Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index

The manuscript proposes two conjectures regarding the supersymmetric phases of large N $\mathcal{N}=4$ super Yang Mills theory, both of which are rather surprising and interesting.

The first, the dressed concentration conjecture, says that the leading SUSY entropy for a set of charges is not necessarily given by that of the the SUSY black hole with those charges. It is instead determined via a maximization procedure that may return the entropy of supersymmetric grey galaxy or DDBH solutions over that of the pure SUSY black hole. The second, the unobstructed saddle conjecture, says that the leading large N behavior of the oscillating sum on the index line is captured in a single maximal degeneracy $n(Z_i)$, for the interacting theory, in that sum.

The two conjectures yield supersymmetric and indicial phase diagrams characterized by different types of supersymmetric grey galaxy, DDBH, and pure black hole solutions in AdS. The result is an impressive update for the supersymmetric phases of large N $\mathcal{N}=4$ super Yang-Mills.

1) In Section 1.2, the authors motivate the 'dressing' of the concentration conjecture by stating (in paragraph starting at line 154) that field theoretic enumerations of supersymmetric states at finite N does not fit well with the concentration explanation. Apart from an analogy between the states found in refs. [17,20,22] and grey galaxy/DDBH solutions, does the dressed concentration conjecture extrapolated to low N lead to a better explanation for the distribution of charges of known fortuitous cohomologies at low values of N? If the situation is yet unclear, the motivating paragraph could be potentially misleading for the reader.

2) From the supersymmetric and indicial phase diagrams (e.g. slices shown in Figures 6 or 14), it appears that a slight perturbation away from the 1/2-BPS limits $H_i$ while $j_1 \neq j_2$ would put one in a rank 2 grey galaxy phase. However one may have expected instead to end up in a rank 2 DDBH phase given that we've perturbed slightly away from a limit where one of the charges $Q_i$ is very large. Is there a physical explanation for why the 1/2-BPS limit (associated with one large charge) is continuously connected to the rank 2 grey galaxy phase (associated with one large angular momentum)?

3) I am curious whether the location of the maximal degeneracy along an index line in the interacting theory could be also inferred by examining the location of the maximal degeneracy of the free BPS partition function refined by five charges at sufficiently large N $\sim O(10)$. This comment is motivated by the findings in supersymmetric SYK of ref. [24], where the Q-cohomology is localized at or near the maximal dimension vector space inside a cochain complex prior to the cohomology. This could be a consistency check for the corresponding locations determined for the interacting theory in the paper using the dressed concentration conjecture, and it would be interesting if there is agreement.

4) As mentioned in Section 6, it would be of interest to know how the indicial phases proposed in the paper are related to those suggested by other approaches to the index. For example, the giant graviton expansion for the N=4 index is known to undergo wall-crossing due to the presence of poles in the space of chemical potentials, see, e.g., discussion around (1.12) of https://arxiv.org/abs/2109.02545. The giant graviton bound-states that contribute to the finite N index can therefore change discontinously depending on the intensive charge space region. Could the boundaries between rank 2 and 4 DDBH and pure black holes be related to walls in the giant graviton expansion? Though this is not necessary for the current submission, a canonical understanding of the indical phase diagram could be helpful for determining potential connections to other approaches to the index.

I would be happy to recommend the paper for publication once the authors get a chance to respond to the above points.

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Building on recent developments by some of the authors and their collaborators—concerning the existence of novel gravitational configurations obtained by dressing known black hole solutions (the core) with either gravitons or dual giants—this paper presents two intriguing conjectures related to the analysis of the superconformal index of 4d $\mathcal{N}=4$ SYM:

• The dressed concentration conjecture.
• The unobstructed saddle conjecture.

In my opinion, and given their potential relevance as well, more refined tests of these conjectures are necessary and of interest. In particular, it is unclear to me how backreaction from the dressing - if present, may affect the final form of the entropy, or of the free energy of the newly conjectured solutions, and consequently it is also unclear how this may affect the final form of the conjectures and of some of the results presented in this paper e.g. the phase diagrams. Leaving that seemingly weak point aside, the paper has a lot of value.

Before recommending it for publication, however, there is one issue I would like the authors to address.

In the discussion section it is suggested that the supersymmetric free energy (not just the entropy) of the dressed black hole configuration is equal to the one of its BPS black hole core \begin{equation} \mathcal{F}=\mathcal{F}[\omega] \end{equation} i.e. the well-known expression quoted in their equation (38).

Precisely, in their words:

a) "The resultant phases all include central black holes whose chemical potentials (approximately) saturate one of the inequalities listed above. These facts suggest that the grand canonical index is indeed given by the usual formula (38) at chemical potentials that obey (68),..."

It is important to note that this suggestion is in clash with the dressed concentration conjecture. Let me explain.

The dressed concentration conjecture states that:

In the microcanonical ensemble the entropy (not the free energy) computed by the superconformal index equals the entropy of the core BPS black holes, i.e. the known entropy formula for the known BPS black holes in $AdS_5\times S_5\,$, \begin{equation} S_{BPS}=S_{BPS}[J_{BH}], \end{equation} evaluated at the core indicial charges $J_{BH}=J_{core}\equiv J_{c}\,$, not at the total indicial charges of the system, $J$. By indicial charges -- borrowing the author's notation, I mean charges that commute with the two supercharges that organize the cohomologies that the superconformal index counts.

The important relation \begin{equation} J_{c}=J_{c}[J] \end{equation} is fixed by the procedure outlined in the dressed concentration conjecture.

As will be explained next, the fact that $J_c[J] \neq J$ implies that the onshell action of the newly proposed configurations should not match the well-known BPS free-energy $\mathcal{F}$ reported in (38). Let me explain this, schematically.

Allow me to start from the following well-known extremization principle (being careless about numerical factors) \begin{equation} \underset{\omega}{ext}\Bigg(\mathcal{F}[\omega]+ \omega J\Bigg)\, =\, S_{BPS}[J] + \text{i} Q[J] \end{equation} and its inverse transform \begin{equation} \underset{J}{ext}\Bigg(S_{BPS}[J] + \text{i} Q[J] -\omega J\Bigg)\, =\, \mathcal{F}[\omega ] \end{equation} where $\mathcal{F}\,$, again, stands for the expression reported in equation (38).

The function of indicial charges $Q=Q[J]$ is defined by the non-linear constraint \begin{equation} \mathcal{P}(J,Q[J])=0 \end{equation} (which it is the same polynomial relation $\mathcal{P}$ or $P$ proposed by the authors as a test function in their diagram in page 43) and $Q$ is the average of the R-charge operator $\widehat{Q}\,$. This is the operator used to represent $(-1)^F$ as $e^{\pi \text{i} \widehat{Q}}\,$. This operator is not an indicial charge (i.e., it does not commute with the supercharge that organizes the cohomology of 1/16 BPS states of interest).

Now the dressed concentration conjecture seems to indicate the substitution of the right-hand side of the first extremization above as follows \begin{equation} S_{BPS}[J] + \text{i} Q[J] \rightarrow S_{BPS}[J_c] + \text{i} Q[J_c] \end{equation} were again $J_{c}= J_{c}[J]$ and \begin{equation} \mathcal{P}(J_c,Q[J_c])=0\,. \end{equation}

At last, going back to grandcanonical ensemble, one obtains

$\underset{J}{ext}\Bigg(S_{BPS}[J_c] + \text{i} Q[J_c] -\omega J\Bigg)=\mathcal{F}_{new} [\omega] = $ $\frac{dJ_c}{dJ} \times \mathcal{F}[\omega_{dressed}]\neq\mathcal{F}[\omega]$

where the dressed chemical potential $\omega_{dressed}$ is defined as \begin{equation} \omega_{dressed} = \omega \frac{dJ}{dJ_c} \end{equation} Note that the extremization procedure fixes $J$ and $\frac{dJ_c}{dJ}$ as functions of the meaningful $\omega\,$. These latter relations can be found, for instance, in perturbations of $\omega$, say about 0 or 1, assuming that one knows $\frac{dJ_c}{dJ}$ for all $J$, or at least in some asymptotic region.

Of course, I do not think it is necessary to perform the precise version of this analysis in this paper - although that would give more value to the paper; my point here is to illustrate as clearly as possible that the observation a) - is in conflict with the conjectured approach.

This is important to highlight because, should a) be correct, then none of the newly conjectured black hole solutions would correspond to new saddles of the index in grandcanonical ensemble -- well within the region (68), and that would be very strange. All known saddle point approaches to this problem have convincingly shown that every known large-$N$ saddle points of the index (in grandcanonical ensemble) has a dual complex gravitational configuration -- with differing on-shell actions, well within the region (68). I expect these solutions -- if they exist, to correspond to new large-$N$ saddles of the superconformal index in grandcanonical ensemble.

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In this manuscript, the authors performed a systematic analysis of the microcanonical phase diagram of the 1/16-th BPS states in the 4d N=4 Super Yang Mills theory. Motivated by recent developments in constructing black hole solutions dressed by rotating objects such as gravitons (“grey galaxies”) and charged objects such as D-branes (“dual dressed black holes”), as well as the explicit construction of their field theory counterparts, the authors conjectured the existence of similarly dressed supersymmetric black holes solutions. Their result presents a dramatic update to the phase diagram of the 1/16-th BPS sector – instead of having only extensive number of states concentrated in a 4-dimensional sheet of 5 possible charges, the dressed black holes cover a 5-dimensional “extensive entropy region”. After carefully analyzing how different regions of the vacuum black hole sheet should be dressed differently, the authors present a method to determine the dominant entropy solution for arbitrary given charges.

Importantly, the outcome of their analysis, under an assumption of “unobstructed saddle conjecture”, suggests new predictions that can be tested against the field theory index. The authors performed such tests numerically in some cases and found moderate support.

The analysis in this manuscript is a remarkable tour de force and represents the state-of-the-art understanding of black hole states in holographic field theories. It opens many possible new directions and questions for future exploration.

Nonetheless, there are some questions/suggestions that I hope the authors can clarify or consider before I recommend this manuscript for publication.

1) In analyzing what are the possible dressings one can add to the black hole, the authors discussed the distinction of the “allowed region” (the possible charges of all operators built out of BPS letters) and the “Bosonic cone” (the possible charges one can arrive at by only including bosonic letters). The authors argued in Appendix B.5 that for charges much greater than one, the supersymmetric gas lie in a region that asymptotes to the bosonic cone. If I understand correctly, the argument uses two facts: (1) fermionic single trace operators can be excited at most once; (2) single trace graviton gas operators contain at most three fermionic letters. However, in Table 2, one sees that there exist bosonic single trace operators that nonetheless contain two fermionic letters. It seems such operators can in principle be excited arbitrary many times (at least when it is somewhat small compared to N^2), which seems to lead to states which lie outside the bosonic cone. One might argue that these will not be the typical gas states, but it is unclear one can invoke entropic arguments since the entropy of the gas modes is negligible in the final solution anyway. Could the authors clarify whether this would affect the analysis?
As an aside, the second paragraph of Appendix B.5 seems to contain several typos, such as “n = 1, 2, …, n”, and the two summations over m are written the same but gives different answers.

2) It seems to me that the “unobstructed saddle conjecture” has relatively less support. In principle, given the entropy S(x) of the solutions the authors found along the direction summed over by the index (parametrized by x), one could perform a simple-minded saddle point analysis of \int e^{S(x)} e^{\pi i n x} with n being an odd integer and check explicitly whether the saddle point contribution agrees with S_{max} (x). It could be that the difference is small enough such that it has not been picked up in the numerical tests. Are there some simple cases where one could work out the expression of S(x) such that this test could in principle be performed?

3) In Figure 17, one observes several bumps in the log plot of the index. Since this is a log plot, such bumps might be viewed as signaling first order competitions between several different saddles, reminiscent of the discussion in https://arxiv.org/abs/2206.15357. Do the authors have any speculations on this feature?

4) Several periods are missing, such as in footnotes 11 and 13 and immediately before footnote 27.

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