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On the exact entropy of N = 2 black holes

by Joao Gomes, Huibert het Lam, Grégoire Mathys

Submission summary

Authors (as Contributors): Grégoire Mathys
Submission information
Preprint link: scipost_202007_00058v2
Date submitted: 2021-03-15 22:48
Submitted by: Mathys, Grégoire
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We study the exact entropy of four-dimensional N = 2 black holes in M-theory both from the brane and supergravity points of view. On the microscopic side the degeneracy is given by a Fourier coefficient of the elliptic genus of the dual two-dimensional N = (0; 4) SCFT and can be extracted via a Rademacher expansion. We show how this expansion is mapped to a modi ed OSV formula derived by Denef and Moore. On the macroscopic side the degeneracy is computed by applying localization techniques to Sen's quantum entropy functional reducing it to a nite number of integrals. The measure for this nite dimensional integral is determined using a connection with Chern-Simons theory on AdS2 x S1. The leading answer is a Bessel function in agreement with the microscopic answer. Other subleading corrections can be explained in terms of instanton contributions.

Current status:
Awaiting resubmission

Author comments upon resubmission

Dear editor and referees,

Thank you for sending us your detailed comments and recommendations in order to improve our work. In the list of changes, we will details how we included all these comments in the new version of our submission.

With kind regards,

Joao Gomes, Huibert het Lam and Grégoire Mathys

List of changes

1 Report 1: 2020-10-14

1. In Eq. (2.6) the electric charges $q_a$ are indeed assumed to vanish. In case of non-vanishing charges, $q_0$ should be replaced by $\hat{q}_0$. Since at this point in the paper we have not
introduced the electric charges yet, we have not mentioned this assumption explicitly.
2. We have added this as a footnote on page 8.
3. We adapted this part as well. We now work with SL(2; Z) instead of PSL(2; Z). This
implies that$ (ST)^3$ = $S^2$ = 1, which implies (2.38).
4. We have added appropriate references between equations (2.53) and (2.54).
5. The referee is correct. The degeneracies cannot be determined by a Schwinger type
computation. We have modi ed this in the text and added the appropriate citations.
6. The referee is right, and we changed the paper appropriately.
7. The referee is correct. The measure factor is indeed a symplectic product of the charges
of the D6 and anti-D6 branes, as can be seen using Eq. (4.26) in Denef(Moore (reference [15] of the paper).
8. The de nition of a special vector is given below Eq. (A.1). Special vectors are integral
vectors x with n components such that Qx is divisible by detQ, i.e. each of its components
is divisible by detQ.
9. We changed the titles to proper English.

2 Report 2: 2020-11-3

1. We have adapted the paper, and stated from the beginning that we are just considering
extreme polar states in our truncation (Eq. (2.68)). This takes away some of the issues the
referee has brought up. For instance, this ensures that the distance between the branes
is always positive. Above (2.96) we indeed meant that this identi cation is satis ed for
a speci c set of (mi, ni) that we have not precisely identi ed. However, we have derived
certain conditions that should be satis ed due to convergence properties. We comment
on having another domain for (mi, ni) in the last sentence of section 2.4.
2. We have removed section 3.2.3. We have also removed the subsequent section because
it was relying heavily on section 3.2.3. We have presented the idea that was contained
in section 3.2.3 in the discussion section. In addition, we have adapted all references to
these sections throughout the paper.
3. We have added the comment about the eld A^. We nevertheless don't think that any important
de nitions are missing, but we would be happy to add any particular occurrences
that the referee would nd missing here.
With kind regards,
Jo~ao Gomes, Huibert het Lam and Gregoire Mathys

Reports on this Submission

Anonymous Report 1 on 2022-7-1 (Invited Report)


The authors have made progress in a very challenging problem in string theory.


The paper does not distinguish clearly what is proved from what is conjectural.


Let me first apologize to the authors for the extremely long delay before they could get a report. As the second referee I have only been asked to have a second look at the paper few months ago.

I acknowledge the fact that section 3.2.3 has been removed. However, the corrections in 2.3 and 2.4 are rather minimal, and do not clarify completely the situation.

In equation (2.68) the set C is still defined as the set of charges with m_a positive with respect to the d norm, whereas it is assumed everywhere as explained below (2.68) that they are moreover extreme polar. The authors change the definition such that there is no non-extreme polar states, but I do not understand what shows that this is the correct assumption. What I understand is that most of the analysis is valid for extreme polar states and the other cases remain to be analysed. In any case I believe the correct equation should be that they define C \subset and not equal to the set of discrete (n,m) with m positive such that the associated states are extreme polar. Talking about (n_i,m_i) bellow (2.68) is actually confusing since the pairs of charges only appear below in (2.69) when rewriting |ZGW|^2 as a sum of instantons.

The author say in the conclusion: we have mapped the Rademacher expansion to the degeneracy as derived by Denef and Moore, but I do not consider they have. They should put some conditional in this sentence, and resume what remains to be clarified.

I believe this paper is interesting and shed some light on the `macroscopic derivation’ of exact black hole degeneracy on N=2 string theory on a Calabi-Yau three-fold. However, I do not consider that the results are as strong as stated by the authors and they should explain clearly what has been proved and what is conjectural. The minimal modification of assuming C to only contain extreme polar states does not make the result more rigorous, whereas the conclusion only admits some points needed to be clarified regarding the supergravity computation. Therefore I will recommend the paper for publication if the authors accept to modify the paper accordingly.

Requested changes

1) (2.47) replace equal sign by approx or sim.

2) (2.52) replace equal sign by approx or sim.

3) (2.68) replace equal sign by subset. Replace (n_i,m_i) by (n,m).

4) State clearly in the conclusion what is proved and what is conjectural and requires further analysis. In particular explain the potential difference between their set C and the definition of Moore and Denef.

  • validity: ok
  • significance: high
  • originality: high
  • clarity: ok
  • formatting: excellent
  • grammar: good

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