Euler Hierarchies and Universal Equations

Abstract

Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for {\it classical} topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of self-dual four dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ``triangular duality'' between certain classes of arbitrary field theories, classical topological field theories and generalised string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalisations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some of the universal equations is given in the simplest cases.