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

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

4. 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

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

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

7. 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 [ ]

8. Strategy 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.

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

10. Extremal Black Holes
Gravitational Input

11. This is too complicated.

12. This looks better.

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

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

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

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

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

18. 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.

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

20. The power of Siegel Modular Forms
Microscopic Side
The power of Siegel Modular Forms

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

22. FIRST INPUT: Justification: 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.

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

24. 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

25. HOWEVER

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

27. 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.

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

29. 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

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

31. Extracting dmicro 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

32. 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.

33. Future Directions
We have lots to do…

34. 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.

35. 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.

36. 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.

37. THANK YOU!