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The Quantum Entropy of Extremal Black Holes

Periodic Reporting for period 1 - BHLOC (The Quantum Entropy of Extremal Black Holes)

Reporting period: 2015-11-01 to 2017-10-31

Quantum Properties of Black Holes are an important and long-standing puzzle in theoretical physics. The overarching aim of the project was to employ holography, in the form of the AdS/CFT correspondence, to explicate properties of black holes, in particular, their entropy.

In related work, we also explored the viability of applying the AdS/CFT correspondence to other long standing and prohibitively difficult problems, such as a formulation of string theory in the extremely high energy ('tensionless') limit, as well as construction of theories of massless fields of spins higher than that of the graviton to arrive at very encouraging results, obtained in the quantum gravitational regime.

A related question regarding AdS/CFT dualities is if they are always true at the quantum level, i.e. can they always be used to study quantum gravity. While investigating this question we found counter-examples. That is to say, AdS/CFT dualities that are true in the classical gravity limit, but appear to break down when we incorporate quantum effects. The dualities involving string theory itself passed these tests successfully.
1. The computation of the quantum entropy function for a particular black hole in string theory and a systematic comparison with the microscopic prediction from string theory. A partial match was found. Further research is required to resolve the discrepancies.

2. The computation of the entropy, with a careful treatment of infinities, of AdS(4) supersymmetric black holes to find a precise match with proposed microscopic results. We showed how this match is a consequence of the very basic rules of the AdS/CFT correspondence.

3. The computation of quantum corrections to the entropy associated only to the event horizon of the black holes of point 2. These results serve as a prediction for putative microscopic computations.

4. The computation of quantum effects in tensionless superstrings, and related non-supersymmetric dualities, along with the development of new techniques to carry out such computations.

All of the above results are available, and free to download, on the arXiv to the general public for dissemination.
The project was able to advance the state of the art of the field in several important directions.

1. By computing the quantum entropy function for a class of supersymmetric black holes, we could find a partial match with microscopic predictions for the entropy of these black holes. The mismatch appears when we push the comparison to an extremely fine grained level, incorporating contributions to the entropy which are almost vanishingly small. Nonetheless the matching should be exact, as a consequence of AdS/CFT duality. We expect that this is an important issue to address within string theory.

2. By computing the on-shell action of supersymmetric black holes in AdS(4) in the supergravity approximation, we provided an extremely universal derivation of the Bekenstein-Hawking formula for entropy for these black holes within the context of the AdS/CFT correspondence. This is particularly important in view of recent results where the entropy for these black holes is computed in the dual CFT. Our result is a single computation which contains in its ambit an extremely wide class, and indeed almost the entire set, of known black hole solutions in AdS(4), and clarifies many conceptual issues in matching gravity results with holographic predictions from the CFT.

3. Further, the computations of logarithmic corrections provided an important insight into the degrees of freedom contained in the event horizon of the black hole. This conclusively demonstrated an important qualitative difference between black holes in AdS(4) and black holes in Minkowskii space, the subject of previous similar analyses in string theory. Namely, that in AdS(4), the degrees of freedom living outside the event horizon contribute significantly to the black hole entropy. This would be an important factor in further developing microscopic descriptions of black holes in Anti-de Sitter space.

4. Further, the subject of constructing interacting higher-spin theories, and tensionless string theory, is a long-standing and difficult problem. It may be expected that if this theory is defined in AdS space, then there would be a dual description of the theory in terms of a CFT. Since typically the CFT description of such theories is easier to construct, it has long been expected that this may be a simpler way to formulate these theories. Our computations represent a concrete application of this approach to study higher-spin theories and tensionless strings at the quantum level.

5. The computations of one-loop free energy for higher-spin/CFT dualities shed important new light on their validity at the quantum level. In particular, having defined an AdS/CFT duality at the classical level, we would like to formulate consistency criteria for its validity at the quantum level. We investigated such criteria in the context of higher-spin/CFT dualities and found that for most well-studied dualities, they are satisfied. Importantly, we found examples where they are violated, signalling an inconsistency for these theories at the quantum level. This firstly emphasizes that these criteria are indeed non-trivial, in the sense that not every duality will pass them. Secondly, we found that these criteria are indeed satisfied by supersymmetric theories. We expect that these results will be of interest to researchers looking to formulate quantum gravity by means of holography in non-supersymmetric settings, an important area of research in its own right.