In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair...

We discretise the Lax pair for reduced Nahm systems and prove its equivalence with Kahan–Hirota–Kimura discretisation procedure. show that these pairs guarantee integrability of discrete providing an invariant. Also, we cannot solve general problem characterisation discretisations.

Recently Lunin and Maldacena used an SL(3, R) transformation of theAdS5×S background to generate a supergravity solution dual to a so-called β-deformation of N = 4 super Yang-Mills theory. We use a T-duality-shift-T-duality (TsT) transformation to obtain the β-deformed background for real β ≡ γ, and show that solutions of string theory equations of motion in this background are in one-to-one co...

We construct a Lax pair for the E (1) 6 q-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices – the q-linear lattice – through a natural generalisation of the big q-Jacobi weight. As ...

In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy 2m (i.e. when y ∼ x for arbitrary values of the integer m) are the same as the determinantal formulae defined by conformal (2m, 1) models. Our approach follows the one developed by Bergère and Eynard in [2] and uses a Lax pair representation of the conformal ...

A new formulation of Calogero-Moser models based on root systems and their Weyl group is presented. The general construction of the Lax-pairs and the proof of the integrability applicable to all models based on the simply-laced algebras (ADE) are given for two types which we will call ‘root’ and ‘minimal’. The root type Lax pair is new; the matrices used in its construction bear a resemblance t...