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Closed forms and multi-moment maps

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Abstract

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.

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References

  1. Baez J.C., Hoffnung A.E., Rogers C.L.: Categorified symplectic geometry and the classical string. Commun. Math. Phys. 293(3), 701–725 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baez J.C., Rogers C.L.: Categorified symplectic geometry and the string Lie 2-algebra. Homol. Homotopy Appl. 12(1), 221–236 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Bär C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)

    Article  MATH  Google Scholar 

  4. Baum H., Friedrich T., Grunewald R., Kath I.: Twistors and Killing Spinors on Riemannian Manifolds. B. G. Teubner Verlagsgesellschaft, Stuttgart, Leipzig (1991)

    MATH  Google Scholar 

  5. Berger M.: Sur les groupes d’holonomie des variétés à connexion affine et des variétés Riemanniennes. Bull. Soc. Math. France 83, 279–330 (1955)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  7. Bonan E.: Sur des variétés riemanniennes à groupe d’holonmie G 2 ou Spin(7). C. R. Acad. Sci. Paris 262, 127–129 (1966)

    MathSciNet  MATH  Google Scholar 

  8. Bryant R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)

    Article  MATH  Google Scholar 

  9. Bryant R.L., Salamon S.M.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cairns G., Jessup B.: New bounds on the Betti numbers of nilpotent Lie algebras. Commun. Algebra 25(2), 415–430 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cariñena J.F., Clemente-Gallardo J., Marmo G.: Reduction procedures in classical and quantum mechanics. Int. J. Geom. Methods Mod. Phys. 4(8), 1363–1403 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cariñena J.F., Crampin M., Ibort L.A.: On the multisymplectic formalism for first order field theories. Differ. Geom. Appl. 1(4), 345–374 (1991)

    Article  MATH  Google Scholar 

  13. Conti D., Salamon S.: Generalized Killing spinors in dimension 5. Trans. Am. Math. Soc. 359(11), 5319–5343 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dixmier J.: Cohomologie des algèbres de Lie nilpotentes. Acta Sci. Math. Szeged. 16, 246–250 (1955)

    MathSciNet  MATH  Google Scholar 

  15. Fernández, M., Gray, A.: Riemannian manifolds with structure group G 2. Ann. Mat. Pura Appl. (4) 132(1), 19–45 (1982)

    Google Scholar 

  16. Freibert M., Schulte-Hengesbach F.: Half-flat structures on decomposable Lie groups. Transform. Groups 17(1), 123–141 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Freibert M., Schulte-Hengesbach F.: Half-flat structures on indecomposable Lie groups. Transform. Groups 17(3), 657–689 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Friedrich T., Kath I., Moroianu A., Semmelmann U.: On nearly parallel G 2-structures. J. Geom. Phys. 23(3–4), 259–286 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery, R.: Momentum maps and classical relativistic fields. Part I: covariant field theory. arXiv:physics/9801019 [math-ph] (1998)

  20. Gran U., Gutowski J., Papadopoulos G.: IIB black hole horizons with five-form flux and KT geometry. J. High Energy Phys. 5, 050 (2011)

    Article  MathSciNet  Google Scholar 

  21. Gray, A.: A note on manifolds whose holonomy group is a subgroup of Sp(n). Sp(1. Mich. Math. J. 16, 125-128 (1969). Corrigendum 17, 409 (1970)

    Google Scholar 

  22. Gray A.: Weak holonomy groups. Math. Z 123, 290–300 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hitchin, N.J.: Stable forms and special metrics. In: Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Contemporary Mathematics, vol. 288, pp. 70–89. American Mathematical Society, Providence, RI (2001)

  24. Hochschild G., Serre J.P.: Cohomology of Lie algebras. Ann. Math. 2(57), 591–603 (1953)

    Article  MathSciNet  Google Scholar 

  25. Joyce D.: Compact riemannian 8-manifolds with holonomy Spin(7). Invent. Math. 123, 507–552 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Karigiannis S.: Deformations of G 2 and Spin(7) structures. Can. J. Math. 57(5), 1012–1055 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Madsen T.B.: Spin(7)-manifolds with three-torus symmetry. J. Geom. Phys. 61(11), 2285–2292 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  28. Madsen, T.B.: Torsion geometry and scalar functions. Ph.D. thesis, University of Southern Denmark (2011)

  29. Madsen T.B., Swann A.F.: Multi-moment maps. Adv. Math. 229, 2287–2309 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Madsen, T.B., Swann, A.F.: Homogeneous spaces, multi-moment maps and 2,3-trivial algebras. In: Proceedings of the XIXth International Fall Workshop on Geometry and Physics, Porto, 6–9 Sept 2010, AIP Conference Proceedings, vol. 1360, pp. 51–62. American Institute of Physics (2011)

  31. Massey, W.S.: Cross products of vectors in higher-dimensional euclidean spaces. Am. Math. Mon. 90(10), 697–701 (1983)

    Google Scholar 

  32. Neeb K.H., Vizman C.: An abstract setting for Hamiltonian actions. Monatsh. Math. 159(3), 261–288 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Poon Y.S., Swann A.F.: Superconformal symmetry and hyperKähler manifolds with torsion. Commun. Math. Phys. 241(1), 177–189 (2003)

    MathSciNet  MATH  Google Scholar 

  34. Reidegeld F.: Spaces admitting homogeneous G 2-structures. Differ. Geom. Appl. 28(3), 301–312 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Reidegeld F.: Special cohomogeneity-one metrics with Q 1,1,1 or M 1,1,0 as the principal orbit. J. Geom. Phys. 60(9), 1069–1088 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Swann A.F.: Aspects symplectiques de la géométrie quaternionique. C. R. Acad. Sci. Paris. 308, 225–228 (1989)

    MathSciNet  MATH  Google Scholar 

  37. Swann A.F.: HyperKähler and quaternionic Kähler geometry. Math. Ann. 289, 421–450 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  38. Whitehead, G.W.: Note on cross-sections in Stiefel manifolds. Comment. Math. Helv. 37, 239–240 (1962–1963)

    Google Scholar 

  39. Witt, F.: Special metric structures and closed forms. Ph.D. thesis, University of Oxford (2004). Eprint arXiv:math/0502443v2[math.DG]

  40. Witt F.: Special metrics and triality. Adv. Math. 219(6), 1972–2005 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wolf, J.A.: The geometry and topology of isotropy irreducible homogeneous spaces. Acta Math. 120, 59–148 (1968). Corrigendum 152(1–2), 141–142 (1984)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Aarhus, Ny Munkegade 118, Bldg 1530, 8000, Aarhus C, Denmark

    Thomas Bruun Madsen & Andrew Swann

  2. CP3-Origins, Centre of Excellence for Particle Physics Phenomenology, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Thomas Bruun Madsen & Andrew Swann

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  1. Thomas Bruun Madsen
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  2. Andrew Swann
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Correspondence to Andrew Swann.

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Madsen, T.B., Swann, A. Closed forms and multi-moment maps. Geom Dedicata 165, 25–52 (2013). https://doi.org/10.1007/s10711-012-9783-4

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  • DOI: https://doi.org/10.1007/s10711-012-9783-4

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