Unfolded form of conformal equations in M dimensions and o(M+2)-modules

TL;DR

This paper introduces a systematic method to formulate linear differential equations invariant under semi-simple Lie algebra symmetries, with applications to classifying conformally invariant equations in Minkowski space, inspired by higher spin gauge theory.

Contribution

It presents a new constructive approach linking invariant differential equations to modules of Lie algebras, facilitating classification of conformal equations in arbitrary dimensions.

Findings

Classified all linear conformally invariant differential equations in Minkowski space.

Provided examples illustrating the application of the method to conformal equations.

Established a framework for potential extension to nonlinear invariant equations.

Abstract

A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential equations are shown to be associated with f-modules that are integrable with respect to some parabolic subalgebra of f. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra o(M,2) to classify all linear conformally invariant differential equations in Minkowski space. Numerous examples of conformal equations are discussed from this perspective.