Skip to main content
SpringerLink
Log in

Mirror Fermât Calabi-Yau threefolds and Landau-Ginzburg black-hole attractors

  • Published:
La Rivista del Nuovo Cimento Aims and scope

Summary

We discuss the “attractor mechanism” for extremal black-holes (BHs) in the context of Maxwell-Einstein supergravity theories. The BH squared mass at the horizon (related to the Bekenstein-Hawking entropy area formula) is determined by the “fixed points” of the so-called “BH potential”. In the considered framework, the scalar fields describe trajectories ending into such fixed points, which only depend on the electric and magnetic BH charges. Thus, the BH appears as a soliton, interpolating between maximally supersymmetric limiting solutions at spatial infinity and at the horizon. The BH entropy depend only on the BH charges, and it is independent of the initial data, i.e. on the values of the scalar fields at spatial infinity. In the considered theories, extremal BHs seem to behave as dynamical systems with fixed points (“attractors”) describing the thermodynamical equilibrium and stability features. An introductory review of the “special Kahler” geometry for scalar manifolds encompassing the BH background is given. A detailed study of the BH attractor equations for one-(complex structure)modulus Calabi-Yau spaces which are the mirror dual of Fermat Calabi-Yau threefolds (CY3’s) is carried out as well. When exploring non-degenerate solutions near the Landau-Ginzburg point of the moduli space of such 4-dimensional compactifications, we always find two species of extremal black-hole attractors, depending on the choice of the Sp(4, ℤ) symplectic charge vector, one \(\frac{1}{2}\)-BPS (which is always stable, according to general results of special Kahler geometry) and one non-BPS. The latter turns out to be stable (local minimum of the “effective black-hole potential” Vbh) for non-vanishing central charge, whereas it is unstable (saddle point of VBH) for the case of vanishing central charge. This is to be compared to the large volume limit of one-modulus CY3-compactifications (of type-IIA superstrings), in which the homogeneous symmetric special Kahler geometry based on cubic prepotential admits (beside the \(\frac{1}{2}\)-BPS ones) only non-BPS extremal black-hole attractors with non-vanishing central charge, which are always stable.

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. Ferrara S., Kallosh R. and Strominger A., N =2 Extremal Black Holes, Phys. Rev. D, 52 (1995) 5412, hep-th/9508072.

    Article  ADS  MathSciNet  Google Scholar 

  2. Strominger A., Macroscopic Entropy of N = 2 Extremal Black Holes Phys. Lett. B 383, (1996) 39, hep-th/9602111

    Article  ADS  MathSciNet  Google Scholar 

  3. Ferrara S. and Kallosh R., Supersymmetry and Attractors Phys. Rev. D 54, (1996) 1514, hep-th/9602136

    Article  ADS  MathSciNet  Google Scholar 

  4. Ferrara S. and Kallosh R., Universality of Supersymmetric Attractors, Phys. Rev. D, 54 (1996) 1525, hep-th/9603090.

    Article  ADS  MathSciNet  Google Scholar 

  5. Sen A., Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity JHEP 09, (2005) 038, hep-th/0506177

    Article  ADS  MathSciNet  Google Scholar 

  6. Goldstein K., Iizuka N., Jena R. P. and Trivedi S. P., Non-Supersymmetric Attractors Phys. Rev. D 72, (2005) 124021, hep-th/0507096

    Article  ADS  MathSciNet  Google Scholar 

  7. Sen A., Entropy Function for Heterotic Black Holes JHEP 03, (2006) 008, hep-th/0508042

    Article  ADS  MathSciNet  Google Scholar 

  8. Tripathy P. K. and Trivedi S. P., Non-Supersymmetric Attractors in String Theory JHEP 0603, (2006) 022, hep-th/0511117

    Article  ADS  MathSciNet  Google Scholar 

  9. Giryavets A. New Attractors and Area Codes, JHEP, 0603 (2006) 020, hep-th/0511215.

    Article  ADS  MathSciNet  Google Scholar 

  10. Goldstein K., Jena R. P., Mandal G. and Trivedi S. P., A C-Function for Non- Supersymmetric Attractors JHEP 0602, (2006) 053, hep-th/0512138

    Article  ADS  MathSciNet  Google Scholar 

  11. Alishahiha M. and Ebrahim H. Non-supersymmetric attractors and entropy function, JHEP, 0603 (2006) 003, hep-th/0601016.

    Article  ADS  MathSciNet  Google Scholar 

  12. Kallosh R. Sivanandam N. and Soroush M., The Non-BPS Black Hole Attractor Equation JHEP 0603, (2006) 060, hep-th/0602005

    Article  ADS  MathSciNet  Google Scholar 

  13. Chandrasekhar B., Parvizi S., Tavanfar A. and Yavartanoo H., Non- supersymmetric attractors in R2 gravities, hep-th/0602022.

  14. Bellucci S., Ferrara S. and Marrani A., On some properties of the Attractor Equations Phys. Lett. B 635, (2006) 172, hep-th/0602161

    Article  ADS  MathSciNet  Google Scholar 

  15. Bellucci S., Ferrara S. and Marrani A., Supersymmetric Mechanics. Vol. 2: The Attractor Mechanism and Space-Time Singularities, Lect. Notes Phys., Vol. 701 (SpringerVerlag, Heidelberg) 2006.

    Book  Google Scholar 

  16. Cardoso G. L., Lüst D. and Perz J., Entropy Maximization in the presence of Higher- Curvature Interactions JHEP 05, (2006) 028, hep-th/0603211

    Article  ADS  MathSciNet  Google Scholar 

  17. Sahoo B. and Sen A., Higher-derivative corrections to non-supersymmetric extremal black holes, hep-th/0603149.

  18. Ferrara S. and Kallosh R. On N = 8 attractors, Phys. Rev. D, 73 (2006) 125005, hep-th/0603247.

    Article  ADS  MathSciNet  Google Scholar 

  19. Alishahiha M. and Ebrahim H., New attractor, Entropy Function and Black Hole Partition Function, hep-th/0605279.

  20. Ferrara S. and Günaydin M. Orbits and attractors for N = 2 Maxwell-Einstein supergravity theories in five dimensions, hep-th/0606108.

  21. Bellucci S., Ferrara S., Günaydin M. and Marrani A., Charge Orbits of Symmetric Special Geometries and Attractors Int. J. Mod. Phys. A 21, (2006) 5043, hep-th/0606209

    Article  ADS  MathSciNet  Google Scholar 

  22. Astefanesei D., Goldstein K., Jena R. P., Sen A. and Trivedi S. P., Rotating Attractors, hep-th/0606244.

  23. Kallosh R. Sivanandam N. and Soroush M., Exact Attractive non-BPS STU Black Holes, hep-th/0606263.

  24. Kaura P. and Misra A., On the Existence of Non-Supersymmetric Black Hole Attractors for Two-Parameter Calabi-Yau’s and Attractor Equations, hep-th/0607132.

  25. Cardoso G. L., Grass V., Lüst D. and Perz J., Extremal non-BPS Black Holes and Entropy Extremization, hep-th/0607202.

  26. Morales J. F. and Samtleben H., Entropy Function and Attractors for AdS Black Holes, hep-th/0608044.

  27. Chandrasekhar B., Yavartanoo H. and Yun S., Non-Supersymmetric Attractors in BI black holes, hep-th/0611240.

  28. Andrianopoli L., D’Auria R., Ferrara S. and Trigiante M., Extremal Black Holes in Supergravity, in String Theory and Fundamental Interactions, edited by Gasperini M. and Maharana J., to be published in Lect. Notes Phys. (Springer, Berlin-Heidelberg) 2007, hep-th/0611345.

    Google Scholar 

  29. Mosaffa A. E., Randjbar-daemi S. and Sheikh-jabbari M. M., Non-Abelian Magnetized Blackholes and Unstable Attractors, hep-th/0612181.

  30. Cardoso G. L., DE Wit B. and Mahapatra S., Black hole entropy functions and attractor equations, hep-th/0612225.

  31. D’Auria R., Ferrara S. and Trigiante M., Critical points of the Black-Hole potential for homogeneous special geometries, hep-th/0701090.

  32. Bellucci S., Ferrara S. and Marrani A., Attractor Horizon Geometries of Extremal Black Holes, contribution to the Proceedings of the XVII SIGRAV Conference, 4-7 September 2006, Turin, Italy, hep-th/0702019.

  33. Andrianopoli L. D’Auria R., Ferrara S. and Trigiante M., Black Hole Attractors in N =1 Supergravity, hep-th/0703178.

  34. Saraikin K. and Vafa C., Non-Supersymmetric Black Holes and Topological Strings, hep-th/0703214.

  35. Bekenstein J. D., Phys. Rev. D, 7 (1973) 2333; Hawking S. W., Phys. Rev. Lett, 26 (1971) 1344; in Black Holes (Les Houches 1972), edited by Dewitt C. and Dewitt B. S. (Gordon and Breach, New York) 1973; Nature, 248 (1974) 30; Commun. Math. Phys., 43 (1975) 199.

    Article  ADS  MathSciNet  Google Scholar 

  36. Gibbons G. W. and Hull C. M., A Bogomol’ny Bound for General Relativity and Solitons in N =2 Supergravity, Phys. Lett. B, 109 (1982) 190.

    Article  ADS  MathSciNet  Google Scholar 

  37. Arnowitt R. Deser S. and Misner C. W., The Dynamics of General Relativity, in Gravitation: an Introduction to Current Research, edited by Witten L. (Wiley, New York) 1962.

    Google Scholar 

  38. Ceresole A., D’Auria R. and Ferrara S., The Symplectic Structure of N = 2 SUGRA and Its Central Extension, talk given at ICTP Trieste Conference on Physical and Mathematical Implications of Mirror Symmetry in String Theory, Trieste, Italy, 5–9 June 1995, Nucl. Phys. Proc. Suppl., 46 (1996), hep-th/9509160.

  39. Ceresole A., D’Auria R., Ferrara S. and van Proeyen A., Duality Transformations in Supersymmetric Yang-Mills Theories Coupled to Supergravity, Nucl. Phys. B, 444 (1995) 92, hep-th/9502072.

    Article  ADS  MathSciNet  Google Scholar 

  40. Ferrara S., Gibbons G. W. and Kallosh R., Black Holes and Critical Points in Moduli Space Nucl. Phys. B 500, (1997) 75, hep-th/9702103

    Article  ADS  MathSciNet  Google Scholar 

  41. Ferrara S. and Günaydin M. Orbits of Exceptional Groups, Duality and BPS States in String Theory, Int. J. Mod. Phys. A, 13 (1998) 2075, hep-th/9708025.

    Article  ADS  MathSciNet  Google Scholar 

  42. Strominger A. and Witten E., New Manifolds for Superstring Compactification, Commun. Math. Phys., 101 (1985) 341.

    Article  ADS  MathSciNet  Google Scholar 

  43. Candelas P., De la Ossa X. C., Green P. S. and Parkes L., A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory, Nucl. Phys. B, 359 (1991) 21.

    Article  ADS  MathSciNet  Google Scholar 

  44. Candelas P., De la Ossa X. C., Green P. S. and Parkes L., An Exactly Soluble Superconformal Theory from a Mirror Pair of Calabi-Yau Manifolds, Phys. Lett. B, 258 (1991) 118.

    Article  ADS  MathSciNet  Google Scholar 

  45. Cadavid A. C. and Ferrara S., Picard-Fuchs Equations and the Moduli Space of Superconformal Field Theories, Phys. Lett. B, 267 (1991) 193.

    Article  ADS  MathSciNet  Google Scholar 

  46. Klemm A. and Theisen S., Considerations of One Modulus Calabi-Yau Compactifications: Picard-Fuchs Equations, Kahler Potentials and Mirror Maps Nucl. Phys. B 389, (1993) 153, hep-th/9205041

    Article  ADS  Google Scholar 

  47. Font A., Periods and Duality Symmetries in Calabi-Yau Compactifications, hep-th/9203084.

  48. Craps B., Roose F., Troost W. and van Proeyen A., The Definitions of Special Geometry, hep-th/9606073.

  49. Craps B., Roose F., Troost W. and van Proeyen A., What is Special Kähler Geometry? Nucl. Phys. B 503, (1997) 565, hep-th/9703082

    Article  ADS  Google Scholar 

  50. Bagger J. and Witten E., The Gauge Invariant Supersymmetric Nonlinear Sigma Model, Phys. Lett. B, 118 (1982) 103.

    Article  ADS  MathSciNet  Google Scholar 

  51. Castellani L., D’Auria R. and Ferrara S., Special Geometry without Special Coordinates, Class. Quantum. Grav., 7 (1990) 1767; Special Kähler Geometry: an Intrinsic Formulation from N = 2 Space-Time Supersymmetry, Phys. Lett. B, 241 (1990) 57.

    Article  ADS  MathSciNet  Google Scholar 

  52. D’Auria R., Ferrara S. and Fré P. Special and Quaternionic Isometries: General Couplings in N = 2 Supergravity and the Scalar Potential, Nucl. Phys. B, 359 (1991) 705; Andrianopoli L., Bertolini M., Ceresole A., D’Auria R., Ferrara S., Fre P. and Magri T. N =2 Supergravity and N =2 Super Yang-Mills Theory on General Scalar Manifolds: Symplectic Covariance, Gaugings and the Momentum Map}, J. Geom. Phys. 23, (1997) 111, hep-th/9605032; Andrianopoli L. Bertolini M. Ceresole A. D’Auria R., Ferrara S. and Fré P., General Matter Coupled N = 2 Supergravity, Nucl. Phys. B, 476 (1996) 397, hep-th/9603004

    Article  ADS  MathSciNet  Google Scholar 

  53. Ferrara S. Bodner M. and Cadavid A. C., Calabi-Yau Supermoduli Space, Field Strength Duality and Mirror Manifolds, Phys. Lett. B, 247 (1990) 25.

    Article  ADS  MathSciNet  Google Scholar 

  54. Cremmer E., Kounnas C., van Proeyen A., Derendinger J. P., Ferrara S., de wit B. and Girardello L., Vector Multiplets Coupled To N = 2 Supergravity: Superhiggs Effect, Flat Potentials And Geometric Structure, Nucl. Phys. B, 250 (1985) 385.

    Article  ADS  Google Scholar 

  55. Cremmer E. and van Proeyen A. Classification of Kahler Manifolds in N = 2 Vector Multiplet Supergravity Couplings, Class. Quantum Grav., 2 (1985) 445.

    Article  ADS  Google Scholar 

  56. Ferrara S. and Louis J., Flat Holomorphic Connections and Picard-Fuchs Identities from N = 2 Supergravity Phys. Lett. B 278, (1992) 240, hep-th/9112049

    Article  ADS  MathSciNet  Google Scholar 

  57. Ceresole A., D’Auria R., Ferrara S., Lerche W. and Louis J., Picard-Fuchs Equations and Special Geometry, Int. J. Mod. Phys. A, 8 (1993) 79, hep-th/9204035; Ceresole A., D’Auria R., Ferrara S., Lerche W., Louis J. and Regge T., Picard-Fuchs Equations, Special Geometry and Target Space Duality, in Mirror Symmetry II, edited by GREENE B. R. and Yau S.-T. (American Mathematical Society - International Press) 1997.

    Article  ADS  MathSciNet  Google Scholar 

  58. Strominger A., Special Geometry, Commun. Math. Phys., 133 (1990) 163.

    Article  ADS  MathSciNet  Google Scholar 

  59. Forsyth A., Theory of Differential Equations, Vol. 4 (Dover Publications, New York) 1959.

    Google Scholar 

  60. Candelas P., Lynker M. and Schimmrigk R., Calabi-Yau Manifolds in Weighted P(4), Nucl. Phys. B, 341 (1990) 383.

    Article  ADS  MathSciNet  Google Scholar 

  61. Greene B. R. and Plesser M. R., (2, 2) and (2, 0) Superconformal Orbifolds, Harvard University Rep. HUTP-89/A043; Duality in Calabi-Yau Moduli Space, Harvard University Rep. HUTP-89-A043A, Nucl. Phys. B, 338 (1990) 15.

    Article  ADS  Google Scholar 

  62. Aspinwall P., Lütken A. and Ross G. G., Construction and Couplings of Mirror Manifolds, Phys. Lett. B, 241 (1990) 373.

    Article  ADS  MathSciNet  Google Scholar 

  63. Giryavets A., Kachru S., Tripathy P. K. and Trivedi S. P., Flux Compactifications on Calabi-Yau Threefolds JHEP 0404, (2004) 003, hep-th/0312104

    Article  ADS  MathSciNet  Google Scholar 

  64. Denef F., Supergravity Flows and D-Brane Stability JHEP 0008, (2000) 050, hep-th/0005049

    Article  ADS  MathSciNet  Google Scholar 

  65. Denef F., On the Correspondence between D-Branes and Stationary Supergravity Solutions of Type II Calabi-Yau Compactifications, hep-th/0010222.

Download references

Acknowledgments

It is a pleasure to acknowledge proofreading by Mrs Suzy Vascotto and Mrs Helen Webster. AY would like to thank the INFN Frascati National Laboratories for the kind hospitality extended to him during the work for the present paper. The work of SB has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 “Constituents, fundamental forces and symmetries of the Universe”. The work of SF has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 “Constituents, fundamental forces and symmetries of the Universe”, in association with INFN Frascati National Laboratories and by D.O.E. grant DE-FG03-91ER40662, Task C. The work of AM has been supported by a Junior Grant of the “Enrico Fermi” Centre, Rome, in association with INFN Frascati National Laboratories. The work of AY was supported in part by the grants NFSAT-CRDF ARPI-3328-YE-04 and INTAS-05-7928.

Author information

Authors and Affiliations

  1. INFN, Laboratori Nazionali di Frascati, Via Enrico Fermi 40, 00044, Frascati, Italy

    S. Bellucci, S. Ferrara, A. Marrani & A. Yeranyan

  2. Physics Department, Theory Unit, CERN, CH 1211, Geneva 23, Switzerland

    S. Ferrara

  3. Department of Physics and Astronomy, University of California, Los Angeles, CA, USA

    S. Ferrara

  4. Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Via Panisperna 89A, 00184, Roma, Italy

    A. Marrani

  5. Department of Physics, Yerevan State University, Alex Manoogian St., 1, Yerevan, 375025, Armenia

    A. Yeranyan

Authors
  1. S. Bellucci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Ferrara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Marrani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Yeranyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Bellucci.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bellucci, S., Ferrara, S., Marrani, A. et al. Mirror Fermât Calabi-Yau threefolds and Landau-Ginzburg black-hole attractors. Riv. Nuovo Cim. 29, 1–88 (2006). https://doi.org/10.1393/ncr/i2007-10013-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1393/ncr/i2007-10013-y

Key words

Search

Navigation