Calogero-Moser System Research Articles

Elliptic solutions to matrix KP hierarchy and spin generalization of elliptic Calogero–Moser model

We consider solutions of the matrix Kadomtsev–Petviashvili (KP) hierarchy that are elliptic functions of the first hierarchical time t1 = x. It is known that poles xi and matrix residues at the poles ρiαβ=aiαbiβ of such solutions as functions of the time t2 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 kth hierarchical time of the matrix KP hierarchy is Hamiltonian with the Hamiltonian Hk obtained via an expansion of the spectral curve near the marked points. The Hamiltonians are identified with the Hamiltonians of the elliptic spin Calogero–Moser system with coordinates xi and spin degrees of freedom aiα,biβ.

Inozemtsev system as Seiberg-Witten integrable system

In this work we establish that the Inozemtsev system is the Seiberg-Witten integrable system encoding the Coulomb branch physics of 4d mathcal{N} = 2 USp(2N) gauge theory with four fundamental and (for N ≥ 2) one antisymmetric tensor hypermultiplets. We describe the transformation from the spectral curves and canonical one-forms of the Inozemtsev system in the N = 1 and N = 2 cases to the Seiberg-Witten curves and differentials explicitly, along with the explicit matching of the modulus of the elliptic curve of spectral parameters to the gauge coupling of the field theory, and of the couplings of the Inozemtsev system to the field theory mass parameters. This result is a particular instance of a more general correspondence between crystallographic elliptic Calogero-Moser systems with Seiberg-Witten integrable systems, which will be explored in future work.

Differential operators and the symmetric groups

In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x 1 , …, x n ] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x 1 , …, x n ] Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.

Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators

We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and 2-parameter rational potential is the best known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.

Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary

We extend duality between the quantum integrable Gaudin models with boundary and the classical Calogero–Moser systems associated with root systems of classical Lie algebras B N , C N , D N to the case of supersymmetric gl(m|n) Gaudin models with m + n = 2. Namely, we show that the spectra of quantum Hamiltonians for all such magnets being identified with the classical particles velocities provide the zero level of the classical action variables.

Quantum integrable systems from supergroup gauge theories

In this note, we establish several interesting connections between the super- group gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Calogero-Moser system by considering certain orbifolded instanton partition functions of their corresponding supergroup gauge theories. We also derive an exotic generalization of \U0001d530\U0001d529(2) XXX spin chain arising from the instanton partition function of SQCD with super- gauge group, and study its Bethe ansatz equation.

Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type. III. Factorized Asymptotics

Abstract In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic Calogero–Moser type. We focused on the 1st steps of the scheme in Part I and on the cases $N=2$ and $N=3$ in Part II. In this paper, we determine the dominant asymptotics of a similarity-transformed function $\textrm{E}_N(b;x,y)$ for $y_j-y_{j+1}\to \infty $, $j=1,\ldots , N-1$ and thereby confirm the long-standing conjecture that the particles in the hyperbolic relativistic Calogero–Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers $N>3$.

The role of commuting operators in quantum superintegrable systems

We discuss the role of commuting operators for quantum superintegrable systems, showing how they are used to build eigenfunctions. These ideas are illustrated in the context of resonant harmonic oscillators, the Krall–Sheffer operators, with polynomial eigenfunctions, and the Calogero–Moser system with additional harmonic potential. The construction is purely algebraic, avoiding the use of separation of variables and differential equation theory.

Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen-Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen-Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero-Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, computing a new unexpected explicit example of a deformed Calogero-Moser system.

Matrix Kadomtsev—Petviashvili Hierarchy and Spin Generalization of Trigonometric Calogero—Moser Hierarchy

We consider solutions of the matrix Kadomtsev-Petviashvili (KP) hierarchy that are trigonometric functions of the first hierarchical time t1 = x and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system at the level of hierarchies. Namely, the evolution of poles xi and matrix residues at the poles a b of the solutions with respect to the kth 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 system with coordinates xi and with spin degrees of freedom α and b . By considering the evolution of poles according to the discrete time matrix KP hierarchy, we also introduce the integrable discrete time version of the trigonometric spin Calogero-Moser system.

Quantum elliptic Calogero-Moser systems from gauge origami

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 obtain the corresponding characteristic polynomial for the doubled version, named the elliptic double Calogero-Moser (edCM) system.

A Universal Construction of Calogero-Moser System with Two Types Spins Interaction

We present universal construction for the Calogero-Moser system with two types spins interaction of trigonometric potential based on the root system of semi-simple Lie algebra. In this formalism, we successfully build up the correct Lax pair as well as the R-matrix for this generalized Calogero-Moser models. Moreover using the property of root system, we make a concise explanation that in the quantized model, the R-matrix takes the same form as the classical one, which is the main new result of this paper.

Real Forms of Elliptic Integrable Systems

We describe the real forms of classical elliptic integrable systems such as the elliptic Calogero-Moser system and the elliptic Euler-Arnold top in the framework of a general scheme for constructing real reductions for a Hitchin system.

Time discretization of the spin Calogero-Moser model and the semi-discrete matrix KP hierarchy

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 Kadomtsev-Petviashvili hierarchy with discrete time. The dynamics of poles is derived using the auxiliary linear problem for the discrete flow.

Nilpotent integrability, reduction of dynamical systems and a third-order Calogero–Moser system

We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: It can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric–algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics) and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent-integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent-integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent-integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a non-trivial set of third-order Calogero–Moser-like nilpotent-integrable equations is obtained.

Geometry of Calogero-Moser systems

The elliptic Calogero-Moser integrable system for an arbitrary root system has a realization as a moduli space of Higgs bundles over an Abelian variety associated with the elliptic curve and with the root system. This paper examines the Fourier-Mukai transform of this, giving an interpretation of the system on a network of elliptic curves. The rational and trigonometric versions of the systems are briefly discussed, and it is shown how they enter as degenerations in this geometric context.

Reflection Functor in the Representation Theory of Preprojective Algebras for Quivers and Integrable Systems

A brief overview of the representation theory of quivers and the associated (deformed) preprojective algebras, as well as of the theories of moduli spaces of these algebras, quiver varieties and a reflection functor, is given. It is proven that a bijection between moduli spaces (in particular, between quiver varieties), which is induced by a reflection function, is the isomorphism of symplectic affine varieties. The Hamiltonian systems on quiver varieties are defined, and the application of a reflection functor to them is described. The review of [1], concerning the case of a cyclic quiver is given, and a role of the reflection functor in this case is clarified. The “spin” integrable generalizations of Calogero–Moser systems and their application to the KP hierarchy generalizations are described.

Spin generalization of the Calogero–Moser hierarchy and the matrix KP hierarchy

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.

Spin Calogero–Moser systems for the cyclic quiver

The method of construction of classical integrable systems on quiver varieties is considered in the case of cyclic quiver. For two types of framing we obtained integrable spin generalisations of the Calogero–Moser systems associated with the complex reflection groups Sn ⋉ (ℤ/mℤ)n. This method gives Lax matrices of the systems in explicit form. In some particular cases the first Hamiltonians of the system are written explicitly. The paper is based on the work of the author with Oleg Chalykh (the University of Leeds).

KP hierarchy for the cyclic quiver

We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra Hk(Zm). This hierarchy depends on m parameters (one of which can be eliminated), with the usual KP hierarchy corresponding to the m = 1 case. Generalising the result of Wilson [Invent. Math. 133(1), 1–41 (1998)], we show that our hierarchy admits solutions parameterised by suitable quiver varieties. The pole dynamics for these solutions is shown to be governed by the classical Calogero–Moser system for the wreath-product Zm≀Sn and its new spin version. These results are further extended to the case of the multi-component hierarchy.