A family of GL(r) multiplicative Higgs bundles on rational base

In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group $GL_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on $\mathbb{P}^{1}$ with prescribed singularities, (ii) moduli spaces of $U(r)$ monopoles on $\mathbb{R}^2 \times S^1$ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d $\mathcal{N}=2$ supersymmetric $A_{r-1}$ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at $\infty \in \mathbb{P}^1$ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in $\mathfrak{gl}_{r}$ and their Darboux parametrization and quantization is well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in Heisenberg algebra (free field realization).


Introduction
Complex completely integrable Hamiltonian systems can be typically constructed starting from a locus M in the moduli space Bun G (Σ) of holomorphic G-bundles or sheaves of certain type on a complex holomorphic symplectic surface Σ with a structure of Lagrangian elliptic fibration Σ → X, where the fibers Σ x are possibly degenerate elliptic curves, and X is an algebraic curve typically called the base curve. Indeed, the symplectic structure on Σ induces the symplectic structure on the space M which becomes the phase space of integrable system, the structure of Lagrangian fibration Σ → X induces the structure of Lagrangian fibration on M, and the fact that the fibers Σ x are abelian varieties (possibly degenerate elliptic curves) induces the structure of abelian varieties on the Lagrangian fibers in M, see [1][2][3][4].
There are three cases to consider depending on whether the elliptic fibers are generically cusped elliptic, nodal elliptic or smooth elliptic.
(1) Fibers are cusped elliptic. If X is an algebraic curve, and Σ → X is a cotangent bundle whose fibers are compactified to cusped elliptic curves, this construction produces algebraic integrable system called Hitchin system on the curve X [5][6][7]. Hitchin system is an example of an abstract Higgs bundle on X valued in an abelian group K over X for the case when the group K is the canonical line bundle on X endowed with natural linear additive group structure in the fiber direction. The Higgs field φ(x) is a holomorphic 1-form valued in the Lie algebra adjoint bundle ad g. The respective integrable system is of additive type in the fiber direction.
(2) Fibers are nodal elliptic. If Σ → X is a fibration whose fibers are nodal elliptic curves, then Bun G (Σ) is equivalently described as a moduli space of multiplicative Higgs bundles mHiggs G (X), that is moduli space of pairs (P, g) where P is a principal G-bundle on X, and Higgs field g(x) is a section of Lie group adjoint bundle ad G. The respective integrable system is of multiplicative type in vertical direction. See [8,9].
(3) Fibers are smooth elliptic. If Σ → X is an elliptically fibered complex surface with generically smooth fibers, the corresponding case was studied in [3,10]. Using Loojienga description of moduli space of G-bundles on a smooth elliptic fiber as a space conjugacy classes in the affine Kac-Moody Lie groupĜ [11], we can also interpret Bun G (Σ) as a moduli space mHiggsĜ(X) of multiplicative Higgs bundles for the affine Kac-Moody groupĜ. The respective integrable system is of elliptic type in vertical direction.
The case (1) of additive Higgs bundles (Hitchin systems) received large amount of attention in the mathematical literature in the context of geometrical Langlands correspondence and in the physical literature in the context of 6d (2,0) superconformal self-dual tensor theory compactified on algebraic complex curve X for G of ADE type [12][13][14][15]. Quantization of additive Higgs bundles on the curve X relates to the theory of Kac-Moody current algebras on X, conformal blocks of W-algebra on X with punctures, D-modules on Bun G (X), and monodromy problems for various related differential equations.
The case (2) of multiplicative Higgs bundles on a complex curve X appeared first in the context of current Poisson-Lie groups G(x) with spectral parameter x ∈ X. A Poisson-Lie group is a Lie group equipped with Poisson structure compatible with the group multiplication law. There is a standard way to equip G(x) with Poisson structure called quadratic Skylanin bracket given a holomorphic no-where vanishing differential 1-form on X (possibly with poles). Quantization of this Poisson structure leads to the theory of quantum groups [16] [17] which have been discovered in the context of the inverse scattering method, quantum integrable spin chains, Yang-Baxter equation and R-matrix with spectral parameter. The standard trichotomy of the rational, trigonometric or elliptic R-matrix corresponds to taking the base X to be the P 1 with 1-form with a single quadratic pole (rational type), the P 1 with 1-form with two simple poles (trigonometric type), or smooth elliptic curve (elliptic type).
For the smooth elliptic base curve X the multiplicative Higgs bundle was studied in [8], following [20,21]. On another hand, multiplicative Higgs bundles on X have been rediscovered as monopoles on real three-dimensional Riemannian manifold X × S 1 via the monodromy map [22][23][24][25][26][27][28]. The relation between quantization of the moduli space of monopoles on R 3 and Yangian has been proposed in [29] and further work in this direction has been in [30]. Recently a quantization of the holomorphic symplectic phase space of the moduli space of monopoles on X × S 1 by a formal semi-holomorphic Chern-Simons functional on X × S 1 × R t , where R t is the time direction, has been studied in [18,19]. For simple Lie groups G of the ADE type these moduli spaces appear as Coulomb branches of the moduli space of vacua of the N = 2 supersymmetric ADE quiver gauge theory on R 3 × S 1 [28,[31][32][33]. Some constructions from the world of additive Higgs bundles have their versions in the world of multiplicative Higgs bundles [34] leading to difference equations and their monodromy problems [35,36], q-geometric Langlands correspondence [37], q-W algebras [38][39][40].
The goal of this paper is to present very concretely a Darboux coordinate system on a moduli space GL r multiplicative Higgs bundles of degree 1 on the rational base X = P 1 x . The base curve X is equipped with a holomorphic one-form dx that has the quadratic pole at x ∞ = ∞. The holomorphic one-form dx together with the Killing form on the Lie algebra induces the quadratic Sklyanin Poisson structure with the classical r-matrix of rational type in the spectral parameter x. Equivalently, we are studying degree 1 symplectic leaves in the rational Poisson-Lie group GL r (K P 1 ), where K P 1 denotes the field of rational functions on P 1 , and degree 1 means that all matrix elements (L ij (x)) 1≤i,j≤r of the multiplicative Higgs field g(x) in the defining representation of GL r by r × r matrices L ij (x) are degree 1 polynomials of x, i.e. linear functions of x.
By concrete presentation we mean introduction of explicit Darboux coordinates (canonically conjugated set of (p, q) = (p I , q I ) variables with {p I , q J } = δ J I ) and presentation of explicit formulae for the matrix elements L ij (x) in terms of (p I , q J ). The complete set of commuting Hamiltonians is obtained from the coefficients of the spectral determinant of L(x). The matrix L(x) valued in functions on the phase space is called Lax matrix and its matrix elements satisfy quadratic Sklyanin Poisson brackets, see in particular [41,42] but also the recent lecture notes [43].
The quadratic Sklyanin Poisson brackets can be also defined as semi-classical limit of the quantum Yang-Baxter equation [44,45]. In this paper we find all rational solutions of degree 1 in the spectral parameter x associated to the classical Yang-Baxter equation defined by the rational gl(r)-invariant r-matrix, cf. [46]. In another note we plan to consider the trigonometric case associated to the base curve being a punctured nodal elliptic curve X = C × x equipped with the holomorphic one-form dx x that has simple pole at x = 0 and x = ∞, a related work appears in [47].
The case of G = GL 2 is well studied. Here the Sklyanin relation admits three different elementary types of non-trivial solutions with matrix elements linear in the spectral parameter x that yield integrable models. These solutions are called the 2 × 2 elementary Lax matrices for the Heisenberg magnet, the DST chain and the Toda chain. For an overview we refer the reader to lecture notes of Sklyanin [48].
For higher rank r, to the best knowledge of the authors the explicit presentation of all linear solutions is missing in the literature. The case regular at the infinity x ∞ ∈ X has been described in [49] and many other places. Some partial cases of Toda like solutions for irregular case have been described in [25,26]. The classifying labels appeared in [50]. In the quantum case some solutions to the Yang-Baxter equation were studied in connection to non-compact spin chains and Baxter Q-operators, in particular for the case of gl r we refer the reader to [51,52]. The solutions relevant for non-compact spin chains can be obtained by realising the quantum R-matrix in terms of an infinite-dimensional oscillators algebra which is also known as free-field realisation, see e.g. [53]. The Lax matrices relevant for Q-operators are certain degenerate solutions in the sense that the term proportional to the spectral parameter is not the identity matrix but a matrix of lower rank. These Lax matrices can be obtained from the non-degenerate case through a limiting procedure as discussed in [54] for gl(2|1), [25,26] for gl 3 or directly from the universal R-matrix as shown in [55] for gl (3). Vice versa to the limiting procedure and as discussed in [52], one can also obtain the Lax matrices of non-compact spin chains by fusing the degenerate solutions relevant for Q-operators.
Here we follow the strategy of fusion in order to construct a family of GL r Lax matrices L(x) whose matrix elements are linear in spectral parameter x.
The discrete data of labels in our family is specified by two partitions λ and µ such that the total size is |λ| + |µ| = r and whose columns λ t i , µ t i are restricted by r. In addition to the discrete partition labels (λ, µ) we have a sequence of complex labels. There is a complex parameter x i assigned to each column λ t i of the partition λ. Geometrically speaking, each pair (λ t i , x i ) describes a type of singularity of the multiplicative Higgs field g(x) at finite point x i ∈ C = P 1 \ {x ∞ } given by the conjugacy class of (x − x i )ω λ t i whereω k denotes k-th fundamental co-weight of GL r : that is the highest weight of the k-th antisymmetric power of the fundamental representation for the Langlands dual group GL r . Such highest weight is encoded by the column of height λ t i in the partition λ. Equivalently, in the neighborhood of the point x i in the spectrum of the r × r Lax matrix L(x) there are exactly λ t i eigenvalues which vanish linearly as x approaches x i , and the remaining r − λ t i eigenvalues are regular non-zero at x i .
The partition µ specifies a dominant co-weight of singularity of the multiplicative Higgs field at the infinity point x ∞ ∈ P 1 , or equivalently the asymptotics of the eigenvalues of the Lax matrix L(x) as x → ∞: given r rows (µ j ) j∈ [1,...,r] of the partition µ, the j-th eigenvalue of the Lax matrix L(x) has asymptotics (x −1 ) µ j −1 as x → ∞.
We remark that the restriction on the total size of two partitions |µ| + i λ t i = r is a consequence of the restriction of the present paper to consider only Lax matrices whose matrix elements are linear functions of x. In the complete classification, if we allow higher degree of x in the matrix elements, which is not in the scope of the present paper, the label of a singularity at any finite point x i is an arbitrary dominant GL r co-weight described by an arbitrary partition λ i , so that if rows of partition λ i are denoted by (λ ij ) j∈ [1,...,r] then j-th eigenvalue of the Lax matrix L(x) behaves as (x − x i ) λ ij as x → x i . We leave for another note the presentation of explicit formulae for complete classification of the symplectic leaves of the degree d whose matrix elements are degree d polynomials of x for |µ| + i λ t i = dr. (By looking at the determinant of g(x) we see that the moduli space is non-empty only if the total size |µ| + i λ t i is integral multiple of rank r, c.f. e.g. [8,28]. This condition means that the total dominant co-weight summed over all singularitiesω tot belongs to the lattice of co-roots. 1 In our solutions we can obtain higher (non-fundamental co-weight) singularities at finite point x * by collision of several fundamental singularities at x i 1 , x i 2 , . . . , x i k which are associated to some columns λ t i 1 , . . . , λ t i k of the partition λ by sending all of them to the common point x * . In this case, generically, the multiplicative Higgs field g(x) develops the singularity at point x * specified by a higher (non-fundamental) co-weight k j=1ωλ t i j . As we will see, all Lax matrices regular at the infinity x ∞ , that is µ = ∅ in the current notations, and arbitrary λ can be obtained by the fusion procedure of the elementary Lax matrices used in the Q-operator construction [52]. Also the case regular at infinity has been described in [49], where it was shown that degree 1 rational symplectic leaves for G = GL r correspond to the co-adjoint orbits in the dual Lie algebra gl * r . The parametrization by Darboux coordinates of the holomorpic symplectic co-adjoint orbits in gl * r identical to the present paper has been proposed in [56].
Then we proceed to build Lax matrices irregular at infinity from the fusion of a certain set of elementary Lax matrices whose irregularity at infinity is of the simplest type.
Let us clarify the geometrical meaning of the procedure that we call fusion in this paper. A Lax matrix L λ,x,µ (x; p, q) with a certain prescribed type of singularities at x = (x i ) and x ∞ describes a system of Darboux coordinates on a finite-dimensional symplectic leaf in the infinite-dimensional Poisson-Lie group G = GL r (K P 1 ) where K P 1 denotes the field of rational functions on P 1 . Suppose we are given a symplectic leaf M λ,x,µ ⊂ G described by Lax matrix L λ,x,µ (x; p, q) and a symplectic leaf . By definition of Poisson-Lie group structure on G the group multiplication map is a Poisson map, i.e. the pushforward of the product Poisson structure on G × G coincides with the Poisson structure on G. The symplectic leaveas M, M ′ are, in particular, co-isotropic submanifolds of G, hence M × M ′ is a co-isotropic submanifold of G × G. Now, since the group multiplication map m in (1.1) is a Poisson map, and since the Poisson map preserves the co-isotropic property of the submanifolds, the image m(M × M ′ ) ⊂ G is a co-isotropic subspace.
The G-elements in the co-isotropic subspace m(M × M ′ ) ⊂ G are represented by Lax matrices and their type of singularities is typically a combination of the types of singularities of (λ, x, µ) and (λ ′ , x ′ , µ ′ ). However, m(M × M ′ ) ⊂ G is not in general a symplectic leaf but a co-isotropic submanifold, and we can further slice it into symplectic leaves by determining the set of Casimir functionsq ′ on m(M × M ′ ) and a set of new conjugated coordinatesp,q. We find that with the canonical transformation dp ∧ dq + dp ′ ∧ dq ′ = dp ∧ dq + dp ′ ∧ dq ′ . (1.4) Notice that the conjugate variablesp ′ to the Casimir functionsq ′ onS do not appear on the right side of (1.2). The Lax matricesLλ ,x,μ (x;p,q) represent elements of G in a new symplectic leaf Mλ ,x,μ covered by Darboux coordinatesp,q. The symplectic leaves M λ,x,µ arise as moduli spaces of multiplicative Higgs bundles of certain type [9], and like additive Higgs bundles (Hitchin system), the symplectic leaves M λ,x,µ support the structure of an algebraic completely integrable system. In fact, the moduli spaces M λ,x,µ can be also interpreted as moduli spaces of U(r) monopoles on 3-dimensional Riemannian space R 2 × S 1 where R 2 ≃ C = P 1 \ {x ∞ }, and consequently [22][23][24]28] as moduli spaces of vacua of certain N = 2 supersymmetric quiver gauge theories on R 3 × S 1 of quiver type A r−1 . The complex parameters x i ∈ C which specify the position of singularities of the Lax matrix L λ,x,µ play the role of the masses of the fundamental multiplets attached to the quiver node λ t i in the A r−1 quiver diagram, and at the same time they play the role of the complex part of the coordinates of the positions of the Dirac singularities of the U(r) monopoles on R 2 × S 1 under identification R 2 ≃ C. If the partition µ is empty, then the corresponding A r−1 quiver gauge theory is N = 2 superconformal theory, and corresponding monopoles on R 2 ×S 1 are regular at infinity. Non-empty partition µ corresponds to monopoles on R 2 × S 1 with non-trivial growth (or charge) at infinity controlled by µ, or to the Coulomb branches of asymptotically-free quiver gauge theories with β-function controlled by µ.
Consequently, the integrable system supported on a symplectic leaf M λ,x,µ is identical to Seiberg-Witten integrable system for a certain A r−1 quiver gauge theory.
The complete set of commuting Hamiltonians functions H ij can be extracted from the spectral determinant of the associated Lax matrix by taking coefficients at the monomials x i y j where the appearing pairs of indices (i, j) can be described by certain profiles like Newton diagrams. The spectral curves (1.5) coincide with the spectral curves of the integrable systems studied in [28,38]. Equivalently, since the determinant can be expanded in terms of the characters tr R k of the k-th external powers of the fundamental representation, the commuting Hamiltonians are expressed as coefficients at powers of x in the characters tr R k L(x). We remark that by switching the role of variables x ∈ C and y ∈ C × (fiber-base duality) the spectral curve (1.5) of multiplicative Higgs bundle on X can be also interpreted as the spectral curve of additive Higgs bundle (Hitchin system) on Y = C × = P 1 0,∞ . This is a peculiarity related to the fact that we are considering the rational case of the base X = P 1 corresponding to the monopoles on R 2 × S 1 and 4d quiver gauge theories rather than the trigonometric or elliptic base X corresponding to the monopoles on R×S 1 ×S 1 or S 1 ×S 1 ×S 1 that relate to 5d or 6d quiver gauge theories compactified on S 1 or S 1 × S 1 , and also that we take the gauge group to be of type GL r . In this situation, the moduli space of U(r) monopoles on R 2 × S 1 with several singularities has alternative presentation (Nahm duality) as GL n Hitchin moduli space on C × with r singularities where n depends on the number and type of the singularities of the multiplicative Higgs bundle on X [22][23][24]28]. Anyways, the fusion method of this paper allows us to analyze the multiplicative Higgs bundles in more general cases, which we leave for a future work, when Nahm duality of multiplicative Higgs bundle to to a Hitchin system is not known. In particular, in the future one can study classification of symplectic leaves with matrix elements of of higher degree in x, one can analyze trigonometric case with the base curve X is C × = P 1 \ {0, ∞} or elliptic case when the base curve X is a smooth elliptic curve like [8] and consider arbitrary complex reductive Lie groups G.
The article is organised as follows. In Section 2 we remind and set notations about Poisson Lie groups, Sklyanin brackets and Lax matrices. In Section 3 we build the Lax matrices for arbitrary partitions λ and empty µ = ∅ from certain elementary building blocks by fusion. Similarly, in Section 4 we build Lax matrices for arbitrary partitions µ with λ = ∅ again employing certain elementary solutions using a slightly modified fusion procedure. In Section 5 we combine the solutions of Section 3 and 4 to write down the Lax matrices for arbitrary λ and µ. In Section 6 we study the spectral determinant of the derived Lax matrices and compare our results with [28]. In Section 7 we say a few words on higher a degree symplectic leaves. In Section 8 we consider the quantization of the algebra of functions and the integrable system.

Acknowledgements.
We would like to thank Chris Elliott and Alexei Sevastyanov for multiple helpful discussions. R.F. is supported by the IHÉS visitor program. The research of V.P. on this project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (QUASIFT grant agreement 677368), V.P. also acknowledges grant RFBR 16-02-01021.

Rational Poisson-Lie group and Sklyanin brackets
Let X = P 1 be the base curve equipped with the differential holomorphic volume form dx that has a single quadratic pole at x ∞ ∈ P 1 . Fix a Killing form tr on g. Then the residue pairing tr induces the metric on g D = g((x)) with respect to which g[[x]] and x −1 g[x −1 ] are isotropic subspaces and we have g D = g + ⊕ g − . This splitting induces the structure of the Lie bialgebra on g + , which means that the space of functions on g + is equipped with the Poisson bracket (induced from the Lie bracket on g − ). The data (g D , g + , g − ) is called Manin triple. The Poisson bracket on the functions on g + can be extended to the Poisson bracket on the functions on the Lie group G + with the Lie algebra g + , and the resulting bracket is called Sklyanin quadratic bracket with the rational r-matrix. The space of rational multiplicative Higgs fields on X = P 1 with a fixed framing of the gauge bundle at x ∞ forms a Poisson-Lie group [9].
In the following we consider gauge group G = GL r and for a Higgs field g(x) we call L(x) the representation of g(x) by r × r matrix valued functions L(x) called Lax matrices. The Lax matrices L(x) satisfy the quadratic Poisson bracket of rational Sklyanin type the quantization of which gives quantum Yang-Baxter equation [45]. Here the I denotes the r × r identity matrix, and the classical rational r-matrix of gl(r) is The bracket on the right-hand-side of (2.2) denotes the commutator [X, Y ] = XY − Y X. In a system of Darboux coordinates (p, q) = (p I , q I ), the Poisson bracket is where we sum over all conjugate variables (p, q) in the Lax matrices L. In index notations, the Poisson bracket of matrix elements (2.3) reads as follows The solutions to the Sklyanin relation (2.2) that appear in this paper are labelled by two partitions The total number |λ| of elements in the partition λ combined with total number |µ| of elements in the partition µ is equal to r. We study solutions L λ,x,µ (x; p, q) whose matrix elements are no higher than of degree 1 in the spectral parameter x. We can assume that (2.7) Here µ t 1 denotes the first column, i.e. the first element in the transposed partition µ t = (µ t 1 , µ t 2 , . . . , µ t r ) and M λ,x,µ (p, q) denotes an r × r matrix which is independent of the spectral parameter x. In total the matrix M λ,x,µ (p, q) contains pairs of variables (p I , q I ), i.e. I = 1, 2, . . . , d λ,µ . Again the transposed partition is denoted as λ t = (λ t 1 , λ t 2 , . . . , λ t r ), and λ t i are called columns. The dimension of the corresponding symplectic leaf or the moduli space of multiplicative Higgs bundles will be given by We fix the singularity of the L(x) at points x → x i to be of the form [g(x)] ∼ (x − x i )ω λ t i , up to a regular factor, whereω k is the k-th fundamental co-weight associated in Young notations to a column of height k, and at infinity x → x ∞ we take the singularity to be Hereω r is a co-weight associated to the column of height r and Figure 1. Single column partition for r = 5 with λ = 1 [3] and µ = 1 [2] . denoting the diagonal homomorphism GL 1 → T GLr where T stands for the maximal torus, that is a co-weight dual to the weight of the determinant line representation.
The determinant of L(x) determined by the partition λ is a polynomial of degree |λ| with roots x i of degeneracy λ t i : The explicit form of the matrices L λ,x,µ (x; p, q) is given in Section 3 for µ = ∅, in Section 4 for λ = ∅ and for arbitrary λ and µ with |λ| + |µ| = r in Section 5.
As explained in the introduction, we can allow parameters x i to collide in which case the dominant co-weightω * of the singularity at the collision point x * is represented by a partition composed of several columns from λ, and is equal to the sum of the fundamental co-weights associated to each individual column in λ. In this way we get a symplectic leaf M λ,x,µ whose singularity type at x * ∈ x is described by a partition λ * ∈ λ. Bearing this in mind, in the following we assume that parameters x i are assigned to individual columns λ t i of a single partition λ.

Degree 1 symplectic leaves regular with fundamental singularity at infinity
In this section we focus on the GL r Lax matrices that correspond to arbitrary partitions λ of size |λ| and a single column µ-partition, µ = 1 [r−|λ|] . In particular if |λ| = r then µ is empty.
Since µ is a single column, the singularity at infinity is specified by a fundamental co-weight. The associated A r−1 quiver gauge theory with fundamental hypermultiplets [28] differs from the conformal class by absence of a single fundamental multiplet in the node µ t 1 . We will assume that each element of the transposed partition λ t , i.e. each column λ t i of the partition λ specifies a singularity of the Lax matrix L λ, the basis dual to the standard basis of weights of the defining representation. This type of GL r Lax matrices can be obtained by fusion of the fundamental solutions associated to a single column λ = 1 [|λ|] and a single column µ = 1 [r−|λ|] . The fundamental solutions are given in Section 3.1, and the fusion is described in Section 3.3. The Lax matrices for arbitrary partitions λ are given in Section 3.2. We closely follow [52] where the elementary building blocks were derived, the factorisation was discussed on the quantum level and a closed formula for the Lax matrices was obtained for the case λ = (r), see also [51].
3.1. Fundamental (λ, µ) orbits. The fundamental building blocks are r × r matrices that correspond to the partition with r = |λ| + |µ|, see Figure 1. They contain |λ| · |µ| pairs of conjugate variables (p ij , q ji ) where 1 ≤ i ≤ |µ| and |µ| < j ≤ r and can be written as Here the upper diagonal block is of the size |µ| × |µ| and the lower one of size |λ| × |λ|. The block matrices on the off-diagonal are parametrized as follows The letter I denotes the identity matrix of appropriate size. In particular we have The matrices L λ,x,µ (x; p, q) satisfy the Sklyanin relation (2.2) as can be verified by a direct computation using Consequently one finds and which is sufficient in order to check the Sklyanin Poisson bracket. It is instructive to see that L λ,x,µ (x; p, q) is factorized into a product of upper diagonal, diagonal and lower diagonal matrices: The determinant is 3.2. Canonical coordinates on regular orbits. In this section we will construct solutions L λ,x,µ (x; p, q) for arbitrary partitions with λ composed of columns λ t i and a single column partition µ t = (µ t 1 ). The columns λ t i are associated to fundamental singularities at x = x i of type λ t i , which means that the singularity of L λ,x,µ (x; p, q) is in the conjugacy class of i.e. distinct λ t i eigenvalues of L λ,x,µ (x; p, q) are vanishing linearly at x = x i . Figure 2. Regular partition with λ = (4, 3, 1) and µ = (1, 1).
The column µ t 1 describes a fundamental singularity at x = ∞ which means that the singularity of L λ,x,µ (x; p, q) at x → ∞ is in conjugacy class of (3.10) We will prove recursively that regular Lax matrices L λ,x,µ (x; p, q) can be parametrized as a block matrix where the upper-left block is of size µ t 1 × µ t 1 and bottom-right block is of size |λ| × |λ|. The matrix elements of block P µ,λ and block Q λ,µ are canonically conjugated variables with (3.12) and the matrix elements of J λ,λ satisfy the algebra of λ × λ-matrices with respect to the Poisson brackets while Poisson commuting with matrix elements of P µ,λ and Q λ,µ . The matrix elements of the |λ| × |λ| matrix J λ,λ have an explicit parametrization in terms of the canonically conjugated coordinates as follows cf. [49,56] and Appendix A. Here X λ denotes the diagonal matrix The corresponding blocks on the diagonal are of the size λ t 1 , . . . , λ t λ 1 . The matrix [P λ,λ Q λ,λ ] + is block upper triangular and reads Here the matricesP ij are of the size λ t i × λ t j and explicitly given bŷ The matrix Q λ,λ is lower triangular and only depends on the variables q while P λ,λ is upper triangular and only depends on the variables p. They read The realization (3.14) of the gl(|λ|) algebra, also known as free field representation, can be constructed as algebra of twisted differential operators on the flag variety G/P λ,+ . Here G = GL(|λ|) and P λ,+ denotes a parabolic subgroup of GL(|λ|) whose Levi is i GL(λ t i ). The big cell of the flag variety G/P λ,+ is identified with the λ t -blocks unipotent subgroup N λ,− whose elements are represented by matrices Q λ,λ as in (3.18). In the classical limit twisted differential operators in J λ,λ form a co-adjoint orbit O X λ in the dual Lie algebra g * for g = gl(|λ|) of the semi-simple element X λ (3.15). See details in Appendix A.
The number of pairs of conjugate variables in the Lax matrix (3.11) agrees with (2.8). There are µ t 1 × |λ| pairs in P µ,λ , Q λ,µ and i<j λ t i λ t j in J λ,λ . Further we verify that the determinant of (3.11) agrees with (2.10).
and another set of variables appearing in the expression for J ′ λ ′ ,λ ′ like in (3.14). The matrix L ′ λ ′ ,x ′ ,µ ′ (x; p ′ , q ′ ) that appears in (3.24) is obtained from the canonical form (3.11) by permutation, that is a conjugation by an element of the Weyl group of GL r , and a canonical transformation in the variables p and q.
In the next step we multiply the matrices (3.22) and (3.24). It was pointed out in [52] for the corresponding solutions of the quantum Yang-Baxter equation that the product can be written as HereLλ ,x,μ (x;p,q) denotes a spectral parameter dependent Lax matrix and CasimirQ ′ is a lower triangular matrix. They are of the form expressed in terms of the new variables The transformation of coordinates (3.29) is a symplectomorphism (i.e. canonical transformation) as we can directly verify. Indeed, computing the differentials we find and hence, after cancellations, we find that the canonical symplectic form is invariant In analogy to the Yang-Baxter equation, the product of two solutions to the Sklyanin relation (2.2) with different sets of conjugate variables (p, q) is again a solution to the Sklyanin relation (2.2).
Therefore the matrix in (3.26) satisfies the Sklyanin bracket when taking the Poisson bracket with respect to the variables (p,q) which denote the elements of the matrices defined in (3.29).
Finally, we note that the result is independent ofP ′ λ ′ ,λ which allows us to strip off the matrixQ ′ from (3.26). Thus we conclude that is a solution of the Sklyanin relation. Here the generatorsJλ ,λ of the gl(|λ ′ | + |λ|) subalgebra are realised asJλ (3.35) Let us remark that here we have chosen a certain order of fusion, but depending on the order we would get different parametrization related by a polynomial choice of variables, see e.g. Appendix G. It would be interesting to explore the resulting cluster structure in more details. β α γ Figure 3. Example of the decomposition in (4.1) for µ = (5, 4, 2, 1, 1). We have α = (1, 1, 1), β = (1, 1) and γ = (4, 3, 1) 3.3.1. Linear fusion. Now to demonstrate the particular parametrization (3.14) forJλ ,λ it is sufficient to assume that λ ′ is a single column partition λ ′ t = (λ ′ t 1 ) while λ is an arbitrary collection of columns. In this case (3.36) Then we find thatJλ ,λ can be again represented in the form As a consequence it follows that the Lax matrices (3.11) satisfy the Sklyanin relation (2.2).

Degree 1 symplectic leaves singular only at infinity
In the following section we focus on the Lax matrices that correspond to λ = ∅ and arbitrary partition µ. Similar to the case labelled by pure λ partitions in Section 3 the present case can be obtained from fusion of the basic building blocks. These basic building blocks are generalisations of the well-known Lax matrix of the Toda chain [41] corresponding to the partition µ = (2). They are introduced in Section 4.1 and 4.2. The Lax matrices for arbitrary partitions µ are presented in Section 4.3. As discussed in Section 4.4 we can apply a similar fusion procedure as in Section 3.2 to derive the general form of the Lax matrices.
To describe the Lax matrices it is convenient to introduce the partitions as shown in Figure 3. The partition µ is then written as µ t = (|α| + |β|, γ t 1 , . . . , γ t γ 1 ). For simplicity we are only considering partitions with µ i ≥ µ j for 1 ≤ i < j ≤ µ t 1 . where µ t 1 = µ t 2 = r 2 and r ∈ 2N. The Lax matrices L µ (x; p, q) = L ∅,∅,µ (x; p, q) are r × r matrices with |α| + |γ| = r whose determinant evaluates to unity. They contain r 2 2 pairs of conjugate variables (p I , q I ) and can be written in the form For later purposes we labeled the upper block by α and the lower block by γ such that the block on the diagonal are of equal size |α| × |α| and |γ| × |γ| respectively. Further we introduced the matrices where [ ] ± denotes the projection on the upper and lower diagonal part respectively and The matrices Q ±,0 are parametrized in terms of the conjugate variables (p, q) as follows and We note that Q + is an upper triangular matrix containing variables q ij with i > j, while Q − is lower triangular containing the variables q ij with i < j. The diagonal matrix Q 0 only contains the exponential function of q ii . All variables p ij are contained in G which is decomposed as the sum of a diagonal, a lower diagonal and an upper diagonal matrix. The Sklyanin relation is equivalent to the commutators Here the latter can be identified with commutators of the gl( r 2 ) algebra, while the parametrization of K α,γ is given in terms of a Gauss decomposition of GL( r 2 ). These relations are verified explicitly in Appendix C.
The expression for the Lax matrix L µ (x; p, q) in (4.14) is in principle valid for any ordering of columns where |α| + |β| denotes the height of the biggest columns and γ the partition that remains after removing that column. If the partition is ordered, i.e. λ i ≥ λ j for i < j, α,α for i < j and D [1] α,α = 0 which simplifies the expressions above.

Fusion procedure for µ partitions.
The formula for the Lax matrices of the µ partitions can be shown in analogy to Section 3.2. We define three partitions µ, µ ′ andμ with |µ| = |µ ′ | = |μ| = r. They are related by fusion via Here we consider a solution to the Sklyanin relation of the form (4.12) written as a 4 × 4 block matrix The blocks on the diagonal are of the size |α|, |β|, |γ ′ | and |γ|, respectively with |α| + |β| + |γ ′ | + |γ| = r. The matrices Q γ,β , Q γ,γ ′ and Pβ ,γ , P γ ′ ,γ are explicitly given in terms of the conjugate variables. They read Furthermore we define a second Lax matrix, cf. (3.24), which also is a solution of the Sklyanin relation. It has the same block structure as (4.25) and reads We got We proceed as in Section 3.2 and multiply the two solutions of the Sklyanin relation. The product can again be written as cf. (3.26). The spectral parameter dependent matrixLμ(x,p,q) and the matrixQ ′ take the formLμ Here we have writtenLμ(x,p,q) in a factorised form and introduced the matrices (4.32) They are expressed in terms of the new variables This is the same change of variables as in (3.29) and therefore it is canonical. Following the same logic as in Section 3.2 we conclude thatLμ(x,p,q) is a solution of the Sklyanin relation.
For convenience we write it in the same form as L µ (x; p, q) such that The size of the block matrices on the diagonal is |α|, |β| and |γ|. We defined the matrices while the remaining elements are given bỹ andDα ,α = Dα ,α D ′α ,α .

Recursion.
We specify the matrix elements in the fusion procedure to describe the fusion of one arbitrary partition µ as proposed in Section 4.3 and an elementary matrix (4.12) corresponding to the partition µ ′ with the restriction α ′ = γ ′ . The resulting partitioñ µ is then written in terms of µ and µ ′ as in (4.24). This can be seen as follows. The primed letters correspond to elements of the Lax matrix corresponding to µ ′ and read The unprimed letters correspond to the partition µ as given in (4.15), (4.16) and (4.17). We find that where similar as for the case of λ-partitions in (3.37) we identifỹ and obtainKγ and Thus we conclude that (4.14) satisfies Sklyanin's quadratic Poisson bracket.

Generic degree 1 symplectic leaves
We will now define the Lax matrices L λ,x,µ (x; p, q) for arbitrary partitions λ and µ. They can be obtained by fusing the Lax matrix for regular partitions (3.11) with the Lax matrix for µ partitions (4.14).

Lax matrix for λ,µ partitions. The Lax matrix for arbitrary partitions λ and µ can compactly be written as
The blocks on the diagonal are of the size |α|, |β| and |γ| + |λ| respectively. Here Q λ,γ and P γ,λ are defined as The remaining matrix elements in (5.1) are given in terms of the components of the Lax matrix for regular partitions (3.11) and the Lax matrix for µ partitions (4.14). We have Again we can check that the number of pairs of conjugate variables agrees with (2.8). First we note that F contains i<j γ t i γ t j + |α| · |γ| and J contains i<j λ t i λ t j pairs. The remaining variables are contained in P β,γλ , Q γλ,β , P α,λ , Q λ,α and P γ,λ , Q λ,γ . By construction, cf. Section 5.2, the determinant of the Lax matrix in (5.1) satisfies (2.10).
The symplectic leaves that we found in the Poisson-Lie group G are orbits of certain representative elements under the dressing action of the dual Poisson-Lie group G * . These representative elements are easily seen as Lax matrices at p = q = 0. Here the Lax matrix (5.1) reduces to a block matrix of the form and X λ denotes the diagonal matrix defined in (3.15). The matrix Σ µ is a permutation matrix containing the elements ±1. This matrix can be block diagonalized such that it contains |α| + |β| blocks of the size µ i , i = 1, . . . , |α| + |β|, corresponding to the rows of the partitions µ. The diagonal of each block i reads diag(0, x, . . . , x) and its remaining elements ±1 correspond to a cyclic permutation of length µ i . For example for a row of µ i = 4 we obtain For µ i = 1 where i = |α| + 1, . . . , |α| + |β|, we obtain a 1 × 1 block containing only the element 1.
5.2. Fusion procedure. The Lax matrix (5.1) can be derived using the factorisation formula in Section 4.4 when substituting the γ block for a λ =λ block as follows Here I is the |α| × |α| identity matrix. The primed elements in the second Lax matrix L ′ are taken to be as defined in (4.14) as Here D ′ is equal to the |α| × |α| zero matrix. This factorisation corresponds to the fusion of the partitions λ, µ and λ ′ , µ ′ expressed in terms of the resulting partitionλ,μ via The final result of the factorisation can be directly read off from (4.34). We conclude that (5.1) is a solution to Skyanin's relation (2.2).

Algebraic completely integrable systems and Coulomb branches of A r−1 quiver gauge theory
The symplectic leaf M λ,x,µ , i.e. the moduli space of multiplicative Higgs bundles with fixed singularities, supports fibration of an algebraic completely integrable system Here H denotes a complete set of independent commuting Hamiltonian functions (also known as conserved charges or action variable or integrals of motion of an integrable Hamilonian dynamical system) and U λ,x,µ denotes the space where the complete set of independent Hamiltonians takes value. The fibers A u = H −1 (u), u ∈ U λ,x,µ of the map (6.1) are abelian varieties which are holomorphic Lagrangians with respect to the holomorphic symplectic structure on M λ,x,µ , so that 3) denote the half-dimension of the symplectic leaf (phase space) M λ,x,µ . In the context of Seiberg-Witten integrable systems [22,23,28,57] the holomorphic symplectic phase space M λ,x,µ is the Coulomb branch of the hyperkähler moduli space of vacua of N = 2 supersymmetric quiver gauge theory on R 3 × S 1 viewed as a holomorphic symplectic manifold at a distinguished point ζ = 0 on the twistor sphere of complex structures. The complex base space U λ,x,µ is the moduli space of vacua of the same N = 2 supersymmetric gauge theory on R 4 called U-plane in the respective context. In terms of action-angle variables, the complex action variables parametrize the base U-plane, and the complex angle variables parametrize the abelian fibers A u . To realize the structure of an algebraic completely integrable system (6.1) we need to construct d λ,x,µ independent Poisson commuting Hamiltonian functions on M λ,x,µ . Like in the case of additive Higgs bundles (Hitchin system), the commuting Hamiltonian functions on multiplicative Higgs bundles (or more general abstract Higgs bundles) can be realized by the abstract cameral cover construction [2]. In the case of additive Higgs bundles on X, the cameral cover construction generates Poisson commuting Hamiltonian functions as coefficients of P (φ(x)) where the Higgs field φ(x) is a section of ad g ⊗ K X and P is an adjoint invariant function on the Lie algebra g. Similarly, in the case of multiplicative Higgs bundles on X, the cameral cover construction generates Poisson commuting Hamiltonian functions as coefficients of χ(g(x)) where χ is an adjoint invariant function on the group G and multiplicative Higgs field g(x) is a section of Ad G on X. The complete set of independent Poisson commuting Hamiltonians for simple G is spanned by the characters χ R i of the fundamental irreducible highest weight modules R k whose highest weight is the fundamental weight ω k for each k in the set of nodes of the Dynkin diagram of g.
If G = GL r , the irreducible highest weight module R k with highest weight ω k associated to the k-th node of the A r−1 Dynkin diagram of the simple factor SL r ⊂ GL r is isomorphic to the k-th external power R k = k R 1 , for k = 1, . . . , r − 1, of the defining r-dimensional representation R 1 , and we set R r = r R 1 to be the determinant 1-dimensional representation. It is convenient to assemble the fundamental characters χ k = χ R k for k = 1, . . . , r into the spectral polynomial det(yI r×r − L(x)) r×r = r k=0 (−1) k y r−k χ k (g) (6.4) for any group element g ∈ G and where ρ k : G → End(R k ) denotes the representation of G in the operators on the vector space R k , and χ k (g) = tr R k ρ k (g). Now we illustrate explicitly the construction of commuting Hamiltonians for the Lax matrices constructed in the previous sections that describe the symplectic leaves M λ,x,µ .
First, for any symplectic leaf M λ,x,µ and its representing Higgs field g λ,x,µ (x) we define its twisted version g λ,x,µ,g L ;g R (x) = g L g λ,x,µ (x)g R (6.5) which represents a symplectic leaf M λ,x,µ;g L ,g R . Here g L ∈ G, g R ∈ G are arbitrary constant (xindependent) Higgs fields. We remark that the symplectic leaves M λ,x,µ;g L ,g R are isomorphic for various g L , g R , and for certain relation between g L and g R they in fact coincide, in this sense the labeling by both g L and g R are redundant. 2 For the following, it is sufficient to take, g L ≡ g ∞ , g R ≡ 1, and define the Lax matrix where ρ 1 (g ∞ ) is r × r matrix representing g ∞ ∈ G. Note that due to the symmetries of the r-matrix in (2.2) the product ρ 1 (g ∞ )L λ,x,µ (x) is a solution to the Sklyanin relation if L λ,x,µ is. Now define the spectral determinant to be a polynomial of two variables x and y W λ,x,µ,g∞ (x, y) = det(y − L λ,x,µ,g∞ (x)) = r k=0 (−1) k y r−k χ k (x) . (6.7) The commuting Hamiltonians are coefficients of the monomials x j y i . With we find that the spectral determinant can be written as i is a polynomial in x, which is independent of the conjugate variables (p, q) of the Lax matrix and which takes the form We note that the number of independent commuting Hamiltonians only depends on the total dominant co-weight represented by the partition obtained by the union of columns of the partitions λ and µ minus the shift by the diagonal co-representation (see below (6.18). The charges are obtained as the coefficients of the expansion cf. Figure 4. The highest coefficients do not depend on the conjugate variables (p I , q I ) but all other coefficients in the expansion do. The total number of linearly independent charges is equal to the number of conjugate pairs in the corresponding Lax matrix cf. (2.8). This relation is shown using Frobenius-like coordinates for the partitions in Appendix D. For given partitions we can plot the non vanishing coefficients of the spectral determinant in a Newton diagram as done in Figure 4. Here we introduce the parameters (6.14) to label the partition and the corresponding Newton diagram. The representation theoretical meaning of the equations (6.11), (6.13) and (6.14) is the following. The partitions λ and µ encode the GL r co-weights of the respective singularity of the multiplicative Higgs field at finite points x and x ∞ . The encoding is in the r-dimensional basis of the dual to the weights of the defining representation that we callě k with k = 1 . . . r. In terms of e i define the simple co-roots α i of SL r to bě and define the fundamental weights to bě The dominant co-weight associated to each singularity x * ∈ x with associated partition λ * = λ * ,1 ≥ λ * ,2 ≥ · · · ≥ λ * ,r = 0 is given by (6.17) and the dominant co-weight associated to the singularity at x ∞ = ∞ with associated partition µ is given byω 18)  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16   10  9  8  7  6  5  4  3  2  1  4  8  9 10 10 10 3 1 1 Figure 5. Representation of the Newton polygon in Figure 4 corresponding to the partition ν = (5, 5, 2, 2, 1, 1, 1) as quiver diagram. Here the integers in the circles denote the number of charges for a given index i indicated below. The parameters m [λ,µ] i are given in the squared boxes.
so that the GL r multiplicative Higgs field behaves up to a multiplication by a regular function as (x − x * )ω * as x → x * and as (1/x)ω ∞ as x → ∞.
Let ρ be the Weyl vector and letω be the sum of the co-weights of all singularities in x and x ∞ . Then we see that the dimension formula (6.13) is equivalent to dim M λ,x,µ;g L ,g R = 2d λ,x,µ = 2(ρ,ω tot ) (6.21) in agreement with the general formula for the dimension of the moduli space of monopoles with Dirac singularities encoded by the total co-weightω tot . The numbers n k and m k for k ∈ [1, r − 1] are the number of colors in the node k and the number of fundamental flavours attached to the node k of the Dynkin quiver [28], including the "deficit" fundamental flavours multiplets of asymptotically free theory which would make it conformal if added. We haveω (6.22) in agreement with (6.11) and (6.14). The position of each singularity x * ∈ x to which we have associated a column λ t * encoding a fundamental co-weightω λ t * is the mass of the fundamental flavour multiplet at the node λ t * . Now the spectral curve can be compared with [28] where a slightly different notation is used. To do so we note that the polynomials Q  Figure 6. Decomposition of the partition λ = (8, 4) for r = 3 into three partitions (3,3), (2,1) and (3) This relation is shown in Appendix E. Setting m [λ,∅] i = 0 for i > |λ| we can now write the spectral determinant (6.9) in the notation used in (7.5) of [28]. We find with i = 1, . . . , r − 1. We note that here the so-called matter polynomials P only depend on the partition λ and not on µ.

Higher degree symplectic leaves
In this section we discuss some symplectic leaves of higher degree n in the spectral parameter x corresponding to the partitions of the total size nr for G = GL r .
(7.1) c.f. [49]. For r = 3 the partitions are of the form λ = (λ 1 , 3n − λ 1 ) with λ 1 ≤ 3n. The Lax matrices for these partitions can be factorized as a product of Lax matrices of partitions λ = (3), λ = (2, 1) and their conjugates. The conjugates correspond to the partitions λ = (3, 3) and λ = (2, 1) respectively and are obtained viā This can be seen as follows. If 3n − λ 1 = 0 we can build the partition from copies of λ = (3) as for the case r = 2. Any such partition is extended to the case where 3n − λ 1 = 1 by adding a partition λ = (2, 1) which again extends to 3n − λ 1 = 2 by adding another partition λ = (2, 1). Now we note that any λ = (λ 1 , 3n − λ 1 ) can be reduced to the cases discussed when stripping off multiples of the partition λ = (3,3). An example is shown in Figure 6. A similar factorization of higher degree leaves to the product of the degree 1 leaves applies to the case of GL 4 . However in the case of GL 5 and higher rank such factorization fails, first time for the n = 2 and the partition λ = (4, 3, 3), i.e. λ t = (3, 3, 3, 1), of the total size |λ| = 10. The Lax matrix associated to this partition is not factorized into a product of degree 1 Lax matrices. However, we can compute L λ= (4,3,3) using the fusion method.

Fusion of degree 2.
In this subsection we present the degree 2 fusion of two elementary Lax matrices and as introduced in (3.2). Here the blocks on the diagonal are of the size k 1 × k 1 , k 2 × k 2 and k 3 × k 3 and the Lax matrices contain k 2 (k 1 + k 3 ) and k 3 (k 1 + k 2 ) pairs of conjugate variables respectively. We find that their product can be decomposed as Here we definedP As in the linear fusion we introduced the new canonical variables (7.10) The final Lax matrix has k 1 k 2 + k 1 k 3 + k 2 k 3 pairs (p, q) and detL(x) = (x − x 1 ) k 1 +k 3 (x − x 2 ) k 1 +k 2 . It corresponds to the partitionλ t = (k 1 + k 3 , k 1 + k 2 ). This can be seen when setting all p and q equal to zero in (7.7). One obtains

Full fusion of degree 2.
We can further multiply the resulting Lax matrix in (7.7) which corresponds to a general regular partition (3.11) with arbitrary λ and µ = 1 [k 1 ] . The blocks on the diagonal are of the size k 1 × k 1 andk 2 = k 2 + k 3 as defined in Section 7.1. One finds Here we definedP ′ 12 = P ′ 12 + P 12 1 . (7.17) The final Lax matrix contains k 1k2 + k J + k J ′ pairs of conjugate variables where k J and k J ′ denote the number of pairs in J and J ′ respectively. It corresponds to the partitioñ λ t = (k 1 + k 3 , k 1 + k 2 , λ t ). Setting p, q = 0 yields the block matrix Here X λ is defined in (3.15) corresponding to an arbitrary partition λ and X ′ λ ′ follows from (7.7) and reads 7.3. Exampleλ = (4, 3, 3). The case GL 5 andλ = (4, 3, 3) we discussed at the beginning of this section corresponds to setting k 1 = 1 andk 2 = 4 in Section 7.2 while setting λ = (2, 1, 1) such that n J = 3. The Lax matrix L ′ is obtained in the case k 1 = 1, k 2 = 2 and k 3 = 2 from the fusion in Section 7.1 and thus yields n J ′ = 4. The half-dimension is 1 2 dim C M 4,3,3 = 11. For p, q = 0 the diagonal of the Lax matrix follows from (7.18) when taking X λ = diag(x 3 , x 3 , x 3 , x 4 ).

Quantization
Notice that the classical Yang-Baxter equation (2.2) is a limit of quantum Yang-Baxter equation where V ≃ C r is the fundamental representation of gl r and A is the quantized algebra of functions on the classical phase space parametrized locally by (p I , q I ). Here the quantum R-matrix is R ∈ End(V ) ⊗ End(V ) and the quantum L-operator isL ∈ End(V ) ⊗ A, that is an r × r matrix valued in operators in A. The quantum R-matrix is where ǫ = −i is the quantization parameter and P is the permutation operator (2.3).
In terms of matrix elementsL ij we have the quantum Yang Baxter equation in A and its classical limit is with the standard convention that can be represented in the algebra of differential operators acting on Hilbert space of states represented by function of q I asq For a polynomial function f (q,p) the normal ordering notation : f (q,p) : means placing all operatorsp I to the right of the operatorsq I in each monomial.
The quantum versionL λ,x,µ (x) of all our classical solutions L λ,x,µ (x) is obtained by replacing all variables (p, q) by the operatorsp,q and assuming normal ordering convention. One can check that such operator valued matrixL λ,x,µ (x) satisfies quantum Yang-Baxter equation. The commuting Hamiltonians are obtained from the expansion of the quantum spectral determinant (quantum spectral curve) as in [58] W x,y = tr A r y − e ǫ∂xL 1 (x) y − e ǫ∂xL 2 (x) · · · y − e ǫ∂xL r (x) = r k=0 (−1) k y r−kχ k (x + ǫ)e ǫk∂x (8.9) where A k is the normalised antisymmetrizer acting on the k-fold tensor product of C r . The quantum characters whose coefficients generate the algebra of quantum commuting Hamiltonians (Bethe subalgebra) arê see also [59]. The definition of the quantum spectral determinant (8.9) is a quantum version of the classical spectral curve (6.7), and there is a quantum version of the factorization (6.8) where the c-valued polynomials arê The quantization of the corresponding integrable systems in the context of the N = 2 supersymmetric quiver gauge theories has been considered in [38], in particular the qcharacter functions appearing in [38] after [60] stand for the eigenvalue of the quantum commuting Hamiltonians (8.10).
The quantized symplectic leavesM λ,x,µ are modules, typically infinite-dimensional, for the dual Yangian algebra Y(gl r ) * which is a quantum deformation algebra of the space of functions on the Poisson-Lie group GL r (K P 1 x ). This representation theory relates to the 'prefundamental' modules of Hernandez-Jimbo [61] associated to the individual singularities at points x i labeled by a fundamental co-weightω λ t i .
Appendix A. Twisted cotangent bundles of generalized flag varieties Let g be a reductive Lie algebra and let g = n − ⊕ h ⊕ n + be a decomposition of g into the Cartan subalgebra h, the negative nilpotent subspace n − = ⊕ α<0 g α and positive nilpotent subspace n + = ⊕ α>0 g α . Here α denote a root of g and g α the α-root subspace of g. Let b + = h + n + and b − = h + n − be the respective Borel subalgebras. If g = gl r then b + (or b − ) is represented by upper (or lower) triangular matrices including the diagonal, and n + (or n − ) is represented by strictly upper (or lower) triangular matrices excluding the diagonal.
Let G, H, N ± , B ± the respective Lie groups with Lie algebras g, h, n ± , b ± , and let x ∈ h * be a weight. Here we record explicit formulas for representation of Ug in x-twisted differential operators on the complete flag manifold G/B + following the approach of Harish-Chandra, Springer, Kostant, Beilinson-Bernstein. We identify the big cell of G/B + with N − and denote elements of N − by Q.
We compute the vector field L X associated to the action of Lie algebra element X on G/B + from the left. Let ε be infinitesimal parameter, and let 1 + εX be a group element corresponding to Lie algebra element X ∈ g. LetQ = Q+εδ X Q denotes a coset representative in G/B + obtained from the action of 1 + εX on Q from the left: where (1 + εn + + εh) is an element of B + that gauges the deformation of Q. We find and thus where [ ] − denotes the projection g → n − . The corresponding vector field and the differential operator on scalar functions on N − is L X = −δ X Q ∂ ∂Q that is where the minus sign comes from the standard convention of defining the vector fields associated to the group actions on manifolds in such a way as to preserve the Lie algebra bracket.
We are actually interested in a more general situation, when the differential operator L X acts not on functions on G/B + but on sections of line bundle induced from the H-bundle G/N + → G/B + by a semi-simple co-weight x ∈ h * , e.g.
The additional connection term is −x(h X ) for the diagonal variation h X in the coset computation (A.1) where [ ] 0 denotes the projection to the diagonal part g → h, we find the differential operator acting on sections of the line bundle on G/B + . Now we fix g = gl r . Let (e ij ) i,j∈ [1,r] denote the standard basis elements of gl r represented by matrices whose (i, j)-entry is equal to 1 and the rest is 0. The upper-triangular Borel subgroup B + preserves the standard full flag 0 ⊂ Ce 1 ⊂ Ce 1 ⊕ Ce 2 ⊂ · · · ⊂ Ce 1 ⊕ Ce 2 · · · ⊕ Ce r (A.9) Further we define the coordinates (q i,j ) with 1 ≤ j < i ≤ r on N − taking the matrix elements of Q ∈ N − in the defining representation of gl r for example, for gl 3 we have We evaluate in coordinates (q) ij the differential operator L x,x associated to each basis element X = e ij in g, and we assemble r × r matrixL valued in twisted differential operatorŝ (A.12) Let us denote with [p ij , q kl ] = δ ik δ jl (A.14) and assemble the upper triangular matrix with only non-zero entries (P ) ij = p ji for i > j. In order to show that the Lax matrix (4.3) satisfies the Sklyanin relation (2.2) we verify (4.8) and (4.9) in the following.
Starting with (4.9) we first note that F γ,γ in (4.4) can be written as where J γ,γ has been defined in (3.14) and satisfies the gl( r 2 ) commutation relations (3.13). It follows that (4.9) is equivalent to where [X, Y ] = XY − Y X denotes the anticommutator. Further we use the notation X 1 = X ⊗ I and X 2 = I ⊗ X and P act as a permutation such that PX 1 = X 2 P. It is convenient to consider different cases. Writing (C.2) in components and taking into account that G ′ and Q − are lower diagonal while [P + Q − ] + is upper diagonal results in the conditions  Here we suppressed the subindeces α and γ. The Latin indices take values i, j, k, l = 1, . . . , r 2 . Using the relation ∂ q K = K(∂ qK )K which follows from (4.5), one finds that the two equations in (C.5) are equivalent. They can be written as In order to show this relation it is again convenient to consider different cases and take into account the dependence of G on p. For a particular choice of the indices i and j we see that (C.6) is equivalent to: These equations can be checked explicitly. The derivatives with respect to q ij for i = j essentially yield delta functions while the derivatives with respect to p ij give the q-dependence.
Appendix D. The number of independent commuting Hamiltonians In order to show the relation (6.13) it is convenient to introduce Frobenius-like coordinates to label the partitions, compare e.g. [62]. As λ is arbitrary it follows that the same relation holds for partitions µ with λ = ∅. Combining the two relations we find that where we used that r = |λ| + |µ|. Another way to show (6.13) is to use the relation (E.1).
Appendix G. Cluster structures In this section we elaborate on the cluster structure of the fusion procedure in Section 3.3. As an example we study the GL 3 case and introduce the Lax matrices In other words, the fusion procedure to obtain the Darboux coordinates is not associative, and the failure of the associativity is described by the symplectomorphism (G.8).