Luen-Chau Li, Zhaohu Nie
In previous work, we introduced a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter, as classified by Etingof and Varchenko for simple Lie algebras. Here the main purpose is to establish the Liouville integrability of these systems by a uniform method.
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The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of R^n to the `angular Calogero-Moser model' on the sphere S^{n-1}. We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a `relative angular Calogero-Moser model', which is analyzed in parallel. We …
A two-dimensional integrable system being a deformation of the rational Calogero-Moser system is constructed via the symplectic reduction, performed with respect to the Sklyanin algebra action. We explicitly resolve the respective classical equations of motion via the projection method and quantize the system.
We examine the Hamiltonian structures of some Calogero-Moser and Ruijsenaars-Schneider N-body integrable models. We propose explicit formulations of the bihamiltonian structures for the discrete models, and field-theoretical realizations of these structures. We discuss the relevance of these realizations as collective-field theory for the discrete models.
Field generalization of elliptic Calogero-Moser system in the form of higher rank Landau-Lifshitz model
本文证明了椭圆Calogero-Moser模型的场泛化与高阶Landau-Lifshitz模型在1+1维时空上的gauge等价性，并给出了变量变换的显式公式。
We give detailed description for continuous version of the classical IRF-Vertex relation, where on the IRF side we deal with the Calogero-Moser-Sutherland models. Our study is based on constructing modifications of the Higgs bundles of infinite rank over elliptic curve and its degenerations. In this way the previously predicted gauge equivalence between L-A pairs of the Landau-Lifshitz type equations and 1+1 field theory generalization of the Calogero-Moser-Sutherland models is described. In this paper the ${\rm sl}_2$ case is studied. Explicit changes of variables are obtained between the rat…
We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero-Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic type, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pa…
Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
本研究将椭圆Calogero-Moser模型推广到1+1维场情形，证明其Lax联络满足Maillet型非超局域r矩阵结构，还拓展了自旋版本及多极推广，并探讨了联系超局域与非超局域r矩阵的IRF-Vertex对应场类比。
The main result of this note is the proof of degenerate quantum integrability of quantum spin Calogero--Moser systems and the description of the spectrum of quantum Hamiltonians in terms of the decomposition of tensor products of irreducible representations of corresponding Lie algebra.
We consider trigonometric solutions of the KP hierarchy. It is known that their poles move as particles of the Calogero-Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution of the poles with respect to the $k$-th hierarchical time of the KP hierarchy is governed by a Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the trigonometric Calogero-Moser hierarchy.
The canonical transformation and its unitary counterpart which relate the rational Calogero-Moser system to the free one are constructed.
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.
We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles of rational solutions to the matrix KP hierarchy with discrete time. The dynamics of poles is derived using the auxiliary linear problem for the discrete flow.
We present a relationship between the Calogero-Moser particles confined in harmonic oscillator potentials and a representation theory of the infinite dimensional Lie algebra which is a semi-direct sum of Virasoro algebra and its module. More precisely, it is a correspondence of excited states of the model and singular vectors in Verma modules over the algebra. This is found by a free field realization of the time evolution operator of the model. We investigate the Verma modules and some explicit example of singular vectors are given.
We show that the configuration in the phase space of the elliptic Calogero-Moser model when particles stick together in pairs is stable under the third Hamiltonian flow of the model. The equations of motion for the pairs coincide with the equations of motion for poles of elliptic solutions to the B-version of the Kadomtsev-Petviashvili equation.
We use Hamiltonian reduction to simplify Falqui and Mencattini's recent proof of Sklyanin's expression providing spectral Darboux coordinates of the rational Calogero-Moser system. This viewpoint enables us to verify a conjecture of Falqui and Mencattini, and to obtain Sklyanin's formula as a corollary.
In this work, we present another example of the Lagrangian 1-form structure for the hy- perbolic Calogero-Moser system both in discrete-time level and continuous-time level. The discrete-time hyperbolic Calogero-Moser system is obtained by considering pole-reduction of the semi-discrete Kadomtsev-Petviashvili (KP) equation. The key relation called the discrete-time closure relation is directly obtained from the compatibility between the temporal Lax matrices. The continuous-time hierarchy of the hyperbolic Calogero-Moser system is obtained through two successive continuum limits. The continuou…
The most general momentum independent dynamical r-matrices are described for the standard Lax representation of the degenerate Calogero-Moser models based on $gl_n$ and those r-matrices whose dynamical dependence can be gauged away are selected. In the rational case, a non-dynamical r-matrix resulting from gauge transformation is given explicitly as an antisymmetric solution of the classical Yang-Baxter equation that belongs to the Frobenius subalgebra of $gl_n$ consisting of the matrices with vanishing last row.
We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n,R). The relation with the Lax formalism is also discussed.
Generalised spin Calogero-Moser systems from Cherednik algebras
1999年Chalykh等人工作中出现了具有非对称势奇点构型的可积自旋Calogero-Moser型系统，利用Cherednik代数表示理论对其进行多种推广。
We construct a class of interacting spin Calogero-Moser type systems. They can be regarded as a many particle system with spin degrees of freedom and as an integrable spin chain of Gaudin type. We prove that these Hamiltonian systems are superintegrable.
We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian $H_k$ of the elliptic Calogero-Moser model which governs the dynamics of poles with respect to the $k$-th hierarchical time. The Hamiltonians $H_k$ are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker-Akhiezer function has essential singularity.
We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In particular, we construct the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition function of the corresponding gauge theory. This is equivalent to the introduction of certain co-dimension two defects. We next generalize our construction to the folded instanton partition function obtained through the so-called "gauge origami" construction and precisely o…
Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by the representations of the reflection groups, which are not the same as those of the corresponding Lie algebras. The latter specify the Lax pairs of Toda models. The Hamiltonians of the elliptic Calogero-Moser models tend to those of Toda models as one of the periods …
Liouville integrability of classical Calogero-Moser models is proved for models based on any root systems, including the non-crystallographic ones. It applies to all types of elliptic potentials, i.e. untwisted and twisted together with their degenerations (hyperbolic, trigonometric and rational), except for the rational potential models confined by a harmonic force.
The possibility of deformation of two body quantum Calogero-Moser-Sutherland models is studied. Obtained are some necessary conditions for the singular locus of the potential function. Such locus is determined if it consists of two, three or four lines. Furthermore, a new deformation of elliptic $B_{2}$ type Calogero-Moser-Sutherland model is explicitly constructed.
We construct integrable generalizations of the elliptic Calogero-Sutherland-Moser model of particles with spin, involving noncommutative spin interactions. The spin coupling potential is a modular function and, generically, breaks the global spin symmetry of the model down to a product of U(1) phase symmetries. Previously known models are recovered as special cases.
Calogero-Moser eigenfunctions modulo $p^s$
本文利用 Matsuo-Cherednik 对偶性和 KZ 方程超几何解的积分表示，构造了有理耦合常数下 Calogero-Moser 本征函数的多项式 $p^s$-截断，并分析了 $n=2$ 时 $s\rightarrow\infty$ 趋于纯 $p$-adic 情形的极限。
Explicit algebraic relations between the quantum integrals of the elliptic Calogero--Moser quantum problems related to the root systems ${\bf A_2}$ and ${\bf B_2}$ are found.
The integrability of the deformed quantum elliptic Calogero-Moser problem introduced by Chalykh, Feigin and Veselov is proven. Explicit recursive formulae for the integrals are found. For integer values of the parameter this implies the algebraic integrability of the systems.
Explicit algebraic relations between the quantum integrals of the elliptic Calogero-Moser quantum problems related to the root systems A_2 and B_2 are found.
Algebraic integrability of the elliptic Calogero-Moser quantum problem related to the deformed root systems A_2(2) is proved. Explicit formulae for integrals are found.
We establish a correspondence between rational solutions to the matrix KP hierarchy and the spin generalization of the Calogero-Moser system on the level of hierarchies. Namely, it is shown that the rational solutions to the matrix KP hierarchy appear to be isomorphic to the spin Calogero-Moser system in a sense that the dynamics of poles of solutions to the matrix KP hierarchy in the higher times is governed by the higher Hamiltonians of the spin Calogero-Moser integrable hierarchy with rational potential.
We compute the full expression of the second Poisson bracket structure for N=2 and N=3 site rational classical Calogero-Moser model. We propose an r-matrix formulation for N=2. It is identified with the classical limit of the second dynamical boundary algebra previously built by the authors.
We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system ($\vee$-system) and we determine all trigonometric $\vee$-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a factorized eigenfunction if and only if it corresponds to the trigonometric $\vee$-system; this inverts a one-way implication observed by Veselov for the rational solutions.
We consider solutions of the matrix KP hierarchy that are trigonometric functions of the first hierarchical time $t_1=x$ and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system on the level of hierarchies. Namely, the evolution of poles $x_i$ and matrix residues at the poles $a_i^αb_i^β$ of the solutions with respect to the $k$-th hierarchical time of the matrix KP hierarchy is shown to be given by the Hamiltonian flow with the Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the spin trigonometric Calogero-Moser…
Sklyanin's formula provides a set of canonical spectral coordinates on the standard Calogero-Moser space associated with the quiver consisting of a vertex and a loop. We generalize this result to Calogero-Moser spaces attached to cyclic quivers by constructing rational functions that relate spectral coordinates to conjugate variables. These canonical coordinates turn out to be well-defined on the corresponding simple singularity of type $A$, and the rational functions we construct define interpolating polynomials between them.
We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type $A_{n-1}$ and $B_n$, as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added.
We prove bispectral duality for the generalized Calogero-Moser-Sutherland systems related to configurations $A_{n,2}(m), C_n(l,m)$. The trigonometric axiomatics of Baker-Akhiezer function is modified, the dual difference operators of rational Macdonald type and the Baker-Akhiezer functions related to both series are explicitly constructed.
It is shown that the Calogero-Moser models based on all root systems of the finite reflection groups (both the crystallographic and non-crystallographic cases) with the rational (with/without a harmonic confining potential), trigonometric and hyperbolic potentials can be simply supersymmetrised in terms of superpotentials. There is a universal formula for the supersymmetric ground state wavefunction. Since the bosonic part of each supersymmetric model is the usual quantum Calogero-Moser model, this gives a universal formula for its ground state wavefunction and energy, which is determined pure…
Classical Calogero-Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an additional set of r-1 algebraically and functionally independent globally defined conserved quantities. At the quantum level, Kuznetsov uncovered the existence of a quadratic algebra structure as an underlying key for superintegrability for the models based on A type root systems. Here we demonstrate in a universal way the quadratic algebra structure for quantum ra…
We discuss a relation between the characteristic variety of the KZ equations and the zero set of the classical Calogero-Moser Hamiltonians.
Various infinite-dimensional versions of the Calogero-Moser operator are discussed. The related class of Jack-Laurent symmetric functions is studied. In the special case when parameter k=-1 the analogue of Jacobi-Trudy formula is given and the relation with representation theory of Lie superlagebra gl(m,n) is discussed.
We consider the generalised Calogero-Moser-Sutherland quantum integrable system associated to the configuration of vectors $AG_2$, which is a union of the root systems $A_2$ and $G_2$. We establish the existence of and construct a suitably defined Baker-Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald-Ruijsenaars type in an explicit form.
We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. It is known that poles $x_i$ and matrix residues at the poles $ρ_i^{αβ}=a_i^αb_i^β$ of such solutions as functions of the time $t_2$ move as particles of spin generalization of the elliptic Calogero-Moser model (elliptic Gibbons-Hermsen model). In this paper we establish the correspondence with the spin elliptic Calogero-Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the $k$-th hierar…
In this paper we construct and prove superintegrability of spin Calogero-Moser type systems on symplectic leaves of $K_1\backslash T^*G/K_2$ where $K_1,K_2\subset G$ are subgroups. We call them two sided spin Calogero-Moser systems. One important type of such systems correspond to $K_1=K_2=K$ where $K$ is a subgroup of fixed points of Chevalley involution $θ: G\to G$. The other important series of examples come from pair $G\subset G\times G$ with the diagonal embedding. We explicitly describe examples of such systems corresponding to symplectic leaves of rank one when $G=SL_n$.
Consistency of the Bäcklund transformation for the spin Calogero-Moser system
本文证明了有理、三角和双曲情形下spin Calogero-Moser系统Bäcklund变换的相容性，通过构造函数证明超定ODE系统在温和假设下恒为零并存在唯一解。
We construct twisted Calogero-Moser (CM) systems with spins as the Hitchin systems derived from the Higgs bundles over elliptic curves, where transitions operators are defined by an arbitrary finite order automorphisms of the underlying Lie algebras. In this way we obtain the spin generalization of the twisted D'Hoker- Phong and Bordner-Corrigan-Sasaki-Takasaki systems. As by product, we construct the corresponding twisted classical dynamical r-matrices and Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of the Lie algebras.
The equivalence between the N-particle Calogero-Moser systems and the integrable sl(N,$\mathbb{C}$)-tops is shown. New rational and trigonometric classical Lax operators for these systems are found. Relations with new solutions of the classical Yang-Baxter equations for sl(N,$\mathbb{C}$) are discussed. Explicit formulae for N=2,3,4 are presented.
We identify the Bethe algebra of the Gaudin model associated to gl(N) acting on a suitable representation with the center of the rational Cherednik algebra and with the algebra of regular functions on the Calogero-Moser space.
Quantum Calogero-Moser spin system is a superintegable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of corresponding simple Lie group. In this paper the underlying Lie group G is a compact connected, simply connected simple Lie group. It has a natural generalization known as quantum Calogero-Moser spin chain. In the first part of the paper we show that quantum Calogero-Moser spin chain is a quantum superintegrable systems. Then we show that the Euclidean multi-time propagator for this model can be written as a partition fun…
Calogero-Moser-Sutherland systems
本文综述一维 Calogero-Moser-Sutherland 型可积多体系统，涵盖经典与量子、非相对论与相对论情形，重点讨论可积性、显式解、作用-角变量与双谱对偶性及其历史与联系。
Various infinite-dimensional versions of Calogero-Moser operator are discussed in relation with the theory of symmetric functions and representation theory of basic classical Lie superlagebras. This is a version of invited talk given by the second author at XVI International Congress on Mathematical Physics in Prague, August 2009.
We define the Dunkl and Dunkl-Heckman operators in infinite number of variables and use them to construct the quantum integrals of the Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple proof of integrability of the deformed quantum CMS systems related to classical Lie superalgebras. We show how this naturally leads to a quantum version of the Moser matrix, which in the deformed case was not known before.
We develop a theory for the Hermite-Krichever Ansatz on the Heun equation. As a byproduct, we find formulae which reduce hyperelliptic integrals to elliptic ones.
We present a construction of a new integrable model as an infinite limit of Calogero models of N particles with spin. It is implemented in the multicomponent Fock space. Explicit formulas for Dunkl operators, the Yangian generators in the multicomponent Fock space are presented. The classical limit of the system is examined.
The integrability of the classical and quantum rational Calogero-Moser systems is verified explicitly via the Lax pair method for the case $n=3$. We provide an extensive survey of reflection groups and root systems. The Olshanetsky-Perelomov operators are constructed for a general root system via Dunkl operators, associated to root systems. The integrability of the quantum rational Calogero-Moser system is discussed via the Olshanetsky-Perelomov operators, which provide a set of commuting integrals of motion. The classical analogues of both the Dunkl and the Olshanetsky-Perelomov operators are…
On the geometry of isomonodromic deformations on the torus and the elliptic Calogero-Moser system
本文研究环面上简单极点联络的等单色变形，建立扩展对称性，并将椭圆 Calogero-Moser 系统纳入该几何框架，证明所引入的扩展辛二形式是闭的。
We obtain kernel functions associated with the quantum relativistic Toda systems, both for the periodic version and for the nonperiodic version with its dual. This involves taking limits of previously known results concerning kernel functions for the elliptic and hyperbolic relativistic Calogero-Moser systems. We show that the special kernel functions at issue admit a limit that yields generating functions of Bäcklund transformations for the classical relativistic Calogero-Moser and Toda systems. We also obtain the nonrelativistic counterparts of our results, which tie in with previous results…
We explicitly construct a supersymmetric $so(n)$ spin-Calogero model with an arbitrary even number $\cal N$ of supersymmetries. It features $\frac{1}{2}{\cal N}n(n{+}1)$ rather than ${\cal N}n$ fermionic coordinates and a very simple structure of the supercharges and the Hamiltonian. The latter, together with additional conserved currents, form an $osp({\cal N}|2)$ superalgebra. We provide a superspace description for the simplest case, namely ${\cal N}{=}2$ supersymmetry. The reduction to an $\cal N$-extended supersymmetric goldfish model is also discussed.
We construct a linear basis for the polynomial eigenfunctions of a family of deformed Calogero-Moser-Sutherland operators naturally associated with hypergeometric polynomials. In our construction the eigenfunctions are obtained as linear combinations of polynomials which generalise the (super) Schur polynomials. As a byproduct, we obtain explicit series representations for the super Jack polynomials.
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