Advertisement

Annals of Global Analysis and Geometry

, Volume 28, Issue 3, pp 285–300 | Cite as

tt*-Geometry and Pluriharmonic Maps

  • Lars SchÄferEmail author
Article

Abstract

In this paper we use the real differential geometric definition of a metric (a unimodular oriented metric) tt*-bundle of Cortés and the author (Topological-anti-topological fusion equations, pluriharmonic maps and special Kähler manifolds) to define a map Φ from the space of metric (unimodular oriented metric) tt*-bundles of rank r over a complex manifold M to the space of pluriharmonic maps from M to {GL}(r)/O(p,q) (respectively {SL}(r)/SO(p,q)), where (p,q) is the signature of the metric. In the sequel the image of the map Φ is characterized. It follows, that in signature (r,0) the image of Φ is the whole space of pluriharmonic maps. This generalizes a result of Dubrovin (Comm. Math. Phys. 152 (1992; S539–S564).

Keywords

tt*-geometry tt*-bundles pluriharmonic maps pseudo-Riemannian symmetric spaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alekseevsky, D. V., Devchand, C. and Cortés, V.: Special complex manifolds, J. Geom. Phys. 42 (2002), 85–S105.CrossRefGoogle Scholar
  2. 2.
    Cecotti, S. and Vafa, C.: Topological-anti-topological fusion, Nuclear. Phys. B 367 (1991), 351–S461.CrossRefGoogle Scholar
  3. 3.
    Cheeger, J. and Ebin, D. G.: Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975.Google Scholar
  4. 4.
    Cortés, V. and Schäfer, L.: Topological-anti-topological fusion equations, pluriharmonic maps and special Kähler manifolds, in: O. Kowalski, E. Musso and D. Perrone (eds), Proceedings of the Conference ‘Curvature in Geometry’ organized in Lecce in honour of Lieven Vanhecke, Progr. Math. 234, Birkhäuser, Basel, 2005.Google Scholar
  5. 5.
    Dubrovin, B.: Geometry and integrability of topological-anti-topological fusion, Comm. Math. Phys. 152 (1992) 539–S564.CrossRefGoogle Scholar
  6. 6.
    Gallot, S., Hulin, D. and Lafontaine, J.: Riemannian Geometry, Springer, Berlin, 1993.Google Scholar
  7. 7.
    Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.Google Scholar
  8. 8.
    Hertling, C.: tt*-Geometry, Frobenius manifolds, their connections and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161.Google Scholar
  9. 9.
    Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. I/II, Interscience, New York, 1964/1969.Google Scholar
  10. 10.
    Sampson, J. H.: Applications of harmonic maps to Kähler geometry, In: Contemp. Math. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 125–134.Google Scholar
  11. 11.
    Schäfer, L.: tt*-Geometrie und pluriharmonische Abbildungen, Diplomarbeit an der Universität, Bonn, 2002.Google Scholar
  12. 12.
    Schäfer, L.: Higgs-Bündel, nicht-lineare Sigma-Modelle und topologische antitopologische Fusion, Diplomarbeit in Physik an der Universität Bonn, July 2004.Google Scholar
  13. 13.
    Schäfer, L.: tt*-Bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps, Differential Geom. Appl., to appear.Google Scholar
  14. 14.
    Schäfer, L.: Harmonic bundles topological–antitoplological fusion and the related puriharmonic maps, J. Geom. Phys., to appear.Google Scholar
  15. 15.
    Simpson, C. T.: Higgs bundles and local systems, Inst. Hautes Etudes Sci. Publ. Math. 75 (1992), 5–95.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität BonnBonnGermany
  2. 2.Institut Élie Cartan de MathématiquesUniversité Henri Poincaré – Nancy 1Vandœuvre-lès-Nancy CedexFrance

Personalised recommendations