Skip to main content
Log in

Abstract

We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kähler 6-manifolds, nearly parallel G 2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our solutions are new instantons on even-dimensional Euclidean spaces, as well as the well-known BPST, quaternionic and octonionic instantons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. E. Corrigan, C. Devchand, D. Fairlie and J. Nuyts, First order equations for gauge fields in spaces of dimension greater than four, Nucl. Phys. B 214 (1983) 452 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. R.S. Ward, Completely solvable gauge field equations in dimension greater than four, Nucl. Phys. B 236 (1984) 381 [INSPIRE].

    Article  ADS  Google Scholar 

  3. S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The geometric universe, S. Hugett et al. eds., Oxford University Press, Oxford U.K. (1998).

    Google Scholar 

  4. S. Donaldson and E. Segal, Gauge theory in higher dimensions, II, arXiv:0902.3239 [INSPIRE].

  5. G. Tian, Gauge theory and calibrated geometry, Ann. Math. 151 (2000) 193 [math/0010015].

    Article  MATH  Google Scholar 

  6. D. Fairlie and J. Nuyts, Spherically symmetric solutions of gauge theories in eight-dimensions, J. Phys. A 17 (1984) 2867 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  7. S. Fubini and H. Nicolai, The octonionic instanton, Phys. Lett. B 155 (1985) 369 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  8. E. Corrigan, P. Goddard and A. Kent, Some comments on the ADHM construction in 4k dimensions, Commun. Math. Phys. 100 (1985) 1.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. T. Ivanova and A. Popov, Selfdual Yang-Mills fields in D = 7, 8, octonions and Ward equations, Lett. Math. Phys. 24 (1992) 85 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. T. Ivanova and A. Popov, (Anti)selfdual gauge fields in dimension d ≥ 4, Theor. Math. Phys. 94 (1993) 225 [INSPIRE].

    Article  MathSciNet  Google Scholar 

  11. J. Broedel, T.A. Ivanova and O. Lechtenfeld, Construction of noncommutative instantons in 4k dimensions, Mod. Phys. Lett. A 23 (2008) 179 [hep-th/0703009] [INSPIRE].

    ADS  Google Scholar 

  12. D. Harland, T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Yang-Mills flows on nearly Kähler manifolds and G 2-instantons, Commun. Math. Phys. 300 (2010) 185 [arXiv:0909.2730] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. D. Harland and A.D. Popov, Yang-Mills fields in flux compactifications on homogeneous manifolds with SU(4)-structure, JHEP 02 (2012) 107 [arXiv:1005.2837] [INSPIRE].

    Article  ADS  Google Scholar 

  14. A.S. Haupt, T.A. Ivanova, O. Lechtenfeld and A.D. Popov, Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons, Phys. Rev. D 83 (2011) 105028 [arXiv:1104.5231] [INSPIRE].

    ADS  Google Scholar 

  15. K.-P. Gemmer, O. Lechtenfeld, C. Nölle and A.D. Popov, Yang-Mills instantons on cones and sine-cones over nearly Kähler manifolds, JHEP 09 (2011) 103 [arXiv:1108.3951] [INSPIRE].

    Article  ADS  Google Scholar 

  16. F. Xu, On instantons on nearly Kähler 6-manifolds, Asian J. Math. 13 (2009) 535.

    MathSciNet  MATH  Google Scholar 

  17. F.P. Correia, Hermitian Yang-Mills instantons on Calabi-Yau cones, JHEP 12 (2009) 004 [arXiv:0910.1096] [INSPIRE].

    Article  Google Scholar 

  18. F.P. Correia, Hermitian Yang-Mills instantons on resolutions of Calabi-Yau cones, JHEP 02 (2011) 054 [arXiv:1009.0526] [INSPIRE].

    Article  Google Scholar 

  19. B.S. Acharya, J. Figueroa-O’Farrill, C. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].

    MathSciNet  Google Scholar 

  20. P. Koerber, D. Lüst and D. Tsimpis, Type IIA AdS 4 compactifications on cosets, interpolations and domain walls, JHEP 07 (2008) 017 [arXiv:0804.0614] [INSPIRE].

    Article  ADS  Google Scholar 

  21. C. Bär, Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993) 509.

    Article  ADS  MATH  Google Scholar 

  22. J.P. Gauntlett and O. Varela, Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions, Phys. Rev. D 76 (2007) 126007 [arXiv:0707.2315] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  23. A. Strominger, Heterotic solitons, Nucl. Phys. B 343 (1990) 167 [Erratum ibid. B 353 (1991) 565] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. J.A. Harvey and A. Strominger, Octonionic superstring solitons, Phys. Rev. Lett. 66 (1991) 549 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. M. Günaydin and H. Nicolai, Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton, Phys. Lett. B 351 (1995) 169 [Addendum ibid. B 376 (1996) 329] [hep-th/9502009] [INSPIRE].

    ADS  Google Scholar 

  26. A. Belavin, A.M. Polyakov, A. Schwartz and Y. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  27. C. Nölle, Homogeneous heterotic supergravity solutions with linear dilaton, J. Phys. A A 45 (2012) 045402 [arXiv:1011.2873] [INSPIRE].

    Article  ADS  Google Scholar 

  28. C.G. Callan Jr., J.A. Harvey and A. Strominger, World sheet approach to heterotic instantons and solitons, Nucl. Phys. B 359 (1991) 611 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. C.G. Callan Jr., J.A. Harvey and A. Strominger, Supersymmetric string solitons, hep-th/9112030 [INSPIRE].

  30. O. Hijazi, A conformal lower bound on the smallest eigenvalue of the Dirac operator and Killing spinors, Commun. Math. Phys. 104 (1986) 151.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. H. Baum, T. Friedrich, R. Grunewald and I. Kath, Twistor and Killing spinors on Riemannian manifolds, Teubner-Verlag, Germany (1991).

    Google Scholar 

  32. S.K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.

    Article  MathSciNet  MATH  Google Scholar 

  33. S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231.

    Article  MathSciNet  MATH  Google Scholar 

  34. K.K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math. 39 (1986) 257.

    Article  MathSciNet  Google Scholar 

  35. K.K. Uhlenbeck and S.-T. Yau, A note on our previous paper, Commun. Pure Appl. Math. 42 (1989) 703.

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002) 303 [math/0102142].

    MathSciNet  MATH  Google Scholar 

  37. I. Agricola, The Srni lectures on non-integrable geometries with torsion, Arch. Math. 42 (2006) 5 [math/0606705].

    MathSciNet  MATH  Google Scholar 

  38. T. Friedrich, I. Kath, A. Moroianu and U. Semmelmann, On nearly parallel G2-structures, J. Geom. Phys. 23 (1997) 259.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. W. Ziller, Examples of riemannian manifolds with non-negative sectional curvature, math/0701389.

  40. L. Verdiani and W. Ziller, Positively curved homogeneous metrics on spheres, Math. Z. 261 (2009) 473 [arXiv:0707.3056] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  41. O. Dearricott, Positive sectional curvature on 3-Sasakian manifolds, Ann. Global Anal. Geom. 25 (2004) 59.

    Article  MathSciNet  MATH  Google Scholar 

  42. C.P. Boyer and K. Galicki, Sasakian geometry, Oxford University Press, Oxford U.K. (2008).

    MATH  Google Scholar 

  43. J.B. Butruille, Homogeneous nearly Kähler manifolds, in Handbook of pseudo-riemannian geometry and supersymmetry, V. Cortés ed., European Mathematical Society (2010), math/0612655.

  44. M. Fernández, S. Ivanov, V. Muñoz and L. Ugarte, Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities, math/0602160.

  45. J. Sparks, Sasaki-Einstein Manifolds, Surveys Diff. Geom. 16 (2011) 265 [arXiv:1004.2461] [INSPIRE].

    MathSciNet  Google Scholar 

  46. D. Fabbri et al., 3D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS 4 /CF T 3, Nucl. Phys. B 577 (2000) 547 [hep-th/9907219] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. L. Castellani, R. D’Auria and P. Fré, SU(3) × SU(2) × U(1) from D = 11 supergravity, Nucl. Phys. B 239 (1984) 610 [INSPIRE].

    Article  ADS  Google Scholar 

  48. J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S 2 × S 3, Adv. Theor. Math. Phys. 8 (2004) 711 [hep-th/0403002].

    MathSciNet  MATH  Google Scholar 

  49. C.P. Boyer and K. Galicki, Sasakian geometry and Einstein metrics on spheres, CRM Proc. Lecture Notes 40 (2006) 47 [math/0505221].

    MathSciNet  Google Scholar 

  50. C.P. Boyer and K. Galicki, Einstein metrics on rational homology spheres, J. Diff. Geom. 74 (2006) 353 [math/0311355].

    MathSciNet  MATH  Google Scholar 

  51. J.P. Gauntlett, D. Martelli, J.F. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys. 8 (2006) 987 [hep-th/0403038] [INSPIRE].

    MathSciNet  Google Scholar 

  52. M. Cvetič, H. Lü, D.N. Page and C. Pope, New Einstein-Sasaki spaces in five and higher dimensions, Phys. Rev. Lett. 95 (2005) 071101 [hep-th/0504225] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. H. Lü, C. Pope and J.F. Vazquez-Poritz, A new construction of Einstein-Sasaki metrics in D ≥ 7,Phys. Rev. D 75 (2007) 026005 [hep-th/0512306] [INSPIRE].

    ADS  Google Scholar 

  54. I. Agricola and T. Friedrich, 3-Sasakian manifolds in dimension seven, their spinors and G 2 structures, J. Geom. Phys. 60 (2010) 326 [arXiv:0812.1651].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].

    MathSciNet  Google Scholar 

  56. J.H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982) 469.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. C. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118 (1994) 109.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. S. Kobayashi and K. Nomizu, Foundations of differential geometry, volume 1, Interscience Publishers, U.S.A. (1963).

  59. T.A. Ivanova, O. Lechtenfeld, A.D. Popov and T. Rahn, Instantons and Yang-Mills Flows on coset spaces, Lett. Math. Phys. 89 (2009) 231 [arXiv:0904.0654] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  60. E. Bergshoeff and M. de Roo, The quartic effective action of the heterotic string and supersymmetry, Nucl. Phys. B 328 (1989) 439 [INSPIRE].

    Article  ADS  Google Scholar 

  61. S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190 [arXiv:0908.2927] [INSPIRE].

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Derek Harland.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harland, D., Nölle, C. Instantons and Killing spinors. J. High Energ. Phys. 2012, 82 (2012). https://doi.org/10.1007/JHEP03(2012)082

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP03(2012)082

Keywords

Navigation