Conformally invariant powers of the Dirac operator in Clifford analysis. (English) Zbl 1201.30065

Eelbode, David; Souček, Vladimír

Math. Methods Appl. Sci. 33, No. 13, 1558-1570 (2010).

Conformal invariance of operators is important, because it is a characteristic of massless field case in physical applications. This paper deals with conformally invariant higher-order operators acting on spinor-valued functions, such that their symbols are given by powers of the Dirac operator. A general classification result proves that these operators are unique, up to a multiplicative constant. A general construction for such invariant operators on manifolds with a given conformal spin structure was described in another paper, generalizing the case of powers of the Laplace operator. Although there is no hope to obtain explicit formulae for higher powers of the Laplace or Dirac operator on a general manifold, it is possible to write down an explicit formula on Einstein manifolds in case of the Laplace operator.
In this paper the question how to explicitly write down the conformally invariant version of powers of the Dirac operator on the sphere is considered. The explicit form of such operators is computed and it is shown that they coincide with operators studied previously by H. Liu and J. Ryan. The methods used in this paper are coming from representation theory combined with traditional Clifford analysis techniques.

Reviewer: Nele De Schepper (Gent)

MSC:

30G35	Functions of hypercomplex variables and generalized variables
35J30	Higher-order elliptic equations

Keywords:

powers of the Dirac operator; conformal invariance; representation theory; Clifford analysis

References:

[1]	Gover R. Laplacian Operators and Q-curvature on Conformally Einstein Manifolds. ArXiv: math/0506037, 2006. · Zbl 1125.53032
[2]	Van Lancker P. Clifford analysis on the unit sphere. Ph.D. Thesis, University of Ghent, 1996. · Zbl 0896.15015
[3]	Liu, Clifford analysis techniques for spherical PDE, Journal of Fourier Analysis and Applications 8 (6) pp 535– (2002) · Zbl 1047.53023
[4]	Ryan, Conformally covariant operators in Clifford analysis, Journal for Analysis and its Applications 14 pp 677– (1995) · Zbl 0841.30039
[5]	Slovak J. Natural operators on conformal manifolds. Masaryk University Dissertation, Brno, 1993.
[6]	Čap, Bernstein-Gelfand-Gelfand sequences, Annals of Mathematics 154 (1) pp 7– (2001)
[7]	Calderbank, Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, Journal fr die Reine und Angewandte Mathematik 537 pp 67– (2001) · Zbl 0985.58002
[8]	Graham, Conformally invariant powers of the Laplacian, I: existence, Journal of London Mathematical Society 46 (2) pp 557– (1992) · Zbl 0726.53010
[9]	Calderbank, Ricci-corrected Weyl, Differential Geometry and its Applications 23 pp 149– (2005)
[10]	Gover, Conformally invariant powers of the Laplacian-a complete non-existence theorem, Journal of American Mathematical Society 17 pp 389– (2004) · Zbl 1066.53037
[11]	Branson T, Hong D.Translation to Bundle Operators. ArXiv: math.DG/0606552, 2006.
[12]	Holland J, Sparling G.Conformally Invariant Powers of the Ambient Dirac Operator. ArXiv math.DG/0112033, preprint.
[13]	Eastwood M, Ryan J. Monogenic Functions in Conformal Geometry, preprint. · Zbl 1133.53032
[14]	Graham, Conformally invariant powers of the Laplacian, I: existence, Journal of London Mathematical Society 46 pp 557– (1992) · Zbl 0726.53010
[15]	Porteous, Topological Geometry (1981)
[16]	Bureš, Dirac operators on hypersurfaces, Commentationes Mathematicae Universitatis Carolinae 34 pp 313– (1993)
[17]	Fulton, Representation Theory: A First Course (1991) · Zbl 0744.22001
[18]	Humphreys, Introduction to Lie Algebra and Representation Theory (1972) · Zbl 0254.17004
[19]	Čap, Parabolic Geometries I. Background Sand General Theory pp 628– (2009)
[20]	Fefferman C, Graham CR. Conformal invariants. Elie Cartan et les Mathématique d’aujourd’hui, Asterisque, Numéro hors Série, 1985; 96-116.