Extreme measures for extremal black holes Alejandra Castro

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

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

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

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

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

Focus Today, I want to focus on building certain statistical systems, which have the right features to describe the black hole. Work in collaboration with A. Belin, J. Gomes, C. Keller [1611. 04588]

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

Macroscopic: Universality of Black Holes Microscopic: Building Partition functions

Extremal Black Holes Gravitational Input

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

Extremality • • • Degenerate horizons: inner = outer. Surface gravity vanishes: zero temperature. Enhancement of symmetry: Rindler Ad. S. Can be supersymmetric: even more symmetry! Horizon area finite: huge residual entropy. Asymptotically flat region Ad. S 2 x S 2 Near horizon region

4 D Reissner-Nordstrom Black Holes Geometry • Static: No rotation (for simplicity). • Dyonic: Carry both electric & magnetic charges. • Extremal: Decoupling of the near horizon geometry. Ad. S 2 Electric Magnetic

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

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

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

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

Microscopic Side The power of Siegel Modular Forms

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

FIRST INPUT: Look within (1+1)-dimensional CFT. Looking for highly degenerate systems. Justification: Success list & Holography In many (but not all) cases, the Ad. S 2 geometry has an Ad. S 3 origin.

CFT 2 Interpretation Gravity CFT 2 Black hole Very entropic state dictionary Energy (L 0) KM current (J 0) central charge For example: And the claim is

CFT 2 Interpretation 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

HOWEVER

CFT 2 Interpretation We had This expression relies on modular transformations, and more importantly on a saddle point approximation which is valid if: This is NOT the gravity regime. Our dictionary and scaling regimes in gravity are

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 CFT 2. Vast majority of known CFTs do not satisfy this condition.

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.

Strategy 1. Define a grand canonical ensemble that sums over central charge. 2. Identify those with exponential growth in the gravity regime. for 3. Extract asymptotic formulas.

Siegel Modular Forms 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

Extracting dmicro Consider To extract the asymptotic behavior, the strategy is: 1. Residue integral: Exploit the zeroes/poles of the SMF. 2. Find the most dominant pole: in the large charge limit this is universal. 3. Saddle point integral of remaining variables.

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.

Future Directions We have lots to do…

Within the land of Siegel Modular Forms: Good news! 1. These modular functions easily capture the leading contribution to the black hole entropy in the desired regime. Highly non-trivial! 2. The entropy can be easily computed and interpreted. The answers are unambiguous and precise. 3. We can build more examples beyond those already known.

Within the land of Siegel Modular Forms: Bad Challenges News… 1. Dictionary between black hole charges and SMF data is crucial: an important task! 2. 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? 3. 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.

And more generally: 1. We postulated a non-trivial symmetry for the statistical system. Is this necessary or sufficient? Perhaps it is just approximate. 1. Can we “bootstrap black holes” via its quantum corrections? Exploit more the hints in the subleading corrections.

THANK YOU!