Advertisement

Journal of High Energy Physics

, 2011:103 | Cite as

Yang-Mills instantons on cones and sine-cones over nearly Kähler manifolds

  • Karl-Philip Gemmer
  • Olaf Lechtenfeld
  • Christoph Nölle
  • Alexander D. Popov
Article

Abstract

We present a unified eight-dimensional approach to instanton equations on several seven-dimensional manifolds associated to a six-dimensional homogeneous nearly Kähler manifold. The cone over the sine-cone on a nearly Kähler manifold has holonomy group Spin(7) and can befoliated by submanifolds with either holonomy group G 2, a nearly parallel G 2-structure or a cocalibrated G 2-structure. We show that there is a G 2-instanton on each of these seven-dimensional manifolds which gives rise to a Spin(7)-instanton in eight dimensions. The well-known octonionic instantons on \( {\mathbb{R}^7} \) and \( {\mathbb{R}^8} \) are contained in our construction as the special cases of an instanton on the cone and on the cone over the sine-cone, both over the six-sphere, respectively.

Keywords

Flux compactifications Solitons Monopoles and Instantons Differential and Algebraic Geometry 

References

  1. [1]
    R. Rajaraman, Solitons and Instantons: An introduction to solitons and instantons in quantum field theory, North-Holland, Amsterdam The Netherlands (1982) [SPIRES].zbMATHGoogle Scholar
  2. [2]
    N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, Cambridge U.K. (2004) [SPIRES].Google Scholar
  3. [3]
    A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [SPIRES].ADSMathSciNetGoogle Scholar
  4. [4]
    M.B. Green,J.H. Schwarz and E. Witten, Superstring theory: Volumes 1 & 2, Cambridge University Press, Cambridge U.K. (1987).Google Scholar
  5. [5]
    E. Corrigan, C. Devchand, D.B. Fairlie and J. Nuyts, First Order Equations for Gauge Fields in Spaces of Dimension Greater Than Four, Nucl. Phys. B 214 (1983) 452 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    R.S. Ward, Completely Solvable Gauge Field Equations in Dimension Greater Than Four, Nucl. Phys. B 236 (1984) 381 [SPIRES].CrossRefADSGoogle Scholar
  7. [7]
    S.K. Donaldson, Anti-self-dual Yang-Mills connections on a complex algebraic surface and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    S.K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    K.K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections on stable bundles over compact Kähler manifolds, Commun. Pure Appl. Math. 39 (1986) 257.CrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Mamone Capria and S.M. Salamon, Yang-Mills fields on quaternionic spaces, Nonlinearity 1 (1988) 517.CrossRefzbMATHADSMathSciNetGoogle Scholar
  11. [11]
    R. Reyes Carrión, A generalization of the notion of instanton, Diff. Geom. Appl. 8 (1998) 1 [SPIRES].CrossRefzbMATHGoogle Scholar
  12. [12]
    G. Tian, Gauge theory and calibrated geometry. I, Ann. Math. 151 (2000) 193 [math/0010015].CrossRefzbMATHGoogle Scholar
  13. [13]
    T. Tao and G. Tian, A singularity removal theorem for Yang-Mills fields in higher dimensions, J. Amer. Math. Soc. 17 (2004) 557 [math/0209352].CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The Geometric Universe, Oxford University Press, Oxford U.K. (1998) http://www.ma.ic.ac.uk/∼rpwt/skd.pdf.
  15. [15]
    S. Donaldson and E. Segal, Gauge Theory in higher dimensions, II, arXiv:0902.3239 [SPIRES].
  16. [16]
    A.D. Popov, Non-Abelian Vortices, super-Yang-Mills Theory and Spin(7)-Instantons, Lett. Math. Phys. 92 (2010) 253 [arXiv:0908.3055] [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  17. [17]
    A.D. Popov and R.J. Szabo, Double quiver gauge theory and nearly Kähler flux compactifications, arXiv:1009.3208 [SPIRES].
  18. [18]
    D.B. Fairlie and J. Nuyts, Spherically symmetric solutions of gauge theories in eight dimensions, J. Phys. A 17 (1984) 2867 [SPIRES].ADSMathSciNetGoogle Scholar
  19. [19]
    S. Fubini and H. Nicolai, The octonionic instanton, Phys. Lett. B 155 (1985) 369 [SPIRES].ADSMathSciNetGoogle Scholar
  20. [20]
    T.A. Ivanova and A.D. Popov, Self-dual Yang-Mills fields in D = 7, 8, octonions and Ward equations, Lett. Math. Phys. 24 (1992) 85 [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  21. [21]
    T.A. Ivanova and A.D. Popov, (Anti)selfdual gauge fields in dimension d ≥ 4, Theor. Math. Phys. 94 (1993) 225 [SPIRES].CrossRefMathSciNetGoogle Scholar
  22. [22]
    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 [SPIRES].ADSGoogle Scholar
  23. [23]
    O. Lechtenfeld, A.D. Popov and R.J. Szabo, Noncommutative instantons in higher dimensions, vortices and topological K-cycles, JHEP 12 (2003) 022 [hep-th/0310267] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    A.D. Popov, A.G. Sergeev and M. Wolf, Seiberg-Witten monopole equations on noncommutative R 4, J. Math. Phys. 44 (2003) 4527 [hep-th/0304263] [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  25. [25]
    O. Lechtenfeld, A.D. Popov and R.J. Szabo, SU(3)-equivariant quiver gauge theories and nonabelian vortices, JHEP 08 (2008) 093 [arXiv:0806.2791] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    T. Rahn, Yang-Mills Equations of Motion for the Higgs Sector of SU(3)-Equivariant Quiver Gauge Theories, J. Math. Phys. 51 (2010) 072302 [arXiv:0908.4275] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    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] [SPIRES].ADSGoogle Scholar
  28. [28]
    F.P. Correia, Hermitian Yang-Mills instantons on Calabi-Yau cones, JHEP 12 (2009) 004 [arXiv:0910.1096] [SPIRES].CrossRefGoogle Scholar
  29. [29]
    F.P. Correia, Hermitian Yang-Mills instantons on resolutions of Calabi-Yau cones, JHEP 02 (2011) 054 [arXiv:1009.0526] [SPIRES].CrossRefGoogle Scholar
  30. [30]
    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] [SPIRES].CrossRefzbMATHMathSciNetGoogle Scholar
  31. [31]
    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] [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  32. [32]
    D. Harland and A.D. Popov, Yang-Mills fields in flux compactifications on homogeneous manifolds with SU(4)-structure, arXiv:1005.2837 [SPIRES].
  33. [33]
    I. Bauer, T.A. Ivanova, O. Lechtenfeld and F. Lubbe, Yang-Mills instantons and dyons on homogeneous G 2 -manifolds, JHEP 10 (2010) 044 [arXiv:1006.2388] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  34. [34]
    A. Strominger, Heterotic solitons, Nucl. Phys. B 343 (1990) 167 [Erratum ibid B 353 (1991) 565] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  35. [35]
    C.G. Callan Jr., J.A. Harvey and A. Strominger, World sheet approach to heterotic instantons and solitons, Nucl. Phys. B 359 (1991) 611 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    C.G. Callan Jr., J.A. Harvey and A. Strominger, Worldbrane actions for string solitons, Nucl. Phys. B 367 (1991) 60 [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    E. Witten, Small Instantons in String Theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    J. Polchinski and E. Witten, Evidence for heterotic-type I string duality, Nucl. Phys. B 460 (1996) 525, [hep-th/9510169] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  39. [39]
    M.R. Douglas, Branes within branes, hep-th/9512077 [SPIRES].
  40. [40]
    M.R. Douglas, Gauge fields and D-branes, J. Geom. Phys. 28 (1998) 255 [hep-th/9604198] [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  41. [41]
    J.A. Harvey and A. Strominger, Octonionic superstring solitons, Phys. Rev. Lett. 66 (1991) 549 [SPIRES].CrossRefzbMATHADSMathSciNetGoogle Scholar
  42. [42]
    T.A. Ivanova, Octonions, selfduality and strings, Phys. Lett. B 315 (1993) 277 [SPIRES].ADSMathSciNetGoogle Scholar
  43. [43]
    D. Harland and C. Nölle, Instantons and Killing spinors, arXiv:1109.3552 [SPIRES].
  44. [44]
    C. Bär, Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993) 509.CrossRefzbMATHADSGoogle Scholar
  45. [45]
    F. Xu, Geometry of SU(3) manifolds, Ph.D. Thesis, Duke University, Durham U.S.A. (2008).Google Scholar
  46. [46]
    R. Cleyton and A. Swann, Cohomogeneity-one G2-structures, J. Geom. Phys. 44 (2002) 202 [math/0111056].CrossRefzbMATHADSMathSciNetGoogle Scholar
  47. [47]
    A. Bilal and S. Metzger, Compact weak G 2 -manifolds with conical singularities, Nucl. Phys. B 663 (2003) 343 [hep-th/0302021] [SPIRES].CrossRefADSMathSciNetGoogle Scholar
  48. [48]
    C.P. Boyer and K. Galicki, Sasakian geometry, holonomy, and supersymmetry, math/0703231 [SPIRES].
  49. [49]
    J.-B.Butruille, Homogeneous nearly Kähler manifolds, math/0612655 [SPIRES].
  50. [50]
    J. Wolf, Spaces of constant curvature, McGraw-Hill, New York U.S.A. (1967).zbMATHGoogle Scholar
  51. [51]
    J.A. Wolf and A. Gray, Homogeneous spaces defined by Lie group automorphisms. I, J. Diff. Geom. 2 (1968) 77.zbMATHMathSciNetGoogle Scholar
  52. [52]
    T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002) 303 [math/0102142].zbMATHMathSciNetGoogle Scholar
  53. [53]
    M. Fernández, S. Ivanov, V. Muñoz and L. Ugarte, Nearly hypo structures and compact Nearly Kähler 6-manifolds with conical singularities, math 0602160 [SPIRES].
  54. [54]
    F.M. Cabrera, SU(3)-Structures on hypersurfaces of manifolds with G 2 -structure, Monatsh. Math. 248 (2006) 29 [math/0410610].CrossRefMathSciNetGoogle Scholar
  55. [55]
    T. Friedrich, Nearly Kaehler and nearly parallel G 2 -structures on spheres, math/0509146.

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Karl-Philip Gemmer
    • 1
  • Olaf Lechtenfeld
    • 1
    • 2
  • Christoph Nölle
    • 1
  • Alexander D. Popov
    • 3
  1. 1.Institut für Theoretische PhysikLeibniz Universität HannoverHannoverGermany
  2. 2.Centre for Quantum Engineering and Space-Time ResearchLeibniz Universität HannoverHannoverGermany
  3. 3.Bogoliubov Laboratory of Theoretical PhysicsJINRDubnaRussia

Personalised recommendations