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On conformal powers of the Dirac operator on Einstein manifolds

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Abstract

We determine the structure of conformal powers of the Dirac operator on Einstein Spin-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the associated Poincaré–Einstein metric, and relies on combinatorial recurrence identities related to the dual Hahn polynomials.

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References

  1. Bourguignon, J.P., Gauduchon, P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144(3), 581–599 (1992)

    Article  MathSciNet  Google Scholar 

  2. Branson, T., Gover, A.R.: Conformally invariant operators, differential forms, cohomology and a generalisation of \(Q\)-curvature. Commun. Partial Differ. Equ. 30(11), 1611–1669 (2005)

    Article  MathSciNet  Google Scholar 

  3. Bär, C., Gauduchon, P., Moroianu, A.: Generalized cylinders in semi-Riemannian and spin geometry. Math. Z. 249(3), 545–580 (2005)

    Article  MathSciNet  Google Scholar 

  4. Branson, T.: Sharp inequalities, the functional determinant, and the complementary series. Trans. Am. Math. Soc. 347, 3671–3742 (1995)

    Article  MathSciNet  Google Scholar 

  5. Dowker, J.S.: Determinants and conformal anomalies of GJMS operators on spheres. J. Phys. A Math. Theor. 44(11), 1–14 (2011)

    Article  MathSciNet  Google Scholar 

  6. Dowker, J.S.: Spherical Dirac GJMS operator determinants, ArXiv e-prints (2013). http://arxiv.org/abs/1310.5563

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

    MathSciNet  Google Scholar 

  8. Fefferman, C., Graham, C.R.: Conformal invariants, pp. 95–116. Astérisque (hors série), Élie Cartan et les mathématiques d’aujourd’hui (1985)

  9. Fefferman, C., Graham, C.R.: The ambient metric (AM-178), no. 178. Princeton University Press, Princeton (2011)

    Google Scholar 

  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. Gover, A.R., Hirachi, K.: Conformally invariant powers of the Laplacian: a complete non-existence theorem. J. Am. Math. Soc. 17(2), 389–405 (2004)

    Article  MathSciNet  Google Scholar 

  12. Graham, C.R., Jenne, R.W., Mason, L., Sparling, G.: Conformally invariant powers of the Laplacian, I: existence. J. Lond. Math. Soc. 2(3), 557–565 (1992)

    Article  MathSciNet  Google Scholar 

  13. Guillarmou, C., Moroianu, S., Park, J.: Bergman and Calderón projectors for Dirac operators, J. Geom. Anal., 1–39, (2012). http://dx.doi.org/10.1007/s12220-012-9338-9

  14. Gover, A.R.: Laplacian operators and \(Q\)-curvature on conformally Einstein manifolds. Math. Ann. 336(2), 311–334 (2006)

    Article  MathSciNet  Google Scholar 

  15. Gover, A.R., Peterson, L.J.: Conformally invariant powers of the Laplacian, \(Q\)-curvature, and tractor calculus. Commun. Math. Phys. 235(2), 339–378 (2003)

    Article  MathSciNet  Google Scholar 

  16. Gover, A.R., Šilhan, J.: Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds, Annales Henri Poincaré, vol. 15. Springer, Berlin (2014)

    Google Scholar 

  17. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. math. 152, 89–118 (2003)

    Article  MathSciNet  Google Scholar 

  18. Holland, J, Sparling, G.: Conformally invariant powers of the ambient Dirac operator, ArXiv e-prints (2001). http://arxiv.org/abs/math/0112033

  19. Juhl, A.: Explicit formulas for GJMS-operators and \(Q\)-curvatures, geometric and functional analysis 23, 1278–1370 (2013). http://arxiv.org/abs/1108.0273

  20. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric orthogonal polynomials and their \(q\)-analogues. Springer monographs in mathematics. Springer, Berlin (2010)

  21. Karlin, S., McGregor, J.L.: The Hahn polynomials, formulas and an application. Scr. Math. 26, 33–46 (1961)

    MathSciNet  Google Scholar 

  22. Slovák, J.: Natural operators on conformal manifolds, Ph.D. thesis. Masaryk University, Brno (1993). http://www.math.muni.cz/~slovak/ftp/papers/vienna.ps

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Authors and Affiliations

  1. E. Čech Institute, Mathematical Institute of Charles University, Sokolovská 83, Prague 8, Karlín, Czech Republic

    Matthias Fischmann & Petr Somberg

  2. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Christian Krattenthaler

Authors
  1. Matthias Fischmann
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  2. Christian Krattenthaler
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  3. Petr Somberg
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Correspondence to Matthias Fischmann.

Additional information

Matthias Fischmann and Petr Somberg Research partially supported by grant GA CR P201/12/G028.

Christian Krattenthaler Research partially supported by the Austrian Science Foundation FWF, Grants Z130-N13 and S50-N15, the latter in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.

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Fischmann, M., Krattenthaler, C. & Somberg, P. On conformal powers of the Dirac operator on Einstein manifolds. Math. Z. 280, 825–839 (2015). https://doi.org/10.1007/s00209-015-1450-7

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  • DOI: https://doi.org/10.1007/s00209-015-1450-7

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