Di erential invariants and curved Bernstein-Gelfand-Gelfand sequences

By D. Calderbank, T. Diemer. 2000

165 citations recorded by the source corpus.

We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential cup product on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A-infinity algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

Subfields: Algebra and Number Theory, Astronomy and Astrophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b1c9c48e14b6091124fa8ad420e119c1c0be9dc5

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Citations in the corpus

PROLONGATIONS OF GEOMETRIC OVERDETERMINED SYSTEMS (2004, methodology): “A more general exposition in a similar context may be found in [3].”
Conformal submanifold geometry I-III (2010, methodology): “This fact is tedious to check directly, but follows easily from [25].”
Ricci-corrected derivatives and invariant differential operators (2003, unknown): “Although in this paper we have applied Ricci-corrected derivatives to the construction of invariant linear differential operators, the same ideas can be expected to yield explicit formulae for multilinear differential operators, such as the operators of [9].”
Correspondence spaces and twistor spaces for parabolic geometries (2001, background): “di¤erential operators intrinsic to a parabolic geometry (see [11] and [5]).”
Higher symmetries of the laplacian (2002, unknown): “This extends (5) to the curved setting.”
CR-Tractors and the Fefferman Space (2006, background): “All these facts are an extremely special case of the machinery of BGG sequences, whose general version has been developed in [12] and [6].”
Infinitesimal automorphisms and deformations of parabolic geometries (2005, methodology): “The following result was first proved in [5].”
Notes on Projective Differential Geometry (2008, methodology): “In the fiat case, this is Zuckerman's translation functor but the method extends to the curved setting [18] and eventually to quite sophisticated machinery [ 7 , 8, 12] the upshot of which is that all the G-invariant operators (as implicitly listed above) admit curved analogues.”
Two constructions with parabolic geometries (2005, unknown): “It follows from the general machinery of BGG–sequences, see [14, 6].”