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Communications in Mathematical Physics

, Volume 348, Issue 3, pp 959–990 | Cite as

Deformations of Nearly Kähler Instantons

  • Benoit Charbonneau
  • Derek Harland
Open Access
Article

Abstract

We formulate the deformation theory for instantons on nearly Kähler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator, and prove that abelian instantons are rigid. As an application, we show that the canonical connection on three of the four homogeneous nearly Kähler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bundle with structure group SU(3).

References

  1. 1.
    Acharya B.S., O’Loughlin M., Spence B.: Higher dimensional analogues of Donaldson–Witten theory. Nucl.Phys. B 503, 657–674 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agricola I.: The Srní lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42(suppl.), 5–84 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Agricola I., Friedrich T.: On the holonomy of connections with skew-symmetric torsion. Math. Ann. 328(4), 711–748 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Agricola I., Friedrich T., Kassuba M.: Eigenvalue estimates for Dirac operators with parallel characteristic torsion. Differ. Geom. Appl. 26(6), 613–624 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Atiyah M.F., Hitchin N.J., Singer I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362(1711), 425–461 (1978)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Baraglia, D., Hekmati, P.: Moduli spaces of contact instantons. Adv. Math. 294, 562–595 (2016)Google Scholar
  7. 7.
    Baulieu L., Kanno H., Singer I.M.: Special quantum field theories in eight and other dimensions. Comm. Math. Phys. 194, 149–175 (1998)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistor and Killing spinors on Riemannian manifolds, vol. 108 of Seminarberichte [Seminar Reports]. Humboldt Universität, Sektion Mathematik, Berlin (1990)Google Scholar
  9. 9.
    Bryant R.L.: Metrics with exceptional holonomy. Ann. of Math. 126(3), 525–576 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bryant R.L., Salamon S.M.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58(3), 829–850 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Butruille J.B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Global Anal. Geom. 27(3), 201–225 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Butruille, J.B.: Homogeneous nearly Kähler manifolds. In: Handbook of pseudo-Riemannian geometry and supersymmetry, vol. 16 of IRMA Lect. Math. Theor. Phys., pp. 399–423. Eur. Math. Soc., Zürich (2010)Google Scholar
  13. 13.
    Capria, M.M., Salamon, S.M.: Yang–Mills fields on quaternionic spaces. Nonlinearity 1(4), 517–530 (1988)Google Scholar
  14. 14.
    Cherkis S.A.: Octonions, monopoles, and knots. Lett. Math. Phys. 105(5), 641–659 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Clarke A.: Instantons on the exceptional holonomy manifolds of Bryant and Salamon. J. Geom. Phys. 82, 84–97 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Correia F.P.: Hermitian Yang–Mills instantons on Calabi–Yau cones. J. High Energy Phys. 2009(12), 004 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Corrigan E., Devchand C., Fairlie D.B., Nuyts J.: First-order equations for gauge fields in spaces of dimension greater than four. Nuclear Phys. B 214(3), 452–464 (1983)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Cortés V., Vásquez J.J.: Locally homogeneous nearly Kähler manifolds. Ann. Global Anal. Geom. 48(3), 269–294 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Donaldson, S., Segal, E.: Gauge theory in higher dimensions, II. In: Surveys in differential geometry, vol. 16 (2011)Google Scholar
  20. 20.
    Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimensions. In: The geometric universe (Oxford, 1996), pp. 31–47. Oxford University Press, Oxford (1998)Google Scholar
  21. 21.
    Dunajski M., Hoegner M.: SU(2) solutions to self-duality equations in eight dimensions. J. Geom. Phys. 62(8), 1747–1759 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fairlie D.B., Nuyts J.: Spherically symmetric solutions of gauge theories in eight dimensions. J. Phys. A 17(14), 2867–2872 (1984)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Foscolo, L., Haskins, M.: New G 2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres (2015). arXiv:1501.07838
  24. 24.
    Friedrich T., Ivanov S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6(2), 303–335 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Fubini S., Nicolai H.: The octonionic instanton. Phys. Lett. B 155(5–6), 369–372 (1985)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Fulton, W., Harris, J.: Representation theory, vol. 129 of Graduate Texts in Mathematics. Springer, New York (1991)Google Scholar
  27. 27.
    Gibbons G.W., Page D.N., Pope C.N.: Einstein metrics on \({S^3,\;{\bf R}^3}\) and \({{\bf R}^4}\) bundles. Commun. Math. Phys. 127(3), 529–553 (1990)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gray, A.: Nearly Kähler manifolds. J. Differen. Geometry 4, 283–309 (1970)Google Scholar
  29. 29.
    Grunewald R.: Six-dimensional Riemannian manifolds with a real Killing spinor. Ann. Global Anal. Geom. 8(1), 43–59 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Günaydin M., Nicolai H.: Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton. Phys. Lett. B 351(1-3), 169–172 (1995)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Harland D., Ivanova T.A., Lechtenfeld O., Popov A.D.: Yang–Mills flows on nearly Kähler manifolds and G 2-instantons. Commun. Math. Phys. 300(1), 185–204 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Harland, D., Nölle, C.: Instantons and Killing spinors. J. High Energy Phys. 082(3), 1–38 (2012) (front matter+37)Google Scholar
  33. 33.
    Haupt A.S., Ivanova T.A., Lechtenfeld O., Popov A.D.: Chern–Simons flows on Aloff–Wallach spaces and Spin(7)-instantons. Phys. Rev. D 83, 105028 (2011)ADSCrossRefGoogle Scholar
  34. 34.
    Haydys A.: Gauge theory, calibrated geometry and harmonic spinors. J. Lond. Math. Soc. 86(2), 482–498 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Humphreys, J.E.: Introduction to Lie algebras and representation theory, vol. 9 of Graduate Texts in Mathematics. Springer, New York, Berlin (1978) (second printing, revised)Google Scholar
  36. 36.
    Jardim, M.B., Menet, G., Prata, D.M., Sá Earp, H.N.: Holomorphic bundles for higher dimensional gauge theory (2015) (to appear in Bulletin of the London Mathematical Society). arXiv:1109.2750
  37. 37.
    Kanno, H.: A note on higher dimensional instantons and supersymmetric cycles. Progr. Theor. Phys. Suppl. 135, 18–28 (1999) [Gauge theory and integrable models (Kyoto, 1999)]Google Scholar
  38. 38.
    Kanno H., Yasui Y.: Octonionic Yang–Mills instanton on quaternionic line bundle of Spin(7) holonomy. J. Geom. Phys. 34(3–4), 302–320 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Knapp, A.W.: Representation theory of semisimple groups, volume 36 of Princeton Mathematical Series. Princeton University Press, Princeton (1986)Google Scholar
  40. 40.
    Lewis, C.: Spin(7) instantons. Ph.D. thesis, University of Oxford (1999). http://people.maths.ox.ac.uk/joyce/theses/LewisDPhil.pdf
  41. 41.
    Menet Grégoire, Nordström Johannes, Sá Earp, Henrique N.: Construction of G 2-instantons via twisted connected sums. October (2015)Google Scholar
  42. 42.
    Moroianu, A., Uwe, S.: The Hermitian Laplace operator on nearly Kähler manifolds. Commun. Math. Phys. 294(1), 251–272 (2010)Google Scholar
  43. 43.
    Muñoz V.: Spin(7)-instantons, stable bundles and the Bogomolov inequality for complex 4-tori. J. Math. Pures Appl. 102(1), 124–152 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Oliveria, G.: Monopoles on the Bryant–Salamon G 2-manifolds. J. Geom. Phys. 86, 599–632 (2014)Google Scholar
  45. 45.
    Popov A.D.: Hermitian Yang–Mills equations and pseudo-holomorphic bundles on nearly Kähler and nearly Calabi–Yau twistor 6-manifolds. Nuclear Phys. B 828(3), 594–624 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Popov A.D.: Non-abelian vortices, super Yang–Mills theory and Spin(7)-instantons. Lett. Math. Phys. 92(3), 253–268 (2010)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Popov A.D., Szabo R.J.: Double quiver gauge theory and nearly kähler flux compactifications. J. High Energy Phys. 2012(2), 1–50 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Reyes Carrión R.: A generalization of the notion of instanton. Differ. Geom. Appl. 8(1), 1–20 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Sá Earp H., Walpuski T.: \({{\mathrm{G}}_2}\)-instantons on twisted connected sums. Geom. Topol. 19(3), 1263–1285 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Sá Earp, H.N.: Generalised Chern–Simons theory and \({\mathrm{G}_2}\)-instantons over associative fibrations. SIGMA 10, 083, 11 (2014)Google Scholar
  51. 51.
    Sá Earp H.N.: \({G_2}\)-instantons over asymptotically cylindrical manifolds. Geom. Topol. 19(1), 61–111 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Tanaka Y.: A construction of Spin(7)-instantons. Ann. Global Anal. Geom. 42(4), 495–521 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Tanaka Y.: A weak compactness theorem of the Donaldson–Thomas instantons on compact Kähler threefolds. J. Math. Anal. Appl. 408(1), 27–34 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Tanaka Y.: A removal singularity theorem of the Donaldson–Thomas instanton on compact Kähler threefolds. J. Math. Anal. Appl. 411(1), 422–428 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Tanaka, Y.: On the moduli space of Donaldson–Thomas instantons. (2015) (to appear in Extracta Mathematicae). arXiv:0805.2192
  56. 56.
    Tao, T., Tian, G.: A singularity removal theorem for Yang–Mills fields in higher dimensions. J. Am. Math. Soc. 17(3), 557–593 (2004) (electronic)Google Scholar
  57. 57.
    Tian G.: Gauge theory and calibrated geometry. I. Ann. of Math. 151(1), 193–268 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Walpuski T.: G2-instantons on generalised Kummer constructions. Geom. Topol. 17(4), 2345–2388 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Ward R.S.: Completely solvable gauge-field equations in dimension greater than four. Nuclear Phys. B 236(2), 381–396 (1984)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    Wolf, J.A.: Spaces of constant curvature, 6th edn. AMS Chelsea Publishing, Providence (2011)Google Scholar
  61. 61.
    Wolf J.A., Gray A.: Homogeneous spaces defined by Lie group automorphisms. I. J. Differ. Geometry 2, 77–114 (1968)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Wolf J.A., Gray A.: Homogeneous spaces defined by Lie group automorphisms. II. J. Differ. Geometry 2, 115–159 (1968)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Xu F.: On instantons on nearly Kähler 6-manifolds. Asian J. Math. 13(4), 535–567 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.School of MathematicsUniversity of LeedsLeedsUK

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