Finite Euler hierarchies and integrable universal equations

Recent work on Euler hierarchies of field theory Lagrangians iteratively constructed from their successive equations of motion is briefly reviewed. On the one hand, a certain triality structure is described, relating arbitrary field theories,classical topological field theories — whose classical solutions span topological classes of manifolds — and reparametrisation invariant theories — generalising ordinary string and membrane theories. On the other hand,finite Euler hierarchies are constructed for all three classes of theories. These hierarchies terminate withuniversal equations of motion, probably defining new integrable systems as they admit an infinity of Lagrangians. Speculations as to the possible relevance of these theories to quantum gravity are also suggested.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.The author would like to thank A. Morozov and especially D. B. Fairlie for a very enjoyable and stimulating collaboration, and the organisers of the Colloquium for their efficient organisation of a most pleasant and informative meeting. This work is supported through a Senior Research Assitant position funded by the S.E.R.C.