Communications in Mathematical Physics

, Volume 351, Issue 2, pp 837–871 | Cite as

SU(N) Transitions in M-Theory on Calabi–Yau Fourfolds and Background Fluxes

  • Hans Jockers
  • Sheldon Katz
  • David R. Morrison
  • M. Ronen Plesser
Article

Abstract

We study M-theory on a Calabi–Yau fourfold with a smooth surface S of AN–1 singularities. The resulting three-dimensional theory has a \({\mathcal{N}=2}\)SU(N) gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional \({\mathcal{N}=1}\)SU(N) gauge theory on the surface S. A variant of the Vafa–Witten equations governs the moduli space of the gauge theory, which—for a trivial SU(N) principal bundle over S—admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi–Yau fourfolds, respectively. We find that the deformed Calabi–Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi–Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi–Yau fourfolds embedded in toric varieties with an AN–1 singularity in codimension two.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Strominger A.: Massless black holes and conifolds in string theory. Nucl. Phys. B 451, 96–108 (1995) arXiv:hep-th/9504090 ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Greene B.R., Morrison D.R., Strominger A.: Black hole condensation and the unification of string vacua. Nucl. Phys. B 451, 109–120 (1995) arXiv:hep-th/9504145 ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Katz S.H., Morrison D.R., Plesser M.R.: Enhanced gauge symmetry in type II string theory. Nucl. Phys. B 477, 105–140 (1996) arXiv:hep-th/9601108 ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Klemm A., Mayr P.: Strong coupling singularities and nonAbelian gauge symmetries in N =  2 string theory. Nucl. Phys. B 469, 37–50 (1996) arXiv:hep-th/9601014 ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Witten E.: Phase transitions in M theory and F theory. Nucl. Phys. B 471, 195–216 (1996) arXiv:hep-th/9603150 ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Morrison D.R., Seiberg N.: Extremal transitions and five-dimensional supersymmetric field theories. Nucl. Phys. B 483, 229–247 (1997) arXiv:hep-th/9609070 ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Katz S.H., Klemm A., Vafa C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173–195 (1997) arXiv:hep-th/9609239 ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Intriligator K.A., Morrison D.R., Seiberg N.: Five-dimensional supersymmetric gauge theories and degenerations of Calabi–Yau spaces. Nucl. Phys. B 497, 56–100 (1997) arXiv:hep-th/9702198 ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    de Wit B., Van Proeyen A.: Potentials and symmetries of general gauged N =  2 supergravity: Yang–Mills models. Nucl. Phys. B 245, 89 (1984)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Cremmer, E., Kounnas, C., Van Proeyen, A., Derendinger, J., Ferrara, S., et al.: Vector multiplets coupled to N =  2 supergravity: superHiggs effect, flat potentials and geometric structure. Nucl. Phys. B 250, 385 (1985)Google Scholar
  11. 11.
    Strominger A.: Special geometry. Commun. Math. Phys. 133, 163–180 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Seiberg N., Witten E.: Electric-magnetic duality, monopole condensation, and confinement in N =  2 supersymmetric Yang–Mills theory. Nucl. Phys. B 426, 19–52 (1994) arXiv:hep-th/9407087 ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in N =  2 supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994) arXiv:hep-th/9408099 ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cremmer E., Ferrara S., Girardello L., Van Proeyen A.: Yang–Mills theories with local supersymmetry: Lagrangian, transformation laws and superHiggs effect. Nucl. Phys. B 212, 413 (1983)ADSCrossRefGoogle Scholar
  15. 15.
    Intriligator K.A., Leigh R., Seiberg N.: Exact superpotentials in four-dimensions. Phys. Rev. D 50, 1092–1104 (1994) arXiv:hep-th/9403198 ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Seiberg, N.: The power of holomorphy: exact results in 4-D SUSY field theories (1994). arXiv:hep-th/9408013
  17. 17.
    Gukov, S., Sparks, J., Tong, D.: Conifold transitions and five-brane condensation in M theory on spin(7) manifolds. Class. Quant. Grav. 20, 665–706 (2003). arXiv:hep-th/0207244
  18. 18.
    Affleck I., Harvey J.A., Witten E.: Instantons and (super)symmetry breaking in (2+1)-dimensions. Nucl. Phys. B 206, 413 (1982)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Aharony O., Hanany A., Intriligator K.A., Seiberg N., Strassler M.J.: Aspects of N =  2 supersymmetric gauge theories in three-dimensions. Nucl. Phys. B 499, 67–99 (1997) arXiv:hep-th/9703110 ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    de Boer J., Hori K., Oz Y.: Dynamics of N =  2 supersymmetric gauge theories in three-dimensions. Nucl. Phys. B 500, 163–191 (1997) arXiv:hep-th/9703100 ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Diaconescu D.-E., Gukov S.: Three-dimensional N =  2 gauge theories and degenerations of Calabi–Yau four folds. Nucl. Phys. B 535, 171–196 (1998) arXiv:hep-th/9804059 ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kapustin A., Strassler M.J.: On mirror symmetry in three-dimensional Abelian gauge theories. JHEP 04, 021 (1999) arXiv:hep-th/9902033 ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Dorey N., Tong D.: Mirror symmetry and toric geometry in three-dimensional gauge theories. JHEP 05, 018 (2000) arXiv:hep-th/9911094 ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Tong D.: Dynamics of N =  2 supersymmetric Chern–Simons theories. JHEP 07, 019 (2000) arXiv:hep-th/0005186 ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Aganagic M., Hori K., Karch A., Tong D.: Mirror symmetry in (2+1)-dimensions and (1+1)-dimensions. JHEP 07, 022 (2001) arXiv:hep-th/0105075 ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Intriligator K., Jockers H., Mayr P., Morrison D.R., Plesser M.R.: Conifold Transitions in M-theory on Calabi–Yau fourfolds with background fluxes. Adv. Theor. Math. Phys. 17, 601– (2013) arXiv:1203.6662 [hep-th]MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Intriligator K., Seiberg N.: Aspects of 3d N =  2 Chern–Simons-matter theories. JHEP 07, 079 (2013) arXiv:1305.1633 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Witten E.: String theory dynamics in various dimensions. Nucl. Phys. B. 443, 85–126 (1995) arXiv:hep-th/9503124 ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Becker K., Becker M.: M-theory on eight-manifolds. Nucl. Phys. B. 477, 155–167 (1996) arXiv:hep-th/9605053 ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Sethi S., Vafa C., Witten E.: Constraints on low dimensional string compactifications. Nucl. Phys. B 480, 213–224 (1996) arXiv:hep-th/9606122 ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Marsano J., Saulina N., Schäfer-Nameki S.: A note on G-fluxes for F-theory model building. JHEP 11, 088 (2010) arXiv:1006.0483 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Marsano J., Saulina N., Schäfer-Nameki S.: G-flux, M5 instantons, and U(1) symmetries in F-theory. Phys. Rev. D 87, 066007 (2013) arXiv:1107.1718 [hep-th]ADSCrossRefMATHGoogle Scholar
  33. 33.
    Braun A.P., Collinucci A., Valandro R.: G-flux in F-theory and algebraic cycles. Nucl. Phys. B 856, 129–179 (2012) arXiv:1107.5337 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Marsano J., Schäfer-Nameki S.: Yukawas, G-flux, and spectral covers from resolved Calabi–Yau’s. JHEP 11, 098 (2011) arXiv:1108.1794 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Krause S., Mayrhofer C., Weigand T.: G 4 flux, chiral matter and singularity resolution in F-theory compactifications. Nucl. Phys. B. 858, 1–47 (2012) arXiv:1109.3454 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Grimm T.W., Hayashi H.: F-theory fluxes, chirality and Chern–Simons theories. JHEP 03, 027 (2012) arXiv:1111.1232 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Krause S., Mayrhofer C., Weigand T.: Gauge fluxes in F-theory and type IIB orientifolds. JHEP 08, 119 (2012) arXiv:1202.3138 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Cvetič M., Grimm T.W., Klevers D.: Anomaly cancellation and abelian gauge symmetries in F-theory. JHEP 02, 101 (2013) arXiv:1210.6034 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Braun A.P., Collinucci A., Valandro R.: Hypercharge flux in F-theory and the stable Sen limit. JHEP 07, 121 (2014) arXiv:1402.4096 [hep-th]ADSCrossRefGoogle Scholar
  40. 40.
    Cvetič M., Klevers D., Peña D.K.M., Oehlmann P.-K., Reuter J.: Three-family particle physics models from global F-theory compactifications. JHEP 08, 087 (2015) arXiv:1503.02068 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Mavlyutov A.R.: Deformations of Calabi–Yau hypersurfaces arising from deformations of toric varieties. Invent. Math. 157, 621–633 (2004) arXiv:math.AG/0309239 ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Mavlyutov A.R.: Embedding of Calabi–Yau deformations into toric varieties. Math. Ann. 333, 45–65 (2005) arXiv:math.AG/0309240 MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Witten E.: On flux quantization in M theory and the effective action. J. Geom. Phys. 22, 1–13 (1997) arXiv:hep-th/9609122 ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Gukov, S., Vafa, C., Witten, E.: CFT’s from Calabi–Yau four folds. Nucl. Phys. B 584, 69–108 [Erratum: Nucl. Phys. B 608, 477(2001)] (2000). arXiv:hep-th/9906070
  45. 45.
    Lüdeling, C.: Seven-dimensional super-Yang–Mills theory in N =  1 superfields (2011). arXiv:1102.0285 [hep-th]
  46. 46.
    Beasley C., Heckman J.J., Vafa C.: GUTs and exceptional branes in F-theory—I. JHEP 01, 058 (2009) arXiv:0802.3391 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Vafa C., Witten E.: A strong coupling test of S duality. Nucl. Phys. B 431, 3–77 (1994) arXiv:hep-th/9408074 ADSMathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Witten E.: Nonperturbative superpotentials in string theory. Nucl. Phys. B 474, 343–360 (1996) arXiv:hep-th/9604030 ADSMathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Jockers, H., Katz, S., Morrison, D.R., Plesser, M.R.: in preparationGoogle Scholar
  50. 50.
    Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994) arXiv:alg-geom/9402002 MathSciNetMATHGoogle Scholar
  51. 51.
    Batyrev, V., Borisov, L.: On Calabi–Yau complete intersections in toric varieties. Higher-Dimensional Complex Varieties (Trento, 1994), pp. 39–65. de Gruyter, Berlin (1996). arXiv:alg-geom/9412017
  52. 52.
    Fulton, W.: Introduction to Toric Varieties. Annals of Math. Studies, vol. 131. Princeton University Press, Princeton (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Hans Jockers
    • 1
  • Sheldon Katz
    • 2
  • David R. Morrison
    • 3
  • M. Ronen Plesser
    • 4
    • 5
  1. 1.Bethe Center for Theoretical PhysicsRheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Departments of Mathematics and PhysicsUniversity of California at Santa BarbaraSanta BarbaraUSA
  4. 4.Weizmann Institute of ScienceRehovotIsrael
  5. 5.IHESBures-sur-YvetteFrance

Personalised recommendations