On the exact entropy of N = 2 black holes

Proposed macroscopic computation of N=2 black hole entropy in string theory, a very challenging problem.

The paper does not distinguish clearly results from conjectures, and some conjectures are rather suspicious.

The authors study the exact degeneracy of N=2 supergravity BPS black holes in type IIA string theory. They derive a Rademacher expansion for the MSW BPS index that they compare to Denef and Moore refined OSV formula. They describe the Sen’s quantum entropy functional using localisation and an uplift to AdS2 x S1. Using several manipulations they claim that the two computations match up to some (very) conjectural identities.

String theory provides an explicite computation of the exact degeneracy for N=4 and N=8 BPS black holes in type II string theory. The situation is more complicated with only N=2 supersymmetry, in part because of wall-crossing and the difficulties to compute Donaldson-Thomas invariants. For N=4 and N=8 it has been shown that one can (up to some string theory inputs) derive the exact entropy using Sen’s quantum entropy functional and localisation techniques. The generalisation of the localisation techniques to N=2 supergravity is a very challenging and this is a very interesting problem in string theory. However, the results of the paper are very speculative and some of the conjectures are not really justified.

The paper starts with a short review of the MSW (4,0) SCFT. They write the BPS index as the scalar product of a vector valued Narain theta function and vector valued holomorphic modular forms. They give an explicit form of the vector representation matrices. They manipulate the Narain theta series to write the index as a Rademacher expansion, which they truncate to the first term c=1 for the rest of the paper. In section 2.3 they describe the refined OSV conjecture of Denef and Moore. They do not use the regularised partition function advocated by Denef and Moore, but take instead a somehow ad hoc truncation of a sum over positive integers in (2.68) that they further refine in the following by assuming convergence properties.

This truncation should be further justified. They say "It is interesting to see whether we obtain a similar truncation as in [15]" while they probably mean, "It would be interesting to check if this truncation is consistent with the prescription proposed in [15]". It is not clear how their truncation compares to [15], is does not seem to be either a stronger or a weaker truncation.

In (2.87) they explicitly find that their criterion is not enough to only get physical charges satisfying to (2.83). The conclusion that such charges do not satisfy to the extreme polar limit is not enough to find no contradiction, as they claim below (2.91). The positivity of the distance between bound-state constituents is a semi-classical formula derived in supergravity that can be generalised to any charges, so it should be satisfied. The use of the limit in reference [15] was to enforce that no further bound states, that are not considered by the authors either, can contribute.

The similar comment below (2.95) is not really encouraging. The sum over nu follows from the Rademacher expansion in (2.52), whereas the sum over m1 and m2 is not really justified. So it is not that it would be very interesting to compare, the authors should prove that they are equivalent up to negligible terms in the approximation scheme. Right after this sentence they conclude that the degeneracy (2.52) is equal to (2.77), whereas they just admitted they don’t understand the sum. I guess they mean that it is satisfied for a specific set of (m_i,n_i) that they have not precisely identified. Just below they write that one can only trust the formula for extreme polar states, which seems in contradiction with the problem found in 2.3, since their sum includes states that are not extreme polar from (2.91).

In Section 3 they discuss the macroscopic entropy using localization techniques. In (3.6) they give a solution with an unfixed \theta with respect to reference [24]. In reference [24] is is said that theta is simply a gauge choice, whereas in the following they need to fix theta to the precise value (3.29). How can it be that a gauge parameter has to be fixed to this specific value, shouldn’t it drop out of the computation? The argument using the Kahler potential in (3.32) is not very convincing, and it seems that the authors have simply fixed theta for their formula to work at next to leading order. Section 3.2.3 is an attempt to propose how instanton corrections could be included. Assuming the Ansatz (3.34), they find that they can reproduce their results of section 2 for a very strange looking prepotential (3.36), if they assume moreover that the prescription (3.32) for fixing theta should be followed. This is really too many if. The prepotential used in (3.10) is only an approximation, and the complete prepotential includes infinitely many higher order terms in both A and the Kahler moduli. Why the AdS2 instantons should dominate the pre-potential corrections? One of the authors himself comments on this difficulty in the first paragraph of page 4 in [58].

In general the paper does not distinguish clearly what is proved from what is conjectural and sometimes just wishful thinking. I believe there are some interesting results in the paper, but there are sections that should not be published as such. Section 3.2.3 in particular should not be published and sections 2.3.2 and 2.4 should be clarified. The generalisation of the localisation formula to N=2 supergravity is a very complicated problem, and I don’t think it would be shameful to restrict to the leading Bessel function in the Rademacher expansion. I think that the paper could be published in SciPost with important amendments along these lines.

1) Sections 2.3.2 and 2.4 should be clarified.

2) Section 3.2.3 should be removed.

3) More definitions should be explained. It would not arm, for example, to recall below (3.6) that A is the Weyl square chiral field and that its contribution in 3.10 gives rise to the Weyl square coupling in the effective action.

1) Detailed analysis of the microscopic and macroscopic aspects of D4-D2-D0 black holes in N=2 string theory compactifications.
2) Derivation of the measure of Sen's quantum entropy functional for this class of black holes.

This article studies the black hole entropy of D4-D2-D0 black holes and their M-theory duals. The authors study the partition function of this class of black holes and its relation to the 2004 conjecture by Ooguri, Strominger and Vafa, which relates black hole entropy to the topological string free energy. The authors further consider Sen's entropy functional and determine the measure in this path integral.

I find the article an interesting contribution, which will be useful in future work in this subject. I recommend it for publication by SciPost.

After reading the manuscript, I have a few comments and suggestions:
1) Eq. (2.6): Are the electric charges $q_a$ assumed to vanish, or should $q_0$ be $\hat q_0$?
2) Below (2.22): I suggest to add that $q_-^2$ and $q_+^2$ vanish both if and only if $q=0$. This is necessary for convergence.
3) Below (2.37): I find (ST)^3=S^2=-I. The elements I and -I are identical in PSL(2,Z), but I and -I act differently on the elliptic variable z of a Jacobi form, isn't it?
4) Above (2.54): I think a reference for the equivalence of D6-D4-D2-D0 partition function and topological string partition function will be helpful for the reader.
5) Below Eq. (2.66): My impression is that the lhs of (2.66) being a product formula is a consequence of the Schwinger calculation, but that the degeneracies $N_{q_,j_L,j_R}$ must be determined separately.
6) In (2.75): Should $\mathbb{R}$ be $\mathbb{R}^{b_2}$?
7) Eq. (2.96): The measure factor appears to resemble a symplectic inner product of the charges of the D6 and anti-D6 branes. Is this indeed the case?
8) Above (A.4): What are the special vectors $l$?
9) Appendices: I would suggest to edit the titles of Appendix A and B