Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
conformal and spin geometry; conformal powers of the Dirac operator; conformal covariance; tractor bundle; tractor D-operator
conformal and spin geometry; conformal powers of the Dirac operator; conformal covariance; tractor bundle; tractor D-operator
Summary:
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.
References:
[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 10.1216/rmjm/1181072333 | MR 1322223 | Zbl 0828.53012
[2] Branson, T.: Conformally covariant equations on differential forms. Comm. Partial Differential Equation 7 (4) (1982), 393–431. DOI 10.1080/03605308208820228 | MR 0652815 | Zbl 0532.53021
[3] Branson, T.: Conformal structure and spin geometry. Dirac Operators: Yesterday and Today, International Press, 2005. MR 2205362 | Zbl 1109.53051
[4] Čap, A., Slovák, J.: Parabolic Geometries: Background and General Theory. vol. 1, American Mathematical Society, 2009. MR 2532439
[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 0880414 | Zbl 0659.53047
[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 2680665 | Zbl 1201.30065
[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 0837196
[8] Fefferman, C., Graham, C.R.: The ambient metric. Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. MR 2858236 | Zbl 1243.53004
[9] Fegan, H.D.: Conformally invariant first order differential operators. Quart. J. Math. Oxford (2) 27 (107) (1976), 371–378. DOI 10.1093/qmath/27.3.371 | MR 0482879 | Zbl 0334.58016
[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), http://arxiv.org/abs/1405.7304 MR 3369353
[12] Gover, A.R.: Laplacian operators and $Q$-curvature on conformally Einstein manifolds. Math. Anal. 336 (2) (2006), 311–334. DOI 10.1007/s00208-006-0004-z | MR 2244375 | Zbl 1125.53032
[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 10.1090/S0894-0347-04-00450-3 | MR 2051616
[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 10.1007/s00220-002-0790-4 | MR 1969732 | Zbl 1022.58014
[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 1190438 | Zbl 0726.53010
[16] Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152 (2003), 89–118. DOI 10.1007/s00222-002-0268-1 | MR 1965361 | Zbl 1030.58022
[17] Guillarmou, C., Moroianu, S., Park, J.: Bergman and Calderón projectors for Dirac operators. J. Geom. Anal. (2012), 1–39, http://dx.doi.org/10.1007/s12220-012-9338-9 DOI 10.1007/s12220-012-9338-9
[18] Hitchin, N.J.: Harmonic spinors. Adv. Math. 14 (1) (1974), 1–55. DOI 10.1016/0001-8708(74)90021-8 | MR 0358873 | Zbl 0284.58016
[19] Holland, J., Sparling, G.: Conformally invariant powers of the ambient Dirac operator. ArXiv e-prints (2001), http://arxiv.org/abs/math/0112033
[20] Juhl, A.: On conformally covariant powers of the Laplacian. ArXiv e-prints (2010), http://arxiv.org/abs/0905.3992
[21] Juhl, A.: Explicit formulas for GJMS-operators and $Q$-curvatures. Geom. Funct. Anal. 23 (2013), 1278–1370, http://arxiv.org/abs/1108.0273 DOI 10.1007/s00039-013-0232-9 | MR 3077914 | Zbl 1293.53049
[22] Kosmann, Y.: Propriétés des dérivations de l’algèbre des tenseurs-spineurs. C. R. Acad. Sci. Paris Sér. A 264 (1967), 355–358. MR 0212712
[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 10.1007/BF00417601 | MR 0433524 | Zbl 0338.53046
[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 1255551 | Zbl 0805.53011
[26] Thomas, T.Y.: On conformal geometry. Proc. Nat. Acad. Sci. U.S.A. 12 (5) (1926), 352–359. DOI 10.1073/pnas.12.5.352
[27] Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka J. Math. 12 (1) (1960), 21–37. MR 0125546 | Zbl 0096.37201
Search
Partner of
