The Multi-field Complex Bateman Equation

TL;DR

This paper introduces a multi-field complex extension of the Bateman equation, exploring its invariance properties, solution forms, and associated Lagrangians, with implications for higher-dimensional string and brane theories.

Contribution

It presents the first multi-field complex generalization of the Bateman equation, revealing its invariance group, implicit solutions, and new Lagrangian formulations.

Findings

Large invariance group identified

Implicit general solutions derived

New inequivalent Lagrangians discovered

Abstract

The multi-field generalisation of the Bateman equation arises from considerations of the continuation of String and Brane equations to the case where the base space is of higher dimension than the target space. The complex extension of this equation possesses a remarkably large invariance group, and admits a very simple implicit form for its general solution, in addition to the special case of holomorphic and anti-holomorphic explicit solutions. A class of inequivalent Lagrangians for this equation is discovered.