We construct local geometric model in terms of Fand M-theory compactification on Calabi-Yau fourfolds which lead to N = 1 Yang-Mills theory in d = 4 and its reduction on a circle to d = 3. We compute the superpotential in d = 3, as a function of radius, which is generated by the Euclidean 5-brane instantons. The superpotential turns out to be the same as the potential for affine Toda theories. ...

The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the complex curve and generating differential. The first derivatives of prepotential or quasiclassical tau-function over the extra times, extending the Toda chain, a...

The Boyer-Finley equation, or SU (∞)-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionless limit of the 2d-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system i...

In 1967, Japanese physicist Morikazu Toda published the seminal papers [78] and [79], exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda’s system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics that sets it apart from the o...

The Boyer-Finley equation, or SU(∞)-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionless limit of the 2d-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is...

Solutions in multidimensional gravity with m p-branes related to Toda-like systems (of general type) are obtained. These solutions are defined on a product of n + 1 Ricci-flat spaces M 0 × M 1 ×. .. × M n and are governed by one harmonic function on M 0. The solutions are defined up to the solutions of Laplace and Toda-type equations and correspond to null-geodesics of the (sigma-model) target-...

Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by t he representations of the reflection groups, which are not the ...