Elliptic Integrable Systems Research Articles

Canonical distributions on Riemannian homogeneousk-symmetric spaces

It is known that distributions generated by almost product structures are applicable, in particular, to some problems in the theory of Monge–Ampère equations. In this paper, we characterize canonical distributions defined by canonical almost product structures on Riemannian homogeneous k-symmetric spaces in the sense of types AF (anti-foliation), F (foliation), TGF (totally geodesic foliation). Algebraic criteria for all these types on k-symmetric spaces of orders k=4,5,6 were obtained. Note that canonical distributions on homogeneous k-symmetric spaces are closely related to special canonical almost complex structures and f-structures, which were recently applied by I. Khemar to studying elliptic integrable systems.

Modifications of bundles, elliptic integrable systems, and related problems

We describe a construction of elliptic integrable systems based on bundles with nontrivial characteristic classes, especially attending to the bundle-modification procedure, which relates models corresponding to different characteristic classes. We discuss applications and related problems such as the Knizhnik-Zamolodchikov-Bernard equations, classical and quantum R-matrices, monopoles, spectral duality, Painleve equations, and the classical-quantum correspondence. For an SL(N,ℂ)-bundle on an elliptic curve with nontrivial characteristic classes, we obtain equations of isomonodromy deformations.

Three-particle integrable systems with elliptic dependence on momenta and theta function identities

We claim that some non-trivial theta-function identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which were introduced in [1,2] and are made from the theta-functions on Jacobians of the Seiberg–Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the Letter. The connection with the Macdonald identities is established. The genus-2 theta-function identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its period matrix. The Lax representations for the two-particle systems are also obtained.

Elliptic Integrable Systems: a Comprehensive Geometric Interpretation

We give a geometric interpretation of all the $m$-th elliptic integrable systems associated to a $k'$-symmetric space $N=G/G_0$ (in the sense of C.L. Terng). It turns out that we have to introduce the integer $m_{k'}$ defined by m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three cases : the primitive case ($m < m_{k'}$), the determined case ($m_{k'}\leq m \leq k'-1$) and the underdetermined case ($m \geq k'$). We prove that we have an interpretation in terms of a sigma model with a Wess-Zumino term. Moreover we prove that we have a geometric interpretation in terms of twistors. See the abstract in the paper for more precisions.

Geometric interpretation of second elliptic integrable systems

In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space G / G 0 can be embedded into the twistor space of the corresponding symmetric space G / H . Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift J taking values in G / G 0 ↪ Σ ( G / H ) . We begin the paper with an example: G / H = R 4 . We also study the structure of 4-symmetric bundles over Riemannian symmetric spaces.

New results from glueball superpotentials and matrix models: the Leigh-Strassler deformation

Using the result of a matrix model computation of the exact glueball superpotential, we investigate the relevant mass perturbations of the Leigh-Strassler marginal ``q'' deformation of N=4 supersymmetric gauge theory. We recall a conjecture for the elliptic superpotential that describes the theory compactified on a circle and identify this superpotential as one of the Hamiltonians of the elliptic Ruijsenaars-Schneider integrable system. In the limit that the Leigh-Strassler deformation is turned off, the integrable system reduces to the elliptic Calogero-Moser system which describes the N=1^* theory. Based on these results, we identify the Coulomb branch of the partially mass-deformed Leigh-Strassler theory as the spectral curve of the Ruijsenaars-Schneider system. We also show how the Leigh-Strassler deformation may be obtained by suitably modifying Witten's M theory brane construction of N=2 theories.

Geometric construction of elliptic integrable systems and N=1* superpotentials

We show how the elliptic Calogero-Moser integrable systems arise from a symplectic quotient construction, generalising the construction for A_{N-1} by Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted) affine Kac-Moody algebras in elliptic Calogero-Moser systems and allows for a natural geometric construction of Lax operators for these systems. We elaborate on the connection of the associated Hamiltonians to superpotentials for N=1* deformations of N=4 supersymmetric gauge theory, and argue how non-perturbative physics generates the elliptic superpotentials. We also discuss the relevance of these systems and the associated quotient construction to open problems in string theory. In an appendix, we use the theory of orbit algebras to show the systematics behind the folding procedures for these integrable models.

Elliptic Sklyanin integrable systems for arbitrary reductive groups

We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin. The Sklyanin integrable systems were originally constructed on symplectic leaves, of a quadratic Poisson structure, on a loop group of type A. The phase space, of our integrable systems, is a group-like analogue of the Hitchin system over an elliptic curve E. We consider the moduli space of pairs (P,f), where P is a principal G-bundle on E, and f is a meromorphic section of the adjoint group bundle. We show: 1) The moduli space admits an algebraic Poisson structure. It is related to the poisson structures on loop groupoids, constructed by Etingof and Varchenko, using Felder's elliptic solutions of the Classical Dynamical Yang-Baxter Equation. 2) The symplectic leaves are finite dimensional. A symplectic leaf is determined by labeling, finitely many points of E, each by a dominant co-character of the maximal torus of G. 3) Each leaf is an algebraically completely integrable system.

Pohlmeyer's transformation and the (2+0) integrable equations of statistical physics

Using the Pohlmeyer transformation in Euclidean space, we obtain an equation which provides the possibility of constructing a wide class of two-dimensional elliptic integrable systems. The problem of the gauge equivalence of these systems with the nonlinear sigma-model and the Getmanov and Bitsadze models is investigated. Exact solutions of some equations are constructed. Bibliography: 34 titles.