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Yang-Mills Detour Complexes and Conformal Geometry

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Abstract

Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.

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References

  1. Atiyah M.F., Hitchin N. and Singer I.M. (1978). Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser A 362: 425–461

    MATH  ADS  MathSciNet  Google Scholar 

  2. Bach R. (1921). Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. 9: 110–135

    Article  MathSciNet  Google Scholar 

  3. Bailey T.N., Eastwood M.G. and Gover A.R. (1994). Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24: 1191–1217

    MATH  MathSciNet  Google Scholar 

  4. Baston, R.J., Mason, L.J.: The conformal Einstein equations, In: Further advances in twistor theory: Volume II: Integrable systems, conformal geometry and gravitation, edited by L.J. Mason, L.P. Hughston, P.Z. Kobak, Essex: Longman, 1995

  5. Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik, 124. Stuttgart: B.G. Teubner Verlagsgesellschaft mbH, 1991

  6. Besse, A.L.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10. Berlin: Springer-Verlag, 1987

  7. Boe B.D. and Collingwood D.H. (1985). A comparison theory for the structure of induced representations, II. Math. Z. 190: 1–11

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Branson, T.: Conformal Structure and spin geometry. In: Dirac operators: yesterday and today. Proceedings of the Summer School and Workshop held in Beirut, August 27–September 7, 2001, edited by J.-P. Bourguignon, T. Branson, A. Chamseddine, O. Hijazi, R.J. Stanton. Somerville, MA: International Press, 2005

  10. Branson T. (2005). Q-curvature and spectral invariants. Rend. Circ. Mat. Palermo (2) Suppl. No. 75: 11–55

    MathSciNet  Google Scholar 

  11. Branson T. and Gover A.R. (2005). Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature. Comm. Part. Differ. Equs. 30: 1611–1669

    Article  MATH  MathSciNet  Google Scholar 

  12. Branson T. and Gover A.R. (2007). The conformal deformation detour complex for the obstruction tensor. Proc. Amer. Math. Soc. 135: 2961–2965

    Article  MATH  MathSciNet  Google Scholar 

  13. Čap, A., Gover, A.R.: Tractor bundles for irreducible parabolic geometries. In: Global analysis and harmonic analysis (Marseille-Luminy, 1999), Sémin. Congr. 4, Paris: Soc. Math. France, 2000, pp. 129–154

  14. Čap A. and Gover A.R. (2002). Tractor calculi for parabolic geometries. Trans. Amer. Math. Soc. 354: 1511–1548

    Article  MathSciNet  MATH  Google Scholar 

  15. Čap A., Slovák J. and Souček V. (2001). Bernstein-Gelfand-Gelfand sequences. Ann. Math. 154: 97–113

    Article  MATH  Google Scholar 

  16. Čap, A., Souček, V.: Subcomplexes in Curved BGG-Sequences. http://arxiv.org/list/math.DG/0508534, 2005

  17. Cartan E. (1923). Les espaces à connexion conforme. Ann. Soc. Pol. Math. 2: 171–202

    Google Scholar 

  18. Chang S.-Y.A. (2004). Non-linear elliptic equations in conformal geometry. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich

    Google Scholar 

  19. Donaldson, S.K.: Floer homology groups in Yang-Mills theory. Cambridge Tracts in Mathematics 147. Cambridge: Cambridge University Press, 2002

  20. Eastwood M. (1996). Notes on conformal differential geometry. Rend. Circ. Mat. Palermo (2) Suppl. No. 43: 57–76

    MathSciNet  Google Scholar 

  21. Eastwood, M.G., Rice, J.W.: Conformally invariant differential operators on Minkowski space and their curved analogues. Commun. Math. Phys. 109, 207–228 (1987); Erratum: Commun. Math. Phys. 144, 213 (1992)

    Google Scholar 

  22. Eastwood M.G. and Slovák J. (1997). Semiholonomic Verma modules. J. Alg. 197: 424–448

    Article  MATH  Google Scholar 

  23. Fefferman, C., Graham, C.R.: Conformal invariants. In: Elie Cartan et les mathématiques d’aujourd’hui, Astérisque 95–116, hors série (Paris: SMF, 1985)

  24. Fefferman C. and Graham C.R. (2002). Q-curvature and Poincaré metrics. Math. Res. Lett. 9: 139–151

    MATH  MathSciNet  Google Scholar 

  25. Friedrich T. (1989). On the conformal relation between twistors and Killing spinors. Rend. Circ. Mat. Palermo (2) Suppl. No. 2: 59–75

    Google Scholar 

  26. Friedrich, T.: Dirac-Operatoren in der Riemannschen Geometrie, Mit einem Ausblick auf die Seiberg-Witten-Theorie. Advanced Lectures in Mathematics. Braunschweig: Friedr. Vieweg & Sohn, 1997

  27. Gover A.R. (1999). Aspects of parabolic invariant theory. Rend. Circ. Mat. Palermo (2) Suppl. No. 59: 25–47

    MathSciNet  Google Scholar 

  28. Gover, A.R.: Almost conformally Einstein manifolds and obstructions. In: Differential geometry and its applications, Prague: Matfyzpress, 2005, pp. 247–260

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Gover, A.R., Leitner, F.: A sub-product construction of Poincare-Einstein metrics. http://arxiv.org/list/math.DG/0608044, 2006

  31. Gover A.R. and Nurowski P. (2006). Obstructions to conformally Einstein metrics in n dimensions. J. Geom. Phys. 56: 450–484

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Gover, A.R., Šilhan, J.: The conformal Killing equation on forms – prolongations and applications, Diff. Geom. Applic., to appear. http://arxiv.org/list/math.DG/0601751, 2006

  34. Graham, C.R., Hirachi, K.: The ambient obstruction tensor and Q-curvature. In: AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys. 8, Zürich: Eur. Math. Soc., 2005, pp. 59–71

  35. Graham C.R., Jenne R., Mason L. and Sparling G. (1992). Conformally invariant powers of the Laplacian, I: existence. J. London Math. Soc. 46: 557–565

    Article  MATH  MathSciNet  Google Scholar 

  36. Graham C.R. and Zworski M. (2003). Scattering matrix in conformal geometry. Invent. Math. 152: 89–118

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Korzynski M. and Lewandowski J. (2003). The normal Cartan connection and the Bach tensor. Class. Quant. Grav. 20: 3745–3764

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Kozameh C., Newman E.T and Tod K.P. (1985). Conformal Einstein Spaces. GRG 17: 343–352

    MATH  MathSciNet  Google Scholar 

  39. Merkulov S. (1984). A conformally invariant theory of gravitation and electromagnetism. Class. Quant. Grav. 1: 349–354

    Article  ADS  MathSciNet  Google Scholar 

  40. Lee J.M. and Parker T.H. (1987). The Yamabe problem. Bull. Amer. Math. Soc. 17: 37–91

    Article  MATH  MathSciNet  Google Scholar 

  41. Penrose, R., Rindler, W.: Wolfgang, Spinors and space-time. Vol. 1 and Vol. 2. Spinor and twistor methods in space-time geometry, Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press, 1987, 1988

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Authors and Affiliations

  1. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand

    A. Rod Gover

  2. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha, Czech Republic

    Petr Somberg & Vladimír Souček

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  1. A. Rod Gover
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  2. Petr Somberg
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  3. Vladimír Souček
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Correspondence to A. Rod Gover.

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Communicated by A. Connes

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Gover, A.R., Somberg, P. & Souček, V. Yang-Mills Detour Complexes and Conformal Geometry. Commun. Math. Phys. 278, 307–327 (2008). https://doi.org/10.1007/s00220-007-0401-5

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  • DOI: https://doi.org/10.1007/s00220-007-0401-5

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