On conformal powers of the Dirac operator on spin manifolds

Fischmann, Matthias

Archivum mathematicum, Tome 50 (2014) no. 4, pp. 237-253

Abstract (VO)

The well known conformal covariance of the Dirac operator acting on spinor fields does not extend to its powers in general. For odd powers of the Dirac operator we derive an algorithmic construction in terms of associated tractor bundles computing correction terms in order to achieve conformal covariance. These operators turn out to be formally (anti-) self-adjoint. Working out this algorithm we recover explicit formula for the conformal third and present a conformal fifth power of the Dirac operator. Finally, we will present polynomial structures for the first examples of conformal powers in terms of first order differential operators acting on the spinor bundle.

Détail

DOI : 10.5817/AM2014-4-237

Classification : 53A30, 53C27
Keywords: conformal and spin geometry; conformal powers of the Dirac operator; conformal covariance; tractor bundle; tractor D-operator

Bibliographie

[1] Bailey, T.N., Eastwood, M.G., Gover, A.R.: Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24 (4) (1994), 1191–1217. | DOI | MR | Zbl

[2] Branson, T.: Conformally covariant equations on differential forms. Comm. Partial Differential Equation 7 (4) (1982), 393–431. | DOI | MR | Zbl

[3] Branson, T.: Conformal structure and spin geometry. Dirac Operators: Yesterday and Today, International Press, 2005. | MR | Zbl

[4] Čap, A., Slovák, J.: Parabolic Geometries: Background and General Theory. vol. 1, American Mathematical Society, 2009. | MR

[5] Eastwood, M.G., Rice, J.W.: Conformally invariant differential operators on Minkowski space and their curved analogues. Comm. Math. Phys. 109 (2) (1987), 207–228. | MR | Zbl

[6] Eelbode, D., Souček, V.: Conformally invariant powers of the Dirac operator in Clifford analysis. Math. Methods Appl. Sci. 33 (13) (2010), 1558–1570. | MR | Zbl

[7] Fefferman, C., Graham, C.R.: Conformal invariants. The mathematical heritage of Élie Cartan (Lyon, 1984). yon, 1984), Astérisque, Numéro Hors Série (1985), 95–116. | MR

[8] Fefferman, C., Graham, C.R.: The ambient metric. Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. | MR | Zbl

[9] Fegan, H.D.: Conformally invariant first order differential operators. Quart. J. Math. Oxford (2) 27 (107) (1976), 371–378. | DOI | MR | Zbl

[10] Fischmann, M.: Conformally covariant differential operators acting on spinor bundles and related conformal covariants. Ph.D. thesis, Humboldt Universität zu Berlin, 2013, http://edoc.hu-berlin.de/dissertationen/fischmann-matthias-2013-03-04/PDF/fischmann.pdf

[11] Fischmann, M., Krattenthaler, C., Somberg, P.: On conformal powers of the Dirac operator on Einstein manifolds. ArXiv e-prints (2014), | arXiv | MR

[12] Gover, A.R.: Laplacian operators and $Q$-curvature on conformally Einstein manifolds. Math. Anal. 336 (2) (2006), 311–334. | DOI | MR | Zbl

[13] Gover, A.R., Hirachi, K.: Conformally invariant powers of the Laplacian - A complete non-existence theorem. J. Amer. Math. Soc. 14 (2) (2004), 389–405. | DOI | MR

[14] Gover, A.R., Peterson, L.J.: Conformally invariant powers of the Laplacian, $Q$-curvature, and tractor calculus. Comm. Math. Phys. 235 (2) (2003), 339–378. | DOI | MR | Zbl

[15] Graham, C.R., Jenne, R.W., Mason, L., Sparling, G.: Conformally invariant powers of the Laplacian, I: Existence. J. London Math. Soc. (2) 2 (3) (1992), 557–565. | MR | Zbl

[16] Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152 (2003), 89–118. | DOI | MR | Zbl

[17] Guillarmou, C., Moroianu, S., Park, J.: Bergman and Calderón projectors for Dirac operators. J. Geom. Anal. (2012), 1–39, | DOI

[18] Hitchin, N.J.: Harmonic spinors. Adv. Math. 14 (1) (1974), 1–55. | DOI | MR | Zbl

[19] Holland, J., Sparling, G.: Conformally invariant powers of the ambient Dirac operator. ArXiv e-prints (2001), | arXiv

[20] Juhl, A.: On conformally covariant powers of the Laplacian. ArXiv e-prints (2010), | arXiv

[21] Juhl, A.: Explicit formulas for GJMS-operators and $Q$-curvatures. Geom. Funct. Anal. 23 (2013), 1278–1370, | arXiv | DOI | MR | Zbl

[22] Kosmann, Y.: Propriétés des dérivations de l’algèbre des tenseurs-spineurs. C. R. Acad. Sci. Paris Sér. A 264 (1967), 355–358. | MR

[23] Ørsted, B.: A note on the conformal quasi-invariance of the Laplacian on a pseudo-Riemannian manifold. Lett. Math. Phys. 1 (3) (1976), 183–186. | DOI | MR | Zbl

[24] Šilhan, J.: Invariant differential operators in conformal geometry. Ph.D. thesis, University of Auckland, 2006.

[25] Slovák, J.: Natural operators on conformal manifolds. Ph.D. thesis, Masaryk University Brno, 1993. | MR | Zbl

[26] Thomas, T.Y.: On conformal geometry. Proc. Nat. Acad. Sci. U.S.A. 12 (5) (1926), 352–359. | DOI

[27] Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka J. Math. 12 (1) (1960), 21–37. | MR | Zbl