Decoding stringy near-supersymmetric black holes

The article is focused on the $\mathcal{N}=4$ $SU(N)$ gauge theory at finite $N$ and tackles the problem of computing the one-loop anomalous dimensions in the subspace of classically BPS-states, i.e. states whose bare dimension satisfies the BPS bound following from the presence of two unbroken supercharges, $Q$ and its hermitian conjugate $Q^\dagger$. The authors construct an explicit matrix representation of the hamiltonian, $H\sim \{Q,Q^\dagger\}$, for various values of $N=2,3,4$, using the observation that at one-loop $Q$ is given by the Noether supercharge derived from the classical interacting lagrangian, while the hermitian conjugation can be computed in the free theory. Diagonalizing this hamiltonian in the subspace of classically BPS-states with some maximal value of the charges, the authors are able to derive the spectrum of non-vanishing anomalous dimensions and the eigenvectors with vanishing anomalous dimension, which are conjectured to remain BPS at arbitrary value of the coupling.

The physical outcomes of this computation are two-folds. On one side one derives the near-BPS spectrum at weak-coupling and finite $N$, from which one can observe an intriguing qualitative similarity with the gapped spectrum that has been predicted from the gravitational path integral at strong coupling. On the other side, one obtains an explicit weak coupling representation of the supersymmetric states that make up the entropy of the 1/16 BPS AdS$_5$ black hole. The authors provide for the first time the full form of a BPS black hole operator that is not of the graviton-gas form, of which only the cohomology class was known.

The details of the computation are explained very accurately and the article contains a number of technical appendices that will prove useful for future developments.

In summary my opinion of the manuscript is very positive, both for its physical relevance and for the methodological rigour. I collect below a list of questions and suggestions for further improving the readability of the article.

1) At the end section 3 (pag. 11) the authors announce that the inner product matrix $T_Y$ will be computed in a "next subsection"; that subsection, however, appears to be missing.

2) Table 1 contains also the results for infinite $N$, but, as far as I understand, the construction of the hamiltonian detailed in section 3 applies only to finite $N$. It would be useful if the authors could give some details on how they obtained the infinite $N$ results, also explaining the difference between the single and multi-trace cases.

3) In the section on the results, it would be useful to add some information on top of the ones already contained in Table 1. For example, are there degeneracies among the non-zero eigenvalues listed in the fifth column of that table, and are there zero eigenvalues corresponding to states that are BPS with respect to supercharges different from the ones in (2.1)? How many BPS states, graviton gas or not, are there in total?

4) How does one understand that the state defined in (5.7), (5.8) is not a graviton-gas state? And, more generally, given a null eigenvector of the Hamiltonian matrix, what is the algorithm one uses to decide if it is a black hole state or a multi-graviton state? This information is probably already contained in the literature, but it would help the reader to recall it in this manuscript.

5) At the end of section 5.1, the authors claim that in the non-BPS case there is no known method to distinguish multi-graviton from black hole states. It would be interesting to have a few more details on the origin of this difficulty. One would naively be tempted to define non-BPS multi-gravitons as the states appearing in the OPE between single-trace operators that are BPS with respect to different supercharges (corresponding to different choices of the index 4 in (2.1)). What would be the problem with such identification?

6) After eq. (3.6): "Let us denote the column reduced matrix by $A'$" should probably be "Let us denote the column reduced matrix by $A$".

7) In the third line of pag. 6, "except $S^+_4$ which has $\Delta=2$" should probably be "except $S^+_4$ which has $\Delta=1/2$".

The main goal of this manuscript is to explore the near-supersymmetric aspects of $\mathcal{N}=4$ super-Yang-Mills theory. The authors computed one-loop anomalous dimension of a large number of near-BPS states. They found hints of a gap in the near-BPS spectrum, which is reminiscent of the spectrum at strong 't Hooft coupling captured by a gravitational path integral.
Specifically, the authors firstly explained their superspace formalism in detail. In doing so, they systematically constructed the one-loop Hamiltonian in the classically-BPS sector. Explicitly diagonalizing the Hamiltonian, they could compute a large number of one-loop anomalous dimensions. They found clear gaps in the smooth histograms of anomalous dimensions for SU(2), SU(3), SU(4) theories. Using their techniques, they also determined the actual weak coupling expression of the smallest BPS black hole operator, which was recently constructed at the level of cohomology.
This paper presents interesting features of the near-supersymmetric spectrum of $\mathcal{N}=4$ super-Yang-Mills theory. It would be desirable to clarify a few points of the paper for publication as given below in "Requested changes."

1. The authors studied the one-loop Hamiltonian of the theory, which is proportional to $g_{\textrm{YM}}^2$. Constructing such Hamiltonian, they used supercharges $Q$ and $Q^\dagger$ using the inner product $\mathbf{T}$ in the free theory $g_{\textrm{YM}}=0$. It would be nice to clarify the convention mapping these free theory quantities at $g_{\textrm{YM}}=0$ to the one-loop Hamiltonian of the order $g_{\textrm{YM}}^2$.

2. The authors presented various Figures in Section 5.1 to show gaps in the near-BPS spectrum for SU(2), SU(3), SU(4) theories. For the SU($\infty$) spectrum, while it was argued that there is no visible gap in the smooth histograms, in Figure 3, $\delta_{\textrm{gap}}$ is explicitly written even for the SU($\infty$) cases. It is desirable to explain the Figures and the notion of "gaps" here in more detail; in particular, how to concretely define, compute and read off the gap $\delta_{\textrm{gap}}$ from the data of anomalous dimensions and each smooth histogram in Figures.