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Journal of High Energy Physics

, 2018:179 | Cite as

Finite deformations from a heterotic superpotential: holomorphic Chern-Simons and an L algebra

  • Anthony Ashmore
  • Xenia de la Ossa
  • Ruben Minasian
  • Charles Strickland-Constable
  • Eirik Eik Svanes
Open Access
Regular Article - Theoretical Physics
  • 100 Downloads

Abstract

We consider finite deformations of the Hull-Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a thirdorder Maurer-Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira-Spencer and holomorphic Chern-Simons theory. The supersymmetric locus of this action is described by an L3 algebra.

Keywords

Superstring Vacua Flux compactifications Superstrings and Heterotic Strings Differential and Algebraic Geometry 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    V. Braun, P. Candelas, R. Davies and R. Donagi, The MSSM spectrum from (0, 2)-deformations of the heterotic standard embedding, JHEP 05 (2012) 127 [arXiv:1112.1097] [INSPIRE].
  2. [2]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A standard model from the E 8 × E 8 heterotic superstring, JHEP 06 (2005) 039 [hep-th/0502155] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    V. Braun, Y.-H. He, B.A. Ovrut and T. Pantev, A heterotic standard model, Phys. Lett. B 618 (2005) 252 [hep-th/0501070] [INSPIRE].
  4. [4]
    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] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing the complex structure in heterotic Calabi-Yau vacua, JHEP 02 (2011) 088 [arXiv:1010.0255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Two hundred heterotic standard models on smooth Calabi-Yau threefolds, Phys. Rev. D 84 (2011) 106005 [arXiv:1106.4804] [INSPIRE].
  7. [7]
    L.B. Anderson, J. Gray, A. Lukas and E. Palti, Heterotic line bundle standard models, JHEP 06 (2012) 113 [arXiv:1202.1757] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stabilizing all geometric moduli in heterotic Calabi-Yau vacua, Phys. Rev. D 83 (2011) 106011 [arXiv:1102.0011] [INSPIRE].
  9. [9]
    C. Hull, Compactifications of the heterotic superstring, Phys. Lett. B 178 (1986) 357.Google Scholar
  10. [10]
    A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (1986) 253 [INSPIRE].
  11. [11]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, Stability walls in heterotic theories, JHEP 09 (2009) 026 [arXiv:0905.1748] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    L.B. Anderson, J. Gray, A. Lukas and B. Ovrut, The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications, JHEP 10 (2011) 032 [arXiv:1107.5076] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    I.V. Melnikov and E. Sharpe, On marginal deformations of (0, 2) non-linear σ-models, Phys. Lett. B 705 (2011) 529 [arXiv:1110.1886] [INSPIRE].
  14. [14]
    X. de la Ossa and E.E. Svanes, Holomorphic bundles and the moduli space of N = 1 supersymmetric heterotic compactifications, JHEP 10 (2014) 123 [arXiv:1402.1725] [INSPIRE].
  15. [15]
    L.B. Anderson, J. Gray and E. Sharpe, Algebroids, heterotic moduli spaces and the Strominger system, JHEP 07 (2014) 037 [arXiv:1402.1532] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    M. Kreuzer, J. McOrist, I.V. Melnikov and M.R. Plesser, (0, 2) deformations of linear σ-models, JHEP 07 (2011) 044 [arXiv:1001.2104] [INSPIRE].
  17. [17]
    M. Bertolini, I.V. Melnikov and M.R. Plesser, Massless spectrum for hybrid CFTs, Proc. Symp. Pure Math. 88 (2014) 221 [arXiv:1402.1751] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Bertolini, Moduli space of (0, 2) conformal field theories, Ph.D. thesis, Duke University, Durham, U.S.A. (2016).Google Scholar
  19. [19]
    M. Bertolini and M.R. Plesser, (0, 2) hybrid models, JHEP 09 (2018) 067 [arXiv:1712.04976] [INSPIRE].
  20. [20]
    P. Candelas, X. de la Ossa and J. McOrist, A metric for heterotic moduli, Commun. Math. Phys. 356 (2017) 567 [arXiv:1605.05256] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J. McOrist, On the effective field theory of heterotic vacua, Lett. Math. Phys. 108 (2018) 1031 [arXiv:1606.05221] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  22. [22]
    E. Witten, New issues in manifolds of SU(3) holonomy, Nucl. Phys. B 268 (1986) 79 [INSPIRE].
  23. [23]
    M.F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957) 181.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    L. Huang, On joint moduli spaces, Math. Ann. 302 (1995) 61.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    S. Gurrieri, A. Lukas and A. Micu, Heterotic on half-flat, Phys. Rev. D 70 (2004) 126009 [hep-th/0408121] [INSPIRE].
  26. [26]
    X. de la Ossa, E. Hardy and E.E. Svanes, The heterotic superpotential and moduli, JHEP 01 (2016) 049 [arXiv:1509.08724] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Ashmore et al., Deformations of the Hull-Strominger system: holomorphic Courant algebroids and effective field theory, in preparation.Google Scholar
  28. [28]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supersymmetric backgrounds and generalised special holonomy, Class. Quant. Grav. 33 (2016) 125026 [arXiv:1411.5721] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    M. Gualtieri, Generalized Kähler geometry, arXiv:1007.3485 [INSPIRE].
  30. [30]
    M. Garcia-Fernandez, R. Rubio, C. Shahbazi and C. Tipler, Canonical metrics on holomorphic Courant algebroids, arXiv:1803.01873 [INSPIRE].
  31. [31]
    M. Garcia-Fernandez, R. Rubio and C. Tipler, Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, arXiv:1503.07562 [INSPIRE].
  32. [32]
    S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685 (2010) 190 [arXiv:0908.2927] [INSPIRE].
  33. [33]
    S. Weinberg, Nonrenormalization theorems in nonrenormalizable theories, Phys. Rev. Lett. 80 (1998) 3702 [hep-th/9803099] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    S. Weinberg, The quantum theory of fields. Volume 3: supersymmetry, Cambridge University Press, Cambridge U.K. (2013).Google Scholar
  35. [35]
    D. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).Google Scholar
  36. [36]
    A. Sen, (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory, Nucl. Phys. B 278 (1986) 289.Google Scholar
  37. [37]
    M. Graña, J. Louis, A. Sim and D. Waldram, E 7(7) formulation of N = 2 backgrounds, JHEP 07 (2009) 104 [arXiv:0904.2333] [INSPIRE].
  38. [38]
    M. Graña, J. Louis and D. Waldram, Hitchin functionals in N = 2 supergravity, JHEP 01 (2006) 008 [hep-th/0505264] [INSPIRE].
  39. [39]
    A.N. Todorov, The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds. I, Commun. Math. Phys. 126 (1989) 325.Google Scholar
  40. [40]
    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    R. Dijkgraaf, S. Gukov, A. Neitzke and C. Vafa, Topological M-theory as unification of form theories of gravity, Adv. Theor. Math. Phys. 9 (2005) 603 [hep-th/0411073] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
  43. [43]
    A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014) 160 [arXiv:1407.7542] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    M. Garcia-Fernandez, Torsion-free generalized connections and heterotic supergravity, Commun. Math. Phys. 332 (2014) 89 [arXiv:1304.4294] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    A. Gray and L.M. Hervella, The sixteen classes of almost hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. IV. 123 (1980) 35.MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    I.T. Jardine and C. Quigley, Conformal invariance of (0, 2) σ-models on Calabi-Yau manifolds, JHEP 03 (2018) 090 [arXiv:1801.04336] [INSPIRE].
  47. [47]
    M. Gualtieri, M. Matviichuk and G. Scott, Deformation of Dirac structures via L algebras, arXiv:1702.08837 [INSPIRE].
  48. [48]
    Y. Fregier and M. Zambon, Simultaneous deformations and Poisson geometry, arXiv:1202.2896.
  49. [49]
    F. Keller and S. Waldmann, Formal deformations of Dirac structures, J. Geom. Phys. 57 (2007) 1015 [math/0606674].
  50. [50]
    D. Roytenberg, A note on quasi Lie bialgebroids and twisted Poisson manifolds, Lett. Math. Phys. 61 (2002) 123 [math/0112152] [INSPIRE].
  51. [51]
    B. Zwiebach, Closed string field theory: quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
  52. [52]
    O. Hohm and B. Zwiebach, L algebras and field theory, Fortsch. Phys. 65 (2017) 1700014 [arXiv:1701.08824] [INSPIRE].
  53. [53]
    D. Roytenberg and A. WEinstein, Courant algebroids and strongly homotopy Lie algebras, math/9802118 [INSPIRE].
  54. [54]
    C.A. Weibel, An introduction to homological algebra, Cambridge University Press, Cambridge U.K. (1994).Google Scholar
  55. [55]
    A. Deser, M.A. Heller and C. Sämann, Extended Riemannian geometry II: local heterotic double field theory, JHEP 04 (2018) 106 [arXiv:1711.03308] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    N.A. Nekrasov, Lectures on curved beta-gamma systems, pure spinors and anomalies, hep-th/0511008 [INSPIRE].
  57. [57]
    R. Thomas, Gauge theory on Calabi-Yau manifolds, Ph.D. thesis, University of Oxford, Oxford U.K. (1997).Google Scholar
  58. [58]
    S. Donaldson and R. Thomas, Gauge theory in higher dimensions, in The geometric universe, S.A. Huggett et al. eds., Oxford University Press, Oxford U.K. (1998).Google Scholar
  59. [59]
    R.P. Thomas, A Holomorphic Casson invariant for Calabi-Yau three folds and bundles on K3 fibrations, J. Diff. Geom. 54 (2000) 367[math/9806111] [INSPIRE].
  60. [60]
    O. Hohm and H. Samtleben, Leibniz-Chern-Simons theory and phases of exceptional field theory, arXiv:1805.03220 [INSPIRE].
  61. [61]
    S. Giusto, C. Imbimbo and D. Rosa, Holomorphic Chern-Simons theory coupled to off-shell Kodaira-Spencer gravity, JHEP 10 (2012) 192 [arXiv:1207.6121] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    K. Costello and S. Li, Quantization of open-closed BCOV theory, I, arXiv:1505.06703 [INSPIRE].
  63. [63]
    E. Witten, Quantization of Chern-Simons gauge theory with complex gauge group, Commun. Math. Phys. 137 (1991) 29 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    M. Alexandrov, A. Schwarz, O. Zaboronsky and M. Kontsevich, The Geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A 12 (1997) 1405 [hep-th/9502010] [INSPIRE].
  66. [66]
    N. Ikeda, Lectures on AKSZ σ-models for physicists, in the proceedings of the Workshop on Strings, Membranes and Topological Field Theory, March 5-7, Tohoku University, Japan (2017), arXiv:1204.3714 [INSPIRE].
  67. [67]
    E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  68. [68]
    E. Witten, Two-dimensional models with (0, 2) supersymmetry: perturbative aspects, Adv. Theor. Math. Phys. 11 (2007) 1 [hep-th/0504078] [INSPIRE].
  69. [69]
    A. Kapustin, Chiral de Rham complex and the half-twisted σ-model, hep-th/0504074 [INSPIRE].
  70. [70]
    A.M. Zeitlin, Perturbed beta-gamma systems and complex geometry, Nucl. Phys. B 794 (2008) 381 [arXiv:0708.0682] [INSPIRE].
  71. [71]
    V. Gorbounov, O. Gwilliam and B. Williams, Chiral differential operators via Batalin-Vilkovisky quantization, arXiv:1610.09657 [INSPIRE].
  72. [72]
    O. Gwilliam and B. Williams, The holomorphic bosonic string, arXiv:1711.05823 [INSPIRE].
  73. [73]
    A. Adams, J. Distler and M. Ernebjerg, Topological heterotic rings, Adv. Theor. Math. Phys. 10 (2006) 657 [hep-th/0506263] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    A.M. Zeitlin, BRST, generalized Maurer-Cartan equations and CFT, Nucl. Phys. B 759 (2006) 370 [hep-th/0610208] [INSPIRE].
  75. [75]
    L.J. Mason and D. Skinner, Heterotic twistor-string theory, Nucl. Phys. B 795 (2008) 105 [arXiv:0708.2276] [INSPIRE].
  76. [76]
    J. McOrist, The revival of (0, 2) linear σ-models, Int. J. Mod. Phys. A 26 (2011) 1 [arXiv:1010.4667] [INSPIRE].
  77. [77]
    I.V. Melnikov, C. Quigley, S. Sethi and M. Stern, Target spaces from chiral gauge theories, JHEP 02 (2013) 111 [arXiv:1212.1212] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    A.M. Zeitlin, Beltrami-Courant differentials and G -algebras, Adv. Theor. Math. Phys. 19 (2015) 1249 [arXiv:1404.3069] [INSPIRE].
  79. [79]
    E. Casali et al., New ambitwistor string theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    A. Clarke, M. Garcia-Fernandez and C. Tipler, Moduli of G 2 structures and the Strominger system in dimension 7, arXiv:1607.01219 [INSPIRE].
  81. [81]
    X. de la Ossa, M. Larfors and E.E. Svanes, The infinitesimal moduli space of heterotic G 2 systems, Commun. Math. Phys. 360 (2018) 727 [arXiv:1704.08717] [INSPIRE].
  82. [82]
    M.-A. Fiset, C. Quigley and E.E. Svanes, Marginal deformations of heterotic G 2 σ-models, JHEP 02 (2018) 052 [arXiv:1710.06865] [INSPIRE].
  83. [83]
    N.J. Hitchin, Stable forms and special metrics, math/0107101 [INSPIRE].
  84. [84]
    N.J. Hitchin, Lectures on special Lagrangian submanifolds, AMS/IP Stud. Adv. Math. 23 (2001) 151 [math/9907034] [INSPIRE].
  85. [85]
    E. Witten, World sheet corrections via D instantons, JHEP 02 (2000) 030 [hep-th/9907041] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    H. Nicolai, D = 11 supergravity with local SO(16) invariance, Phys. Lett. B 187 (1987) 316 [INSPIRE].
  87. [87]
    M. Graña, J. Louis and D. Waldram, SU(3) × SU(3) compactification and mirror duals of magnetic fluxes, JHEP 04 (2007) 101 [hep-th/0612237] [INSPIRE].
  88. [88]
    A. Ashmore and D. Waldram, Exceptional Calabi-Yau spaces: the geometry of \( \mathcal{N}=2 \) backgrounds with flux, Fortsch. Phys. 65 (2017) 1600109 [arXiv:1510.00022] [INSPIRE].
  89. [89]
    A. Ashmore, M. Petrini and D. Waldram, The exceptional generalised geometry of supersymmetric AdS flux backgrounds, JHEP 12 (2016) 146 [arXiv:1602.02158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    W. Siegel, Manifest duality in low-energy superstrings, hep-th/9308133 [INSPIRE].
  92. [92]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  93. [93]
    M. Graña, R. Minasian, M. Petrini and A. Tomasiello, Supersymmetric backgrounds from generalized Calabi-Yau manifolds, JHEP 08 (2004) 046 [hep-th/0406137] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  94. [94]
    P. Gauduchon, La l-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984) 495.MathSciNetCrossRefzbMATHGoogle Scholar
  95. [95]
    S. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986) 257.MathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    S.T. Yau and J. Li, Hermitian-Yang-Mills connections on non-Kähler manifolds, World Scientific, London U.K. (1987).Google Scholar
  98. [98]
    X. de la Ossa and E.E. Svanes, Connections, field redefinitions and heterotic supergravity, JHEP 12 (2014) 008 [arXiv:1409.3347] [INSPIRE].CrossRefGoogle Scholar
  99. [99]
    A. Aeppli, On the cohomology structure of Stein manifolds, in the proceedings of the Minnesota Conference on Complex Analysis (COCA), March 16-21, University of Minnesota, Minneapolis U.S.A. (1965).Google Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordU.K.
  2. 2.Institut de Physique ThéoriqueUniversité Paris Saclay, CEA, CNRSGif-sur-YvetteFrance
  3. 3.School of MathematicsUniversity of Edinburgh, James Clerk Maxwell BuildingEdinburghU.K.
  4. 4.School of Physics, Astronomy and MathematicsUniversity of HertfordshireHatfieldU.K.
  5. 5.Department of PhysicsKing’s College LondonLondonU.K.
  6. 6.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

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