Universal Field Equations with Covariant Solutions D.B. Fairlie

TL;DR

This paper introduces metric-independent sigma models that generalize membrane theories, featuring a hierarchical structure of equations of motion culminating in a universal, potentially integrable equation, independent of initial Lagrangians.

Contribution

The paper presents a novel class of metric-independent sigma models with a hierarchical structure leading to a universal equation of motion, extending to multiple fields.

Findings

Existence of a hierarchical structure of equations of motion.

Derivation of a universal, possibly integrable, equation.

Generalization to multiple fields.

Abstract

Metric independent $\sigma$ models are constructed. These are field theories which generalise the membrane idea to situations where the target space has fewer dimensions than the base manifold. Instead of reparametrisation invariance of the independent variables, one has invariance of solutions of the field equations under arbitrary functional redefinitions of the field quantities. Among the many interesting properties of these new models is the existence of a hierarchical structure which is illustrated by the following result. Given an arbitrary Lagrangian, dependent only upon first derivatives of the field, and homogeneous of weight one, an iterative procedure for calculating a sequence of equations of motion is discovered, which ends with a universal, possibly integrable equation, which is independent of the starting Lagrangian. A generalisation to more than one field is given.