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

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.

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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.

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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

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Key words

  • PACS 04.65.+e–Supergravity
  • PACS 04.70.Dy–Quantum aspects of black holes, evaporation, thermodynamics.