Advertisement

\(L^{2}\) harmonic forms on complete special holonomy manifolds

  • Teng HuangEmail author
Article

Abstract

In this article, we consider \(L^{2}\) harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form \(\omega \). The main result is that if \((X,\omega )\) is a complete \(G_{2}\)- (or \(\textit{Spin}(7)\)-) manifold with a d(linear) \(G_{2}\)- (or \(\textit{Spin}(7)\)-) structure form \(\omega \), then the \(L^{2}\) harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on \((X,\omega )\) only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over \((X,\omega )\).

Keywords

\(L^{2}\) harmonic form \(G_{2}\hbox {- } (\textit{Spin}(7)\hbox {-})\)manifold d(linear)-form Gauge theory 

Notes

Acknowledgements

I would like to thank the anonymous referee for careful reading of my manuscript and helpful comments. I would like to thank Professor Verbitsky for kind comments regarding his article [36]. Also I would like to thank Yuguo Qin for further discussions about this work. This work is supported by Nature Science Foundation of China No. 11801539 and Postdoctoral Science Foundation of China No. 2017M621998, No. 2018T110616.

References

  1. 1.
    Bauer, I., Ivanova, T.A., Lechtenfeld, O., Lubbe, F.: Yang–Mills instantons and dyons on homogeneous \(G_{2}\)-manifolds. JHEP 2010(10), 1–27 (2010)zbMATHGoogle Scholar
  2. 2.
    Bryant, R.: Metrics with exceptional holonomy. Ann. Math. 126(2), 525–576 (1987)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bryant, R.: Some remarks on \(G_{2}\)-structures. In: Proceedings of Gökova Geometry-Topology Conference, pp. 75–109 (2005)Google Scholar
  4. 4.
    Cao, J.G., Frederico, X.: Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature. Math. Ann. 319, 483–491 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Carrión, R.R.: A generalization of the notion of instanton. Differential Geom. Appl. 8(1), 1–20 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Corrigan, E., Devchand, C., Fairlie, D.B., Nuyts, J.: First order equations for gauge fields in spaces of dimension great than four. Nucl. Phys. B. 214(3), 452–464 (1983)Google Scholar
  8. 8.
    Dodziuk, J., Min-Oo, M.: An \(L_{2}\)-isolation theorem for Yang–Mills fields over complete manifolds. Compos. Math. 47, 165–169 (1982)zbMATHGoogle Scholar
  9. 9.
    Donaldson, S.K., Thomas, R.P.: Gauge Theory in Higher Dimensions, pp. 31–47. The Geometric Universe, Oxford (1998)zbMATHGoogle Scholar
  10. 10.
    Donaldson S. K., Segal E.: Gauge theory in higher dimensions, II. arXiv:0902.3239 (2009)
  11. 11.
    Escobar, J.F., Freire, A., Min-Oo, M.: \(L^{2}\) vanishing theorems in positive curvature. Indiana Univ. Math. J. 42(4), 1545–1554 (1993)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fubini, S., Nicolai, H.: The octonionic instanton. Phys. Lett. B. 155(5), 369–372 (1985)MathSciNetGoogle Scholar
  13. 13.
    Gemmer, K.P., Lechtenfeld, O., Nölle, C., Popov, A.D.: Yang–Mills instantons on cones and sine-cones over nearly Kähler manifolds. JHEP 9, 103 (2011)zbMATHGoogle Scholar
  14. 14.
    Gerhardt, G.: An energy gap for Yang–Mills connections. Comm. Math. Phys. 298, 515–522 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Green, M.B., Schwarz, J.H., Witten, E.: Supperstring Theory. Cambridge University Press, Cambridge (1987)Google Scholar
  16. 16.
    Gromov, M.: Kähler hyperbolicity and \(L_{2}\)-Hodge theory. J. Differential Geom. 33, 263–292 (1991)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Harland, D., Ivanova, T.A., Lechtenfeld, O., Popov, A.D.: Yang–Mills flows on nearly Kähler manifolds and \(G_{2}\)-instantons. Comm. Math. Phys. 300(1), 185–204 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hitchin, N.J.: \(L^{2}\) cohomology of hyper-Kähler quotients. Comm. Math. Phys. 211, 153–165 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hitchin, N.J.: The geometry of three-forms in six and seven dimensions. J. Differential Geom. 55(3), 547–576 (2003)zbMATHGoogle Scholar
  20. 20.
    Huang, T.: Instanton on cylindrical manifolds. Ann. Henri Poincaré 18(2), 623–641 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Huang, T.: Stable Yang–Mills connections on special holonomy manifolds. J. Geom. Phys. 116, 271–280 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Huang T.: Asymptotic behaviour of instantons on cylinder manifolds. arXiv:1801.06959v4
  23. 23.
    Ivanov, S.: Connections with torsion, parallel spinors and geometry of \(Spin(7)\) manifolds. Math. Res. Lett. 11, 171–186 (2004)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ivanova, T.A., Popov, A.D.: Instantons on special holonomy manifolds. Phys. Rev. D 85, 10 (2012)Google Scholar
  25. 25.
    Ivanova, T.A., Lechtenfeld, O., Popov, A.D., Rahn, T.: Instantons and Yang–Mills flows on coset spaces. Lett. Math. Phys. 89(3), 231–247 (2009)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jost, J., Zuo, K.: Vanishing theorems for \(L^{2}\)-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry. Comm. Anal. Geom. 8, 1–30 (2000)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Joyce D.: Compact Riemannian \(7\)-manifolds with holonomy \(G_{2}\), I,II. J. Differ. Geom. 43(2), 291–328. 329–375 (1996)Google Scholar
  28. 28.
    Joyce, D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  29. 29.
    Karigiannis, S., Leung, N.C.: Hodge theory for \(G_{2}\)-manifolds: intermediate Jacobians and Abel–Jacobi maps. Proc. Lond. Math. Soc. 99(3), 297–325 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kovalev, A.: Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lee, J.H., Leung, N.C.: Geometric structures on \(G_{2}\) and \(Spin(7)\)-manifolds. Adv. Theor. Math. Phys. 13(1), 1–31 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Min-Oo, M.: An \(L_{2}\)-isolation theorem for Yang–Mills fields. Compos. Math. 47, 153–163 (1982)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differential Geom. 36, 417–450 (1992)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Verbitsky, M.: An intrinsic volume functional on almost complex \(6\)-manifolds and nearly Kähler geometry. Pacific J. Math. 235(2), 323–344 (2008)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Verbitsky, M.: Hodge theory on nearly Kähler manifolds. Geom. Topol. 15, 2111–2133 (2011)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Verbitsky, M.: Manifolds with parallel differential forms and Kähler identities for \(G_{2}\)-manifolds. J. Geom. Phys. 61(6), 1001–1016 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Ward, R.S.: Completely solvable gauge field equations in dimension great than four. Nucl. Phys. B. 236(2), 381–396 (1984)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesSoochow UniversitySuzhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

Personalised recommendations