The symmetry structure of the heavenly equation

TL;DR

This paper explores the symmetry structure of the heavenly equation, revealing two reduced equations with distinct physical implications, including gravitational instantons and novel meron-like configurations, and links to the Schr{"o}der equation.

Contribution

It uncovers the symmetry-generated excitations of the heavenly equation, introduces a new equation with meron-like solutions, and connects it to the Schr{"o}der equation in bootstrap and renormalization theories.

Findings

Identification of symmetry operators generating excitations

Derivation of a Liouville-type equation for gravitational instantons

Introduction of a new equation with fractional topological charge

Abstract

We show that excitations of physical interest of the heavenly equation are generated by symmetry operators which yields two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi-Hanson and Gibbons-Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the socalled Schr{\"o}der equation, which plays a crucial role in the bootstrap model and in the renormalization theory.