Elliptic Top Research Articles

Relativistic Interacting Integrable Elliptic Tops

We propose relativistic generalization of integrable systems describing $M$ interacting elliptic ${\rm gl}(N)$ tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic ${\rm GL}(M)$ Ruijsenaars-Schneider model for $N=1$ case, while in the $M=1$ case they turn into relativistic integrable ${\rm GL}(N)$ elliptic tops. The Lax pairs with spectral parameter on elliptic curve are constructed.

Generalized model of interacting integrable tops

We introduce a family of classical integrable systems describing dynamics of M interacting glN integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GLNR-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In N = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for glNM -valued Lax pair with spectral parameter and its classical dynamical r-matrix are obtained. Possible applications are briefly discussed.

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.

Noncommutative extensions of elliptic integrable Euler–Arnold tops and Painlevé VI equation

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang–Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of the reduced elliptic top and verify that the latter constraints are fulfilled in this case. The construction of matrix extensions is naturally generalized to the monodromy preserving equation. In this way we get matrix extensions of the Painlevé VI equation and its multidimensional analogues written in the form of non-autonomous elliptic tops. Finally, it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra. At the end of the paper we also describe special elliptic Gaudin models which can be considered as matrix extensions of the ( reduced) elliptic top.

New integrable systems as a limit of the elliptic SL(N, ℂ) top

We consider the scaling limit of an elliptic top. This limit is a combination of a scaling of the elliptic top variables, an infinite shift of the spectral parameter, and the trigonometric limit. We give general necessary constraints on the scaling of the variables and examples of such a degeneracy. A certain subclass of limit systems is integrable in the Liouville sense, which can also be shown directly.

Reduction of the elliptic top

We propose a relation between the elliptic top and Toda systems and obtain a new class of integrable systems in a specific limit of the elliptic top. The relation is based on the Inozemtsev limit and a symplectic map from the elliptic Calogero–Moser system to the elliptic top. In the case when N = 2 we use an explicit form of a symplectic map from the phase space of the elliptic Calogero–Moser system to the phase space of the elliptic top and show that the limiting tops are equivalent to the Toda chains. In the case when N > 2 we generalize the above procedure using only the limiting behavior of Lax matrices. In a specific limit we also obtain a more general class of systems and prove the integrability in the Liouville sense of a certain subclass of these systems. This class is described by a classical r-matrix obtained from an elliptic r-matrix.

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL (N, C) and explain the Symplectic Hecke correspondence for these systems.

Lectures on Classical Integrable Systems and Gauge Field Theories

In these lectures we consider Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimension. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N, C) and explain the Symplectic Hecke correspondence for these systems.

Integrable Model of Interacting Elliptic Tops

We suggest a method for constructing a system of interacting elliptic tops. It is integrable and symplectomorphic to the Calogero-Moser model by construction.

Universality of Calogero-Moser Model

In this review we explain interrelations between the Elliptic Calogero-Moser model, the integrable Elliptic Euler-Arnold top, monodromy preserving equations and the Knizhnik-Zamolodchikov-Bernard equation on a torus.