Integrable systems and complex geometry

Matrix Models , Complex Geometry and Integrable Systems . II ∗

We consider certain examples of applications of the general methods, based on geometry and in-tegrability of matrix models, described in [1]. In particular, the nonlinear differential equations, satisfied by quasiclassical tau-functions are investigated. We also discuss a similar quasiclassical geometric picture, arising in the context of multidimensional supersymmetric gauge theories and the A...

Matrix Models , Complex Geometry and Integrable Systems . I ∗

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. ...

Discrete Integrable Systems and Geometry

Integrable Systems in Symplectic Geometry

Quaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space n =Sp(n+ 1)/Sp(1)×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry on n modelled on (spn+1, sp1 × spn). The integrability structure is shown to be geometrically encoded by a Poisson– Nijenhuis stru...

Algebraic geometry and stability for integrable systems

In 1970s, a methodwas developed for integration of nonlinear equations bymeans of algebraic geometry. Starting froma Lax representationwith spectral parameter, the algebro-geometricmethod allows to solve the system explicitly in terms of theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular ...