A geometry for multidimensional integrable systems

A deformed differential calculus is developed based on an associative ★-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one obtains a geometric description of the operators. A dual theory is also possible, based on deformation of differential forms. This calculus is applied to a number of multidimensional integrable systems such as the KP hierarchy, thus obtaining a geometrical description of these systems. The limit in which the deformation disappears correspond to taking the dispersionless limit in these hierarchies