Differential Invariants of Self-Dual conformal structures

TL;DR

This paper analyzes the differential invariants of self-dual conformal structures, computing the quotient of the self-duality equation under diffeomorphisms and characterizing the invariants that distinguish generic structures.

Contribution

It provides explicit calculations of the quotient space, Hilbert polynomial, and Poincaré function, advancing the understanding of invariants in self-dual conformal geometry.

Findings

Computed the quotient of the self-duality equation under diffeomorphisms

Determined the Hilbert polynomial and Poincaré function for invariants

Described the field of rational differential invariants for generic orbits

Abstract

We compute the quotient of the self-duality equation for conformal metrics by the action of the diffeomorphism group. We also determine Hilbert polynomial, counting the number of independent scalar differential invariants depending on the jet-order, and the corresponding Poincar\'e function. We describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action, resolving the local recognition problem for self-dual conformal structures.