Universal Field Equations with Reparametrisation Invariance

TL;DR

This paper introduces universal, reparametrisation invariant field equations for d-branes in (d+1)-dimensional space, featuring a hierarchical structure and invariance properties, with a specific solution for the 2-dimensional case.

Contribution

It constructs a new class of universal, reparametrisation invariant field equations for d-branes, with a hierarchical derivation and invariance properties, including a solved case for d=2.

Findings

Derived new reparametrisation invariant d-brane equations

Established a hierarchical structure of equations from covariant Lagrangians

Solved the specific case of a reparametrisation invariant string in 3 dimensions

Abstract

New reparametrisation invariant field equations are constructed which describe $d$-brane models in a space of $d+1$ dimensions. These equations, like the recently discovered scalar field equations in $d+1$ dimensions, are universal, in the sense that they can be derived from an infinity of inequivalent Lagrangians, but are nonetheless Lorentz (Euclidean) invariant. Moreover, they admit a hierarchical structure, in which they can be derived by a sequence of iterations from an arbitrary reparametrisation covariant Lagrangian, homogeneous of weight one. None of the equations of motion which appear in the hierarchy of iterations have derivatives of the fields higher than the second. The new sequence of Universal equations is related to the previous one by an inverse function transformation. The particular case of $d=2$, giving a new reparametrisation invariant string equation in 3 dimensions is solved.