Finite Euler Hierarchies And Integrable Universal Equations

TL;DR

This paper reviews Euler hierarchies in field theory, introduces finite hierarchies leading to universal equations that may define new integrable systems, and discusses potential implications for quantum gravity.

Contribution

It constructs finite Euler hierarchies for various theories, resulting in universal equations that could represent novel integrable systems.

Findings

Finite Euler hierarchies terminate with universal equations.

Universal equations potentially define new integrable systems.

Speculations on relevance to quantum gravity.

Abstract

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, {\it classical\ts} 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, {\it finite} Euler hierarchies are constructed for all three classes of theories. These hierarchies terminate with {\it universal\ts} 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.