Towards generalized mirror symmetry for twisted connected sum G2 manifolds

Abstract

We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted connected sum G2 manifolds. For a given G2 manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of G2 manifolds, the other is a duality between Type II strings with different chiralities for another pair of mirror manifolds. We clarify the role of the B-field in the construction, and check that the corresponding massless spectra are respected by the generalized mirror maps. We discuss hints towards a homological version based on BPS spectroscopy. We provide several novel examples of smooth, as well as singular, mirror G2 backgrounds via pairs of dual projecting tops. We test our conjectures against a Joyce orbifold example, where we reproduce, using our geometrical methods, the known mirror maps that arise from the SCFT worldsheet perspective. Along the way, we discuss non-Abelian gauge symmetries, and argue for the generation of the Affleck-Harvey-Witten superpotential in the pure SYM case.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. [2]

    S.L. Shatashvili and C. Vafa, Superstrings and manifold of exceptional holonomy, Selecta Math. 1 (1995) 347 [hep-th/9407025] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    B.S. Acharya, Dirichlet Joyce manifolds, discrete torsion and duality, Nucl. Phys. B 492 (1997) 591 [hep-th/9611036] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    J.M. Figueroa-O’Farrill, A note on the extended superconformal algebras associated with manifolds of exceptional holonomy, Phys. Lett. B 392 (1997) 77 [hep-th/9609113] [INSPIRE].

    ADS  Article  Google Scholar 

  5. [5]

    B.S. Acharya, On mirror symmetry for manifolds of exceptional holonomy, Nucl. Phys. B 524 (1998) 269 [hep-th/9707186] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    B.S. Acharya, On realizing N = 1 super Yang-Mills in M-theory, hep-th/0011089 [INSPIRE].

  7. [7]

    S. Gukov, S.-T. Yau and E. Zaslow, Duality and fibrations on G 2 manifolds, hep-th/0203217 [INSPIRE].

  8. [8]

    R. Roiban, C. Romelsberger and J. Walcher, Discrete torsion in singular G 2 manifolds and real LG, Adv. Theor. Math. Phys. 6 (2003) 207 [hep-th/0203272] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    M.R. Gaberdiel and P. Kaste, Generalized discrete torsion and mirror symmetry for G 2 manifolds, JHEP 08 (2004) 001 [hep-th/0401125] [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    J. de Boer, A. Naqvi and A. Shomer, The topological G 2 string, Adv. Theor. Math. Phys. 12 (2008) 243 [hep-th/0506211] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    K. Becker, D. Robbins and E. Witten, The α expansion on a compact manifold of exceptional holonomy, JHEP 06 (2014) 051 [arXiv:1404.2460] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003) 125.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    A. Corti, M. Haskins, J. Nordström and T. Pacini, Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds, Geom. Topol. 17 (2013) 1955.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    A. Corti, M. Haskins, J. Nordström and T. Pacini, G2 -manifolds and associative submanifolds via semi-Fano 3-folds, Duke Math. J. 164 (2015) 1971 [arXiv:1207.4470] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    J. Halverson and D.R. Morrison, The landscape of M-theory compactifications on seven-manifolds with G 2 holonomy, JHEP 04 (2015) 047 [arXiv:1412.4123] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    J. Halverson and D.R. Morrison, On gauge enhancement and singular limits in G 2 compactifications of M-theory, JHEP 04 (2016) 100 [arXiv:1507.05965] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  17. [17]

    A.P. Braun and M. Del Zotto, Mirror symmetry for G 2 -manifolds: twisted connected sums and dual tops, JHEP 05 (2017) 080 [arXiv:1701.05202] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    T.C. d.C. Guio, H. Jockers, A. Klemm and H.-Y. Yeh, Effective action from M-theory on twisted connected sum G 2 -manifolds, arXiv:1702.05435 [INSPIRE].

  19. [19]

    A.P. Braun and S. Schäfer-Nameki, Compact, singular G 2 -holonomy manifolds and M/Heterotic/F-theory duality, arXiv:1708.07215 [INSPIRE].

  20. [20]

    D.D. Joyce, Compact Riemannian 7-manifolds with holonomy G 2 . I, J. Diff. Geom. 43 (1996) 291.

  21. [21]

    D.D. Joyce, Compact riemannian 7-manifolds with holonomy g 2 . II, J. Diff. Geom. 43 (1996) 329.

  22. [22]

    D. Joyce, Compact manifolds with special holonomy, Oxford mathematical monographs, Oxford University Press, Oxford U.K. (2000).

    Google Scholar 

  23. [23]

    A.P. Braun, Tops as building blocks for G 2 manifolds, JHEP 10 (2017) 083 [arXiv:1602.03521] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    I. Affleck, J.A. Harvey and E. Witten, Instantons and (super)symmetry breaking in (2 + 1)-dimensions, Nucl. Phys. B 206 (1982) 413 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. [25]

    S.H. Katz and C. Vafa, Geometric engineering of N = 1 quantum field theories, Nucl. Phys. B 497 (1997) 196 [hep-th/9611090] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    J. de Boer, K. Hori, Y. Oz and Z. Yin, Branes and mirror symmetry in N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 502 (1997) 107 [hep-th/9702154] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    H. Nicolai and H. Samtleben, Chern-Simons versus Yang-Mills gaugings in three-dimensions, Nucl. Phys. B 668 (2003) 167 [hep-th/0303213] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  28. [28]

    S. Gukov and D. Tong, D-brane probes of G 2 holonomy manifolds, Phys. Rev. D 66 (2002) 087901 [hep-th/0202125] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  29. [29]

    S. Gukov and D. Tong, D-brane probes of special holonomy manifolds and dynamics of N = 1 three-dimensional gauge theories, JHEP 04 (2002) 050 [hep-th/0202126] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. [30]

    X. de la Ossa, M. Larfors and E.E. Svanes, Infinitesimal moduli of G 2 holonomy manifolds with instanton bundles, JHEP 11 (2016) 016 [arXiv:1607.03473] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  31. [31]

    X. de la Ossa, M. Larfors and E.E. Svanes, The infinitesimal moduli space of heterotic G 2 systems, Commun. Math. Phys. (2017) [arXiv:1704.08717] [INSPIRE].

    Google Scholar 

  32. [32]

    X. de la Ossa, M. Larfors and E.E. Svanes, Restrictions of heterotic G 2 structures and instanton connections, arXiv:1709.06974 [INSPIRE].

  33. [33]

    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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. [34]

    E. Bergshoeff et al., New formulations of D = 10 supersymmetry and D8-O8 domain walls, Class. Quant. Grav. 18 (2001) 3359 [hep-th/0103233] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    G. Papadopoulos and P.K. Townsend, Compactification of D = 11 supergravity on spaces of exceptional holonomy, Phys. Lett. B 357 (1995) 300 [hep-th/9506150] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  36. [36]

    M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  37. [37]

    B.S. Acharya and E. Witten, Chiral fermions from manifolds of G 2 holonomy, hep-th/0109152 [INSPIRE].

  38. [38]

    E. Witten, Anomaly cancellation on G 2 manifolds, hep-th/0108165 [INSPIRE].

  39. [39]

    S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4 − D N = 2 gauge theories: 1., Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    S. Cecotti, S. Ferrara and L. Girardello, Geometry of Type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  42. [42]

    V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  43. [43]

    C. Vafa, Modular invariance and discrete torsion on orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  44. [44]

    R. Harvey and H.B. Lawson Jr., Calibrated geometries, Acta Math. 148 (1982) 47.

    MathSciNet  Article  MATH  Google Scholar 

  45. [45]

    R.C. Mclean, Deformations of calibrated submanifolds, Commun. Anal. Geom. 6 (1996) 705.

    MathSciNet  Article  MATH  Google Scholar 

  46. [46]

    D.R. Morrison, On the structure of supersymmetric T 3 fibrations, arXiv:1002.4921 [INSPIRE].

  47. [47]

    M. Gross, Mirror symmetry and the Strominger-Yau-Zaslow conjecture, Curr. Devl. Math. 1 (2012) 133 [arXiv:1212.4220].

    Article  MATH  Google Scholar 

  48. [48]

    P.S. Aspinwall, D.R. Morrison and M. Gross, Stable singularities in string theory, Commun. Math. Phys. 178 (1996) 115 [hep-th/9503208] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  49. [49]

    M. Gross, Special lagrangian fibrations II: geometry, math/9809072.

  50. [50]

    P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, hep-th/9404151 [INSPIRE].

  51. [51]

    V.V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 111.

    MathSciNet  MATH  Google Scholar 

  52. [52]

    K. Becker et al., Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4 folds, Nucl. Phys. B 480 (1996) 225 [hep-th/9608116] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. [53]

    R. Blumenhagen and V. Braun, Superconformal field theories for compact G 2 manifolds, JHEP 12 (2001) 006 [hep-th/0110232] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  54. [54]

    R. Roiban and J. Walcher, Rational conformal field theories with G 2 holonomy, JHEP 12 (2001) 008 [hep-th/0110302] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  55. [55]

    J. Gutowski and G. Papadopoulos, Moduli spaces and brane solitons for M-theory compactifications on holonomy G 2 manifolds, Nucl. Phys. B 615 (2001) 237 [hep-th/0104105] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  56. [56]

    M. Atiyah and E. Witten, M theory dynamics on a manifold of G 2 holonomy, Adv. Theor. Math. Phys. 6 (2003) 1 [hep-th/0107177] [INSPIRE].

    MathSciNet  Article  MATH  Google Scholar 

  57. [57]

    S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].

  58. [58]

    S. Cecotti and M. Del Zotto, 4d N = 2 gauge theories and quivers: the non-simply laced case, JHEP 10 (2012) 190 [arXiv:1207.7205] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  59. [59]

    B. Fiol, The BPS spectrum of N = 2 SU(N ) SYM and parton branes, JHEP 02 (2006) 065 [hep-th/0012079] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  60. [60]

    S. Cecotti, Categorical tinkertoys for N = 2 gauge theories, Int. J. Mod. Phys. A 28 (2013) 1330006 [arXiv:1203.6734] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. [61]

    S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  62. [62]

    S. Cecotti, M. Del Zotto and S. Giacomelli, More on the N = 2 superconformal systems of type D p(G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  63. [63]

    A. Kovalev and J. Nordström, Asymptotically cylindrical 7-manifolds of holonomy g2 with applications to compact irreducible G 2 -manifolds, Ann. Global Anal. Geom. 38 (2010) 221 [arXiv:0907.0497].

    MathSciNet  Article  MATH  Google Scholar 

  64. [64]

    J. Nordström, Deformations and gluing of asymptotically cylindrical manifolds with exceptional holonomy, Ph.D. Thesis, Cambridge University, Cambridge U.K. (2008).

  65. [65]

    G. Curio and D. Lüst, A class of N = 1 dual string pairs and its modular superpotential, Int. J. Mod. Phys. A 12 (1997) 5847 [hep-th/9703007] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  66. [66]

    V.V. Nikulin, Discrete reflection groups in lobachevsky spaces and algebraic surfaces, in the proceedings of the International Congress of Mathematicians (ICM 1986), August 3-11, Berkeley, U.S.A. (1986).

    Google Scholar 

  67. [67]

    W. Nahm and K. Wendland, A Hiker’s guide to K3: aspects of N = (4, 4) superconformal field theory with central charge c = 6, Commun. Math. Phys. 216 (2001) 85 [hep-th/9912067] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  68. [68]

    A.P. Braun, R. Ebert, A. Hebecker and R. Valandro, Weierstrass meets Enriques, JHEP 02 (2010) 077 [arXiv:0907.2691] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  69. [69]

    J. Conway and N. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften. Springer, Germany (1998).

    Google Scholar 

  70. [70]

    P. Candelas and A. Font, Duality between the webs of heterotic and type-II vacua, Nucl. Phys. B 511 (1998) 295 [hep-th/9603170] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  71. [71]

    E. Perevalov and H. Skarke, Enhanced gauged symmetry in type-II and F theory compactifications: Dynkin diagrams from polyhedra, Nucl. Phys. B 505 (1997) 679 [hep-th/9704129] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  72. [72]

    V.S. Kulikov, Degenerations of K3 surfaces and enriques surfaces, math. USRR-Izv. 11 (1977) 957.

    Article  MATH  Google Scholar 

  73. [73]

    U. Persson and H. Pinkham, Degeneration of surfaces with trivial canonical bundle, Ann. Math. 113 (1981) 45.

    MathSciNet  Article  MATH  Google Scholar 

  74. [74]

    R. Davis et al., Short tops and semistable degenerations, arXiv:1307.6514.

  75. [75]

    A.P. Braun and T. Watari, Heterotic-type IIA duality and degenerations of K3 surfaces, JHEP 08 (2016) 034 [arXiv:1604.06437] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  76. [76]

    D. Huybrechts, Moduli spaces of hyperkaehler manifolds and mirror symmetry, math/0210219.

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Braun, A.P., Del Zotto, M. Towards generalized mirror symmetry for twisted connected sum G2 manifolds. J. High Energ. Phys. 2018, 82 (2018). https://doi.org/10.1007/JHEP03(2018)082

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Keywords

  • String Duality
  • Supersymmetry and Duality