Skip to main content
SpringerLink
Log in

Critical Points and Supersymmetric Vacua, III: String/M Models

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold X with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas [AD] and Denef-Douglas [DD1] are given, together with van der Corput style remainder estimates.

Supersymmetric vacua are critical points of certain holomorphic sections (flux superpotentials) of a line bundle \(\mathcal{L} \to \mathcal{C}\) over the moduli space of complex structures on X × T 2 with respect to the Weil-Petersson connection. Flux superpotentials form a lattice of full rank in a 2 b 3(X)-dimensional real subspace \(\mathcal{S} \subset H^0(\mathcal{C}, \mathcal{L})\). We show that the density of critical points in \(\mathcal{C}\) for this lattice of sections is well approximated by Gaussian measures of the kind studied in [DSZ1,DSZ2,AD,DD1].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ashok S., Douglas M.R. (2004). Counting Flux Vacua. JHEP 0401:060

    Article  ADS  MathSciNet  Google Scholar 

  2. Berline N., Getzler E., Vergne M. (1992). Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften 298. Springer-Verlag, Berlin

    MATH  Google Scholar 

  3. Bleher P., Shiffman B., Zelditch S. (2000). Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2):351–395

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bleher P., Shiffman B., Zelditch S. (2001). Correlations between zeros and supersymmetry. Commun. Math. Phys. 224(1):255–269

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. BP. Bousso R., Polchinski J. Quantization of four-form fluxes and dynamical neutralization of the cosmological constant. J. High Energy Phys. 2000, no 6, Paper 6

  6. Candelas P., Horowitz G., Strominger A., Witten E. (1985). Vacuum configurations for superstrings. Nucl. Phys. B 258(1):46–74

    Article  ADS  MathSciNet  Google Scholar 

  7. CO. Candelas P., de la Ossa XC. Moduli space of Calabi-Yau manifolds. Nucl. Phys. B 355(2), 455–481 (1991); also appeared in Strings ’90 (College Station, TX, 1990). River Edge, NJ: World Sci. Publishing, 1991, pp. 401–429

  8. Denef F., Douglas M.R. (2004). Distributions of flux vacua. JHEP 0405:072

    Article  ADS  MathSciNet  Google Scholar 

  9. Denef F., Douglas M.R. (2005). Distributions of nonsupersymmetric flux vacua. JHEP 0503:061

    Article  ADS  MathSciNet  Google Scholar 

  10. DeWolfe O., Giryavets A., Kachru S., Taylor W. (2005). Enumerating Flux Vacua with Enhanced Symmetries. JHEP 0502:037

    Article  ADS  MathSciNet  Google Scholar 

  11. Douglas M.R. (2003). The statistics of string/M theory vacua. JHEP 0305:046

    Article  ADS  Google Scholar 

  12. Douglas M.R., Shiffman B., Zelditch S. (2004). Critical Points and supersymmetric vacua I. Commun. Math. Phys. 252(1–3): 325–358

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Douglas M.R., Shiffman B., Zelditch S. (2006). Critical Points and supersymmetric vacua II: Asymptotics. and external metrics. J. Differential Geom. 72(3):381–427 (arxiv: math. (CV/0406089))

    MathSciNet  MATH  Google Scholar 

  14. DO. Duke, W., Imamoglu, O.: Lattice points in cones and Dirichlet series. IMRN 53 (2004)

  15. Fyodorov Y.V. (2004). Complexity of Random Energy Landscapes, Glass Transition and Absolute Value of Spectral Determinant of Random Matrices. Phys. Rev. Lett. 92:240601; Erratum: ibid. 93, 149901 (2004)

    Article  PubMed  ADS  MathSciNet  Google Scholar 

  16. Giddings S.B., Kachru S., Polchinski J. (2002). Hierarchies from fluxes in string compactifications. Phys. Rev. D (3) 66(10):106006

    Article  ADS  MathSciNet  Google Scholar 

  17. Giryavets A., Kachru S., Tripathy P.K. (2004). On the taxonomy of flux vacua. JHEP 0408:002

    Article  ADS  MathSciNet  Google Scholar 

  18. Griffiths P., Harris J. (1978). Principles of Algebraic Geometry. Wiley-Interscience, New York

    MATH  Google Scholar 

  19. Gromov M., Milman V.D. (1983). A topological application of the isoperimetric inequality. Amer. J. Math. 105(4):843–854

    Article  MathSciNet  MATH  Google Scholar 

  20. Gross M., Huybrechts D., Joyce D. (2003). Calabi-Yau Manifolds and Related Geometries. Springer Universitext, Springer, New York

    MATH  Google Scholar 

  21. Gukov S., Vafa C., Witten E. (2000). CFT’s from Calabi-Yau four-folds. Nucl. Phys. B 584(1–2):69–108

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. HW. Hardy G.H., Wright E.M. (1979). An introduction to the theory of numbers (5thed). The Clarendon Press New York Oxford University Press, New York

  23. Hlawka E. (1950). Über Integrale auf konvexen Körpern I, II. Monatsh. Math. 54(1–36):81–99

    Article  MathSciNet  Google Scholar 

  24. Hörmander L. (2003). The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin

    MATH  Google Scholar 

  25. Kachru S., Kallosh R., Linde A., Trivedi S.P. (2003). De Sitter vacua in string theory. Phys. Rev. D 68:046005

    Article  ADS  MathSciNet  Google Scholar 

  26. Kallosh R., Linde A. (2004). Landscape, the scale of SUSY breaking, and inflation. JHEP 0412:004

    Article  ADS  MathSciNet  Google Scholar 

  27. Klemm A., Lian B., Roan S.S., Yau S.T. (1998). Calabi-Yau fourfolds for M- and F-theory compactifications. Nucl. Phys. B 518:515

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Lu Z. (2001). On the Hodge metric of the universal deformation space of Calabi-Yau threefolds. J. Geom. Anal. 11(1):103–118

    MathSciNet  MATH  Google Scholar 

  29. Lu Z., Sun X. (2006). On the Weil-Petersson volume and the first Chern Class of the moduli space of Calabi-Yau manifolds. Commun. Math. Phys. 261:297–322

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Lu Z., Sun X. (2004). Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds. J. Inst. Math. Jussieu 3(2):185–229

    Article  MathSciNet  MATH  Google Scholar 

  31. NR. Nechayeva, M., Randol, B.: Approximation of measures on S n by discrete measures (math.NT/0601230)

  32. Pommerenke CH. (1959). Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5:227–257

    MathSciNet  Google Scholar 

  33. Randol B. (1966). A lattice-point problem. Trans. Amer. Math. Soc. 121:257–268

    Article  MathSciNet  MATH  Google Scholar 

  34. Si. Silverstein E. AdS and dS Entropy from String Junctions. In: From Fields to Strings: Circumnavigating Theoretical Physics. Ian Kogan Memorial collection, Vol. 3*, Shifman, M. (ed.,) River Edge, NJ: World Scientific, pp. 2005 1848–1863

  35. Strominger A. (1990). Special geometry. Commun. Math. Phys. 133(1):163–180

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. St2. Strominger A. (1999). Kaluza-Klein compactifications, supersymmetry and Calabi-Yau manifolds. In: Deligne P. et al. (eds) Quantum fields and strings: a course for mathematicians. Providence, RI: Amer. Math. Soc.; Providence, RI: Institute for Advanced Study (IAS), Vol. 2, pp. 1091–1115

  37. Sullivan D. (1977). Infinitesimal computations in topology. Inst. Hautes Études Sci., Publ. Math. No. 47:269–331

    Article  MathSciNet  MATH  Google Scholar 

  38. Sus. Susskind, L.: The anthropic landscape of string theory. http://arxiv.org/list/hep-th/0302219, 2003

  39. Talagrand M. (1996). A new look at independence. Ann. Probab. 24(1):1–34

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang C.-L. (1997). On the incompleteness of the Weil–Petersson metrics along degenerations of Calabi–Yau manifolds. Math. Res. Lett. 1:157–171

    ADS  Google Scholar 

  41. Wang C.-L. (2003). Curvature properties of the Calabi–Yau moduli. Doc. Math. 8:577–590

    MathSciNet  MATH  Google Scholar 

  42. WB. Wess J., Bagger J. (1992). Supersymmetry and supergravity (2nd ed). Princeton Series in Physics. Princeton University Press, Princeton, NJ

    Google Scholar 

  43. Ze1. Zelditch, S.: Angular distribution of lattice points, In preparation

  44. Ze2. Zelditch, S.: Counting Flux vacua, in “Mathametical Physics of Quantum Mechanics, selected and Refereed Lectures from QMath9,” Springer Lecture Notes in Physics 690 (2006). Available at http: mathnt.mat.jhu.edu/zelditch/preprints/preprints.html, 2006

Download references

Author information

Authors and Affiliations

  1. NHETC and Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, 08855–0849, USA

    Michael R. Douglas

  2. I.H.E.S., Bures-sur-Yvette, France

    Michael R. Douglas

  3. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Bernard Shiffman & Steve Zelditch

Authors
  1. Michael R. Douglas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernard Shiffman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Steve Zelditch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael R. Douglas.

Additional information

Communicated by N.A. Nekrasov

Research partially supported by DOE grant DE-FG02-96ER40959 (first author) and NSF grants DMS-0100474 (second author) and DMS-0302518 (third author).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Douglas, M.R., Shiffman, B. & Zelditch, S. Critical Points and Supersymmetric Vacua, III: String/M Models. Commun. Math. Phys. 265, 617–671 (2006). https://doi.org/10.1007/s00220-006-0003-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-0003-7

Keywords

Search

Navigation