Black Hole Entropy Function, Attractors and Precision Counting of Microstates (0708.1270v4)

Abstract: In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N=4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centered black holes as well.

• The paper introduces the entropy function formalism, streamlining the computation of black hole entropy even with higher derivative corrections.
• It demonstrates that the attractor mechanism fixes the near-horizon geometry, making the entropy independent of asymptotic moduli variations.
• The paper achieves precise microstate counting for quarter BPS dyons, validating macroscopic entropy through string theory dualities and Siegel modular forms.

Overview of the Paper on Black Hole Entropy, Attractors, and Microstate Counting

The paper authored by Ashoke Sen is a comprehensive exploration of the intriguing topic of black hole entropy, particularly focusing on extremal black holes within the framework of string theory. It introduces and elaborates on the entropy function formalism, a method that provides significant insights into the attractor mechanism and facilitates precise counting of microstates for a class of quarter BPS dyons in $\mathcal{N} = 4$ supersymmetric string theories. The manuscript explores both the classical and statistical perspectives on black hole entropy, addressing crucial aspects such as duality symmetries, higher derivative corrections, and comparisons between black hole and statistical entropies.

Key Components and Results

1. Entropy Function Formalism:
   • The paper provides a detailed exposition of the entropy function formalism, which relates the black hole entropy to the extremum of a function $E = 2\pi (eq^i - f)$, where $f$ is derived from the near-horizon geometry.
   • This approach streamlines the computation of black hole entropy, especially with higher derivative corrections in the low-energy effective actions of string theory.
2. Attractor Mechanism:
   • Sen discusses the attractor mechanism, which implies that for a large class of extremal black holes, the near-horizon geometry and scalar configurations are fully determined by the charges of the black hole, independent of the asymptotic moduli.
   • The paper highlights the importance of this mechanism in stabilizing the near-horizon geometry and ensuring the invariance of black hole entropy under variations of asymptotic moduli.
3. Precision Microstate Counting:
   • A critical component of the paper is the precise enumeration of microstates for quarter BPS dyons using string theory realizations, with the degeneracy expressed in terms of the Siegel modular forms.
   • The relation between statistical entropy and black hole entropy is analyzed, showcasing how microscopic counts corroborate the macroscopic calculations, particularly demonstrating the efficacy of duality symmetries in validating these counts across different regimes.
4. Impact of Higher Derivative Corrections:
   • Analysis is provided on the influence of higher derivative terms, such as the Gauss-Bonnet term, elucidating their role in the corrections to black hole entropy.
   • The paper demonstrates that the inclusion of specific terms retains duality invariance and affects vanishing corrections for particular charge configurations, underpinning the robustness of the entropy function formalism.
5. Walls of Marginal Stability:
   • The manuscript addresses how the spectrum of BPS states and the associated entropy can exhibit discontinuities as one crosses walls of marginal stability in the moduli space.
   • The formalism provides a method to track these changes and maintains an invariant characterization of different domains in the moduli space.

Implications and Future Directions

The research intricately ties into the broader discourse on quantum gravity, contributing significantly to the understanding of how quantum microstates give rise to thermodynamic properties of black holes. It suggests pathways to extend these insights to non-supersymmetric configurations, potentially paving the way towards a deeper comprehension of quantum aspects of gravity. The role of symmetry principles and duality in determining the non-perturbative dynamics of string compactifications is also underscored, hinting at their potential utility in broader quantum field theory contexts.

In the field of theoretical developments, one can anticipate future advancements involving further explorations into non-supersymmetric counterparts, small black holes, and generalizations to other compactification schemes, potentially offering more rigorous validation against statistical mechanics predictions and insights into the resolution of singularities through string theory frameworks.