Unfolded form of conformal equations in M dimensions and o(M+2)-modules
O. V. Shaynkman, I. Yu. Tipunin, M. A. Vasiliev
TL;DR
<p>We develop a universal unfolded framework to classify and construct conformally invariant linear differential equations in ${M}$-dimensional space-times by embedding them into covariant-constancy conditions for modules of the conformal algebra ${\mathfrak o}(M{+}2)$. The method reduces the problem to the cohomology ${H}^{p}({\mathfrak t}(M),{\mathfrak M})$ of a parabolic (with Abelian radical) subalgebra acting on an ${\mathfrak o}(M{+}2)$-module ${\mathfrak M}$, and uses generalized Verma modules to organize primitive and non-primitive systems via singular and subsingular vectors. When specialized to Minkowski space, the authors classify all conformally invariant linear equations, including Klein–Gordon, Dirac, conformal p-forms, and higher-spin generalizations, and provide explicit constructions for primitive and non-primitive cases, such as conformal Maxwell theory and its higher-spin extensions. The work connects jet-bundle/invariant differential-operator perspectives with generalized Verma module structures, offering a foundation for nonlinear deformations and interacting theories in a conformal/unfolded framework.</p>
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.