Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models

TL;DR

This paper reviews the construction and applications of homological evolutionary vector fields on infinite jet spaces across various models like Korteweg-de Vries, Liouville, and Maxwell, highlighting their role in geometric and variational formalisms.

Contribution

It introduces a unified framework for homological evolutionary vector fields and explores their applications in integrable systems, hyperbolic systems, and gauge theories.

Findings

Application of homological vector fields to KdV and Liouville equations

Insights into the geometry of hyperbolic systems and gauge theories

Formulation of open problems in the field

Abstract

We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.