consider \(H\in {\mathfrak {H}}\) defined by

$$\begin{aligned} H= \Lambda _R^*(h) \quad \hbox {with}\quad h(b):= (e^{2\gamma } -1) \mathrm{tr} (bb^\dagger ). \end{aligned}$$
(4.14)

Then, the evolution equation on \({{\mathcal {M}}}_0^\mathrm{reg}\) corresponding to the vector field \(Y_H^0\) (4.8) with \({{\mathcal {Z}}}=0\) can be written explicitly as follows:

$$\begin{aligned}&\frac{1}{2} {\dot{q}}_j := \frac{1}{2{\mathrm{i}}} Y_H^0[Q_j] Q_j^{-1}= F_{jj}, \end{aligned}$$
(4.15)
$$\begin{aligned}&{\dot{v}}(\alpha )_i := Y_H^0[v(\alpha )_i]=- \sum _{j\ne i} F_{ij} v(\alpha )_j V\left( \frac{q_j -q_i}{2}\right) , \end{aligned}$$
(4.16)

where \(F = \sum _\alpha v(\alpha ) v(\alpha )^\dagger \) and the ‘potential function’ V reads

$$\begin{aligned} V(x)= \cot x - \cot (x-{\mathrm{i}}\gamma ). \end{aligned}$$
(4.17)

These formulae reproduce the spin RS equations of motion (1.2) by setting \(x_i = q_i/2\) and imposing the additional reality condition (1.4).

Proof

In this case the definition (4.1) gives

$$\begin{aligned} {{\mathcal {V}}}(L) = 2{\mathrm{i}}(e^{2\gamma }-1) L. \end{aligned}$$
(4.18)

Equation (4.15) follows immediately from (4.8) since \({{\mathcal {V}}}(L)_{jj} = 2 {\mathrm{i}}F_{jj}\) by (3.28) and we have \(Q_j = \exp ({\mathrm{i}}q_j)\). Taking advantage of Hermite’s cotangent identity,

$$\begin{aligned} \cot (z-a_1) \cot (z-a_2) = -1 + \cot (a_1-a_2)\cot (z-a_1) + \cot (a_2-a_1) \cot (z-a_2), \nonumber \\ \end{aligned}$$
(4.19)

it is not difficult to re-cast the off-diagonal matrix function K(QL(Qv)) (4.7) in the form

$$\begin{aligned} K_{kl} = F_{kl} \left[ \cot \left( \frac{q_k- q_l}{2}\right) - \cot \left( \frac{q_k- q_l}{2} + {\mathrm{i}}\gamma \right) \right] , \end{aligned}$$
(4.20)

for all \( k\ne l\). This gives (4.16) with (4.17). The validity of the last sentence of the corollary can also be checked directly. \(\square \)

Remark 4.4

The restriction of the Hamiltonian H (4.14) to \({{\mathcal {M}}}_0\) gives

$$\begin{aligned} H(Q,v) = (e^{2\gamma } -1) \text {tr}( L(Q,v)) = \sum _{j=1}^n F_{jj}. \end{aligned}$$
(4.21)

We shall confirm in Sect. 5.2 that the ensuing reduced Hamiltonian generates the equations of motion (1.8) via the reduced Poisson structure described in coordinates using the gauge fixing condition (1.7).

The Reduced Poisson Structure

The main purpose of this section is to present the explicit form of the reduced Poisson structure in terms of the variables that feature in the equations of motion (1.8). The first subsection contains a couple of auxiliary lemmae, in which we provide explicit formulae for the Poisson brackets of the half-dressed and dressed spins, and the matrix entries of \(g_R\) and L. These permit us to establish that the \({\mathrm{U}}(n)\) invariant integrals of motion (3.24) form a closed polynomial Poisson algebra on the unreduced phase space, which automatically descends to the reduced phase space. This interesting algebra is given by Proposition 5.5. In the second subsection, we utilize the Poisson brackets of another set of \({\mathrm{U}}(n)\) invariant functions in order to characterize the reduced Poisson structure. We shall rely on the fact that the restriction of the Poisson brackets of \({\mathrm{U}}(n)\) invariant functions to a gauge slice in the ‘constraint surface’ must coincide with the Poisson brackets of the restricted functions calculated from the reduced Poisson structure.

All calculations required by this section are straightforward, but they are quite voluminous and not enlightening. We strive to give just enough details to provide the gist of these calculations, and so that an interested reader may reproduce them. Some of these details are relegated to “Appendix B and C.”

Some Poisson Brackets Before Reduction

Using the results from Sect. 2, the Poisson structure \(\{ \ , \ \}_{{\mathcal {M}}}\) on \({{\mathcal {M}}}\) can be described in terms of the (complex-valued) functions returning the entries of the matrices \((g_R,b_R,w^1,\ldots ,w^d)\) and their complex conjugates. Namely, we can use (2.7)–(2.8) with \(K=g_R\) or \(K=b_R\), then (2.18)–(2.19) to characterize the Poisson structure restricted to functions on the Heisenberg double; for fixed \(\alpha =1,\ldots ,d\), the Poisson brackets involving \(w^\alpha \) are given by (2.28)–(2.29). The Poisson brackets between functions of \(w^\alpha \) and functions of \(g_R\) and \(b_R\) vanish. Our aim is to translate these relations to the matrices \( (g_R, L=b_R b_R^\dagger , v(1)=b_R v^1,\dots , v(d)=b_R v^d)\), which are more convenient to understand the reduced phase space \({{\mathcal {M}}}_{\text {red}}\), see Sect. 3. As a first step, we express the Poisson structure on the half-dressed spins \(v^\alpha =b_1\cdots b_{\alpha -1}w^\alpha \) defined in (3.3). We let \(\{ \ , \ \}:=\{ \ , \ \}_{{\mathcal {M}}}\) for the rest of the section.

Lemma 5.1

The Poisson brackets of the half-dressed spins are given by the following formulae

$$\begin{aligned}&\{ v^\alpha _i,v^\beta _k \} =-{\mathrm{i}}\, \mathrm{sgn}(k-i) v_k^\alpha v_i^\beta +{\mathrm{i}}\, \mathrm{sgn}(\beta - \alpha ) v_k^\alpha v_i^\beta \,, \end{aligned}$$
(5.1)
$$\begin{aligned}&\{ v^\alpha _i,{\bar{v}}^\beta _k \} = {\mathrm{i}}\delta _{ik} v_i^\alpha \bar{v}_k^\beta +2{\mathrm{i}}\delta _{ik}\sum _{r>k}v_r^\alpha \bar{v}_r^\beta +{\mathrm{i}}\delta _{\alpha \beta } v^\alpha _i{\bar{v}}^\beta _k + 2 {\mathrm{i}}\delta _{\alpha \beta }\sum _{\mu <\alpha } v_i^\mu \bar{v}_k^\mu + 2 {\mathrm{i}}\delta _{ik}\delta _{\alpha \beta } \,, \nonumber \\ \end{aligned}$$
(5.2)

where \(1\le i,k \le n\) and \(1\le \alpha ,\beta \le d\). In particular, this defines a Poisson structure on \({\mathbb {C}}^{nd} \simeq {\mathbb {R}}^{2nd}\).

This result is proved in Appendix B. From the reality of the Poisson bracket, we have

$$\begin{aligned} \{ {\bar{v}}^\alpha _i,{\bar{v}}^\beta _k \}=+{\mathrm{i}}\, \text {sgn}(k-i) {\bar{v}}_k^\alpha {\bar{v}}_i^\beta -{\mathrm{i}}\, \text {sgn}(\beta - \alpha ) {\bar{v}}_k^\alpha {\bar{v}}_i^\beta \,. \end{aligned}$$
(5.3)

Remark 5.2

If we complexify the formulae of Lemma 5.1 by introducing \(a_{i\alpha }:=(v^\alpha _i)^{{\mathbb {C}}}\) and \(b_{\alpha i}:=({\bar{v}}^\alpha _i)^{{\mathbb {C}}}\), we get a complex holomorphic Poisson structure on \({\mathbb {C}}^{2nd}\) given by

$$\begin{aligned} \{ a_{i\alpha },a_{k \beta } \}= & {} -{\mathrm{i}}\, \text {sgn}(k-i) a_{k\alpha } a_{i\beta } +{\mathrm{i}}\, \text {sgn}(\beta - \alpha ) a_{k\alpha } a_{i\beta }\,, \end{aligned}$$
(5.4)
$$\begin{aligned} \{ b_{\alpha i},b_{\beta k} \}= & {} +{\mathrm{i}}\, \text {sgn}(k-i) b_{\alpha k} b_{\beta i} -{\mathrm{i}}\, \text {sgn}(\beta - \alpha ) b_{\alpha k} b_{\beta i}\,, \end{aligned}$$
(5.5)
$$\begin{aligned} \{ a_{i\alpha },b_{\beta k} \}= & {} {\mathrm{i}}\delta _{ik} a_{i\alpha } b_{\beta k} +2{\mathrm{i}}\delta _{ik}\sum _{r>k} a_{r\alpha } b_{\beta r} +{\mathrm{i}}\delta _{\alpha \beta } a_{i\alpha } b_{\beta k}\nonumber \\&+ 2 {\mathrm{i}}\delta _{\alpha \beta }\sum _{\mu <\alpha } a_{i\mu } b_{\mu k} + 2 {\mathrm{i}}\delta _{ik}\delta _{\alpha \beta } .\quad \end{aligned}$$
(5.6)

After appropriate rescaling, this reproduces the minus Poisson bracket introduced by Arutyunov and Olivucci in their treatment of the complex holomorphic spin RS system by Hamiltonian reduction [5]. Considering the analogous construction with the variables \(v_{+,i}^\alpha :=v_i^{d-\alpha +1}\) instead, we obtain the plus Poisson bracket introduced in [5].

From now on, we let \(b=b_R,g=g_R\). Using Lemma 5.1 and the Poisson structure of the Heisenberg double, we can easily write the Poisson brackets involving the entries \(v(\alpha )_i\) of the dressed spins \(v(\alpha )=b_R v^\alpha \).

Lemma 5.3

The Poisson brackets of the dressed spins are given by the following formulae

$$\begin{aligned}&\{ v(\alpha )_i,v(\beta )_k \} =-{\mathrm{i}}\, \mathrm{sgn}(k-i) v(\alpha )_k v(\beta )_i +{\mathrm{i}}\, \mathrm{sgn}(\beta - \alpha ) v(\alpha )_k v(\beta )_i\,, \end{aligned}$$
(5.7)
$$\begin{aligned}&\{ v(\alpha )_i,\overline{v}(\beta )_k \} = {\mathrm{i}}\delta _{ik} v(\alpha )_i \overline{v}(\beta )_k+2{\mathrm{i}}\delta _{ik}\sum _{r>k}v(\alpha )_r \overline{v}(\beta )_r +{\mathrm{i}}\delta _{\alpha \beta } v(\alpha )_i\overline{v}(\beta )_k \nonumber \\&\qquad \qquad \qquad \quad \ + 2 {\mathrm{i}}\delta _{\alpha \beta }\sum _{\mu <\alpha } v(\mu )_i \overline{v}(\mu )_k + 2 {\mathrm{i}}\delta _{\alpha \beta } (b b^\dagger )_{ik} \,. \end{aligned}$$
(5.8)

The Poisson brackets of the dressed spins and the matrices bg are given by the following formulae

$$\begin{aligned}&\{ v(\alpha )_i,g_{kl} \} = -{\mathrm{i}}\delta _{ik} v(\alpha )_i g_{kl} - 2 {\mathrm{i}}\delta _{(i<k)} v(\alpha )_k g_{il}\,, \end{aligned}$$
(5.9)
$$\begin{aligned}&\{ \overline{v}(\beta )_i,g_{kl} \}= -{\mathrm{i}}\delta _{ik} \overline{v}(\beta )_i g_{kl} - 2 {\mathrm{i}}\delta _{ik}\sum _{r>i} \overline{v}(\beta )_r g_{rl}\,, \end{aligned}$$
(5.10)
$$\begin{aligned}&\{ v(\alpha )_i,b_{kl} \} = 2{\mathrm{i}}\delta _{(k<i)} v(\alpha )_k b_{il} + {\mathrm{i}}\delta _{ik} v(\alpha )_k b_{il} - 2 {\mathrm{i}}\sum _{s<l} b_{ks}v_s^\alpha b_{il} - {\mathrm{i}}b_{kl} v_l^\alpha b_{il}\,, \end{aligned}$$
(5.11)
$$\begin{aligned}&\{ \overline{v}(\beta )_i,b_{kl} \} = -{\mathrm{i}}\delta _{ik} \overline{v}(\beta )_i b_{kl} - 2 {\mathrm{i}}\delta _{ik}\sum _{r>k} \overline{v}(\beta )_r b_{rl} + {\mathrm{i}}{\bar{b}}_{il} {\bar{v}}_l^\beta b_{kl} + 2 {\mathrm{i}}\sum _{s<l} {\bar{b}}_{is}{\bar{v}}_l^\beta b_{ks}\,. \nonumber \\ \end{aligned}$$
(5.12)

Finally, we can express the Poisson structure in terms of the elements \( (g, L, v(1),\dots , v(d))\) where \(L=b b^\dagger \). This is a direct application of Lemma 5.3 and the relations (2.7)–(2.8), (2.18)–(2.19) of the Heisenberg double by using that \(L_{kl}=\sum _r b_{ks}{\bar{b}}_{ls}\). Alternatively, one may use the formula (2.23) to derive equations (5.14)–(5.15).

Lemma 5.4

The Poisson brackets involving L are given by the following formulae

$$\begin{aligned} \{ v(\alpha )_i,L_{kl} \}= & {} -{\mathrm{i}}(2\delta _{(k>i)}+\delta _{ik}) v(\alpha )_k L_{il} + {\mathrm{i}}\delta _{il} v(\alpha )_i L_{kl} + 2 {\mathrm{i}}\delta _{il} \sum _{r>l} v(\alpha )_r L_{kr} \,, \end{aligned}$$
(5.13)
$$\begin{aligned} \{ g_{ij},L_{kl} \}= & {} {\mathrm{i}}(\delta _{ik}+\delta _{il}) g_{ij} L_{kl} + 2 {\mathrm{i}}\delta _{(k<i)} g_{kj} L_{il} + 2 {\mathrm{i}}\delta _{il} \sum _{r>i} L_{kr}g_{rj}\,, \end{aligned}$$
(5.14)
$$\begin{aligned} \{ L_{ij},L_{kl} \}= & {} {\mathrm{i}}[2 \delta _{(i>k)}+\delta _{ik}-2\delta _{(j>l)}-\delta _{lj}] L_{il}L_{kj} \nonumber \\&+ {\mathrm{i}}(\delta _{il}-\delta _{jk})L_{ij}L_{kl} + 2 {\mathrm{i}}\delta _{il} \sum _{r>i} L_{kr} L_{rj} - 2 {\mathrm{i}}\delta _{jk} \sum _{r> k} L_{ir} L_{rl}\,. \end{aligned}$$
(5.15)

Now we present an interesting application of the above auxiliary results. Recall that our ‘free Hamiltonians’ (2.40) Poisson commute with the functions \(I_{\alpha \beta }^k\) defined in (3.24), and hence, they Poisson commute with the elements of the polynomial algebra

$$\begin{aligned} {\mathcal {I}}_L={\mathbb {R}}[\text {tr}L^k,\mathfrak {R}(I_{\alpha \beta }^k),\mathfrak {I}(I_{\alpha \beta }^k) \mid 1\le \alpha ,\beta \le d,\, k \ge 0]\,. \end{aligned}$$
(5.16)

The algebra \({\mathcal {I}}_L\) is finitely generated as a consequence of the Cayley–Hamilton theorem for L. We also note that for an arbitrary non-commutative polynomial \(\text {P}\) obtained as a linear combination of products of the matrices L and \(v(\alpha )v(\beta )^\dagger \), \(1\le \alpha ,\beta \le d\), we have that \(\text {tr}(\text {P})\in {\mathcal {I}}_L\) in view of the identity

$$\begin{aligned} \text {tr}\left( L^{a_0}v(\alpha _0)v(\beta _1)^\dagger L^{a_1} v(\alpha _1)v(\beta _2)^\dagger \cdots L^{a_{l}}v(\alpha _l)v(\beta _0)^\dagger L^{a_{l+1}}\right) =I_{\alpha _0 \beta _0}^{a_0+a_{l+1}} I_{\alpha _1 \beta _1}^{a_1} \cdots I_{\alpha _l \beta _l}^{a_l}\,.\nonumber \\ \end{aligned}$$
(5.17)

A key property of \({\mathcal {I}}_L\) is that it is a Poisson subalgebra of \(C^\infty ({{\mathcal {M}}})\). This follows from the next result, which can be proved by direct calculation.

Proposition 5.5

For any \(M,N\ge 0\) and \(1\le \alpha ,\beta ,\gamma ,\epsilon \le d\),

$$\begin{aligned} \begin{aligned} \{ I_{\alpha \beta }^M,I_{\gamma \epsilon }^N \}=&\,\,2 {\mathrm{i}}\delta _{\alpha \epsilon } I_{\gamma \beta }^{M+N+1} - 2 {\mathrm{i}}\delta _{\gamma \beta } I_{\alpha \epsilon }^{M+N+1} \\&+{\mathrm{i}}(\delta _{\alpha \epsilon } - \delta _{\gamma \beta }) I_{\alpha \beta }^M I_{\gamma \epsilon }^N +2 {\mathrm{i}}\delta _{\alpha \epsilon } \sum _{\mu<\alpha } I_{\gamma \mu }^N I_{\mu \beta }^M -2{\mathrm{i}}\delta _{\gamma \beta } \sum _{\lambda <\beta } I_{\alpha \lambda }^M I_{\lambda \epsilon }^N \\&+{\mathrm{i}}\, \mathrm{sgn}(\gamma -\alpha ) I_{\gamma \beta }^M I_{\alpha \epsilon }^N -{\mathrm{i}}\, \mathrm{sgn}(\epsilon -\beta ) I_{\gamma \beta }^N I_{\alpha \epsilon }^M \\&+{\mathrm{i}}\left( \sum _{b=0}^{M-1}+\sum _{b=0}^{N-1} \right) \left( I_{\gamma \beta }^b I_{\alpha \epsilon }^{M+N-b} - I_{\gamma \beta }^{M+N-b} I_{\alpha \epsilon }^b \right) \,. \end{aligned} \end{aligned}$$
(5.18)

Using that \(\overline{I_{\alpha \beta }^M} = I_{\beta \alpha }^M\), one can verify that the complex Poisson brackets in (5.18) enjoy the property

$$\begin{aligned} \{ \overline{I_{\alpha \beta }^M},\overline{I_{\gamma \epsilon }^N} \} =\overline{\{ I_{\alpha \beta }^M,I_{\gamma \epsilon }^N \}}, \end{aligned}$$
(5.19)

which together with (5.18) implies that \({\mathcal {I}}_L\) (5.16) is indeed a real Poisson algebra. Since the elements of \({\mathcal {I}}_L\) are invariant with respect to the \({\mathrm{U}}(n)\) action on \({{\mathcal {M}}}\), they descend to the reduced phase space. We shall further inspect these integrals of motion in Sect. 6.

The Reduced Poisson Bracket in Local Coordinates

In this subsection, we shall derive explicit formulae for the reduced Poisson structure, restricting ourselves to an open dense subset \(\check{{\mathcal {M}}}_\text {red}^\text {reg}\) of the reduced phase space. More precisely, it will be more convenient to work on a covering space of \(\check{{\mathcal {M}}}_\text {red}^\text {reg}\) that supports residual \(S_n\) gauge transformations.

We start by introducing the open dense subset \(\check{{\mathcal {M}}}^\text {reg}_0\subset {{\mathcal {M}}}^\text {reg}_0\) (3.62), which is defined as

$$\begin{aligned} \check{{\mathcal {M}}}^\text {reg}_0:= \{ (Q, L(Q,v), v)\in {{\mathcal {M}}}_0^{\text {reg}} \mid \sum _{1\le \alpha \le d} v(\alpha )_i \ne 0 \text { for }i=1,\ldots ,n\, \}\,. \nonumber \\ \end{aligned}$$
(5.20)

The corresponding open dense subset of the reduced phase space is

$$\begin{aligned} \check{{\mathcal {M}}}_\text {red}^\text {reg}:= \check{{\mathcal {M}}}^\text {reg}_0/{{\mathcal {N}}}(n)\,. \end{aligned}$$
(5.21)

Using that \(S_n={{\mathcal {N}}}(n)/{\mathbb {T}}^n\), we can take the quotient in two steps. Thus, similarly to (3.64), we have

$$\begin{aligned} \check{{\mathcal {M}}}_0^\text {reg}/{{\mathcal {N}}}(n) =\left( \check{{\mathcal {M}}}_0^\text {reg}/{\mathbb {T}}^n\right) /S_n. \end{aligned}$$
(5.22)

For our purpose, we choose to identify \( \check{{\mathcal {M}}}_0^\text {reg}/{\mathbb {T}}^n\) with the following subset of \(\check{{\mathcal {M}}}_0^\text {reg}\)

$$\begin{aligned} \check{{\mathcal {M}}}_{0,+}^\text {reg}:= \{ (Q, L(Q,v), v)\in {{\mathcal {M}}}_0^\text {reg}\mid \sum _{1\le \alpha \le d} v(\alpha )_i >0 \ \hbox {for}\ i=1,\dots , n\}.\nonumber \\ \end{aligned}$$
(5.23)

Indeed, if \((Q, L, v)\in \check{{\mathcal {M}}}^\text {reg}_0\) we can write the vector \(\sum _{1\le \alpha \le d} v(\alpha )\) as \((U_1 e^{{\mathrm{i}}\phi _1}, \ldots , U_n e^{{\mathrm{i}}\phi _n})^T\) with \(U_i >0\), \(\phi _i \in {\mathbb {R}}\) for all i. Acting on (QLv) by \(\tau = \text {diag}(e^{-{\mathrm{i}}\phi _1}, \ldots ,e^{-{\mathrm{i}}\phi _n})\in {\mathbb {T}}^n\) yields an element of \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\), and it is clear that \(\tau \) is the unique element of \({\mathbb {T}}^n\) with this property. The upshot is the identification

$$\begin{aligned} {\check{{{\mathcal {M}}}}}_\text {red}^\text {reg}\equiv {\check{{{\mathcal {M}}}}}_{0,+}^\text {reg}/S_n. \end{aligned}$$
(5.24)

The main reason for introducing the particular gauge slice \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) for the free \({\mathbb {T}}^n\) action on \(\check{{\mathcal {M}}}_0^\text {reg}\) is that \(S_n\) still acts on it in the obvious manner, by permuting the n entries of Q and the components of each column vector \(v(\alpha )\in {\mathbb {C}}^n\). Similar ‘democratic gauge fixing’ was employed in the previous papers dealing with holomorphic systems [4], [5], [10]. The relation between the spaces just defined and those given in Sect. 3 is summarized in Fig. 1.

Fig. 1
figure 1

(From right to left.) \({{\mathcal {M}}}_0\) is the subspace (3.26) of the constraint surface \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) where each point \((Q,b_R,v)\) satisfies that \(Q \in {\mathbb {T}}^n\). \({{\mathcal {M}}}_0^{\text {reg}}\subset {{\mathcal {M}}}_0\) is the subspace (3.39) where \(Q\in {\mathbb {T}}^n_{\text {reg}}\), \(\check{{\mathcal {M}}}^\text {reg}_0\subset {{\mathcal {M}}}_0^{\text {reg}}\) is the subspace (5.20) where the vector \(\sum _{\alpha =1}^d v(\alpha )\) has only nonzero entries, while \(\check{{\mathcal {M}}}^\text {reg}_{0,+}\subset \check{{\mathcal {M}}}^\text {reg}_0\) is the subspace (5.23) obtained by imposing to the vector \(\sum _{\alpha =1}^d v(\alpha )\) to have positive entries. The spaces appearing on the second line are the sets corresponding to the \({\mathrm{U}}(n)\)-orbits inside \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\)

Let

$$\begin{aligned} \xi : \check{{\mathcal {M}}}_{0,+}^\text {reg}\rightarrow {{\mathcal {M}}}\end{aligned}$$
(5.25)

be the tautological inclusion. General principles of reduction theory [25], [41] ensure that the pull-back \(\xi ^* \Omega _{{\mathcal {M}}}\) is symplectic and satisfies \(\xi ^*\Omega _{{\mathcal {M}}}= {{\check{\pi }}}^* (\Omega _\text {red})\), where \({{\check{\pi }}}: \check{{\mathcal {M}}}_{0,+}^\text {reg}\rightarrow {{\mathcal {M}}}_\text {red}\) is the canonical projection and \(\Omega _\text {red}\) is the reduced symplectic form. We let \(\{\ ,\ \}_\text {red}\) denote the Poisson bracket on \(C^\infty (\check{{\mathcal {M}}}_{0,+}^\text {reg})\) that corresponds to \(\xi ^* \Omega _{{\mathcal {M}}}\) (2.37), and note that it possesses the key property

$$\begin{aligned} \xi ^* \{ F_1, F_2\} = \{ \xi ^* F_1, \xi ^* F_2\}_\text {red}, \qquad \forall F_1, F_2 \in C^\infty ({{\mathcal {M}}})^{{\mathrm{U}}(n)}, \end{aligned}$$
(5.26)

where \(\{F_1,F_2\} := \{ F_1, F_2\}_{{\mathcal {M}}}\) is the Poisson bracket associated with \(\Omega _{{\mathcal {M}}}\) (2.37). We shall determine the form of this reduced Poisson bracket by applying the identity (5.26) to a judiciously chosen set of invariant functions.

Remark 5.6

The bracket \(\{\ ,\ \}_\text {red}\) is also known as the Dirac bracket [25] associated with the gauge slice \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\). To avoid any potential confusion, we stress that our notation \(\{\ ,\ \}_\text {red}\) involves a slight abuse of terminology, since not all elements of \(C^\infty (\check{{\mathcal {M}}}_{0,+}^\text {reg})\) arise as restrictions of elements of \(C^\infty ({{\mathcal {M}}})^{{\mathrm{U}}(n)}\) (which carries the reduced Poisson algebra in the strict sense). For example, all those restricted \({\mathrm{U}}(n)\) invariants are \(S_n\) invariant function on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\). However, the Poisson algebra \((C^\infty (\check{{\mathcal {M}}}_{0,+}^\text {reg}), \{\ ,\ \}_\text {red})\) encodes all information about \((C^\infty ({{\mathcal {M}}})^{{\mathrm{U}}(n)}, \{\ ,\ \})\), since \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) projects onto a dense open subset of \({{\mathcal {M}}}_\text {red}\). We shall see shortly that it underlies the Hamiltonian interpretation of the spin RS equations of motion given by (1.8).

In order to implement the above ideas, now we introduce the following \({\mathrm{U}}(n)\) invariant elements of \(C^\infty ({{\mathcal {M}}})\)

$$\begin{aligned} f^{\alpha \beta }_m:= \text {tr}(v(\alpha ) v(\beta )^\dagger g_R ^m) = v(\beta )^\dagger g_R^m v(\alpha )\,, \quad f_m:= \text {tr}(g_R^m)\,, \quad m \in {\mathbb {N}},\,\, 1\le \alpha ,\beta \le d\,.\nonumber \\ \end{aligned}$$
(5.27)

Lemma 5.7

For any \(M,N \in {\mathbb {N}}\) and \(1\le \alpha ,\beta \le d\) we have the following Poisson bracket relations in \(C^\infty ({{\mathcal {M}}})^{{\mathrm{U}}(n)}\):

$$\begin{aligned}&\{ f_M,f_N \}=0\,, \quad \{ f_M, {\bar{f}}_N \}=0\,, \quad \{ {\bar{f}}_M, {\bar{f}}_N \}=0\,, \end{aligned}$$
(5.28)
$$\begin{aligned}&\{ f_M^{\alpha \beta },f_N \}=-2{\mathrm{i}}N f_{M+N}^{\alpha \beta }\,. \end{aligned}$$
(5.29)

Furthermore, letting \(\phi ^{\mu \nu }(a,c):=\mathrm{tr}[v(\mu )v(\nu )^\dagger g_R^a L g_R^c]\) for \(a,b \in {\mathbb {N}}\), we have

$$\begin{aligned} \{ f_M^{\alpha \beta },f_N^{\gamma \epsilon } \}= & {} \,\,2{\mathrm{i}}\left( \sum _{a=1}^M - \sum _{a=1}^N \right) f_a^{\alpha \epsilon } f_{M+N-a}^{\gamma \beta } -{\mathrm{i}}f_M^{\alpha \epsilon } f_N^{\gamma \beta } + {\mathrm{i}}f_N^{\alpha \epsilon } f_M^{\gamma \beta }\nonumber \\&+{\mathrm{i}}\mathrm{sgn}(\gamma -\alpha ) f_N^{\alpha \epsilon } f_M^{\gamma \beta } -{\mathrm{i}}\mathrm{sgn}(\epsilon -\beta ) f_M^{\alpha \epsilon } f_N^{\gamma \beta } +{\mathrm{i}}(\delta _{\alpha \epsilon } - \delta _{\gamma \beta }) f_M^{\alpha \beta } f_N^{\gamma \epsilon } \nonumber \\&+2{\mathrm{i}}\delta _{\alpha \epsilon }\sum _{\mu<\alpha } f_N^{\gamma \mu } f_M^{\mu \beta } - 2 {\mathrm{i}}\delta _{\gamma \beta } \sum _{\lambda <\beta } f_M^{\alpha \lambda } f_N^{\lambda \epsilon } \nonumber \\&+2{\mathrm{i}}\delta _{\alpha \epsilon } \phi ^{\gamma \beta }(M,N) - 2 {\mathrm{i}}\delta _{\gamma \beta } \phi ^{\alpha \epsilon }(N,M)\,. \end{aligned}$$
(5.30)

Proof

The identities (5.28) are well known. To establish (5.29), we use the decomposition (here \(g=g_R\))

$$\begin{aligned} \begin{aligned} \{ f_M^{\alpha \beta },f_N \}=&N\sum _{ijkl}\{ v(\alpha )_i ,g_{kl} \} \overline{v}(\beta )_j g^M_{ji}g^{N-1}_{lk} + N\sum _{ijkl}\{ \overline{v}(\beta )_j,g_{kl} \} g^M_{ji}v(\alpha )_i g^{N-1}_{lk}\\&+N \sum _{ijkl}\sum _{b=0}^{M-1}\{ g_{ij},g_{kl} \} (g^{M-b-1}v(\alpha )v(\beta )^\dagger g^b)_{ji}g^{N-1}_{lk}\,, \end{aligned} \end{aligned}$$
(5.31)

then use for these three terms (5.9), (5.10) and (2.7), respectively. Some obvious cancellations yield (5.29). Finally, (5.30) only requires some of the Poisson brackets gathered in §5.1 and it can be proved in a way similar to (5.29). \(\square \)

Convenient variables on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) are provided by the evaluation functions \(Q_j=e^{{\mathrm{i}}q_j} \in \text {U}(1)\) and the real and imaginary parts of the \(v(\alpha )_j\in {\mathbb {C}}\). The latter are not all independent, since they obey the gauge fixing conditions

$$\begin{aligned} {{\mathcal {U}}}_j=\mathfrak {R}({{\mathcal {U}}}_j)>0\,, \quad \text { with } \quad {{\mathcal {U}}}_j:=\sum _{1\le \alpha \le d}v(\alpha )_j\,, \quad 1\le j \le n\,. \end{aligned}$$
(5.32)

It is clear that all these functions belong to \(C^\infty (\check{{\mathcal {M}}}_{0,+}^\text {reg})\) and their mutual Poisson brackets completely determine \(\{\ ,\ \}_\text {red}\).

The pull-backs of the functions (5.27) can be written in the local variables on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) as

$$\begin{aligned} \xi ^* f^{\alpha \beta }_m =\sum _{i=1}^n v(\alpha )_i Q_i^m \overline{v}(\beta )_i\,, \quad \xi ^* f_m = \sum _{i=1}^n Q_i^m\,, \end{aligned}$$
(5.33)

and we note that

$$\begin{aligned} \sum _{\beta } \xi ^* f^{\alpha \beta }_m =\sum _{i=1}^n {{\mathcal {U}}}_i v(\alpha )_i Q_i^m\,, \quad \sum _{\alpha } \xi ^* f^{\alpha \beta }_m =\sum _{i=1}^n {{\mathcal {U}}}_i \overline{v}(\beta )_i Q_i^m\,, \quad \sum _{\alpha ,\beta } \xi ^* f^{\alpha \beta }_m =\sum _{i=1}^n {{\mathcal {U}}}_i^2 Q_i^m\,.\nonumber \\ \end{aligned}$$
(5.34)

In conjunction with Lemma 5.7 and Eq. (5.26), these expressions can be used to determine the reduced Poisson brackets of the variables \(Q_j\) and \(v(\alpha )\). To state the result, we introduce the \(n\times n\) matrix-valued functions \(S^0\) and \(R^\alpha \), \(1\le \alpha \le d\), whose entries are given by

$$\begin{aligned}&S^0_{ij}=\frac{1}{4}\sum _{\mu ,\nu } \text {sgn}(\nu -\mu ) v(\nu )_i v(\mu )_j -\frac{1}{4} \sum _{\mu } v(\mu )_i \overline{v}(\mu )_j\nonumber \\&\quad -\frac{1}{2} \sum _{\nu }\sum _{\mu <\nu } v(\mu )_i \overline{v}(\mu )_j - \frac{d}{2} L_{ij},\quad \qquad \end{aligned}$$
(5.35)
$$\begin{aligned}&R^\alpha _{ij}=L_{ij}-\frac{1}{2} \sum _{\kappa } \text {sgn}(\kappa -\alpha )v(\kappa )_i v(\alpha )_j +\frac{1}{2} v(\alpha )_i \overline{v}(\alpha )_j + \sum _{\kappa < \alpha } v(\kappa )_i \overline{v}(\kappa )_j.\quad \nonumber \\ \end{aligned}$$
(5.36)

We also define the matrix S with entries \(S_{ij}=S^0_{ij}-\overline{S}^0_{ij}\).

Theorem 5.8

In terms of the functions \((Q_j=e^{{\mathrm{i}}q_j},v(\alpha )_j)\) defined on \(\check{{\mathcal {M}}}_{0,+}^\mathrm{reg}\), and using the formulae (3.29) for L and (5.32) for \({{\mathcal {U}}}_j\), we can write the reduced Poisson bracket as

$$\begin{aligned}&\{ q_i,q_j \}_\mathrm{red}=0\,, \quad \{ v(\alpha )_i,q_j \}_\mathrm{red}=-\delta _{ij} v(\alpha )_i\,, \\&\{ v(\alpha )_i,v(\gamma )_j \}_\mathrm{red}= {\mathrm{i}}\, \mathrm{sgn}(\gamma -\alpha ) v(\alpha )_j v(\gamma )_i + {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i}\frac{v(\gamma )_j}{{{\mathcal {U}}}_j} S_{ij} + {\mathrm{i}}\frac{v(\gamma )_j}{{{\mathcal {U}}}_j} R^\alpha _{ij} - {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i} R^\gamma _{ji} \nonumber \\&\quad +\frac{1}{2} {\mathrm{i}}\delta _{(i \ne j)} \frac{Q_i+Q_j}{Q_i-Q_j}\left[ 2 v(\alpha )_j v(\gamma )_i+ v(\alpha )_iv(\gamma )_j -\frac{{{\mathcal {U}}}_i}{{{\mathcal {U}}}_j}v(\alpha )_j v(\gamma )_j - \frac{{{\mathcal {U}}}_j}{{{\mathcal {U}}}_i} v(\alpha )_i v(\gamma )_i \right] \,, \end{aligned}$$
(5.37)
$$\begin{aligned}&\{ v(\alpha )_i,\overline{v}(\epsilon )_j \}_\mathrm{red}= {\mathrm{i}}\delta _{\alpha \epsilon } \left( v(\alpha )_i \overline{v}(\epsilon )_j + 2 \sum _{\kappa < \alpha } v(\kappa )_i \overline{v}(\kappa )_{j}+2L_{ij}\right) \nonumber \\&\quad +\frac{1}{2} {\mathrm{i}}\delta _{(i \ne j)} \frac{Q_i+Q_j}{Q_i-Q_j}\left[ - v(\alpha )_i \overline{v}(\epsilon )_j +\frac{{{\mathcal {U}}}_i}{{{\mathcal {U}}}_j}v(\alpha )_j \overline{v}(\epsilon )_j + \frac{{{\mathcal {U}}}_j}{{{\mathcal {U}}}_i} v(\alpha )_i \overline{v}(\epsilon )_i \right] \end{aligned}$$
(5.38)
$$\begin{aligned}&- {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i}\frac{\overline{v}(\epsilon )_j}{{{\mathcal {U}}}_j} S_{ij} -{\mathrm{i}}\frac{\overline{v}(\epsilon )_j}{{{\mathcal {U}}}_j} R^\alpha _{ij} - {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i} \overline{R}^\epsilon _{ji}.\nonumber \end{aligned}$$
(5.39)

The bracket \(\{ -,- \}_\mathrm{red}\) is invariant under simultaneous permutations of the n components of the variables q and \(v(\alpha )\) for \(\alpha =1,\dots , d\).

The proof of this result is the subject of “Appendix C.” Let us already mention that the reader can check the reality condition \(\{ \overline{v}(\alpha )_i,v(\epsilon )_j \}_\text {red}= \overline{\{ v(\alpha )_i,\overline{v}(\epsilon )_j \}}_\text {red}\).

We know from Corollary 4.3 that the projection of the Hamiltonian vector field of \(H= (e^{2\gamma }-1)\, \text {tr}(L)\) onto the gauge slice \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) leads to the equations of motion (1.8). Of course, the corresponding reduced Hamiltonian must generate the same evolution equations via the reduced Poisson bracket. The reduced Hamiltonian is encoded by the pull-back \({{\mathcal {H}}}:= \xi ^*H\) on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\). Thus, the next result shows the consistency of the computations performed in Sects. 4 and 5.

Corollary 5.9

Consider the reduced Hamiltonian

$$\begin{aligned} {{\mathcal {H}}}(Q,v)=(e^{2\gamma }-1)\, \mathrm{tr} (L(Q,v)) = \sum _{k=1}^n F_{kk}, \qquad F_{kk} = \sum _{\alpha =1}^d v(\alpha )_k \overline{v}(\alpha )_k, \nonumber \\ \end{aligned}$$
(5.40)

on the gauge slice \(\check{{\mathcal {M}}}_{0,+}^\mathrm{reg}\). Then, the Hamiltonian vector field generated by \({{\mathcal {H}}}\) via the reduced Poisson bracket of Theorem 5.8 reproduces the equations of motion (1.8)–(1.9).

Proof

We get from (5.37) that

$$\begin{aligned} \dot{q}_j := \{ q_j,{{\mathcal {H}}} \}_\text {red}= \sum _k \{ q_j, F_{kk} \}_\text {red}=2 F_{jj}\,, \end{aligned}$$
(5.41)

which is just (1.8). To compute \({\dot{v}}(\alpha )_i\), we use that \(F_{kl}=\sum _{\alpha =1}^d v(\alpha )_k \overline{v}(\alpha )_l\) together with Theorem 5.8 in order to obtain

$$\begin{aligned} \begin{aligned}&\{ v(\alpha )_i,F_{kk} \}_\text {red}={\mathrm{i}}\delta _{(i \ne k)} v(\alpha )_k \left[ F_{ik} + \frac{Q_i+Q_k}{Q_i-Q_k}F_{ik}+2L_{ik} \right] \\&\quad -\frac{1}{2}{\mathrm{i}}\delta _{(i \ne k)} v(\alpha )_i \frac{{{\mathcal {U}}}_k}{{{\mathcal {U}}}_i} \left[ \left( F_{ik} + \frac{Q_i+Q_k}{Q_i-Q_k}F_{ik}+2L_{ik}\right) + \left( F_{ki} + \frac{Q_k+Q_i}{Q_k-Q_i}F_{ki}+2L_{ki}\right) \right] . \end{aligned}\nonumber \\ \end{aligned}$$
(5.42)

Noticing the identity

$$\begin{aligned} {\mathrm{i}}\left( F_{ik} + \frac{Q_i+Q_k}{Q_i-Q_k}F_{ik}+2 L_{ik}\right) =-F_{ik} V\left( \frac{q_k-q_i}{2}\right) \,, \end{aligned}$$
(5.43)

where V(x) is the potential (1.5), this allows us to write

$$\begin{aligned} \begin{aligned} \{ v(\alpha )_i,F_{kk} \}_\text {red}=&-\delta _{(i \ne k)} v(\alpha )_k F_{ik}V \left( \frac{q_k-q_i}{2}\right) \\&+\frac{1}{2}\delta _{(i \ne k)} v(\alpha )_i \frac{{{\mathcal {U}}}_k}{{{\mathcal {U}}}_i} \left[ F_{ik}V\left( \frac{q_k-q_i}{2}\right) + F_{ki} V\left( \frac{q_i-q_k}{2}\right) \right] \,. \end{aligned}\nonumber \\ \end{aligned}$$
(5.44)

Summing over k precisely gives \({\dot{v}}(\alpha )_i\) in (1.8) with (1.9). \(\square \)

As a second consequence of Theorem 5.8, we can write down the reduced Poisson brackets of the ‘collective spins’ \((F_{ij})\), which can be found in “Appendix D.” By using Eq. (3.29), then we can obtain the formula for the Poisson brackets of the entries of the Lax matrix on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\), which implies that the symmetric functions of L are in involution. This is in agreement with the fact that \({\mathfrak {H}}\) given in (2.40) is an Abelian Poisson subalgebra of \(C^\infty ({{\mathcal {M}}})\). To present the desired formula, we use the matrix S defined before Theorem 5.8. We also define

$$\begin{aligned}&\! r_{12}:=\sum _{a\ne b} \frac{{\mathrm{i}}Q_b}{Q_a-Q_b} E_{aa} \otimes \left( E_{bb} - \frac{{{\mathcal {U}}}_b}{{{\mathcal {U}}}_a} E_{ba} \right) -\sum _{a\ne b} \frac{{\mathrm{i}}Q_a}{Q_a-Q_b} E_{ab} \otimes \left( \frac{{{\mathcal {U}}}_a}{{{\mathcal {U}}}_b} E_{bb} -2E_{ba} \right) \nonumber \\&\qquad +\sum _{a,b} {\mathrm{i}}\frac{S_{ab}}{{{\mathcal {U}}}_a {{\mathcal {U}}}_b}\, E_{aa}\otimes E_{bb} + {\mathrm{i}}\sum _a E_{aa}\otimes E_{aa}\,, \nonumber \\ \end{aligned}$$
(5.45)

and

$$\begin{aligned}&\! s_{12}:=\sum _{a\ne b} \frac{{\mathrm{i}}Q_a}{Q_a-Q_b} E_{aa} \otimes \left( \frac{{{\mathcal {U}}}_b}{{{\mathcal {U}}}_a} E_{ab}-E_{bb} \right) + \sum _{a\ne b} \frac{{\mathrm{i}}Q_a}{Q_a-Q_b}\frac{{{\mathcal {U}}}_a}{{{\mathcal {U}}}_b} E_{ab} \otimes E_{bb} \nonumber \\&\qquad -\sum _{a,b} {\mathrm{i}}\frac{S_{ab}}{{{\mathcal {U}}}_a {{\mathcal {U}}}_b}\, E_{aa}\otimes E_{bb} + \frac{1}{2} {\mathrm{i}}\sum _a E_{aa}\otimes E_{aa}\,, \end{aligned}$$
(5.46)

where \(E_{ab}\) is the \(n\times n\) elementary matrix with only nonzero entry equal to \(+1\) in position (ab).

Proposition 5.10

On the gauge slice \(\check{{\mathcal {M}}}_{0,+}^\mathrm{reg}\) (5.23), the entries of the Lax matrix L (3.29) satisfy

$$\begin{aligned} \{ L_1,L_2 \}_\mathrm{red}=r_{12} L_1L_2 + L_1L_2 t_{12} - L_1 s_{21} L_2 + L_2 s_{12} L_1 \,, \end{aligned}$$
(5.47)

where \(t_{12}=-s_{12}+s_{21}-r_{12}\). This relation implies that the functions \(\mathrm{tr} (L^k)\) are in involution.

In (5.47), we used the standard notations \(L_1=L \otimes {{\varvec{1}}}_n\), \(L_2={{\varvec{1}}}_n \otimes L\), and \(\{ L_1,L_2 \}_\text {red}=\sum _{ijkl} \{ L_{ij},L_{kl} \}_{\text {red}} E_{ij}\otimes E_{kl}\), where the entries of L are seen as evaluation functions on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\).

Remark 5.11

The formulae of Theorem 5.8 exhibit an interesting two-body structure in the sense that the Poisson brackets of the basic variables with particle labels i and j close on this subset of the variables. This is consistent with the fact that the Hamiltonian (1.10) is the sum of one-body terms, while the equations of motion (1.8)–(1.9) reflect two-body interactions. It should be stressed that this interpretation is based on viewing \(q_i\) and the dressed spin \(v(-)_i\) as degrees of freedom belonging to particle i. The same features hold in the complex holomorphic spin RS models as well [4], [5], [10]. It is also worth noting that the formulae of Theorem 5.8 enjoy a nice homogeneity property. Namely, let us define a \({\mathbb {Z}}^n\)-valued weight \(\text {wt}[-]\) by setting

$$\begin{aligned} \text {wt}[1] = \text {wt}[ q_j] =\text {wt}[ Q_j ] = 0\,, \quad \text {wt}[ \overline{v}(\alpha )_j ] = \text {wt}[ v(\alpha )_j] ={\mathbf {e}}_j\,, \quad \text {for } 1\le j \le n \,,\nonumber \\ \end{aligned}$$
(5.48)

where \({\mathbf {e}}_j\in {\mathbb {Z}}^n\) is \(+1\) in its j-th entry and zero everywhere else. Extending this weight by \( \text {wt}[ f g ] = \text {wt}[ f ] + \text {wt}[ g ] \) for homogeneous elements fg, we easily get that

$$\begin{aligned} \text {wt}[ {{\mathcal {U}}}_j^{\pm 1} ] =\pm {\mathbf {e}}_j\,, \quad \text {wt}[ F_{ij} ] ={\mathbf {e}}_i+{\mathbf {e}}_j\,, \quad \text {wt}[ L_{ij} ] ={\mathbf {e}}_i+{\mathbf {e}}_j\,. \end{aligned}$$
(5.49)

We can then observe from (5.37)–(5.39) that the reduced Poisson bracket preserves this weight.

Degenerate Integrability of the Reduced System

We discussed the degenerate integrability of the unreduced free system in Sect. 2.3 and now wish to show that this property is inherited by the reduced system. This is expected to hold not only in view of the earlier results on holomorphic spin RS systems [5], [10] and related models [44], [45], [46], but also on account of a general result in reduction theory. In fact, it is known (Theorem 2.16 in [58], see also [27]) that the integrability of invariant Hamiltonians on a manifold descends generically to the reduced space of Poisson reduction. However, the pertinent spaces of group orbits are typically not smooth manifolds. The existing results provide strong motivation, but do not help us directly to establish integrability in our concrete case.

Our goal is to prove the degenerate integrability of the reduced system in the real-analytic category by explicitly displaying the required integrals of motion. Specifically, we wish to show that the n reduced Hamiltonians arising from the functions

$$\begin{aligned} \text {tr}(L^k), \qquad k=1,\dots , n, \end{aligned}$$
(6.1)

are functionally independent and that one can complement them to \((2nd -n)\) independent functions using suitable reduced integrals of motion that arise from the real and imaginary parts of the \({\mathrm{U}}(n)\) invariant functions

$$\begin{aligned} I^k_{\alpha \beta } = v(\beta )^\dagger L^k v(\alpha ). \end{aligned}$$
(6.2)

These integrals of motion appeared before in Proposition 5.5. As throughout the paper, we assume that \(d\ge 2\).

The independence of functions means linear independence of their exterior derivatives at generic points, and this can be translated into the non-vanishing of a suitable Jacobian determinant. For real-analytic functions, the determinant at issue is also real-analytic, and hence, it is generically nonzero if it is nonzero at a single point. Thus, by patching together analytic charts, one sees that on a connected real-analytic manifold independence of real analytic functions follows from the linear independence of their derivatives at a single point. We can use this observation since we know (see Remark 3.6 and Corollary 3.15) that \({{\mathcal {M}}}_\text {red}\) is a connected real-analytic manifold.

Construction of Local Coordinates

Our first goal below is to construct local coordinates around certain points of the reduced phase space in which the formulae of the integrals of motion become simple. The coordinates will involve the eigenvalues of L, whereby the Hamiltonians \(\text {tr}(L^k)\) acquire a trivial form. We start by noting that the moment map constraint admits solutions for which only a single one of the vectors \(v(\alpha )\) is nonzero. Concerning those elements of \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\), the following useful result can be obtained from (the proof of) Lemma 5.2 of [19].

Lemma 6.1

Consider any \(y \in {\mathbb {R}}^n\) whose components \( y_1,\dots , y_n\) satisfy the inequalities

$$\begin{aligned} y_i> e^{2\gamma }y_{i+1} \quad \forall i=1,\dots , n\quad \hbox {with}\quad y_{n+1}:= 0. \end{aligned}$$
(6.3)

Then, there exists \((g^0,L^0,v^0)\in \Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) such that \(L^0=\mathrm{diag}(y_1,\ldots ,y_n)\) and \(v(\alpha )^0=0\) for each \(1\le \alpha <d\) (where \(d\ge 2\) and \(\gamma >0\)). For such elements, all components of the vector \(v(d)^0\) are nonzero.

Proof

Given \(L^0=\text {diag}(y_1,\ldots ,y_n)\) and \(v(1)^0=\ldots =v(d-1)^0=0\), we have to find \(g^0\in {\mathrm{U}}(n)\) and \(v(d)^0\in {\mathbb {C}}^n\) such that the moment map constraint (3.17) holds. Using (3.19), this means that

$$\begin{aligned} e^{2\gamma } (g^0)^{-1} L^0 g^0 = L^0 + v(d)^0 (v(d)^0)^\dagger . \end{aligned}$$
(6.4)

This is equivalent to the requirement that there exists \(v(d)^0\in {\mathbb {C}}^n\) such that \(L^0 + v(d)^0 (v(d)^0)^\dagger \) and \(e^{2\gamma }L^0\) have the same spectrum. But this holds if and only if we have the equality of polynomials in \(\lambda \)

$$\begin{aligned} \det (L^0 + v(d)^0 (v(d)^0)^\dagger -\lambda {{\varvec{1}}}_n)=\det ( e^{2\gamma }L^0 -\lambda {{\varvec{1}}}_n) = \prod _{k=1}^n (e^{2\gamma } y_k-\lambda )\,.\nonumber \\ \end{aligned}$$
(6.5)

We can expand the left-hand side as follows :

$$\begin{aligned} \begin{aligned} \det (L^0 + v(d)^0 (v(d)^0)^\dagger -\lambda {{\varvec{1}}}_n) =&\det (L^0-\lambda {{\varvec{1}}}_n) [1+(v(d)^0)^\dagger (L^0-\lambda {{\varvec{1}}}_n)^{-1} v(d)^0] \\ =&\prod _{k=1}^n (y_k-\lambda ) + \sum _{j=1}^{n}|v(d)_j^0|^2 \prod _{k\ne j} (y_k-\lambda )\,. \end{aligned}\nonumber \\ \end{aligned}$$
(6.6)

Thus, we seek \(v(d)^0\in {\mathbb {C}}^n\) such that

$$\begin{aligned} \prod _{k=1}^n (e^{2\gamma } y_k-\lambda )-\prod _{k=1}^n (y_k-\lambda ) = \sum _{j=1}^{n}|v(d)_j^0|^2 \prod _{k\ne j} (y_k-\lambda )\,. \end{aligned}$$
(6.7)

Evaluating this identity at \(\lambda =y_l\) yields

$$\begin{aligned} |v(d)_l^0|^2= (e^{2\gamma }-1)y_l \prod _{k\ne l} \frac{e^{2\gamma } y_k-y_l}{y_k-y_l}\,, \end{aligned}$$
(6.8)

which is positive due to (6.3). It now suffices to pick \(v(d)^0\) whose components have moduli given by (6.8), while we pick for \(g^0\) any unitary matrix diagonalizing \(L^0 + v(d)^0 (v(d)^0)^\dagger \) into \(e^{2\gamma } L^0\). \(\square \)

Remark 6.2

Notice that a completely gauge fixed normal form of the elements appearing in Lemma 6.1 can be obtained by requiring all components of the vector \(v(d)^0\) to be positive. We also note in passing that in the \(d=1\) case the set of possible (ordered) eigenvalues of L in \((g_R,L,v)\in \Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) is given [19] by the polyhedron in \({\mathbb {R}}^n\) specified by the conditions \(y_i\ge e^{2\gamma }y_{i+1}\) for all \(i=1,\dots , n-1\) and \(y_n>0\).

Now we introduce two subsets of the inverse image of the ‘constraint surface.’

Definition 6.3

Denote

$$\begin{aligned} {{\mathcal {S}}}=\{(g_R,L,v) \in \Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n) \mid L=\text {diag}(y_1,\ldots ,y_n),\, y_i>y_{i+1},\, v(1)_i>0 \,\, \forall i \}.\nonumber \\ \end{aligned}$$
(6.9)

The open subset \({{\mathcal {S}}}_1\subset {{\mathcal {S}}}\) is defined by imposing the further condition that the matrix

$$\begin{aligned} L_1:= L+\sum _{\alpha =1}^{d-1} v(\alpha )v(\alpha )^\dagger \end{aligned}$$
(6.10)

is conjugate to \(\text {diag}(\mu _1,\dots , \mu _n)\), where the \(\mu _i\) satisfy the inequalities

$$\begin{aligned} e^{2\gamma } y_i> \mu _i> e^{2\gamma }y_{i+1} \quad \forall i=1,\dots , n-1\quad \hbox {and}\quad e^{2\gamma } y_n > \mu _n. \end{aligned}$$
(6.11)

Note that \({{\mathcal {S}}}\) is non-empty since we can apply the analogue of Lemma 6.1 to obtain elements of \(\Lambda ^{-1}(e^\gamma {{\varvec{1}}}_n)\) for which only v(1) is nonzero, and \({{\mathcal {S}}}_1\) is non-empty since for those elements \(L_1=L\). It is clear that \({{\mathcal {S}}}\) can serve as a model of an open dense subset of the reduced phase space. Below, we provide a full characterization of the elements of \({{\mathcal {S}}}_1\).

Taking y and \(\mu \) subject to the inequalities in (6.11), define \({{\mathbf {V}}}(y,\mu ) \in {\mathbb {R}}^n\) by

$$\begin{aligned} {{\mathbf {V}}}_l(y,\mu ):= \left[ (e^{2\gamma }y_l-\mu _l) \prod _{k\ne l} \frac{e^{2\gamma } y_k-\mu _l}{\mu _k-\mu _l}\right] ^{\frac{1}{2}}\, \quad \forall l=1,\dots , n. \end{aligned}$$
(6.12)

Observe that the function under the square root is positive, and the positive root is taken.

Lemma 6.4

For any \((g_R, L, v)\in {{\mathcal {S}}}_1\) pick a matrix \(g_1 \in {\mathrm{U}}(n)\) for which

$$\begin{aligned} g_1 L_1 g_1^{-1} = \mathrm{diag}(\mu _1,\dots , \mu _n) \end{aligned}$$
(6.13)

with \(\mu \) satisfying (6.11). Then, v(d) is of the form

$$\begin{aligned} v(d) = g_1^{-1} \mathrm{diag}(\tau _1,\dots , \tau _n) {{\mathbf {V}}}(y,\mu ) \quad \hbox {with some}\quad \tau \in {\mathbb {T}}^n. \end{aligned}$$
(6.14)

Furthermore, \(g_R\) is of the form

$$\begin{aligned} g_R = \mathrm{diag}(\Gamma _1,\dots , \Gamma _n) g_R^0 \quad \hbox {with some}\quad \Gamma \in {\mathbb {T}}^n, \end{aligned}$$
(6.15)

where \(g_R^0 \in {\mathrm{U}}(n)\) is a fixed solution of the constraint equation

$$\begin{aligned} g_R^{-1} e^{2\gamma } L g_R = L_1 + v(d) v(d)^\dagger . \end{aligned}$$
(6.16)

Conversely, take any positive definite \(L=\mathrm{diag}(y_1,\dots , y_n)\) and \({\mathbb {C}}^n\) vectors \(v(1),\dots , v(d-1)\) such that L and \(L_1\) given by (6.10) satisfy the spectral conditions (6.11), and all components of v(1) are positive. Choose a diagonalizer \(g_1\) according to (6.13) and define \(v(d)\in {\mathbb {C}}^n\) by the formula (6.14) using an arbitrary \(\tau \in {\mathbb {T}}^n\). Then, Eq. (6.16) admits solutions for \(g_R\), the general solution has the form (6.15) with arbitrary \(\Gamma \in {\mathbb {T}}^n\), and all so obtained triples \((g_R, L, v)\) belong to \({{\mathcal {S}}}_1\).

Proof

By using the definitions (3.25) of L and (6.10) of \(L_1\), we can always recast the moment map constraint (3.19) in the form (6.16), which implies the equality of characteristic polynomials

$$\begin{aligned} \det ( e^{2\gamma } L -\lambda {{\varvec{1}}}_n)= \det (L_1 + v(d) v(d)^\dagger -\lambda {{\varvec{1}}}_n). \end{aligned}$$
(6.17)

Since L is diagonal for \((g,L,v)\in {{\mathcal {S}}}_1\), we have

$$\begin{aligned} \det (e^{2\gamma } L-\lambda {{\varvec{1}}}_n) = \prod _j (e^{2\gamma } y_j - \lambda ). \end{aligned}$$
(6.18)

By using (6.13) and introducing

$$\begin{aligned} {\tilde{u}}:=g_1 v(d), \end{aligned}$$
(6.19)

we can write the polynomial on the right-hand side of (6.17) as

$$\begin{aligned} \begin{aligned} \det (L_1-\lambda {{\varvec{1}}}_n) [1+v(d)^\dagger (L_1-\lambda {{\varvec{1}}}_n)^{-1} v(d)] =&\prod _{k=1}^n (\mu _k-\lambda ) \, \left[ 1+ \sum _{j=1}^n \tilde{u}^\dagger _j \frac{1}{\mu _j-\lambda }\tilde{u}_j \right] \\ =&\prod _{k=1}^n (\mu _k-\lambda ) + \sum _{j=1}^{n}|\tilde{u}_j|^2 \prod _{k\ne j} (\mu _k-\lambda )\,. \end{aligned}\nonumber \\ \end{aligned}$$
(6.20)

Thus, (6.17) evaluated at \(\lambda =\mu _l\) yields

$$\begin{aligned} |\tilde{u}_l|^2= (e^{2\gamma }y_l-\mu _l) \prod _{k\ne l} \frac{e^{2\gamma } y_k-\mu _l}{\mu _k-\mu _l} = {{\mathbf {V}}}_l(y,\mu )^2, \end{aligned}$$
(6.21)

which is positive due to (6.11). We conclude from this and Eq. (6.19) that v(d) has the form (6.14). The claim (6.15) about the form of \(g_R\) follows from (6.16) since L is diagonal and has distinct eigenvalues.

The converse statement is proved by utilizing that the equality of the polynomials in \(\lambda \) (6.17) is equivalent to the existence of a unitary matrix \(g_R\) that solves the constraint equation (6.16). Then, we simply turn the above arguments backward. The crux is that the spectral assumption (6.11) ensures the positivity of the expression in (6.12), whence v(d) can be constructed starting from the vector \({\tilde{u}} = \text {diag}(\tau _1,\dots , \tau _n){{\mathbf {V}}}(y,\mu )\). \(\square \)

From now on, we write

$$\begin{aligned} v(\alpha )_j = v(\alpha )_j^\mathfrak {R}+ {\mathrm{i}}v(\alpha )_j^\mathfrak {I}\quad \hbox {for}\quad \alpha =2,\dots , d-1, \end{aligned}$$
(6.22)

with real-valued \(v(\alpha )_j^\mathfrak {R}\), \(v(\alpha )_j^\mathfrak {I}\). In the next statement, we summarize how Lemma 6.4 gives us coordinates on \({{\mathcal {S}}}_1\).

Corollary 6.5

Via the formulae of Lemma 6.4 for v(d) and \(g_R\), the elements of \({{\mathcal {S}}}_1\) are uniquely parametrized by the \(2n(d-1)\) variables

$$\begin{aligned} y_j, v(1)_j, v(\alpha )_j^\mathfrak {R}, v(\alpha )_j^\mathfrak {I}, \quad j=1,\dots , n, \quad \alpha =2,\dots , d-1\,, \end{aligned}$$
(6.23)

together with the 2n variables

$$\begin{aligned} \tau _j \in {\mathrm{U}}(1), \quad \Gamma _j \in {\mathrm{U}}(1), \quad j=1,\dots , n\,. \end{aligned}$$
(6.24)

The variables (6.23) take values in an open subset of \({\mathbb {R}}^{2n(d-1)}\). The matrix elements of \(g_1\) (6.13) can be chosen to be real-analytic functions of the \(2n(d-1)\) variables (6.23), and then, the components of v(d) (6.14) are also real-analytic functions of these variables and the \(\tau _j\). Likewise, the matrix elements of \(g_R^0\) can be chosen to be real-analytic functions of the variables (6.23) and the \(\tau _j\). Consequently, the variables (6.23) together with \(t_j\) and \(\gamma _j\) in \(\tau _j=e^{{\mathrm{i}}t_j}\) and \(\Gamma _j = e^{{\mathrm{i}}\gamma _j}\) define a coordinate system on the open submanifold of the reduced phase space corresponding to \({{\mathcal {S}}}_1\).

Proof

The variables (6.23) run over an open set simply because the eigenvalues of \(L_1\) depend continuously on them. This dependence is actually analytic since those eigenvalues are all distinct. Regarding the dependence of \(g_1\) and \(g_R^0\) on the variables, we use the well-known fact that the eigenvectors of regular Hermitian matrices can be chosen as analytic functions of the independent parameters of the matrix elements. \(\square \)

Degenerate Integrability

The reduced integrals of motion arising from (6.1) and (6.2) take a simple form in terms of our coordinates on \({{\mathcal {S}}}_1\). Relying on this, we shall inspect the following \(2n(d-1)\) reduced integrals of motion:

$$\begin{aligned} \begin{aligned} \text {tr}(L^k)=\sum _j y_j^k\,, \quad&I_{1,1}^k=\sum _j v(1)_j^2 y^{k}_j\,, \\ \mathfrak {R}[I_{\alpha ,1}^k]=\sum _j v(1)_j y^{k}_j v(\alpha )_j^\mathfrak {R},\quad&\mathfrak {I}[I_{\alpha ,1}^k]=\sum _j v(1)_j y^{k}_j v(\alpha )_j^\mathfrak {I}\,, \end{aligned} \end{aligned}$$
(6.25)

where \(k=1,\dots , n\) and \(\alpha =2,\dots , d-1\), and the additional 2n integrals of motion supplied by the real and imaginary parts of

$$\begin{aligned} I^k_{d,1}= \sum _j v(1)_j y_j^k v(d)_j \quad \hbox {with}\quad v(d) = g_1^{-1} \text {diag}(\tau _1,\ldots ,\tau _n) {{\mathbf {V}}}. \end{aligned}$$
(6.26)

Proposition 6.6

The \(2n (d-1)\) reduced integrals of motion (6.25), which include the n reduced Hamiltonians \(\mathrm{tr}(L^k)\), are functionally independent on \({{\mathcal {S}}}_1\). On each connected component of \({{\mathcal {S}}}_1\), n further integrals of motion may be selected from the real and imaginary parts of the functions (6.26) in such a way that together with (6.25) they provide a set of \(2nd - n\) independent functions.

Proof

We are going to prove functional independence by inspection of Jacobian determinants using the coordinates on \({{\mathcal {S}}}_1\) exhibited in Corollary 6.5. Let us first consider the functions given by (6.25). If we order the \(2n(d-1)\) functions as written in (6.25) and also order the \(2n(d-1)\) coordinates as written in (6.23), then the corresponding Jacobian matrix J takes a block lower-triangular form, with \(n\times n\) blocks. The first diagonal block, \((\partial \text {tr}L^k/\partial y_j)\), is given by \(Y\in {\text {Mat}}_{n\times n}({\mathbb {R}})\) with \(Y_{kj}= k y_j^{k-1}\), while all other diagonal blocks are given by \(XD_1\) with \(X_{kj}=y^k_j\) and \(D_1=\text {diag}(v(1)_1,\ldots , v(1)_n)\), except the second one, \((\partial I^k_{1,1}/\partial v(1)_j)\), which equals \(2 X D_1\). By the definition of \({{\mathcal {S}}}_1\), the coordinates \(y_j\) are positive and distinct, while the \(v(1)_j\) are positive, so that X, Y and \(D_1\) are invertible. Hence, J has rank \(2n(d-1)\).

To continue, consider the 2n functions

$$\begin{aligned} \mathfrak {R}(I_{d,1}^k),\, \mathfrak {I}(I_{d,1}^k) \quad \hbox {with}\quad 1\le k \le n. \end{aligned}$$
(6.27)

It is clear that any function G taken from (6.25) satisfies \(\partial G/\partial t_j=0\). So our claim will follow if there exists a subset of n functions \(F_1,\ldots ,F_n\) from those in (6.27) for which the Jacobian matrix \(\left( \frac{\partial F_k}{\partial t_l}\right) _{kl}\) is invertible.

Note from Lemma 6.4 and Corollary 6.5 that

$$\begin{aligned} \frac{\partial v(d)_j}{\partial t_l}={\mathrm{i}}(g_1^{-1})_{jl} e^{{\mathrm{i}}t_l} {{\mathbf {V}}}_l\,, \end{aligned}$$
(6.28)

since \({{\mathbf {V}}}\) and \(g_1\) depend only on the variables (6.23). In particular, the matrix

$$\begin{aligned} \frac{\partial v(d)}{\partial t} := \left( \frac{\partial v(d)_j}{\partial t_l}\right) _{1\le j,l\le n} \end{aligned}$$
(6.29)

is invertible, because so are \(g_1\), \(\text {diag}( e^{{\mathrm{i}}t_1},\ldots , e^{{\mathrm{i}}t_n})\) and \(\text {diag}({{\mathbf {V}}}_{1},\ldots ,{{\mathbf {V}}}_{n})\).

If the \(2n\times n\) real matrix

$$\begin{aligned} \left( \frac{\partial \Big (\mathfrak {R}(I_{d,1}^k), \mathfrak {I}(I_{d,1}^k)\Big ) }{\partial t_l}\right) _{1\le k,l\le n} \end{aligned}$$
(6.30)

has rank n, then we are done. Assume by contradiction that this matrix has rank less than n. From (6.26), we have

$$\begin{aligned} \frac{\partial I_{d,1}^k}{\partial t_l} = \sum _j v(1)_j y_j^k \frac{\partial v(d)_j}{\partial t_l}\,, \end{aligned}$$
(6.31)

and therefore we can write the following equality of complex matrices

$$\begin{aligned} \frac{\partial I_{d,1}}{\partial t} := \left( \frac{\partial I_{d,1}^k }{\partial t_l}\right) _{1\le k,l\le n} = X \text {diag}(v(1)_1,\ldots ,v(1)_n) \frac{\partial v(d)}{\partial t}\,, \end{aligned}$$
(6.32)

where X is given by \(X_{kj}=y_j^k\) as before. We have already established that all three factors in the above product of matrices are invertible. Thus, \(\partial I_{d,1}/\partial t \) is invertible and hence has rank n.

To finish the proof, it suffices to remark that the complex matrix \(\partial I_{d,1}/\partial t\) is a complex linear combination of the rows of the matrix given in (6.30). If the latter matrix has rank strictly less than n, then so does \(\partial I_{d,1}/\partial t\), which gives a contradiction. \(\square \)

Let us recall from Proposition 5.5 that the unreduced phase space supports the polynomial Poisson algebra \({\mathcal {I}}_L\) (5.16), whose Poisson center contains the polynomial algebra

$$\begin{aligned} {\mathfrak {H}}_{\text {tr}}:={\mathbb {R}}[\text {tr}L^k, k \ge 0]\,. \end{aligned}$$
(6.33)

Since these Poisson algebras consist of \({\mathrm{U}}(n)\) invariant functions, they engender corresponding Poisson algebras \({\mathcal {I}}_L^\text {red}\) and \({\mathfrak {H}}_{\text {tr}}^\text {red}\) over the reduced phase space \({{\mathcal {M}}}_\text {red}\). Our final result is a direct consequence of Proposition 6.6.

Theorem 6.7

The reduced polynomial algebras of functions \({\mathfrak {H}}_\mathrm{tr}^\mathrm{red}\) and \({\mathcal {I}}_L^\mathrm{red}\) inherited from \({\mathfrak {H}}_\mathrm{r}\) (6.33) and \({\mathcal {I}}_L\) (5.16) have functional dimension n and \(2nd - n\), respectively. In particular, on the phase space \({{\mathcal {M}}}_\mathrm{red}\) of dimension 2nd, the Abelian Poisson algebra \({\mathfrak {H}}_\mathrm{tr}^\mathrm{red}\) yields a real-analytic, degenerate integrable system with integrals of motion \({\mathcal {I}}_L^\mathrm{red}\).

Proof

Let us consider \({\mathcal {I}}_L^\text {red}\) and its Poisson center \({{\mathcal {Z}}}({\mathcal {I}}_L^\text {red})\). Denote r and \(r_0\) the functional dimensions of these polynomial algebras of functions. Observe from Proposition 6.6 that

$$\begin{aligned} r\ge (2nd - n) \quad \hbox {and}\quad r_0 \ge n. \end{aligned}$$
(6.34)

The second inequality holds since \({\mathfrak {H}}_{\text {tr}}^\text {red}\) is contained in \({{\mathcal {Z}}}({\mathcal {I}}_L^\text {red})\), and Proposition 6.6 implies that the functional dimension of \({\mathfrak {H}}_{\text {tr}}^\text {red}\) is n.

In a neighborhood \(U_0\) of a generic point of \({{\mathcal {M}}}_\text {red}\), we can choose a system of coordinates given by 2nd functions \(F_1,\dots , F_{2nd}\) such that the first r functions belong to \({\mathcal {I}}_L^\text {red}\), of which the first \(r_0\) belong to \({{\mathcal {Z}}}({\mathcal {I}}_L^\text {red})\). In terms of such coordinates, the Poisson matrix \(P=(\{ F_i,F_j \})_{i,j}\) can be decomposed into blocks as

$$\begin{aligned} P=\left( \begin{array}{ccc} 0_{r_0\times r_0}&{} 0_{r_0\times (r-r_0)}&{} B \\ 0_{(r-r_0)\times r_0}&{} A&{} *\\ -B^T&{}\,\,*\,\, &{} *\end{array} \right) . \end{aligned}$$
(6.35)

This matrix must be non-degenerate since the reduced phase space is a symplectic manifold. In particular, this implies that the \(r_0\) rows of B must be independent. Then the number of independent columns of B must be also \(r_0\), which cannot be bigger than the number of columns. This gives \(r_0 \le (2nd -r)\), or equivalently

$$\begin{aligned} r_0 + r \le 2nd. \end{aligned}$$
(6.36)

By combining (6.34) with (6.36), we obtain that \(r_0 = n\) and \(r = (2nd -n)\).

\(\square \)

We see from the above proof that \({{\mathcal {Z}}}({\mathcal {I}}_L^\text {red})\) and \({\mathfrak {H}}_{\text {tr}}^\text {red}\) have the same functional dimension. Since \({\mathfrak {H}}_{\text {tr}}^\text {red}\subseteq {{\mathcal {Z}}}({\mathcal {I}}_L^\text {red})\), we expect that these polynomial algebras of functions actually coincide.

Remark 6.8

Let us explain that our coordinates on \({{\mathcal {S}}}_1\) are very close to action-angle variables. To start, we recall that the joint level surfaces of the integrals of motion of the unreduced free system are compact, because (with the help of the variables \((g_R, L, W)\)) they can be identified with closed subsets of \({\mathrm{U}}(n)\). This compactness property is inherited by the reduced system. If we restrict ourselves to the open subset of the reduced phase space parametrized by \({{\mathcal {S}}}_1\), then the connected components of the joint level surfaces of the elements of \({\mathcal {I}}_L^\text {red}\) (5.16) are the n-dimensional ‘\(\Gamma \)-tori’ obtained by fixing all variables in (6.23) and (6.24) except the \(\Gamma _j\). Both the gauge slice \({{\mathcal {S}}}\) (6.9) and its subset \({{\mathcal {S}}}_1\) are invariant under the flow (2.42) of the Hamiltonian \(H_k:= \frac{1}{2k} \text {tr}(L^k)\), for every \(k=1,\dots , n\), which gives the following linear flow on the \(\Gamma \)-torus:

$$\begin{aligned} \Gamma _j(t) = \exp ( {\mathrm{i}}y_j^k t) \Gamma _j^0,\quad \forall j=1,\dots , n, \end{aligned}$$
(6.37)

where \(\Gamma _j^0\) refers to the initial value. This statement holds since for \(H_k(L) \equiv h_k(b_R)\) one has \(D h_k(b_R) = {\mathrm{i}}L^k\). The flow (6.37) entails that on \({{\mathcal {S}}}_1\) the variables \({{\hat{p}}}_j:=\frac{1}{2} \log y_j\) are canonical conjugates to the angles \(\gamma _j\) in \(\Gamma _j = e^{{\mathrm{i}}\gamma _j}\), i.e., they satisfy \(\{ \gamma _j, {{\hat{p}}}_l\}_\text {red}= \delta _{jl}\).

Conclusion

In this paper, we investigated a trigonometric real form of the spin RS system (1.2) introduced originally by Krichever and Zabrodin [32] and studied subsequently in [5], [10] in the complex holomorphic setting. We have shown that this real form arises from Hamiltonian reduction of a free system on a spin extended Heisenberg double of the \({\mathrm{U}}(n)\) Poisson–Lie group and exploited the reduction approach for obtaining a detailed characterization of its main features. In particular, we presented two models of dense open subsets of the reduced phase space where the system lives. The model developed in Sect. 3.3 led to an elegant description of the reduced symplectic form (Theorem 3.14), while the equations of motion and the corresponding Hamiltonian are complicated in the pertinent variables based on the ‘primary spins.’ On the other hand, the model studied in Sect. 3.2 and in Sects. 4 and  5 allowed us to recover the spin RS equations of motion (1.8) from the projection of a free flow (Corollary 4.3), but the reduced Poisson brackets (Theorem 5.8) take a relatively complicated form in the underlying ‘dressed spin’ variables. In our framework, the solvability of the evolution equations by linear algebraic manipulations emerges naturally (Remark 4.2), and we also proved their degenerate integrability by explicitly exhibiting the required number of constants of motion (Theorem 6.7).

A basic ingredient of the unreduced phase space that we started with was a \({\mathrm{U}}(n)\) covariant Poisson structure on \({\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}\) that goes back to Zakrzewski [56], for which we found the corresponding moment map (Proposition A.3) and symplectic form (Proposition A.6).

We finish by highlighting a few open problems related to our current research. As always in the reduction treatment of an integrable Hamiltonian system, one should gain as complete an understanding of the global structure of the reduced phase space as possible. The basic point is that the projections of free flows are automatically complete, but only on the full reduced phase space. In the present case, one should actually construct two global models of the reduced phase space: one fitted to the system that we have studied and another one that should be associated with its action-angle dual. Without going into details, we refer to the literature [20], [23], [44], [49] where it is explained that the integrable many-body systems usually come in dual pairs, and the same holds for their several spin extensions. In our case, the commuting Hamiltonians of the dual system are expected to arise from the reduction of the Abelian Poisson algebra \({\hat{{\mathfrak {H}}}} = \Xi _R^*(C^\infty ({\mathrm{U}}(n)))\), which is in some sense dual to \({\mathfrak {H}}\) (2.40) on which our system was built.

It could be interesting to explore generalizations of the construction employed in our study. For example, one may obtain new variants of the trigonometric spin RS model by replacing some or all of the primary spins \(w^\alpha \) by \(z^\alpha \) subject to the Poisson bracket described at the end of Appendix A (Remark A.7). Generalization of our reduction in which the Heisenberg double is replaced by a quasi-Hamiltonian double of the form \({\mathrm{U}}(n)\times {\mathrm{U}}(n)\) [2], and the primary spins are also modified suitably, should lead to compactified spin RS systems. It should be possible to uncover a reduction picture behind the hyperbolic real form of the spinless and spin RS models, too. All these issues, as well as the questions of quantization and the reduction approach to elliptic spin RS models, pose challenging problems for future work.

Notes

  1. This form of the chiral RS Hamiltonian is the one found in [50]. Different Darboux variables, which avoid the appearance of square roots in the Hamiltonian, are also often used in the literature.

  2. In full generality, the concept of Poisson–Lie moment map goes back to Lu [36]; our conventions are slightly different from hers.

  3. This means that the exterior derivatives of the elements of \({\mathfrak {H}}\) span an r-dimensional subspace of the cotangent space for generic points of \({{\mathcal {M}}}\), which form a dense open submanifold.

  4. More precisely, \({{\mathcal {M}}}_0^\text {reg}/{\mathbb {T}}^n\) is a principal fiber bundle with structure group \(S_n\) over the base \({{\mathcal {M}}}_\text {red}^\text {reg}\).

  5. This difficulty evaporates in the scaling limit discussed in Remark 3.16.

  6. This is a symplectic gradient associated with the standard symplectic form, \(\omega (\xi ,\eta )=\mathfrak {I}(\xi ^\dagger \eta )\), on \({\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}\).

  7. In our convention, the wedge does not contain \({\textstyle {\frac{1}{2}}}\) and \(dH = \Omega (\,\cdot \,, V_H)\) with the Hamiltonian vector field \(V_H\).

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Acknowledgements

We wish to thank J. Balog for useful remarks on the manuscript. We are also grateful to A. Alekseev and R. Sjamaar for advise regarding proper moment maps. The research of M.F. was partly supported by the travel grant ECR-1819-01 of the London Mathematical Society and a Rankin-Sneddon Research Fellowship of the University of Glasgow. The work of L.F. was supported in part by the NKFIH research grant K134946.

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Communicated by Nikolai Kitanine.

Appendices

Properties of the primary spin variables

In this appendix, we first elaborate the properties of the primary spin variables that were summarized in Proposition 2.1. As was already mentioned, the pertinent Poisson structure on \({\mathbb {C}}^n \simeq {\mathbb {R}}^{2n}\) is a special case of the \({\mathrm{U}}(n)\) covariant Poisson structures due to Zakrzewski [56]. Nevertheless, to make our text self-contained, we shall also verify its Jacobi identity and covariance property. Then, we present the corresponding moment map and symplectic form, which have not been considered in previous work.

For any real function \(F\in C^\infty ({\mathbb {C}}^n)\), we define its \({\mathbb {C}}^n\)-valued ‘gradient’ \(\nabla F\) by the equalityFootnote 6

$$\begin{aligned} \mathfrak {I}\left( (\nabla F(w))^\dagger V\right) := \left. \frac{\hbox {d}}{\hbox {d}t}\right| _{t=0}F(w + t V), \qquad \forall w, V\in {\mathbb {C}}^n, \end{aligned}$$
(A.1)

where the elements of \({\mathbb {C}}^n\) are viewed as column vectors. We note that any real linear function on the real vector space \({\mathbb {C}}^n\) is of the form

$$\begin{aligned} F_\xi (w):= \mathfrak {I}(\xi ^\dagger w), \end{aligned}$$
(A.2)

for some \(\xi \in {\mathbb {C}}^n\), and for such function \(\nabla F_\xi = \xi \). Next we give a convenient presentation of Zakrzewski’s Poisson bracket.

Proposition A.1

For real functions \(F, H\in C^\infty ({\mathbb {C}}^n)\), let \(\xi (w):= \nabla F(w)\) and \(\eta (w):= \nabla H(w)\). Then, the following formula

$$\begin{aligned} \{F, H\}(w)= & {} \mathfrak {I}\left( \xi (w)^{\dagger }(w \eta (w)^{\dagger })_{{\mathfrak {u}}(n)} w - {\textstyle {\frac{1}{2}}}\xi (w)^{\dagger }w\eta (w)^{\dagger }w - {\textstyle {\frac{1}{2}}}\xi (w)^{\dagger }w w^{\dagger }\eta (w)\right. \nonumber \\&\left. - \xi (w)^{\dagger }\eta (w) \right) , \end{aligned}$$
(A.3)

where the notation (2.4) is used, defines a Poisson bracket on \(C^\infty ({\mathbb {C}}^n)\). Equivalently to the formula (A.3), the Hamiltonian vector field \(V_H\) associated with \(H\in C^\infty ({\mathbb {C}}^n)\) is given by

$$\begin{aligned} V_H(w) = (w \eta (w)^{\dagger })_{{\mathfrak {u}}(n)} w - \eta (w) -{\textstyle {\frac{1}{2}}}(\eta (w)^{\dagger }w + w^{\dagger }\eta (w))w, \qquad \eta (w)= \nabla H(w). \nonumber \\ \end{aligned}$$
(A.4)

Extending the real Poisson bracket to complex functions by complex bilinearity, the Poisson brackets of the component functions \(w\mapsto w_i\) satisfy the explicit formulae (2.28) and (2.29).

Proof

The anti-symmetry of the last two terms of (A.3) is obvious, while the anti-symmetry of the sum of the first and second terms is seen from the identity

$$\begin{aligned} \mathfrak {I}\left( \xi ^{\dagger }(w \eta ^{\dagger })_{{\mathfrak {u}}} w - {\textstyle {\frac{1}{2}}}\xi ^{\dagger }w\eta ^{\dagger }w\right) = {\textstyle {\frac{1}{2}}}\mathfrak {I}\,\text {tr}\left( (w \eta ^{\dagger })_{{\mathfrak {u}}} (w \xi ^{\dagger })_{\mathfrak {b}}-(w \xi ^{\dagger })_{{\mathfrak {u}}} (w \eta ^{\dagger })_{\mathfrak {b}}\right) ,\nonumber \\ \end{aligned}$$
(A.5)

where we used constant \(\xi \) and \(\eta \) for simplicity. Here and below, the subscripts \({\mathfrak {u}}\) and \({\mathfrak {b}}\) stand for \({\mathfrak {u}}(n)\) and \({\mathfrak {b}}(n)\).

Regarding the Jacobi identity, it is enough to verify it for linear functions \(F_\xi \), \(F_\eta \) and \(F_\zeta \) for arbitrary \(\xi ,\eta ,\zeta \in {\mathbb {C}}^n\). In this verification, we may use the formula (A.4), since this expresses the identity \(\{F,H\}(w) = \mathfrak {I}(\xi (w)^\dagger V_H(w))\) and does not rely on the Jacobi identity.

We start by calculating the gradient of \(\{ F_\xi , F_\eta \}\) from (A.3) and find

$$\begin{aligned} \left( \nabla \{ F_\xi , F_\eta \}(w)\right) ^\dagger = \xi ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}- \eta ^{\dagger }(w\xi ^{\dagger })_{\mathfrak {u}}+ {\textstyle {\frac{1}{2}}}\eta ^{\dagger }w\xi ^{\dagger }- {\textstyle {\frac{1}{2}}}\xi ^{\dagger }w \eta ^{\dagger }- {\textstyle {\frac{1}{2}}}(w^{\dagger }\eta )\xi ^{\dagger }+ {\textstyle {\frac{1}{2}}}(w^{\dagger }\xi )\eta ^{\dagger }.\nonumber \\ \end{aligned}$$
(A.6)

Combining this with \(V_{{F_\zeta }}(w)\) from (A.4), we have to inspect

$$\begin{aligned} \begin{aligned} {{\mathcal {J}}}(w):&= \{\{ F_\xi , F_\eta \}, F_\zeta \}(w) + \hbox {cycl. perm.} \\&=\mathfrak {I}\left[ \left( \nabla \{ F_\xi , F_\eta \}(w)\right) ^\dagger \left( (w\zeta ^{\dagger })_{\mathfrak {u}}w - \zeta - {\textstyle {\frac{1}{2}}}(\zeta ^{\dagger }w)w - {\textstyle {\frac{1}{2}}}(w^{\dagger }\zeta )w\right) \right] + \hbox {c.p.} \end{aligned}\nonumber \\ \end{aligned}$$
(A.7)

By spelling this out, we obtain

$$\begin{aligned} \begin{aligned}&{{\mathcal {J}}(w)}= \mathfrak {I}\Bigl [ \eta ^{\dagger }(w\xi ^{\dagger })_{\mathfrak {u}}\zeta - \xi ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}\zeta + {\textstyle {\frac{1}{2}}}(\xi ^{\dagger }w)\eta ^{\dagger }\zeta \\&\quad - {\textstyle {\frac{1}{2}}}(\eta ^{\dagger }w)\xi ^{\dagger }\zeta + {\textstyle {\frac{1}{2}}}(w^{\dagger }\eta )\xi ^{\dagger }\zeta - {\textstyle {\frac{1}{2}}}(w^{\dagger }\xi )\eta ^{\dagger }\zeta + \hbox {c.p.}\Bigr ] \\&\quad + \mathfrak {I}\Bigl [\xi ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}(w\zeta ^{\dagger })_{\mathfrak {u}}w - \eta ^{\dagger }(w\xi ^{\dagger })_{\mathfrak {u}}(w\zeta ^{\dagger })_{\mathfrak {u}}w + {\textstyle {\frac{1}{2}}}\eta ^{\dagger }w\xi ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}w - {\textstyle {\frac{1}{2}}}\xi ^{\dagger }w \eta ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}w \\&\quad + {\textstyle {\frac{1}{2}}}(w^{\dagger }\xi )\eta ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}w - {\textstyle {\frac{1}{2}}}(w^{\dagger }\eta ) \xi ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}w +\hbox { c.p.} \\&\quad - {\textstyle {\frac{1}{2}}}(\zeta ^{\dagger }w)\xi ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}w + {\textstyle {\frac{1}{2}}}(\zeta ^{\dagger }w)\eta ^{\dagger }(w\xi ^{\dagger })_{\mathfrak {u}}w - \textstyle {\frac{1}{4}}(\eta ^{\dagger }w)(\xi ^{\dagger }w)(\zeta ^{\dagger }w) + \textstyle {\frac{1}{4}}(\xi ^{\dagger }w) (\eta ^{\dagger }w)(\zeta ^{\dagger }w) \\&\quad + \textstyle {\frac{1}{4}}(w^{\dagger }\eta )(\xi ^{\dagger }w)(\zeta ^{\dagger }w) - \textstyle {\frac{1}{4}}(w^{\dagger }\xi )(\eta ^{\dagger }w)(\zeta ^{\dagger }w) + \hbox {c.p.} \\&\qquad - {\textstyle {\frac{1}{2}}}(w^{\dagger }\zeta )\xi ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}w + {\textstyle {\frac{1}{2}}}(w^{\dagger }\zeta )\eta ^{\dagger }(w\xi ^{\dagger })_{\mathfrak {u}}w - \textstyle {\frac{1}{4}}(w^{\dagger }\zeta )(\eta ^{\dagger }w)(\xi ^{\dagger }w) + \textstyle {\frac{1}{4}}(w^{\dagger }\zeta ) (\xi ^{\dagger }w)(\eta ^{\dagger }w) \\&\quad + \textstyle {\frac{1}{4}}(w^{\dagger }\eta )(w^{\dagger }\zeta )(\xi ^{\dagger }w) - \textstyle {\frac{1}{4}}(w^{\dagger }\xi )(w^{\dagger }\zeta )(\eta ^{\dagger }w) + \hbox {c.p.}\Bigr ] \end{aligned} \end{aligned}$$

After making several self-evident cancellations, and using cyclic permutations to reorganize terms in a convenient way, we get

$$\begin{aligned} \begin{aligned} {{\mathcal {J}}(w)}&=\mathfrak {I}\Bigl (\zeta ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}\xi - \xi ^{\dagger }(w\eta ^{\dagger })_{\mathfrak {u}}\zeta + {\textstyle {\frac{1}{2}}}(\xi ^{\dagger }w - w^{\dagger }\xi )\eta ^{\dagger }\zeta - {\textstyle {\frac{1}{2}}}(\eta ^{\dagger }w - w^{\dagger }\eta )\xi ^{\dagger }\zeta + \hbox {c.p.}\Bigr )\\&\quad +\,\mathfrak {I}\,\text {tr}\Bigl (w\xi ^{\dagger }\bigl [(w\eta ^{\dagger })_{\mathfrak {u}},(w\zeta ^{\dagger })_{\mathfrak {u}}\bigr ] + (\eta ^{\dagger }w) w\xi ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}- (\xi ^{\dagger }w) w\eta ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}+ \hbox {c.p.}\Bigr ). \end{aligned} \end{aligned}$$

It is not difficult to see that the first line gives zero. Rearranging the second line, we have

$$\begin{aligned} {{\mathcal {J}}(w)}= & {} \mathfrak {I}\,\text {tr}\Bigl (w\xi ^{\dagger }\bigl [(w\eta ^{\dagger })_{\mathfrak {u}},(w\zeta ^{\dagger })_{\mathfrak {u}}\bigr ] + w\eta ^{\dagger }w\xi ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}- w\xi ^{\dagger }w\eta ^{\dagger }(w\zeta ^{\dagger })_{\mathfrak {u}}+ \hbox {c.p.}\Bigr ) \\= & {} \mathfrak {I}\,\text {tr}\Bigl (-w\xi ^{\dagger }\bigl [(w\eta ^{\dagger })_{\mathfrak {b}},(w\zeta ^{\dagger })_{\mathfrak {u}}\bigr ] + \hbox {c.p.}\Bigr )\\= & {} - \mathfrak {I}\,\text {tr}\Bigl ((w\xi ^{\dagger })_{\mathfrak {u}}\bigl ( [(w\eta ^{\dagger })_{\mathfrak {b}},(w\zeta ^{\dagger })_{\mathfrak {u}}] + [w\eta ^{\dagger },(w\zeta ^{\dagger })_{\mathfrak {b}}]\bigr ) \\&+ (w\xi ^{\dagger })_{\mathfrak {b}}\bigl ( [(w\eta ^{\dagger })_{\mathfrak {b}},(w\zeta ^{\dagger })_{\mathfrak {u}}] + [(w\eta ^{\dagger })_{\mathfrak {u}}, w\zeta ^{\dagger }]\bigr )\Bigr )\\= & {} -\mathfrak {I}\,\text {tr}\Bigl ( (w\xi ^{\dagger })_{\mathfrak {u}}\bigl ( [(w\eta ^{\dagger })_{\mathfrak {b}},(w\zeta ^{\dagger })_{\mathfrak {u}}] + [(w\eta ^{\dagger })_{\mathfrak {u}},(w\zeta ^{\dagger })_{\mathfrak {b}}] + [(w\eta ^{\dagger })_{\mathfrak {b}},(w\zeta ^{\dagger })_{\mathfrak {b}}]\bigr )\\&+ (w\xi ^{\dagger })_{\mathfrak {b}}\bigl ( [(w\eta ^{\dagger })_{\mathfrak {b}},(w\zeta ^{\dagger })_{\mathfrak {u}}] + [(w\eta ^{\dagger })_{\mathfrak {u}},(w\zeta ^{\dagger })_{\mathfrak {u}}] + [(w\eta ^{\dagger })_{\mathfrak {u}},(w\zeta ^{\dagger })_{\mathfrak {b}}]\bigr )\Bigr )\\= & {} \mathfrak {I}\,\text {tr}\bigl (w\xi ^{\dagger }[ w\eta ^{\dagger }, w\zeta ^{\dagger }]\bigr ) +\hbox {cycl. perm.} =0.\\ \end{aligned}$$

Having verified the Jacobi identity, it remains to calculate the Poisson brackets of the components of w and their complex conjugates. Let \(e_k\) \((k=1,\dots , n)\) denote the canonical basis of \({\mathbb {C}}^n\). One obtains by tedious calculation that the Hamiltonian vector fields of the linear functions given by the real and imaginary parts of the components \(w_k\) have the following form:

$$\begin{aligned} V_{\mathfrak {R}w_k}(w) = \left\{ \begin{aligned}&{\mathrm{i}}\mathfrak {R}(w_k) w_k e_k+ {\mathrm{i}}\sum _{r>k}(w_kw_re_r+ |w_r|^2e_k) + {\mathrm{i}}e_k- {\textstyle {\frac{1}{2}}}{\mathrm{i}}(w_k-{\overline{w}}_k)w\qquad k<n,\\&{\mathrm{i}}\mathfrak {R}(w_n)w_ne_n+ {\mathrm{i}}e_n- {\textstyle {\frac{1}{2}}}{\mathrm{i}}(w_n-{\overline{w}}_n)w \quad \qquad \qquad \qquad \qquad \qquad \qquad k=n. \end{aligned}\right. \end{aligned}$$

and

$$\begin{aligned} V_{\mathfrak {I}w_k}(w) = \left\{ \begin{aligned}&{\mathrm{i}}\mathfrak {I}(w_k) w_ke_k+ \sum _{r>k}(w_kw_re_r- |w_r|^2e_k) - e_k- {\textstyle {\frac{1}{2}}}(w_k+{\overline{w}}_k)w\qquad k<n,\\&{\mathrm{i}}\mathfrak {I}( w_n) w_ne_n- e_n- {\textstyle {\frac{1}{2}}}(w_n+{\overline{w}}_n)w \,\,\,\, \qquad \qquad \qquad \qquad \qquad \qquad k=n. \end{aligned}\right. \end{aligned}$$

By using these, one can check that the formulae (2.28) and (2.29) follow. If desired, the reader can supply the details. \(\square \)

The bracket (A.3) has the nice property that the natural action of \({\mathrm{U}}(n)\) on \({\mathbb {C}}^n\) is Poisson [56], and this can also be checked using linear functions \(F_\xi \). To this end, for any \(g\in {\mathrm{U}}(n)\) and \(w\in {\mathbb {C}}^n\) we define the functions \(F_\xi (g\,\cdot \,)\in C^\infty ({\mathbb {C}}^n)\) and \(F_\xi (\,\cdot \,w)\in C^\infty ({\mathrm{U}}(n))\) by

$$\begin{aligned} F_\xi (g\,\cdot \,)(w) = F_\xi (gw) = F_\xi (\,\cdot \,w)(g). \end{aligned}$$
(A.8)

Then, an easy calculation gives that

$$\begin{aligned} \{F_\xi ,F_\eta \}(gw) - \{F_\xi (g\,\cdot \, ),F_\eta (g\,\cdot \, )\}(w) \end{aligned}$$
(A.9)

is equal to

$$\begin{aligned} \mathfrak {I}\text {tr}\bigl (gw\xi ^{\dagger }(gw\eta ^{\dagger })_{{\mathfrak {u}}(n)} - w\xi ^{\dagger }g (w\eta ^{\dagger }g)_{{\mathfrak {u}}(n)}\bigr ), \end{aligned}$$
(A.10)

which in turn is equal to the value at g of the Poisson bracket (2.10) of the functions \(F_\xi (\,\cdot \, w)\) and \(F_\eta (\,\cdot \,w)\) on \({\mathrm{U}}(n)\). The last equality follows using \(D F_\xi (\,\cdot \,w)(g)= (g w \xi ^{\dagger })_{{\mathfrak {b}}(n)}\) and elementary manipulations. Thus, we have

$$\begin{aligned} \{F_\xi ,F_\eta \}(gw) = \{F_\xi (g\,\cdot \, ),F_\eta (g\,\cdot \, )\}(w) + \{F_\xi (\,\cdot \,w),F_\eta (\,\cdot \,w)\}_U(g), \end{aligned}$$
(A.11)

which means that the map \({\mathrm{U}}(n) \times {\mathbb {C}}^n \ni (g,w) \mapsto gw \in {\mathbb {C}}^n\) is indeed a Poisson map.

Let us recall the diffeomorphism

$$\begin{aligned} b \mapsto b b^\dagger \end{aligned}$$
(A.12)

from the group \(\text {B}(n)\) to the space \({{\mathfrak {P}}}(n)\) of positive definite Hermitian matrices. By this diffeomorphism, the Poisson structure (2.11) on \(\text {B}(n)\) is converted into a Poisson structure on \({{\mathfrak {P}}}(n)\), which is given by the first term of (2.23), i.e. ,

$$\begin{aligned} \{f, h\}_{{\mathfrak {P}}}(L)=4 \langle L d f(L), \left( L d h (L)\right) _{{\mathfrak {u}}(n)} \rangle \end{aligned}$$
(A.13)

for all \(f,h \in C^\infty ({{\mathfrak {P}}}(n))\). Here, the \({\mathfrak {u}}(n)\)-valued derivatives df and dh are defined by (2.22).

Proposition A.2

With respect to the brackets (A.3) and (A.13), the map

$$\begin{aligned} \Phi :w\mapsto {{\varvec{1}}}_n+ ww^{\dagger } \end{aligned}$$
(A.14)

from \({\mathbb {C}}^n\) to \({{\mathfrak {P}}}(n)\) is Poisson.

Proof

Let \(X,Y\in {\mathfrak {u}}(n)\) and consider the pull-backs \(\Phi ^*(f_X)\) and \(\Phi ^*(f_Y)\) of the functions \(f_X(L):=\langle X,L\rangle \) and \(f_Y(L) := \langle Y,L\rangle \). We have

$$\begin{aligned} \Phi ^*(f_X)(w) = \mathfrak {I}(w^{\dagger }Xw) + \mathfrak {I}\text {tr}(X)\quad \hbox {and}\quad \Phi ^*(f_Y)(w) = \mathfrak {I}(w^{\dagger }Yw) + \mathfrak {I}\text {tr}(Y).\nonumber \\ \end{aligned}$$
(A.15)

Using the formula (A.3) with \((\nabla \Phi ^* (f_X))(w) = - 2 Xw\) and similar for \(f_Y\), we can compute

$$\begin{aligned} \{\Phi ^*(f_X),\Phi ^*(f_Y)\}(w)= & {} 4 \mathfrak {I}\Bigl ( w^{\dagger }X(ww^{\dagger }Y)_{\mathfrak {u}}w + w^{\dagger }X Yw \nonumber \\&- {\textstyle {\frac{1}{2}}}w^{\dagger }Xww^{\dagger }Yw + {\textstyle {\frac{1}{2}}}w^{\dagger }X ww^{\dagger }Y w\Bigr ) \nonumber \\= & {} 4\mathfrak {I}\text {tr}\Bigl ( ({{\varvec{1}}}_n+ww^{\dagger })X\bigl (({{\varvec{1}}}_n +ww^{\dagger })Y\bigr )_{{\mathfrak {u}}(n)} \Bigr ) = \{ f_X, f_Y\}_{{\mathfrak {P}}}(\Phi (w)).\nonumber \\ \end{aligned}$$
(A.16)

Here, we have taken into account that, for example, \(\mathfrak {I}\text {tr}(XY) =0\) for \(X,Y\in {\mathfrak {u}}(n)\). The statement follows since the linear functions of the form \(f_X\) can serve as coordinates on \({{\mathfrak {P}}}(n)\). \(\square \)

Let \({{\mathbf {b}}}: {\mathbb {C}}^n \rightarrow \text {B}(n)\) be the map determined by the condition

$$\begin{aligned} \Phi = {{\mathbf {b}}}{{\mathbf {b}}}^\dagger . \end{aligned}$$
(A.17)

It follows from Proposition A.2 that this is a Poisson map with respect to the Poisson brackets (A.3) on \({\mathbb {C}}^n\) and (2.11) on \(\text {B}(n)\).

Proposition A.3

The map \({{\mathbf {b}}}\) defined by (A.17) with (A.14) is the moment map for the Poisson action (2.26) of \({\mathrm{U}}(n)\) on \({\mathbb {C}}^n\). According to (2.27), this means that we have

$$\begin{aligned} \mathfrak {I}\left( (\nabla F(w))^\dagger Xw\right) = \mathfrak {I}\mathrm{tr} \left( X \{F, {{\mathbf {b}}}\}(w) {{\mathbf {b}}}(w)^{-1}\right) , \quad \forall X\in {\mathfrak {u}}(n),\,w\in {\mathbb {C}}^n,\, F\in C^\infty ({\mathbb {C}}^n). \nonumber \\ \end{aligned}$$
(A.18)

Proof

For ease of notation, we verify the relation for linear functions \(F_\xi \) on \({\mathbb {C}}^n\), which is sufficient. For this, we have to calculate the \({\mathfrak {b}}(n)\)-valued function

$$\begin{aligned} \beta _F := \{ {{\mathbf {b}}}, F\} {{\mathbf {b}}}^{-1}\,, \quad F:=F_\xi . \end{aligned}$$
(A.19)

Since (A.12) is a diffeomorphism, \(\beta _F\) is uniquely determined by

$$\begin{aligned} \{ \Phi , F\} = \beta _F \Phi + \Phi \beta _F^\dagger , \end{aligned}$$
(A.20)

and this can be calculated as follows. First, we rearrange the expression (A.4) of the Hamiltonian vector field in the form

$$\begin{aligned} V_F(w)= {\textstyle {\frac{1}{2}}}(\xi ^{\dagger }w - w^{\dagger }\xi )w -\xi - (w\xi ^{\dagger })_{{\mathfrak {b}}(n)} w. \end{aligned}$$
(A.21)

Then, as \((\xi ^{\dagger }w - w^{\dagger }\xi )\in {\mathrm{i}}{\mathbb {R}}\), we obtain

$$\begin{aligned} \{ \Phi , F\}(w)= & {} V_F(w) w^{\dagger }+ w (V_F(w))^{\dagger }\\= & {} -(w\xi ^{\dagger })_{{\mathfrak {b}}(n)} ww^{\dagger }- ww^{\dagger }(w\xi ^{\dagger })_{{\mathfrak {b}}(n)}^{\dagger }- \xi w^{\dagger }- w\xi ^{\dagger }\nonumber \\= & {} -(w\xi ^{\dagger })_{{\mathfrak {b}}(n)}\Phi (w) - \Phi (w)(w\xi ^{\dagger })_{{\mathfrak {b}}(n)}^{\dagger }+ \bigl ((w\xi ^{\dagger })_{{\mathfrak {b}}(n)}-w\xi ^{\dagger }\bigr )\nonumber \\&+ \bigl ((w\xi ^{\dagger })_{{\mathfrak {b}}(n)}^{\dagger }-\xi w^{\dagger }\bigr ).\nonumber \end{aligned}$$
(A.22)

But the last two terms cancel, and hence we see that

$$\begin{aligned} \beta _F(w) = - \left( w \xi ^\dagger \right) _{{\mathfrak {b}}(n)}. \end{aligned}$$
(A.23)

By using this, the right-hand-side of (A.18) becomes

$$\begin{aligned} - \mathfrak {I}\text {tr}( X \beta _F(w)) = \mathfrak {I}\text {tr}(X w \xi ^\dagger ) = \mathfrak {I}(\xi ^\dagger X w), \end{aligned}$$
(A.24)

whereby the proof is complete. \(\square \)

Remark A.4

We had no need for the explicit formula of \({{\mathbf {b}}}(w)\) in the above, but in some other calculations it is needed. The reader can verify directly that it obeys equation (2.31).

Remark A.5

The maximal torus \({\mathbb {T}}^n < {\mathrm{U}}(n)\) is a Poisson subgroup with vanishing Poisson bracket, and therefore, the restriction of the \({\mathrm{U}}(n)\) action to \({\mathbb {T}}^n\) gives an ordinary Hamiltonian action. One can identify the dual Poisson–Lie group of \({\mathbb {T}}^n\) with \(\text {B}(n)_0\), the group of positive diagonal matrices, with zero Poisson bracket. Then, the corresponding group valued moment map is provided by \(w \mapsto {{\mathbf {b}}}(w)_0\), which is the diagonal part of \({{\mathbf {b}}}(w)\). Writing

$$\begin{aligned} {{\mathbf {b}}}(w)_0 = \exp (\phi (w)), \end{aligned}$$
(A.25)

we get the ordinary moment map \(w \mapsto \phi (w) \in {\mathfrak {b}}(n)_0\), where \({\mathfrak {b}}(n)_0\) (the space of real diagonal matrices) is identified with the linear dual of \({\mathfrak {u}}(n)_0\).

The following proposition represents one of the side results of the paper.

Proposition A.6

The Poisson bracket (A.3) is symplectic and, with \({{\mathcal {G}}}_j= 1 + \sum _{k=j}^n \vert w_j\vert ^2\), the corresponding symplectic form on \({\mathbb {C}}^n\) is given by

$$\begin{aligned} \Omega _{{\mathbb {C}}^n} = \frac{{\mathrm{i}}}{2}\sum _{k=1}^n\frac{1}{{{\mathcal {G}}}_k} dw_k\wedge d{\overline{w}}_k + \frac{{\mathrm{i}}}{4}\sum _{k=1}^{n-1}\frac{1}{{{\mathcal {G}}}_k{{\mathcal {G}}}_{k+1}} d{{\mathcal {G}}}_{k+1} \wedge \left( {\overline{w}}_k dw_k - w_k d{\overline{w}}_k\right) .\nonumber \\ \end{aligned}$$
(A.26)

Proof

We start from the coordinate form of the Poisson bracket, copied here for convenience:

$$\begin{aligned} \begin{aligned} \{w_i,w_k\}&= {\mathrm{i}}\,\text {sgn}(i-k)w_iw_k\\ \{w_i,{\overline{w}}_l\}&= {\mathrm{i}}\delta _{il}(2+|w|^2) + {\mathrm{i}}w_i{\overline{w}}_l + {\mathrm{i}}\delta _{il}\sum _{r=1}^n\text {sgn}(r-l) |w_r|^2. \end{aligned} \end{aligned}$$
(A.27)

We shall first invert the Poisson tensor on the dense open submanifold on which all \(\vert w_j \vert >0\), where we can use the parametrization \(w_j = e^{{\mathrm{i}}\varphi _j} \vert w_j\vert \).

Let us consider

$$\begin{aligned} \{w_i, |w_k|^2\}= & {} {\mathrm{i}}\,\text {sgn}(i-k)|w_k|^2w_i +{\mathrm{i}}|w_k|^2w_i +{\mathrm{i}}\delta _{ik}(2+|w|^2)w_i\nonumber \\&+{\mathrm{i}}\delta _{ik}w_i\sum _{r=1}^n\text {sgn}(r-k)|w_r|^2, \end{aligned}$$
(A.28)

from which we easily obtain

$$\begin{aligned} \{|w_i|^2,|w_k|^2\}=0. \end{aligned}$$
(A.29)

Using this, and restricting now to our submanifold, the relation (A.28) implies

$$\begin{aligned} \{e^{{\mathrm{i}}\varphi _i},|w_k|^2\} = \{ \frac{w_i}{|w_i|} , |w_k|^2 \} = {\mathrm{i}}[1-\delta _{ik}+\text {sgn}(i-k)]|w_k|^2 e^{{\mathrm{i}}\varphi _i} + 2{\mathrm{i}}\delta _{ik} {{\mathcal {G}}}_k e^{{\mathrm{i}}\varphi _i}. \nonumber \\ \end{aligned}$$
(A.30)

Plainly, we have the identity

$$\begin{aligned} \{ w_j , w_k \} + e^{2{\mathrm{i}}\varphi _j} e^{2{\mathrm{i}}\varphi _k} \{ {\overline{w}}_j , {\overline{w}}_k \} = 2 \vert w_j w_k \vert \{ e^{{\mathrm{i}}\varphi _j} , e^{{\mathrm{i}}\varphi _k} \}. \end{aligned}$$
(A.31)

The left-hand side can be checked to vanish, and thus, we get

$$\begin{aligned} \{ e^{{\mathrm{i}}\varphi _j} , e^{{\mathrm{i}}\varphi _k} \}=0. \end{aligned}$$
(A.32)

It is convenient to change variables, noting that the map \((|w_1|^2,\dots ,|w_n|^2) \mapsto ({{\mathcal {G}}}_1,\dots ,{{\mathcal {G}}}_n)\) is invertible. Then, it is elementary to derive from (A.30) the relation

$$\begin{aligned} \{e^{{\mathrm{i}}\varphi _i}, {{\mathcal {G}}}_k\} = \left\{ \begin{aligned}&2{\mathrm{i}}{{\mathcal {G}}}_k e^{{\mathrm{i}}\varphi _i},\quad k\le i\\&0, \ \ \ \quad \qquad k>i\end{aligned}\right. \end{aligned}$$
(A.33)

that can be also written as

$$\begin{aligned} \{ \varphi _i, \ln {{\mathcal {G}}}_k\} = \left\{ \begin{aligned}&2,\quad k\le i\\&0,\quad k>i \end{aligned} \right. \end{aligned}$$
(A.34)

This means that the matrix of Poisson brackets, in the variables \( \varphi _i, \ln {{\mathcal {G}}}_k \) has the form

$$\begin{aligned} P = 2\left( \begin{array}{cc}0 &{}\quad A \\ -A^T &{}\quad 0\end{array}\right) \end{aligned}$$
(A.35)

with

$$\begin{aligned} A = {{\varvec{1}}}_n + B + B^2 + \dots +B^{n-1}, \end{aligned}$$
(A.36)

where B is the nilpotent matrix having the entries \(B_{ik} = \delta _{i,k+1}\). Both A and P are invertible, and their inverses are

$$\begin{aligned} A^{-1}={{\varvec{1}}}_n-B \quad \hbox {and}\quad P^{-1} = \frac{1}{2} \left( \begin{array}{cc}0 &{} -(A^{-1})^T \\ A^{-1} &{} 0\end{array}\right) . \end{aligned}$$
(A.37)

Consequently, we obtain the symplectic formFootnote 7 (\(x^\alpha \) represent the local variables \(\varphi _i\) and \(\ln {{\mathcal {G}}}_k\))

$$\begin{aligned} \Omega= & {} \frac{1}{2} \sum _{\alpha ,\beta =1}^{2n}(P^{-1})_{\alpha \beta } dx^\alpha \wedge dx^\beta \nonumber \\= & {} {\textstyle {\frac{1}{2}}}\sum _{k=1}^{n-1} [d\ln {{\mathcal {G}}}_k-d\ln {{\mathcal {G}}}_{k+1}]\wedge d\varphi _k + {\textstyle {\frac{1}{2}}}d\ln {{\mathcal {G}}}_n\wedge d\varphi _n. \end{aligned}$$
(A.38)

If we now substitute the identities

$$\begin{aligned} d \ln {{\mathcal {G}}}_k - d \ln {{\mathcal {G}}}_{k+1}= \frac{ {{\mathcal {G}}}_{k+1} d \vert w_k \vert ^2 - \vert w_k\vert ^2 d {{\mathcal {G}}}_{k+1}}{{{\mathcal {G}}}_k {{\mathcal {G}}}_{k+1}} \end{aligned}$$
(A.39)

and

$$\begin{aligned} d\varphi _k = (2{\mathrm{i}}|w_k|^2)^{-1}({\overline{w}}_k dw_k - w_kd{\overline{w}}_k), \end{aligned}$$
(A.40)

then \(\Omega \) (A.38) takes the form

$$\begin{aligned} \Omega = \frac{{\mathrm{i}}}{2}\sum _{k=1}^n\frac{1}{{{\mathcal {G}}}_k} dw_k\wedge d{\overline{w}}_k + \frac{{\mathrm{i}}}{4}\sum _{k=1}^{n-1}\frac{1}{{{\mathcal {G}}}_k{{\mathcal {G}}}_{k+1}} d {{\mathcal {G}}}_{k+1} \wedge \left( {\overline{w}}_k dw_k - w_k d{\overline{w}}_k\right) . \nonumber \\ \end{aligned}$$
(A.41)

It is clear that both the original Poisson tensor corresponding to (A.27) and \(\Omega \) (A.41) are regular over the whole of \({\mathbb {C}}^n\). As a result, their inverse relationship extends from the dense open submanifold (where \(|w_j|>0\) for all j) to the full phase space. \(\square \)

Remark A.7

The image of the map \(w\mapsto {{\varvec{1}}}_n + w w^\dagger \) is the union of the \({\mathrm{U}}(n)\) orbits in \({{\mathfrak {P}}}(n)\) passing through the degenerate diagonal matrices

$$\begin{aligned} \text {diag}(1 + R^2, 1\dots , 1), \qquad R\ge 0. \end{aligned}$$
(A.42)

For any fixed \(R>0\), the orbit is a symplectic leaf in \({{\mathfrak {P}}}(n)\) of dimension \(2(n-1)\); \(R=0\) corresponds to a trivial symplectic leaf. The union of the orbits consisting of the conjugates of the matrices

$$\begin{aligned} \text {diag}(1 - r^2, 1,\dots , 1), \qquad 0 \le r < 1 \end{aligned}$$
(A.43)

is the image of the map

$$\begin{aligned} z \mapsto {{\varvec{1}}}_n - z z^\dagger \end{aligned}$$
(A.44)

from

$$\begin{aligned} {{\mathcal {B}}}(1):= \{ z \in {\mathbb {C}}^n \mid |z|^2 <1 \} \end{aligned}$$
(A.45)

to \({{\mathfrak {P}}}(n)\). In fact, the open ball \({{\mathcal {B}}}(1)\), identified as a subset of \({\mathbb {R}}^{2n}\), can be equipped with the Poisson bracket

$$\begin{aligned} \begin{aligned} \{z_i,z_k\}&= {\mathrm{i}}\,\text {sgn}(i-k)z_iz_k\\ \{z_i,{\overline{z}}_l\}&= {\mathrm{i}}(|z|^2 -2)\delta _{il} + {\mathrm{i}}z_i{\overline{z}}_l + {\mathrm{i}}\delta _{il}\sum _{r=1}^n\text {sgn}(r-l)|z_r|^2 \end{aligned} \end{aligned}$$
(A.46)

with respect to which the map (A.44) is Poisson. This is also a special case of the Poisson structures found in [56]. The analogue of Proposition A.3 holds for the map \({{\mathbf {b}}}_-: {{\mathcal {B}}}(1) \rightarrow \text {B}(n)\) defined by

$$\begin{aligned} {{\varvec{1}}}_n - z z^\dagger = {{\mathbf {b}}}_-(z) {{\mathbf {b}}}_-(z)^\dagger . \end{aligned}$$
(A.47)

The Poisson map \({{\mathbf {b}}}_-\) can be used to introduce variants of our reduction. Concretely, one may replace one or more of the \({{\mathbf {b}}}\) factors in (3.1) by \({{\mathbf {b}}}_-\) and study the reduced system. The restriction \(\gamma >0\) in the moment map constraint (3.17) then might not be necessary. Let us also note that one obtains a Poisson pencil on \({\mathbb {C}}^n\) if one replaces the last term of (A.3) by \(-\lambda \mathfrak {I}\left( \xi (w)^\dagger \eta (w)\right) \) for any real parameter \(\lambda \), and the formula (A.46) corresponds to \(\lambda =-1\).

Proof of Lemma 5.1

In this section, we work over \(({\mathbb {C}}^{n\times d}, \{\ ,\ \}_{{\mathcal {W}}})\) with the primary spins \((w^\alpha )\), see Sect. 2.2. We set \(\{ \ , \ \}:=\{\ ,\ \}_{{\mathcal {W}}}\) to simplify notations.

As noted in Sect. 3.1, the half-dressed spins \(v^\alpha \) can be defined in \({\mathbb {C}}^{n\times d}\) in terms of the primary spins. It is convenient to introduce the matrices \(b_\alpha ={{\mathbf {b}}}(w^\alpha )\) and \(B^\alpha =b_1\cdots b_\alpha \), so that

$$\begin{aligned} v^\alpha = B^{\alpha -1} w^\alpha \,. \end{aligned}$$
(B.1)

Remark that \(B^\alpha \) is related to the matrix \(B_\alpha \) introduced in (3.2) by \(B_\alpha = b_R B^\alpha \). We also note the following lemma, which follows from Proposition 2.1 by straightforward computations.

Lemma B.1

For any \(1\le \alpha \le d\), \(1\le i , j , l \le n\),

$$\begin{aligned}&\{ w^\alpha _i,(b_\alpha )_{jl} \}=\,{\mathrm{i}}\,[\delta _{ij}+2 \delta _{(i>j)}]\,w^\alpha _j (b_\alpha )_{il} \,, \end{aligned}$$
(B.2)
$$\begin{aligned}&\{ \overline{w}^\alpha _i,(b_\alpha )_{jl} \}=-{\mathrm{i}}\delta _{ij} \overline{w}^\alpha _i (b_\alpha )_{il} -2{\mathrm{i}}\delta _{ij} \sum _{k=j+1}^l \overline{w}^\alpha _k (b_\alpha )_{kl} \,. \end{aligned}$$
(B.3)

Furthermore, the Poisson bracket evaluated on \(((b_\alpha )_{ij},(\overline{b}_\alpha )_{ij})\) is given by (2.7)–(2.8).

Next, we need to describe the Poisson brackets between the matrix entries of \((B^\alpha ,w^\alpha )\), which appear in the decomposition (B.1). To write them down, we introduce the matrices

$$\begin{aligned} B^{\alpha ;\gamma }=b_\alpha \cdots b_\gamma \,, \quad 1\le \alpha \le \gamma \le d\,, \end{aligned}$$
(B.4)

which are such that \(B^{1;\alpha }=B^\alpha \) and \(B^{\alpha ;\alpha }=b_\alpha \). We also set \(B^{\alpha +1;\alpha }:={{\varvec{1}}}_n\) and \(B^0:={{\varvec{1}}}_n\).

Lemma B.2

For any \(1\le \alpha ,\beta \le d\), \(1\le i ,k,l \le n\),

$$\begin{aligned}&\{ w^\alpha _i,B^\beta _{kl} \}= - {\mathrm{i}}\delta _{(\alpha \le \beta )} B^{\alpha -1}_{ki} w_i^\alpha B^{\alpha ;\beta }_{il} +2{\mathrm{i}}\delta _{(\alpha \le \beta )} \sum _{k' \le i} B^{\alpha -1}_{kk'} w_{k'}^\alpha B^{\alpha ;\beta }_{il} \,, \end{aligned}$$
(B.5)
$$\begin{aligned}&\{ \overline{w}^\alpha _i,B^\beta _{kl} \}= +{\mathrm{i}}\delta _{(\alpha \le \beta )} B^{\alpha -1}_{ki} \overline{w}_i^\alpha B^{\alpha ;\beta }_{il} - 2{\mathrm{i}}\delta _{(\alpha \le \beta )} B^{\alpha -1}_{ki} \sum _{i\le u} \overline{w}_u^\alpha B^{\alpha ;\beta }_{ul} \,. \end{aligned}$$
(B.6)

Proof

By construction, for \(\beta \ne \alpha \) we have \(\{ w^\alpha _i,w^\beta _k \}=0\), hence \(\{ w^\alpha _i,(b_\beta )_{kl} \}=0\). We get that

$$\begin{aligned} \{ w^\alpha _i,B^\beta _{kl} \}=0,\,\, \alpha <\beta \,; \quad \{ w^\alpha _i,B^\beta _{kl} \}=\sum _{k \le l' \le l} \{ w^\alpha _i,B^\alpha _{kl'} \}B^{\alpha +1;\beta }_{l'l}\,, \,\, \beta > \alpha \,. \end{aligned}$$
(B.7)

When \(\beta =\alpha \), \(\{ w^\alpha _i,(b_\alpha )_{kl} \}\) is given by (B.2) and we get

$$\begin{aligned} \{ w^\alpha _i,B^\alpha _{kl} \}= -{\mathrm{i}}B^{\alpha -1}_{ki} w_i^\alpha (b_\alpha )_{il} +2{\mathrm{i}}\sum _{k' \le i} B^{\alpha -1}_{kk'} w_{k'}^\alpha (b_\alpha )_{il}\,, \end{aligned}$$
(B.8)

from which the first identity can be obtained. The second case is proved in the same way. \(\square \)

Lemma B.3

For any \(1\le \alpha ,\beta \le d\), \(1\le i,j,k,l\le n\),

$$\begin{aligned} \begin{aligned} \{ B^\alpha _{ij},B^\beta _{kl} \}{\mathop {=}\limits ^{\alpha \leqslant \beta }}&- 2 {\mathrm{i}}B^{\alpha }_{kj}\sum _{r> j}B^{\alpha }_{ir} B^{\alpha +1;\beta }_{rl} - {\mathrm{i}}B^{\alpha }_{kj} B^{\alpha }_{ij} B^{\alpha +1;\beta }_{jl} + 2 {\mathrm{i}}\delta _{(i>k)} B^\alpha _{kj} B^\beta _{il}+ {\mathrm{i}}\delta _{ik} B^\alpha _{kj} B^\beta _{il}\,, \\ \{ B^\alpha _{ij},\overline{B}^\beta _{kl} \}{\mathop {=}\limits ^{\alpha \leqslant \beta }}&-{\mathrm{i}}B_{ij}^\alpha \overline{B}_{kj}^\alpha \overline{B}_{jl}^{\alpha +1;\beta } - 2 {\mathrm{i}}\sum _{s<j} B_{is}^\alpha \overline{B}_{ks}^\alpha \overline{B}_{jl}^{\alpha +1;\beta } +{\mathrm{i}}\delta _{ik} B_{ij}^\alpha \overline{B}_{kl}^\beta + 2 {\mathrm{i}}\delta _{ik} \sum _{r>k} B_{rj}^\alpha \overline{B}_{rl}^\beta \,, \\ \{ B^\alpha _{ij},\overline{B}^\beta _{kl} \}{\mathop {=}\limits ^{\alpha \geqslant \beta }}&-{\mathrm{i}}B_{il}^\beta \overline{B}_{kl}^\beta B_{lj}^{\beta +1;\alpha } - 2 {\mathrm{i}}\sum _{s<l} B_{is}^\beta \overline{B}_{ks}^\beta B_{lj}^{\beta +1;\alpha } +{\mathrm{i}}\delta _{ik} B_{ij}^\alpha \overline{B}_{kl}^\beta + 2 {\mathrm{i}}\delta _{ik} \sum _{r>k} B_{rj}^\alpha \overline{B}_{rl}^\beta \,. \end{aligned} \end{aligned}$$

Proof

For the first equality, we have for \(\alpha \le \beta \) that

$$\begin{aligned} \{ B^\alpha _{ij},B^\beta _{kl} \} =\sum _{1\le \gamma \le \alpha } \sum _{i',j',k',l'}(B^{\gamma -1})_{ii'} (B^{\gamma -1})_{kk'} \{ (b_\gamma )_{i'j'},(b_\gamma )_{k'l'} \} B^{\gamma +1;\alpha }_{j'j} B^{\gamma +1;\beta }_{l'l}\,.\nonumber \\ \end{aligned}$$
(B.9)

A similar expansion holds for \(\{ B^\alpha _{ij},\overline{B}^\beta _{kl} \}\). It then suffices to use Lemma B.1.

\(\square \)

Note that in the case \(\beta =\alpha \), the Poisson brackets from Lemma B.3 take the usual form (2.7)–(2.8) on \(\text {B}(n)\). We can also see that we can use \(\beta =0\) in Lemma B.2 and \(\alpha ,\beta =0\) in Lemma B.3, since in such cases the Poisson bracket vanishes on \(B^0={{\varvec{1}}}_n\).

We can now prove Lemma 5.1 using Lemmae B.2 and B.3 . We will use the definition of the half-dressed spins given by (B.1). To show (5.1) we need to write

$$\begin{aligned} \begin{aligned} \{ v^\alpha _i,v^\beta _k \}=&\sum _{j,l}\{ B^{\alpha -1}_{ij} w^\alpha _j, B^{\beta -1}_{kl} w^\beta _l \} \\ =&\sum _{j,l}\{ B^{\alpha -1}_{ij} , B^{\beta -1}_{kl} \} w^\alpha _j w^\beta _l +\sum _{j,l}\{ B^{\alpha -1}_{ij} , w^\beta _l \} w^\alpha _j B^{\beta -1}_{kl} \\&+\sum _{j,l}\{ w^\alpha _j, B^{\beta -1}_{kl} \} B^{\alpha -1}_{ij} w^\beta _l +\sum _{j,l}\{ w^\alpha _j, w^\beta _l \} B^{\alpha -1}_{ij} B^{\beta -1}_{kl} \end{aligned} \end{aligned}$$
(B.10)

where we assume \(\alpha \le \beta \) without loss of generality. We can then use Lemmae B.2 and B.3 to show that

$$\begin{aligned} \{ v^\alpha _i,v^\alpha _k \}=-{\mathrm{i}}\, \text {sgn}(k-i) v_{k}^\alpha v_i^\alpha \,; \quad \{ v^\alpha _i,v^\beta _k \} =-{\mathrm{i}}\, \text {sgn}(k-i) v_k^\alpha v_i^\beta +{\mathrm{i}}v_k^\alpha v_i^\beta \,,\,\, \alpha < \beta \,.\nonumber \\ \end{aligned}$$
(B.11)

By anti-symmetry, (B.11) implies that (5.1) holds.

The Poisson bracket (5.2) is computed in the same way and requires to remark in the case \(\alpha =\beta \) that

$$\begin{aligned} \sum _{s} B^{\gamma }_{is} \overline{B}^{\gamma }_{ks} =\sum _{\mu =1}^\gamma v^\mu _i \bar{v}^\mu _k + \delta _{ik}\,. \end{aligned}$$
(B.12)

This identity is equivalent to \(B^\gamma (B^\gamma )^\dagger =\sum _{\mu =1}^\gamma v^\mu (v^\mu )^\dagger + {{\varvec{1}}}_n\), which is obtained by induction on \(\gamma \) using (2.33); it becomes (3.7) when \(\gamma =d\).

Proof of Theorem 5.8

Recall that we work over the gauge slice \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) (5.23) and wish to compute the reduced Poisson brackets \(\{\ ,\ \}_\text {red}\) of the basic evaluation functions \(Q_j=e^{{\mathrm{i}}q_j} \in {\mathrm{U}}(1)\) and \(v(\alpha )_j\in {\mathbb {C}}\), where the latter obey the relations (5.32). Our fundamental tool will be the identity (5.26), which concerns \({\mathrm{U}}(n)\) invariant functions on \({{\mathcal {M}}}\) and their pull-backs on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\). Knowing the left-hand side of (5.26), we will be able to determine the reduced Poisson brackets. In the particular case at hand, we consider the invariant functions \(f_m,f_m^{\alpha \beta }\in C^\infty ({{\mathcal {M}}})\) defined by (5.27). Their Poisson brackets on \({{\mathcal {M}}}\) are given by Lemma 5.7, and their restrictions (pull-backs) to \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\) are displayed in (5.33). The point is that the right-hand side of (5.26) can be also expressed through the reduced Poisson brackets of the basic variables on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\), which enables us to derive the explicit formulae of Theorem 5.8.

We begin by giving an auxiliary lemma, which will be used below.

Lemma C.1

The \(n\times n\) matrices \({{\mathcal {E}}},\tilde{{{\mathcal {E}}}}\) given by

$$\begin{aligned} {{\mathcal {E}}}_{kl}=Q_l^k \quad \text {and} \quad \tilde{{{\mathcal {E}}}}_{kl}=Q_l^k {{\mathcal {U}}}_l \end{aligned}$$
(C.1)

are invertible on \(\check{{\mathcal {M}}}_{0,+}^\mathrm{reg}\).

Proof

We can write that \({{\mathcal {E}}}=V Q\) with \(Q=\text {diag}(Q_1,\ldots ,Q_n)\) and \(V=(V_{kl})\), \(V_{kl}=Q_k^{l-1}\), which is a Vandermonde matrix. Since \(Q\in {\mathbb {T}}^n_{\text {reg}}\) on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\), both V and Q are invertible. We also have that \(\tilde{{{\mathcal {E}}}}={{\mathcal {E}}}D\) where \(D=\text {diag}({{\mathcal {U}}}_1,\ldots ,{{\mathcal {U}}}_n)\). As \({{\mathcal {U}}}_j>0\) on \(\check{{\mathcal {M}}}_{0,+}^\text {reg}\), \(\tilde{{{\mathcal {E}}}}\) is also invertible. \(\square \)

Deriving (5.37)

Lemma C.2

For any \(i,j=1,\ldots ,n\), \(\{ q_i,q_j \}_\mathrm{red}=0\).

Proof

From (5.28) and (5.33), we get for any \(M,N \in {\mathbb {N}}\),

$$\begin{aligned} 0=\xi ^*\{ f_M,f_N \} = \{ \xi ^*f_M,\xi ^*f_N \}_{\text {red}}=-MN\,\sum _{i,j=1}^n e^{{\mathrm{i}}M q_i}e^{{\mathrm{i}}N q_j} \{ q_i,q_j \}_{\text {red}}\,. \end{aligned}$$

Considering this equality for \(M,N=1,\ldots ,n\), this is equivalent to

$$\begin{aligned} {{\mathcal {E}}}\, \hat{U}^{(0)} \, {{\mathcal {E}}}^T=0_{n \times n}\,, \end{aligned}$$

where \(\hat{U}^{(0)} \in {\text {Mat}}_{n \times n}({\mathbb {C}})\) is given by \(\hat{U}^{(0)}_{kl}=\{ q_k,q_l \}_{\text {red}}\). By Lemma C.1, \({{\mathcal {E}}}\) is invertible on \(\check{{\mathcal {M}}}_{0,+}^{\text {reg}}\) so that \(\hat{U}^{(0)}\) is the zero matrix. \(\square \)

Lemma C.3

For any \(i,j=1,\ldots ,n\),

$$\begin{aligned} \{ {{\mathcal {U}}}_i,q_j \}_\mathrm{red}=-\delta _{ij}{{\mathcal {U}}}_i\,, \quad \{ v(\alpha )_i,q_j \}_\mathrm{red}=-\delta _{ij} v(\alpha )_i\,, \quad \{ \overline{v}(\alpha )_i,q_j \}_\mathrm{red}=-\delta _{ij} \overline{v}(\alpha )_i\,.\nonumber \\ \end{aligned}$$
(C.2)

Proof

From (5.29), after summing over all \(\alpha ,\beta \) we get for any \(M,N \in {\mathbb {N}}\)

$$\begin{aligned}&\sum _{i,j}\{ {{\mathcal {U}}}_i^2 e^{{\mathrm{i}}M q_i}, e^{{\mathrm{i}}N q_j} \}_{\text {red}} = \sum _{\alpha ,\beta }\{ \xi ^*f_M^{\alpha \beta },\xi ^*f_N \}_{\text {red}} = -2{\mathrm{i}}N \sum _{\alpha ,\beta } \xi ^*f_{M+N}^{\alpha \beta } \\&\quad = -2{\mathrm{i}}N \sum _i {{\mathcal {U}}}_i^2 e^{{\mathrm{i}}(M+N) q_i}\,. \end{aligned}$$

Using Lemma C.2, we obtain

$$\begin{aligned} \sum _{i,j} e^{{\mathrm{i}}M q_i}{{\mathcal {U}}}_i e^{{\mathrm{i}}N q_j}\{ {{\mathcal {U}}}_i , q_j \}_{\text {red}} = -\sum _i {{\mathcal {U}}}_i^2 e^{{\mathrm{i}}(M+N) q_i}\,. \end{aligned}$$

We can rewrite this for \(N,M=1,\ldots ,n\) as

$$\begin{aligned} \tilde{{{\mathcal {E}}}} \, \hat{U}^{(1)} \, {{\mathcal {E}}}^T=\tilde{{{\mathcal {E}}}} \, {U}^{(1)} \, {{\mathcal {E}}}^T\,, \end{aligned}$$

where the \(n\times n\) matrices are given by \(\hat{U}^{(1)}_{kl}=\{ {{\mathcal {U}}}_k,q_l \}_{\text {red}}\), \({U}^{(1)}_{kl}=-\delta _{kl}{{\mathcal {U}}}_k\). By Lemma C.1, both \({{\mathcal {E}}}\) and \(\tilde{{{\mathcal {E}}}}\) are invertible. Hence, \(\hat{U}^{(1)}={U}^{(1)}\).

For the second identity, we use (5.29) with summation over all \(\beta \), and we get for any \(M,N \in {\mathbb {N}}\)

$$\begin{aligned} \sum _{i,j}\{ {{\mathcal {U}}}_i v(\alpha )_i e^{{\mathrm{i}}M q_i}, e^{{\mathrm{i}}N q_j} \}_{\text {red}} = -2{\mathrm{i}}N \sum _i v(\alpha )_i{{\mathcal {U}}}_i e^{{\mathrm{i}}(M+N) q_i}\,. \end{aligned}$$

Now that the first identity is proved, we can use it to get

$$\begin{aligned} \sum _{i,j}e^{{\mathrm{i}}M q_i}{{\mathcal {U}}}_i e^{{\mathrm{i}}N q_j} \{ v(\alpha )_i , q_j \}_{\text {red}} = -\sum _i v(\alpha )_i {{\mathcal {U}}}_i e^{{\mathrm{i}}(M+N) q_i}\,. \end{aligned}$$

As before, we write this for \(N,M=1,\ldots ,n\) as

$$\begin{aligned} \tilde{{{\mathcal {E}}}} \, \hat{U}^{(2)} \, {{\mathcal {E}}}^T=\tilde{{{\mathcal {E}}}} \, {U}^{(2)} \, {{\mathcal {E}}}^T\,, \end{aligned}$$

where the \(n\times n\) matrices are given by \(\hat{U}^{(2)}_{kl}=\{ v(\alpha )_k,q_l \}_{\text {red}}\), \({U}^{(2)}_{kl}=-\delta _{kl}v(\alpha )_k\). Again by invertibility of \({{\mathcal {E}}}\) and \(\tilde{{{\mathcal {E}}}}\), we get \(\hat{U}^{(2)}={U}^{(2)}\).

The last identity follows from the second one by complex conjugation. \(\square \)

From now on, we do not provide complete proofs of the different results that are stated. They can be successively obtained by direct computations in the same way as we got Lemmas C.2 and C.3 .

Deriving (5.38)

We first need two preliminary lemmae.

Lemma C.4

For any \(i,j=1,\ldots ,n\),

$$\begin{aligned} \begin{aligned} \{ {{\mathcal {U}}}_i,{{\mathcal {U}}}_j \}_\mathrm{red}=&\,\,\frac{1}{2} {\mathrm{i}}\delta _{(i\ne j)} \frac{Q_i+Q_j}{Q_i-Q_j}{{\mathcal {U}}}_i{{\mathcal {U}}}_j +\frac{1}{4} {\mathrm{i}}\sum _{\mu ,\nu } \mathrm{sgn}(\nu -\mu ) \left[ v(\nu )_i v(\mu )_j - \overline{v}(\nu )_i \overline{v}(\mu )_j \right] \\&+\frac{1}{4} {\mathrm{i}}\sum _{\mu }\left[ v(\mu )_i \overline{v}(\mu )_j - v(\mu )_j \overline{v}(\mu )_i \right] + \frac{d}{2} {\mathrm{i}}(L_{ij}-L_{ji}) \\&+\frac{1}{2} {\mathrm{i}}\sum _{\nu }\sum _{\mu <\nu }\left[ v(\mu )_i \overline{v}(\mu )_j - v(\mu )_j \overline{v}(\mu )_i \right] \,. \end{aligned} \end{aligned}$$
(C.3)

Proof

It suffices to use (5.30) where we sum over all \(\alpha ,\beta ,\gamma ,\epsilon \). After elementary manipulations, we arrive at

$$\begin{aligned} \begin{aligned} \sum _{i,j} Q_i^M{{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \{ {{\mathcal {U}}}_i,{{\mathcal {U}}}_j \}_{\text {red}} =&\sum _{i,j}Q_i^M{{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \, {U}^{(3)}_{ij} \,, \end{aligned} \end{aligned}$$
(C.4)

where \({U}^{(3)}_{ij}\) is the right-hand side of (C.3). We can then write the equalities with \(N,M=1,\ldots ,n\) as

$$\begin{aligned} \tilde{{{\mathcal {E}}}} \, \hat{U}^{(3)} \, \tilde{{{\mathcal {E}}}}^T=\tilde{{{\mathcal {E}}}} \, {U}^{(3)} \, \tilde{{{\mathcal {E}}}}^T\,, \end{aligned}$$
(C.5)

where the \(n\times n\) matrix \(\hat{U}^{(3)}\) is given by \(\hat{U}^{(3)}_{kl}=\{ {{\mathcal {U}}}_k,{{\mathcal {U}}}_l \}_{\text {red}}\). By invertibility of \({\tilde{{{\mathcal {E}}}}}\), this proves the claim (C.3). \(\square \)

Lemma C.5

For any \(i,j=1,\ldots ,n\),

$$\begin{aligned} \begin{aligned} \{ v(\alpha )_i,{{\mathcal {U}}}_j \}_\mathrm{red}=&\,\,\frac{1}{2} {\mathrm{i}}\delta _{(i\ne j)} \frac{Q_i+Q_j}{Q_i-Q_j}v(\alpha )_j {{\mathcal {U}}}_i +\frac{1}{2} {\mathrm{i}}\sum _\kappa \mathrm{sgn}(\kappa -\alpha ) v(\alpha )_j v(\kappa )_i \\&-\frac{1}{4} {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i} \sum _{\mu ,\nu } \mathrm{sgn}(\nu -\mu ) \left[ v(\nu )_i v(\mu )_j + \overline{v}(\nu )_i \overline{v}(\mu )_j \right] \\&+\frac{1}{2} {\mathrm{i}}v(\alpha )_i \overline{v}(\alpha )_{j} -\frac{1}{4} {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i}\sum _{\mu }\left[ v(\mu )_i \overline{v}(\mu )_j + v(\mu )_j \overline{v}(\mu )_i \right] \\&+ {\mathrm{i}}\sum _{\kappa< \alpha } v(\kappa )_i \overline{v}(\kappa )_j -\frac{1}{2} {\mathrm{i}}\frac{v(\alpha )_i}{{{\mathcal {U}}}_i}\sum _{\nu }\sum _{\mu <\nu } \left[ v(\mu )_i \overline{v}(\mu )_j + v(\mu )_j \overline{v}(\mu )_i \right] \\&+\frac{1}{2} {\mathrm{i}}\left[ 2L_{ij} - d \frac{v(\alpha )_i}{{{\mathcal {U}}}_i} (L_{ij}+L_{ji})\right] \,. \end{aligned} \end{aligned}$$
(C.6)

Proof

It suffices to use (5.30) after summing over \(\beta ,\gamma ,\epsilon \). We arrive at

$$\begin{aligned} \begin{aligned} \sum _{i,j} Q_i^M {{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \{ v(\alpha )_i , {{\mathcal {U}}}_j \}_{\text {red}} =\sum _{i,j}Q_i^M{{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \, {U}^{(4)}_{ij}\,, \end{aligned} \end{aligned}$$
(C.7)

where \({U}^{(4)}_{ij}\) is the right-hand side of (C.6). We can then write the equalities with \(N,M=1,\ldots ,n\) as

$$\begin{aligned} \tilde{{{\mathcal {E}}}} \, \hat{U}^{(4)} \, \tilde{{{\mathcal {E}}}}^T= \tilde{{{\mathcal {E}}}} \, {U}^{(4)} \, \tilde{{{\mathcal {E}}}}^T\,, \end{aligned}$$

where the \(n\times n\) matrix \(\hat{U}^{(4)}\) is given by \(\hat{U}^{(4)}_{kl}=\{ v(\alpha )_k,{{\mathcal {U}}}_l \}_{\text {red}}\). By invertibility of \({\tilde{{{\mathcal {E}}}}}\), we obtain the equality (C.5). \(\square \)

Summing over \(\beta ,\epsilon \) in (5.30) and using the previous results, we can get

$$\begin{aligned} \sum _{i,j} Q_i^M {{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \{ v(\alpha )_i , v(\gamma )_j \}_{\text {red}} =\sum _{i,j}Q_i^M{{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \,{U}^{(5)}_{ij}\,, \end{aligned}$$
(C.8)

where \({U}^{(5)}_{ij}\) is the right-hand side of (5.38). We can then write the equalities (C.8) with \(N,M=1,\ldots ,n\) as

$$\begin{aligned} \tilde{{{\mathcal {E}}}} \, \hat{U}^{(5)} \, \tilde{{{\mathcal {E}}}}^T=\tilde{{{\mathcal {E}}}} \, {U}^{(5)} \, \tilde{{{\mathcal {E}}}}^T\,, \end{aligned}$$

where the \(n\times n\) matrix \(\hat{U}^{(5)}\) is given by \(\hat{U}^{(5)}_{kl}=\{ v(\alpha )_k,v(\gamma )_l \}_{\text {red}}\). By invertibility of \({\tilde{{{\mathcal {E}}}}}\), this implies that (5.38) holds.

Deriving (5.39)

By anti-symmetry and complex conjugation, we get \(\{ {{\mathcal {U}}}_i,\overline{v}(\epsilon )_j \}_{\text {red}}\) from Lemma C.5. We can then use the previous results as well as (5.30) after summing over \(\beta ,\gamma \) in order to get

$$\begin{aligned} \sum _{i,j} Q_i^M {{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j \{ v(\alpha )_i , \overline{v}(\epsilon )_j \}_{\text {red}} =\sum _{i,j}Q_i^M{{\mathcal {U}}}_i Q_j^N {{\mathcal {U}}}_j {U}^{(6)}_{ij}\,, \end{aligned}$$
(C.9)

where \({U}^{(6)}_{ij}\) is the right-hand side of (5.39). We can then write the equalities (C.9) with \(N,M=1,\ldots ,n\) as

$$\begin{aligned} \tilde{{{\mathcal {E}}}} \, \hat{U}^{(6)} \, \tilde{{{\mathcal {E}}}}^T=\tilde{{{\mathcal {E}}}} \, {U}^{(6)} \, \tilde{{{\mathcal {E}}}}^T\,, \end{aligned}$$

where the \(n\times n\) matrix \(\hat{U}^{(6)}\) is given by \(\hat{U}^{(6)}_{kl}=\{ v(\alpha )_k,\overline{v}(\epsilon )_l \}_{\text {red}}\). By invertibility of \(\tilde{{{\mathcal {E}}}}\), we can conclude that (5.39) holds.

Poisson Brackets of Collective Spins

Recall the matrix \((S_{ij})\) defined before Theorem 5.8. The reduced Poisson brackets of the so-called collective spins F (3.29) can be computed in the following form.

Lemma D.1

Denoting \(q_{ab}:=q_a-q_b\), the following identity holds on \(\check{{\mathcal {M}}}_{0,+}^\mathrm{reg}\)

$$\begin{aligned} \begin{aligned} \{&F_{ij},F_{kl} \}_\mathrm{red} = \,\,{\mathrm{i}}\left( \frac{S_{ik}}{{{\mathcal {U}}}_i {{\mathcal {U}}}_k}-\frac{S_{lj}}{{{\mathcal {U}}}_l {{\mathcal {U}}}_j} + \frac{S_{kj}}{{{\mathcal {U}}}_k {{\mathcal {U}}}_j} - \frac{S_{il}}{{{\mathcal {U}}}_i {{\mathcal {U}}}_l} \right) F_{ij} F_{kl} \\&+\frac{1}{2} \left[ \delta _{(i \ne k)} \cot (\frac{q_{ik}}{2}) + \delta _{(j\ne l)} \cot \left( \frac{q_{jl}}{2}\right) + \delta _{(k \ne j)} \cot \left( \frac{q_{kj}}{2}\right) + \delta _{(i \ne l)} \cot \left( \frac{q_{li}}{2}\right) \right] \, F_{ij}F_{kl}\\&+\left[ \delta _{(i \ne k)} \cot \left( \frac{q_{ik}}{2}\right) + \delta _{(j \ne l)} \cot (\frac{q_{jl}}{2}) - \cot \left( \frac{q_{jk}}{2}-{\mathrm{i}}\gamma \right) + \cot \left( \frac{q_{li}}{2} - {\mathrm{i}}\gamma \right) \right] \, F_{il}F_{kj} \\&+\frac{1}{2} \left[ \delta _{(k \ne i)} \cot \left( \frac{q_{ki}}{2}\right) - \cot \left( \frac{q_{li}}{2}-{\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_k}{{{\mathcal {U}}}_i} F_{ij}F_{il}\\&+\frac{1}{2} \left[ \delta _{(j \ne k)} \cot \left( \frac{q_{jk}}{2} \right) + \cot \left( \frac{q_{lj}}{2}-{\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_k}{{{\mathcal {U}}}_j} F_{ij} F_{jl} \\&+\frac{1}{2} \left[ \delta _{(i\ne k)}\cot \left( \frac{q_{ki}}{2}\right) + \cot \left( \frac{q_{jk}}{2}-{\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_i}{{{\mathcal {U}}}_k} F_{kj} F_{kl}\\&+\frac{1}{2} \left[ \delta _{(i \ne l)} \cot \left( \frac{q_{il}}{2} \right) - \cot \left( \frac{q_{jl}}{2} - {\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_i}{{{\mathcal {U}}}_l} F_{lj}F_{kl} \\&+\frac{1}{2} \left[ \delta _{(i \ne l)} \cot \left( \frac{q_{il}}{2} \right) - \cot \left( \frac{q_{ik}}{2} - {\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_l}{{{\mathcal {U}}}_i} F_{ij} F_{ki}\\&+\frac{1}{2} \left[ \delta _{(l \ne j)} \cot \left( \frac{q_{lj}}{2} \right) + \cot \left( \frac{q_{jk}}{2} - {\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_l}{{{\mathcal {U}}}_j} F_{ij} F_{kj} \\&+\frac{1}{2} \left[ \delta _{(j \ne k)} \cot ( \frac{q_{jk}}{2} ) + \cot \left( \frac{q_{ki}}{2} - {\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_j}{{{\mathcal {U}}}_k} F_{ik} F_{kl}\\&+\frac{1}{2} \left[ \delta _{(j \ne l)} \cot \left( \frac{q_{lj}}{2} \right) - \cot \left( \frac{q_{li}}{2} - {\mathrm{i}}\gamma \right) \right] \frac{{{\mathcal {U}}}_j}{{{\mathcal {U}}}_l} F_{il} F_{kl} \end{aligned} \end{aligned}$$

This follows from Theorem 5.8 by direct calculation. The reader can easily check the reality condition \(\{ F_{ji},F_{lk} \}_{\text {red}}=\{ \overline{F}_{ij},\overline{F}_{kl} \}_{\text {red}}=\overline{ \{ F_{ij},F_{kl} \} }_{\text {red}} \). Taking \(i=j\) and \(k=l\) in Lemma D.1, everything cancels out except for the third line, which can be rewritten as follows:

$$\begin{aligned} \{ F_{jj},F_{kk} \}_{\text {red}} = F_{jk} F_{kj} \frac{2\cot (\frac{q_{jk}}{2})}{1+\sinh ^{-2}(\gamma )\,\sin ^2(\frac{q_{jk}}{2})}\,, \quad \text {for }j\ne k\,. \end{aligned}$$
(D.1)

Let us now assume that \(d=1\), so that \(F_{jk} F_{kj} = F_{jj} F_{kk}\). Note that the formula of L (3.29) shows that \(F_{jj} >0\). Motivated by the form of the equations of motion (1.8) and the spinless Hamiltonian (1.11), we make the change of variables

$$\begin{aligned} F_{jj} = e^{2\theta _j}\prod _{i\ne j} \left[ 1+ \frac{\sinh ^2\gamma }{1 + \sin ^2\frac{q_i - q_j}{2}}\right] ^{\frac{1}{2}}. \end{aligned}$$
(D.2)

Using (5.37) and (D.1), it turns out that \((q_j, \theta _j)\) are Darboux variables, and we recover the standard chiral RS Hamiltonian (1.11) for \({{\mathcal {H}}}= \sum _j F_{jj}\).

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Fairon, M., Fehér, L. & Marshall, I. Trigonometric Real Form of the Spin RS Model of Krichever and Zabrodin. Ann. Henri Poincaré 22, 615–675 (2021). https://doi.org/10.1007/s00023-020-00976-4

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