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Extreme measures for extremal black holes

Explore extremal black holes and their unique properties, focusing on building statistical systems to describe them. Learn about extremal and non-extremal horizons, decoupling near and far regions, and computing black hole entropy. Discover quantum gravity, holography, and Siegel Modular Forms in relation to black hole mechanics and thermodynamics.

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Extreme measures for extremal black holes

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  1. Extreme measures for extremal black holes Alejandra Castro University of Amsterdam New Directions in Theoretical Physics 2, 2017

  2. Thermodynamics Black hole Mechanics Mass Area horizon Surface Gravity Energy Entropy Temperature Gravity knows about thermodynamics

  3. As a (string) theorist, my personal obsession is: Gravity knows about thermodynamics. Quantum Gravity knows about statistical mechanics. What are the features of the statistical system? And can we write

  4. This is an old question…. … and I’m supposed to talk about new directions

  5. Quantum Gravity knows about statistical mechanics. String theory provides examples of this equality Higher derivative corrections Quantum corrections

  6. Today, I want to focus on building certain statistical systems, which have the right features to describe the black hole. Focus Work in collaboration with A. Belin, J. Gomes, C. Keller [1611.04588]

  7. 1. Inspiration from known examples in String Theory. 2. Exploit geometrical symmetries of the black hole. Strategy 3. Exploit holography. 4. Exploit number theory: crafting suitable partition functions.

  8. Macroscopic: Universality of Black Holes Microscopic: Building Partition functions

  9. Extremal Black Holes Gravitational Input

  10. This is too complicated.

  11. This looks better.

  12. Extremal black holes have the unique feature: we can isolate the horizon. Far away Non-extremal Horizon Extremal Geometry develops a throat. We can decouple the near horizon from the far region.

  13. Degenerate horizons: inner = outer. • Surface gravity vanishes: zero temperature. • Enhancement of symmetry: RindlerAdS. • Can be supersymmetric: even more symmetry! • Horizon area finite: huge residual entropy. Asymptotically flat region Extremality Near horizon region AdS2 x S2

  14. 4D Reissner-Nordstrom Black Holes • Static: No rotation (for simplicity). • Dyonic: Carry both electric & magnetic charges. • Extremal: Decoupling of the near horizon geometry. Geometry S2 AdS2 Electric Magnetic

  15. How to compute black hole entropy and its corrections: Quantum Entropy Function. [A. Sen] PI over all fields including metric. Black Hole Entropy Boundary conditions that focus on black hole. Includes local corrections (Wald entropy), and quantum corrections.

  16. How to compute black hole entropy and its corrections: Quantum Entropy Function. [A. Sen] Statistical Interpretation (using holography) Black Hole Entropy Entropy

  17. Semi-classical regime: Einstein-matter theory describes BH physics Near horizon is weakly curved & Single centered black hole is dominant Valid for Scaling & Predictions Controlled by massless particles.

  18. Challenge: find statistical systems with huge amounts of residual entropy. Opportunity: given a statistical system, we have two universal pieces.

  19. Microscopic Side The power of Siegel Modular Forms

  20. String theory progress report Success Story [A. Sen et al; F. Larsen et al; …]

  21. FIRST INPUT: Look within (1+1)-dimensional CFT. Looking for highly degenerate systems. Justification: Success list & Holography In many (but not all) cases, the AdS2 geometry has an AdS3 origin.

  22. Gravity CFT2 Black hole Very entropic state Energy (L0) dictionary KM current (J0) central charge CFT2 Interpretation For example: And the claim is

  23. In the CFT it is easy to evaluate d(c,E,J) in the Cardy regime. Consider the partition function of the CFT where The asymptotic growth of the Fourier coefficients are mostly determined by symmetries CFT2 Interpretation

  24. HOWEVER

  25. We had This expression relies on modular transformations, and more importantly on a saddle point approximation whichis valid if: CFT2 Interpretation This is NOT the gravity regime. Our dictionary and scaling regimes in gravity are

  26. Question: How to access the gravitational regime? We want partition functions that have exponential growth in this regime. This is a highly non-trivial property of the CFT2. Vast majority of known CFTs do not satisfy this condition.

  27. SECOND INPUT: Average over theories. Add extra symmetry – Siegel Modular Forms. Justification: It gives me what I want. It was the key for the known examples.

  28. Define a grand canonical ensemble that sums over central charge . 2. Identify those with exponential growth in the gravity regime. 3. Extract asymptotic formulas. Strategy for

  29. A Siegel Modular Form (SMF) is a generating function Siegel Modular Forms which in addition to the modular properties of Zc, it is also invariant under This enhances the symmetries to

  30. Consider To extract the asymptotic behavior, the strategy is: Residue integral: Exploit the zeroes/poles of the SMF. Find the most dominant pole: in the large charge limit this is universal. Saddle point integral of remaining variables. Extracting dmicro

  31. Once the dust settles, and one gathers all these results, we found a family of SMFs that in the regime we have a leading asymptotic growth that resembles Cardy and a logarithmic correction that is govern by the degree of the pole in SMF and the weight.

  32. Future Directions We have lots to do…

  33. Within the land of Siegel Modular Forms: Good news! These modular functions easily capture the leading contribution to the black hole entropy in the desired regime. Highly non-trivial! The entropy can be easily computed and interpreted. The answers are unambiguous and precise. We can build more examples beyond those already known.

  34. Within the land of Siegel Modular Forms: Bad News… Challenges Dictionary between black hole charges and SMF data is crucial: an important task! At the moment we have more examples of SMF than black holes. Are there new BHs that we have not discovered? Or are our examples ill? We need a CFT interpretation of the SMFs that fall outside the favorite string theory examples. This will elucidate properties in building a gravity dual.

  35. And more generally: We postulated a non-trivial symmetry for the statistical system. Is this necessary or sufficient? Perhaps it is just approximate. Can we “bootstrap black holes” via its quantum corrections? Exploit more the hints in the subleading corrections.

  36. THANK YOU!

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