Unfolded Form of Conformal Equations in M Dimensions and 𝔬(m + 2)-Modules

Abstract

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. Contents

Asymptotically locally flat spacetimes and dynamical nonspherically-symmetric black holes in three dimensions

“…Their conformal invariance was originally shown by identifying those Poincaré irreps that lift to conformal ones [21,23]. In terms of EOM symmetries this follows from the results of [27], while a manifestly local and conformal formulation of these bosonic spinning singletons in terms of curvatures was constructed in [14]. This completes our discussion of possible conformal invariance of bosonic gauge fields on Minkowski spacetime.…”

“…on (quasi-)invariant differential operators [15,[24][25][26]. In particular, a technique to classify linear partial differential equations for which Poincaré lifts to conformal symmetry was developed in [27].…”

A note on higher-derivative actions for free higher-spin fields

“…In d ≥ 4 , a free action possessing the above symmetry involves the square of the higher-spin Weyl tensor (the traceless part of the higher-spin curvature), and thus contains 2s derivatives for a given spin s . For d = 4 , it coincides with the conformal higher-spin theory 3 of [22][23][24] (see also [25][26][27][28][29]). Although one can directly consider 2s-derivative theories and study their ghost exorcising mechanisms, one can still wonder whether interesting theories exist whose number of derivatives lies between 2 and 2s .…”

On conformal higher spin wave operators

“…In particular, the actions of Weyl gravity and conformal supergravity, together with their corresponding wave equations, have been studied in great detail [1][2][3][4][5][6][7][8][9][10][11] as natural extensions of ordinary gravity and supergravity theories. Interest has been also devoted to the corresponding higher spin generalizations [12][13][14][15][16][17][18], not just because of the intriguing role of conformal symmetry. Flat space higher spin (HS) fields are namely naturally endowed with higher derivative linearized curvatures [19] that play a key role in conformal gauge theories.…”