Numerical Hermitian Yang-Mills connections and vector bundle stability in heterotic theories

  • Lara B. Anderson
  • Volker Braun
  • Robert L. Karp
  • Burt A. Ovrut


A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.


Superstrings and Heterotic Strings Differential and Algebraic Geometry Superstring Vacua 


  1. [1]
    D.J. Gross, J.A. Harvey, E.J. Martinec and R. Rohm, The Heterotic String, Phys. Rev. Lett. 54 (1985) 502 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    P. Hořava and E. Witten, Heterotic and type-I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209] [SPIRES].ADSGoogle Scholar
  4. [4]
    P. Hořava and E. Witten, Eleven-Dimensional Supergravity on a Manifold with Boundary, Nucl. Phys. B 475 (1996) 94 [hep-th/9603142] [SPIRES].ADSGoogle Scholar
  5. [5]
    E. Witten, Strong Coupling Expansion Of Calabi-Yau Compactification, Nucl. Phys. B 471 (1996) 135 [hep-th/9602070] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  7. [7]
    R.Y. Donagi, Principal bundles on elliptic fibrations, alg-geom/9702002.
  8. [8]
    A. Lukas, B.A. Ovrut and D. Waldram, Non-standard embedding and five-branes in heterotic M-theory, Phys. Rev. D 59 (1999) 106005 [hep-th/9808101] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    R. Donagi, B.A. Ovrut, T. Pantev and D. Waldram, Standard-model bundles, Adv. Theor. Math. Phys. 5 (2002) 563 [math/0008010]. = MATH/0008010;
  10. [10]
    R. Donagi, A. Lukas, B.A. Ovrut and D. Waldram, Holomorphic vector bundles and non-perturbative vacua in M-theory, JHEP 06 (1999) 034 [hep-th/9901009] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    R. Blumenhagen, S. Moster and T. Weigand, Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds, Nucl. Phys. B 751 (2006) 186 [hep-th/0603015] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    R. Blumenhagen, S. Moster, R. Reinbacher and T. Weigand, Massless spectra of three generation U(N) heterotic string vacua, JHEP 05 (2007) 041 [hep-th/0612039] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    A. Lukas, B.A. Ovrut and D. Waldram, On the four-dimensional effective action of strongly coupled heterotic string theory, Nucl. Phys. B 532 (1998) 43 [hep-th/9710208] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    A. Lukas, B.A. Ovrut, K.S. Stelle and D. Waldram, The universe as a domain wall, Phys. Rev. D 59 (1999) 086001 [hep-th/9803235] [SPIRES].MathSciNetADSGoogle Scholar
  15. [15]
    A. Lukas, B.A. Ovrut, K.S. Stelle and D. Waldram, Heterotic M-theory in five dimensions, Nucl. Phys. B 552 (1999) 246 [hep-th/9806051] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    R. Donagi, B.A. Ovrut, T. Pantev and D. Waldram, Standard models from heterotic M-theory, Adv. Theor. Math. Phys. 5 (2002) 93 [hep-th/9912208] [SPIRES].MathSciNetGoogle Scholar
  17. [17]
    R.Y. Donagi, J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Visible branes with negative tension in heterotic M-theory, JHEP 11 (2001) 041 [hep-th/0105199] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    R. Donagi, Y.-H. He, B.A. Ovrut and R. Reinbacher, Moduli dependent spectra of heterotic compactifications, Phys. Lett. B 598 (2004) 279 [hep-th/0403291] [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    V. Braun, B.A. Ovrut, T. Pantev and R. Reinbacher, Elliptic Calabi-Yau threefolds with Z(3) × Z(3) Wilson lines, JHEP 12 (2004) 062 [hep-th/0410055] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    R. Donagi, Y.-H. He, B.A. Ovrut and R. Reinbacher, The spectra of heterotic standard model vacua, JHEP 06 (2005) 070 [hep-th/0411156] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  21. [21]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [SPIRES].MathSciNetADSGoogle Scholar
  22. [22]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, Vector Bundle Extensions, Sheaf Cohomology and the Heterotic Standard Model, Adv. Theor. Math. Phys. 10 (2006) 4 [hep-th/0505041] [SPIRES].MathSciNetGoogle Scholar
  23. [23]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, The exact MSSM spectrum from string theory, JHEP 05 (2006) 043 [hep-th/0512177] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    V. Bouchard and R. Donagi, An SU(5) heterotic standard model, Phys. Lett. B 633 (2006) 783 [hep-th/0512149] [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    L.B. Anderson, J. Gray, Y.-H. He and A. Lukas, Exploring Positive Monad Bundles And A New Heterotic Standard Model, JHEP 02 (2010) 054 [arXiv:0911.1569] [SPIRES].CrossRefGoogle Scholar
  26. [26]
    M. Ambroso and B.A. Ovrut, The B-L/Electroweak Hierarchy in Smooth Heterotic Compactifications, Int. J. Mod. Phys. A 25 (2010) 2631 [arXiv:0910.1129] [SPIRES].ADSGoogle Scholar
  27. [27]
    M. Ambroso and B. Ovrut, The B-L/Electroweak Hierarchy in Heterotic String and M-theory, JHEP 10 (2009) 011 [arXiv:0904.4509] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  28. [28]
    P. Candelas and S. Kalara, Yukawa couplings for a three generation superstring compactification, Nucl. Phys. B 298 (1988) 357 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  29. [29]
    P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [SPIRES].CrossRefADSGoogle Scholar
  30. [30]
    B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys. 173 (1995) 559 [hep-th/9402119] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  31. [31]
    V. Braun, Y.-H. He and B.A. Ovrut, Yukawa couplings in heterotic standard models, JHEP 04 (2006) 019 [hep-th/0601204] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    R. Donagi, R. Reinbacher and S.-T. Yau, Yukawa couplings on quintic threefolds, hep-th/0605203 [SPIRES].
  33. [33]
    L.B. Anderson, J. Gray, D. Grayson, Y.-H. He and A. Lukas, Yukawa Couplings in Heterotic Compactification, Commun. Math. Phys. 297 (2010) 95 [arXiv:0904.2186] [SPIRES].CrossRefADSGoogle Scholar
  34. [34]
    M.B. Green, J.H. Schwarz, and E. Witten, Superstring Theory. vol.1: Introduction, in Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, Cambridge U.K. (1987).Google Scholar
  35. [35]
    S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978) 339.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    K. Uhlenbeck and S.-T. Yau., On the existence of Hermitian Yang-Mills connections in stable bundles, Comm. Pure App. Math. 39 (1986) 257.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    S. Donaldson, Anti Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundles, Proc. London Math. Soc. 3 (1985) 1.CrossRefMathSciNetGoogle Scholar
  38. [38]
    S.K. Donaldson, Scalar curvature and projective embeddings. II, Q. J. Math. 56 (2005) 345 math/0407534.
  39. [39]
    S.K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001) 479.MATHMathSciNetGoogle Scholar
  40. [40]
    S.K. Donaldson, Some numerical results in complex differential geometry, math/0512625.
  41. [41]
    G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990) 99.MATHMathSciNetGoogle Scholar
  42. [42]
    X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005) 253.MATHMathSciNetGoogle Scholar
  43. [43]
    M.R. Douglas, R.L. Karp, S. Lukic and R. Reinbacher, Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic, JHEP 12 (2007) 083 [hep-th/0606261] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  44. [44]
    M. Headrick and T. Wiseman, Numerical Ricci-flat metrics on K3, Classical Quantum Gravity 22 (2005) 4931 [hep-th/0506129] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  45. [45]
    C. Doran, M. Headrick, C.P. Herzog, J. Kantor and T. Wiseman, Numerical Kähler-Einstein metric on the third del Pezzo, Commun. Math. Phys. 282 (2008) 357 [hep-th/0703057] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  46. [46]
    M. Headrick and A. Nassar, Energy functionals for Calabi-Yau metrics, arXiv:0908.2635 [SPIRES].
  47. [47]
    M.R. Douglas and S. Klevtsov, Black holes and balanced metrics, arXiv:0811.0367 [SPIRES].
  48. [48]
    V. Braun, T. Brelidze, M.R. Douglas and B.A. Ovrut, Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 05 (2008) 080 [arXiv:0712.3563] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  49. [49]
    V. Braun, T. Brelidze, M.R. Douglas and B.A. Ovrut, Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds, JHEP 07 (2008) 120 [arXiv:0805.3689] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  50. [50]
    M.R. Douglas, R.L. Karp, S. Lukic and R. Reinbacher, Numerical Calabi-Yau metrics, J. Math. Phys. 49 (2008) 032302 [hep-th/0612075] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  51. [51]
    C. Okonek, M. Schneider and Heinz Spindler, Vector Bundles on Complex Projective Spaces, Birkhauser Verlag, Boston U.S.A. (1988).Google Scholar
  52. [52]
    L.B. Anderson, Y.-H. He and A. Lukas, Heterotic compactification, an algorithmic approach, JHEP 07 (2007) 049 [hep-th/0702210] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  53. [53]
    L.B. Anderson, Y.-H. He and A. Lukas, Monad Bundles in Heterotic String Compactifications, JHEP 07 (2008) 104 [arXiv:0805.2875] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  54. [54]
    D. Huybrechts and M. Lehn, The geometry of the Moduli Spaces of Sheaves, Aspects of Mathematics, vol. 31 (1997).Google Scholar
  55. [55]
    E.R. Sharpe, Kähler cone substructure, Adv. Theor. Math. Phys. 2 (1999) 1441 [hep-th/9810064] [SPIRES].MathSciNetGoogle Scholar
  56. [56]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, The Edge Of Supersymmetry: Stability Walls in Heterotic Theory, Phys. Lett. B 677 (2009) 190 [arXiv:0903.5088] [SPIRES].MathSciNetADSGoogle Scholar
  57. [57]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stability Walls in Heterotic Theories, JHEP 09 (2009) 026 [arXiv:0905.1748] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  58. [58]
    L.B. Anderson, J. Gray and B. Ovrut, Yukawa Textures From Heterotic Stability Walls, JHEP 05 (2010) 086 [1001.2317] [SPIRES].CrossRefGoogle Scholar
  59. [59]
    Y. Sano, Numerical algorithm for finding balanced metrics, Osaka J. Math. 43 (2006) 679.MATHMathSciNetGoogle Scholar
  60. [60]
    M. Headrick and T. Wiseman, Numerical Ricci-flat metrics on K3, Class. Quant. Grav. 22 (2005) 4931 [hep-th/0506129] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  61. [61]
    P. Griffiths and J. Harris, Principles of algebraic geometry, in Pure and Applied Mathematics, Wiley-Interscience (John Wiley & Sons), New York U.S.A. (1978).Google Scholar
  62. [62]
    L.B. Anderson, Heterotic and M-theory Compactifications for String Phenomenology, arXiv:0808.3621 [SPIRES].
  63. [63]
    B.A. Ovrut, T. Pantev and J. Park, Small instanton transitions in heterotic M-theory, JHEP 05 (2000) 045 [hep-th/0001133] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  64. [64]
    E. Buchbinder, R. Donagi and B.A. Ovrut, Vector bundle moduli and small instanton transitions, JHEP 06 (2002) 054 [hep-th/0202084] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  65. [65]
    J. Gray, Y.-H. He and A. Lukas, Algorithmic algebraic geometry and flux vacua, JHEP 09 (2006) 031 [hep-th/0606122] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  66. [66]
    J. Gray, Y.-H. He, A. Ilderton and A. Lukas, STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology, Comput. Phys. Commun. 180 (2009) 107 [arXiv:0801.1508] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  67. [67]
    G.-M. Greuel, V. Levandovskyy, and H. Schönemann, Singular::Plural 2.1, A Computer Algebra System for Noncommutative Polynomial Algebras, Centre for Computer Algebra, University of Kaiserslautern, 2003
  68. [68]
    D. Kaledin and M. Verbitsky, Non-Hermitian Yang-Mills connections, Selecta Math. 4 (1998) 279 [alg-geom/9606019].
  69. [69]
    L.B. Anderson, J. Gray, and B.A. Ovrut, Stability walls and the connected web of heterotic vacua, to appear.Google Scholar
  70. [70]
    L.B. Anderson, V. Braun, and B.A. Ovrut, Numerical connections and Kähler cone substructure, to appear.Google Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • Lara B. Anderson
    • 1
  • Volker Braun
    • 2
  • Robert L. Karp
    • 3
  • Burt A. Ovrut
    • 1
  1. 1.Department of PhysicsUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Dublin Institute for Advanced StudiesDublin 4Ireland
  3. 3.Department of PhysicsVirginia Polytechnic Institute and State UniversityBlacksburgU.S.A.

Personalised recommendations